D ynam ics o f vo rtex R ossb y w aves in trop ica l cyclon es

by

Lidia N ik itin a

A thesis submitted to

the Faculty of Graduate and Postdoctoral Affairs

in partial fulfillment of

the requirements for the degree o f

Doctor of Philosophy

in

Mathematics

Carleton University

Ottawa, Ontario

© 2013

Lidia Nikitina

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A b stract

This thesis describes an analytical study of vortex Rossby waves in tropical cy­

clones. Observational analyses of hurricanes in the tropical atm osphere indicate the

existence of spiral rainbands which propagate outwards from th e eye and affect the

structure and intensity of the hurricane. These disturbances may be described as

vortex Rossby waves.

The aim of this research is to study the propagation of vortex Rossby waves in

tropical cyclones and wave-mean-flow interactions near the critical radius where the

mean flow angular velocity matches the phase speed of the waves. Depending on

the wave magnitude, the problem can be linear or nonlinear. Analytical techniques

including Laplace transforms, multiple scaling and asym ptotic expansions are used to

obtain approximate solutions of the governing linear and nonlinear equations. In this

study we carry out asymptotic analyses to examine the evolution of the interactions

near the critical radius in some two-dimensional configurations on an /-p lane and a

/?-plane.

The results are used to explain some features of the tropical cyclone’s develop­

ment, namely, the change of angular wind in the critical layer, the secondary eyewall

formation and the eyewall dynamics.

A ck n ow led gem en ts

I would like to express my deepest gratitude to my supervisor, Prof. Lucy Camp­

bell, and thank her for her perm anent support and patience, for her wisdom and

kindness, for her high-level of professionalism both in research and teaching, for the

great time I spent at the School of M athematics and S tatistics during the course of

my thesis work.

I appreciate the time and effort the external examiner Prof. Simon Clarke and the

committee members Prof. Dave Amundsen, Prof. Arian Novruzi, and Prof. Abhijit

Sarkar put into reading and reviewing my thesis in spite of the ir busy schedules.

I would like to thank Dr. G ilbert Brunet for bringing this interesting problem to

our attention. I also would like to thank him for helpful discussions.

This thesis project was supported by N atural Sciences and Engineering Council

of Canada (NSERC) for three years with Alexander G raham Bell Canada Graduate

Scholarship.

C ontents

T itle page i

A bstract ii

A cknow ledgm ents iii

List o f Tables v i

List o f F igures v ii

List o f A ppendices v iii

1 In troduction 1

1.1 Motivation for the study and overview of the t h e s i s .................................. 1

1.2 Study of tropical cyclones .................................................................................. 4

2 B asic con cep ts o f geophysica l fluid dyn am ics 13

2.1 Basic equations of fluid m echan ics........................................................................ 14

2.2 Effects of ro ta t io n ................................................................................................... 16

2.3 The Coriolis effect for a thin layer on a ro tating s p h e r e ................................ 21

2.4 Waves in the ocean and the a tm osphere ..............................................................23

iv

V

3 V ortex dynam ics in th e a tm osph ere 26

3.1 Barotropic vorticity eq u a tio n ................................................................................... 26

3.2 Barotropic Rossby w a v e s .......................................................................................... 34

3.3 Vortex Rossby w a v e s .................................................................................................36

3.4 Previous theoretical studies of tropical cyclone d y n a m ic s ............................ 39

4 Steady linear problem 46

4.1 Steady linear solution for some special c a s e s .....................................................46

5 T he linear tim e-d ep en d en t prob lem 64

5.1 Laplace transform of the linear time-dependent e q u a t i o n ............................ 65

5.2 Evaluation of the inverse Laplace transform .....................................................75

5.3 Time-dependent solution close to the critical la y e r ...........................................88

5.4 The inner layer solution .......................................................................................... 91

5.5 Discussion of the linear time-dependent s o lu t io n ..............................................94

6 Effect o f non linearity and /3-effect 97

6.1 Weakly-nonlinear a n a l y s i s .......................................................................................97

6.2 Effect of the variation of the Coriolis f o r c e ...................................................... 101

6.3 Combined effects of nonlinearity and the /3-effect............................................ 109

6.4 Evolution of the mean flow in the inner la y e r ................................................... 114

7 C onclusions 118

7.1 S u m m a r y ....................................................................................................................118

A H ypergeom etric fu n ction s 122

A .l Generalized hypergeometric fu n c tio n s .................................................................122

A.2 The solutions of the hypergeometric equation ................................................123

vi

B N onlinear evo lu tion o f th e m ean flow 128

C T he phase sh ift for th e stead y so lu tion 131

B ibliography 134

List o f Tables

6.1 Wavenumbers corresponding to the term s in equation (6.49)

List o f F igures

1.1 Tropical cyclone s t r u c t u r e ................................................................................. 7

1.2 Hurricane Ivan 2004, secondary eyewall fo rm a tio n .........................................11

2.1 Vector in a rotating f r a m e .....................................................................................17

2.2 Centripetal acceleration ........................................................................................ 19

2.3 Local Cartesian c o o rd in a te s ................................................................................. 21

4.1 The domain of the problem ................................................................................. 48

4.2 The angular velocity profile ................................................................................. 50

4.3 Intervals for the steady s o lu t io n ...........................................................................57

4.4 The am plitude of the steady solution....................................................................62

4.5 Contour plots for the steady s o lu t io n .................................................................63

5.1 The contour of integration in the complex s-plane. Case 1 and 2. . . . 69

5.2 The contour of integration in the complex s-plane. Case 3 ..........................70

6.1 Influence of /3-effect on the vortex Rossby wave propagation ..................106

6.2 Schematic diagrams for w avenum bers...............................................................I l l

6.3 Dynamics of the critical radius location............................................................. 117

A .l The hypergeometric functions............................................................................... 127

viii

ix

List o f A ppendices

A Hypergeometric functions.........................................................................................122

A 1 Generalized hypergeometric functions .........................................................122

A 2 The solutions of the hypergeometric equation ..........................................123

B Nonlinear evolution of the mean flow .................................................................. 128

C The phase shift for the steady solution .............................................................. 131

C hapter 1

Introduction

1.1 M otivation for th e s tu d y and overv iew o f th e

th esis

The effect of the Coriolis force gives rise to large scale oscillations in the atmosphere

and ocean which are known as Rossby waves. These waves affect weather and cli­

mate and play a role in various observed phenomena including the development of

hurricanes or tropical cyclones. The dynamics of tropical cyclones have been studied

extensively during the past few decades using observational data, theoretical mod­

eling and numerical simulations. The motivation for understanding and simulating

hurricanes is based on the profound effects th a t hurricanes can have on human life as

one of the most devastating disasters. But a full theory of hurricane dynamics is far

from complete.

Vortex Rossby waves propagating within tropical cyclones are known to play an

im portant part in cyclone development and evolution. The waves affect the strength,

structure and dynamics of a cyclone. In particular, vortex Rossby waves are believed

1

C H APTER 1. INTRO D U CTIO N 2

to play a role in the formation of the secondary eyewall, an outer ring of intense

thunderstorms th a t develops outside the initial eyewall of the hurricane and eventually

replaces the original eyewall.

In this thesis we present a simple two dimensional configuration th a t represents

Rossby wave propagation in a cyclonic vortex and describes a possible mechanism for

the secondary eyewall formation. Approximate analytic solutions are derived using

asymptotic methods and weakly-nonlinear analyses. Previous analytic investigations

of this mechanism have been based on linearized equations th a t did not include the

effects of the gradient of the Coriolis force. The addition of nonlinearity and Coriolis

effects are im portant features of this thesis project.

An overview of the thesis is as follows. The rest of this chapter gives background

information about tropical cyclones. It includes a description of cyclones and the

physical processes involved in the development of the tropical cyclones, as well some

basic information about the structure of cyclones, their life cycle, intensity. In partic­

ular, the secondary eyewall replacement cycle is described.

Chapter 2 gives a mathem atical description of the basic features of geophysical

fluid dynamics, including the effect of the E arth ’s rotation, and a description of Rossby

waves.

Chapter 3 deals with analysis of vortex dynamics in the atmosphere. This chapter

includes the derivation of the basic equations th a t are solved in the thesis. These

equations describe tropical cyclones as well as the propagation of vortex Rossby waves

in tropical cyclones. Chapter 3 also gives information about some basic approaches

to the study of the dynamics of tropical cyclones.

Chapters 1-3 present introductory m aterial on hurricanes, geophysical fluid dy­

namics, and Rossby waves based on standard texts and papers on these topics. The

derivation of the basic equations follows K undu and Cohen (2004), Holton (1992),

C H APTER 1. INTRO DU CTION 3

Landau and Lifshitz (1953), and Gill (1982). The equations for the vortex waves are

based on the studies of Montgomery and Kallenbach (1997) an d Brunet and Mont­

gomery (2002).

The description of my original work sta rts in C hapter 4. I represent a tropical

cyclone as a vortex in a two-dimensional configuration. The vortex Rossby waves

are considered as small perturbations to the basic rotation of the cyclone, which

are periodic in the azimuthal direction and have am plitude th a t varies in the radial

direction. I derive a solution of the governing equation for a configuration in which

the wave am plitude is steady. The waves propagate on an /-p lane , a horizontal plane

in which the Coriolis force is taken to be a constant. The waves are assumed to be

of small amplitude and the governing equation is linearized. T he problem is defined

in term s of polar coordinates. The waves are generated by a boundary condition,

periodic both in time and in the azimuthal direction, a t some fixed distance from the

centre of the vortex.

The steady linear problem is governed by an ordinary differential equation which

describes the wave amplitude in term s of the radial variable. T he equation is singular

at the radius where the cyclone angular velocity is equal to the phase speed of the

waves. This is called the critical radius and the region close to this radius is called

the critical layer. The solution of the ordinary differential equation is expressed in

terms of hypergeometric functions.

In Chapter 5 a Laplace transform is used to derive an approxim ate time-dependent

solution for the linear problem. I derive an approxim ate solution valid in the outer

region, far from the critical layer, as well as in the inner region inside the critical layer.

In Chapter 6 I include the effects of nonlinearity and the “/3-effect” which arises

from the latitudinal variation of the Coriolis force. In section 6.1 I carry out a weakly-

nonlinear analysis of the nonlinear equation considering th e nondimensional wave

C H APTER 1. INTRO D U CTIO N 4

amplitude e as a small param eter w ithout taking into account th e /3-effect. In section

6.2 the /3-effect is included in the linear problem and in section 6.3 I discuss the effect

of including both nonlinearity and the /3-effect. The nonlinear interaction between

vortex waves and the mean flow changes the angular m om entum and changes the

mean flow speed in the inner layer. This is discussed in section 6.4. I also give some

explanation for the secondary eyewall formation and the eyewall replacement cycle.

Appendix A describes the properties of the hypergeometric equation, its singular

points, and its solutions th a t I use in my thesis. Appendix B gives the derivation of

the angular momentum equation th a t I use in Chapter 6 to examine the nonlinear

evolution of the angular velocity of the mean flow. Appendix C explains the phase

shift of a singular term in the steady solution.

1.2 S tu d y o f trop ica l cyclon es

A cyclone is a system of ro tating fluid flow w ith sustained w inds in the form of an

axisymmetric vortex. In this section a brief description of tropical cyclones, their

development and their structure is given following Holton (1992), Ogawa (1992) and

the review article of Emanuel (2003). The eyewall replacem ent cycle is described.

Tropical cyclones

A tropical cyclone is a cyclone th a t originates over tropical oceans and is driven by

heat transfer from the ocean. Tropical cyclones are categorized according to their

maximum wind speed, defined as the maximum speed of th e wind a t an altitude of

10 m. Tropical cyclones w ith maximum wind speeds in the range 1 5 -1 7 ms-1 , are

known as tropical depressions; when their speeds are in the range of 18 to 32 ms-1 ,

they are called tropical storms. Tropical cyclones with m aximum winds of 33 ms-1 or

C H APTER 1. INTRO D U CTIO N 5

greater are called hurricanes in the western North A tlantic and eastern North Pacific

regions; in the South Pacific and Indian Oceans they are usually called cyclones; in

East Asia, e.g. Japan and China, they are usually called typhoons. O ther names used

include “cordonazo” in Mexico, “bagyo” in the Philippines, and “tainos” in Haiti.

The kinetic energy of these large scale vortex motions is abou t 1018 Joules. Trop­

ical cyclones are among the most spectacular and deadly geophysical phenomena.

Tropical cyclones th a t cause extreme destruction are rare, although when they occur,

they can cause great amounts of damage or thousands of fatalities. The 1970 Bhola

cyclone is the deadliest tropical cyclone on record, killing more than 300,000 people

and potentially as many as 1 million after striking the densely populated Ganges

Delta region of Bangladesh on 13 November 1970. The N orth Indian cyclone basin

has historically been the deadliest. The G reat Hurricane of 1780 is the deadliest A t­

lantic hurricane on record, killing about 22,000 people in the Lesser Antilles in the

Caribbean. A tropical cyclone does not need to be extremely strong to cause memo­

rable damage; deaths can be caused by rainfall, flood, or mudslides. The Galveston

Hurricane of 1900 is the deadliest natural disaster in the U nited States, killing an

estimated 6,000 to 12,000 people in Galveston, Texas. Hurricane K atrina is estimated

as the costliest tropical cyclone worldwide, causing $ 81.2 billion in property damage

with overall damage estimates exceeding $ 100 billion. K atrina killed a t least 1,836

people after striking Louisiana and Mississippi as a major hurricane in August 2005.

Damage from Hurricane Sandy in 2012 is estim ated as $ 50 billion. Hurricane An­

drew (1992) is also one of the most destructive tropical cyclones in U.S history, with

damages totaling $ 40.7 billion.

Up until World War II, detection of tropical cyclones relied on reports from coastal

stations, islands, and ships a t sea. At this time, it is likely th a t many storms escaped

detection entirely, and some of them were observed only once or a few times during

C H APTER 1. INTRO DU CTION 6

the course of their life cycle. During World War II, m ilitary aircraft were tasked with

finding tropical cyclones tha t could pose a danger to naval operations. An aircraft

was able to penetrate the interior of strong storm s for the first tim e in July 27, 1943,

flying into the eye of a Gulf of Mexico hurricane. At about th e same time, the first

radar images revealed the structure of tropical cyclones including the eye and spiral

rainbands. A great step forward came in 1960, when the first image of a tropical

cyclone was transm itted from a polar orbiting satellite. Now all tropical cyclones are

recorded by satellite based measurements.

T h e s t ru c tu re o f a tro p ic a l cyc lone

The radius of a tropical cyclone is generally between 100 and 2000 km. A cyclone of

high intensity tends to develop an eye, an area of relative calm (and lowest atmospheric

pressure) a t the center of circulation. The eye is often visible in satellite images as a

small, circular, cloud-free spot. Surrounding the eye is the eyewall, an area about 16

km to 80 km wide in which the strongest thunderstorm s and winds circulate around the

storm ’s center. The maximum sustained winds in the strongest tropical cyclones have

been estimated at about 85 ms-1 . Maximum upflow occurs in the eyewall. In intense

storms, clouds over the eyewall may form a nearly concentric ring. There may be two

or even three eyewalls present a t the same time, evolving through a characteristic cycle

in which the outer eyewalls contract inward and replace dissipating inner eyewalls.

The earliest radar images of tropical cyclones showed th e outward-propagating

spiral rainbands tha t accompany them. They consist of bands of cumulus clouds,

some tens of kilometers wide with radial propagation speeds of about 4 ms-1 , some­

times connecting with the eyewall. Outside the eyewalls the w ind speed and intensity

decrease gradually at first and then fall off faster a t large radius.

Tropical cyclones are formed in four m ajor phases. The first stage occurs when a

CHAPTER 1. INTRO D U CTIO N 7

D*m* Cirrus Ovsrcsst

Figure 1.1: Tropical cyclon e stru ctu re. The main parts of a tropical cyclone are the eye, the eyewall, and the rainbands. This figure is taken from http://serc.carleton.edu/research education/katrina/understanding.htm l.

collection of small thunderstorm s forms over the tropical ocean. The hurricanes th a t

threaten the East and Gulf coasts of the United States, for example, are the product

of storms th a t first form over the coast of western Africa and head west across the

Atlantic Ocean. The cyclones th a t strike East Asia and A ustralia form over the central

Pacific. Australia is also affected by cyclones which form in th e North-W estern Indian

Ocean and Torres Strait. Tropical cyclones are formed between 5° and 20° latitude,

because of the high humidity, light winds, and warm sea surface tem peratures present

in these locations. Cyclones and hurricanes can only form w hen the ocean water is

around 26.5°C or higher, and these conditions are present in these latitudes during

certain months of the year.

Weather conditions in the central Atlantic and Pacific Oceans are most favorable

for hurricane formation during the summer and early fall m onths, which is why the

period of June to October is often referred to as “Hurricane Season” , in the north­

C H APTER 1. INTRO D U CTIO N 8

ern hemisphere. The opposite situation occurs in the southern hemisphere since the

tropical waters of Australia are warmest from December to April.

A tropical disturbance officially enters the second stage of development called a

tropical depression once the sustained wind speed w ithin the storm reaches 15 ms-1 .

The term tropical depression refers to the falling surface pressures measured in the

region surrounding the storm. The pressure drop occurs as w ater vapor within the

storm condenses into water droplets and releases latent heat into the atmosphere.

Latent heat is defined as the energy released when a substance changes phases, such

as gas to liquid in this case.

This process becomes a chain reaction th a t pulls hot, humid air from the surface of

the ocean up to high altitude where the air becomes cold and w ater vapor condenses

into thick clouds. This growing air mass becomes increasingly dense causing the

atmospheric pressure to grow. The increasing pressure pushes the growing mass of

clouds outward away from the center to create the spiraling bands of clouds th a t

hurricanes are known for. As the air mass spirals outward, its pressure decreases and

the dense air plunges back towards the ocean surface where i t started. It now picks

up vapor again from the warm waters below and is sucked back into the center of the

depression to begin its journey anew. As the cycle continues, the surface pressure at

the center drops lower, causing the circulation of the air to strengthen and the winds

to grow increasingly stronger.

Once the sustained wind speed increases to 18 ms-1 , the tropical depression enters

the third stage of development called a tropical storm. Until now, the storm has not

had any formal designation. Tropical depressions are usually numbered in a sequential

order as they appear, but it is not until they become tropical storm s tha t they receive

an official name. The tradition of naming storms dates back to the arrival of the

Europeans in the Americas, bu t the current m ethod of system atically naming tropical

CH APTER 1. INTRO DU CTION 9

storms was not formalized until 1953. Female names were used exclusively until the

late 1970s when the use of both male and female names in an alternating format was

adopted.

It is also during the tropical storm phase th a t the storm begins to take on its

characteristic appearance as a ro tating spiral w ith bands of clouds circulating around

the point of lowest surface pressure a t its center.

When the wind speed reaches 33 m s-1 , the tropical storm is officially classified as

a tropical cyclone or hurricane. This is the fourth stage of the hurricane development.

The storm now develops an eye a t the center of the circulating spirals where there

is very little wind, few clouds, and extremely low atm ospheric pressure. The eye

may extend from 8 to 65 km in diam eter. Surrounding the eye is an area called the

eyewall tha t circulates around the eye and contains the highest clouds. The eyewall

is the region containing the strongest winds, heaviest precipitation, and most intense

thunderstorms of any part of the hurricane. The most destructive region of the cyclone

is the portion of the eyewall ro tating in the direction of the forward motion of the

storm since the force of the circulating winds is combined w ith the motion of the

storm. Radiating outward from the center are the spiraling bands of clouds tha t

produce heavy rain and wind but are still considerably weaker than the eyewall.

Cyclones typically move westward a t a speed of about 4 m s-1 during the early

stages of formation. The storms are steered in this direction by the trade winds tha t

occur near the equator and blow towards the west. As a sto rm moves further from

the equator, its course is turned towards the poles and the cyclone curves towards the

north in the northern hemisphere or towards the south in th e southern hemisphere.

Tropical storms tha t form in the northern hemisphere always ro tate in a counter­

clockwise direction due to the Coriolis force induced by the E arth ’s rotation, while

tropical storms of the southern hemisphere ro tate clockwise.

C H APTER 1. INTRO D U CTIO N 10

E yew all rep la cem en t c y c le

Eyewall replacement cycles occur naturally in intense tropical cyclones. Fortner (1958)

was the first to document the concentric eyewall and eyewall replacement cycle in

Typhoon Sarah in 1956. W hen a tropical cyclone reaches its highest intensity, the

asymmetric ring of outer rainbands contracts and strengths and organizes into a ring

of thunderstorm s called an outer eyewall, or secondary eyewall, around the existing

inner eyewall (Willoughby et al., 1982). As a result, the inner eyewall weakens and is

eventually replaced by the more intense outer eyewall. This process takes place on the

order of a day or two (Sitkowski et al., 2011). This process of weakening followed by

reintensification is called the eyewall replacement cycle. Recent measurements have

shown tha t nearly half of all tropical cyclones attaining a t least 60 ms-1 wind speeds

undergo the concentric eyewall replacement cycle (Hawkins and Helveston, 2008).

In these hurricanes the asymmetric outer rainbands form the ir own convective ring

(secondary eyewall) in coincidence w ith a local tangential wind maximum around the

inner eyewall (Willoughby et al., 1982). Figure 1.2 shows a rad ar image of Hurricane

Ivan (2004) with a developed secondary eyewall.

The motivation to understand and forecast the eyewall replacement cycle is high,

because the process can have very serious consequences arising from dram atic intensity

and structure changes. The formation of an outer eyewall can cause a decrease in the

maximum wind speed at the location of the initial eyewall, while the outer wind

maximum tends to broaden the hurricane wind field which can lead to an increase of

integrated kinetic energy (Maclay et al., 2008). This has profound impacts both near

and far from coastal regions. A sudden expansion of hurricane force winds near landfall

can impact a larger coastal area while reducing preparation tim e. W hen a hurricane

is farther from shore, an increased hurricane wind field is likely to lead to a greater

C H APTER 1. INTRO D U CTIO N 11

04Q9MI Ivan• 4 • t - s t ’i j I T ' *c '3JS 38* '*>»>, /WO•1- 1; -.m r t f ’~& ■-. > ’ PM *•!«. i - - h r * * ‘ e t j r r t * ^*«wnr»

Secondary eyewall

Primary eyewall

Figure 1.2: H u rr ic a n e Ivan . Radar image of Hurricane Ivan (2004) showing the secondary eyewall formation around the prim ary eyewall.This image is taken from the website of the NOAA H urricane Research Departm ent http://www.aoml.noaa.gov/hrd/Storm_pages/ivan2004/i040909il640_1700c.jpg

C H APTER 1. INTRO D U CTIO N 12

storm (Irish et al., 2008). In addition, the contraction of an outer eyewall near the end

of an eye replacement cycle can sometimes lead to rapid intensification resulting in a

more intense hurricane than when the process of eyewall form ation began. This was

the case for Hurricane Andrew (1992), which intensified to a category 5 hurricane as

it approached the southeast coast of Florida during the end of a n eyewall replacement

cycle (Willoughby and Black, 1996). Observations of the hurricane season of 2005

(Houze et al., 2007) showed th a t the winds in the secondary (or outer) eyewall were

initially weaker than those in the original prim ary (or inner) eyewall, bu t as the

secondary eyewall contracted, the storm reintensified. The processes occurring in the

moat region between the old and new eyewalls became dynam ically similar to those

in the eye.

C hapter 2

Basic concepts o f geop h ysica l fluid

dynam ics

In this chapter I give a basic description of geophysical fluid dynam ics and its appli­

cation to wave and vortex processes in the atmosphere.

The term geophysical fluid dynamics refers to the dynam ics of fluid flows, partic­

ularly air and water, in the E arth ’s environment. The discipline includes the study of

both fluid phases: liquids (waters in the ocean) and gases (in th e E arth ’s atmosphere

and in the atmospheres of other planets). Usually geophysical fluid dynamics deals

with large-scale flows, and the rotation of the E arth or density differences (warm and

cold air, fresh and saline waters) are im portant for these processes. Typical problems

in geophysical fluid dynamics concern the variability of the atm osphere (weather and

climate dynamics), of the ocean (waves, vorticity and currents),the motions in the

E arth ’s interior responsible for the magnetic dynamo effect, vortices on other planets

(such as Jupiter’s G reat Red Spot), and convection in stars (the Sun, in particular).

We live within the atmosphere and are ever affected by th e weather and its rather

chaotic behaviour. The motion of the atmosphere is connected with th a t of the ocean,

13

C H APTER 2. BA SIC CO NCEPTS OF G EO PH YSICAL FLUID D YN AM IC S 14

with which it exchanges fluxes of momentum, heat and m oisture, and it makes the

dynamics of the ocean as im portant as the atmosphere.

The two features th a t distinguish geophysical fluid dynam ics from other areas of

fluid dynamics are the rotation of the E arth and the vertical density stratification of

a medium because of the vertical gradient of tem perature or salinity.

The effect of the rotation of the E arth takes the form of th e Coriolis force and

gives rise to large scale oscillations in the atmosphere and ocean which are known

as Rossby waves. The effects of stratification and the gravitational force give rise to

internal gravity waves and surface gravity waves. These waves have profound effects

on weather and climate and play a role in various observed phenomena including

the development of tropical cyclones. This thesis deals with vortex Rossby waves

which propagate in a tropical cyclone and have an influence on the cyclone strength,

structure and dynamics.

The motion of fluid in the atmosphere and the ocean is naturally studied in a

coordinate frame rotating with the Earth. In this chapter we discuss the basic theory

of Rossby waves in a geophysical fluid th a t is affected by th e Coriolis force. We

describe these effects in the governing equations following Landau and Lifshitz (1953),

Kundu and Cohen (2004), Holton (1992), and Gill (1982).

2.1 B asic equations o f fluid m ech an ics

The study of fluid mechanics is based on the laws of conservation of mass, momentum,

and energy.

C H APTER 2. BA SIC CO NCEPTS OF GEO PH YSICAL FLUID D YN AM IC S 15

C on serva tion o f m ass

The conservation of mass equation is

^ + V • (pu) — 0, (2.1)

where p is the density and u is the fluid velocity. This equation is called the continuity

equation and expresses the differential form of the principle of conservation of mass.

C on serva tion o f m o m e n tu m

The momentum equation

Dui dp d~ + P 9 i +Dt dxj dxj

. dui duj . _ _+ 3 ' , v ' u '5”

(2 .2 )

is known as the Navier-Stokes equation. The notation refers to the full derivative,

D d d dxi .D t = dt + d x i~ d t ' ^

Here p is the pressure, g is the gravity acceleration, p is the coefficient of dynamic

viscosity, i and j correspond to coordinate axes, the repeated indices mean summation

over i — 1 ,2 ,3 or j = 1,2,3. For an incompressible fluid the momentum equation is

written in the form:DUi d p cy

p ^ t = ~ a i i + l,9- + 'i V ( 2 ' 4 )

or in vector form asD u 0

p— = - V p + pg + p V u. (2.5)

For atmospheric dynamics the viscous term in the m om entum equation is small

CH APTER 2. BA SIC CO NCEPTS OF G EO PH YSICAL FLUID D YN AM IC S 16

relative to other terms. For hurricane dynamics the length scale is L ~ 105m and

the velocity scale is U ~ 10 m s-1 . The kinematic viscosity of the atmosphere is

u = n /p ~ 10~5m2s_1. Hence, the Reynolds number which is a measure of the ratio

of inertial forces to viscous forces Re = U L /u ~ 1011 is very large for hurricane-scale

flows, and we can neglect the viscous term s in the momentum equation

D u _ , ,p— = - Vp + pg. (2.6)

C o n serv a tio n o f e n e rg y

The conservation of energy equation is

p|5 = - „ ( V . u ) + 4 > - | | , ( 2 .7 )

where e is the internal energy of the system, vector q is the h ea t flux through a unit

area, (j> is the rate of viscous dissipation. This is called the therm al energy equation.

The set of equations (2.1), (2.2) and (2.7) is used as the s ta rting point for studies of

fluid dynamics.

2.2 E ffects o f ro ta tion

In studying the motion of the fluid in the atm osphere we often have to take into

account the rotation of the Earth. The E arth ’s rotation is im portan t when the scales

of the processes under consideration are large and comparable w ith the E arth ’s radius.

Following the description given in Kundu and Cohen (2004) and Holton (1992), we

consider a frame of reference ( a q , :^ ,^ ) ro tating w ith a uniform angular velocity f I

CH APTER 2. BA SIC CO NCEPTS OF G EO PH YSICAL FLUID D YN AM IC S 17

Figure 2.1: V ector in a ro ta tin g fram e. The vector i ro ta tes with the reference frame. O is the angular velocity of the frame rotation, a is th e angle between the vector i and the axis of rotation. The change in the angle of ro tation d9 causes the change in the vector di.

with respect to a fixed frame. Any vector P is represented in th e rotating frame by

P — P iii + P 2I2 + ^313- (2 .8 )

To a fixed observer the directions of the rotating unit vectors i i , i2 and i3 change with

time. To this observer the time derivative of P is

dPi dP2 dPi dt 12 dt 13 dt

where we have used the notation dij /dt , j = 1,2,3, to represent the rates of change

of the unit vectors. To an observer in the ro tating frame, th e rate of change of P is

CH APTER 2. BA SIC CO NCEPTS OF G EO PH YSICAL FLUID D YN AM IC S 18

the sum of the first three terms in (2.9), so th a t

(f~)F = ( ^ ) c Pl + P2 + Ps ’ ( 2 ' 1 0 )

where the subscript of F refers to the fixed frame and the subscript of R to the

rotating frame.

Each unit vector i traces a cone with a radius of sin a, where a is a constant angle

between the vector and axis of rotation (Figure 2.1). The m agnitude of the change of i

in time dt is |di| = sin adO, which is the length traveled by the t ip of i. The magnitude

of the rate of change is therefore |d i/d i| = sin a{d6fdt) = |Q| s in a , and the direction

of the rate of change is perpendicular to the (fi, i)-plane. Thus, (d ijd t ) = SI x i for

any rotating unit vector i. The sum of the last three term s in (2.10) can thus be

w ritten as P\Pt x i t + P2Pt x i2 + P^Yl x i3 = f t x P . E quation (2.10) then becomes

( S ) , - (? )„ * » -We can use this argument to derive an expression for the Coriolis force. Consider

the point in the atmosphere with position vector r. The application of the rule (2.11)

to the vector r gives

u f = u jj + 11 x r (2-12)

where u F = ( ^ ) F and u F = ( ^ ) R- Applying (2.12) to the velocity vector u F gives

du .p \ { d u p \ . .+ £2 x u F. (2.13)

dt ) F y dt j ft

C H APTER 2. BA SIC CO NCEPTS OF G EO PH YSICAL FLUID D YN AM IC S 19

Q

R -Q2R

Figure 2.2: C e n tr ip e ta l a c c e le ra tio n . The centripetal acceleration — fl2R is per­pendicular to the axis of rotation. f2 is the angular velocity of the rotating frame, r is the position vector of a point in the frame, a is the angle between the vector f and the axis of rotation, R is the projection of r onto the plane perpendicular to the axis of rotation.

We then substitute the expression (2.12) into the right-hand side of (2.13) to get

So the acceleration in the fixed frame is given in term s of th a t in the rotating frame

— ) + n x u f l + f l x Q x r .d t J R

f i x r ) + f2 x (u r + SI x r)R

(2.14)

by

= a# + 2 fI x u r + f i x (f i x r). (2.15)

The last term in (2.15) can be w ritten in terms of a vector R drawn perpendicular

C H APTER 2. BASIC CO NCEPTS OF G EO PH YSICAL FLUID D YN AM IC S 20

to the axis of rotation (Figure 2.2). Clearly from Figure 2.2, f t x r = O x R because

|0 x r | = |0 | | r | s i n a = |0 | |R |. (2.16)

Using the vector identity A x (B x C) = B (A C ) — C (A B ) we can rewrite the last

term in (2.15) as

n x (ft x R ) = f t( f tR ) - |0 |2R . (2.17)

Since O and R are perpendicular, f t ■ R = 0, and equation (2.15) becomes

ap = + 2O x u/j — |0 |2R . (2.18)

So the inertial acceleration in the fixed frame ap equals the acceleration measured

in the rotating system, plus the Coriolis acceleration 2f2 x u and the centripetal

acceleration — |0 |2R.

In the momentum equation (2.2) we replace the term p ^ f - by p™ + 2 0 x u —

|0 |2R . The centripetal acceleration —|0 |2R is included in gravity for the geopotential

surface of geoid. So in the ro tating frame of reference, the m om entum equation (2.6)

can be written as

= ~ p V p ~~ 2Q X U + g ’ 2 ' 19^

where the term 2 0 x u is defined as the Coriolis acceleration.

C H APTER 2. BA SIC CO NCEPTS OF G EO PH YSICAL FLU ID D YN AM IC S 21

noflhpole

Figure 2.3: L ocal C a r te s ia n c o o rd in a te s . The local coordinate system rotates with the Earth. Q is the angular velocity of the E arth ro tation. 0 is the latitude of the origin of the local coordinate system, x is the west-east horizontal variable, y is the south-north horizontal variable, z is in the vertical direction. This figure is taken from Gill (1982).

2.3 T he C oriolis effect for a th in layer on a ro ta tin g

sphere

The atmosphere and the ocean are very th in layers in which the depth scale H of

the flow is a few kilometers, but the horizontal scale L is of the order of hundreds

or thousands of kilometers. We consider a coordinate frame ro ta ting with the Earth.

The Earth rotates with an angular speed |f2| — Q. — 0.73 • 10-4 s-1 . In the local

coordinate system (x , y , z ), the angular velocity is Cl = (0, f lc o s# , fls in # ), where 6 is

latitude (see Figure 2.3). The Coriolis force F is

F — 2pfl x u = 2p(wQcos9 — u fls in# , u fls in # , — uQcosO). (2.20)

C H APTER 2. BA SIC CO NCEPTS OF G EO PH YSICAL FLUID D YN AM IC S 22

Since the vertical component of the velocity is small com pared with the horizontal

components w v, w u, the Coriolis force can be approxim ated as

2 pfl x u « p(—2vfl sin 0, 2 ufi sin 0, —2uflcos 0).

The quantity f = 2Q sin 0 is called the planetary vorticity, or th e Coriolis parameter.

B a ro tro p ic flow

A barotropic flow is a fluid flow in which the pressure is a function of the density only,

i.e. p — p(p). It is a flow in which isobaric constant pressure surfaces are isopycnic

surfaces (surfaces of constant density). A two-dimensional flow on a horizontal plane

where density variations are neglected is an example of a barotropic flow.

T he /-p la n e approxim ation

The Coriolis param eter / = 2 fis in# varies w ith latitude 0. The variation is im portant

only for processes having very long time scales or very long length scales. For small

scale processes the Coriolis param eter / can considered to b e constant: f = fo =

2flsin0o where 0O is a fixed latitude under consideration. T his is called the /-p lane

approximation.

T he /3-plane approxim ation

A better approximation for the Coriolis param eter / which takes into account the

variation of / w ith latitude can be obtained by linearizing / ab o u t some fixed latitude

do. We consider a plane tangent to the surface of the E arth a t the latitude On and

define Cartesian coordinates x going from west to east and y going from south to

C H APTER 2. BASIC CO NCEPTS OF G EO PH YSICAL FLUID D YN A M IC S 23

north on this plane. We write

/ = 20 sin d — 20. sin[(0 — 0O) + 0O] = 2 0 [sin(0 — 60) cos do +• eos(0 — do) sin 0O] •

(2 .21 )

For latitudes d close to d0, d — d0 is a small angle, so cos(0 — do) ~ 1 and sin(0 — do) ~ 0.

We use the arclength formula to express the south-north coordinate on the plane in

terms of the latitude by y = Rsin(d — do) where R is the radius of the Earth. The

Coriolis param eter / can then be approxim ated as

/ = 2f2(sin 0O + ^ cos d0) = fo + @y, (2.22)

where we define the constants fo and ,6 as

fo = 20sindo, (2.23)

l i

This is called the ^-approximation. I t approximates the curved surface of the Earth

by a horizontal plane.

2.4 W aves in th e ocean and th e a tm osp h ere

Barotropic vortex Rossby waves are examples of waves th a t occur in the atmosphere.

M athematically they are represented as sinusoidal perturbations with specified wave­

lengths and amplitudes. We write a sinusoidal one-dimensional wave in the form

CH APTER 2. BASIC CO NCEPTS OF G EO PH YSICAL FLUID D YN AM IC S 24

where the argument 2 n (x —ct)/X is called the phase of the wave, and a is the amplitude.

The param eter A is called the wavelength. The wavenumber is defined as

k = y , (2.26)

which is the number of complete waves in a length of 2tt. The period T of a wave is

the tim e required for the wave to travel one wavelength:

T = (2.27)

The number of oscillations per unit time at a fixed point in space is the frequency,

given by

v = (2.28)

Clearly, c — Xv. The quantity

oo — 2it v — kc (2.29)

is called the circular frequency. The speed of propagation of th e wave is given by

c = y - (2.30)k

This is called the phase speed, as it is the ra te a t which the ‘phase’ of the wave (i.e.

the crests and troughs) propagates. In general the circular frequency is a function

depending on the wavenumber oo = oo(k). The relation between oo and k is called

a dispersion relation oo = oo(k). If the phase speed oo/k is not constant then the

wave is called dispersive. If we consider a wavepacket, which comprises a spectrum of

waves of different frequencies and wavenumbers, dispersive waves from a wavepacket

propagate with different speed depending on the frequency, and the observed pulse

C H APTER 2. BASIC CO NCEPTS OF G EO PH YSICAL FLUID D YN AM IC S 25

changes shape. If ui/k is constant, the wave is nondispersive, and all waves in a

wave packet propagate with the same speed. For the wavepacket we define the group

velocity as

c» = | (2.31)

For non-dispersive waves the group velocity cg is equal to the phase velocity c. For

dispersive waves cg ^ c.

C hapter 3

V ortex dynam ics in th e atm osphere

3.1 B arotropic v o rtic ity eq u ation

The goal of this thesis is to investigate the dynamics of hurricanes and vortex waves

using a mathematical framework based on the equations of m otion for a barotropic

fluid flow on a horizontal plane tangent to the surface of the E arth . The model com­

prises a vortex with Rossby waves which are generated near th e centre and propagate

outwards. It includes time dependence, the effect of the Coriolis force and nonlinear

interactions between the vortex and the waves. However, it is simple enough to allow

us to derive approximate analytical solutions which give us som e insight into possible

mechanisms tha t can lead to the secondary eyewall formation and replacement cycle.

In this chapter we derive the governing equations for our model, i.e. the equa­

tions for barotropic vortex Rossby waves, in both rectangular and polar coordinates.

We also discuss previous work on the subject of vortex Rossby waves and hurricane

dynamics.

Analyses of tropical dynamics (Holton, 1992, section 11.2) show th a t in the absence

of condensation heating, the synoptic-scale dynamics of the tropical atmosphere is

26

C H APTER 3. VO RTEX D YN A M IC S IN T H E A T M O SP H E R E 27

approximately barotropic when the vertical scale is comparable with the scale height

of the atmosphere. So in this thesis we shall describe the hurricane dynamics by the

equations of motion for a barotropic fluid flow. We derive the equations in terms of

Cartesian coordinates, x and y in a horizontal plane tangent to the surface of the

Earth, and z in the vertical direction. The corresponding com ponents of the velocity

u are u, v, and w.

We derive the potential vorticity equation following H olton (1992) and Kundu

and Cohen (2004). Large-scale atmospheric flow can be considered as barotropic and

incompressible (see, e.g., Holton, 1992, section 4.5). For such a flow the continuity

equation (2.1) simplifies todu dv dw „ ,Tx + Ty + l T z = 0 - ^

We consider a shallow layer of fluid of depth h, where h is a function of x, y, and t. Let

y (x ,y , t ) be the vertical displacement of a fluid particle in th e flow, and h ( x , y , t ) =

7y2(x,y, t) — rji(x ,y ,t) , where rp is the vertical displacement o f the liquid particle at

the bottom of the layer, and % is the vertical displacement a t its surface. In the

shallow wave approximation it is assumed th a t if there are waves in the flow, their

wavelength A is much larger than h, the depth of the layer (h/X <C 1). In this

case the vertical velocities are much smaller than the horizontal velocities. Then the

acceleration d w /d t is negligible in the vertical component of th e momentum equation

(2.6). The pressure distribution is assumed to be hydrostatic,

P = Po + P9V, (3.2)

where po is the pressure a t some reference height. The horizontal pressure gradients

CH APTER 3. V O RTEX D YN A M IC S IN T H E A T M O SP H E R E 28

are therefore

1 dp dp p d x 9 d x ’ 1 dp dp p d y 9 d y '

(3-3)

Holton showed (Holton, 1992, section 11.2) th a t in the tropical troposphere a

typical scale of the vertical velocity is 0.003 ms-1 . For the hurricane dynamics a

typical horizontal velocity scale U is 10 ms-1 . The vertical length scale H is 104m,

equation is W /H which is 3 x 10-7s-1 . Scale of the horizontal derivatives and

is U /L which is 10-4s-1 .

So we see tha t W /H -C U /L and can consider to be close to zero, and trea t

the vertical velocity as being independent of 2 . Hence, from th e continuity equation

(3.1) the sum ^ is independent on 2 as well. Integrating (3.1) with respect to 2

across the layer of fluid from z — p 1 a t the bottom to 2 = % a t the surface, we obtain

a t the bottom of the layer. The difference in velocity between the surface and at the

bottom is given by

and the horizontal length scale L is 105 m. So the scale of the te rm “ in the continuity

k d i + n&y + ~ = ° ’(3-4)

where w(p2 ) is the vertical velocity a t the surface and w(pi) is the vertical velocity

w(p2) - w(pi) =P ( V 2 ( x , y , t ) - p i(x ,y , t ) ) = D h (x ,y , t ) = dh

D t D t d td h dh.

+ U — b V-

CH APTER 3. V O R T E X D YN A M IC S IN TH E A T M O SP H E R E 29

and the sum of equations (3.4) and (3.5) can be rew ritten as

^ + - k iKh] + d i lvh] = 0- <3'6)

Writing the pressure in term s of rj (3.4) for a fluid shallow flow in a rotating frame,

we obtain the horizontal components of the momentum equation (2.19) in the form

du du du „ dn ,a i + u a i + v B y ^ v = - s 8 i ' (3'7)dv dv dv „ duM + u a x + v a i + f u = ~ % - (38)

A vorticity equation can be derived by differentiating (3.7) w ith respect to y , (3.8)

with respect to .x, and subtracting. We denote the vertical com ponent of the relative

vorticity V x u bydv du

i = Tx -ay- ( 3 ' 9 )

In the /3-plane approximation / depends on y only, / = /o + Py- Then from (3.7),

(3.8) and (3.9) we obtain

<3 >°>

We can write equation (3.6) as

D h , / du d v \D i + h \ d i + d ^ J = ° ( 3 U )

and then use it to eliminate the horizontal divergence d u /d x + d v /d y from (3.10) to

C H APTER 3. V O RTEX D YN A M IC S IN T H E A T M O SP H E R E 30

obtain

We make use of

to write (3.12) as

D t h D t

This equation can be w ritten in the compact form

DC (C + f ) D h _ B fm = — T - D i ~ v T v <312>

~Dt = ~dy (3' 13)

D(C + / ) _ (C + /)D fc

D t \ h

This is the potential vorticity conservation law in the shallow-water approximation.

If the flow is horizontal or at least approxim ately horizontal then we can consider

w = 0 and from (3.4) obtain tha t d u /d x + d v /d y = 0. In this case (3.10) is simplified

to

I + ”| = °- <3-16>

and the barotropic potential vorticity equation is

^ ( C + / ) = 0 . (3.17)

This equation could also be derived by using the baroclinic potential vorticity equation

(Holton, 1992, section 4.6):

§ t (Q + f ) + (C» + / ) v , - u = - k • V„ x , (3.18)

where k is the unit vector in 2-direction, 9 is the potential tem perature, and the

CH APTER 3. VO RTEX D YN A M IC S IN TH E A T M O SP H E R E 31

subscript 9 means tha t the derivatives are taken on a surface w ith constant 9. The

potential tem perature is by definition the tem perature tha t a parcel of dry air at

pressure p and temperature T would acquire if it were expanded or compressed adia-

batically to the reference pressure ps = 1000 mb:

0 = t ( ^ J (3.19)

where k = R /c p, R is the gas constant for the dry air, and cv is the specific heat at

constant pressure. The notation ^

D & „ ,_ _ _ _ + u . Ve (3.20)

is the to tal derivative on a surface where 9 = constant.

If we make the assumption th a t the flow is barotropic, equation (3.18) simplifies

to the equation (3.15) which we derived using the shallow w ater approximation. To

show this, we note tha t

£^((0 + /) + (Ce + f ) ^ e ■ u = + / ) • (3.21)

We consider a tropical cyclone to be a two-dimensional vortex in a horizontal plane.

For an ideal gas, 9 is constant on a barotropic surface. For th is case the operator V#

is simplified to the horizontal component of the operator V, and the vertical vorticity

Q — £. We consider adiabatic flow, so D 9 /D t — 0. In th is case, equation (3.18)

reduces to equation (3.15)

J^ (C + / ) = 0 . (3.22)

We will use this equation to obtain an equation for the propagation of barotropic

C H APTER 3. VO RTEX D YN A M IC S IN TH E A T M O SP H E R E 32

Rossby waves.

We will consider a tropical cyclone as a vortex where the vertical scale H is smaller

than the horizontal scale L, H /L <C 1. As discussed above in this section, tropical

dynamics can be considered to be a barotropic flow and we can neglect the vertical

variation of the flow and consider the problem as two-dimensional. So the continuity

equation can be approximated by

where the subscripts denote partial differentiation. This allows us to define a stream-

function 'S>(x,y,t) by

ux + vy = 0, (3.23)

u = - tyy , v = \&x. (3.24)

The vorticity can then be expressed in terms of the stream function as

C = VX ~ U y = V 2<£ . (3.25)

We can rewrite equation (3.22) in term s of the stream function as

# ( V 2* + / ) QDt

(3.26)

On a /3-plane the Coriolis param eter f — fo + fly, so we can write

(3.27)

or in the form

(3.28)

CH APTER 3. VO RTEX D YN A M IC S IN TH E A T M O SP H E R E 33

This equation is called the barotropic vorticity equation. We will use it to describe

barotropic Rossby waves.

To simplify further consideration of the problem let us consider the nondimensional

form of the barotropic vorticity equation. Let us denote all dimensional variables by

an asterisk * and write the dimensional barotropic vorticity equation as:

v (■W ) " ar A ) + s ^ v U f ) = - / r ( 3 ' 2 9 )

Suppose the characteristic length scale of our problem is L and the characteristic

velocity is U . Then the characteristic time scale is T = L /U and the characteristic

streamfunction is ^ = UL. We define the nondimensional variables in terms of the

corresponding dimensional ones as follows:

This means th a t the derivatives in term s of the non-dimensional variables will be:

d I d dx* L dx ’A - J Ady* L dy ’ d_ _ \^d_

dt* ~ T d t

(3.31)

(3.32)

(3.33)

We write equation (3.29) in term s of the non-dimensional variables as

After simplification, we obtain the nondimensional barotropic vorticity equation in

CH APTER 3. VO RTEX D Y N A M IC S IN TH E A T M O SP H E R E 34

the form

V 2* , - tfvV 2# x + 'M V 2'^ ) - (3.35)

The parameter f3 in (3.35) is the nondimensional gradient of th e Coriolis param eter.

To determine the magnitude of /3 we note th a t the dimensional /?*, according to (2.24),

is

3 ’ = ^ 3 3 c o Se0, (3.36)f i 'E a r th

where 6q is the latitude of the centre of the vortex. Since the vortex is located in the

tropics, Oq is close to zero. So, with REarth ~ 6.3 x 106km an d the angular velocity

of the E arth ’s rotation P lE a r th ~ 7 x 10"5 s” 1, the value of (3* is close to 2.2 x 1CT11

m _1s_1. The dimension of /3* is m -1s-1 . So, the characteristic value of (3 is B — U /L 2

where U is the characteristic velocity of a hurricane, and L is its radius.

A typical radius of the vortex L is about 2 — 3 x 105 m, and a typical speed of the

tangential wind U is about 30 — 100 m s-1 . So the nondimensional j3 is

/3 2QE a r th L 2 2 f r , o ^ n ,

v j i ? = 77 ' (3-37)

We can therefore consider j3 to be a small param eter in our problem , i.e. j3 <C 1.

3.2 B arotropic R ossb y w aves

The barotropic vorticity equation (3.28) can be used to describe the propagation of

waves th a t are known as barotropic Rossby waves. The stream function T and vorticity

V 2T axe considered to be the to tal stream function and to ta l vorticity, respectively,

each comprising a mean quantity corresponding to the background fluid flow, and a

perturbation quantity representing the waves. If the wave am plitude is considered

C H APTER 3. V O RTEX D YN A M IC S IN TH E A TM O SP H E R E 35

small relative to the magnitude of the mean flow, then the equation can be linearized.

If the background flow is taken to be the zonal-mean of th e zonal wind and the

mean speed assumed to be a constant f/, then the waves can be represented in the

form

$ = A e i { k x + l y - * t ) (3 .38)

where x and y are rectangular coordinates, the wavenumbers k and I determine the

direction of phase propagation in the x and y directions, A is th e wave amplitude, and

u is the wave frequency. Substituting this form of the wave into the vorticity equation

(3.28) and linearizing it we obtain the dispersion relation for barotropic Rossby waves

“ = V k - J h - <M9>

The group velocity is the gradient of to in the wavenumber space

. dui . du>Cs = l l d k + i2 ~dl• (3'40)

The phase speed isuikij. + Ii2)

c = e + p (3-41)

and the phase speed in the x-direction can be w ritten in the form

kM U k2 (3k2Cx~ k 2 + P ~ k 2 + P (fc2 + /2)2'

(3.42)

The negative sign of the /3-term shows th a t the phase propagation is westward relative

to the mean flow. If the eastward current cancels the westward phase speed, giving

cx = 0, stationary waves are formed.

C H APTER 3. VO RTEX D YN A M IC S IN TH E A T M O SP H E R E 36

3.3 V ortex R ossb y w aves

For studying vortex dynamics it is convenient to use polar coordinates, r and A, where

r is the radial distance from the centre of the vortex, and A is the angle measured

from a reference direction, usually to the east. The polar coordinates are defined

as r = y /x 2 + y2, A = arc tan(x /y ), or x — r cos A and y = rsinA . The velocity

components are vradiai and vaziTnuthai in the directions of r and A respectively. In polar

coordinates the continuity equation

V u = 0 (3.43)

is w ritten as1 9 1 Q~ ~a~(j”Vradial) 4 ~^rVazimuthal 0- (3.44)r a r r o A

The relations between the stream function T (r, A, t) and the velocity components are

1 TVradiai * A 5r

Vazimuthal ^ r- (3.45)

The x and y derivatives of the stream function can be w ritten in polar coordinates as

4/^ — v — Vradiai cos A “I- sin A 4/a cos A “I- sin A,r

t y y = - u = vazimuthai sin A - Vradiai cos A = 4>r sin A + - ' F a c o s A. (3.46)

Substituting (3.46) into (3.28), we can write the barotropic vorticity equation in polar

coordinates as

V 2^ t - - ^ AV 2^ r + - * rV 2tfA - —4/a sin A + /M r cos A = 0. (3.47)r r r

CH APTER 3. V O RTEX D YN A M IC S IN TH E A T M O SP H E R E 37

The basic flow in a cyclonic vortex is generally taken to be a uniform rotation of

the vortex depending on the radial variable r and independent of the azimuthal angle

A and time t, and it can thus be w ritten as (0, v(r)). The stream function ip(r) for the

basic flow is given by

v(r) = (3.48)

where the prime denotes differentiation w ith respect to r. We also define the angular

velocity D(r) of the basic flow by v = f2(r)r.

The total streamfunction ^ ( r , A, t) and angular and radial velocity components are

each decomposed into a contribution from the steady basic flow and a time-dependent

perturbation representing the waves as follows,

\&(r, A, t) = ^ ( r ) + e^ (r, \ , t ) , (3.49)

where

V a z i m u t h a l ( r , A, t) = v{r) + £v(r, A, t ) (3.50)

and

Vradial{r, A, t ) = eu(r, A, t ) . (3.51)

It is assumed tha t u, v and ip are 0 (1 ), so th a t the param eter e gives a measure

of the magnitude of the waves relative to th a t of the basic flow. Using (3.49) - (3.51),

the equation (3.47) can be w ritten in terms of the stream function perturbation as

d v d \ 2 , 1 / ® /- l - \ P , ■ i a i \ P - x— + V ip ip \Tr(vr + - v ) W\ sin A -I- +p-ipr cos A—v cos A =at r oX J r or r r e

- ip \V 2ipr)- (3.52)

The f term in (3.52) is present because the basic flow velocity v(r) does not satisfy

C H APTER 3. VO RTEX D YN A M IC S IN TH E A T M O SP H E R E 38

equation (3.47) with the /3 term s included.

We s tart our analyses of equation (3.47) by considering th e case of waves on an

/-p lane, where the Coriolis param eter / is approxim ated by / 0, a nonzero constant.

In this configuration we set /3 = 0 in equation (3.52) and obtain

d d d A 1 d . 1 , s . r* o , . .— -I- ) V ip ii)xw-{vr + - v ) = — (VvV ipr). (3.53)at r oX J r dr r r

It assumed also th a t the wave am plitude is small relative to the magnitude of the

background flow. So the param eter e <C 1. This allows us to linearize equation (3.53)

and obtain

( l + IOv2 ^ ({v+ )=o' <3'54)

We derive approximate analytical solutions to this linear equation first for the case of

waves with an amplitude th a t does not depend on time (chapter 4), and then for the

case where the wave am plitude evolves with tim e (chapter 5). In the time-dependent

configuration we derive asymptotic solutions th a t are valid for late time. In the limit

of infinite time these time-dependent solutions approach th e corresponding steady

solutions derived in chapter 4. Section 6.1 gives a weakly-nonlinear analysis of the

nonlinear equation (3.53) in which e is considered to be a sm all param eter but the

/3-effect is neglected. This corresponds to the situation where /3 -C e -C 1.

The /3-effect is included in sections 6.2 and 6.3. The goal is to include both

nonlinearity and the /3-effect; however as a first step, we investigate the effect of

adding /3 terms to the linear problem (section 6.2). This corresponds to a situation

where e /3. In tha t case the term proportional to | is larger than the 0 (1 ) terms

in the solution and can not satisfy the specified 0 (1 ) boundary conditions. In order

to derive a solution in this case we need to neglect this term. We can do so by adding

an extra term B(r, A) = —-ncos(A ) to (3.47). The main focus of this thesis is on the

C H APTER 3. VO RTEX D Y N A M IC S IN TH E A T M O SP H E R E 39

configuration where /3 <§; e which is the more likely situation to occur in our problem

where /? ~ 1CT2 for hurricanes with a length scale L ~ 105m. In th a t case the 0(/3/e)

term in equation (3.52) gives rise to an extra term in the solution but does not change

the qualitative behaviour of the solution. We will discuss the influence of 0 (/? /e) term

on the solution in section 6.3.

3.4 P revious th eo retica l stu d ies o f tro p ica l cyclon e

dynam ics

Several theories have been advanced to describe hurricane dynam ics, the physics of

outward propagating hurricane bands, secondary eyewall form ation and other features

of the hurricane structure. Some theories describe the spiral bands as inertia-gravity

waves (Abdullah, 1966, K urihara and Tuleya, 1974, K urihara, 1976). Some more

recent papers describe them as potential-vorticity disturbances (Guinn and Shubert,

1993), while another theory hypothesizes th a t symmetric instability plays an impor­

tan t role in the spiral band formation (Willoughby at ah, 1984).

MacDonald (1968) was the first to propose and qualitatively describe the spiral

bands as vortex Rossby waves. Montgomery and Kallenbach (1997) extended the the­

ory of vortex wave propagation by unifying the physics of barotropic vortex axisym-

metrization and vortex wave propagation in rapidly ro tating vortices. Montgomery

and Kallenbach (1997) developed an inviscid model where the waves propagate radially

outward from the original eyewall region as spiral bands and participate in wave-mean

flow interactions. They integrated the linearized vorticity equation on an /-p lane for

the case of a symmetric vortex with a vorticity profile decreasing monotonically with

radius. They derived an exact solution for the wave com ponent corresponding to

C H APTER 3. V O RTEX D YN A M IC S IN TH E A T M O SP H E R E 40

wavenumber one and an approximate solution for the higher wave numbers.

This solution was extended by Schecter and Montgomery (2004). They studied the

conditions under which the potential vorticity in the core of a tropical cyclone becomes

vertically aligned and horizontally axisymmetric. They showed th a t the amplitude of

vortex Rossby waves decays exponentially outside the core of the cyclone, and the

decay rate depends on the gradient of potential vorticity a t th e critical radius.

Brunet and Montgomery (2002) studied the linear initial-value problem for waves

propagating on a smooth circular vortex on an /-p lane. They developed a theory for

a barotropic configuration for a vortex of finite depth w ith a nonzero Rossby radius

where p is the density, D is the depth of the flow, and / is the Coriolis parameter. It

was shown tha t the non-dimensional evolution equation for the potential vorticity per­

turbation depends on one param eter, involving the azimuthal wavenumber, the basic

state potential vorticity gradient and the Rossby radius of deformation. The time-

dependent initial-value problem for this configuration was solved in Hankel transform

space.

Brunet and Montgomery (2002) and Schecter and M ontgomery (2004) discussed

the influence of inertia-buoyancy oscillations, i.e. gravity waves, on the vortex dynam­

ics. They concluded tha t these oscillations are im portant for cyclones in the middle

latitudes. Schecter and Montgomery (2004) showed th a t the influence of vortex waves

exceeds the effect of gravity wave propagation on the vortex dynamics. Hendricks et al.

(2010) included the inertia-gravity waves in their numerical sim ulations of the vortical

motion in a hurricane. They concluded th a t inertia-gravity waves are insignificant in

the process of intensification or decay of the vortex.

Martinez et al. (2010a), M artinez et al. (2010b), M artinez et al. (2011) used

the empirical normal mode technique to isolate vortex waves from other datasets.

of deformation. The Rossby radius of deformation is proportional to lr

C H APTER 3. VO RTEX D YN A M IC S IN TH E A TM O SP H E R E 41

They studied the impact of vortex waves on the hurricane dynam ics and intensity,

concentric eyewall genesis, and other main features of a m ature tropical cyclone. The

results of the analyses of their simulations led them to conclude th a t the main features

of tropical cyclones can be explained by vortex Rossby waves.

In summary, the conclusions of the above mentioned studies in the hurricane dy­

namics suggest th a t vortex Rossby waves play a more significant role than inertia-

gravity waves. This justifies (at least as a first approxim ation) the representation of

the hurricane dynamics using a barotropic model th a t includes propagating Rossby

waves on a horizontal plane bu t no inertia-gravity waves or o ther three-dimensional

effects.

The subject of secondary eyewalls and their formation is one of the most im portant

research topics in the dynamics of tropical cyclones. However, there is as yet no unified

theory to explain secondary eyewall formation. Kossin and Sitkowski (2009), Sitkowski

et al. (2011) developed empirical models based on observational datasets to predict the

hurricane dynamics and the appearance of the secondary wind maximum. Willoughby

and Shapiro (1982) proposed a model where the eyewall form ation is the response of

a tropical cyclone to circularly symmetric, convective heat sources. Willoughby et

al. (1982) presented numerical simulations which showed th a t the tangential wind

increases rapidly inside the radius of maximum wind and decreases inside the eye near

the central axis of the vortex. According to their numerical simulations and aircraft

observations, the secondary eyewall often contracts as it intensifies. A secondary

eyewall is frequently observed to become narrower and replace a pre-existing eyewall.

Montgomery and Kallenbach (1997) hypothesized th a t th e secondary eyewall for­

m ation may take place because of vortex wave propagation in tropical cyclones. The

waves transfer angular momentum to the mean flow at the critical radius where their

phase velocity matches the mean flow angular velocity.

C H APTER 3. VO RTEX D YN A M IC S IN T H E A T M O SP H E R E 42

Another theory is based on the work of Nong and Em anuel (2003) in which it is

proposed tha t the instability mechanism leading to the secondary eyewall formation

results from wind-induced surface heat exchange.

Kuo et al. (2008) and M artinez et al. (2011) used numerical simulations to

investigate the mechanisms by which vorticity perturbations around a strong tropical

cyclone may form a ring of enhanced vorticity. Martinez e t al. (2011) applied a space­

time empirical normal mode statistical technique to study the genesis of a secondary

eyewall in a simulated hurricane.

High-resolution, full-physics numerical simulations have also been used to study

secondary eyewall formation by Terwey and Montgomery (2008) who hypothesized

tha t the secondary eyewall is the result of an anisotropic upscale energy cascade of

convectively generated vorticity anomalies and vortex Rossby wave propagation.

Abarca and Corbosiero (2011) carried out numerical sim ulations of hurricanes R ita

and K atrina and investigated the secondary eyewall formation. The results of their

numerical simulations support the idea th a t vortex Rossby waves play an im portant

role in the process. Secondary eyewall formation in their model occurs a t a radius of 65

(80) km in K atrina (Rita) close to the hypothesized critical radius of the vortex waves.

The secondary eyewall formation in their simulations is characterized by maxima in

convective activity and potential vorticity m axim a in the lower troposphere. Their

results support the notion th a t secondary eyewall formation is intimately related to

bands of high precipitable water and potential vorticity variance emanating from the

primary eyewall.

The debate about the mechanisms for the secondary eyewall formation is still very

active, as seen for example in the recent polemic paper of Terwey et al. (2012) where,

based on the numerical simulations in Terwey and M ontgomery (2008), the authors

argue against some conclusions of Ju d t and Chen (2010) an d present arguments to

C H APTER 3. V O RTEX D YN A M IC S IN TH E A T M O SP H E R E 43

show the importance of vortex Rossby waves in the secondary eyewall formation.

One of the key points to explain the secondary eyewall form ation mechanism is

the interaction between vortex Rossby waves and the mean flow of the cyclone. Wave-

mean flow interactions generally take place when waves reach a location in the atmo­

sphere where the background wind speed equals the wave phase speed. This location

is called a critical latitude or critical level, or in the case of vortex waves, a critical

radius. Montgomery and Kallenbach (1997) developed a balanced numerical model

based on vortex wave propagation and their experiments dem onstrated th a t the in­

teraction between the vortex waves and the mean flow at the critical radius may be

the primary mechanism for the increase in the angular wind in the vortex. They also

suggested th a t vortex wave dynamics may be a key factor to explain the secondary eye­

wall formation. Since th a t work, the concept of the critical layer interaction has been

used to explain some features of tropical cyclone dynamics. B runet and Montgomery

(2002) discussed critical layer interaction in their solution an d showed its influence

on the cyclone dynamics. Martinez et al. (2011) used the em pirical normal-mode ap­

proach to demonstrate th a t the maximum cyclonic angular m om entum is transported

to the location where the secondary eyewall forms. The fact th a t the critical radius

for some modes of vortex waves is contained inside the region where the secondary

eyewall forms led researchers to suggest th a t a wave-mean flow interaction mechanism

may be suitable to explain dynamical aspects of concentric eyewall genesis.

In this thesis we will use the concept of the critical layer interaction to investigate

the secondary eyewall formation. Interest in critical layer interactions has been stimu­

lated by the suggestion th a t this mechanism can explain certain phenomena observed

in the atmosphere and ocean. For example, Holton and Lindzen (1972) argued tha t

critical layer absorption of gravity waves might be the cause of the changes in mean

flow momentum tha t produce the quasi-biennial oscillation o f easterly and westerly

C H APTER 3. VO RTEX D YN A M IC S IN TH E A T M O SP H E R E 44

winds tha t are observed in the middle atmosphere.

From a mathematical point of view, the critical layer occurs a t the point where the

steady linear inviscid equation for the wave am plitude has a singularity. The singu­

larity can be avoided by introducing to the critical layer one or more the effects th a t

have been neglected, namely time-dependence, nonlinearity, an d viscosity. The intro­

duction of time-dependence was first considered by Booker and Bretherton (1967) in

their study of the vertical propagation of gravity waves in a stratified shear flow. They

considered a linear, inviscid model and showed th a t near the critical layer the waves

are absorbed and are unable to penetrate above the layer. Analyses by Stewartson

(1978) and W arn and W arn (1976, 1978) used asym ptotic m ethods to investigate a

nonlinear Rossby waves critical layer regime defined by a balance between tempo­

ral evolution and nonlinearity. There have since been a num ber of analytical and

numerical studies on the development of Rossby waves critical layers. For example,

Campbell and Maslowe (1998) and Campbell (2004) which exam ined a configuration

with a spatially-localized Rossby wave packet.

These investigations all involved Rossby waves in rectangular geometry. A recent

study by Caillol (2012) deals w ith vortex waves in a rapidly rotating vortex. His

simplified analytical model is based on the propagation of a vortex Rossby wave in

a barotropic and axisymmetric vortex on an /-p lane. The wave-flow interaction is

described in the critical layer where the viscous term is included in the governing

equation. This study describes the change of the mom entum flux and predicts the

existence of multiple vortices in the critical layer.

My thesis investigates the role of vortex Rossby waves in tropical cyclones dynam­

ics and includes:

• a mathematical description for vortex Rossby wave propagation in a cyclonic

two-dimensional vortex,

C H APTER 3. VO RTEX D YN A M IC S IN TH E A TM O SP H E R E 45

• solutions for the linear steady problem for waves on an /-p lane in term s of

hypergeometric functions,

• solutions for the linear time-dependent problem for waves on an /-p lane far

away from the critical layer (the outer region) and in th e critical layer (the

inner region),

• a weakly-nonlinear analysis of the wave-mean flow interaction for waves on an

/-plane,

• an analysis of the linear equation for waves on a /3-plane, i.e. including the effect

of the variation of the Coriolis param eter with latitude,

• an analysis of the combined effects of nonlinearity an d the variation of the

Coriolis param eter (the /3-effect) on the vortex dynamics,

• an investigation of the change of the mean angular velocity in the critical layer

which can explain the formation of the secondary eyewall and its dynamics.

C hapter 4

Steady linear problem

4.1 S tead y linear so lu tio n for som e sp ec ia l cases

In this chapter we derive solutions for the linear problem involving vortex waves on

an /-plane. The governing equation for this configuration is (3.54):

We consider waves with the stream function rp(r, A, t) generated a t some fixed distance

r = ri from the centre of the vortex, corresponding to th e prim ary eyewall, and

propagating outwards in the direction of increasing r. The dom ain for the problem is

defined in terms of polar coordinates r and A by r i < r < oo, 0 < A < 2-7T with r = 0

corresponding to the centre of the vortex (see Figure 4.1). T h e waves are generated

by a steady oscillatory boundary condition for the pertu rbed streamfunction of the

(4.1)

which can be written in term s of the angular velocity fl(r) = v ( r ) / r as

(4.2)

46

C H APTER 4. S T E A D Y LIN E A R P R O B L E M 47

form

iP(n, A, t) = A (ei(fcA- wt) + e - i{kX~u t ) ) , (4.3)

where A is a constant, k is the azim uthal wave number, k > 0, and cu is the circular

frequency of the wave. We require t h a t T is finite as r —>■ oo. T he to tal streamfunction

ip(r, \ , t ) is, as defined in (3.49),

V ( r , \ , t ) = tp(r) + e ip (r , \ , t ) , (4.4)

where e « l and ip(r, A, t) is 0 (1 ). Thus A is taken to be 0 (1 ), and we can set A = 1.

W ith the boundary condition (4.3) we can derive a normal m ode solution to equa­

tion (4.2) in the form

ip(r, A, t) — ^ (r)e ^ fcA_w + c.c., (4.5)

where c.c. denotes the complex conjugate of the preceding term and <p(r) is the

complex wave amplitude. By adding the complex conjugate we ensure th a t is real

although <f> generally has a nonzero im aginary part. The form (4.5) assumes tha t the

wave amplitude is a function of the radial variable only and does not evolve with time.

In chapter 5 we consider the case where the am plitude evolves with time.

After substituting this normal mode form into the linear equation (4.2) we obtain

an ordinary differential equation for the wave am plitude <i>(r) ,

S' e *(fi”(r) + 3S2'/r)0 + 7 ~ ^ + — z r r m — * = 0 ’ <4-6>

with boundary conditions d>(r\) = 1 and <p(r) finite as r —» oo.

We see th a t the equation is singular a t r = r c, where f2(rc) = This radius is

the radius where the angular velocity of the vortex Ll(r) equals the phase speed of the

waves and is called the critical radius, and the region surrounding the critical radius

CHAPTER 4. S T E A D Y LIN E A R PR O BLE M 48

/* ~ » r = r ,r?f

1 s /» » * * * *

Figure 4.1: T h e d o m a in o f th e p ro b le m . The domain is r x < r < oo, 0 < A < 2-7T with r = 0 corresponding to the centre of the vortex, r and A are polar coordinates. Waves are generated a t the prim ary eyewall r = r x.

is called the critical layer.

We only consider cases where the mean flow angular velocity is an analytic function

of r. This means tha t a solution of equation (4.6) in the vicinity of the critical radius

could be derived as a series in powers of (r — r c) using the m ethod of Frobenius. A

second solution would include a logarithmic term log(r — rc).

We examine some specific profiles of angular velocity Ll(r) — v / r which allow

an exact solution of (4.6) to be obtained. These profiles are approximations of the

observed profile of the angular velocity in a tropical cyclone and have been studied

by previous researchers by analytical methods and numerical simulations.

We first consider the case where fl(r) is constant and the re is no critical radius.

We then consider the case where Ll(r) is a quadratic function of r. This is the profile

tha t will be used throughout the thesis.

C H APTER 4. S T E A D Y LIN EAR P R O B L E M 49

T h e case w h ere f2 = f2o, a c o n sta n t

If the angular velocity Q is constant, then the equation for the wave am plitude d>(r)

(4.6) becomes

r r2(4.7)

which is a Cauchy-Euler equation. Its solution is

4> = C xr k + C2r~k (4.8)

where C\ and C2 are constants. In order for (f> to be bounded a t large r, C\ = 0. The

boundary condition 4>{ri) = 1 tells us th a t C2 = T and so th e solution is wave with

amplitude decreasing with distance from the hurricane center

T h e ca se w h ere Q is a q u a d ra tic fu n c tio n o f r

Martinez et al. (2010a), M artinez et al. (2010b) used an exponential profile of the

basic flow angular velocity in their numerical simulations of hurricane dynamics. This

profile is based on typical flows observed in tropical cyclones. Brunet and Mont­

gomery (2002) argued th a t the angular velocity of a cyclone can be approximated by

a quadratic function of r. Based on th a t, we will use

as the angular velocity profile. This gives an approxim ation for the exponential profile

Ll0e~ar2 for r < 1 / -J a (see Figure 4.2).

(4.9)

Q(r) — Q0( l — £*r2), a > 0, (4.10)

C H APTER 4. S T E A D Y LIN E A R P R O BLE M 50

co/k

1/Va r

Figure 4.2: T h e a n g u la r v e lo c ity p ro file . The angular velocity Ct(r) = fio(l — otr2) used in our investigation is an approxim ation for the exponential profile Q(r) = Q0e~ar , r i is the location of the prim ary eyewall, rc is the critical radius and corre­sponds to the point where f2(rc) = oo/k.

W ith the angular velocity profile (4.10) the equation (4.6) becomes

8kaD0b + — — —<p

oj - D o k (I ~ otr2)0 = 0. (4.11)

This equation has regular singular points a t the center of the vortex where r = 0 and

at r —> oo. The regular singular point a t r = 0 is not in the domain of our problem

and does not need to be considered. There may also be a regular singular point at

the critical radius,

where the denominator of the last term in (4.11) becomes zero. The critical radius is

located where the angular velocity of the vortex equals the phase speed of the waves

u> = flofc(l — otr2). (4.13)

CH APTER 4. S T E A D Y L IN E A R PR O BLE M 51

The form of the solution of (4.11) depends on the choice o f u>, fi0, and k. There

are three possibilities: u = Qok, u> > Ll0 k, and u < Ll0 k.

T h e ca ses w h ere u = Qok a n d to > Qok

If uj = Llok, then equation (4.11) becomes

aj k 2 + 8

4>" 4--------------^— 4> = 0, 0 < ri < r < oo, (4-14)

which is an Euler equation. There are no singular points in the dom ain of the problem.

The general solution is

<f>{r) = C lr'/WT* + (4.15)

In order for the solution to be bounded for large r, Ci — 0 and the solution satisfying

the boundary condition <j){ri) = 1 is

cj) = ( r / r 1) - ' /P T I. (4.16)

If u) > Q0 k, then again there are no singular points in the dom ain of the problem. If

rq is small then we can examine the behaviour of the solution for small r. In tha t

case we can write equation (4.11) as

+ £ _ ( V + ~ ~ b T r2) ^ + ° ( r2) = ° ‘ (4 ' 17)r \ u — ilok J

So a t leading-order, the solution can be expressed in term s of modified Bessel functions

of the first and second kind

\ t ( I SkaLlo \ —T ts ( I SkaLt0 \ .m = C J t (V z r ^ k T) + c *K k n < r «*• (418)

C H APTER 4. S T E A D Y L IN E A R P R O B L E M 52

T h e case w h ere u < Qok

The most interesting case for our problem is th a t in which ui < Q 0 k. This is the focus

of our investigation for the rest of the thesis. In this case equation (4.11) has a regular

singular point atf k n 0 - u \ 1 / 2

r° V k a ^o ) '

We will show tha t equation (4.11) can be transformed into a hypergeometric equation.

In order to do so we rewrite it as

(uj — Llok + Qqkar2) ^ <j>" + —— ~2 ^ j ~ = 0, (4.20)

A A „ + *; _ 8(i = 0> m i }or

( - -\ kaO 0 J \ r r

and then make the change of variables

f = T/ r ‘ = (4-22)

where r c is the critical radius given by (4.19). We also define a new variable <p by

4>{r) = 4>{t).

The derivatives of 4> in term s of the new variables are

0 '(r) = —0 '(r), (f>"(r) = \<j>”{r). (4.23)rc r*

In terms of <j> and r, equation (4.21) can be w ritten as

( - l + r2) ( V + f - ^ < A - 8 0 = 0 (4.24)

C H APTER 4. S T E A D Y LIN E A R P R O B L E M 53

or

+ J - ^ = ° ' (4'25)

This equation has regular singular points at r = 1 where the denom inator in the last

term of equation (4.11) is zero, corresponding to the critical radius r = rc.

Following Erdelyi (1953), we seek a solution for (4.25) in th e form

(f>(r) = rpf ( f 2), (4.26)

in order to reduce equation (4.20) to a hypergeometric equation. The first and the

second derivatives of (f)(f) with respect to r are

4>'(r) — prv~ 1 f { f 2) + 2 rp+lf ' ( r 2) (4-27)

and

4>"(r) = p(p — l ) r p~2/ ( r 2) + 2(2p + l ) r p/ ' ( r 2) + 4 r p+2 f " ( r 2), (4.28)

where the primes on the right-hand side represent derivatives with respect to f 2.

Substituting (4.26)-(4.28) into (4.24) we obtain

r 2 (r 2 — l)[p(p — 1 ) f p ~ 2 f { f 2) + 2(2 p + l ) r p/ ' ( r 2) + 4rp+2/ " ( r 2)]

+ f 2( r2 - l)[pfp~2f (r2) + 2rp/ ' ( r 2)] - k 2 rp(r 2 - 1 ) / ( r 2) - 8r p+2/ ( r 2) = 0. (4.29)

After some simplifications we get

4f4(f2 - l ) / " ( r 2) + 4 r2(r2 - 1 ) (p + 1 ) / ' ( r 2) + r 2[p2 - k 2 - 8] / ( r 2)

+ [ k 2 - p 2] f ( f 2) - 0. (4.30)

C H APTER 4. S T E A D Y LIN E A R PR O B LE M 54

This equation can be transformed into a hypergeometric equation if the last term is

set to be zero. This means

p2 = k 2 (4.31)

and so p is either k or — k.

We also introduce the new variable 2 = f 2, so the equation for / can be w ritten as

z(z - 1 ) f ' { z ) + (z - l)(p + 1 ) f ( z ) - 2f ( z ) = 0, (4.32)

where the primes denote differentiation with respect to 2 . T his is a hypergeometric

equation. A hypergeometric equation is generally w ritten in the form (see Abramowitz

and Stegun, 1964)

2(1 — z ) f " + (c — (a + b + 1 ) z ) f — (ab)f = 0. (4.33)

Writing equation (4.32) in the form (4.33)

2(1 - z ) f " + ((p + 1) - (p + 1)2) / ' - ( - 2 ) / = 0, (4.34)

we see that

a + b + 1 = p + 1 and ab = —2. (4.35)

k + y/k 2 + 8 , k - V k 2 + 8 , ,If p = k, a = , b — --------- , an d c = k + 1. (4.36)

tc » - k + y/k 2 + 8 —k - \Jk 2 + 8If p — —k, a = --------- , b — , and c = —k + 1. (4.37)

Equation (4.34) has three singular points, a t 2 = 0, 2 = 1 and 2 —» 00. As already

noted, 2 = 0 corresponds to r = 0 and it is not in the dom ain of the problem. The

solution of the hypergeometric equation close to point 2 = 1 can be written in the

C H APTER 4. S T E A D Y L IN E A R P R O B LE M 55

form of hypergeometric functions (see the detailed description of the hypergeometric

equation and its solutions in Appendix A). Two linearly independent solutions of the

hypergeometric equation (4.34) for z close to 1, 0 < \z — 1| < 1, are

f m (z) = (1 - z )F(a + 1, 6 + 1; 2; (1 - z)) (4.38)

and

/ 2(i)(z) = (1 - z ) F { a + 1, 6 + 1; 2 ; (1 - z ))lo g (l - z) +

( 1 _ z ) p ^ ± ^ _ p u ( 1 _ z r

(x(a + 1 + n) - x(a + 1) + x (6 + 1 + n) - x(b + 1) -

X(2 + n) + x(2) - x ( n + 1) + x ( l ) ) + (4.39)

where (a)n = II”r01(a + i) is the Pochham mer symbol, and F is the hypergeometric

function (see Appendix A) given by

o o ,

F ( a , b , c , z ) = J 2 ^ f z n. (4.40)^ cW n!

The function x(z) is

x W = M M i . (4.41)

The solutions fi(i)(z) and f 2 (\){z) are valid for 0 < \z — 1| < 1. Following Abramowitz

and Stegun (1964) we use the subscripts 1(1) and 2(1) to indicate th a t the solutions

(4.38) and (4.39) are valid close to the singular point z — 1. So with the constants a,

b, c given by (4.37) and (4.36), the two linearly independent solutions of (4.32) can

C H APTER 4. S T E A D Y L IN E A R PR O B LE M 56

be written in terms of r 2/ r 2 as

/l(l) 1 21 -

/±fc+\/fc2+ 8

£n=0+ l)(n) r 2\ "

0 < < 1 (4.42)

and

/ 2(i) ( ^2

+ 1r* \ soo (, g + J n ) (

± k - V k * + 82-'n=0 (2)(n)n!

+ !)(») ^

'±fc-v/fc +8

(2)(n)n!

/ ±fc + V F + 8 , \ ( ± k + \ / F + 8 ,X o + 1 + n - x --------- =---------+ 1

( ± k - \ / F T 8 , ^+X ---- ----o-----------h 1 + n

[ ± k - V k * T 8 \ .x I --------- + 1 I — x (2 + n)

+ X ( 2 ) - x ( « + l) + X(l)].

0 < 1 - — < 1(4.43)

These solutions are valid in the intervals r\ < r < rc and rc < r < \/2 rc (see Figure

4.3).

Recall th a t we searched for a solution of the form of <f>{r) = <p(r) = fp/ ( r 2)

(4.26). Thus, the solution of equation (4.20) close to r = rc is proportional to a linear

combination of / 1(1) and / 2(q

<p(r) =±k

am f m + a 2( i) /2(i) (4.44)

where aqj) and a2(i) are constants. Applying the boundary condition th a t 4>{r\) = 1

C H APTER 4. S T E A D Y L IN E A R PR O B LE M 57

/fl(=c)5 f-2 (x)

2 2Figure 4.3: Intervals for th e s tea d y so lu tion . The solutions / i (i)(^2) and / 2( i)(^ ), given by (4.43) and (4.42), are valid for ry < r < \ /2 rc, r ^ rc. T he solutions /poo) (( 5) and / 2(oo)(^i), given by (4.51) and (4.52), are valid for r > r c. In the black interval, rc < r < \ / 2r c, the solutions overlap.

we can write the solution in the form

f r \ ± k f m ( g ) + * 1/ 2(1) ( s )0( r ) = r 7 - 7 ^ — ’ ( 445)

' x' f m ( j i ) + a xf 2(i) y i j

where a\ = The solutions /p i) and /p p are valid for 0 < |1 — ^ | < 1, i.e. in the

intervals r x < r < rc and r c < r < y/2rc. The leading-order term s of the solution are

C H APTER 4. S T E A D Y LIN E A R P R O B L E M 58

or, taking into account that

tog 1 - -o = loe (r c - r ) + log(rc + r) - log(r^), (4.47)

we can write the leading-order term s of (4.44) in the form

4 >(r) ~ r±k [0 (1) + 0 ((rc - r) log(rc - r) + 0 (rc — r ) ] ,

0 <1 c

< 1. (4.48)

The logarithmic term in (4.45) is (1 — 4 )k> g(l — ^ ) where

log 1 = log 1 + % arg ( 1

For r < rc, arg (l — 4 ) = 0; for r > r c, a rg (l - T-^) = ± 7r. T he sign of the argumentr c r c

is determined by considering the tim e-dependent problem (see Appendix C of this

thesis). It is found tha t arg(l — rj) = —7t. So there is a phase shift of —7r across ther c

critical layer from r < rc to r > rc and the am plitude of the solution is discontinuous

across the critical layer. This is analogous to the conclusions of previous researchers

for the problem of forced Rossby waves in a rectangular dom ain (e.g. W arn and Warn,

1978).

For z > 1 the solution of (4.32) is proportional to a linear combination of the

functions defined in (A. 17) and (A. 18):

/i(oc)(z) = 2 aF (a ,a - c + I; a - b + 1; - ) , |z | > 1, (4.49)

h(oo){z) = ^ hF(b,b - c + 1 ;6 - a + 1; - ) , |z | > 1. (4.50)

C H APTER 4. S T E A D Y LIN E A R PR O B LE M 59

In term s of our problem these two solutions are

oo / -fc-Vfc^+ix /fe-vF+gx^ /"»V hi f r . ( 4 52)

' n ! ( - > / F T 8 + l)„ V r ? ' ’ ;

r*2 \ -«2

The solution of equation (4.20), valid in the interval |^-| > 1, or r > r c (see Figure

4.3), is proportional to a linear combination of /i(oo) and / 2(oo)

<Kr ) = r,.

±k«l(oo)/l(oo) ( r 2 j + a2(oo)/2(oo) f 2 , r > rc, (4.53)

where ai(oo) and a2(oc) are constants. The leading-order term s in the solution are

±k1 /l(oo) I ,

±k / 2r \ ( rrc

fe) 1 + 0

1 + 0- 2

- 2

r > r c, (4.54)

and

±fc

/2(oo) (0 ±fe x 2N i * * v p > *r \ ( r ' 2

1 + 0

1 + 0

+ P +8

r > r. (4.55)

The term with the positive power is divergent for r —¥ oo. So th e coefficient <22(00) = 0

CH APTER 4. S T E A D Y LIN E A R P R O B L E M 60

and the solution is

^(r ) ~ ( r l al(oo)/l(oo) ( 2 O ©-v/P+8'

r > rc,

as expected from (4.16), and satisfies the condition <f>(r) -> 0 as r —> oo.

We need to match the solution (4.56) with the solution (4.48). The solution (4.48)

is valid in the intervals ri < r < rc and rc < r < y/2rc (see Figure 4.3), while the

solution (4.56) is valid for r > rc . The two solutions should m atch in the overlapping

interval rc < r < y/2rc (the black interval on Figure 4.3). The solution (4.56) decreases

in this interval with increasing r , so the solution (4.48) should also be decreasing in

the interval. To satisfy this condition we have to choose the factor r~k in (4.48).

W ith the negative sign of k, the wave am plitude (4.45) in th e interval r\ < r < rc

and rc < r < y/2 rc is

4>{r)a x

0 < < 1, (4.56)

where Qj is the denominator of (4.45)

a i - /i(i) + a i / 2(i) ^ (4.57)

So the solution of (4.20) is

C H APTER 4. S T E A D Y L IN E A R PR O B LE M 61

where 0 i(r) is

rr, «i(i)/i(i) + 02(i)/2(i) , (4.59)

where and / 2(i) are given by (4.42) and (4.43) with the negative sign of k. Func­

tion (f>2 (r) is given by

/ „ \ — Vfca+8 oo ( -fc+>/fc2+ 8 \ ( k + \ / k 2 + 8 \ / 2 \ —71, r > r c . (4,0)

n=0 n

The solution for the steady-state case is plotted in figures 4.4 and 4.5 for a normal

mode with wavenumber k = 2. From the graph it is evident th a t the amplitude of

the wave decreases across the critical layer from r < rc to r > r c where the waves are

partially absorbed.

C H APTER 4. S T E A D Y LINEAR PR O B LE M 62

<f>( r)1.4

1

0 r rl c

Figure 4.4: T h e a m p litu d e o f th e s te a d y so lu tio n . Profile of the wave amplitude 4>{r) (4.45) of vortex waves which is the solution of (4.20) w ith k — 2, ai = 1. The eyewall location r\ is taken r i = 0.1 r c. The dashed line shows th a t the solution is singular in the inner region close to the critical radius rc.

CH APTER 4. S T E A D Y LIN E A R PR O B LE M 63

rc

.

71

r l2710

Figure 4.5: C o n to u r p lo ts fo r th e s te a d y so lu tio n . Contour plots for the steady solution ip(r, A) — (j>{r)etkX+ c.c., where the am plitude of vortex waves <j>(r) (4.45) is the solution of (4.20) with k = 2, a\ — 1. The eyewall location ri is taken ri = 0.1 r c. The contour plots are presented in a rectangular domain (the upper panel) and a circular domain (the lower panel) for r\ < r < V 2 rc, 0 < A < 2 ir. T he plots dem onstrate the phase shift in the solution across the critical level a t r = rc an d resulting attenuation of the wave amplitude for r > rc.

C hapter 5

T he linear tim e-d ep en d en t problem

In this chapter and the rest of the thesis we consider the tim e-dependent problem in

which the wave amplitude is allowed to vary w ith time. In th is chapter we solve the

linear time-dependent problem on an /-p lane, i.e. we neglect nonlinearity and the

/^-effect. The problem is given by equation (3.54)

( I + + 18*= ° (5-i)

in the domain r i < r < o o , 0 < A < 27t, t > 0. As in chapter 4 the boundary

condition is

i ;(ru \ , t ) = Aei{kX- “t ) +c.c., (5.2)

where A is a constant th a t we set to 1. We also require th a t ^>(r, A, t) and its derivatives

be bounded as r —> oo. The initial condition is th a t the wave vorticity V 2ip = 0 at t =

0. As before, the mean flow is given by the angular velocity profile fi(r) = fl0(l ~ a r 2)

and the azimuthal velocity v ( r ) = Cl(r)r. Since the governing equation (5.1) is linear,

64

C H APTER 5. THE LIN E A R TIM E-D EPEN D EN T P R O B L E M 65

the boundary condition (5.2) means th a t the solution of (5.1) m ust take the form

V>(r, A, t) — (f){r, t )elkX + c.c. (5.3)

Substituting this into the governing equation (5.1) gives an equation for the wave

amplitude <p(r,t),

^ 4>rr + i <j>r — + &ikQo®4> = 0 , (5.4)

where the subscripts of r denote partial differentiation with respect to r. The bound­

ary conditions are <j>(ri,t) = 4 >(r,t) —> 0 as r —» oo and the initial condition is

r(f>r + r 2 (f>rr — k 2 (f> = 0 at t = 0. We solve this problem by m aking use of a Laplace

transform.

5. 1 Laplace transform o f th e linear tim e-d ep en d en t

equ ation

Let (f>(r,s) be the Laplace transform of the function 0 (r, t), defined as

poo

4>(r,s) = I <j)(r,t)e~sidt, (5.5)Jo

where s is a complex variable. Taking the Laplace transform of (5.4) and applying

the zero initial condition on the vorticity we obtain

- ( ~ I ~ k 2 ~\(s -(- ikLl) ( 4>rr -I— <f>r (j> J + 8 ikLloa<t> = 0. (5.6)

C H APTER 5. THE LIN EAR TIM E -D E P E N D E N T P R O B L E M 6 6

The boundary condition for s) is

r ° ° 1I e ’" ‘e std t = —. (5.7)

Jo S + ZUJ

We can rewrite equation (5.6) in the form

( ~ 1 - k 2 ~\(s + ikLlo — ikPloar2) f cj)rr H— 4>r ------<f> j + 8 ikQ 0 acf> = 0. (5-8)

Using the fact th a t ui — kCl(rc) — A:ST0( 1 — a f 2), we can also w rite (5.8) in term s of uj

( _ I - k 2 ~\(s + iu> + ik f lo a(r2 — r 2)) ( <f>rr -\— 4>r ----- (f> J + 8 ikQ,0 a<P = 0. (5.9)

Following the same procedure we used for the steady-state case in chapter 4 we intro­

duce a new variable r, which now depends on s,

f { s ) = { t t S c = (510)

where. . iPLoOtk

*•> = i r k k - <511>We also define <j>{r,s) = <p(r,s). In term s of the wave angular frequency u>, we can

write

= s + J , n: t k a r r ( 5 ' 1 2 )

Note th a t for s = —iui, we obtain

7 ( - i u ) = (5.13)r c

C H APTER 5. THE LIN EAR T IM E-D EPEN D EN T P R O B L E M 67

In terms of the new variables r and 4>, we can rewrite equation (5.8) as

(5.14)

where the subscripts denote partial derivatives of <fi w ith respect to f.

Equation (5.14) has the same form as the steady-state equation (4.25) and the

solution is of the same form, <f>{r) = r p/ ( f 2) with p = —k, as for the steady-state case.

We have to consider different regions in the complex s-plane to find the solution of

this equation and evaluate the inverse Laplace transform. E quation (5.14) has three

singularities.

• r = 0, or = 0. This case corresponds to r —» 0 an d we do not consider

it because it is not in the domain of our problem.

• r = 1. Near this singular point, the solution is proportional to a linear combi­

nation of /i( i)(r2) and f 2 (i){f2), where /q q and f 2(\) a re given by (A .15) and

(A. 16) in Appendix A. So the solution is

where aqi) and a2(i) are functions of s th a t can be determ ined from the boundary

conditions. This solution is valid for 0 < |1 — r 2| < 1 (see Abramowitz and

Stegun, 1964) which corresponds to the region

4>(r,s) = (V 7 (s)r) k [a i( i)(s)/i(1)(7 (s )r2) + a2(i) (s ) /2(i)(7 (s )r2)] , (5.15)

s + iilok ' s + iilok ^(5.16)

or

s + > -Qo&kr2, r ^ rc. (5.17)

C H APTER 5. TH E LIN E A R TIM E-D EPEN D EN T P R O B L E M 6 8

In the complex s-plane this is the region in the complex s-plane outside the

circle of radius | Lloakr2 centred a t s = — as shown in Figures 5.1 and 5.2.

We shall refer to this as region I.

• f -> oo. For r » 1 the solution of equation (5.14) is proportional to a linear

combination of the two functions /q o o )^ 2) and / 2 (<x>)ir2), where /i(oo) and / 2(oo)

are given by two functions (A .17) and (A .18) in A ppendix A. So the solution is

4>{r, s) = (V 7 (s)r)~k [a1(oo)( s ) /1(oo)(7 (s )r2) + a2 (oc){s)f2 (oo)(l(s)r2)] , (5.18)

where oo) and a 2(oo) are functions of s th a t can be determ ined by matching

this solution to (5.15).

This solution is valid for |r2| = [7 (s )r2| > 2 (see Abramowitz and Stegun, 1964).

|s + zfl()fc| < ctkr2. (5.19)

This is the region in the complex s-plane inside the circle of radius ^Ll0 a k r 2

centred a t s — —iTlnk (the grey shaded region in Figures 5.1 and 5.2). We shall

refer to this as region II.

To obtain <p(r, t ) we need to compute the inverse Laplace transform of tp{r. s). In

doing so, we have to take into account the fact th a t there a re two different regions

in the complex s-plane th a t need to be considered separately. Region I is outside of

the circle |s + iQok\ > ^Doakr 2 where the solution has the form (5.15), and region II

is the region inside the circle |s + iQ0 k\ < | f 20cxkr2 where th e solution has the form

(5.18) (see Figures 5.1 and 5.2).

CH APTER 5. THE LIN E A R T IM E-D EPEN D EN T P R O B L E M 69

Im(s)Im(s)

Re(s)Re(s)

s— -iOok(l-ar2) s= -i<a = -iQ0k(l-arc2)

- s—iQokCl-ar2)- s=-k>= -ifiok(l-arc2)

■ s-ifio k Q -ari2)

Figure 5.1: T h e contour o f in tegration in th e com p lex s -p la n e for th e inverse Laplace transform for th e cases w here th e p o le is in reg ion I.The shaded region is region II where the solution (5.40) is valid. The region outside the circle is region I where the solution (5.39) is valid. The pole P, s — —ioj, is in region I. It gives the steady term of the tim e-dependent solution in the form (4.59), which is valid for r\ < r < V 2 rc, r ^ rc.(a) If r > \ f i r i, the solution (5.39) has two branch points, s = — iQ0 k ( l — a r 2) and s = — iLlok(l — a r 2), which are surrounded by the contours C 3 and C2.(b) If y/2ri < r < \ /2 rc, r ^ rc, the solution (5.39) has one branch point in region I, s = — iLlok(l — a r 2), surrounded by contour C3.

C H APTER 5. TH E LIN E A R TIM E-D EPEND ENT P R O B L E M 70

Im(s)

- s= -iQok(l-ar2)

s= -ioo = -iQok (l-arc2)

Figure 5.2: T h e c o n to u r o f in te g ra tio n in th e co m p lex s -p la n e fo r th e in v e rse L ap lace tra n s fo rm fo r th e case w h e re th e p o le is in r e g io n II .The shaded region is region II where the solution (5.40) is valid. The region outside this disc is region I where the solution (5.39) is valid.The pole P, s = —iu>, is in region II. It gives the steady term in the time-dependent solution in the form (4.60), which is valid for r > y/2rc. T here is one branch point, in region I, s = — Q0&(1 ~~ <*r2), surrounded by contour C3.

C H APTER 5. THE LIN EAR TIM E-D EPEN D EN T P R O B L E M 71

R eg io n I

In region I (outside the grey disc in Figures 5.1 and 5.2), where 0 < j 1 — 7 (s )r2| < 1,

the solution is proportional to a linear combination of the two linearly independent

solutions (see Appendix A, A. 15 and A. 16):

OO / - f c W f c j + 8 , - | \ , - k - V k i + 8 , I ' l

Aii) (7 ( s y ) = (1 - 7 ( V ) E 2------ ^ (1 - 7 M - Tn = 0 K * H n ) n .

(5.20)

and

h { i) (7 (s)r2) = ~

+ (1 _ 7 (s )r2) £ + +n=0

(1 - 7 (s )r2)n log ( l - 7 (s )r2)

+ (1 - 7(*)r2) E ( 2 g ; ( )n , 2------ ^ (1 - 7 ( s ) r T71— 1

( r H v /F + 8 1 , - k + \ Z W + %( x( j + ! + «)- x( g +

/ —& - \/A:2 + 8 — fc - \ / k 2 + 8+X ( j + ! + « ) “ x( g + X) “ X(2 + n)

+ X ( 2 ) -X ( « + 1 ) + X ( 1 ) ) - (5.21)

Both series (5.20) and (5.21) converge for all r in the interval 0 < 11 — 7 (s )r2| < 1

(see Abramowitz and Stegun, 1964). The solution is

4>i(r,s) = (V*7 (s)r) *[a1(i ) /1(i)(7 r 2) + a 2(i)/2(i)(7^2)], (5.22)

where ai(i) and a2(i) are functions of s which can be determ ined by boundary condi­

tions.

C H APTER 5. THE L IN E A R T IM E-D EPEN D EN T P R O B L E M 72

The leading-order terms of / 1(1) and / 2(i) are

fn i ) (r , s ) ~ (1 - l ( s ) r 2) + 0 ( ( 1 ~ l ( s ) r 2)2) , |1 - 7 ( s ) r2| < 1, (5.23)

s ) ----- ^ + (1 - 7 (s )r2) log(l - 7 (s)r2) + 0(1 - 7 (s )r2),

|1 - 7 ( s ) r 2| < 1. (5.24)

For small values of j 1 — 7 (s)r2| where |1 — 7 (s )r2| < 1, the solution /^ j) <C / 2(i)- The

leading-order term s for the solution are

M r < s ) ~ ( V l ( s ) r )~k + (1 - 7 {s)r2) log (1 - 7 (s )r2) + 0 (1 - 7 (s )r2)^ ,

|1 - 7 (s )r2| < 1. (5.25)

After applying the boundary condition <j>(r\,s) = , we obtain

~ 1 f r \ ~ k - 5 + (1 - 7 ( s ) r 2) lo g (l - n / j s y 2) + 0(1 ~ i ( s ) r 2)1 r,S s + fc u V ri/ - | + (1 - 7 (s)rf) log(l - 7 (s)rf) + 0(1 - 7 ( s ) r f ) ’

|1 - 7 (s )r2| < 1. (5.26)

This solution has two singular points, a t s = —iQ0 k( 1 — a r 2) where 1 — 7 (s)r 2 = 0

and s = — a r 2) where 1 — 7 (s )r2 = 0 .

R eg io n II

Region II is the grey shaded disc shown in Figure 5.1, where |7 (s )r2| > 2, or js +

< \Ll0 a k r 2. The point s — —iQ0k is the centre of the disc. At the point

s = —iQ0 k , 7 (s) —> 00 and for s ^ —iQ0 k, 7 (s) — is finite.

C H APTER 5. THE LIN EAR TIM E-D EPEND ENT P R O B L E M 73

For s = —iQok equation (5.6) becomes

0rr + ~4>r ~ - - ^ - 0 = 0 (5.27)r r*

and the solution is

4>(r, - iQ0 k ) = a 1r ' /PT1 + a2r - ' /P T i (5.28)

where a\ and a2 are constants. The solution should be finite as r —» oo, so we must

set ai = 0. To find a2 we apply the boundary condition (5.7) a t r = rj:

4 >(ru - i f l 0 k)-iQ0k + iu iQ0 a k r %

and hence obtainI / r \ - V f c 2 + 8

M r . - i d o = - <5-29)

Equation (5.27) has no singularities in the domain of our problem, so s = —iQok is

not a singular point for the inverse Laplace transform.

For s 7 —ikQo, when y /y (s)r is large bu t finite, the solution of (5.6) has a form

similar to the steady-state solution for large r (4.53):

4>n(r,s) = ^-z ( v /;:Ks)^)“ *:[ai(oo)(s)/i(oo)(7 (s )r2) + a 2(oo)( s ) /2(oo)(7 (s )r2)],

|7 (s )r2| > 2, (5.30)

where a^oo) and a2(oo) are functions of s, and /i(oo) and / 2(oo) are the functions given

by (A. 17) and (A. 18) in Appendix A:

/To—— °° ( — 8 \ / fc+\/fc2+8

h U < ‘ P ) = h ( s y ) ~ - g „,2(^ ) : 8 + ^ " ( v T M r r .

|7 (s )r2| > 2 , (5.31)

CH APTER 5. THE LIN EAR T IM E-D EPEN D EN T P R O B L E M 74

. ^ /TJTZ 00 ( ~k-V&+&\ / fc -y ^ + 8 \A ( . ) h W = 2,," n!( —vA;2 + 8 + 1)„

|7 (s )r2| > 2. (5.32)

So in region II the solution 0 //(r , s) is a linear combination of th e functions

( y / r ( s j r ) kf\(oo)('y(s)r2) ~ (VtCs)*-) k{ l ( s ) r 2) k ~ ( y / r ( s ) r ) -v/F+i

|7 (s )r2| > 2 , (5.33)

and

{ V l ( s ) r ) - kf 2 (oo)(7 (s)r2) ~ ( \ / l < s ) r ) - k( j ( s ) r 2 ) h±:/f I i ~ (■V CK *)r)WE3+*,

|7 (s )r2| > 2. (5.34)

The function (5.34) is divergent for r —>• 00. So the coefficient <22(00) (s) = 0 and the

solution (5.30) is

4>n(r,s) ~ (vS < s)r) *ai(oc)/i(oo)(7 (s )r2)0 0 f - k - V k 2 + 8 \ / f c - y 4 ? + 8 \

~ « . M ( ^ W r ) +- , r a E - K - ^ + S + l) , ( 7 ( , ) r 2 r "n=0 n

/ f s + i k Q ( ^ m ) n )( ^ ± g ) u / a + t fcn 0 1 \ "Gl(°°) i t o f l o r y ^ n ! ( - \ A 2 + 8 + 1)„ \ r 2 /

|s + ikLlo| < i/ufloQ;?'2- (5.35)

CH APTER 5. THE LIN E A R TIM E-D EPEN D EN T P R O B L E M 75

After applying the boundary condition (5.7) we obtain

/ — k — \/fc2 +8 \ f k — v /k2 -4-8 \ / \ n. . -V P +8 r 00 ( " 1 ....)n’( ' 2 ( s+ ik Q p 1 \

7 , , 1 / V \ Z ^ n = 0 n ! ( - V f c 2 + 8 + l ) „ V i k a Q o r 2 J<Pn(r,s) — 1 1

s + i u ) \ r j ( -A t^ E tj)wi(t z ^ ± i )n / S+Ifcn„ x \ » !Z ^ n = 0 r i! ( -v 'fe i + 8 + l ) „ \ tfcaUo r f /

|s + iA;Oo| < - k f l o a r 2. (5.36)

The leading-order terms of the solution are

.) ~ ^ (iV U + o ( ^ i ) + O

fo r|s + zfcfi0| < -^kLl^cur2. (5.37)

s + i u ) \ r i J V V ikaClo r

We observe th a t this solution gives us (5.29) in the limit s -» —iLl0 k. So the solution

4>ii(r j s) is continuous in region II where it is defined. We have now obtained the

solution 4 >(r, s ) everywhere in the complex s-space and can find <fi(r, t) by making use

of an inverse Laplace transform.

5.2 E valuation o f th e inverse L aplace transform

In section 5.1 we derived the solution of equation (5.6) in different regions in complex

space,

4>[(r,s), for |s -I- ifcflol > ^kLtoar 2 (Region I),H e s) = •{ _ l (5.38)

i|i/;(r,s), for \s + ikLlo\ < -kL lo ^ f 2 (Region II),

C H APTER 5. TH E LIN EAR TIM E-D EPEN D EN T P R O B L E M 76

where

~ 1 / r \ * - § + ( ! - 7(g)?~2) log(l ~ l { s ) r 2) + Q (( 1 - 7 (s )r2))

/r,S s + i u Vr , / - 5 + (l-7(s)»'i)log(l-7(s)r?) + 0(( l -7(s)r?))

To find <j>(r,t), the solution of equation (5.1), we evaluate the inverse Laplace

integration will lie to the right of all the singularities of the integrand. The integral is

evaluated along the closed Bromwich contour in the complex s-plane shown in Figures

5.1 and 5.2. The contour of integration is closed around the pole of the integrand and

deformed to go around any branch points of the integrand. Following the standard

procedure, the radius of the large semicircle is allowed to become infinite while the

radii of the circles around the branch points go to zero. M aking use of the residue

theorem the integral is found to be equal to the sum of the contributions from the

singularities.

We evaluate the inverse Laplace transform in the complex s-plane for each value

of r , rq < r < oo. Depending on the value of r , we have three different configurations

in the complex s-plane. Case 1 for rq < r < i /2 r i, Case 2 for y/r\ < r < y/2rc, r ^ rc,

1 - 2(1 - g g g .r3 ) log(l - ^ r 2) + Q (( l - g g f e H ) )ikQ q q 2

s+ ik Q o 1ik f t q q 2 s-Mfcf2o 1

tfcOpq 2 s+ikClo 1

and

Vf?+8

+ o (s -b ikfyo 1

ika^lo) ) - ( 5 .4 0 )

transform of 4>(r, s) using the Laplace inversion integral

(5.41)a —ioc

where cj>(r, s ) is given by (5.38). Here a is a real number chosen so th a t the contour of

CHAPTER 5. THE LIN EAR T IM E-D EPEN D EN T P R O B L E M 77

and Case 3 for r > rc. The union of the intervals in Case 1 and Case 2 is the interval

ri < r < V2rc, r ^ rc, which is where the steady solution (4.59) is valid. And we

shall see tha t in the limit as t -* oo the time-dependent solution th a t is valid in this

interval converges to the steady solution (4.59). The interval in Case 3 is r > \ /2 rc

which is where the steady solution (4.60) is valid. And we shall see th a t in the limit

as t -> oo, the time-dependent solution th a t is valid in this interval converges to the

steady solution (4.60). The three cases are summarized below and are represented in

Figures 5.1 and 5.2.

C ase 1 . ri < r < y/2r\

In this case the integrand in (5.41) has

• a pole P at s — —iu in region I, outside the grey disc,

• a branch point a t s — —iQok(l — a r 2), o r s = — i u + ikQoa(r 2 — r 2),

• a branch point at s = — iQok(l — a r 2), or s = —iuj -I- ik f lo ^ i^ i ~ r2).

All the points lie outside the grey disc, so they are in region I. This is shown in Figure

5.1a.

C ase 2 . y/2r\ < r < y/2rc, r ^ rc

In this case the integrand in (5.41) has

• a pole P a t s — — iu in region I, outside the grey disc,

• a branch point a t s = —rf20A:(l — a r 2).

The point (s — —iflok(l — a r 2)), in this case, lies inside the grey disc, i.e. in region

II. Since it is not in region I, it does not contribute to the solution. This is shown in

Figure 5.1b.

CH APTER 5. TH E LIN EAR TIM E-D EPEND ENT P R O B L E M 78

C ase 3. r > s /2 rc

In this case the integrand in (5.41) has

• a pole P at s — — iu which is now in the region II,

• a branch point of the solution (5.39) a t s = —iQok(l — a r 2).

As in case 2, the point (s — —ifl 0 k( 1 — a r 2)) lies inside the grey disc, i.e. in region

II. Since it is not in region I, it does not contribute to the solution. This is shown in

Figure 5.2.

We now describe the evaluation of the contributions from the pole and branch

points.

T h e resid u e a t th e p o le P , s = —iu

In all 3 cases the integrand in (5.41) has a pole at s = — iu = — iQ0 k ( l — a r 2). The

residue at the pole is

For cases 1 and 2, where < r < V2rc, the pole P is in region I so we use

R e s ^ ^ ) = lim^ (j>(r, s)est j = 0 (r, - i u ) e lwt (5.42)

<f>(r,s) = cf)j(r,s) in (5.42). Recalling tha t 'y(-iuj) = (5.13), we obtainr c

Res {s=- iu)( ^ ( r , s ) ) = Ress=_iW(<^/(r, s))

(5.43)

For cases 1 and 2 the residue corresponds to the steady-state solution (4.59) which is

valid for r\ < r < s/2rc, r / rc. For case 3, where r > s/2.rc, th e pole —iu is in region

CH APTER 5. THE LIN EAR TIM E-D EPEND ENT P R O B L E M 79

II (Figure 5.2) so we use <j>(r,s) = <fin(r,s) in (5.42) and therefore

Res(*=_*,)(<A(r,s)) = Ress=-tw(4>n(r, s))t — k — \/fc2 + 8 \ / k — \/fc +8 \ / -> \ 1

-VF+8 r ° ° (----2------)"■(—*1---- )"r \ /s n = 0 n ! ( - \ / f c 2+ 8 + l ) „ \ r 2)

n ) 7 r ? \ W2^n=0 n^-v^S+g+i^ ^

This residue corresponds to the steady-state solution which is valid for r > y/2rc.

T h e c o n tr ib u tio n s from th e b ran ch p o in ts

To evaluate the inverse Laplace transform we integrate (5.39) along the contours

shown in Figures 5.1 and 5.2. Let us consider the integral for the case 1 where the

integrand has a pole and two branch points. For case 1 the contour consists of the 7

contours C 1 -C 7 which are shown in Figure 5.1a.

So the solution is

4>(r, t ) = Res(s=_iw) ( j : <ft(r, s) \ - ~ V ] f s)estds. (5.45)\ s T iuj J 2m J c . s + iuj

T h e co n to u rs C \, C2 , an d C3

The contour C\ can be described in polar coordinates as

s ^ a + R e 16, | < 9 < Y ■ (5-46)

So

3 tt/2

— —~ 4 >{r, s)estds =iCx

(5.44)

O / l / £,

L ^ {r’S)e“ d S = I a + R l - + ^ ' W ' ]‘^ ' a + R ‘ ', ) m < t d S ‘ (5-47)7r/2

C H APTER 5. TH E LIN EAR T IM E-D EPEN D EN T P R O B L E M 80

As R —> oo, the integrand in (5.47) is proportional to eRe'a or eRcos8, and cos#

is negative since | < 0 < So the integral along the semicircular contour C\

converges to zero as the radius of the semicircle becomes infinite.

The contour C2 is the circle surrounding the branch point s = —iQ0 k( l — a r 2). It

can be described as

and the integral converges to zero when e —> 0 .

The contour C3 is the circle surrounding the branch point s = —ifl0^ ( l — a r 2).

The integral along C3 takes a similar form to (5.49) and converges to zero as the

radius of the circle converges to zero.

So the inverse Laplace transform for case 1 is equal to th e contribution from the

residue at the pole (5.43) plus the contributions from the contours C4 and C5 near the

branch point .s = —iQ0 k( 1 — a r 2) and from the contours C6 and C 7 near the branch

point s = — iQok(l — a r 2).

s — —iLlok(l — a r 2) + ee10, 7r < 6 < 37r. (5.48)

So

/ 4 >(r, —iQ0 k( 1 — a r 2) + ee10) —iQok(l — a ( r 2 — r^)) + ee*0&

-in 0fc(l-a(r2- r 2))+ee‘s eei e ^

7T

T h e co n to u rs C4 and C5

The contours C4 and C5 correspond to cuts to the branch point s = —i i \ k ( l —a r 2). or

s — —iui + ikLl0 a (r 2 — r 2). Along the contour C4, s = s + = pein — iu> + ikLloa(r2 — r2),

C H APTER 5. THE LIN EAR TIM E-D EPEN D EN T P R O B L E M 81

and p changes from oo to 0. Along the contour C5, s = = pe %1X—iuj+ikQoa(r2—r 2),

and p changes from 0 to oo. Hence

{La Sc) -iut^ikQ.QCt(r2 — r 2)t (- k

+ I I (0 (r > s)est)ds ~ e/c4 Jo*,/ Vr l

f°° 1 1 — 2(1 — 7 (s )r2) log(l — 7 (s )r2) + 0 (1 — 7 ( s ) r2)

Jo/o (s + i u ) l - 2(1 - 7 (s)r?) log(l - 7 (s)rf) + 0 (1 - 7 (s)r?)e ptdp. (5.50)

For large t, the dominant contribution to this integral comes from small p because of

the factor of e~~pt, so we can obtain an approxim ation valid for large t. Let us estimate

the integrand for small values of p. We can write

1 1 1

s + iu ik f t0a(r + rc)(r — rc) — p ikDoa(r + r c)(r — rc) 1 ----- e-

1 A 1

ikfloa(r + rc)(r — rc) \ ikLl0a ( r + rc) r — rc

(5.51)

and expand it as a geometric series

1 1

s + iuj ikLl0a(r + rc)(r — rc)

O O / \ 7

(5.52)

This expansion is valid for j -j <C 1. The other term s in the integrand (5.50) can also

be approximated for small p. Along the contours C4 and CV,, 7 (5) can be expressed

in terms of p as

7 (5 )if loak

ikCl0a r 2 — p (5.53)

C H APTER 5. THE LIN EAR TIM E-D EPEN D EN T P R O B L E M 82

In the numerator of the integrand in (5.50) we have

1 — 7 (s)r2 = -------- —------ ~ ------- —— (1 H-------—-------1-n ' ikQ0a r 2 - p ikD0a r 2 V ikD0a r 2 ’) ’

p « 1. (5.54)

In the denominator of the integrand in (5.50) we have

2 f r 2 - r? pem \ f pen 2,+ + , « 1 . (5.55)

Thus, the denominator of the integrand in (5.50) is

1 - 2(1 - 7 (s)rJ) log(l - 7 (s)r?)

~ 1 - 2 ^ log (d ^ i) _ 2£_d (log (d_d) + t) + 0 ( p 2 ) ,

p < 1. (5.56)

The sum of the integrals along C4 and C5 is proportional to th e phase shift between

the two branches of the logarithm,

log(l — 7 (s+)r2) — log(l — 7 ( s - ) r 2) = —27ri, (5.57)

and the sum of the integrals is

CH APTER 5. THE LIN E A R TIM E-D EPEN D EN T P R O B L E M 83

where

= 2 hi(r)1 (ikLl0a ) 2(r + rc)r2 ’

__ h2(r)2 (ikLloa)3(r + rc)r4 ’

r2 _ „2 /„2 _ „2\ \ - 1r i i / r r iM r ) = ( 1 - 2— log J J , (5.60)

h2(r) = 2hi(r) - 2r 2 + 2 /i i ( r ) ^ |( r 2 - r 2) ^log ~ ^ + 1 ^ . (5.61)

The expansion (5.58) is convergent for \T0.Tc\ < 1 and so we can interchange the order

of the integration and summation in (5.58). For each n, we can express the integral

in terms of the gamma function T(n) as

fJopn~1e~ptdp — (5.62)

and we evaluate the integral (5.50) along the contours C4 and C5 as

( f + I ) r’ s)eStds\ J c 4 J C i /

/ \ —fc 4 OO 4~ 2 ( ^ j - ^ a j ( n + l ) ( i ( r _ ^ ) )n , fo r |r - rc\t > l.(5.63)

T h e co n to u rs Cq and C7

The contours C§ and Cj correspond to cuts to the branch poin t s = —iQ0k ( l — a r \ ),

or s = —iu — i£lQOtk{r . — rf). Along the contour C$, s = s+ = pet7r — iLl0k( 1 — a rf ) ,

and p changes from 00 to 0. Along the contour C7, s = s - = pe~t7r — iQok(l — a r 2),

C H APTER 5. TH E LIN E A R TIM E-D EPEN D EN T P R O B L E M 84

and p changes from 0 to oo. Hence

+ J ^ 0 (r, s)estds ~ e- ^ t eikQoa(rj-r^t

f ° ° 1 1 ~ 2(1 - 7 (s )r2) log(l - 7 (s )r2) + 0 (1 - 7 ( s ) r 2) _ p t ( .Jo (s + iu) 1 - 2(1 - 7 (s)r^)log (l - j( s ) r? ) + 0 (1 - y (s )rf)

The dominant contribution to the integrand comes from small p because of the factor

of e~pt. So for t 1 we can approximate the integrand by considering small p. The

term ^4— in the integral is approxim ated by

1 1 , 2 ' + O 0 > 2) V (5.65)s + itu ikLl0a(r j — r%) y ikQ.Qa{r^ — r\)

Along the contours C6 and C7, 7 (5) is w ritten in term s of p as

iD0a k7 = I t ? w r 7 (5 66)

and for small p it can be approxim ated as

7(s) = ~ (7 ( 1 " i , / i k n °ar2 ) + 0 { f ? ) ■ (5 67)

We also make the approximation

(1 - 7 (»)r2) ~ + ( l + + 0 (p 2) ) , (5.68)

So the num erator of the integrand (5.64) becomes

1 — 2(1 — 7 (s )r2) log(l — 7 (s )r2) ~

1 - o- ( ^ h +

C H APTER 5. THE LIN E A R TIM E-D EPEN D EN T P R O B L E M 85

The sum of the integrals along the contours and C 7 depends on the phase shift

between two branches of the logarithm. So again we have

log(l - 7 (s+)^i) - log(l - 7 (« - ) r i) = (5.70)

So for small p we can write the leading-order terms of the sum of the integrals along

C 6 and C 7 in the form

(.L+D4>(r, s)estds

g - i u t g i k Q o a ( r j - r % ) t

where

( y j ( - 2in) J ptdp, p<^i 1, (5.71)

6l(r) ( i k n 0a )2(rj - r l ) r \ (5'?2)

^3(r ) = 2 ( l — 2^-i—2— log ( 9— ^ \ (5.73)

with

So the sum of integrals along C 6 and C 7 is

( / + / ) 4>{r,s)estds ~ - 2i7re - iu;V fcnoa(r2- r")t ( y j , t » (5.74)

So we have obtained approxim ations for the tim e-dependent solution, valid for

t » 1, by evaluating the inverse Laplace transform in each of the cases shown in

Figures 5.1 and 5.2.

C H APTER 5. THE LIN E A R TIM E-D EPEN D EN T P R O B L E M 86

C ase 1, 7~i < r < \f2r\

For ri < r < \ /2 r i , the configuration for the inversion of the Laplace transform in the

complex s-plane is shown in Figure 5.1a. The pole is in region I and there are two

branch points. In this case the solution is

(j>(r,t) ~ $ oc(r)e_lwt

+e- ^ t eiknoa(r>~r*c)t ( l Y " ( 2h2(r)\ r i / \ t 2(ikQaa)2(r2 — r2)r2 t3{ikLloa)3{r2 — r2)2rA

-iuit ik S lo a (r \—rj?)t { ^ \ ( ^ 3 (0 , h 2 (r^)+e~ e 0 Kl~ c> — ^ ----------------;rr—« +r i ) \ t 2(ikf l0a)2(r2 — r2)r\ t3(ikQ0a ) 3(r2 — r2)2r \ /

t » 1,(5.75)

where <Lco(r) is the steady-state solution (4.59).

C ase 2, \ / 2 ri < r < \ /2 rc

For \/2 r i < r < y/2r c, r ^ rc, the configuration for the inversion of the Laplace

transform in the complex s-plane is shown in Figure 5.1b. T he pole is in region I.

There is only one branch point and the solution takes the form

<j>(r,t) ~ 4>oo(r)e lut

+ e -i*teifenoa(ra-r2)t ( L V V + 2M 0r i / \ t 2{ikQ,oa)2{r2 — r2)r2 t3(ikf loa)3(r2 — r2)2r4 /

1,(5.76)

where <Foc(r) is the steady-state solution (4.59).

CH APTER 5. TH E LINEAR TIM E-D EPEND ENT P R O B L E M 87

C ase 3, r > y/2rc

For r > V2rc, the configuration for the inversion of the Laplace transform in the

complex s-plane is shown in Figure 5.2. The pole is in region II. There is only one

branch point and the solution is

4>(r,t) ~ $oo(r)e ''twt

2fti(r)________ 2 h2(r) \t2{ikLlaOt)2{r2 — r 2)r2 t 3 (ikQoa)3 (r2 — r 2)2r4 )

t > 1(5.77)

where <F0c(r) is the steady-state solution of the form (4.60).

We observe th a t each of these solutions (5.75)-(5.77) comprises a term 4>00(r)e~'*u,t,

where $ 00(r) is the corresponding steady-am plitude solution from chapter 4, and one

or two time-dependent terms with frequencies coi = u — kl laa ( r 2 — r2) — k d 0a r 2 and

u>2 = u) — kLloa{r\ —r2) — kLl0a r 2.

The solutions (5.75), (5.76) and (5.77) give a complete description of the solution

of the linear time-dependent problem (5.1) for r\ < r < 1 / y / a , r ^ rc, the interval

shown in Figure 4.2. The solution (5.76) is valid around th e critical radius r = r c,

but not a t r — rc. In order to find a solution a t r = r c, we need to consider an inner

region, defined by a critical layer around the point r = rc.

In deriving the solution (5.76) we considered < 1 in order to make the

approximation (5.52). Integrating the geometric series (5.52) term by term gave us

series of term s proportional to [(r — rc)t]~n which is convergent for \r — rc\t > 1. Thus,

the solution (5.76) is valid for t » 1 and |r — rc\t > 1, outside of a layer of thickness

I centred a t r = rc. On the edge of this layer where (r — rc)t ~ 0 (1 ) and inside the

layer where |r — rc\t < 1, the solution (5.76) is not valid. T he region \r — rc\ > | is

_|_g-iwteifeno«(r2—r1)t

C H APTER 5. THE LINEAR T IM E-D EPEN D EN T P R O B L E M 88

considered to be the outer region.

In the inner layer where |r — rc\t ~ 0 (1 ) we can not use th is solution because of

the singularity The solution in the inner layer will be considered in the next

section.

5.3 T im e-d ep en d en t so lu tio n close t o th e critical

layer

To find the solution a t the edge of the critical layer where |r — r c\t ~ 0 (1 ), we return

to the evaluation of the integral (5.58). In the outer region we considered 1

and approximated by

-— E f V ) " e~ptdP> I— r < 1- (5-78)s + iuj r ~ r c ./o \ r ~ rcJ |r - rc|

Close to the critical layer |r — rc\t ~ 0 (1 ) and ~ 0 (1 ), so we need to retain all

the terms in the expansion (5.78).

We also expand f2(r) = fl0(1 — o r 2) in powers of r — rc. Since Ll(r) is analytic at

r c, we can use its Taylor series expansion

Q(r) ~ D(rc) + Q'(rc)(r - rc) + O ( (r - r c)2) ~ ^ - 2aTt0rc{r - rc) + O ((r - rc)2) .

(5.79)

Recall, tha t u / k — f i(rc).

C H APTER 5. THE LIN E A R TIM E-D EPEN D EN T P R O B L E M 89

Now we can write (5.58) in the form

/ + 1 H r ' s )eStdsJcA JCsJ/ \ ^ /»0O / z t x 7T

iuit e ikU 0a ( r 2- r 2 )t ( £_ \ (2z7r) J E°° 0e er j / Jo n \ i k Q o a 2 r c( r — r c)

( 2 p h \ ( r ) + p 2h j ( r ) \ g\(zA:fioCi)2r 22rc(r — r c) (ifcf2oo03r 42rc(r — r c) / ’

p < 1. (5.80)

where

/i4(r) = hx(r) - 4hf(r)(zfi0a ) 2r?2rc(r - r c) ^log - - - - - - + 1^ . (5.81)

So

( [ + [ } H C s ) e Std S ~ e - i “ t e - 2 i k n 0a r c (rc - r ) t (£ _ j\Vc'4 Jcs /

k

Jr ( 2ki v

Jo ^(^A:flott)2r 2(rc - r) ^(p )n+1

(zA:floQ:)2r2(r c — 0 (ikD0a2rc{rc — r))n

2hn (p)”"*~2 \( ikf l0a )3rA2rc(rc — r ) ■*—' (ifcQ0o:2rc(rc — r ) )n J 6 P> P ^ (5 82)

Again making use of (5.62), we can write (5.82) in term s of th e gamma function as

( f + f ) H r , s ) e stds ~ ( - ) — - - (2mt)\ J c 4 J c 5/ \ r i / ( i k n 0a r 2)

f 2 hi _______ r(n + 2)____________1 ikLlootr2 “ J (2z/d2oCM"c)n+1(rc — r )n+1fn+2

2/i4 (r) r (n + 3) t ^ > 1 (5 83){ikDoOi)2rA (2ifcf20a r c)n+1(rc — r )n+1t n+3 J

C H APTER 5. THE LINEAR TIM E-D EPEN D EN T P R O B L E M 90

Close to the critical layer where |r — rc\t ~ 0 (1 ), the solution is

( ^ J + J s)estds ~ e- ^ e- 2tfen°^c(rc-r)t k 2 mr\ J (i k Q 0a r 2)

2h i(r) y - r ( n + 2)( 9 . H c O n f y r A n + 1 ( r ^ — « O n - t - l / n + likQo(xr2t (2ikDoarc)n+1(rc — r )n+1tT1

2 hi{r) ^ T(n + 3)V 1 P L 1 - 1 (A 84'!(ikLl0ct)2r4t2 ^ (2iA:f2oQ!rc)n+1(rc — r ) n+1£n+1y

Note th a t all the terms in both series are of the same order in th e critical layer where

|r — rc\t ~ 0 (1 ). Following Campbell and Maslowe (1998), the series in (5.84) can be

expressed in term s of exponential integrals which are defined as

ro o - z u

E£(z) = I —k~du, Re(z) > 0 (5.85)J i u

andro o zu

E k ( z ) = / — du, Re(2:) > 0 . J i u

After integrating by parts we get

(5.86)

< - >

So close to the critical layer where (r — rc)t ~ 0 (1 ), the solution is

0 (r , f) ~ - e - “ ‘ ( ^ ( ^ ± - E y ^ ‘ \ 2ikr?Q |rc _ r |()

2 h j{r) j?egn(r-rc) (0i{ikLl0a r 2)2t2

E sgn(r '•c)(2tfcr*a|rc _ r |^ + q , (5.88)

where hi and are defined by (5.60) and (5.81) and 4>oc O'") is th e steady-state solution

which was derived in chapter 4. Close to the critical layer, the solution ^ ^ ( r ) is in the

C H APTER 5. TH E LIN EAR TIM E-D EPEN D EN T P R O B L E M 91

form (4.59). The solution (5.88) consists of the steady part and term s proportional to

negative powers of t. As t —> oc the solution converges to the steady state solution.

So, we have found the solution (5.76) valid for |r — rc\t » 1, t » 1, and the

solution (5.88) valid for |r — rc\t ~ 0 (1 ), 1.

5.4 T he inner layer so lu tio n

The time-dependent solution (5.76) and its derivatives with respect to r are singular

a t r — rc. The steady term includes a term proportional to (1 — 4 ) lo g ( l _ r |)r c r c

which is singular a t r = rc and the time-dependent term s of solution (5.76) include

terms proportional to ~ r - In order to find a solution th a t is nonsingular a t r — rc

we have to consider a so-called inner layer around r = rc.

We can write the solution (5.88) for r close to r c in terms of (r — rc) as

« r , t) ~ * _ ( * - « - ( T ) " ( j M L B r ^ \ 2 , k r ^ K - r\t)

2hi(r):E.'lgn{r Tc){2ikr2ca\rc - r\t) + O ( , (5.89)

(ikLl0a r 2)2t2 c \ t 3

where $oo(r ) is defined by (4.59). The leading-order term s of $ 00(r') also can be

w ritten in terms of (r — rc) as

< M r ) ~ e - ia,% ( r c) ( y )-k r 4

~ r ) lo§ ( ~ ) + ° ( E ~ r )2 ri rc

1 - ^ ( r c - r ) lo g (rc - r)

, |r - r c| < 1, (5.90)

wherer 2

M r c) = ( ^ l - 2 ^ 1 —^ J l o g ^ l - ^ J J . (5.91)

We wish to find a nonsingular solution valid in the inner region, where |r — rc| <

C H APTER 5. THE LIN E A R TIM E-D EPEN D EN T P R O B L E M 92

for t 1. On a time scale of t ~ 0{j i 1), where / i d , the thickness of the critical

layer a t time t is \r — rc\ ~ O(fi). So we define a new radial variable

Z = - ( r - r c) (5.92)

and a slow-time variable

T - nt. (5.93)

At the edge of the critical layer \Z \T ~ 0 (1 ), and inside the critical layer \Z\T < 0 (1 ).

We rewrite (5.88) in terms of Z, T and /r as

<f>(Z, T , t) = e~lujth\ (r) ( l + n \ o g i ^ ( —\ Z + Z + — Z lo g (-Z )\ \^*c/ \ \ ^ c TCJ rc

- R V 2 ( t k D 02 a r t:jZ\T) + 0 ( ^ 2). (5.94)

Making use of the following property of the exponential integral E<i (Abramowitz and

Stegun, 1964, equation 5.1.12) tha t

OO / \nE f { z ) ~ ± ( z \ o g ( ± z ) + (~fE - l ) z ) ~ ...............7 ".> (5-95)

where 7e is Euler’s constant, we can rewrite (5.94) as

r \ k /4 /r i 2<f>(Z,T,t) ~ J e l“l ^ /q ( rc) + /ulog/r J Z + /j, lo g — j Z

+ R ^ Z log( - Z ) - , 1 (ik2Q0a r cZ 'I ') lo g ( -z fc 2 fW cZT)

+“ l ^ F ^ ~ 1)l-ik2^ r‘Z T ) +

2/ii(rc) ^ (—2ikQ0a r cZ T ) 2 \ Ofifik S l^ lT Z . ' ( „ - ! ) „ ! + ° ^ ) | - (5-96>

w c ~ _ n * > - ^ 1 v / /

C H APTER 5. TH E LIN EAR TIM E-D EPEN D EN T P R O B L E M 93

After some simplifications, (5.96) becomes

4>{Z, T, t ) ~ e - l“l hi{rc) + filogfj, ( Z + n log — 1 Z

+/z— Z lo g ( -Z ) - n - — ^ - Z log(2ikQ0a r cT) - fJ -h--~^- Z \ o g ( ~ Z ) rr. r, r.

4hi(rc) 2hi(rc) E72—0, 711

( - 2 ik n 0a rcZ T ) T (n — l)n! + o (m2)

\ZT\ < 1. (5.97)

We see th a t the terms with Z log( - Z ) cancel each other and so we obtain a nonsingular

inner solution

fanner(Z,T,t) ~ e \ e lwth x (rc) ( l + n log / i — Z + ( — log — ) Z\ r xJ \ tc \ r c rc

4 4—fi—Z lo g (—2ikLloarcT) — n —(7 j5 — 1 )Z

Tr Tr

+/iT

2 + 8 ik n 0a Z 2T 2 + Ak2Q2° al rcZ3T 3 + o ( z 4r 4)ikLloar^

,(5.98)

In term s of the variables (r — r c) and t, (5.98) can be w ritten as

-kf a n n e r ( r , t) ~ e lut J e luJth x{rc) ^1 + logH— {rc - r)

4 2 \ 4 4— log — ) (rc - r) 4 (rc - r) log( - 2 ikfiQ0a r ct) + — (7^ - l) ( rc - r)+

+ - ikQ0ar%+ 8iA;floc*(rc ~~ r) t +

Ak2PtQa2rc{rc — r)3t3+ 0 ( ( r c - r) t ) ( 5.99)

This solution is valid for \Z \T < 1, or |( r — r c) |t < 1. In contrast with the outer

region where the solution tends to a steady state as t —> 00, th e inner solution grows

like log(f). From this we note th a t the qualitative behaviour of the solution for this

linear configuration for vortex waves with /3 — 0 is the sam e as for Rossby waves

CH APTER 5. THE LINEAR T IM E-D EPEN D EN T P R O B L E M 94

(with /3 ^ 0) in a configuration w ith rectangular geometry (Dickinson, 1970, Warn

and Warn, 1976).

In the next chapter we introduce nonlinearity and the /3-effect to the vortex wave

problem.

5.5 D iscussion o f th e linear tim e-d ep en d en t so lu ­

tion

We discuss the linear time-dependent solution we have obtained here and compare

with the conclusions made by other researchers who have exam ined similar configura­

tion. The problem of wave-mean flow interactions a t a critical layer has been studied

by many previous researchers in the context of barotropic Rossby waves in a rectangu­

lar domain (Dickinson, 1970, W arn and W arn, 1976, W arn and W arn, 1978, Campbell

and Maslowe, 1998, Campbell, 2004). We see some similarities between the solutions

we have derived in this chapter and those obtained in these previous studies. In the

rectangular configuration as well, the tim e-dependent solution comprises the steady

solution and time-dependent term s which are arises from th e branch points of the

integrand in the inverse Laplace transform . One branch point arises from the effect of

the mean flow, the other from the effect of the boundary condition. In both problems

in the outer layer time-dependent term s decrease w ith time as 1/t, and the inner layer

solution is valid for |r — rc\t < 1. We note th a t in the rectangular configuration these

effects arise because of the variation of Coriolis force, in our problem they arise even

in the /-p lane configuration.

The problem of deriving analytical solutions for vortex wave propagation in cy­

clones was considered by other researchers (see, e.g., M ontgomery and Kallenbach,

CH APTER 5. TH E LIN E A R TIM E-D EPEN D EN T P R O B L E M 95

1997 and Brunet and Montgomery, 2002). We can compare ou r results with the re­

sults obtained by Brunet and Montgomery (2002) whose configuration is the most

closely related to ours. They solved the initial-value problem on an /-p lane and used

an angular velocity profile in the form of a quadratic function, sim ilar to the profile we

used. They solved the initial-value problem using a Hankel transform . Their solution

for large r in the notation of this thesis is w ritten as

4>{r,t) ~ (1 + i f )-/i/4_1 (—r 2( l + i t ) )k/2 M ( ^ ^ + 8 , k + 1, - r 2(l + it)), (5.100)z z

where M(a, b, z) is Rum m er’s function.

Let us compare this result with the solution we derived in th is thesis. For negative

values of the real part of z, as \z\ —> oo, Rum m er’s function has the approximation

(Abramowitz and Stegun, 1964, equation 13.1.5)

M ( a , M ~ W ^ o ) {~ Zra[1 + 0 ( |Z |" 1)] (5’101)

for \z\ —> oo, where Re(z) < 0. So the asymptotic behaviour o f the solution for large

values of |r 2(l + i t ) | is

<f)(r, t) ~ O ^ [ —r 2(l + it)]k^2 [—r 2( l -I- it)] 2 2 ^ ~ O ^ r ” '/fc2+8[l + i t ^ ■

(5.102)

For large t this solution is proportional to 0 (r~ v'fc2+8) and it has the same qualita­

tive behaviour as the steady-state solution (4.56) we obtained for r > rc which is

proportional to r -v,fe2+8.

To figure out the reason for the constant of 8 in this expression, we have to go

CH APTER 5. THE LIN EAR TIM E-D EPEN D EN T P R O B L E M 96

back to the beginning of chapter 4, to the equation we solved for the steady case

6' k2 - 4 - { v ' + v / r )

The numerator in the last term of equation (5.103) is + v/ r) . This expression

can be written in terms of the stream function of the mean flow ip(r) as

- S - ( v ' + v / r ) = -^-{iprr + ~i>r) = (5.104)r or r or r r or

which is proportional to the radial gradient of the vorticity of the mean flow. W ith

our choice of the angular velocity profile Cl = fl0( l — a r 2), we have v(r) = fi0(r — a r 3)

and so the gradient of the mean flow vorticity (5.103) is —8f2oar. This is the reason

for the 8 th a t appears in the power of r in the solution. We conclude th a t the rate

of attenuation of the vortex wave am plitude depends on the wave number k and the

gradient of the vorticity of the mean flow.

C hapter 6

Effect o f nonlinearity and /3-effect

6.1 W eakly-nonlinear an alysis

The linear time-dependent solutions derived in chapter 5 can b e considered as a first

approximation to the solution of the nonlinear equation (3.53)

J ; + = - - ( ^ r V 2^A - ^ AV V r)- (6.1)ot r oX J r or r r

To obtain a better approximation we can consider (3.53) an d carry out a weakly-

nonlinear analysis for £ < 1 . We can express the solution of (6.1) in powers of e

4>(r,\,t) ~ ip(0\ r , X , t ) + e ^ (1)(r, A,t) + 0 ( £ 2), (6.2)

where X,t) ~ 0 (1 ). The leading-order term satisfies th e boundary condition

^ ° ) ( r i , A,t ) = elkX + c.c. and can be thus w ritten as

tp ^ ( r , X, t ) = t)elkx + c.c. (6.3)

97

C H APTER 6. EFFECT OF N O N L IN E A R IT Y A N D 0 -E F F E C T 98

When the expansion (6.2) is substituted into the nonlinear equation (6.1), at 0 (1 ) we

obtain the linear equation (3.54) which tells us th a t ^*0)(r, X,t) is the linear solution

(5.76) tha t was derived in chapter 5.

At 0(e) we obtain

d v d \ 2 m 1 m d , 1 .Wt + r T \ ) V 'P d - C ' + C

= - - ( ^ w f - (6.4)

with the boundary conditions ip^(r-i, A, t) = 0. Substituting (6.3) into (6.4) gives

d v d \ 2 m 1 (i) d , 1 .7J7 + V i)( ] - - 1p x w - ( v r + - v ) ot r o X J r or r

( + W tT + V>«<°> - 0,<o|« + «(0>4? + V(0VS>ik» . T t t t t • t r t r r _ ~ t t t t t - t t t t t t - t t * __r \ \ r r

„ 2 i k A ( ( 0 ( O ) ) 2 + ^ ( O ) 0 ( O ) _ I 0 ( O ) 0 ( O ) _ V > 0 £ >

r r

_ e - 2 i k X ( ( ^ ( 0 } ) 2 + 0 * ( O ) ^ ( O ) _ ^ * ( 0 ) ^ ( 0 ) _ I ^ * ( O ) 0 J 4 O )

( 21m - e2lfe* ( ( 0 W)2 + - V > < ^ > - V 0> ^ ) '

_ e ~ 2 i k \ f { € ( 0 ) ) 2 + ^ ( O ) 0 * ( O ) _ I ^ * ( O ) 0 * ( O ) _ I ^ ( O ) 0 ( ; o A \ ( g 5 )

This implies th a t takes the form

rp^\r , A, t) = </>^(r, t) + <p^(r, t)e2lkX + c.c., (6 -6)

where 4>^(r, t) and <f> (r, t) are complex functions. The first te rm in (6.6) is the zero-

wavenumber component which arises from the wave-mean flow interactions. This term

satisfies

Jl V V > = ?i*im( 2 .W (»>^"))). (6.7)

CH APTER 6. EFFEC T OF N O N L IN E A R IT Y A N D (3-EFFECT 99

The evolution of this component is discussed in more detail in section 6.4. The

second term in (6.6) and its complex conjugate are the leading-order contributions to

the second harmonic. The function (f> (r, t) satisfies

( d v d \ _2 ,m 1 ,(i) d , 1 .{ a t + r d - \ ) V * A

= - ( § ( ( 4°>)2 - ). (6 .8 )

In the outer region where \r — rc\t 1, for t 1, the linear solution t) is (5.76)

-iu>t

, iknoarH ( r \ k ( 2^1_(r)_______ 2^2 (r)_______v i / \ t 2(ifcf7oQ:)2)(^2 — r 2)r2 t3(ikDoa)3(r2 — r 2)2r 4

+ c«h w ?. ( l Y N _____Mr)______ + j ( 6 . 9 )\ r i / \ f 2(?A;f2oa:)2) ( r2 — r 2) r 2 t3 ( ikf loa)3 (r2 — r 2)2r\ )

where 4>oo(r) is the steady-state solution. In the inner region where |r — rc\t <C 1, the

linear solution is

4>{0}(r,t) = cf>inner(r,t) ~ e ^ h i ( r c) ^1 + ^ (r _ rc)

4 4 (r - rc) log( - 2 ikQ0arct ) {^B - l ) ( r - r c)

+ S i/M M r - ^ ) 2i2 - ~ rc)3^ + 0 ((r - rc)V )ikLioar2 (.6 .10)

The nonlinear terms on the right-hand side of (6.5) contain derivatives with respect

to r. In the outer region where 0 ^ ( r , t) is given by (6.9), we observe tha t each time

th a t t) is differentiated with respect to r, the term s w ith the factor of elkn°ar2t

are multiplied by a factor of t. This means th a t for < > 1, th e leading-order terms on

C H APTER 6. EF FEC T OF N O N L IN E A R IT Y A N D (3-EFFECT 1 0 0

the right-hand side of (6.5) are proportional to

„jfc00Q(r2— r*) 0 (^2)

In the inner layer where <f>(°\r,t) is given by (6.10), we observe th a t (p^ (r , t ) is

0 ( l ) + 0 ( r —r c) + 0 ( lo g t) (r — rc) (5.99) and so the first derivative of (p(0) w ith respect

to r is 0 (1 ) 4- O (logt). We can write it in the form

4/ij(rc) / . 2 1(j>i0)(r, t ) -------1— E- ( lo g ------ 1 + t -ei k Q o 2 a rc( r ~ r c)t

rc V rc ikQ,Q2arc{r — rc)t

+ log( - ikQ 02arct) + (1 - 7b ) t 1

n = 1

ikLlo2arc{r — rc)t

rc)t)n (n + 1)!

(—ikQo2arc(r — rc)t)n \ . . . .

In the first derivative (6.11) there is the term proportional to eika-°a2rc{T-rc)t_

(Pr is differentiated with respect to r, it is multiplied by a factor of t. So, the second

derivative with respect to r is 0( t) , and the th ird derivative is 0 ( t 2).

Thus, for large t, the term s on the right-hand side of equation (6.1) are 0 ( t 2).

For large t the nonlinear term s increase as 0 ( t 2) and become 0 (1 ) on a time-scale

of t ~ 0 ( e ” ,//2). We can conclude tha t the nonlinear expansion (6.3) is valid for

t < 0{e~1/2). To examine the late-time nonlinear solution we would have to define a

slow-time variable r = el/2t.

All the solutions derived up to this point are for vortex wave propagation on an

/-plane where the Coriolis param eter is assumed to be a nonzero constant. In the

next section we reintroduce the /5-effect into the governing equations and examine

first a linear and then, in section 6.3, the nonlinear configuration.

C H APTER 6. EFFECT OF N O N L IN E A R IT Y AN D /3-EFFECT 101

6.2 Effect o f th e variation o f th e C oriolis force

The vortex Rossby wave equation with the /3-effect included is (3.52)

( d v d \ o , I , d 1 P , ■ i 0 - x a , x) V ' lP V\w-{Vr + - v ) sin A + -u c o s A + /3^y cos A =\ o t r oX J r or r r e- - ( t f v V 2^ A- ^ A V V r ) . ( 6 .12)

r

The problem is governed by two nondimensional param eters (3 and e. In section

3.1 we demonstrated th a t (3 can be considered as a small param eter given the typical

length scale for our problem, which gives /3 ~ 10~2. In general, wave amplitudes

are small relative to the magnitude of the mean flow so the param eter e can also be

considered to be small. T hat justifies the use of weakly-nonlinear analyses such as

the analysis th a t was carried out in section 6.1. The first step in our investigation

of the /3-effect shall be to examine the linear problem with th e /3-effect added. This

corresponds to a configuration where /3 e and so the nonlinear O(e) term s in (6.12)

appear at higher order than the 0(/3) terms. However this scaling means th a t the

0 ( /3e~1) term in (6.12) becomes the dominant term in the equation. Since this term

cannot satisfy the specified 0 (1) boundary conditions, the linear problem is not well-

defined in th a t case. In order to overcome this issue we would have to neglect the

0(/3e_1) term. We can do this by adding an additional term B(r, A) = —|u co sA to

the governing equation as discussed in section 3.1. In section 6.3 we consider the full

nonlinear equation (6.12) with both param eters e and /3 included. We shall assume

there tha t /3 <gC e which is the more likely physical configuration.

The linear equation with the addition of the term B becomes

C H APTER 6. EFFECT OF N O N L IN E A R IT Y A N D f3-EFFECT 1 0 2

To solve (6.13) we look for a solution of the form

ip(r, A, t) = ^ (0)(r, A, t) + /30(1)(r, A, t) + 0(/32), (6.14)

where tp^ (r , A, t) is the linear solution of (3.54) obtained in chapter 5 for the /-p lane

configuration. As in section 6.1, we write 0(°)(r, A, t) as

A, t) = < ^ ( r , t)etkX + c.c. (6.15)

After substitution of (6.14) into the governing equation (6.13) we obtain at 0(0):

+ I V V (1) ~ = ~ COS \ (6.16)d v d dt + r d A

with the boundary conditions -i//b(r, A,t) — 0. W riting cos A = (ezX + e zX)/2 and

sin A = (elX — e~zX)/2i and substituting (6.15) into (6.16), we obtain

+ ) V V (1) - ^ v )d v d dt + r d A _ 1

~ 2V ° > - ^ 0)) e*(fe+1)A _ °) + 0 (0)(r ) ) + C.C. (6.17)

This tells us tha t A, t) can be w ritten as

0 (1)(r, A, t) = ^ (la)(r, A, t) + 0 (16)(r, A, t ), (6.18)

where ^ la> satisfies

1 ( k2 \ r

0 ( ° ) - 0 (o ) )eW ) A+ c.c . (6.19)

C H APTER 6. EFFECT OF N O N L IN E A R IT Y A N D /3-EFFECT 103

and satisfies

^ V > + 4 0)( r ) ) e ^ - ^ + c.c., (6 .20)

with boundary conditions tp^la\ r i , A, t) — 0 and ip(lb\ r i , A , t ) = 0.

Each of the equations (6.19) and (6.20) is a nonhomogeneous differential equation,

so we can write the general solution in each case as the sum of the general solution

of the corresponding homogeneous equation and a particular solution of the nonho­

mogeneous equation, which is proportional to e^fc±1 A -f c.c. Recall th a t the function

is the sum of (f>00(r)e lujt and two time-dependent term s th a t go to zero as

t oo. Therefore, a particular solution of each of (6.19) and (6.20) is proportional

to e*((fc±1)A- wt) for t 1. The zero boundary condition at r = rq tells us tha t the

solution of the homogeneous equation is also proportional to eI((fc±1)A- wt) for f > i.

Thus the solutions of the homogeneous equations corresponding to each of (6.19) and

(6.20) are of the same form as the solution bu t w ith the wavenumber k replaced

by k ± 1. For f > 1 the solution of the homogeneous equations a t leading order are

(6 .21 )

and

(6 .22 )

where <^la) and 4>(lb> satisfy equations

CH APTER 6. EF FEC T OF N O N L IN E A R IT Y A N D (3-EFFECT 104

and

*<>»" + _ A l l > > » _ ? 5 £ 2 _ ---------- = 0 . (6.24)r H (H0(l — a r 2)(k — 1) — uj)

Equation (6.23) has a singular point a t the value of r where uj = (k + l)flo (l — a r 2),

i.e. the mode with wavenumber k + 1 has a critical radius at

1 ^ - n 1 ' • (6.25)c*+1 a 1/2 \ (A; + l)fl0/

Similarly, the mode w ith wavenumber (k — 1) from (6.24) has a critical radius at

1 / \ 1/21 / . uj \ '

rCk = — I 1

The wavenumber k is the number of wavelengths in a full circle 0 < A < 2tt, s o k is a

positive integer. We can consider two cases k = 1 and A; > 1. If k = 1, then k + 1 = 2

and A: — 1 = 0. This means th a t a zerowavenumber (non-oscillatory) component is

generated by the /3-effect. This component represents a change in the mean flow and

will be discussed in section 6.4. If k > 1 then k + 1 and A: — 1 are both nonzero and

there is no effect on the mean flow.

The critical radii for the 3 modes are related by

rCk_, < r c < rCk+1 (6.27)

as shown in Figure 6.1. All three waves (with wavenumbers A;, A: + 1 and k — 1)

experience critical layer absorption a t their respective critical radii. This means tha t

as the waves propagate outwards from the eye of the vortex, the (A; — 1) mode is

absorbed first at rCJc_1 and its am plitude is greatly reduced while the other waves

C H APTER 6. EF FEC T OF N O N L IN E A R IT Y A N D (3-EFFECT 105

continue to propagate. Then the k mode is absorbed a t rc while the (k + 1) mode

continues to propagate and is absorbed a t rCk+l. This m eans th a t a t r c there are

only two components of the solution with non-negligible am plitude, corresponding to

wavenumbers k and (A; + 1). In the critical layer around r c, th e inner layer solution

for the mode k is valid, but to obtain information about the effect th a t the (k + 1)

mode has on the evolution of the critical layer around r c, we need an outer solution

for the (k + 1) mode.

In the critical layer near r c, the (k + 1) mode is described by its outer solution.

We write it as

^ la\ r , A, t) ~ [^(T O e-*** + Yparticular] ei(fc+1)\ t » 1, (6.28)

where 0 x ( r ) is the solution of (6.23), and can be w ritten as a linear combination of

the hypergeometric functions of the form

/ 2 ( 1 ) W J*

-{k+1)

- f c - i

Cfc+1+ 0 1 -

Cfc+1

(6.29)

. (6.30)

To find Y p a r t i c u l a r we return to the nonhomogeneous equation (6.19) and substi­

tu te in the inner solution for 0 °* (r, t) since we are considering the critical layer near

r c. The inner solution for 0 (o (r, t) is (5.99). Substituting this into the right-hand side

CH APTER 6. EF FEC T OF N O N L IN E A R IT Y A N D (3-EFFECT 106

\ j ' c (m o d e 'k + 1 )

le k-1

Figure 6.1: S ch em a tic d ia g ra m o f th e k, k — 1, k + 1 m o d e s p ro p a g a tio n .Schematic diagram of the propagation of the waves w ith wavenumber k (arising from the boundary forcing a t r — r x), and wavenumber k ± 1 (arising from the /3-effect). The shaded regions show the inner layers for the critical radii o f each wave. The arrows show propagation of waves up to their critical radii. T he wave with wavenumber k + 1 is influenced by (3 effect at the critical layer for wave with wavenumber k.

C H APTER 6. EFFECT OF N O N L IN E A R IT Y A N D /3-EFFECT 107

of (6.19) gives

(f)(0) __ 0(0) ^ e - i u td ) ( r ^ 1^ ~ C'W /ll(rc)~ ( r ~ rc) + C (t)hi{rc) + 0 ( l / t )

+ Cn[(r - rc)(t)]n ) , t » 1, (6.31)n=l

where

C (t) = — ( \ o g ( - i k n 0ar^t) + (7^ - 1)) .T* n

(6.32)

Then we use the variation of param eters formula to obtain th e particular solution

Y p a r t i c u l a r ( r , t ) ~ ( ^ J j y ^ y / l ( l ) /2(1)

^ y / 2( i ) ( ^ (0) - / 1(1), t » 1, (6.33)

where W(r) is the Wronskian of the two functions / ip ) and f 2(i). So, the general

solution for the nonhomogeneous equation can be w ritten in th e form

^ y / i ( i ) ( V 0) - 4 0) )dr + A 2 ) / 2(1)

- ( / Mi) ( V 0) ~ (t>{r ] d r - f m , t > 1. (6.34)

Since we are evaluating the solution near r c, we write (6.29) and (6.30) in terms

of (r — rc) and get

/i(p ~ r {k+1) 1 - 21og 1 - 1 - + O I -T' f 1 4”' c fc+l / \ c* + 1 / \ cfc+l

~ r~ (fe+1)[a0 + a ^ r _ r c)], |r - r c| < 1 (6.35)

C H APTER 6. EFFECT OF N O N L IN E A R IT Y A N D (3-EFFECT 108

and

/ 2(i) ~ r {k+1) ( 1

where

c k + 1

~ r (*+1)[a2 + a3(r - r c)], |r - r c| < 1,

Cfc+ 1 ck+ ir-2

ai = ^ - ( 21og(lCk+1 Ck+1

0,2 — 1

03

c*+i2r c

r 2C f c + 1

The Wronskian for the functions /i(i) and / 2(2) is

W(r)/l(l) / 2(1)

/ l( l) /2(1)

~ r 2(fc+0 (aoa3 _ a ia 2), |r — r c| < 1.

is

— icot [/(ao + a ^ r - r c ) ) ^ 2^ 2) ( k u , s k------------------------- C2 h+i) ~ h n r ) - C { t ) h i - ( r - rcaoa3 — Oia2 r ^ k+1> \ r r

+C(t)hi + ^ Cn((r — rc)t)n J dr + A\

I71— 1

r fc [a2 + a3(r - r c)]

(a2 + a3(r — ^c)) r^2k+2'> ( k , . . . k~ ----- “ ------- ~(2 k+i) r - C < r - r c + C i /ma0a3 — a ia 2 r'.2'c+1) \ r r

+ Y 2 cn((r - rc)t)n \ d r + A 2n=l

r [a0 + a i ( r - r c) ] , |r — rc\ < 1, t » 1.

(6.36)

(6.37)

(6.38)

)

(6.39)

C H APTER 6. EFFECT OF N O N L IN E A R IT Y A N D (3-EFFECT 109

To find (^la*(r) we have to evaluate the integrals in (6.39). Note th a t

/ j T c n[ ( r - r c)t}ndr = [ cn[ ( r - r c)t}ndr = y ^ < [ ( r - r c)t]n ~ O f y") . (6.40)^ n—1 •* n= 1 n=2 ' '

After integrating (6.39) we obtain

<j>{la\ r , t ) ~ e-»wfr (_fe) [a2 + a3(r - r c)] - A2[a0 + Gi(r — rc)] + ^ r2C'(i )^ >

(6.41)

or,

<f la\ r , t ) ~ [0(1) 4 - 0 ( r — r c) + 0 (lo g t)]e_lwt. (6.42)

This solution represents the leading-order contribution to the solution near the crit­

ical layer (near rc), arising from the variation of the Coriolis param eter. The wave

amplitude increases as log t because of the /3-effect, and it is th e result of interaction

between the k mode and the k + 1 mode in the critical layer for the k mode.

6.3 C om bined effects o f n o n lin ea r ity and th e /3-

effect

We now consider equation (3.52) which includes both nonlinearity and the /3-effect.

3 , » 9 ) „ 2 i 1 ^ /- . * 0 i -v , 0,Tr: + - TTT v ^ -i ’\ T r ( vr + ~v) V’a sin A + - v cos A -I- /3^y cos Aat r o A / r or r r e

= - - ( ^ V V a - ^ aV V t-)- (6.43) r

The param eters e and (3 are both small, the solution of (6.43) can be w ritten as

0 ~ + £ipH) 4. (3i(j(la) 4- 4- 0 ( e 2) -f 0 ( e ( 3) 4- 0(/32), (6.44)

CHAPTER 6. EFFECT OF N O N L IN E A R IT Y A N D 0 -E F F E C T 110

where i p ^ \ ip(la\ i p ^ are functions of r , A, and t.

As noted in chapter 3 and section 6.2, w ith the reference length scale for our

problem it is reasonable to consider the situation where « £ < 1. In th a t case

0e~l <C 1 and so the 0e~ l term th a t arises in the solution is not a leading order term.

We first investigate the 0(e) and 0 ( 0 ) term s in the solution an d then a t the end of

this section we will add the 0e~ l term to the analysis.

We can rewrite equation (6.43) in the form

L(ip) + 0B(xp) = — eN(i>, ip), (6.45)

where L, B, and N are the partial differential operators given by

. . ( d v d \ 2 1 d 1

m = \ m + r T \ r ^ - r ^ r ^ + r ^

N t y u l h ) = ~ ?/hAV2t M ,

B(ip) = ~ —ip\ sin A + "i/y cos A = -^r. tp\(elX - e~lX) + \ ipr (elX + e~lX) r 2i 2

Substituting (6.44) into equation (6.45), we obtain

L(ip{0)) + 0L(ip{la)) + 0L(ip{lb)) + eL(ipw ) + 0B(ip{o)) + e0B(iP{1))

= - e N ( ip i0), i p ^ ) - e 0 N ( ^ \ ^ ) - £0N('4>{o\'4>W)

- e 2N ( ^ ° \ 0<x>) + O ( e 20 ) + O(e02) + 0 ( e 3) + O ( 0 3). (6.49)

To analyze the order of the different term s in (6.49), their wavenumbers and their

possible interactions, we first note th a t a t leading order L ( i p ^ ) = 0 , where i p ^ is

the linear solution defined in chapter 5 w ithout the /1-effect and corresponds to the

modes ± k specified a t the forced boundary r = r j. We note also th a t the 0(e) term

(6.46)

(6.47)

(6.48)

C H APTER 6, EFFEC T OF N O N L IN E A R IT Y AN D /3-EFFECT 1 1 1

corresponds to wavenumbers 0 and ± 2 k, as shown in section 6.1. The 0(/3) term

corresponds to wavenumbers ± (k + 1) and ± ( k — 1), as shown in section 6.2, the

0 ( e 2) term corresponds to wavenumbers L k and ± 3 k, the 0 ( e 8 ) term corresponds

to wavenumbers ±1, ± (2 k + 1) and ± (2 k — 1), and the 0(/32) term corresponds to

wavenumbers ±fc, L (k + 2) and ± ( k — 2). This information is summarized in Table

6 . 1.

If f3 ~ e, e <C 1 and 8 -C 1, then e and j3 are the same order of magnitude and the

solution up to 0 (e ) and 0 (8 ) is simply the sum of the solutions obtained respectively

in sections 6.1 and 6.2.

Table 6.1: Wavenumbers corresponding to the term s in equation (6.49)

order terms wavenumber wavenumber special case k = 1

£ N ( ^ ° 8 i ’(0)y, L (ip^) 0 ;± 2 k 0, ±2

8 T ( ^ la>); L(-0<16>); B (^ (0)) ± (k — 1); ±(/c + 1) 0, ±2

82 B (^ (la)); 5 ( ^ (lb)) ± ( k - 2 ) ] ± k ; ± ( k + 2 ) ±1, ±3

e8£ ( ^ (1>) ±1; ± (2 k — 1); ±(2k + 1) ± 1 ,± 3

iV('0(o), ^ (la)); yv(^(0), ^ (lb)) ± l ;± ( 2 f c - l ) ;± (2 fc + 1) ± 1 ,± 3

£■2 ± 3 k ± 1 ,± 3

-k-2 -k-1 -k+1 -k+2 k-1 k+1 a

...........

Figure 6.2: S ch em atic d ia g ra m for w av en u m b e rs . Schematic diagram for the wavenumbers in equation (6.49) of the term s of 0 (e ) and 0(/3), the case where k > 0, k ^ 1. The 0 ( e ) terms include a component with zero wave number, but 0 (8 ) do not. There is no interaction between the e and 8 term s in th e mean flow.

C H APTER 6. EFFEC T OF N O N L IN E A R IT Y A N D /3-EFFECT 1 1 2

If k = 1, then the 0 (e ) and 0 (3 ) term s both correspond to the same wavenumbers,

0 and ±2.

If /? -C £ 1 then the solution can be w ritten in the form

rjj ~ ^(°) + + /3ip<'la'1 + + 0(ef3) + 0 (e 2) + 0(/?2). (6.50)

Substituting this equation into (6.49) we obtain

0 (1 ) ; L(^°>) = 0,

0 ( e ) : L ( ^ ) = - iV ( ^ (0), ^ (0)),

0(13): L (^ (la)) + L(x/j{lb)) = - £ ( ^ (0)),

0(e/3) : L(4>{2eO) = - iV ( ^ (0), ^ (la)) - Ar(^(o),-0(16)) - B(ip{1)),

0 ( e 2) : L ( ^ {2e)) = -7 V(^(o),-0(1)) - N ^ l ' i p W ) ,

0((32) : L ( ^ ) = -B ( i / i {la) + ^ u)). (6.51)

If A; > 1, then, up to the orders shown in the table, there are no interactions

between the terms of each order because they all correspond to different wavenumbers.

The larger the value of k is, the further apart the wavenumbers for the 0 (e ) and 0(/3)

terms are. This situation is illustrated in Figure 6.2. Continuing to higher orders of

e and (3 the solution can be represented as a discrete spectrum of contributions from

several wavenumbers. Since the 0 (e ) term includes a zero wavenumber component, it

means tha t nonlinearity affects the mean flow. W ith k > 1, however, the 0 (3 ) term

has no zero wavenumber component, so the /3-effect does not affect the mean flow.

The 0(/32) and 0 ( e 2) include term s corresponding to the fundam ental wavenumber

± k . All the 0(/32), 0 (e /3) and 0 ( e 2) term s contribute higher wavenumbers to the

solution. Figure 6.2 demonstrates th a t there are no further contributions to the zero

CHAPTER 6. EF FEC T OF N O N L IN E A R IT Y A N D /3-EFFECT 113

wavenumber component until higher orders. So we can conclude th a t if k > 1 the

mean flow is influenced only by nonlinearity.

The only situation where term s of different orders have the same wavenumbers is

the special case where k = 1. In th a t case the O(e) and 0((3) term s both correspond

to the wavenumbers k = 0 and k — 2, and so the solution u p to this order is the

sum of both effects. Since the 0{e) and 0(/3) term s both have zero wavenumber

components, we conclude th a t both nonlinearity and the /3-effect influence the mean

flow. The evaluation of the mean flow is discussed in section 6.4.

We now investigate the 0{/3e~l ) term in the solution

ip ~ _|_ e^(!) + /3t/;(la) + /3'ip(lb) + 0 { e2) + 0(s/3) + 0(/32). (6.52)e

The function A, t) satisfies the equation

0(J3/e) : L (^ > ) = - V- { e iX + e ^ ) . (6.53)

Thus ip(B\ r , A, t) corresponds to a wavenumber of 1 and m ust take the form

= <pB(r, t)e tX + c.c. (6.54)

with the boundary condition tfiB (rx,t) = 0. The solution o f the corresponding ho­

mogeneous equation will take the same general form as tha t obtained for the leading

order solution and can be expressed in term s of hypergeometric functions as it was

done in chapter 4. The solution has the same qualitative form as the steady solution

4>oo. Since the nonhomogeneous term is independent of tim e, we can find a steady

particular solution to this equation. Because of the zero boundary condition, the

homogeneous solution should be steady as well. So the function <pB is steady and

CH APTER 6. EF FEC T OF N O N L IN E A R IT Y A N D (5-EFFECT 114

satisfies the equation

& + & - V + ^ r(Br = - f ■ (6.55)r z uj — i lk 2r

Its solution has only a quantitative effect on the solution for th e modes ±1. If k = 1,

<pB has the same wavenumber as the fundam ental mode. If k ^ 1, then the 0 (3 e ~ l )

term does not interact with the fundamental mode or any of th e 0(e) terms (which

correspond to wavenumbers 0, ±fc, ± 2 k).

6.4 E volu tion o f th e m ean flow in th e inner layer

From equation (6.7) we see th a t the mean flow changes w ith tim e due to the nonlinear

interaction. We also saw in section 6.3 th a t in the case where k = 1, the /3-effect

contributes to the mean flow change. In th a t case the change in the mean angular

velocity is the sum of the changes due to the nonlinear and /3 effects. Let us consider

first the change of the mean flow th a t results from the nonlinear wave-mean flow

interaction.

According to (3.49) the to tal stream function is

^ ( r , A, t ) = xj;(r) + etp(r, A, t ) . (6.56)

The perturbation eip(r, A, t) now includes a zero wave num ber component which we

will call ipo(r, t). Thus the mean stream function a t time t is th e sum of the initial mean

streamfunction 4>(r) and the time-dependent mean stream function rp0(r, t) induced by

the nonlinear interaction. We let /iptotai('i',t) be the mean stream function at time t.

Then we can write

Aotai (r , t) = i ) { r ) + -ip0{ r , t ) . (6.57)

CH APTER 6. EFFEC T OF N O N L IN E A R IT Y A N D [3-EFFECT 115

Similarly, the mean azimuthal velocity is

Vtotai(r, t) = v(r) + v0(r, t ) (6.58)

where v0(r, t) is the wave-induced mean velocity. We can ob ta in an equation for the

evolution of the mean flow either from equation (6.7) or d irectly from the azimuthal

momentum equation as done in Appendix B. In either case, we end up with (B.16):

^ r = e2h ^ {2kr ta^<0)« <0|)>' <6'59)

where Vtotai is the azimuthal velocity of the main flow.

To find the change in the azim uthal velocity in the inner layer where |r - rc\t < 1

we use the leading-order solution 0 ^ ( r , t) (5.99)

^(°) ^ e-«wt hi ^ + bi(rc - r ) - ib2^ ( r c - r)

+b2(rc - r) log(f) - i y + ib4t(r - rc)2 + 0 ( ( r - r c)3t2)^ (6.60)

where 6, for i = 1 ,2 ,3 ,4 are real numbers which are defined as

4

&i = —(log(kO0otrl) + - 1),

kQ0ar% ’64 = 8kCloa. (6.61)

CH APTER 6. EFFEC T OF N O N L IN E A R IT Y A N D 0 -E F F E C T 116

The derivative of 0 * ^ w ith respect to r in the inner layer is

# ( 0) ^ e- t ^ h i ^ _ 6i _ ib^ _ _ h lQg(t)

- 2 ib4(rc - r )t + 0 ( ( r - rc)2t2)) . (6.62)

Substituting this into (6.59) gives

* > - ■“ ( - t , : + + 0 « r - r j V ) I . (6.63)dt r% \ r i ) \ 2 t

So the time derivative of the mean azim uthal velocity satisfies

9 V t o t a l

dt~ e2 (o(l) + O + 0 ( t 2(r - r c)2) ) . (6.64)

Integrating with respect to tim e from the initial tim e to tim e t gives an equation for

the azimuthal velocity of the mean flow in the critical layer as a function of tim e t,

W , t ) ~ e(r) - ( j - l t - - ^ ( l o g ( ) 2) • (6.65)

So we see tha t the azimuthal velocity decreases w ith time.

We saw in section 6.3 th a t in the case where k = 1, nonlinearity and the /5-effect

both influence the mean flow. In this case we find th a t

Oi? f 4 7T \Vtotai(r,t) ~ v(r) - e2— h\ I — ~ t + O (logf)2 I + £0O{logt)elult + c.c. (6.66)

rc \ r c 4 J

So the mean flow change consists of two parts: a term decreasing with tim e and

some oscillations with am plitude proportional to log t. This change in the azimuthal

velocity means tha t the mean angular velocity Cltotai (r, t) also decreases with time in

the critical layer and hence the location of the critical radius changes with time.

C H APTER 6. EFFECT OF N O N L IN E A R IT Y A N D (3-EFFECT 117

For any initial Ototai(r) profile th a t decreases monotonically with r (such as the

profile Qtotai(r, 0) = Cl(r) = f2o(l _ £*r2) used in our study), a decrease in Ototai(r,t)

means tha t the critical radius moves inwards, towards the centre of the vortex. This

is seen in Figure 6.3. This result is in agreement w ith hurricane observations where

the secondary eyewall moves toward the centre of the hurricane (see, e.g., Sitkowski

et al., 2011).

» • • • *

r'r r, r

Figure 6.3: D ynam ics o f th e critica l radius location . T h e initial critical radius is given by £2 — uo/k, where £2(r) is the initial angular velocity. If the angular velocity decreases in the critical layer near r c, as shown by the dashed line, the critical radius moves towards the centre of the vortex, from location r c to r'c < rc.

C hapter 7

C onclusions

7.1 Sum m ary

In this thesis we presented approxim ate asym ptotic solutions for a problem represent­

ing vortex Rossby waves propagation in a tropical cyclone. The cyclone was considered

on a horizontal plane defined by polar coordinates r and A. T he cyclone was repre­

sented as a vortex with angular velocity 0 ( r ) in a domain given by t\ < r < oo and

0 < A < 2n. At r = r lt a sinusoidal wave was generated by a boundary condition

— e^kX wtK We considered waves propagating outw ards in the cyclone in

the form 0 (r, \ , t ) = 4>(r,t)etkX + (p*(r,t)e~lkX. We derived equations for the wave

amplitude starting from the governing equations for fluid dynam ics in a coordinate

frame rotating with the Earth. Several different configurations were examined.

First, we considered the linear problem where the wave am plitude 0 is independent

on t. In this case 0 is given by an ordinary differential equation in r, which is singular

at rc, where the phase speed of the wave mode is equal to the angular velocity of

the cyclone rotation. This equation was solved for a quadratic profile of the angular

velocity. The solution dem onstrates the attenuation of the waves in the cyclone at

118

CHAPTER 7. CONCLUSIONS 119

the critical radius.

Next we studied the linear time-dependent problem where th e wave amplitude <f>

depends on both r and t. We derived the solutions valid for la te time for the different

regions in the tropical cyclone: the region between the eyewall and the critical layer,

the region inside the critical layer, and the region outside the critical layer, far from

the centre of the vortex. These solutions were expressed in te rm s of hypergeometric

functions. Each solution comprises a term with steady am plitude and some time-

dependent terms th a t go to zero as t —> oo.

To find a nonsingular solution near the critical radius r = rc, we introduced an

inner layer and an inner variable Z — (r — rc)jfx w ith a corresponding slow time

variable T = fit, where p is the thickness of the inner layer. W e found th a t the inner

solution contains term s proportional to log(t). Then we reintroduced the nonlinear

terms to the problem and observed th a t these term s would grow like t2 in both the

outer and inner solutions. This means th a t the nonlinear solution is valid for t <

0 (e~ x/2). The nonlinear effects arise in the wave mode with wavenumber 2k and in

the mean flow with wavenumber zero.

We also added a /3-term into consideration to evaluate the effects of the variation of

the Coriolis force on the vortex Rossby wave dynamics. Our analyses showed tha t the

/3 term gives rise to wave modes with wavenumbers (fc+ l) and (/c —1). All three waves,

modes (k + 1), k and (k — 1), experience critical layer absorption a t their respective

critical radii. While the waves propagate outwards from the eye of the cyclone, the

(k — 1) mode is absorbed first at its critical layer rCk , and its am plitude is reduced

a t this radius. The k mode is absorbed a t rc while the (k + 1) mode continues to

propagate and is absorbed a t rCk+\. So there are two modes, with wavenumbers k

and (k + 1), in the critical layer r c and the wave am plitude o f the k + 1 mode grows

like log(f) in the vicinity of the critical layer r c. We also considered the more general

C H APTER 7. CONCLUSIONS 1 2 0

configuration th a t includes both nonlinearity and the /3-effect.

We concluded tha t if the fundamental mode has wavenumber k > 1, the /3-effect

does not influence the mean flow. If k — 1 then the mean flow is influenced by both

effects. The azimuthal velocity of the mean flow decreases w ith time because of the

nonlinear interaction, and the /3-effect adds oscillations to the mean flow.

The main features of our solutions, namely the critical layer absorption of the

waves, the development of concentric rings of high wave activity and the changes in

the location of these rings with time, can be used to explain some of the observed char­

acteristics of tropical cyclones, such as the formation and evolution of the secondary

eyewall.

An im portant issue in the study of vortex Rossby wave dynamics in tropical cy­

clones is the interaction between the waves and the mean flow in the vortex. The

problem of wave-mean flow interaction has been studied by m any previous researchers

in the context of barotropic Rossby waves in a rectangular domain (Dickinson, 1970,

Warn and Warn, 1976, Warn and Warn, 1978, Campbell and Maslowe, 1998, Camp­

bell, 2004). We see some similarities in our conclusions com pared with these previous

studies. They found th a t wave-mean flow interactions occur in the critical layer, and

the thickness of the critical layer is 0 ( e -1/2) which is similar to our conclusions for

the vortex problem. In both problems there is also the phase shift in the logarithmic

term of the solution which defines the attenuation in the wave amplitude. The time-

dependent solution also has similar features. The mean flow changes with time due

to nonlinearity.

It is interesting to note th a t in rectangular coordinates these features of the solu­

tion arise because of the /3-effect. In the context of vortex dynamics these effects are

obtained even without the inclusion of the /9-effect.

Our results give some insight into the secondary eyewall cycle. We have identified

CH APTER 7. CONCLUSIONS 1 2 1

at least two mechanisms tha t could result in changes in the in itia l eyewall configura­

tion. One is the development of multiple concentric rings of critical layers a t different

orders, due to the inclusion of the /3-effect; these could be seen as concentric secondary

eyewalls. Another observation is the modification of the mean flow by the wave mo­

mentum flux divergence which changes the location of the critical radius for the 0 (1 )

wave (the strongest of the concentric eyewalls) for tim e scales u p to t ~ 0 (e ~ 1/2). For

longer time scales when the nonlinear term s have become the sam e order as the linear

terms, a late-time solution may be possible through the introduction of a “slow” time

variable. This is one of the possibilities for further work on th is problem. Numerical

simulations can also be used to shed further light on the evolution of the waves and

their effects on the mean flow.

A possible further extension of this problem would be the introduction of a verti­

cal coordinate, making the problem three-dimensional and allowing the introduction

of internal gravity waves in addition to vortex Rossby waves. There has been con­

siderable debate in the past few decades as to which type of waves, vortex Rossby

waves or vortex gravity waves, play a greater role in hurricane dynamics. Brunet and

Montgomery (2002), and Schecter and Montgomery (2004) discussed the influence

of inertia-buoyancy oscillations, i.e. gravity waves, on the vortex dynamics. They

conclude tha t these oscillations are im portant for cyclones in the middle latitudes.

Schecter and Montgomery (2004) showed th a t the influence of vortex waves exceeds

the effect of gravity wave propagation on the vortex dynamics. Hendricks et al. (2010)

included the inertia-gravity waves in their numerical simulations of the vortical mo­

tion in a hurricane. They concluded th a t inertia-gravity waves are insignificant in the

process of intensification or decay of the vortex. An investigation th a t includes both

types of waves could provide some insight to help address th is question.

A ppendix A

H ypergeom etric functions

A .l G eneralized hyper geom etr ic fu n ctio n s

The generalized hypergeometric function has the form (see Erdelyi, 1953)

pFq(au a2...,ap-b1,b2,...,bq;z) = (A>1)^__0 V°1 ) n - - \ P q ) n n .

where 2 is a complex variable, a*,, with k — 1 , . . . ,p, and bj w ith j = 1, ,q, are

complex constants. The notation (a)„ is the Pochhammer symbol,

(a)„ = n ”T01(a + f), (A.2)

and (a)0 = 1, (0)0 = 1. For the case where p — 2 and q — 1, the generalized

hypergeometric function is called simply the hypergeometric function F and is given

by

F 2 Fy{au a2-,by\z) = E (l )V i " / ‘ A -3)„=0

1 2 2

AP P E N D IX A. H YP E R G E O M E T R IC FU NCTIONS 123

A .2 T he so lu tion s o f th e h yp erg eo m etr ic equ ation

The hypergeometric function (A.3) is a solution of the hypergeometric differential

equation

where 2 is a complex variable, and a, b and c are complex constants. This equation

has three regular singular points, a t 2 = 0 ,1 ,00 (see Abramowitz and Stegun, 1964).

The solutions depend on the values of the constants a, b, an d c. In our problem

(chapter 4, p.55), c — a + b + l — p + 1 , ab = —2. We find th a t p = —k or p = k so

the values of a and b are

2(1 — z ) f " + (c — (a + b + 1 ) z ) f ' — ab f = 0, (A-4)

ak + V W + %

2

b =f c - N / F + 8

2

c — k + 1 (A-5)

for p = k. And

—k + \ / k 2 + 8a =

2- j f c - -n/ P T 8

c = —k + 1 (A-6)

for p = —k. Near the singular point 2 = 0 equation (A.4) has two linearly independent

solutions valid for \z\ < 1 (equations 15.5.18 and 15.5.19 from Abramowitz and Stegun,

APPEND IX A. H YP E R G E O M E T R IC FU NCTIONS 124

1964). These solutions are

t r \ 7“t/ I 1 . I \ jr fk + \ / k 2 + 8 k - y /k2 + 8/i(o)(2) = F (a , 6; 1 + fc, z) = F ( ------- , , 1 + k, z)

V - (A .7)^ ( l+ fc )„ n ! '

/ 2(o) = F{a, b-l + k, z) log 2 +

[X(a + n) - * (a) + x(& + n ) ~ x{b)~

X { k + l + n) + x (k + 1) - x (n + 1) + x ( l) ]

^ ( 1 - a U l - » C ■ (A 8 )71=0

F(a\ 6; c; z) in solutions (A.7) and (A.8) is a hypergeometric function which is defined

by the formula

F(a;b;c;z) = f ) (A.9)n=0 1 /"

where (a)„ is the Pochhammer symbol,

{a)n = n ”^ 1 (a + i ), (A. 10)

with (a)o = 1, (0)o = 1. In our problem the singular point 2 = 0 is not in the domain

of the problem. So we do not need solutions (A.7) and (A.8).

Near the other singular point 2 = 1 we introduce a new variable Zi = 1 — 2 and

write (A.4) as

21(1 - z x) f " + (0 - { - k + 1 ))21/ / - 2 / = 0. (A.11)

In this case the solution can be w ritten following equations (15.5.20) and (15.5.21) in

APPEND IX A. H YP ERG EO M ETRIC FU NCTIONS 125

Abramowitz and Stegun (1964) for the case where c = 0. For p — —k two linearly

independent solutions of the hypergeometric equation (4.34), valid for \z — 1| < 1, are

/i( i) ( l ~ z ) = = (1 - z )F (a + 1, b + 1; 2; (1 - z)) (A.12)

and

where

/ 2(i)(l ~ z ) = h (\)(z i) = (1 - z)F (a + 1, b + 1; 2; (1 - z)) log(l - z)

+ ( 1 ~ z ) h — s a — ( 1 - z)[x(a + 1 + n) - x(a + 1) + x ( b + 1 + n ) - x ( b + 1 )-

x(2 + n) + x(2) - x (n + 1) + x(l)] + (A.13)

x W _ (a .i4)

These solutions are valid for 0 < \z\ < 2, \z\ ^ 1. In our problem a and b are defined

by (A.5) and (A.6). For p = these solutions can be w ritten as

° ° / drfc+VA44-8 I 1 A ( i i k —V A 4+8 i -I A

- ^) = (1 - 0 2 3 -— 3------ o f 2------ ± - 4 M ( l - 2)“ (A.15)71=0

APPEND IX A. H YP E R G E O M E TR IC FU NCTIONS 126

and

/ 2 ( i ) ( l - z) = ~ 2

oo f ± k + V k 2+ 8 , 1\ / ± k - V k * + 8 , i \+(1 - z) E ( ~T^ ^ (1 - *>" iog(i - z ) +

t ' o (2)(")n!00 ( ± k + V W +8 | n /-±fc-y^+8 | i \(i _ 2) g 2 ± i M _ J ±1>M(1 _ 2)»

( » l ^ f ± g + 1 + n)- x (±t + f ^ + 1)

±A: — Vfe2 + 8 . ±fc — \/fc2 + 8 . . .+X ( j + ! + « ) - X( j + ^ ~ X( + n )+

x ( 2 ) - x ( r a + l ) + x ( l ) ) . ( A . 16)

These two functions /m ) and / 2(i) are plotted in Figure A .l for z = ^ . We see th a t

the function / ; is an order of m agnitude smaller than / 2(i)-

The third singularity of equation (A.4) is the singular po in t a t infinity. The so­

lutions for z —> 00 are given by equations (15.5.7) and (15.5.8) in Abramowitz and

Stegun (1964):

fi(oo)(z) = z~aF(a, a - c + 1; a - b + 1; g ) , (A.17)

f 2 (ao){z) = z~hF(b,b - c + 1; 6 — a + 1; g ) . (A.18)

W ith the values of a, b, and c for our problem (A.5) and (A.6) and w ith p = ± k these

solutions are

j - t . V n f z A + V j i E ± I ( k + V k ^ + 8 \

/,,„,(*) = ( Z ) * ^ Y , M - (A.19)

APPEND IX A. H YP E R G E O M E TR IC FUNCTIONS 127

1.5

1

f 2 (1)

0.5

f l ( l )

0rl rc

Figure A .l: T h e h y p e rg e o m e tr ic fu n c tio n s . The hypergeometric functions/i( i)(r2/r;?) and h(\){r2/ r l ) in the interval r x < r < rc, k = 2, r x = 0.1 rc.

and

r - x — 00 ( - k - s / k 2+ 8 \ / k — \ / k ? + & \

/ „ - , ( * ) = ,A ,0 ,

These solutions are valid for \z\ > 1.

A ppendix B

N onlinear evolu tion o f th e m ean

flow

The angular velocity of the background flow changes with tim e due to the momentum

flux divergence at the critical layer (Montgomery and Kallenbach, 1997). To derive the

equation for the time evolution of the angular velocity let us w rite the A component

of the momentum equation (2.6) in term s of the stream function ^ ( r , A, t),

d ^ r ( 1 _ d . _ . _ \ 1 dp . .' ^ ( r t f r ) + — (B .l)d t \ r 2 dr r ) p d X

We take the mean of each term in (B .l) in the azimuthal direction by integrating one

with respect to A from 0 to 2n and then dividing by 2ir. This gives

d'V 1 d 1r - - 2 ^ ! - ( r t f r ) + - t f r ¥ rA (B.2)dt r2 dr r

where the bar denotes the azim uthal mean. Using the fact th a t

1 T 9 / T . 1 d . T T . 1 T T“2 ^ A« - ( r ^ r ) = U r W r ^ r^A r (B.3)r 2 dr r l dr r

128

AP PEN D IX B. NO NLINEAR E V O LU TIO N OF TH E M E A N F LO W 129

we can rewrite equation (B.2) in the form

dt r2 dr

The mean of -4 fr \I/rA

d ^ r 1 dr'^r'Hx 2 T T _ .A + - * r t t rA. (B.4)

i r i T T „ i i , r .2 „ , i /■ i T x » / x , x- 1 = - - (*.) IS' - - 1 -Wr dx. ( )

The first term on the right-hand side of (B.5) is zero. So

f 27r 12 / - m ry rXd \ = 0, (B.6)

Jo r

and we can rewrite equation (B.4) in the form

dt r2 dr

Recall from (3.49) tha t the stream function consists of 2 parts

(B-7)

^ ( r ,X , t ) = ip(r) + eip(r, A,<). (B.8)

Let ifi(r, A, t ) be the wave

ip(r, A, t) = 0 ( r ,t )e ifeA + c.c. = <f>(r,t)eikX + 4>*(r,t)e~ikX. (B.9)

Derivatives of ip(r, A, t) with respect to r and A are

ipr(r, A, t ) = 0 r (r, t)etfcA + </>*(r, f)e~tfcA (B.10)

APPEND IX B. NO N LIN EAR E V O LU TIO N OF TH E M E A N F L O W 130

and

A, t) = ikcf)(r, t)elkX — ikcf)*(r, t)e~lkX. (B .ll)

Hence,

iprtpa = (4>v{r,t)elkX + 4>*(r,t)e~lkX) (ik(f>{r,t)elkX - ik<t>*(r,t)e~lkX)

= ik - <j>*r<f>) = - 2 k \ m ( M l ) . (B.12)

In section 6.4 we discussed the time evolution of the mean flow by representing the

mean streamfunction as a sum

4>totai(r,t) = $ (r ) +ipo(r,t) (B.13)

where 4>(r) is the initial mean flow, and ipo(r, t) is the tim e-dependent mean stream­

function induced by nonlinearity. The corresponding azim uthal velocity is

Vtotai(r,t) = v(r) + v0(r,t) (B.14)

where, .v d^toha{r,t)vtotai{r,t) = ------—---------. (B.15)

So from (B.7) and (B.12) the equation for the evolution of the m ean azimuthal velocity

is

^ Ira(^ :)) (B 16)

or, for the mean angular velocity,

^ = £2M ( 2 t r Im ('w ’; , ) - (B 17)

A ppendix C

T he phase shift for th e stead y

solution

Here we will show tha t the phase shift of the steady solution (4.59) in the critical layer

is —7r. The steady-state solution (4.59), which is valid in the interval r\ < r < y/2rc,

r > rc, 9 = ±7r, and following Miles (1961), we can determine th e sign by considering

the inverse Laplace transform in the tim e-dependent problem th a t was considered in

chapter 5.

We can write the logarithmic term in the tim e-dependent solution (5.26) in the

s-plane in the form

r rc, or 1 — ^ < 1 , includes the term (1 — r 2/ r 2)lo g (l — r 2/ r 2). This term is2

singular close to the critical radius where r = rc. We observe th a t log(l — h ) =r c

2log [ 1 — + i&, where 0 is the a r g u m e n t F o r r < rc, 9 = 0. For

r cFor r < rc, 9 = 0. For

131

APPEND IX C. THE PH ASE SH IFT FOR TH E S T E A D Y SO L U T IO N 132

log(l - 7 (s )r2)

= log

, „ ikDoar2 \ , / ikQ o& ^iReis) — zlm (s) — ikD0)log 11 - r r - n = r I = log ( 1 ------------------ + (im(s) + knay

kQ, o«r2R e(s)1 +

s ikD0 j yA:Q0o^2(Ini(s) + kQ0)

Re(s)2 + (Im(s) + kQo)2 Re(s)2 4- (Im (s) + kQ0)2( C . l )

Consider the complex-valued function

c w = 1 +M2o(*r2(Im(s) + kQ0)

Re(s)2 + (Im(s) + k f l0)2 _fcfl0o:r2Re(s)

Re(s)2 4 (Im (s) 4- kD0)2

We note tha t Re(s) > 0 along the contour of integration (a — zoo, a 4- zoo) since a > 0.

So Im(£(s)) < 0. This means th a t

—7r < arg{Q < 0.

The steady solution corresponds to the pole s = — iui. We consider the limit s —» —iu,

which means tha t Re(s) —> 0 and Im(s) -» — to. As s —> — iuj,

k t t0a r 2(— to + ktto) _ kQ0a r 2 _Q[S) ( - u + kQ0)2 “ “ kD0a r 2 ~

This means tha t log(£(s)) —> log ^1 — which can be w ritten as

r 2 *c

log|C(s)l + *arg(C(s)) log + zarg 0 4 )

(C.2)

(C.3)

Taking this limit allows us to determ ine the correct sign of th e argument of ^1 — ^ j

APPEND IX C. THE PH ASE SH IFT FOR TH E S T E A D Y SO LU T IO N 133

in the steady-state solution (4.59). We noted in chapter 4 th a t

r 2\ I 0, if r < rc, a r g ( l - i i ) = < (C.4)

±7r = 9, if r > rc.

Since —7r < arg(£) < 0, we conclude th a t as s —> —iou, arg(C(s)) —>■ 0 if r < rc and

arg(£(s)) —> —7r if r > rc. This means th a t the phase shift 9 = — it.

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