phase and group velocity

Note: NOT vector components! (“trace speeds”)

Simplest case: Ignoring f, why not align x axis with waves (no equation 2a). Form a vorticity equation from 1a, 3a (eliminating p’)Use mass continuity (4a) to eliminate u.

(wxx + wzz)tt = N2 wxx

Assume wavelike solutions exp(i (kx+mz-st)) w2 = N2 k2 (k2 + m2)-2

w = N cos q

• freq = N cosq• (slope of parcel

motion paths)• http://www.atm

os.ucla.edu/~fovell/DTDM/

• http://www.atmos.ucla.edu/~fovell/AOS101/sgw.html

Note: momentum is being fluxed (u’w’)

Mountain waves

Courtesy: Geraint Vaughan

Inertio-gravity waves• These rather forbidding equations hide a pleasing elegance to the

gravity-wave solutions which become more evident if x is taken to be the direction of propagation of the wave. Then ℓ=0 and V = -iUf/ω. V and U are in quadrature, causing the wind vector to rotate elliptically – clockwise for upward energy propagation and anticlockwise for downward. For oscillations near to the inertial frequency (which are common in the lower stratosphere) the ellipses are nearly circles, as shown in the example below measured by a VHF radar at Aberystwyth, Wales.

Geraint Vaughan

Taylor-Goldstein eq• Don’t assume wavelike in z, only in x & t• Permit U(z), and r(z) (Anelastic)

Taylor-Goldstein eq• Don’t assume wavelike in z, only in x & t• Permit U(z), and r(z) (Anelastic)

Critical level, where u = cor intrinsic frequency 0

• dumps westerly momentum below critical level– think QBO (downward propagation of u)

“overreflection”: ideal for wave

trapping/ducting

“Importance” of gravity waves1. For middle atmopshere: Systematic upward

flux of zonal momentum [u’w’]– goes as (wind amplitude) x (tilt of motions)

• tilt goes with frequency ( w = N cosf) – high frequencies are highly weighted in this importance

metric

– systematic E/W asymmetry• asymmetric sources

– form drag in flow over obstacles (mtns or cloud tops)

• asymmetric filtering on the way up (critical levels etc.)– even w/ symmetric sources, but base state flows/ shears

– detectability an issue for global assessment• vertical wavelengths for satellite kernels, etc.

“Importance” of gravity waves1. Notes: w’T’ = 0 so they don’t carry heat flux

vertically 2. often viewed in terms of w field which

emphasizes short wavelength and high frequency waves

3. These ideas can almost close off from view another important role for long (hydrostatic) waves: as an adjustment mechanism

1. carrying heat away from convection horizontally2. adjusting stratification profile

1. to moist adiabatic

“Importance” of gravity waves 2.

1. adjusting stratification: vertical displacement– strong interactions with convection– goes as (w amplitude) x (period or duration)

• low frequency components especially important– like from net heating events (zero frequency limit)

– a whole different slice of (k,l,m,w) space• which is only really 3D since dispersion relation holds:

M&R Bores: horizontally moving wavefronts• Parsons

ppt on 663 course web page

Back to simple waves, in resting Boussinesq fluid.Hydrostatic, or low frequency limit

(small k & l, i.e. m >> k,l)

Aligning x with the wave (l=0), w = N k/mc = cg = N/m = vert. wavelength/ BV period“Horizontally propagating vertical modes”

Vertical modes

• w=0 BC at top and bottom of stratified layer discretizes allowable vertical wavenumbers m– w = sin(kz) – wave numbers ½, 1, 1.5, ... of the layer– waveguide or duct – no vertical propagation, or upward + downward

propagation are equal (reflections) yielding standing oscillations (vertical modes)

– The higher you place the lid, the closer together the wavelengths permitted

Example: response to deep convective heat sources

• Boussinesq, hydrostatic, constant-N, nonrotating, resting basic state– w = N k/m is dispersion relation for waves

• Heating is forcing, confined to ‘troposphere’– lower portion of deep stratified fluid under lid

• Heating turns on at t=0 and then maintained • 2 modes of tropospheric layer

– convective (sin(z) all positive)– stratiform (sin(2z) vertical dipole)

w solution 20 hours later• radiating

upward• slope of ray

path0 set by 20h timescale since heating began

acts like virtual dipole source in infinte fluid since BC is a symmetry line (w=0)

Mapes JMSJ 1998

Lids cause errors: interference w/artificially reflected waves

• Here steep rays are those set by k,m

• Their arrival peapods w solution

Mapes JMSJ 1998

displacement (T’) field is far less wrong than w’ field, even for 10km lid! Vertical propagation and reflection error misses the point.

Vertically propagating waves don’t carry heat upward

• but they did carry it outward

Mapes JMSJ 1998

Let’s just look at tropopase lid case

• here for f=0, 6h heating event 16 hours ago

w field at t=16h, 10h after a 6h heating event happened at the origin.

T field for same case

What does f do? • Traps heat within a Rossby deformation radius

of the origin, held back by thermal wind shear

inertio-gravity waves

b trapped

inertio-gravity waves (zero-sum waves) escape

Solution procedure

€

˙ Φ n (x, t)∝ Q(x,z, t)sin(2πz /Lzn )dz0

Z top

∫ ; Lzn = 2ztop /n

Fourier transform in zalong with B.C.s

Unknowns: u(x,z,t), w(x,z,t), T(x,z,t), p(x,z,t)

Known: Q(x,z,t)

Shallow water system (1 for each Fourier mode: n = 1, 2,3,..)

Since N is const.

Governing equations

€

∂un∂t +∂φn

∂x=0 (1)

€

∂φn∂t +cn

2∂un∂x=˙ Φ n (2)

Stefan Tulich ppt

Sketch of solution*

€

w ~ ±sin(πz /D) → Lzn = 2D

Vertical velocity w at times ti < t < tf

cn-cn

€

cn = ± LznN 2π = ± DN π ≅ 50m s−1

*References: Nicholls et al. 1991; Mapes 1998

Stefan Tulich ppt

Sketch of solution

Horizontal velocity u at time t < tf

50 m/s-50 m/s

Stefan Tulich ppt

Sketch of solution

Temperature T at time t < tf

50 m/s-50 m/sWarm

Stefan Tulich ppt

Sketch of solution

Times after the heating is switched off (t > tf)

WarmWarm 50 m/s-50 m/s

Stefan Tulich ppt

Sketch of solution

Times after the heating is switched off (t > tf)

WarmWarm-50 m/sWarm

50 m/sWarm-50 m/sWarm

-50 m/sWarm

50 m/sWarm 50 m/sWarm 50 m/sWarm-50 m/sWarm

“Modes”? Convective and Stratiform

Example: 2 radar echo (rain) maps (w/ VAD circles)

200 km

Convective & stratiform “modes”

Con

Con

Strat

Strat

In pure simplest theory case

Con: sin(z)

Strat: sin(2z)

Houze 1997 BAMS

Addition of a stratiform-like source

Spatial structure:

cooling

warming

€

Qst ∝−sin(2πz /D) → Lzn = D

tDCi tDCf = tStitime

QDC0 QST0

tStf

Temporal structure:

Stefan Tulich ppt

Response to stratiform heating

Just before the heating is switched off

50 m/swarmwarm

cold

25 m/s

Stefan Tulich ppt

Longer vertical wavelengths travel faster horizontally

A complex convective event in a salt-stratified tank excites many vertical wavelengths in the surrounding fluid (photo inverted to resemble a cloud).

Strobe-illuminated dye lines are displaced horizontally, initially in smooth, then more sharply with time.

Mapes 1993 JAS

early

late

Glimpses of invisible env. flow

Fourier spectra of QDC and QSt for different lid heights

Ztop = DZtop = 2DZtop = 4DZtop ∞

€

˙ Φ n (x, t)∝ Q(x,z, t)sin(2πz /Lzn )dz0

Z top

∫ ; Lzn = 2ztop /n

positive coeff. implies warmingresponse near the surface

negative coeff. implies cooling response near the surface

Lidded solutions are crude approximations to continuous solution

Multiple wave packets are excited

Stefan Tulich ppt

Summary of wave/mode background

• The flow of stratified clear air outside convective clouds is dispersive– longer vertical wavelength components

expand faster/farther away from source horizontally

• Any vertical profile, e. g. divergence, can be expressed as a spectrum, w/ axis labeled by phase speed. – lid discretizes spectrum; bands robust

Is all this sin(z) ghost/mode stuff realistic? (or kinda kooky?)

• Need: modes of a realistic atmosphere (actual stratification profiles)–Ready: Fulton and Schubert 1985

• Need: realistic heating (divergence) profiles–Ready: many many VAD measurements

What about for real atmospheres where N is not constant?

We can still use the same procedure but a more general (“vertical mode”) transform must be used:

€

˙ Φ n (x, t)∝ f [Q(x,z, t)]ψ n (z)ρdz0

Z top

∫

The vertical structure functions n (and their associated phase speeds cn) are obtained as numerical solutions to the vertical structure problem: an eigenvalue problem with dependence on N

Stefan Tulich ppt

Vertical modes associated with the CRM’s basic state atmosphere*

*calculated using the algorithm of Fulton and Schubert (1985)

Lzn ≈ 28 kmLzn ≈ 14 km“ Lzn” ≈ 11 km“

cos(2z/Lzn)

Structure functions of both u and

Spectrum of average VAD divergencefrom many profiles in tropical rain

different lid pressures ->

different discretizations,

bands robust

Hey -- what’s this?

Mapes 1998

T response when

observed mean VAD

divergence is used as a

mass source in observed

mean stratification

Shallow water solved for

each mode, then sum it up

Mapes and Houze 1995

Top-heavy C+S: spectrum & response

Melting: forcing is

localized in z, response is localized in

wavenumber!

Melting mode

Mapes and Houze 1995

Raw data: Snow melts,

whole troposphere

shivers

(wavelength set by melting layer

thickness?)spectral view not quite so

kooky?

m=1

m=3/2

Does this exist?

m=1/2

Re: kookinessAre convective and stratiform really dynamical modes?

Rare, but compelling

(great data quality)

Jialin Lin

Rare, but compelling(5h of data, from front to back of

storm)

Aboard the R/V BrownJASMINE project

considerable front-back

cancellation

May 22, 1999(figs from U. of Washington webpages on JASMINE)

~15 m/s

Webster et al. 2003, Zuidema 2003

In a storm notable for fast, long-distance propagation

diurnal

Kousky - Janowiak - Joyce (NOAA CPC)

ship

Re: kookinessnumerical modeling, with advection and no lid

Pandya and Durran 1996

u

u later

Re: kookiness

Wavefront 2 stays vertical and coherent despite advection by sheared winds nearly half the wave speed!

Pandya and Durran 1996

Re: kookinessmore numerical modeling

Even convective cells appear to be gravity waves!?

Yang and Houze 1995

This stuff hasn't totally sunk in to the convection community (myself

included!)

Spectral questions

• Where do the observed modes come from ultimately?

Modal (band) responses seen away

from convection

• Yes, Convective and stratiform “modes” seen in T fluctuations, but

• ~15 m/s also prominent

Fast ghosts zipping everywhere - only

statistics are available reliably

?

A fundamental source for c ~ 15 m/s

radiative

cooling

12km

moist adiabat runs dry

8km

spectrum of square Qrad forcingobs.

strat.

NO fundamental source for c ~ 25 m/s ("stratiform mode")

• Apparently excited by processes internal to convective cloudiness

– half-troposphere depth cumulus congestus rainclouds

– precipitating stratiform anvil clouds

No fundamental source -> GCMs failLack of stratiform processes, or of cumulus showers?

GCM

Deep convectionheating in GCM

Lee Kang Mapes 2001

20N-20S cooling

Deep convectionheating

obs

Earth

Mapes 2000

Cloud resolving model has it...

Tulich Randall Mapes 2006

shallow cu (SC) & stratiform (ST)

opposed

SC only in lower half of

mode

Revisiting the outline• (Intro: white lumps, invisible

environs)• Spectral laws of stratified flow• “Modes” of convection • The life cycle: why grow just to

die?

– A question of coupling between the 2 halves of convective circulations

»(the white part (clouds) + spectral (dispersive by m) environmental flow)

Stratiform instability• Recall d(KE + PE) = [b’Q’] • Heating where it’s hot drives all disturbance energy• But buoyancy driven convection should favor cold

places.• Low-level b key

– Inhibition control

http://journals.ametsoc.org/doi/full/10.1175/1520-0469 (Mapes 2000)

Occurs on many scalesbigger<-> longer lived, suggests a velocity

scale

Mapes Tulich Lin Zuidema 2006

Clean: 4000 km rain waves in a 2D model

(All the followingwork by Stefan Tulich)

cc3

The life and death

of cc3

a multicellular

entity

shallow

deep

strat.

Why die? Why do new cells fail?

1 km warm T’

BUOYANCY OF LIFTED AIR PARCELSFROM LOW LEVELS

env warm

(& dried)

cell-killing warm wedge: a downward displacement in a wave

warm T’ cold pools slide

under, but new cu fail

What does the LS wave look

like?

a larger version of

cc3, of course!

cu in front

deep

strat.

LS wave motion to right

Note T’ no bigger in heated areas -

equilibrated wave

Front edge: wave forces cu clouds

cu heating nestled in low T’, which keeps falling

But why does the large scale wave exist?Must go back to origins

(different model run - main wave went R->L)

widening riverof wave

amplitudeas events trigger

next events

Key mechanism: short vertical wavelength mode

change it via radiative cooling

depth and/or lapse rate

changed wavelength spectrum

rain wave speed

changes accordingly

Conclusions• Illusion of clouds as substantial is

visually compelling–Must be resisted with rationality

• Motions of embedding environment are inseparable, and spectral–Longer vert. waves travel faster –chromatography of outgoing signals–sloped destabilization by incoming signals

Not kooky, but a little spooky

• Artifice of upper lid not too bad–believe bands not modes

• (but mode is a convenient word)• Neglect of advection not too bad–wavefronts remain upright & coherent

even in strong shear• how ??

–secondary circs?

Where does wave-1 of troposphere “mode” come from?

• Precipitating stratiform anvils force it• Cumulus congestus showers force it

» lower half only

• These cancel on average - there is no physically fundamental source

»large-scale models can miss it via parameterization errors

Convective & stratiform

–Inevitable microphysical outcomes of bubble ascent (rain, ice, etc)?

–Or dynamical modes of motion?•What governs downdraft depth for example?

»rain could just saturate air & stop evaporating if descent didn’t agree with the ambient airflow...

Leading edge of the life cycle

• Is this 2000 km / 20 hour wedge scale governed by the cumulus dynamics of moisture buildup?

• Or does wave cooling invite (by buoyancy) or demand (for balance) a certain heating?

» Sensitivity to precipitation efficiency of cu?

shallow cu heating

Is the MCS just another convectively coupled wave type?

• small scale, large amp., but qualitatatively...

What’s up with this?

Substantial, very repeatable

deviation from a moist adiabat.

CRMs don’t get it.

microphys (e.g. ice?)

small cu effects?

LS (trades) crucial?

Discussion welcomedmapes @ miami.edu

Thank you!