Contents · Contents 1 Atmos & Ocean Thermodynamics 5 ... These notes emerged from a graduate course which I used to teach at the Uni-versity of California Los Angeles. The course

Contents

1 Atmos & Ocean Thermodynamics 51.1 Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Systems & States . . . . . . . . . . . . . . . . . . . . . . 81.1.2 Thermodynamic Processes & Equilibrium . . . . . . . . . 101.1.3 Temperature . . . . . . . . . . . . . . . . . . . . . . . . 101.1.4 Equations of State . . . . . . . . . . . . . . . . . . . . . 11

1.2 The First Law and its Consequences . . . . . . . . . . . . . . . . 151.2.1 The Calculus . . . . . . . . . . . . . . . . . . . . . . . . 171.2.2 Some Consequences of the First Law . . . . . . . . . . . 21

1.3 The Second Law and its Consequences . . . . . . . . . . . . . . . 251.3.1 Carnot Cycles . . . . . . . . . . . . . . . . . . . . . . . . 251.3.2 The Second Law . . . . . . . . . . . . . . . . . . . . . . 261.3.3 The Clausius Inequality . . . . . . . . . . . . . . . . . . 271.3.4 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.3.5 Some Consequences of the Second Law . . . . . . . . . . 31

1.4 Phase Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . 351.4.1 Clausius Clapeyron . . . . . . . . . . . . . . . . . . . . . 371.4.2 Condensate . . . . . . . . . . . . . . . . . . . . . . . . . 391.4.3 Moist Enthalpy . . . . . . . . . . . . . . . . . . . . . . . 401.4.4 Equivalent Potential Temperature . . . . . . . . . . . . . 401.4.5 Static energies . . . . . . . . . . . . . . . . . . . . . . . 42

2 Thermodynamic Processes, and Convection 472.1 Thermodyamic Diagrams . . . . . . . . . . . . . . . . . . . . . . 48

2.1.1 Pseudo-Adiabats . . . . . . . . . . . . . . . . . . . . . . 522.1.2 Polytropic Processes . . . . . . . . . . . . . . . . . . . . 532.1.3 Soundings . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.2 Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.2.1 Saturation by isobaric mixing . . . . . . . . . . . . . . . 56

1

2 CONTENTS

2.2.2 Buoyancy Reversal . . . . . . . . . . . . . . . . . . . . . 572.2.3 Mixing Diagrams . . . . . . . . . . . . . . . . . . . . . . 59

2.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592.3.1 Moist convective instability . . . . . . . . . . . . . . . . 612.3.2 CAPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.3.3 Oceanic convection . . . . . . . . . . . . . . . . . . . . . 65

3 Boundary Layers 693.1 Laminar Boundary Layer Theory . . . . . . . . . . . . . . . . . . 69

3.1.1 Flat Plate Boundary Layer . . . . . . . . . . . . . . . . . 703.1.2 Ekman Layer . . . . . . . . . . . . . . . . . . . . . . . . 743.1.3 Secondary Circulations . . . . . . . . . . . . . . . . . . . 77

3.2 Atmospheric Boundary Layers . . . . . . . . . . . . . . . . . . . 773.2.1 Mixing Length Theory . . . . . . . . . . . . . . . . . . . 783.2.2 Outer and Inner Layers . . . . . . . . . . . . . . . . . . . 79

3.3 Buoyancy Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 813.3.1 Monin-Obukhov Similarity . . . . . . . . . . . . . . . . . 823.3.2 Convective Boundary Layers . . . . . . . . . . . . . . . . 853.3.3 Diurnal Evolution . . . . . . . . . . . . . . . . . . . . . . 89

Preface

These notes emerged from a graduate course which I used to teach at the Uni-versity of California Los Angeles. The course was ten weeks and tried to set thefoundations for subsequent studies of convection and small scale processes in theatmosphere. In this version I try to capture some of the main ideas from thesenotes, augmented by material on ocean thermodynamics. My emphasis is on a the-oretical development of the subject matter. The phenomenology is developed in aparallel class Atmospheric & Oceanic Sciences 200B. Sources for these notes andsuggestions for further reading are provided in the section on “Further Reading” atthe end of each Chapter.

3

4 CONTENTS

Chapter 1

Atmosphere and OceanThermodynamics

1.1 Structure

The atmosphere, or air, as we experience it, is a multi-component gas in whicha great variety (if not great amount) of finescale particulate matter is suspended.The gas phase constituents include several major gases (Nitrogen, Oxygen, Argon)which through the current era have existed in a relatively fixed proportion to oneanother. To a large degree these determine the thermodynamic properties of whatwe called dry air, that is an ideal mixture composed of 78.11% N2, 20.96% O2

and 0.93% Ar. Real air contains slightly less of each of these constituents so as toaccommodate variable vapors such as carbon dioxide and water, along with a hostof seemingly minor gases (e.g., Neon, Helium, Methane, Nitrous Oxide, Ozone)some of which can be important for determining the radiative properties of the at-mosphere and the quality of the air we breath. Of the variable constituents, water isthe most striking as it ranges from abundances of nearly zero to as much as 4% byvolume. Because of its proclivity to change phase and the manner in which thesephase changes affect the local temperature on the one hand, and foster diverse inter-actions with radiant energy on the other, water has the capacity to strongly interactwith atmospheric flows over a range of timescales, from minutes to millennia orlonger. In this sense the simplest, accurate description of the dynamic atmosphereis best formed by viewing the working fluid as a two-component fluid: dry air andwater.

The structure of the atmosphere, as measured by its temperature and pressureexhibits regularity in space and time, which has fostered a nomenclature of dis-tinct layers (troposphere, stratosphere, mesosphere, thermosphere) based largely

5

6 CHAPTER 1. ATMOS & OCEAN THERMODYNAMICS

stratosphere

mesosphere

thermosphere

troposphere

mesopause

stratopause

tropopause

Figure 1.1: Structure of atmosphere as a function of height (plotted on abscissa). From leftto right, temperature, pressure, and density. Density of air (solid) and water vapor (dashed)shown normalized by their surface values of 1.191 kg m−3 and 0.014 kg m−3 respectively.Data taken from the averaged mid-latitude summer (McClatchey) temperature structurebelow 60 km. and from the US standard atmosphere above 60 km.

on the thermal structure as shown in Fig. 1.1. A nomenclature based on the de-gree of ionization of the gases leads to the identification of an ionosphere above75 km; one based on the atmospheric composition demarcates the atmosphere intoa homosphere below roughly 80 km where the composition of dry air is relativelyfixed, and a heterosphere above. The subject of dynamic meteorology is almostexclusively concerned with the homosphere, especially its lowest most portion thetroposphere, hence our definition of dry air.

Through the depth of the troposphere density and pressure varies by an order ofmagnitude, and temperature varies almost 100C, from well below to well abovefreezing. From Fig. 1.1 it is evident that the density and pressure of that atmospherefall off almost exponentially with height, hence permitting a description in the form

f(z) = e−z/λ (1.1)

where f can be pressure or density, z is height above the surface and λ is a scaleheight. For the profiles in Fig. 1.1 λ = 8.8, 7.5 and 1.4 km respectively for pres-sure, density and water vapor density. Thus while pressure and density are com-mensurate with the scale of the troposphere the water-vapor scale-height is muchsmaller. Although not shown the temperature structure of the troposphere alsovaries greatly at the surface with maximum temperatures approaching 50C andminima of -50C. Similarly water vapor amounts vary from a trace to as much as40 gkg−1 over warm tropical oceans.

For the purposes of dynamic oceanography the ocean is best thought of as anideal solution of various ions (e.g., chlorine , sodium, magnesium, sulfur, calciumand potassium) in water. Together they comprise what we call sea-salt whose con-centration by mass is called the salinity. The salinity, s of water is approximately

1.1. STRUCTURE 7

dep

th [m

]1000

4000

5000

0 5 10 15 35 36Temperature [deg C] Salinity [PSU]

thermocline halocline

dee

p (a

bys

sal)

oce

an

upper oceansurface mixed layer

Figure 1.2: Structure of the ocean, showing temperature and salinity versus depth. Herethe data is taken from Levitus 98 for a point in the north Atlantic (40N, 50W) in the winterseason

3.5%, or 35 h, where the practical salinity unit or PSU is often used instead ofper mil. Treating the various salts as one composite quantity is only useful becausetheir proportions are relatively fixed with respect to one another (55.4% Chlorine,30.8% Sodium, 3.7% Magnesium, 2.6% Sulfur, 1.2% Calcium and 1.1% Potas-sium). Like the atmosphere the ocean contains a variety of other constituents whichregulate its properties. These range from biological matter which regulates its ra-diative properties, to other chemical constituents (including carbon dioxide) whichwhen cycled among the land and atmosphere help determine the state of the cli-mate system. In summary, the ocean, like the atmosphere is best idealized as atwo component system, but unlike the atmosphere its important minor constituent(while variable, but in both cases less than 4% by mass) does not undergo changesin phase—although its major constituent does.

The structure of the ocean also exhibits regularity with well defined layers ap-parent in both the temperature and salinity (Fig. 1.2). The upper 10-200m of theocean is usually associated with a surface mixed layer (shaded in the figure) inwhich temperature and salinity are relatively uniform. Below this layer both T ands change markedly through layers called the thermo and halocline respectively.The abyssal, or deep, ocean (which typically resides below 1km and contains mostof the ocean mass) tends to have a more uniform temperature and salinity structure.The pressure of the ocean varies by four orders of magnitude, between surface pres-sures near 105 Pa to pressures near 4×107 Pa at its average depth1 of about 3800m.At the location shown in Fig. 1.2 temperatures range from 17C to near zero, al-though in select locations temperatures over 30C are not uncommon. Salinity in

1The deepest point in the ocean is thought to be near 11 km and located in the Marianas Trench,the bottom pressure at this location would be around 1.1× 108 Pa


abyssal ocean varies relatively less that at the surface. In the figure the upper wa-ters are more saline than at depth, as is characteristic of the Atlantic, where surfacewaters have salinities approaching 37 PSU in the subtropics. Generally speakingthe surface waters of the Pacific, with salinities falling to less than 34 PSU nearnear the ice margins. In enclosed seas the differences are much more extreme. Forinstance the Baltic is relatively fresh, with a salinity of between 10 and 15 PSU,while Europe’s other enclosed sea, the Mediterranean, is quite salty, with salinityvalues nearer 40 PSU.

1.1.1 Systems & States

Quantities like pressure, density, and temperature are thermodynamic quantities.In using these quantities to describe the atmosphere and ocean we are attempting amacroscopic description of a system. What do we mean by this? The idea is that asystem need not necessarily be specified in terms of the location and momenta ofeach of its components, but can be usefully described in terms of fields of locallyhomogeneous fields of macroscopic quantities, i.e., things like Temperature, pres-sure, density, salinity, etc. In some general sense thermodynamics is the study ofhow relationships among these variables, or if you will, the behavior of systems, isconstrained by two laws of nature: The first and second law of thermodynamics.

p,T,VSystem

Surroundings

Figure 1.3: The Piston-Cylinder System. A canon-ical thermodynamic system.

More formally, when we speak of a system wehave in mind a spatially compact, simply connected,quantity of matter. For our purposes we usually con-sider homogeneous systems, i.e., the properties of thesystem are invariant to translations in space. In sys-tems in which gravity plays an important role suchtranslations are often restricted to directions perpen-dicular to the gravitational acceleration. In the atmo-sphere or ocean thermodynamic quantities also varyin space and time; however, in this case we can speakof locally homogeneous systems, i.e., parcels of air orwater whose properties to all extents and purposes canbe considered homogeneous. This is an idealization ofcourse, but a useful one. As illustrated in Fig. 1.3 we can distinguish between openand closed systems on the basis of whether they interact with their surroundings.Closed systems are isolated from their surroundings. Field theories of fluids aredeveloped around the idea of fluid parcels, which are necessarily open in that theyexchange energy with their neighbors. For primarily pedagogical purposes, whendeveloping thermodynamic principles it proves useful to keep in mind the classicalexpression of a closed thermodynamic system: a piston containing some volume

1.1. STRUCTURE 9

of gas, otherwise known as a piston cylinder system.Systems are characterized by their state. Any two systems with an identical

state are equivalent. Hence when we refer to the state of the system we are actu-ally referring to some minimal set of measures necessary to uniquely characterizethe system. These variables are thus called state variables, or thermodynamic co-ordinates. In the piston-cylinder system of Fig. 1.3 the pressure, p, Temperature,T, volume, V, and mass, M, are all state variables. In more complex systems ad-ditional state variables might be necessary to completely describe the system Forinstance in a two component system the mass fraction of the second component isnecessary. In the atmosphere the mass of the primary variable constituent, watervapor, is measured by the specific humidity:

qv ≡mv

m(1.2)

wheremv denotes the mass of water vapor, whilem denotes the mass of the system.An alternative, and often used analog is the water vapor mixing ratio rv ≡ mv/md

where md denotes the mass of dry air, i.e., md = m−mv. Hence rv = qv/qd. Forsea water the additional constituent is the specific salinity:

s ≡ ms

m(1.3)

where ms denotes the mass of salts. For more complex systems other quantitiesmight be necessary. In the atmosphere in the presence of condensate one oftentracks the mass of different classes of hydrometeors, for instance ql and qi denotethe specific liquid and specific ice mass, in which case qt = qv + qc + qi is thespecific mass of all (or the total) water in the system. Another example is the studyof ionized media where the specification of the system state rests additionally onthe characterization of its degree of ionization.

Both qv and s are all examples of intensive variables, in that they are indepen-dent of the amount of matter in the system. They can be contrasted with extensivevariables, such as the volume, V, or the mass,M,which depend in a precise way onthe amount of matter in the system. Specifically, any homogeneous function2 (ofdegree one) of an extensive variable is itself an extensive variable. Homogeneousfunctions of degree one are a subset of linear functions, hence extensive variablesdepend linearly on the amount of matter in the system. It follows that the ratio oftwo extensive quantities define an intensive quantity. For instance, density, is anintensive quantity. Similarly, this implies that normalizing any extensive variableby the amount of mass in the system produces an intensive variable. Intensive vari-ables defined in this way are usually prefaced by the adjective specific, hence the

2Recall that a homogeneous function of degree, n satisfies f(αx) = αnf(x)


name specific humidity for the ratio of the mass of humidity in the system to thetotal mass of the system. Throughout we endeavor to represent specific versions ofextensive variables by lower case, although the use of the symbol α to denote thespecific volume, V/m, and the use ofm to denote the total mass are notable excep-tions. Lastly, we note that while we often equate specific quantities with extensivequantities normalized by the mass of the system, in certain contexts it proves moreuseful to think of quantities normalized by the number of molecules in the system,i.e., molar specific quantities.

1.1.2 Thermodynamic Processes & Equilibrium

The concept of equilibrium is essential in our subsequent discussions. We speak ofa system being in equilibrium if its state is not changing in time. Almost always weequate equilibria with stable equilibria, by which we mean that systems perturbed asmall distance from their equilibrium will return to that state. Equilibrium systemslend themselves most readily to a thermodynamic description. If a system is inequilibrium it can be represented by a point in its state space. That is, if we thinkgeometrically of the thermodynamics coordinates of a system describing a space,then the state of a system is characterized by a point in that space.

A thermodynamic process refers to the change in the state of a system — usu-ally this change takes place in time. A reversible thermodynamic process is onein which a system changes its state without departing from equilibrium. Thus areversible thermodynamic process can be described by a line in the state-space ofthe system.

Clearly the concept of a reversible thermodynamic processes poses somethingof a paradox. To constitute a process the system’s state must be changing, yet toremain in equilibrium a system’s state must not change. This paradox is resolvedthrough the depth of time. If the state of the system is being changed sufficientlyslowly relative to the time it takes a system to adjust to a change we can consider areversible processes as a sequence of infinitesimal changes to the state of a system.Strictly speaking a reversible process defined in this manner should take infinitetime. That is why the idea of a reversible process is a concept best expressedas a limit, one in which time looses meaning. Equilibrium thermodynamics isatemporal.

1.1.3 Temperature

Temperature and pressure are basic thermodynamic variables which make readyreference to our experiences and are commonplace in our study of the atmosphere.Pressure has a ready interpretation in terms of ideas from classical mechanics,

1.1. STRUCTURE 11

namely forces and areas. Temperature, although seemingly familiar is more am-biguous. In that it it has no microscopic interpretation, like entropy it is also amore particularly thermodynamic quantity. Temperature can be defined in a purethermodynamic sense by a consideration of ideal gases, and the efficiency of heatengines. Neither of which make reference to the nature of matter underpinning thethermodynamic system. However, most of us are familiar with models of matterthat allow a physical interpretation of temperature, based on a working model ofthe matter. For instance in the context of the kinetic theory the internal energy, U

of a monatomic gas is proportional to the temperature, T,

U =32NkT. (1.4)

Here N is the number of atoms, k = 1.38× 10−23 J K−1 is Boltzmann’s constant,essentially a proportionality factor, which converts one measure of energy (Joules)to another (K). For the special case of a monotonic gas the internal energy is equalto the kinetic energy contained in the random motion of the individual atoms com-prising the gas. The correspondence between the temperature and the averagekinetic energy of the gas molecules expresses the inherently statistical nature ofthe former. Because kinetic and statistical mechanical descriptions of thermody-namic systems are relatively recent a variety of temperature scales, many of whichare unphysical as measures of energy (for instance because they permit negativevalues) continue to linger in our culture. As it turns out the Kelvin temperaturescale is consistent with less aggregated descriptions of a gas. Because incrementson the Kelvin scale are equal to those on the Celsius scale these two scales are usedinterchangeable in the study of atmospheric and oceanic phenomena. By conven-tion the triple point of water is 273.16 K on the Kelvin scale, because this is 0.01K above water freezing point (the zero point of the Celsius scale) there exists anoffset of 273.15 between the two scales.

1.1.4 Equations of State

The equilibria of thermodynamic systems are described by an equation of state.This equation reduces the dimensionality of the system by stating that equilibriaof the system exists as a surface in the space of the system’s ostensibly inde-pendent thermodynamic variables. For instance, if p, α and T are taken as thethermodynamic coordinates of an ideal gas, the existence of an equation of stateamounts to the statement that equilibria of the system can be associated with themaxima or minima of some suitably defined function of these coordinates. Thisconstraint means that in equilibrium, only two of the three coordinates are inde-pendent (thereby reducing the dimensionality of the system from three to two),


and the third can always be defined in terms of the other two—by the equation ofstate.

The best known equation of state, and one that proves quite useful in describ-ing the atmosphere, is the ideal gas law, credited to a synthesis of pre-existinglaws by the French physicist and engineer, Benoıt Paul Emile Clapeyron. It statesthat, the product of the pressure and volume of a perfect gas is proportional to itstemperature, such that

pV = NkT. (1.5)

For a multi-component gas, such as the atmosphere, it follows that for a perfectmixture (i.e., each component behaves independently of the others)

piV = NikT, (1.6)

where pi is the partial pressure of the ith component, so that p =∑pi and N =∑

Ni. A specific form of this expression can be defined by dividing by the massof the individual components, so that

piαi =Ni

niMikT, (1.7)

where in the denominator on the rhs we have written the mass mi of the ith con-stituent as the sum of the number, ni, of moles of that constituent times its molarmass, Mi. From this it follows readily that for a multi-component system the idealgas law can be written as

p = ρ∑

i

qiRiT, where Ri =NA

Mik, (1.8)

is the specific gas constant, with Na = Ni/ni the Avogadro constant. Here ρ =m/V is the density of the system, and qi is the specific mass of each component.

Excluding water vapor, within the homosphere the major constituents of theatmosphere exist in fixed proportion to one another. Hence it proves useful todefine the atmosphere as being composed of something called dry air, and watervapor, in which case its equation of state can, in so far as it behaves like an idealgas, be written as

p = ρ(qdRd + qvRv)T, (1.9)

where Rd = 287.03 J kg−1K−1, is the specific gas constant for dry air. Is just themass weighted sum of the gas constants of its constituents, and , Rv the specificgas constant for the water vapor is 461.4 J kg−1K−1,

(1.10)

1.1. STRUCTURE 13

Allowing for variable vapor concentrations implies an effective specific gasconstant which varies with the specific humidity of the atmosphere as

R = Rdqd +Rvqv. (1.11)

In the absence of condensate qd = 1− qv allowing us to rewrite (1.11)

R = Rd(1 + εqv) (1.12)

whereε ≡ Rv

Rd− 1 ≈ 0.608 (1.13)

To avoid working with a gas constant which depends on the composition of the fluidMeteorologists often work in terms of something called the virtual temperature,which absorbs the compositional dependence in R, in the absence of condensate itis given as

Tv = T (1 + εqv). (1.14)

In terms of Tv Eq. 1.8 is simply p = ρRdTv. Thus Tv can be interpreted as thetemperature required of dry air for it to have the same density as moist air at somepressure and temperature.

Equation 1.8 describes how a gas behaves in the limit as the ratio of the inter-molecular distance over the mean molecular size becomes infinite. Real gases ofcourse depart from this limit in important ways. For instance the ideal gas law isincapable of predicting phase changes, and thus is a bad approximation for watervapor near saturation. It also fails to describe the relationship among state variablesin a liquid. To describe the behavior of gases nearer saturation other equations ofstate have been proposed. Perhaps the most famous is the van der Waals Equationof state: (

p+a

α2

)(α− b) = RT. (1.15)

Other equations of state include the Clausius equation of state, and the Beattie-Bridgman equation:

p =RT

α

[(1− ε)(α+B)

α

]− A

α2. (1.16)

Note that both the Beattie-Bridgman and the van der Waals Equation of state reduceto the ideal gas law in the limit when their additional parameters vanish. For thevan der Waals equation the constant a accounts for the effect of inter-molecularforces, while the constant b accounts for non-vanishing molecular sizes. Thus forp→ 0 we expect that a = b = 0.


Yet another common equation of state is the Virial Equation which expands thepressure-volume product in a power series in p

pα =∑n=0

An(T )pn. (1.17)

Here the sum can be carried out to the extent that measurements permit. Note thatthe coefficients in the summation are functions of temperature, with the first virialcoefficient being A0 = RT. This equation should make clear the empirical basisof most equations of state. One job of statistical mechanical descriptions of matterwould is to predict these relationships based on an underlying description of thematter composing the system.

Although the more complex equations of state are necessary to predict the be-havior of vapor near saturation, we are fortunate that for most purposes in theatmospheric sciences the ideal gas law describes the behavior of the atmosphereto a good degree of approximation. The oceanic sciences are not so fortunate, theequation of state used by many ocean models is empirical and very complex. Al-though even the relatively complex expressions are themselves simplifications of astandard called the UNESCO International Equation of State (or IES 80) which isderived as a complicated curve fit to very precise measurements. An example ofone such equation is

ρ =p+ p1

p2 + 0.7028423(p+ p1)(1.18)

where

p1(T, s) = c1,0 + c1,1T + c1,2T2 + c1,3T

2 + d1,1s

p2(T, s) = c2,0 + c2,1T + c2,2T2 + d2,1s+ e2,1s

where T is the temperature in C, s is the salinity in PSU, and p is the pressurein bars (105 Pa). The coefficients appearing in the expressions for p0 and p1 areprovided in Table 1.1. However, even this simplification is far too unwieldy foranalytical work.

Although only accurate near its reference state, a yet more approximate equa-tion, designed to isolate essential nonlinearities in the more accurate expressions,is as follows:

α = α0 [1 + βt(T − T0)− βs(s− s0)− βp(p− p0)]

+ α0

[γ∗(p− p0))(T − T0) +

β∗t2

(T − T0)2]. (1.19)

Here βT denotes the thermal expansion coefficient, βs the salinity contraction co-efficient, βp the isothermal compressibility coefficient. Non-linearities are denoted

1.2. THE FIRST LAW AND ITS CONSEQUENCES 15

c1,0 5884.8170366 c2,0 1747.4508988c1,1 39.803732 c2,1 11.51588c1,2 -0.3191477 c2,2 -0.046331033c1,3 0.0004291133 d2,1 -3.85429655d1,1 2.6126277 e2,1 -0.01353985

Table 1.1: Coefficients for the Wright (1997) equation of state for seawater as describedin Eq. 1.18

by starred terms, with γ∗ being called the thermobaric parameter and β∗T the sec-ond coefficient of thermal expansion. Values of these parameters can be calculatedfrom (1.18) and is left as an exercise, there physical interpretation is discussed inthe next section.

1.2 The First Law and its Consequences

The First Law of Thermodynamics can be stated in a variety of equivalent ways:

• Heat and work are equivalent

• Energy is conserved

• A perpetual motion machine of the first kind is impossible

If we let U denote the internal energy of the system3, then the first law statesthat the difference in internal energy

∆U = Ub − Ua (1.20)

between the two equilibrium states b and a is equal to the heat Q (whose specificvalue is not to be confused with the mass fractions of the various phases of water,e.g., qv) added to the system plus the work done by the system in going fromone state to the next. If we denote by W the work done by the system on itssurroundings in moving from a to b the first law can be written as

∆U = Q−W. (1.21)

3Througout we use the Fraktur typsetting for classical thermodynamic symbols which have dualmeanings in the context of atmospheric and oceanic fluid dynamics. So for instance letting s denotethe entropy allows us to maintain the convention of s representing salinity. Likewise for u the specificinternal energy, compared with u the zonal velocity.


b

a

..

VV Va b

p

dw’dw

Figure 1.4: Relation between work and path for systems that can be represented ona (p, v) diagram.

This relation establishes the equivalence of work and heat. It is best understood inthis sense by realizing the historical context in which it was introduced: For a longtime work and heat were thought of as physically distinct concepts.

Work is defined as the product of the force F exerted on a mass in the directionof its displacement and this displacement l, i.e.,

W =∫

F · dl . (1.22)

In terms of the system in Fig. 1.3 then the force, F is simply pA the product of thepressure and the upper surface area A of the piston. In this situation, for a smalldisplacement ∆` of the piston

W = F∆l = pA∆` = p∆V. (1.23)

Consequently the specific work done W in a small expansion or compression isp∆v. This expression for work allows us to write the first law in the form:

∆U = Q− p∆V. (1.24)

Recall that for a system (such as an ideal gas) whose state can be representedby a (p, v) diagram, any reversible transformation can be represented by a line onthis diagram. This means that the work done on a system through the course ofa reversible transformation depends on the nature of the process. For instance inFig. 1.4 we identify two processes with the two paths connecting states a and b.Denoting the work performed by going along the dashed path by W′ =

∫ ba pdV′

and the work along the solid path by W =∫ ba pdV we note that W′ > W. In a


cyclic process a → b → a work (W < 0) is done on the system for a counter-clockwise loop, while for clockwise loops work is done by the system, in whichcase W > 0.

1.2.1 The Calculus

Exact Differentials

The previous discussion highlights an important concept. Work is not a state vari-able. If work were a state variable we could speak of Wa and Wb as the work ofstate a and the work of state b. Moreover we could think of a differential amountof work dW implied by an infinitesimal change in the state as a unique quantity.However, because the differential amount of work depends on the processes (orpath) doing so would be incorrect, which is another way of saying that work isnot a state variable. Heat is also not a state variable. While we can speak of thetemperature of a system, the pressure of a system, or its specific volume, we cannot speak of the heat of a system. Instead we speak about the work done by (or on)the system, or the heating of a system during a change in state, but never the heator the work of the system.

To discriminate between differentials of state variables and vanishingly smallquantities of work and heat the concept of an exact (or total) differential is oftenintroduced. An exact differential has the property that its integral over a closedpath vanishes. That is dχ is an exact (or total) differential if:∮

dχ = 0. (1.25)

Thus differentials of state variables are exact differentials. That is they do notdepend on the path of integration, consequently for systems in equilibrium thevalue of a state variable depends only on the systems state. Sometimes a differentialchange in the system’s state can result in a certain amount of heat absorbed, orwork done by the system. In such circumstances there is the tendency to think ofthis heat, or work, as a differential amount of heat (dq), or work (dw), which isdone, in accord with the differential change in the systems state. This tendencyoften leads to the mistaken identification of dq and dw with exact differentials.This is incorrect and leads to confusion. To avoid this confusion it is important toremember that while small amounts of heat, q, may be added, or work, w, done inassociation with a differential change in the state of the system, it is non-ensicalto speak of the differential heating of the system, or the differential working of thesystem.

Although Q and W are not state variables, the internal energy U (or the specificinternal energy u) is. That is we can speak of the internal energy of the system, and


in a cyclic process ∆U =∮

dU = 0. Thus in a cyclic process the first law impliesthat Q = W. The heat flowing into the system is equal to the work done by thesystem. This encourages an alternative formulation of the first law, which is worthbearing in mind when the second law is introduced, namely:

A transformation whose only final result is to transform heat intowork can not expend more energy in work than it absorbs in heat.

Partial Derivatives

Again, let us restrict ourselves to systems that can be represented on the plane(i.e., by two state variables). In this case we note that all such systems empiricallysatisfy an equation of state that can be written in the form of F(xi) = 0 whereF depends on the form of the system, but is analytic (which is another way ofsaying it allows a power series representation) and where xi = x1, x2, . . . , xNare N state variables. As stated previously this implies that only N − 1 of the statevariables are independent.

In general for thermodynamic systems desried by three variables (x, y, z) con-nected by an equation of state we can write

x = f(y, z), y = g(x, z). (1.26)

This implies that

dx =∂f

∂ydy +

∂f

∂zdz, and dy =

∂g

∂xdx+

∂g

∂zdz. (1.27)

Substituting for dy from the latter, into the former,

dx =∂f

∂y

(∂g

∂zdz +

∂g

∂xdx)

+∂f

∂zdz, (1.28)

or (1− ∂f

∂y

∂g

∂x

)dx =

(∂f

∂y

∂g

∂z+∂f

∂z

)dz. (1.29)

But since dz and dx are independent the coefficients multiplying them on theleft and right hand sides of (1.29) must vanish which implies the following rela-tionships among the partial derivatives:

∂f

∂y=(∂g

∂x

)−1

and∂f

∂y

∂g

∂z= −∂f

∂z. (1.30)


the latter equality can be written as

∂f

∂y

∂g

∂z

(∂f

∂z

)= −1 −→

(∂x

∂y

)z

(∂y

∂z

)x

(∂z

∂x

)y

= −1. (1.31)

where the latter follows from the definitions of f, g and f3 above. The point beingthat knowledge of the equation of state (say f ) imposes broad constraints on thepartial derivatives of the system, e.g., g. In the case when an equation of state isnot available, great insight into the behavior of the system can be obtained simplyby knowledge of its partial derivatives, which in most cases can be determinedempirically.

If we always chose to describe our system with the same thermodynamics co-ordinates, then there would be no ambiguity in the partial derivatives. However,because ones choice of coordinates are often suited to a particular purpose, themeaning of the partial derivatives can be ambiguous. For instance, if the specificinternal energy u is being used to characterize the system, then ∂u/∂p is ambigu-ous depending on whether u is defined as a function of p and α or as a function ofp and T. To avoid confusion, the following notation is common

∂u(p, α)∂p

≡(∂u

∂p

)α

. (1.32)

The subscripts remind the reader of the thermodynamic coordinates being used.Often this notation is generalized to any measure of change, i.e., (dT )p denotesthe isobaric change in temperature. In anticipation of the entropy a subscript s isused to denote isentropic processes, i.e., (dT )s refers to an isentropic differentialchange in temperature.

Two partial derivatives which are readily measured and often used to deter-mine the behavior of a system are the coefficient of thermal expansion βT and thecoefficient of isothermal compression, βp defined as follows:

βT ≡ α−1

(∂α

∂T

)p

(1.33)

−βp = −α−1

(∂α

∂p

)T

. (1.34)

For saline solutions the salinity s should also be held constant in the partial deriva-tives above. Likewise such solutions permit the definition of another material prop-erty, the salinity contraction coefficient, βs,

βs = −α−1

(∂α

∂s

)p,T

= (1.35)


For systems, such as ideal gases, with simple closed form equations of state,specifying βT and βp is trivial. However, in more complicated systems it is oftenuseful to be able to characterize the system in terms of empirically determinedcoefficients, as for instance was the case for seawater (cf., Eq. 1.19).

Another important property of a substance is its heat capacity, or specific heatcapacity. Consider adding a measurable amount of heat Q to our system of Fig. 1.3.If, after equilibration, we find that its temperature has increased by an amount ∆Twe can linearly relate the heat added to the temperature change,

C =Q

∆T. (1.36)

However because Q is not a state function, it can not be written as a function ofother state variables, which means that

limQ→0

Q

∆T(1.37)

depends on the path. Because the heat added depends on the path, the value of thislimit depends on the process. That is there are an infinite number of heat capac-ities, each corresponding to a different process. Because the limit is only definedfor specified processes, it is only meaningful to speak of the heat capacity withrespect to one or the other process. The heat capacities corresponding to two com-mon processes are thus commonly introduced, Cp and Cv being the isobaric andisometric (constant volume) heat capacity respectively; their specific counterpartsare denoted cp and cv. Given that we denote the specific volume by α we should inprincipal denote cv as cα however the use of cv is by now so universally adoptedthat we tolerate this notational inconsistency in our subsequent discussion. As weshall see in the next section, given cv and (βp, βT ), cp can be calculated directly(as can specific heats associated with other processes for that matter).

Just as one task of deeper theories of thermodynamics, i.e., the kinetic theoryor statistical mechanics, is to predict the equation of state of a system, (and hencethe βs) given a model of the microscopic structure of the system, one would likethese theories to also predict the values of the specific heats. The kinetic theorywas moderately successful at this. By assuming that energy is partitioned equallyamong all the degrees of freedom of a molecule it follows from (1.4) that for anideal gas

cv =d

2R, (1.38)

where d is the degrees of freedom. For a monatomic gas d = 3 corresponds tothe three components of velocity, while for a diatomic gas d = 7 accounts for two


additional rotational and two vibrational modes of motion.4 This would predictthat cv = 7

2R. By experience we know that for diatomic gases cv ≈ 52R is a

better model. This crisis can only be resolved by quantum mechanics which showsthat the vibrational modes are not accessible until diatomic gases (such as O2 andN2, of which the atmosphere is mostly composed) reach much higher temperaturesthan those found in the lower atmosphere. This also explains why at very lowtemperatures (where the quantized rotational modes are not accessible) cv ≈ 3

2R.

1.2.2 Some Consequences of the First Law

Specific Heats

Here we show how the first law can be used to interpret the specific heats dis-cussed in the previous section, and to derive relationships among specific heatscorresponding to different processes. In particular the first law allows a ready in-terpretation of cv in terms of the specific internal energy, q. Choosing α and T asthermodynamic coordinates and noting that u is a function of state, i.e., u(α, T ),hence

du =(∂u

∂α

)T

dα+(∂u

∂T

)α

dT. (1.39)

Using this form of du in the expression for the first law yields

q =(∂u

∂T

)α

dT +(p+

∂u

∂α

)T

dα, (1.40)

where here we denote the amount of heat added through these differential changesin the working fluids state by q. From (1.40) we see that for any isometric processthe second term vanishes so that the isometric specific heat capacity is readilyinterpreted in terms of the internal energy:

cv ≡(∂u

∂T

)α

. (1.41)

which follows naturally from (1.4). This interpretation in terms of the microscopicstate should make clear that the effective value of cv for a composite system shouldbe given as a mass weighted sum of the individual components, i.e., cv = qdcv,d +qvcv,v where cv,d is the isometric specific heat of dry air, and cv,v is its counter partfor water vapor. Eq. (1.41) also allows us to write the first law as

q = cv dT +(p+

∂u

∂α

)T

dα. (1.42)

4In principle there should be eight degrees of freedom, but for the idealization of point massesrotation about the common axis of a diatomic molecule is meaningless


Equation (1.42) can be used to relate cp to cv in terms of other properties ofthe system. For instance, given an isobaric process q = cp(dT )|p (by definition),substituting for q above yields

cp( dT )p = cv( dT )p +[p+

(∂u

∂α

)T

]( dα)p (1.43)

which upon rearrangement and use of the definition of βT yields

cp = cv +[p+

(∂u

∂α

)T

]βTα =⇒

(∂u

∂α

)T

=cp − cvβTα

− p (1.44)

which yields an expression of the first law in terms of well known material proper-ties

q = cv dT +(cp − cvβTα

)dα. (1.45)

For an ideal gas βTα = R/p so that

cp = cv +R (1.46)

and the expression for the first law takes a somewhat more familiar form:

q = cp dT − α dp. (1.47)

Enthalpy

An additional state function, which has some correspondence with the internalenergy, is the enthalpy h :

h = u + pα =⇒ dh = du + d(pα). (1.48)

The first law can be expressed in terms of h as follows:

q = du + pdα = du + d(pα)− α dp =⇒ dh = q + α dp. (1.49)

This implies that the change of enthalpy is equal to the heat added to the system inan isobaric process, hence h is sometimes called the heat function. Its introductionobviates the need to use the word “heat” as a noun and can be though of in thatsense, i.e., as the heat function. Furthermore, because q = cp( dT )p

cp =(∂h

∂T

)p

, (1.50)


which further illustrates that the enthalpy is the isobaric equivalent to the isometricinternal energy. For moist air in the absence of condensate (md +mv)h = mdhd +mvhv. From (1.50) we can calculate the effective heat capacity of the moist systemas

cp = qd

(∂hd

∂T

)p

+ qv

(∂hv

∂T

)p

(1.51)

= qdcp,d + qvcp,v (1.52)

where cp,v is the isobaric specific heat of water vapor and cp,d is the correspondingspecific heat for dry air. This expression for cp is in correspondence with ourexpectation based on our experiences with R and cv. Further explorations of h andcp will be conducted after the introduction of entropy below.

Adiabatic Processes

A matter of considerable interest in the atmosphere is how state variables changein processes that do not involve an exchange of heat with the surroundings. Theseare called adiabatic processes. Here we consider the behavior of an adiabatic trans-formation of a single component ideal gas, i.e., dry air. Starting from the adiabaticform of the first law for an ideal gas, and substituting for α from the equation ofstate yields the following expression for the first law:

cp dT − RT

pdp = 0 (1.53)

=⇒ d(Tp−R/cp) = 0, or Tp−R/cp = constant. (1.54)

Note, that these define curves on a (p, T ) diagram, which can be compared toisotherms and isobars on a similar diagram. These curves are called adiabats, or inanticipation of entropy, isentropes.

These relations help define what is called potential temperature and denoted byθ. Instead of describing the state of a parcel by its temperature, which varies in apredictable way with pressure, the potential temperature allows us to characterizethe thermal state of a parcel in a way that does not depend on its current pressure.Physically this is because θ is defined to be that temperature a parcel would haveif adiabatically brought to some reference pressure, p0. Clearly adiabatic displace-ments of a parcel will change its temperature, but not this potential (or reference)temperature. An expression for θ can be derived from (1.54) by noting that if a par-cel at some initial temperature T is brought adiabatically from its current pressurep to a reference pressure p0 its new temperature is given as

θ = T

(p0

p

)R/cp

. (1.55)


It is straightforward to show (and we do so subsequently) that (1.55) can be readilygeneralized to a multi-component fluid, simply by replacing R and cp by theireffective values. Its generalization to a fluid which permits adiabatic changes inphase of one of its components will also be dealt with later.

Similarly, and of practical use in oceanography is the concept of a potentialdensity, ρθ. Like θ, ρθ is the density at a reference state. For instance, if forseawater we write

ρ = ρ(s, T, p)

thenρθ ≡ ρ(s, θ; p0).

For dry air this implies that ρθ = p0/(Rdθ), which because p0 and Rd are fixedis simply proportional to the inverse of θ. In the oceans ρθ also accounts for theeffect of salinity on density and thus is a better measure of the static stability (whichdepends on density differences) of a water column than simply the density. Becausethe density varies so little in the ocean often a perturbation density, defined as

σθ = ρθ − 1000 (1.56)

is used. Unlike in the atmosphere, where 1000 hPa is almost unanimously adoptedfor the the reference pressure, p0, different reference pressures may be used in thedefinition of ρθ and hence σθ. To indicate this the symbol σn is used in lieu ofσθ where n denotes the reference pressure in hbars. For instance, σ2 denotes thedeviation potential density defined with respect to a reference pressure of 200 bars(roughly corresponding to a depth of 2km).

For the same reasons that the concept of potential density proves useful fordescribing the ocean one might think that it would be similarly adopted by me-teorologists when describing moist, but unsaturated flows, for which the specifichumidity affects the density. However, to capture this effect atmospheric scientiststraditionally work in terms of the virtual potential temperature

θv ≡ θR

Rd= θ(1 + εqv) (1.57)

where ε is related to the ratio of the gas constants for dry air and water vapor andwas defined in Eq. 1.13. Clearly θv is an analog to Tv defined in Eq. 1.14. Because

ρθ =p0

Rθ=

p0

Rdθv(1.58)

ρθ is proportional to the reciprocal of θv. Thus working in terms of θv captures theadvantages of ρθ. Physically it can be thought of as the temperature dry air wouldneed to have to have the same density as moist air at a reference pressure p0.

1.3. THE SECOND LAW AND ITS CONSEQUENCES 25

1.3 The Second Law and its Consequences

T1

T2

T2

T1

p

v

C

B

A

D

insulatorinsulator

A−B B−C C−D D−A

Figure 1.5: A piston taken through a Carnot Cycle. The cycle is shown on the(p, α) plane on the left.

1.3.1 Carnot Cycles

Consider a fluid whose state can be represented on a (p, α) diagram. Then a cyclethat is composed of two isothermal and two adiabatic legs is called a Carnot Cycle.Fig. 1.5 illustrates the basic Carnot Cycle in terms of a diagram on the (p, α) planeand in terms of a piston and some temperature reservoirs. The cycle has four steps:

1. In going from A to B the system does work and absorbs heat a fixed temper-ature T2

2. In going from B to C the system does work and cools adiabatically to atemperature T1 < T2. By the first law this work must come at the expense ofthe systems internal energy.

3. Along segment C toD work is done on the system and heat is removed fromit at fixed a temperature T1.

4. The final segment formD toA completes the cycle. Here work is again doneon the system and this work goes into the internal energy of the system as itwarms from T1 back to T2.

The Cycle we have described is called a Carnot Cycle, or Heat, Engine. In thedirection we have described it work is done by the system:

W = Q1 + Q2. (1.59)


Here Q2 and Q1 denote the amount of heat added and expelled from the systemrespectively. In order for the work to have the right sign Q1 < 0 and Q2 > 0.

The efficiency of the heat engine, χ is defined as the fraction of heat added tothe system which is used to do work, i.e., χ = W/Q2. This implies that

χ =Q1 + Q2

Q2(1.60)

Nothing we have said prevents us from considering the cycle in reverse. In thiscase work is done on the system. A quantity of heat |Q1| is added to the system ata temperature T1 and a greater quantity of heat |Q2| is extracted from the system ata temperature T2. A cycle operating in this fashion is called a Carnot Refrigerator.

1.3.2 The Second Law

Kelvin’s postulate of the second law is:

A transformation whose only final result is to transform into work,heat extracted from a source which is at the same temperature through-out, is impossible.

While Clausius postulates that:

A transformation whose only final result is to transfer heat froma body at a given temperature to a body at a higher temperature isimpossible.

Q2Q2

Q2 Q1

T1

T2

W

Figure 1.6: Let the dashed lines enclose the system.

It turns out that these statements are equivalent expressions of what has cometo be called the 2nd law of thermodynamics. This can be shown by noting thatthe negation of one implies the negation of the other. Suppose Clausius’ statementwere false, then we could transfer Q2 units of thermal energy (heat) from a reser-voir at temperature T1 to a reservoir at some higher temperature T2. This heat could


be used to do work via a Carnot Cycle. Since the warm reservoir (at T2) would re-main unchanged this would lead to a situation in violation of Kelvin’s statement.This is illustrated schematically in Fig. 1.6 which shows that assuming Clausius’postulate is false allows one to couple two cycles whose only function it to extractheat from a reservoir at temperature T1 to do work W. Kelvin’s statement of thesecond law can also be read: “a perpetual motion machine of the second kind isimpossible.”

Efficiencies of Heat Engines:

• No heat engine operating in cycles between two reservoirs at constant tem-perature can have a greater efficiency than a reversible engine operating be-tween the same two reservoirs.

• All reversible engines operating between two reservoirs at constant temper-atures have the same efficiency.

These follow because in both cases if the less efficient engine is reversible itcould be driven by the more efficient engine operating as a refrigerator. This wouldresults in a situation that violates Clausius’s statement of the second law.

Absolute Temperature Scales:

The efficiency of heat engines can be used to define a new temperature scale. Thistemperature scale has the property that:

f(T1)f(T2)

=Q1

Q2. (1.61)

Kelvin suggested that a temperature scale be defined such that f(T ) = T, so thatT1/T2 = Q1/Q2. This defines an absolute temperature scale which is independentof the working fluid. Sometimes it is called an absolute thermodynamic tempera-ture scale. It turns out that such a scale is compatible with the temperature scaledefined by the ideal gas thermometers discussed in Chapter 1.

1.3.3 The Clausius Inequality

Consider a cycle composed of a sequences of N processes that exchange heat witha reservoir. Using i to index the processes we denote by Qi the heat exchanged withthe ith reservoir at temperature Ti. Note that because the work done by a system isdefined to be positive the first law dictates that the heat added to a system must alsobe positive. Given this arbitrary cycle we can superimpose n Carnot cycles on the


Q0,1

T0

WQ0,3

T3

Q3’3

Q0,2

T1

Q1’TQ2’

2

W 2W1 WQ0,M

TM

QM’M W

Q0,M+1

QM+1’M+1

TM+1

Figure 1.7:

system to form a complex system in which each subsystem consisting of a Carnotcycle exchanges heat between a reservoir at temperature T0 and Ti as sketched inFig. 1.7. Hereafter, once cycle of the complex system consists of one cycle each ofthe N Carnot cycles, and one cycle of the original system.

From the property of the absolute thermodynamic temperature, for each of theN superimposed Carnot cycles we can write

Q′0,i =

T0

TiQ′

i. (1.62)

Denoting the net amount of heat surrendered by the reference reservoir by Q0 suchthat:

Q0 =N∑

i=1

Q′0,i = T0

N∑i=1

Q′i

Ti(1.63)

we find that because the net heat exchanged in the cyclic process must equal thework done, the second law requires that

W =N∑

i=1

(Qi + Q′

i + Q′0,i

)≤ 0. (1.64)

However, because Qi + Q′i = 0 by construction, substitution for Q′

0,i from (1.62)shows us that this in turn implies that

N∑i=1

Qi

Ti≤ 0. (1.65)

This is Clausius’ inequality, it expresses the requirement that Q0 ≤ 0. If this werenot the case then the only final result of our system would be to extract heat (Q0)from a reservoir at some constant temperature T0 to do work.

If our cycle were to operate in the reverse direction (which is only possi-ble for reversible cycles) all the Qi would change sign, which would imply that


T0

C1 C3

CM

C2

CM+1

p

v

Figure 1.8: Decomposition of an arbitrary cycle into a sequence of Carnot cycles.

∑Ni=1(Qi/Ti) ≥ 0. The only way this result can be reconciled with our reasoning

is if the equality sign holds for reversible cycles.This argument is made schematically in Fig. 1.8 now for an arbitrary process

spanned by N M Carnot cycles. The decomposition becomes exact in thelimit as the component cycles become arbitrarily small, so that the net heat addedthrough the course of the original process is exactly canceled by that taken up inthe N Carnot cycles.

1.3.4 Entropy

Clausius’ inequality, which holds for a cyclic process, essentially states that thereis some quantity, essentially Qi/Ti, that when summed over a reversible and cyclicprocess, is conserved, hence this quantity describes a property of the system. Wecall it the entropy, S, whose specific value we denote by s. Traditionally the spe-cific entropy is denoted by s but we use s to avoid confusion with the specificsalinity. Like other state variables the entropy can be represented as a perfect dif-ferential, ∮

ds = 0. (1.66)

i.e., it makes sense to ask how much the entropy of a system changes for an infin-tesimal change of state, irrespective of the path the system takes in arriving at itsnew state. In other words, given some reference entropy s0, the entropy,

s(A) = s0 +∫ A

Ods, (1.67)


where O denotes the reference state. It follows that

s(B)− s(A) =∫ B

Ad( q

T

)=∫ B

A

du

T+∫ B

A

p

Tdα (1.68)

The above describes how the entropy changes for reversible transformations.How about irreversible transforms? For these it is straightforward to show thatthe ratio of the heat exchanged to the temperature of the resevoir at which it isexchanged, must be bounded by the change in entropy:

s(B)− s(A) ≥iB∑

i=iA

qi

Ti. (1.69)

This result follows directly from (1.66) whereby we note that any irreversible trans-formation can have a reversible transformation superimposed on it so as to bringthe system back to the original state, thereby defining a cycle which is subject tothe constraint of Clausius’ inequality:

0 ≥∑

i

qi

Ti

∣∣∣∣∣cyclic

=iB∑

i=iA

qi

Ti

∣∣∣∣∣∣irrev

+∫ A

Bds (1.70)

=iB∑

i=iA

qi

Ti

∣∣∣∣∣∣irrev

+ s(A)− s(B), (1.71)

which is equivalent to (1.69). Here we have represented a general process as asummation, only writing reversible process (for which a path is clear) as integrals.

For a completely isolated system qi = 0 and hence s(B) ≥ s(A) where weinterpret B as succeeding A. Thus for isolated systems the only states accessibleare those for which the entropy is non-decreasing. From this argument if followsthat the state of maximum entropy is an equilibrium state of an isolated system.

The introduction of the specific entropy s allows us to reformulate the first lawas follows:

Tds ≥ q = du + w (1.72)

where the equality holds for reverisible processes. This inequality is more readilyinterpreted as a bound on the work a system can do, namely

w ≤ Tds− du, (1.73)

the emergence of entropy, as a property of a system that bounds the work it can do,makes its interpretation (given Kelvin’s or Clausisus’ statement of the second law)almost intuitive.


1.3.5 Some Consequences of the Second Law

Potential Temperature

It proves useful to reconsider our concept of potential temperature from the per-spective of entropy. For a moist system in the absence of condensate (i.e., water inthe vapor phase only) the properties of ideal gasses allows us to write the first andsecond laws as

mTds ≥ mdcv,ddT +mvcv,vdT + pdV. (1.74)

For an ideal mixture the pressure is the sum of the partial pressures i.e., p = pd + ewhere we follow the atmospheric convention and express the vapor pressure, pv bye. Hence we can express the work as

pdV = d(pdV + eV )− V dp = (mdRd +mvRv)dT − V dp, (1.75)

which allows us to write (1.74) as

ds ≥ cp d lnT − pα

Td ln p = cp d lnT −R d ln p. (1.76)

This expression is identical to the first law for a dry, single component system,except that now R = qdRd + qvRv and cp = qdcp,d + qvcp,v depend on the com-position of the working fluid.

Integrating ds along an isentrope connecting the given state to a referencepressure and temperature (p0, T0) with the same entropy, s, constrains the relativechanges in pressure and temperature:

0 = cp lnT

T0−R ln

p

p0, (1.77)

which follows from (1.76) because, by definition, for an isentropic process ds = 0.Hence

T0(s, p0) = T

(p0

p

)R/cp

. (1.78)

T0 is the temperature a system in a given state would have if it were brought isen-tropically to some reference pressure. By standardizing the reference pressure to1000 hPa we can define a function, θ(s) that is only a function of the given state.This function is just the potential temperature. For the moist fluid:

θ(s) ≡ T0(s; p0) = T

(p0

p

)R/cp

; (1.79)

it represents the temperature the system would obtain if transformed isentropicallyto the specified state. For a given value of qv it defines a line in the state-spaceof the system. For a dry atmosphere R = Rd and c = cp,d in which case (1.79)reduces to the familiar form for the potential temperature given by (1.55).


Free Energies

In analogy to a purely mechanical system, wherein the external work which can beperformed during a transformation is bounded by the energy in the system

W ≤ −∆U (1.80)

the concept of free energies is often introduced to bound the amount of work thatcan be done during isothermal transformations in thermodynamic systems. Freeenergies, sometimes called thermodynamic potentials, are particular to thermody-namic systems wherein not all of the energy is available to do work.

In an isothermal transformation the amount of heat added is bounded by theentropy difference between the final and starting states. From (1.73), the amountof work a system can do for an isothermal tranformation between some stateA andB is

w ≤ − (u(B)− u(A)) + T [s(B)− s(A)] . (1.81)

From this we note that the function f = u−T s yields the desired analogy to (1.80),i.e., for isothermal transformations

w ≤ −∆f. (1.82)

That is f the free energy, or more precisely the Helmholtz free energy, bounds theamount of work the system can do in an isothermal transformation.

The Helmholtz free energy is also called the isometric or isochoric thermody-namic potential. This is because in an isometric (or isochoric) transformation

w ≡ 0 =⇒ 0 ≤ −∆f. (1.83)

In other words f(B) ≤ f(A). Thus stable configurations of a dynamically isolatedsystem in contact with a heat reservoir at some fixed temperature must be in a stateof stable equilibrium when f is a minimum. That is an isolated system can not ac-cess any other state since doing so would imply f(B) > f(A). Thus the concept ofa thermodynamic potential is developed in analogy to a system’s potential energywhich is a minimum for mechanical systems in stable equilibrium.

Analogously we can introduce the concept of a thermodynamic potential forisobaric (rather than isochoric) isothermal transformations. In such processes

w = p[α(B)− α(A)] 6= 0 =⇒ p∆α ≤ −∆f, (1.84)

or0 ≤ −∆f− p∆α. (1.85)


Definingg = f + pα = u− T s + pα = h− T s (1.86)

implies that for isothermal, isobaric, processes

g(B) ≤ g(A). (1.87)

g is called the specific Gibbs function, or isobaric thermodynamic potential. For asystem in stable equilibrium (at constant pressure in contact with a heat reservoirat temperature T ) it must, in analogy to f, be a minimum.

Thermodynamic potentials have many uses, particularly in the study of chemi-cal reactions and phase changes.

Relations among partial derivatives

Here we again explore some implications of the entropy for systems whose equilib-ria can be displayed in the (p, α) plane. Choosing α and T as our thermodynamiccoordinates

du =(∂u

∂T

)α

dT +(∂u

∂α

)T

dα (1.88)

the expression of the first law in (1.72) allows us to write

ds =1T

(∂u

∂T

)α

dT +1T

[(∂u

∂α

)T

+ p

]dα. (1.89)

However because s is also a function of state

ds =(∂s

∂T

)α

dT +(∂s

∂α

)T

dα, (1.90)

from which it follows that(∂s

∂T

)α

=1T

(∂u

∂T

)α

(1.91)(∂s

∂α

)T

=1T

[(∂u

∂α

)T

+ p

]. (1.92)

Taking the partial of the former with respect to α and the partial of the latter withrespect to T and equating implies that(

∂u

∂α

)T

= T

(∂p

∂T

)α

− p = TβT

βp− p. (1.93)

Note that for an ideal gas T (∂p/∂T )α = p in which case the RHS vanishes.Consequently the internal energy for an ideal gas is a function of T only. For anon-ideal gas (1.93) relates (∂u/∂α)|T to the measured material properties of thesystem, i.e., βT and βp.


Maxwell Relations

Given the definition of the free energies the first law can be written in a variety offorms, for instance:

Tds = du + pdα (1.94)

Tds = dh− αdp (1.95)

−sdT = df + pdα (1.96)

−sdT = dg− αdp (1.97)

If we consider the first of these, and specify (s, α) to be the thermodynamiccoordinates, the first law can be written as:

Tds =(∂u

∂s

)α

ds +[(

∂u

∂s

)α

+ p

]dα. (1.98)

Because s and α are independent this implies that(∂u

∂s

)α

= T and(∂u

∂α

)s

= −p. (1.99)

Because∂2u

∂α∂s=

∂2u

∂α∂s(1.100)

cross differentiation of the partial derivatives in (1.99) implies that(∂T

∂α

)s

= −(∂p

∂s

)α

. (1.101)

In addition to providing an interesting definition of temperature and pressure thisexercise provides useful relations among the partial derivatives of various statefunctions. These relations are known as Maxwells’ relations. Three more suchrelations can be derived by following a similar procedure for the other forms of thefirst law: (

∂T

∂p

)s

=(∂α

∂s

)p

(1.102)(∂s

∂α

)T

=(∂p

∂T

)α

(1.103)(∂s

∂p

)T

= −(∂α

∂T

)p

(1.104)

1.4. PHASE CHANGES 35

T1 T2

p

v

Ta

Td

Tc

Tb

p

v

Figure 1.9: Left Panel: cartoon illustrating ideal gas like behavior in a systemwhere T2 > T1. Right Panel: behavior of a real, or van der Waals gas in thevicinity of a phase change.

1.4 Phase Changes

Imagine a piston cylinder containing a single constituent gas in thermal equilib-rium with a heat reservoir, similar to Fig. 1.3. Now imagine measuring the specificvolume of the cylinder at different values of the specified pressure. At high tem-peratures we would measure isotherms that look similar to those shown in Fig. 1.9.The extent to which we get straight lines in the p, α−1 plane reflecting the degreeto which our gas behaves ideally.

As we repeat this experiment at increasingly low temperatures we notice de-partures from an ideal gas, and eventually something strange happens. The depar-tures from an ideal gas become more pronounced, and then even discontinuous.That is the smallest change in p leads to a discontinuous change in α. This is illus-trated in Fig. 1.9 where we show a sequence of isotherms at decreasing temperatureTd < Tc < Tb < Ta.

If we focus for the moment on the Td isotherm we notice that at small pres-sures (starting to the right in the right panel of Fig. 1.9) the volume decreases withincreasing pressure about like we would expect. But at some point the cylindercontracts more markedly. Thereafter further changes in pressure have a remark-ably small change on the volume. Our working fluid behaves quite differentlyfrom an ideal gas. If we looked inside the cylinder we might further notice differ-ences in the appearance of the fluid. This change in the working fluid is called aphase change. It comes about because at some critical value of the state parame-ters the substance spontaneously reorganizes itself into what is called a new state.Typically we think of three states of matter — but there are actually very manystates, or patterns of organization, and not all substances behave similarly. H2Ois known to have at least VII different solid states while He exists in two liquidsates. Because different states correspond to different forms of organization of the


p

TTc

Triple Point

Critical Point

liquid

pc

vapor

solid

gas

Figure 1.10: Cartoon depicting lines of equilibria between phases.

molecular structure of the substance, only one gas state can exist if we associatethe concept of a gas with disorder.

For substances which have an equation of state of the form f(p, α, T ) = 0 thestate of the system can be represented by a surface in (p, α, T ) space. Fig. 1.10shows the projection of this surface on the (p, T ) plane for a substance like waterthat expands on freezing. Here we find ideal gas like behavior at large temper-atures, and low pressures. At higher pressures or lower temperatures molecularforces and the shapes and sizes of the molecules become important and the sub-stance aggregates into different states. The fact that states other than the gaseousstate depend on the molecular properties of the the working matter makes a generaltreatment of liquids and solids very difficult.

Some interesting aspects of Fig. 1.10 include:

1. A vapor is a gas that is capable of condensing isothermally.

2. The critical point demarcates the isotherm bounding gases from vapors. Attemperatures above the triple point no separation into two phases of differentdensities occurs in isothermal compression from large volumes.

3. The transition from a liquid to a solid corresponds to an increase in α.


1.4.1 Clausius Clapeyron

Note that the boundaries between phases on Fig. 1.10 appear as lines. This is aconsequence of Gibbs’ Phase Rule, whereby equilibria between two phases im-poses a constraint on the system that eliminates a degree of freedom. Becauseequilibrium between three phases puts two constraints on a system the triple pointis independent of both temperature and pressure.

Because such profound changes occur across phase boundaries it is interestingto derive equations for them. The Clausius-Clapeyron Equation is just such anequation. Most commonly it is applied to the line separating the liquid from thevapor phase, and this is the situation we shall consider here.

It can be derived by a consideration of the state of the system in the (p, α) planeas shown in Fig. 1.9. Again we focus on the Td isotherm in under the dashed curve.Here α is multi-valued reflecting the fact that our system consists of liquid andvapor in equilibrium. Liquid has values of α on the left side of the line and vaporhas values of α on the right side of the line. Values of α in between correspond toour system being comprised of different relative proportions of liquid and vapor.

For a system in this multi-phase state the total mass of the system is the sumof the masses in each phase, m = ml + mv. Consequently the internal energy ofthe system is the sum of the internal energies of the components, similarly for thevolume:

U = mlul(Td) +mvuv(Td) (1.105)

V = mlαl(Td) +mvαv(Td). (1.106)

Consider the evaporation of a small amount of condensate, so that the vapor massincreases, m′

v = mv + δm and the condensate mass decreases, m′l = ml − δm.

The associated change in U and V are

δU = [uv(Td)− ul(Td)] δm (1.107)

δV = [αv(Td)− αl(Td)] δm (1.108)

which by the first law implies the heating,

Q = δm (uv(Td)− ul(Td) + e[αv(Td)− αl(Td)]) (1.109)

=⇒ Q

δm= uv(Td)− ul(Td) + e [αv(Td)− αl(Td)] . (1.110)

where e denotes the vapor pressure i.e., pv.The specific heating, Q/δm is called the specific enthalphy of vaporization,5

and denoted by L or Lv. Its name comes from the fact that it measures the differ-ence between the vapor and liquid enthalpy at saturation, as is apparent by casting

5In the meteorological literature it is often called the latent heat.


(1.110) in terms of the enthalpies:

Q

δm= Lv ≡ hv(Td)− hl(Td) (1.111)

In the classical thermodynamics, Lv is an empirical property of the system thatmay depend on temperature.

The ratio of the change in u to the change in volume for such a transformation,defines, in the limit of small perturbations, the partial derivative of the specificenergy with respect to the specific volume at the constant temperature, T = Td.From the above, this quantity is readily expressed in terms of Lv :

∂u

∂α

∣∣∣∣T=Td

= limδ−→0

δU

δV=[

Lv

αv(Td)− αl(Td)− e

]. (1.112)

With the aid of (1.93), this constraint can be expressed as one on changes in satu-ration pressure and temperature:

∂es∂T

∣∣∣∣α

=Lv

T (αv − αl). (1.113)

This is the Clausius-Clapeyron Equation. It describes how the saturation vaporpressure changes with temperature through the course of a phase change. Usuallythis partial derivative is written total derivative because es is the saturation pres-sure, which does not depend on α. It tells us how the saturation pressure changeswith temperature, i.e, it describes the spacings of the isobaric isotherms under thedashed curve in Fig. 1.9. For the common place case whereby αv αl we have:

desdT

' Lv

Tαv=

Lv

T (RvT/es)(1.114)

which, for Lv constant is readily integrable:

es(T ) = es(T0)e−Lv

Rv

“1T− 1

T0

”. (1.115)

A more accurate expression, which accounts for the manner in which the enthalpyof vaporization Lv depends on temperature, is

es(T ) =8∑

i=0

ciTi, (1.116)

with the constants tabulated in Table 1.2


c0 0.611213476e+03 c4 0.196237241e-05c1 0.444007856e+02 c5 0.892344772e-08c2 0.143064234e+01 c6 -0.373208410e-10c3 0.264461437e-01 c7 0.209339997e-13c4 0.305930558e-03

Table 1.2: Coefficients for the Flatau et al., expression for saturation vapor pressure

1.4.2 Condensate

Phase changes in the atmosphere lead to the suspension (and sometimes precip-itation) of liquid or solid water whose specific masses we denote ql and qi re-spectively. Hence qt = qv + ql + qi recalling that qt is the specific mass of thetotal water, and qv is the specific humidity. If we continue to restrict ourselvesto an equilibrium description, where the partitioning of the water mass among itsphases is determined by the Clausius-Clapeyron equation, and neglect processessuch as the surface tension across droplets or charge accumulation on condensatethen the state of our system can continue to be described by three state variables,i.e., qt, T, p. The volume of this system may be written in terms of the specificvolumes of its individual components:

V = mvαv +mlαl +miαi, (1.117)

That V is independent of αd above, follows from the assumption of an ideal mix-ture of ideal gases: both gases access the same volume so that the volume of thegas phases, Vg = mvαv = mdαd, can be given by either. Dividing by the totalmass of the system leads to an expression for the specific volume:

α = qvαv + qlαl + qiαi (1.118)

≈ qvαv (1.119)

The approximation follows because the density of both liquid and ice is roughlythree orders of magnitude larger than that of vapor, and because both qi and ql aretypically smaller than qt which in turn in much less than unity. Hence to a verygood degree of approximation (better than one part in 100 000) in (1.119) we canneglect the specific volume of the condensed phases in comparison to the specificvolume of the gases. Substituting for αv above noting that

αv =RvT

eand e =

(qvRv

qdRd + qvRv

)p (1.120)


yields for the equation of state:

α =(qdRd + qvRv)T

p=RT

p, (1.121)

which is identical to the equation of state derived in section 1.1.4 for a moist air inthe absence of condensate. However, in this case qd = 1−qv−ql−qi which impliesthat the virtual temperature (and by analogy the virtual potential temperature) is

Tv = T (1 + εqv − ql − qi). (1.122)

The specific humidities of the condensed phases are sometimes called the liquidand ice loading terms. They do not appear in the traditional form of Tv as derivedin (1.14). To distinguish this form of Tv from that other expression some authorsdescribe it as the density temperature and denote it Tρ.

1.4.3 Moist Enthalpy

In deriving the effective heat capacity of our system we made use of the enthalpyof an ideal mixture of water vapor and dry air. One property of ideal mixtures ofperfect gases is that pressure is given by the sum of the partial pressures thus hasan extensive character. The additive property of extensive quantities simplifies thetreatment of composite systems.

Consider the enthalpy. For a system containing liquid water, dry air and watervapor we have

H = Hd + Hv + Hl (1.123)

= mdhd +mvhv +mlhl (1.124)

= (mdcp,d +mvcp,v +mlcl)T. (1.125)

Because H = (md +mt)h and

Lv = hv − hl (1.126)

the specific enthalpy of this moist system is simply

h = qdhd + qvLv + qthl = (qdcp,d + qtcl)T + qvLv. (1.127)

1.4.4 Equivalent Potential Temperature

For a system which allows phase changes between vapor and liquid one can alsodevelop the concept of a potential temperature as the temperature air would have


if isentropically brought to a reference state. However, in addition to specifyingthe reference state pressure, one must also specify the disposition of the water inthis reference state. Commonly one of two reference states are chosen, either areference state in which all the water is in the liquid form, so that the referencestate entropy is simply:

s0 = qdsd,0 + qtsl,0 (1.128)

or an all vapor reference state. The isobaric specific heat in such a reference stateis simply qdcp,d + qtcl and denoted by cp,0. In general, given a composite systemconsisting of vapor, liquid and dry air its specific entropy may be written as

s = qdsd + qlsl + qvsv; (1.129)

where the entropies for the individual subsystems are,

sd = sd,0 + cp,d lnT

T0−Rd ln

pd

p0(1.130)

sv = sv,0 + cp,v lnT

T0−Rv ln

e

p0(1.131)

sl = sl,0 + cl lnT

T0. (1.132)

Replacing ql by qt − qv in (1.129) allows us to express the specific entropy as

s = qdsd + qtsl + qv(sv − sl), (1.133)

which upon equating with s0 for an all liquid reference state yields an expressionof the form

0 = (qdcp,d + qtcl) lnT

T0− qdRd ln

pd

p0+ qv(sv − sl). (1.134)

Solving for T0 which we denote by θe, and call the equivalent potential temperatureyields:

θe(s) ≡ T0 = T

(p0

pd

)qdRd/cp,0

exp[qv

(sv − sl

cp,0

)]. (1.135)

The term which remains in the exponential of (1.135) can be cast in terms ofthe thermodynamic coordinates (T, e, p) by first noting that

sv − sl = sv − ss + ss − sl (1.136)

= sv − ss +Lv

T, (1.137)


i.e., the difference between the saturated and liquid enthalpies is equal to the en-thalpy differences between the phases (denoted by Lv); and second that the entropydifferences for elements of the same phase and temperature

sv − ss = −Rv

(ln

e

p0− ln

esp0

)︸︷︷︸

ln(e/es)

. (1.138)

are measured by differences in the partial pressures. Substituting from above into(1.135), and noting that

p0

pd=p0

p

p

pd=p0

p

(qdRd + qvRv

qdRd

), (1.139)

yields a more interpretable expression for the equivalent potential temperature:

θe = T

(p0

p

)(qdRd/cp,0)

Ωe exp(qvLv

cp,0T

), (1.140)

where

Ωe =(

1 +qvRv

qdRd

)(qdRd/cp,0)

H−(qvRv/cp,0) (1.141)

captures the relative humidity (H = e/es) dependence as well as the slight modifi-cation that arises to (1.135) when we express the pressure dependence in terms of pinstead of pd. It is typically near unity so that to a reasonably good approximation

θe ≈ θ exp(qvLv

cp,dT

). (1.142)

1.4.5 Static energies

Because fluid displacements in the atmosphere and ocean result in pressure changeswhich are, to a good degree of approximation, hydrostatic (α dp ' −gdz), adia-batic transformations of fluid parcels conserve (to the same degree of approxima-tion) the sum of their moist enthalpy and their potential energy. That is, from thefirst law, for adiabatic transformations:

0 = dh− α dp ' dh + gdz. (1.143)

Hence the first law, and hydrostatics together conspire to conserve the quantity

h + φ, (1.144)


where φ = gz denotes the geopotential, otherwise known as the specific gravita-tional potential energy. Most generally this property of the system is called thestatic energy, depending on the disposition of moisture it takes various forms. Forinstance, defining the moist enthalpy following (1.127), defines a quantity

he = (qdcp,d + qtcl)T + qvLv + gz, (1.145)

usually called the moist static energy, which is conserved under phase changes foradiabatic transformations in which the pressure changes hydrostatically.

Alternatively, if one substitutes for qv using the relation qv = qt − ql in thedefinition of the moist enthalpy, so that

h = qdhd + qthv − qlLv, (1.146)

it follows that an analogous quantity,

hl = (qdcp,d + qtcl)T + qvLv + gz, (1.147)

called the liquid-water static energy is also conserved. Note the parallelism be-tween θe and he, or between θl and hl. Likewise, he dry static energy, sd =cp,dT + gz, follows as an analog to the dry potential temperature for processesinvolving only dry air.

Further Reading

Much of the classical thermodynamics is developed following the outline by Fermi,(1936) Thermodynamics. Fermi’s book is brief and some of the ideas are developedfrom a broader perspective in the classic text Statistical Physics, Part I (3rd edi-tion) by Landau and Lifshitz. A development of the subject which might be moreamenable to those with an engineering background is provided in An Introductionto Thermodynamics, The Kinetic Theory of Gases, and Statistical Mechanics (2ndedition) by F. W. Sears (1953). As its title promises the latter also provides a niceintroduction to the ideas of kinetic theory and statistical mechanics, ideas which arethen developed more systematically by Landau and Lifshitz. My development ofocean thermodynamics draws heavily on unpublished course notes by G. Vallis ofPrinceton University. Further references on the idiosyncrasies of atmospheric ther-modynamics include the now out of print book Atmospheric Thermodynamics byIribarne and Godson, or the in print book of the same title by Bohren and Albrecht(1998). The latter gives a more modern, and rather personal, treatment. A moreadvanced, and rather concise, treatment is also provided in the book AtmosphericConvection by K. Emanuel (1994). Foreman Williams classic text Combustion


Theory also provides an advanced review of thermodynamics (in an appendix) thatI find useful in helping frame the thermodynamics in somewhat more general (lessmeteorological terms).

Exercises

1. If the salinity of a sheltered sea is determined only by the net surface evap-oration (i.e, evaporation minus precipitation) and exchange with the openocean, estimate the rate of water mass exchange between the Baltic and theNorth Atlantic on the one hand, versus the Mediterranean and the Subtrop-ical Atlantic on the other. Hint: start by looking up the net rate of surfaceevaporation in a climatology text, using this and the geometry of the basinyou should be able to solve for the net inflow and outflow of water throughthe channels connecting the sheltered seas to the ocean.

2. Given the definition of an extensive variable, prove that the ratio of extensivevariables define an intensive variable.

3. Derive Eq. 1.8 from Eq. 1.7.

4. What would be the temperature of air at 900 hPa if its temperature at 1000hPa were 300 K and it were brought to its final temperature adiabatically?(i) Assume that the water vapor content of the air is zero and the molecularproperties of the air are characteristic of the Earth’s atmosphere. (ii) As-sume a water vapor content of 10 g kg−1 but that the air remains unsaturatedthrough the expansion. (iii) Assume that the air was composed of 80% N2

and 20% Ar.

5. What is the practical salinity unit and how does it differ from a mass specificquantity such as the specific salinity. Read about the practical salinity scale,perhaps starting with the Wikpedia entry on salinity.

6. Calculate the compressibility of sea-water by comparing the volume of 1kgof water at the surface of the ocean to the volume of 1 kg of water at themean ocean depth of 3800 m. How much does this depend on salinity?What would the effective pressure change in the atmosphere be to effect acommensurate change in volume?

7. Calculate the coefficients of Eq. 1.19 given Eq. 1.18 and a reference state ofT = 0C, p = 0 bar and s = 35ppt.


8. Suppose thatdz = M(x, y) dx+N(x, y) dy.

Show that dz is an exact differential if ∂M∂y = ∂N

∂x . If these latter relationsare not satisfied what does this imply about the functional dependence of z.Interpret your results in terms of the first law.

9. Derive the difference between cp and cv for sea water using Eqs.(1.44) and(1.93) and the equation of state for sea-water.

10. Draw an adiabat and an isotherm on a (p, V ) diagram, explain physicallywhy they differ.

11. Derive the profile of density as a function of height (through the troposphere)for an adiabatic uplift of dry air assuming that the atmospheric pressure de-pendence on height is given by the hydrostatic equation dp = −ρgdz, andthat the density of the atmosphere conspires to remain the same as the parcelbeing lifted through it. Use T = 20C, and p = 1000 hPa for your boundaryconditions.

12. In 200 words or less tell me something interesting about the Lieb and Yng-vason article on entropy. (Physics Today, April 2000, page 32). What rolsare played by the concepts of “System” and “Equilibrium”

13. If

θ ≡ T

(p0

p

)R/cp

is the potential temperature, find a relationship between changes in it and theentropy. Use the second law.

14. Show that ∆g = gv − gl can be expresse in terms of the logarithm of therelative humidity e/es.

15. Derive the Clausius Clapeyron equation by noting that for a reversible trans-formation is a saturated system, dgv = dgl where g is the specific Gibbsfunction

16. Following the procedure used to derive θe, derive an expression for θl thetemperature a parcel of air would have in a reference state where all of itswater mass was in the vapor form at some fixed reference pressure p0. Inter-pret physically.


17. Given the equilibrium state of a system by θe, qt and p can you derive aprocedure for calculating the amount of condensate, if any, that would beexpected in the system?

18. The potential temperatures were derived without regard to the ice phase.Can they be generalized to incorporate freezing and sublimation processes?If not, why not?

19. It turns out that θe is nearly conserved under precipitation. Calculate ∂θe/∂qlto estimate how precipitation affects θe. How much would the introductionof 1 g kg−1 of condensate change θe?

20. Summarize the different temperatures used to describe the atmosphere.

21. Use Kirchhoff’s equation to derive a more accurate formula (than the expo-nential form given in the notes) for the saturation vapor pressure, p∗ as afunction of temperature. Use the fact that cpw = 4200 J kg−1 K−1 and cpv =1850 Jkg−1 K−1 and that at at T0 = 0 C p∗(T0) = 6.11 hPa, Lv(T0) = 2.5MJkg−1. Compare this to the more exact expression given in the notes. Atabout what temperature (above freezing) will the relative error be 1%.

Chapter 2

Thermodynamic Processes, andConvection

In the previous chapter we discussed how best to represent the thermodynamic stateof atmospheric or oceanic fluids. We paid particular attention to the description ofthese fluids as two component systems. In the atmosphere, phase changes in theminor constituent (water vapor), add complexity to the thermodynamic descriptionof the system.

Another source of complexity is that, while the thermodynamics has been de-veloped for equilibrium systems, the atmosphere is generally far from equilibrium:pressure, temperature and density change continuously as one moves through thisfluid. To make use of the thermodynamics we often conceptualize the atmosphereor ocean in terms of fluid parcels in local equilibrium. To do so such fluid parcelsmust be large-enough to allow a continuum description, i.e., much larger than themean-free path of their molecular constituents, but much smaller than the scaleover which changes in the intensive properties of the fluid are apparent. This typi-cally is not a bad assumption, as in the lower atmosphere the mean free path is tensof nanometers and, away from small particles (cloud droplets, particulate matter,or ice-crystals), diffusive processes are effective at homogenizing the thermody-namic properties of the air on scales of millimeters. Fluid parcels then allow oneto develop a more nearly kinetic conceptualization of the atmosphere (or ocean),in which fluid parcels atomize the fluid. This conceptualization is worth bearing inmind as we develop the ideas in this and subsequent chapters.

47

48 CHAPTER 2. THERMODYNAMIC PROCESSES, AND CONVECTION

2.1 Thermodyamic Diagrams

In our development of the thermodynamics we saw a glimpse of the power ofthermodynamic diagrams. They can be used to represent the state of a system,a process, or sequence of processes. If properly constructed they can also yieldimportant quantitative information, such as the work done during a process.

For the atmosphere and ocean, thermodynamic diagrams are useful both forthese reasons, and because they provide a way of obtaining a glimpse of the stateof the fluid in its entirety. Unlike for classical systems, which the thermodynamicsdescribes as a single entity, or system, and hence a point in the thermodynamicspace, the atmosphere or ocean is best thought of as at an intermediate level ofaggregation, as an ensemble of points. Because pressure plays such an importantrole in the atmosphere, diagrams in this space often usefully describe the verticalstructure of the fluid. The distribution of these points then can hint at the processesthat act to set the vertical structure of the system, or how it may evolve under aperturbation.

A variety of thermodynamic diagrams are used to study the vertical structureof the atmosphere, in which context they are called aerological diagrams. Themost common being the Skew-T diagram, and its cousin the Tephigram. Otheraerological diagrams you might have heard of include the Clapeyron, the Ema-gram or Neuhoff, and the Stuve diagram. Ocean structure and circulation are oftendescribed with the aid of temperature-salinity diagrams. Diagrammatic methodsare also useful for describing mixing processes in the atmosphere and ocean, suchmixing diagrams, are often formulated in terms of trace-gases or species.

T

ln Θ

p

T

(a) (b) (c)

−ln p

T

−ln

Figure 2.1: Thermodynamic coordinates for: (a) tephigram; (b) emagram; (c)skew-emagram (ln P Skew-T ).

The aerological diagrams have in common their projection of the atmosphericstate onto the plane. They differ in their choice of thermodynamic coordinates, i.e.,the axes of the plane, and the manner in which they orient these axes. Our focus

2.1. THERMODYAMIC DIAGRAMS 49

here will be on the Skew-T diagram, in which the thermodynamic coordinates areTemperature along the abscissa and log pressure along the ordinate. The basicstructure of this diagram, as well as for its cousins the Emagram and the Tephigramare all plotted in Fig. 2.1.

From the perspective of the thermodynamic coordinates, the Skew-T and theEmagram are identical, they only differ in that the coordinate axes are not orthogo-nal in the latter. Because isotherms will be parallel to the ln p axis and isobars willbe parallel to the temperature axis, changing the orientation of the former leads to agreater distinction between isotherms and adiabats. On the Skew-T they are almostperpendicular. In both, ln p is used as a thermodynamic coordinate (as opposed top) because the height of an isothermal atmosphere is proportional to ln p ratherthan p. As a result the vertical coordinate of such a diagram is closely related to theheight above the surface. The Tephigram differs from the Emagrams in that pres-sure is not a coordinate axis. However from the first law we have for the specialcase of dry adiabats

ln θ − lnT = −Rd

cpln p+ const. (2.1)

Thus associating the y-axis with ln θ and the x-axis with T we find that isobarsare lines satisfying y = lnx + C. In the range of practical interest the curvaturein these lines ends up being small, so that if the diagram is appropriately rotatedisobars become approximately horizontal. Thus on a practical level Tephigramsand Skew-T s are very similar, the chief difference being that isobars are curved onthe former and adiabats are slightly curved on the latter.

Isobars on a Tephigram and isentropes on a Skew-T are examples of fundamen-tal lines. That is they are not thermodynamic coordinates, but are isolines whoseshape is decided by the thermodynamic coordinates. Other examples of fundamen-tal lines include equisaturation curves, pseudo adiabats, and isotherms. Note thatby a coordinate transform the fundamental lines of one thermodynamic diagramcan serve as the thermodynamic coordinates of another.

Although the great variety of fundamental lines that can be added to a diagramcan make it seem rather confusing and busy, they aid the experienced eye in decid-ing at a glance how the state of the system will change under a given process. Toillustrate this consider Fig. 2.2 where we show a standard Skew-T diagram, withthe April 16th Brownsville Sounding superimposed. On this figure we note thefollowing:

• Pressure decreases logarithmically upwards, leading to an almost linear re-lation between height and distance along the ordinate — as indicated by theblack numbers which denote height in meters just to the right of the pressurevalues.


Figure 2.2: USAF Skew-T , ln p Diagram, with the Brownsville, Texas Soundingfor 1200 UTC, 16 April, 2001


• Isotherms are plotted in 10 C bands, which are shaded and converted toFahrenheit where they intersect the abscissa.

• Adiabats are thin lines with negative slope and slight curvature which inter-sect the isotherms nearly perpendicularly.

• Equisaturation curves, or curves of constant mixing ratio (recall that the mix-ing ratio rs = qs/qd) which satisfy the relation

rs =es(T )

p− es(T )

(Rd

Rv

),

are inclined to the left of the isotherms. This follows because if we reducethe pressure along an isotherm we would expect rs to increase.

• What we call moist pseudo-adiabats are almost perpendicular to the abscissanear the surface and curve to become parallel to dry adiabats as p decreases.

moist adiabat

dry adiabat

isotherm

equisaturationpseudoadiabat saturated adiabat

isobar

Figure 2.3: Relative orientation of fundamental lines. Adapted from Fig. VI-4 ofIribarne and Godson.

The relative orientation of the fundamental lines are illustrated more abstractlyin Fig. 2.3. Among other things this figure illustrates slight differences in the slopeof the dry versus the moist adiabat. Both are given by (1.79), except for the dryadiabat we set R′ and c′p to Rd and cpd respectively. That is the modification ofthe heat capacity and specific gas constant due to water vapor in our system is notaccounted for. Because for moist systems R′/c′p < Rd/cpd we expect the temper-ature to fall off less rapidly along a moist adiabat than it does along a dry adiabat.In general the difference between the moist and the dry adiabats will depend onthe moisture content of the air. For this reason moist adiabats are not fundamentallines on the diagrams we have discussed. They could be represented as fundamen-tal surfaces on three dimensional diagrams which had moisture content markingthe third dimension, but this becomes difficult to visualize.


2.1.1 Pseudo-Adiabats

For the same reason that moist adiabats do not constitute fundamental lines, sat-urated adiabats, such as lines of constant θe also are not uniquely representablein the p − T plane. However, by neglect the contribution of liquid water to theheat capacity of the system it is possible to define a nearly adiabatic process thatis representable in the p − T plane. Such a processes is called a pseudo adiabat.This process is equivalent to assuming that upon condensation any liquid water thatforms is removed (precipiated) from the parcel. Because this process is irreversibleit is not isentropic. But because differences between this and the reversible processunderling the definition of θe are small we can consider it as a pseudo-isentropicprocess, and call the lines corresponding to it pseudo-isentropes, or pseudo adia-bats – the latter being more common.

To understand the differences between pseudo-adiabats and saturated isen-tropes (adiabats) consider that for an isentropic process

dh = α dp, (2.2)

where h = (cp,dqd + clqt)T + Lvqs is the moist (saturated) enthalpy. It differsfrom the moist enthalpy given by Eq. (1.127) only through the substitution of qsfor qv, that is the assumption of saturation. Because qs = f(T, p) its differentialcan be related to differentials of T and p, the latter then can be grouped with therhs of (2.2) and related to height differentials by the hydrostatic equation. Hencegiven the definition of h is is straight-forward to show that the lapse rate followinga saturated isentropic process is

Γm ≡ −dTdz

∣∣∣∣h

= g

1/cp∗︷︸︸︷[1 + qsLv

qdRdT

qdcpd + qscpv + qlcl + qsL2v(1+qsRv/qdRd)

RvT 2

]. (2.3)

Note that (2.3) differs from the dry adiabatic lapse rate Γd ≡ g/cp by the ratioof cp to cp∗, the latter being an effective isobaric specific heat capacity that takesinto account the enthalpy differences between the phases. Thus given the temper-ature at any height the temperature at other heights (pressures) can be calculatedby integrating Γm. Because Γm depends on ql it is not representable on the p− Tplane; however the psuedo-adiabatic lapse rate, γm| ql = 0 is. In principal T at anyheight z can be calculated for a pseudo-adiabat by integrating Γm subject to theconstraint that ql = 0. The temperature at 1000 hPa, following the pseudo-adiabat,is called the pseudo equivalent potential temperature and sometimes confused withthe equivalent potential temperature given by (1.140). In contrast to the latter, the


former has no known analytic expression, and hence its quantiative use relies onapproximate formulae.

A straightforward calculation of the difference between pseudo-adiabats andadiabats shows that they differ most appreciably in the upper troposphere. This isto be expected because T decreases with height, hence qs decreases and ql increasescorrespondingly. Overall the differences in T tend to be less than 0.5 K below 400hPa increasing to as much as 3-5K between 100 and 200 hPa. Because in theupper troposphere ice processes play an increasingly important role the lack ofcorrespondence between Γm and Γmp is less of a concern because the relevance ofΓm is questionable here in the first place.

2.1.2 Polytropic Processes

One of course is not limited to drawing adiabats on a thermodynamic diagram. Ifthe diabatic processes are well behaved they can be accounted for by compensatedadiabats. An example of such a process is Polytropic expansion or compression.

Here we consider the change in state of a parcel that is cooling or heating ata fixed rate. Such a process is particularly relevant for the subsiding regions ofthe atmospheric circulation, such as are commonly found in the sub-tropics. Inthese regions the atmosphere descends steadily and cools radiatively as it does so.This radiative cooling should be viewed as a diabatic process whose magnitude wedenote by cpdλδt (i.e., where λ is a heating rate) such that the appropriate form forthe first law is

cpdλδt = cpd dT − RdT

pdp, (2.4)

where in the interest of clarity we develop the theory for the case of a single compo-nent fluid which we take to be dry air. If we look for a relation between temperatureand pressure such that

Tp−κq = const., (2.5)

where κq is a parameter to be determined, we find upon differentiation of (2.5) andsubstitution into the first law, that

cpdλδt = const.(κqcpd −Rd)p(κq−1) dp. (2.6)

Substituting for dp using the hydrostatic equation and substituting for p using (2.5)yields

cpdλδt = −(κqcpd −Rd)

Rdgdz =

(1− κq

κd

)g dz, where κd =

Rd

cpd, (2.7)


which can be solved for κq :

κq = κd

(1− Q

g dz

). (2.8)

This result suggests that a potential-like temperature can be used to specify Tas a function of pressure, κd is modified according to the heat absorbed per unitymass and unit length of transit. If we assume that the parcels change in position isuniform in time (δz/δt = w) then the coefficient κq can be expressed as

κq = κd

(1− cpλ

gw

). (2.9)

That is for a polytropic process we can write

T = T0

(p

p0

)κq

, (2.10)

where κp is given by (2.9). In other words the lapse-rate of an atmosphere subsid-ing and cooling at a fixed rate, is simply

Γd −λ

w. (2.11)

2.1.3 Soundings

Soundings in the atmosphere are measurements of its state as a function of altitude.Generally they are made with a balloon which drifts with the mean wind as itrises and measures the, temperature and relative humidity along its trajectory, thetrajectory itself can be differentiated to yield the wind vector. An example of sucha sounding is shown in Fig. 2.2. Plotted are two lines, the right-most one denotingthe temperature structure of the atmosphere, and the leftmost one being the dew-point temperature, Td. The dew-point temperature is defined as the temperatureat which a vapor, with the given vapor pressure, would be saturated, i.e., it canbe found by solving qs(Td; p) = qv(p) for Td. Physically one can think of it asbeing the temperature to which the atmosphere must be isobarically cooled beforecondensation takes place. The wind is plotted with wind-barbs on the far-right,wherein the magnitude of the wind is denoted by the barbs and the direction by theorientation. Some things worth noting about the sounding are the following:

• The wind is primarily from the west at upper levels, and reaches a maximumof approximately 75 knots near 200 hPa.

• Surface winds are from the east with a southerly jet at 850 hPa.

2.2. MIXING 55

• For the most part the temperature profile fluctuates between a rate of increasebounded by Γd and Γm, tending to the latter between 600 and 200 hPa. Ex-ceptions include a layer between 925 and 750 hPa in which the temperatureis more isothermal than moist adiabatic, and a layer above 200 hPa wherethe profile is again more isothermal.

• The minimum temperature at 200 hPa denotes the tropopause. The warmestair is at the surface, although if the air at 750 hPa were brought to the surfacefollowing a dry adiabat, it would be very much warmer, nearly 40C.

• If air at the surface could be mechanically forced (along a saturated adiabat)to an altitude of approximately 650 hPa it would begin to be warmer than itsenvironment. Its subsequent trajectory is denoted by the dashed line.

• The moisture structure of the atmosphere is varied. A moist layer can befound at the surface but the warm air just above this is very dry, as is the airin the upper troposphere.

We will return to some of these features when we discuss atmospheric stabilityin sectino 2.3.

2.2 Mixing

Mixing is the last anisentropic (diabatic) process that we wish to consider. Themain point we wish to make in this context is that non-linearity in the thermody-namics can lead to rather un-expected behavior, and that in the atmosphere moistconserved variables have great utility in such circumstances. Although the non-linear equation of state permits a variety of exotic behavior in the ocean we herefocus for the moment on the atmosphere. Regarding un-expected behavior we il-lustrate this with the aid of two processes. The first illustrates how condensationcan result from mixing two air-masses which are unsaturated. The second exam-ines how the temperature of a mixture of two air-masses need not be bounded bythe temperature of either of its constituent air masses.

For both cases we consider isobaric mixing. Although not reversible, nor isen-tropic, such a process is adiabatic. Because the process is both isobaric and adia-batic, the moist enthalpy

h = cpT + qvLv, (2.12)

does not change. Although cp depends on the composition of the system, andLv depends on the temperature, as a first approximation we neglect these effectswhen we discuss different mixing scenarios below. They are straightforward, albeit


algebraically cumbersome, to include, an assertion which is left for the reader toexplore in the exercises.

h, qt, m

h1, q

t,1, m

1

h2, q

t,2, m

2

Figure 2.4: Schematic of mixing.

Before exploring specific mixing scenarios we first develop some results andnotation. Given two homogeneous air masses as depicted in Fig. 2.7 we denotethe mixed air mass as being of some mass m = m1 +m2. Introducing the mixingfraction χ = m2/m and denoting by ψ properties which mix linearly we note thatin general:

ψ = ψ1 + χ∆ψ, where ∆ψ = ψ2 − ψ1. (2.13)

Such a rule applies to any specific property of the system, such as h the specificenthalpy, or the total-water specific humidity, i.e.,

h = h1 + χ∆h and qt = qt,1 + χ∆qt. (2.14)

These mixing rules will be used extensively in the subsequent discussion.

2.2.1 Saturation by isobaric mixing

Here following Fig. 2.7 we consider two air masses indexed by i ∈ 1, 2. Bothare subsaturated, with the degree of subsaturation being given by δiq. Thus withoutloss of generality we may write qt,i = qs(Ti) − δiq, where by our subsaturationassumption δiq ≥ 0. Here we explore the possibility of the mixed state being justsaturated, in which case qt = qs(T ). That is we consider when

qt = qt,1 + χ∆qt = qs(T ), given T = T1 + χ∆T, (2.15)

where the latter follows because the linear mixing of qt and h implies that T mixeslinearly in systems without condensate. Expanding qs in a Taylor series about T1,

2.2. MIXING 57

yields the following relations:

qs(T ) = qs(T1) +∞∑

n=1

γn(T − T1)n (2.16)

= qs(T1) +∞∑

n=1

γn(χ∆T )n (2.17)

and

qs(T2) = qs(T1) +∞∑

n=1

γn(∆T )n. (2.18)

The latter implies that

∆qt = (qs(T2)− δ2q)− (qs(T1)− δ1q) (2.19)

=∞∑

n=1

γn(∆T )n − (δ2q − δ1q). (2.20)

Substituting the above relations into (2.15), neglecting terms greater than ordertwo in the Taylor Series and rearranging yields the following condition for themixture to be just saturated:

δ2q = (1− χ)[γ2(∆T )2 − δ1q

χ

]. (2.21)

Which illustrates that condensation in the mixture is readily obtained so long asthe condensation deficit in the respective components of the mixture is sufficientlysmall relative to the saturation change implied by the temperature difference. Notethat this effect depends critically on the non-linearity in qs(T ). If this equationwere linear in T then γn = 0 for n > 1 and saturation by mixing of unsaturatedparcels would be impossible.

2.2.2 Buoyancy Reversal

Another interesting phenomenon, thought to have broad consequences in the at-mosphere is the idea of buoyancy reversal. Here the isobaric mixing of two sys-tems can, through non-linearities in the equation of state, lead to a mixture whosetemperature is not bounded by the temperatures of the constituent air masses. Toillustrate this possibility we first consider the somewhat simpler situation wherebytwo air masses can mix to produce a mixed airmass whose temperature is less theneither constituent. Physically one can imagine this happening when a saturatedairmass (cloud) mixes with a subsaturated (dry) airmass resulting in evaporative


cooling which cools the mixture. Clearly the maximum evaporative potential is re-alized when the mixed parcel is just saturated, implying that all of the liquid waterin the cloudy parcel has been evaporated.

Once again, we take as our starting point the two parcels in Fig. 2.7. The twostates are given by (h1, qt1) and (h2, qt2) but chosen such that qt,1 > qs(T1, p),qt,2 < qs(T2, p), and T2 > T1. In words parcel 2 is relatively cool and saturated.Parcel 1 is subsaturated and relatively warm. The moist enthalpy of the systembefore and after mixing isobarically is the same, that is

h1 + χ∆h = h. (2.22)

which if the mixture remains saturated (and we neglect compositional effects on cpand take L to be independent of temperature), can be expanded as follows

cpT1 + Lvqs(T1, p) + χ∆h = cpT + Lvqs(T, p). (2.23)

For small temperature changes, which we denote δT = T − T1,

qs(T, p)− qs(T1, p) '∂qs∂T

∣∣∣∣T1

δT =qsLv

RvT 21

(p

p− es(T1)

)δT. (2.24)

Substituting this expression for δqs into the previous expression yield a relationshipbetween the perturbation to the temperature under mixing to the differences in theenthalpies of the two states:(

cp +L2

vqs(T1, p)RvT 2

1

p

p− es(T1)

)δT = χ∆h. (2.25)

Hence temperature can reverse itself, T < T1 < T2 if ∆h < 0. The conditionfor the latter, given that ∆T > 0, is approximately that −∆q > (cp/Lv)∆T. Inwords the warmer parcel must be sufficiently subsaturated to offset its temperaturedifference.

Note the effective specific heat, given by the prefactor to δT on the lhs of theabove, is similar to that which arises due to the phase change effect in the derivationof the moist adiabat given by Eq. 2.3, the difference being in this case the effectsof p on qs, which show up in the numerator of (2.3), play role. Not surprisinglythis condition ∆h < 0 can also be expressed as the condition on the moist staticenergy or the equivalent potential, namely that ∆h < 0 or Θe < 0

The condition (2.25) for temperature reversal can be readily extended to thecase of buoyancy reversal by examining under what conditions δTv < 0. In de-riving a condition on this, the additional complication comes from the fact that Tv

depends on both temperature and moisture. Similarly more exact expressions canbe found by restoring the compositional dependencies on the specific heats, andaccounting for the temperature dependence of Lv given by Kirchoff’s relation.

2.3. STABILITY 59

2.2.3 Mixing Diagrams

Clearly moist conserved variables provide a convenient framework for evaluatingmixing processes. To make best use of this framework mixing diagrams are oftenintroduced. These diagrams are constructed using variables that measure both theheat content of the air, and the moisture content, for instance hm and qt are obviousand popular choices. Then by plotting the air at different layers as points on thediagram one can evaluate the role of mixing in setting the structure of the atmo-sphere. For instance, on such a diagram intermediate layers whose thermodynamicstate is set by the mixing of air from bounding layers should lie along a line whichconnects the states of the bounding layers. This would not be true if the state of theintermediate layers was set by differential advection, or other processes.

Although, qt and h (or its equivalent θe) are the most common choice for ther-modynamic coordinates for mixing diagrams, their lack of orthogonality makesthem sub-optimal. Because θe is strongly influenced by qt such a diagram does notseparate states as strongly as it could. To maximize the separation of states a better

2.3 Stability

The static stability of the atmosphere is governed by the rate at which densitydecreases with height. We can see this heuristically by considering the force on aninitially static parcel of air with some density ρ(z) in an environment characterizedby the density ρ(z) – here we use un-decorated symbols to denote the environmentand hats to denote the parcel –

dwdt

=d2z′

dt2= − ρ

′

ρ0g where ρ′ = ρ− ρ, (2.26)

where here z′ can be viewed as a parcel displacement z′ = z− z0 from a referenceheight z = 0. For a parcel starting with the environmental value of ρ at somereference height zref after a small displacement z′ we can evaluate its accelerationas

d2z′

dt2≈ −g

ρ

(dρdz− dρ

dz

)z0

z′ ≈ −g(

1ρ

dρdz− 1ρ

dρdz

)z′. (2.27)

Where this expression arises from taking the leading order of the Taylor seriesexpansion of ρ and ρ about zref . Hence both approximations are increasingly validin the limit of small δz.

Equation 2.27 is just the oscillator equation with frequency

N2 = g

(1ρ

dρdz− 1ρ

dρdz

). (2.28)


It implies that the condition for stability of parcels to density perturbations is thatN2 > 0. If the density of a parcel decreases with height more rapidly than does itsenvironment, i.e., if

dρdz

<dρdz, (2.29)

then N2 < 0 and (2.27) implies exponentially growing modes, or instability. Thisbuoyancy frequency, N2, is called the Brunt-Vaisalla frequency, and it provides asimple measure of the static stability of the atmosphere. In an inviscid atmosphereN2 < 0 is a sufficient condition for instability, whereas N2 > 0 characterizes thegravity wave frequency.

With the aid of the ideal gas lawN2 can be phrased in terms of the temperaturelapse rate, which upon neglect of pressure fluctuations is proportional to the densitylapse rate:

N2 = −g ddz

(ln T − lnT

). (2.30)

However, by the first law the change of temperature of a parcel with height isgoverned by the dry adiabatic lapse rate, Γd, from which it follows that

N2 =g

T

(dTdz

+ Γd

), (2.31)

where here T is the environmental temperature. Thus another way to phrase thecondition for inviscid static instability is to require the environmental temperatureto decrease faster with height than does the dry adiabatic lapse rate. Alternativelywe can work in terms of potential temperature. So doing allows us to express theN2 entirely in terms of the environmental conditions:

N2 =g

θ

dθdz. (2.32)

The simplicity of this equation originates in the conservation properties of θ; itsvariations vanish for adiabatic processes.

In a moist (but unsaturated) atmosphere density variations depend also on vari-ations of the specific humidity (or water vapor mixing ratio) with height, while ina saturated atmosphere the effects of latent heat release must also be accounted for.Consider the saturated case with (2.28) as our starting point. For density pertur-bations to a saturated atmosphere which obeys the saturated equation of state wehave

dρρ

= −dTT

+Rd

Rdqt −

(qsRv

qdRd

)Lv

RvT

dTT

(2.33)

= −dTT

(1 +

qsLv

qdRdT

)+ dqt

(Rd

R

), (2.34)

2.3. STABILITY 61

where the second component of the dT term arises from relating isobaric variationsin qs to perturbations in T through the Clausius-Clapeyron equation, and R =qsRv + qdRd is the gas constant for the composite system. Substituting the abovein (2.28) and noting that

dqtdz

= 0 anddTdz

≡ −Γm, (2.35)

yields the following expression for the static stability in a saturated atmosphere:

N2m =

g

T

(dTdz

+ Γm

)(1 +

Lv

qdRdT

)− gRd

R

dqtdz

, (2.36)

where we use subscript m (referring to moist, actually saturated) to delineate itfrom the dry static stability. N2

m differs from N2 in that it must account forchanges in the environmental lapse rate of qt as well as the effect of condensa-tion/evaporation on the temperature lapse rates.

2.3.1 Moist convective instability

isobar

absolute stability

absolute instabilityinversion

conditional

instabilityisotherm

saturated

adiabat

dry

adiabat

ΓmΓd

Figure 2.5: Different regions of atmospheric sta-bility, as delineated by the fundamental lines ona Skew-T diagram. Adapted from Iribarne andGodson.

Except for the few tens of me-ters nearest the ground duringa warm day, the static stabil-ity of the atmosphere is almostnever negative. This is evidentin both Fig. 2.2 and Fig. 2.8where the environmental tem-peratures decrease everywhereless rapidly than does the dryadiabat. Yet the overturning ofthe atmospheric column man-ifest in many severe stormsmust be an expression of someform of instability. What mightthis be? It turns out that whilethe atmosphere is rarely stati-cally unstable, it is frequently conditionally unstable. By conditionally unstablewe mean that the atmosphere is unstable if the displacements are of saturated air.Thus instead of N2 being the determining parameter, the sign of N2

m is decisive.These different measures of atmospheric stability are delineated with the help ofFig. 2.5 below.


The conditional instability of the atmosphere is conditioned on the saturationof the parcel. That is the instability will be realized only if saturated parcels aredisplaced. This property of the moist system allows a kind of finite amplitude in-stability not found in dry atmospheres. By finite amplitude we mean that while theatmosphere might be stable to infinitesimal perturbations, larger perturbations arepotentially unstable. Consider the sounding in Fig. 2.2. Air very near the surfaceis nearly saturated. If it is lifted adiabatically its temperature will decrease along adry adiabat and its dew-point will increase along a line of constant saturation mix-ing ratio until they meet, at approximately 1000 hPa. This level is called the liftingcondensation level or LCL.

Further displacements of the parcel will be along a moist adiabat. Here wesee that such displacements are stable through a layer of 350 hPa. However, ifsomehow enough work were done on a surface parcel, to lift it to approximately650 hPa, further displacements would result in a parcel which is warmer than itsenvironment. At this point, instead of having to do work on the environment to liftthe parcel, the environment will do work on the parcel, and the parcel will begin toaccelerate upwards. At this point, which we refer to as the level of free convection(or LFC), we speak of the parcel being unstable.

In the process just described surface parcels of air were stable to small per-turbations, but not with respect to deeper perturbations. In the above case theatmosphere is unstable to perturbations which mechanically, and adiabatically liftparcels to pressures less than 650 hPa. Note however that at the level of free con-vection the atmosphere is not absolutely unstable, i.e., unstable to perturbations tothe air at this level. This is because the air at 650 hPa is far from saturation, as it isconsiderably drier than air that is adiabatically lifted from the surface. Thus, unlikea dry atmosphere in which any finite amplitude instability implies an infinitesimalinstability, a moist atmosphere can be truly stable to infinitesimal perturbations,but unstable to finite amplitude ones.

2.3.2 CAPE

To measure the amount of work the atmosphere is capable of doing on a parcellifted to its level of free convection the concept of convective available potentialenergy (CAPE) is introduced (and denoted by A) :

A =∫ znb

zfc

b dz, (2.37)

where b is the buoyancy (or reduced gravity) and has units of acceleration, zfc isthe level of free convection and znb is the level of neutral buoyancy (LNB). This isthe level at which the parcel lifted along the specified adiabat, or pseudo-adiabat,

2.3. STABILITY 63

ceases to be buoyant relative to the environment. Most of the time znb is just abovethe tropopause.

From our development of the Boussinesq equations we note that the buoyancyfor a moist fluid can, under certain circumstances, be expressed in terms of tem-perature fluctuations:

b = −g ρ′

ρ0≈ g

T ′vTv0

. (2.38)

Substituting (2.38) into (2.37) and using the hydrostatic equation yields

A =∫ pfc

pnbRdT

′v d(ln p). (2.39)

Thus on the Skew-T diagram (e.g., Fig. 2.2) A is proportional to the area betweenthe environmental temperature and the dashed line.

CAPE as defined by (2.37) of course depends on the parcel being lifted andthe manner in which it is lifted. Small differences in the initial state of a parcelcan lead to large differences in A. For instance, the sounding plotted in Fig. 2.2is taken from just before sunrise. During the day we expect the surface air to heatup. If it heats 5C and we recalculate theA for a parcel with the identical moisturecontent, but which is 5C warmer we will find that it increases almost 100%, from1 400 J kg−1 to 2 350 J kg−1.

Given that A describes the work the atmosphere can do on a parcel, or alterna-tively the potential energy available to a convecting parcel, it can be related to themaximum kinetic energy. That is it bounds the amount of kinetic energy a parcelcould have, thus defining a velocity scale:

wmax =√

2A (2.40)

which is another measure of the intensity of convection.

Other CAPE-like Measures

CAPE is the most common measure of the potential instability of the atmosphericcolumn. While it constitutes a necessary condition for instability, many other fac-tors can come into play. The amount of work one must generally do to access theCAPE of a sounding is highly variable. In some cases one does not have to do alarge amount of work on a parcel before the atmosphere starts returning the favor.To measure this aspect of the atmosphere another parameter, called Convective In-hibition (or CIN) is sometimes introduced. It is the analog to CAPE, but measuresthe amount of work that must be done to lift a parcel from some reference level to


its level of free convection:

CIN = −∫ pfc

pref

RdT′v d(ln p). (2.41)

Thus the potential instability of the atmosphere depends not on CAPE alone, butalso on other factors such as CIN.

Tw,pir

A

D

T

Td

Tsat

Figure 2.6: Illustration of different types of CAPE using the Brownsville soundingof Fig. 2.2

Another CAPE like measure of the atmosphere is called down-draft CAPE,or D. It is illustrated in reference to regular CAPE in Fig. 2.6. D measures thestability of saturated downward displacements of air first brought to its wet-bulbtemperature by evaporation of falling rain. The wet-bulb temperature (sometimesdenoted Tw) is bounded by the dew-point temperature and the actual temperature.It is the temperature that one gets by isobarically bringing air to saturation throughevaporation. Because water is evaporated into the air, the air both cools and moist-ens increasing its mixing ratio while decreasing its temperature.

Thus we can write

D(pi) =∫ psfc

pi

RdT′v d(ln p), (2.42)

where here Tv(pi) = Tw and T ′v is the difference between the virtual temperature ofa saturated parcel initially at Tv(pi) and the environmental value of Tv for parcelsfollowing saturated adiabats. Physically D measures the stability of air to evapo-ration of rain. In environments with large values of D vigorous down-drafts can

2.3. STABILITY 65

be formed by evaporating water (from precipitation) into a dry ambient environ-ment. This is analogous to the process discussed in Chapter 6, whereby negativelybuoyant parcels can be created through isobaric mixing.

Caveats on CAPE

As alluded to above the actual CAPE of a given atmospheric sounding is not anumber without ambiguity. This ambiguity stems mostly from the varied defini-tions associated with it. Above we have defined CAPE to be the positive area onthe thermodynamic diagram. Others define it to be the net positive area associ-ated with a parcel lifted dry adiabatically from the surface to the level of neutralbuoyancy. Others compute the CAPE associated with a parcel characterized by themean properties of the lower 10-50 hPa of the atmosphere. Yet others adjust thesurface properties of the sounding to reflect what they anticipate conditions will belike at the height of the day.

The thermodynamic processes which govern the evolution of the state of theparcel above the lifting condensation level also play a large role in determiningCAPE. For instance parcels in which ice forms will have different values of CAPEthan parcels assumed to follow a pseudo-adiabat. Thus CAPE can be seen as afunction of the state of a parcel, the environment, and the specified type of thermo-dynamic process. These ambiguities do not diminish the value of a measure likeCAPE, but they do indicate that if it is to be used quantitatively one must be clearas to what one means when referring to it.

2.3.3 Oceanic convection

For the most part convection in the ocean does not exhibit the exotic behavior as-sociated with moist atmospheric processes. However, two processes, thermobaricconvection and cabbeling, are analogous to conditional instability and buoyancyreversal respectively, and their discussion is useful to help provide perspective onthese phenomena.

Thermobaric Effects

Recall the approximate equation of state for seawater given by (1.19).

α = α0 [1 + βt(T − T0)− βs(s− s0)− βp(p− p0)]

+ α0

[γ∗(p− p0)(T − T0) +

β∗t2

(T − T0)2]

(2.43)

The thermobaric parameter is denoted by γ.


Cabbeling

A

B

T [d

egC]

s [psu]

15

30

38373635

Figure 2.7: Isopycs for seawater, as a functionof temperature and salinity. Also shown is themixing line between two states (denoted byAand B denoted by the dashed line).

The process of cabbeling is best il-lustrated by a consideration of twoparcels whose states are denotedby the points A and B on Fig. 2.7.Both have the same density at theirgiven pressure, but have differenttemperatures and salinities. Mix-tures of the two parcels would beexpected to have some intermedi-ate temperature and salinity andbe confined to a mixing line asdenoted by the dashed line. Be-cause the dashed line lies below (athigher density) than the isopyc ofits parent mixture it will tend tobe convectively unstable and de-scend. Hence as was the case forbuoyancy reversal, where mixtures of two parcels could have a density greater thaneither constituent, this process in the ocean may generate convective motions. Thisprocess may be involved in the formation of bottom water in the Weddell Sea.

Further Reading

Some of this material is covered in a more cursory fashion in Section 9.5 of Anintroduction to Dynamic Meteorology by Holton. More extensive treatments arealso given in Atmospheric Thermodynamics by Iribarne and Godson, the book ofthe same title by Bohren and Albrecht (1998) and in Atmospheric Convection byK. Emanuel (1994).

Exercises

1. Construct a Tephigram plotting at least one iso-line of pressure, equisatura-tion, and pseudo-potential temperature.

2. On the standard Skew-T heights are written adjacent to the isobars. Arethese heights for an isothermal atmosphere, and adiabatic atmosphere, orsomething in between? Justify your answer with a calculation.

2.3. STABILITY 67

3. Derive (2.3) stating all assumptions.

4. Does the temperature along a saturated adiabat decrease more or less rapidlywith height than it does along a pseudo-adiabat. Justify your answer physi-cally.

5. Read section 6.4 in the Ocean Circulation text and answer question 6.11 onpage 232 of that text.

6. Discuss the sounding in Fig. 2.8. What processes might explain the structureof the atmosphere at the various heights.

7. In regions of deep convection the temperature of the atmosphere, both withinthe convecting and nearby non-convecting atmosphere, quite closely followsa saturated adiabat. For the non-convecting atmosphere this temperaturestructure results from a combination of radiative cooling and subsidence.If the rate of radiative cooling is 2 K day−1, what is the subsidence rate,assuming a moist-adiabatic lapse rate of 6 K km−1. If the average upwardvelocity in the convective clouds is 1 m s−1, what ratio do you expect to findbetween the convecting and non-convecting area?

8. Use the appropriate moist conserved variable and a Skew-T to estimate thedifference between temperatures for a parcel following a reversible saturatedadiabat and a pseudo-adiabat. Let the parcel be just saturated at 0.1 MPa ata Temperature of 20C.

9. What is the dew-point lapse rate? Given the dew-point lapse rate and the dryadiabatic lapse rate, can you estimate the lifting condensation level given thedew-point depression?

10. Neglecting compositional effects on the specific heats, find the condition forbuoyancy reversal δTv < 0 given ∆Tv > 0. Repeat this derivation but nowaccount for the compositional effects on the specific heats.

11. Express N2m in terms of moist conserved variables.

12. Plot the Emanuel, Del Rio Texas sounding on a Skew-T diagram and discussits stability. Identify the pressures corresponding to the LCL, LFC and LNBfor air lifted from the surface.

13. Repeat the previous question for the Emanuel tropical west-Pacific sounding.

14. Read the articles: The convective cold top and quasi equilibrium by Hol-loway and Neelin (J. Atmos. Sci, 2007, 64, page 1467) and Is the tropical


atmosphere conditionally unstable? by Xu and Emanuel (Mon. Wea. Rev.,1989, 117, 1471-1479). For the former feel free to neglect the discussion ofthe convective cold-top. What is the main point of these articles? How doesthe former improve on the latter?

Figure 2.8: 15 April 2001 (1200 UTC) sounding from Kaui.

Chapter 3

Boundary Layers

Atmospheric boundary layers play an important role in a wide variety of processes.Not only are boundary layers important in determining the net drag at the surface,they also mediate the transfer of heat and matter between the earths surface andfree atmosphere. Like many of the topics discussed in these notes, boundary layertheory is relatively mature and wide-ranging. And while a systematic study ofboundary layer processes is beyond our purview can address some basic questions.Such as what are boundary layers, why they arise and what important results haveemerged from their study.

3.1 Laminar Boundary Layer Theory

U

δ

no slipfree slip x

z

Figure 3.1: Flat plate boundary layer.

Although atmospheric boundary layersare often turbulent, we begin our studywith a review of classical laminar bound-ary layer theory. This theory proves use-ful in shaping our thinking about bound-ary layers in general and helps set thestage for our later discussions. We fo-cus on two examples which should helpillustrate the basic idea of what a boundary layer is, why it arises, and wherein itderives its name. The first example is that of Blasius, associated with the boundarylayer that develops over a flat plate, the second is Ekman’s solution for a boundarylayer in a rotating fluid.

69

70 CHAPTER 3. BOUNDARY LAYERS

3.1.1 Flat Plate Boundary Layer

Imagine a free flow that suddenly impinges on an infinitesimally thin flat plate, asdepicted in Fig. 3.1. The flow going over the flat plate “notices” the presence ofthe plate in so far as its velocity must vanish at the surface.

To understand how the flow might respond we analyze the incompressibleNavier-Stokes equations in two dimensions for a fluid of constant density:

∂u

∂x+∂w

∂z= 0 (3.1)

∂u

∂t+ u

∂u

∂x+ w

∂u

∂z= −1

ρ

∂p

∂x+ ν

(∂2u

∂x2+∂2u

∂z2

)(3.2)

∂w

∂t+ u

∂w

∂x+ w

∂w

∂z= −1

ρ

∂p

∂z+ ν

(∂2w

∂x2+∂2w

∂z2

)(3.3)

subject to the boundary conditions

u = w = 0 at z = 0 and u = U at z = ∞. (3.4)

Here U is the undisturbed fluid velocity, and p is the dynamic pressure.Recall from the previous chapters that in order for the viscous term to be im-

portant at otherwise large Re (thereby allowing the lower boundary conditions tobe satisfied) a new length-scale, associated with viscosity and the no-slip boundarycondition, must emerge. This length-scale, which we denote δ should scale with(ντ)1/2 where τ is an advective timescale. At large Re we expect δ/L = ε << 1where L is an advective length-scale. Thus admitting a role for viscosity introducesa second length-scale into our problem. This length-scale is expected to be relevantto velocity derivatives perpendicular to the plate, while the advective length-scaleL should be relevant to stream-wise velocity derivatives, such that

∂

∂z=

1εL

∂

∂zand

∂

∂x=

1L

∂

∂x, (3.5)

where tilde denotes a non-dimensional quantity.Applying (3.5) to the continuity equation and using the outer velocity, U to

scale u implies that w must scale with εU if the stream-wise velocity divergencesis not to vanish at leading order. In addition if we assume that the pressure gradientin the boundary layer is that “impressed” by the outer flow, then because the outer-flow must satisfy:

∂U

∂t+ U

∂U

∂x= −1

ρ

∂P

∂x, (3.6)

3.1. LAMINAR BOUNDARY LAYER THEORY 71

the pressure density ratio must scale with the velocity squared

P ρ

P ρ∝ U2. (3.7)

Based on this scaling our governing equations may be written non-dimensionallyas follows:

∂u

∂x+∂w

∂z= 0 (3.8)

∂u

∂t+ u

∂u

∂x+ w

∂u

∂z= −1

ρ

∂P

∂x+

1Re

(∂2u

∂x2+

1ε2∂2z

∂u2

)(3.9)

ε∂w

∂t+ εu

∂w

∂x+ εw

∂w

∂z= −ε

ρ

∂P

∂z+

1Re

(ε∂2w

∂x2+

1ε

∂2w

∂z2

), (3.10)

where the scaling of the ∂P/∂z term has been determined by the scaling of all ofthe other terms in (3.10).

Equation (3.9) reaffirms our earlier arguments, namely that if viscosity is tohave a role it must be in a layer whose relative scale ε scales with Re−1/2. Underthis scaling (3.8)-(3.10) imply

∂u

∂x+∂w

∂z= 0 (3.11)

∂u

∂t+ u

∂u

∂x+ w

∂u

∂z= −1

ρ

∂P

∂x+ ν

∂2u

∂z2. (3.12)

at leading order. These are Prandtl’s boundary equation. They can be written as asingle equation in terms of the stream function

ψ(x, z) where u =∂ψ

∂z, and w = −∂ψ

∂x. (3.13)

Prandtl’s student Blasius solved these equations for the special case of steadyflow with no pressure gradients. In terms of the stream function Blasius equationsare:

∂ψ

∂z

(∂2ψ

∂x∂z

)− ∂ψ

∂x

(∂2ψ

∂z2

)= ν

∂3ψ

∂z3. (3.14)

In addition to the x (the distance from the plates leading edge) and z (the heightabove the plate) both the free stream velocity U and ν might be expected to controlthe nature of the solution. That is we might expect to look for solutions of the form:

ψ = g(x, z, ν, U). (3.15)


Based on the dimensions of the parameters of the problem the Π theorem sug-gests that the non-dimensional stream function should be a function of no morethan two dimensionless numbers. Here we take the free stream Reynolds num-ber Ux/ν and z/x and we non-dimensionalize ψ by the free-stream velocity andlength scales U and x :

ψ = xUg(z/x,Re). (3.16)

If we look for solutions exhibiting incomplete similarity in Re such that

ψ =xU

Reαf( zxReα

), (3.17)

we find that solutions satisfying (3.14) do exist for α = 1/2. This means that

ψ = (νxU)1/2f(η) where η = z

(U

νx

)1/2

, (3.18)

should be a solution of (3.14). Thus instead of looking for a solution in terms ofa function ψ(x, z) (or f(z/x,Re)) of two variables our problem becomes one ofsolving for a function, f(η), of one variable. By reducing the order of our systemby one we go from solving a PDE to an ODE. The fact that the problem exhibitsincomplete similarity in Re leads to a drastic simplification, as upon substitutionof (3.18) into (3.14) our problem reduces to solving the equation

f(η)d2f

dη2+ 2

d3f

dη3= 0, (3.19)


f(0) =df

dη

∣∣∣∣η=0

= 0 anddf

dη

∣∣∣∣η=∞

= 1. (3.20)

Solutions to (3.20) can be obtained by series methods and are plotted in Fig. 3.2,where excellent agreement with theory is found. Because the flow asymptoticallyadjusts to the outer flow the boundary layer thickness is not an un-ambiguous quan-tity. If one equates the boundary layer thickness, hwith the point where u ≈ 0.99Uwe find that this corresponds to η ≈ 5,

h ≈ 5(νxU

)1/2. (3.21)

Alternatively one can measure the thickness of the boundary layer in terms of inte-gral scales. One such scale is the displacement height

δ1 ≡1U

∫ ∞

0[U − u(z)] dz ≈ 1.721

(νxU

)1/2, (3.22)


1.0

0.8

0.6

0.4

0.2

0.0

1.0 2.0 3.0 4.0 5.0 6.0

η

u/U

Re

z

Theory

Blasius

Figure 3.2: Velocity distribution in a laminar boundary layer on a flat plate at zeroincidence, adapted from Schlichting Fig. 7.9.

another is the momentum thickness

δ2 ≡1U2

∫ ∞

0u(z) [U − u(z)] dz ≈ 0.664

(νxU

)1/2. (3.23)

All of these measures of boundary layer depth differ in their precise values of theO(1) proportionality constant multiplying

√νx/U , but are otherwise similar.

The Blasius solutions can also be integrated to yield the net drag on the flowthrough some distance L of flow over a plate of some width b:

D = b

∫ L

0ν

(∂u

∂z

)z=0

dx = f ′′(0)νU(U

νx

)1/2

≈ ν1/2U3/2

3x1/2. (3.24)

A remarkable feature of this solution is that the drag scales with U3/2 rather thanU as would be expected in the absence of a boundary layer. This ability of bound-ary layers to qualitatively change the drag regime of a flow is one of their mostimportant effects.


Overall Blasius’s problem illustrates how a boundary layer, associated with aviscous length scale, develops in the flow. It also shows that this boundary layerdevelops self-similarly, which means that at each point downstream of the platesleading edge the flow is non-dimensionally equivalent to the flow at each and everyother point. The interesting twist in this solution is that rather than the solutionbeing scaled by what one might expect in the absence of a boundary layer, i.e., U,the appropriate velocity scale is actually a combination of both the external velocityscale and a diffusive velocity scale, i.e., (Uν/x)1/2. The Blasius problem neatlyillustrates that the concept of a boundary layer is intrinsically bound up with thedevelopment of an additional length-scale in the flow, in this case one associatedwith viscous effects. It also provides a powerful illustration of more advancedsimilarity concepts.

3.1.2 Ekman Layer

Another laminar boundary layer of some interest is the Ekman layer. For an Ek-man layer one is concerned with how viscous effects and a no-slip lower boundarymanifest themselves in the presence of rotation. The framework we take is for arotating plane with the outer-flow in geostrophic balance such that

−fVg = −1ρ

∂P

∂x(3.25)

fUg = −1ρ

∂P

∂y. (3.26)

For this flow we look for u(z) and v(z) for a subset of Prandtl’s equations ona rotating plane:

−∂w∂z

=∂u

∂x+∂v

∂y(3.27)

−fv = −1ρ

∂P

∂x+ ν

∂2u

∂z2(3.28)

fu = −1ρ

∂P

∂y+ ν

∂2v

∂z2. (3.29)


(u(0), v(0)) = (0, 0) and (u(∞), v(∞)) = (Ug, Vg). (3.30)

Here (−fv, fu) is the apparent force associated with our non-inertial referenceframe. In addition to adding the f terms the above equations differ from the orig-inal Prandtl equations in that time-derivatives and advective terms are neglected,furthermore the inclusion of f forces us to consider the third dimension.


If we restrict ourselves to problems wherein d2Ug/dz2 = d2Vg/dz

2 = 0 thensubstituting for the pressure gradients using (3.25) and (3.26) allows us to rewrite(3.28) and (3.29) in terms of the velocity defects (u− Ug) and (v − Vg) :

νd2

dz2(u− Ug) + f(v − Vg) = 0 (3.31)

νd2

dz2(v − Vg)− f(u− Ug) = 0. (3.32)

Or equivalently[d2

dz2− i

f

ν

]φ = 0 where φ = (u− Ug) + i(v − Vg), (3.33)

which was obtained by first multiplying (3.32) by i and then adding it to (3.31).Solutions take the form

φ = φ+ exp

[(if

ν

)1/2

z

]+ φ− exp

[−(if

ν

)1/2

z

]. (3.34)

If we note that

i = eiπ/2 =⇒√i = eiπ/4 =

1√2(1 + i), (3.35)

it becomes apparent that to satisfy the boundary conditions at z = ∞ requiresφ+ = 0 while satisfaction of the boundary conditions at z = 0 requires that φ− =−Ug − iVg. Thus by taking the real and imaginary parts of φ, we obtain solutionsfor u and v :

u = Ug

[1− e−κz cos(κz)

]− Vge

−κz sin(κz) (3.36)

v = Vg

[1− e−κz cos(κz)

]+ Uge

−κz sin(κz), (3.37)

where κ =√f/2ν in the northern hemisphere (f > 0).

Solutions for (3.36) and (3.37) for Vg = 0 are plotted in Fig. 3.3. As was thecase for the Blasius problem they show that the action of viscosity leads to the de-velopment of a new length-scale in the flow. This length-scale defines the Ekmanboundary layer. As we did for the Blasius problem we can define the boundarylayer depth in terms of the displacement thickness. For the case where our coordi-nate system is aligned with the geostrophic wind (so that Vg vanishes)

δ1 ≡1Ug

∫ ∞

0[Ug − u] dz =

∫ ∞

0e−κz cos(κz) dz =

(ν

2f

)1/2

. (3.38)


Thu Apr 27 15:07:33 2000

0 0.5 10

0.5

1

1.5

u/G, v/G

z/[π

(2ν/

f)1/

2 ]

Thu Apr 27 15:11:20 2000

0 0.5 1

0

0.5

1

u/G

v/G

Figure 3.3: Ekman Boundary Layer Solutions for Ug = G. In the left plane weplot the u and v winds. In the right panel we plot the hodograph [the parameterizedcurve u(z), v(z) in the u− v plane], this illustrates the famous Ekman spiral.

Alternatively some define the boundary layer thickness h to be the level where(u, v) is aligned with (Ug, Vg) in which case h = π

√2ν/f . In either case we find

that the depth of the boundary layer increases with viscosity and decreases withrotation.

Other things to notice about the Ekman solution is that the wind at the surface isrotated π/4 radians relative to the geostrophic wind so that in a coordinate systemaligned with the geostrophic wind the surface wind is has an equal magnitude inboth the x and y directions. Unlike the Blasius solutions the drag at the surface ofthe Ekman boundary layer

ν

(du

dz+dv

dz

)=[(U2

g + V 2g )

8ν3

f

]1/2

(3.39)

scales with the magnitude of the geostrophic wind, and the three-halfs power of vis-cosity. Because the Ekman boundary layer allows steady solutions the drag at thesurface must be balanced by pressure gradients which accelerate the flow withinthe boundary layer. These pressure gradients act on the unbalanced (sub/super -geostrophic) component of the flow, i.e., the velocity defect. Physically we canthink of surface drag decelerating the flow to the point where the velocity defectsbecome large enough such that the unbalanced part of the pressure gradient bal-ances the drag. As rotation weakens, the acceleration is increasingly less effectiveand thus must occur over an increasingly deep layer. In the limit as f → 0 δ →∞which is another way of saying that steady solutions are no longer possible.

3.2. ATMOSPHERIC BOUNDARY LAYERS 77

3.1.3 Secondary Circulations

Both the Ekman and the Blasius boundary layers have residual vertical circulations.For the Blasius boundary layer the convergence of the stream-wise wind drives alarge-scale circulation away from the plate:

w =12

(νU

x

)1/2 [ηf ′ − f

]. (3.40)

In contrast the Ekman boundary layer only has an implied vertical circulation inthe case when the forcing varies spatially. By continuity, and the requirement thatthe undisturbed (outer) flow is divergence free,

w(z) =∫ z

0

(∂Vg

∂x− ∂Ug

∂y

)e−κz sin(κz) dz, (3.41)

where we note that the term in the brackets is just the geostrophic vorticity, ωg. Inthe case where ωg is constant with height it can be taken outside of the integral sothat

w(z) = ωg

∫ z

0e−κz sin(κz) dz,= δωg. (3.42)

This indicates that the vertical motion is positive for cyclonic (fωg > 0) motionand negative for anti-cyclones. Physically we can see the effect of the boundarylayer is to turn the flow in the direction of the pressure gradient. This leads to crossisobaric flow which is divergent in high pressure regions and convergent in lowpressure regions.

These residual circulations are important because they are thought to interactwith the large scale flow. For instance in the Ekman boundary layer it is the resid-ual circulation which organizes the tea leafs into the center of the cup upon stirring.It is also this residual circulation which is responsible for stretching or contractingvortex tubes, thereby either spinning up or spinning down a large-scale circulationon a timescale much faster than the viscous timescale L/ν2. The effects of residualcirculations in atmospheric analogs to Ekman flows have also been suggested as amechanism for organizing convection in the atmosphere, and through this processhave been implicated in a large-scale instability called CISK (Convective Instabil-ity of the Second Kind).

3.2 Atmospheric Boundary Layers

The previous section illustrates with the help of exact solutions how the NavierStokes equations allow for the development of a boundary layer, a thin layer wherein


the flow adjusts to its boundary condition. Because this layer scales with ν1/2 itgenerally is of order L/

√Re ≈

√Lν/U. Taking a 10 m s−1 velocity scale im-

plies that even for very large-scale flows that one might expect to find a viscousboundary layer with h < 1m. Likewise we expect from Ekman’s solutions thath = π

√2ν/f ≈ 1 m in the mid-latitudes. One might ask what relevance the above

solutions have. Such questions might be amplified when one recalls that the Earth’ssurface is far from smooth, and flow in the atmospheric boundary layer is rarelylaminar, and the effects of buoyancy can almost never be neglected. Moreover at-mospheric boundary layers often have clouds, vegetation and complex topographyover land, or moving surfaces and waves over the ocean. Most of these effects wewill not address at all. In the remainder of this section we consider the effects ofturbulence and roughness on neutrally stratified boundary layers. Buoyancy effectson the atmospheric boundary layer are treated in §3.3

The fact that the atmospheric boundary layer is almost always turbulent and thesurface is almost always rough leads to even the simplest atmospheric boundarylayers having a multiplicity of scales. To see this requires us to examine the be-havior of turbulent flows, something for which we have yet to develop the requiredmachinery. The main result we need being the mixing length theory developed byPrandtl.

3.2.1 Mixing Length Theory

Recall that for turbulent flows we are interested in the behavior of the mean hori-zontal wind,

∂U

∂t+ Uj

∂U

∂xj− V f =

−1ρ0

∂P

∂x+ ν∇2U − ∂

∂xj

⟨u′u′j

⟩. (3.43)

where here we use uppercase to denote the averages, so that U ≡ 〈u〉 . The form ofthis equation differs from the starting point for the Ekman and Blasius equations bythe presence of the Reynolds stress. Rewriting the above using Prandtl’s boundarylayer assumptions leads to

∂U

∂t+ Uj

∂U

∂xj− V f =

−1ρ0

∂P

∂x+ ν∇2U − ∂

∂z

⟨u′w′

⟩. (3.44)

Further analysis of this system requires some model, or closure for the Reynoldsstress. The most common, and simplest is given by Prandtl’s mixing length theory.

Prandtl hypothesized that 〈u′w′〉 could, despite arising from advective interac-tions, behave diffusively. To see why consider a purely zonal mean flow as depictedin Fig.3.4. Thus if u′(z1) is associated with fluid being transported bodily a dis-tance l up, or down, the velocity gradient, then we expect eddies of scale l to mix

3.2. ATMOSPHERIC BOUNDARY LAYERS 79

fluid with the properties of the flow at z1 + l or z1 − l to the level z1. From theTaylor’s series expansion of the flow, the momentum at these locations is just

u(z1 + l) = u(z1) + ldU

dz+O(l2) =⇒ u′(z1) = l

dU

dz+O(l2). (3.45)

z1

z1+l

z1-l

z

U(z)

U(z1)

Figure 3.4: Mixing Length Theory.

By continuity u′ andw′ should be thesame order so that⟨

u′w′⟩≈ −l2

∣∣∣∣dUdz∣∣∣∣ dUdz +O(l3).

(3.46)Noting that l2‖dU/dz‖ has units of dif-fusivity allows us to write

⟨u′w′

⟩= −Km

dU

dz. (3.47)

This is Prandtl’s mixing-length hypothe-sis. The term Km is a property of theflow, as it depends both on the assump-tion of a mixing length l and the velocitygradients. It has the units of diffusivity (dynamic viscosity). Historically Km wascalled the Austausch coefficient (German for Exchange) and was denoted by Athese days it is often called the eddy viscosity — an unfortunately confusing namewhich originates from its analogy to the molecular viscosity which acts to transportmomentum down the mean gradient. Unlike ν which is a property of the fluid Km

is a property of the flow. The above arguments of course do not hold if the scale ofthe mixing length is similar to the scale of the velocity gradients, which is why thetheory is often associated with (and certainly best justified for) “small” eddies.

3.2.2 Outer and Inner Layers

If one accepts (3.47) as a model of the turbulent stress Prandtl’s boundary layerequation for U becomes

∂U

∂t+ Uj

∂U

∂xj− V f =

1ρ0

∂P

∂x+

∂

∂z

(Km

∂U

∂z

), (3.48)

where because Km is typically very much larger than ν the direct effects of molec-ular effects have been omitted. This equation for the mean wind is similar in formto the starting point for the Blasius and Ekman solutions, the only difference be-ing that Km need not be constant. If as a first approximation one takes Km as


being constant the results from the laminar boundary layer theory can be appliedto turbulent atmospheric boundary layers, with one caveat: care must be taken inmatching (3.48) to the lower boundary.

U(z)

zs

h

d

Outer (Ekman) Layer

Inner (Surface or Log) Layerln z

z0

f

-<u'w'> = u*u

*

Figure 3.5: Schematic of Inner/Outer LayerSeparation.

Clearly the validity of Km cannot extend all the way to the sur-face, particularly if Km is notblended with ν in a very thinlayer just near the surface, or ifthe flow at the surface is turbulentwith much of the drag being dueto pressure effects rather than di-rect viscous stresses. The presenceof a rough surface characterizedby displacements of some depthd usually encourages another ap-proach. Rather than trying to solvemodified Ekman or Blasius prob-lems in which the velocity field is integrated to the surface, the existence of asurface layer of some height zs is postulated. This situation is depicted in Fig. 3.5,wherein we analyze the inner and outer layers separately and match them at theirboundaries.

This separation has merit if zs h as this implies that we can neglect rotationin the inner layer (this is corroborated by Fig. 3.3 wherein we note that as thesurface is approached the wind stops turning with height), If zs d we can alsoneglect the height d of the roughness elements. Neglecting f h and d allows us touse the similarity arguments of §?? to derive a relation between the surface flux ofmomentum (u2

∗) and the velocity profile. This yields the so-called law of the wall:

u(z) = k−1u∗ ln(z/z0), where d z h (3.49)

k = 0.4 is von Karman’s constant and z0 is an empirical quantity (integrationconstant) called the aerodynamic roughness length. We define z0 to be the heightwhere the extrapolated velocity profile vanishes. Although it is found empirically,i.e., by measuring u(z) at various heights, for simple rough surfaces it can be shownto be a function of the height and spatial density (i.e., closely packed versus sparse)of the roughness elements. Typical values for z0 are listed in Table 3.1. Note thatz0 is typically much smaller than the size d of the roughness elements

For the outer layer we note that the law of the wall implies a surface stress ofu2∗, combining this with the velocity gradient at the top of the layer as implied by

(3.49) leads to an expression for the eddy viscosity valid at the top of the surface

3.3. BUOYANCY EFFECTS 81

surface z0water 0.0001 - 0.001bare soil 0.001 - 0.01crops 0.005 - 0.05forest 0.5

Table 3.1: Approximate roughness heights for various surfaces.

layer:

Km = u2∗

(du

dz

)−1

= kzsu∗. (3.50)

This viscosity can then be used in the Ekman solutions to derive h and other pa-rameters. For instance, it implies an Ekman depth

h ≈ π

√2Km

f= π

√2kzsu∗f

. (3.51)

In the mid-latitudes f ≈ 10−4 so even though h ∝ √zs the fact that u∗ is order onestill allows us to maintain the original separation whereby zs h. For instancefor a bare soil, or fallow crop land we expect d < 10cm. If u∗ = 1 m s−1 andzs = 10m we find h ≈ 100m > zs = 10m d = 10m. Other arguments aresometimes used to estimate the Ekman depth at h ≈ cu∗/|f |. With c ≈ 0.25 thisyields h ≈ 2 km, which is significantly larger.

One downfall of the above arguments is that h seems to depend on how highwe take our surface layer to be, and there is some arbitrariness in this. In reality thestress at some height z will not be equal to the surface stress, and Km may betterbe modeled as varying with height. Relaxing the above solutions leads to gener-ally more acceptable models of atmospheric Ekman layers, but the basic pattern,wherein we have separation between inner and outer layers, each with their ownscaling and matching at the boundaries, does not change.

3.3 Buoyancy Effects

Heretofore we have only discussed momentum boundary layers. In such layersturbulence exists to transport momentum. Turbulence arises in the first place be-cause of the means shear that develops in the fluid. When turbulence productionis associated with a mean shear we often speak of mechanically produced turbu-lence. This is in contrast to the buoyancy generation of turbulence. In addition to


adding a production mechanism for turbulence, buoyancy effects can also stabilizethe fluid, thereby destroying turbulent motions. Buoyancy also gives turbulenceanother job, that of transferring heat. In addition to modifying the local structureof the turbulence the presence of buoyancy can lead to thermal boundary layerswhose depth is controlled more by thermal instabilities and the stratification ofthe free atmosphere and less by viscous processes and the momentum stress at thesurface. These varying effects of buoyancy are discussed in the next section.

3.3.1 Monin-Obukhov Similarity

Recall that the development of the log layer was based on a similarity argumentthat admitted a single length-scale z and a single velocity scale u∗. On dimensionalgrounds we posited that the non-dimensional velocity gradient must be universal.When we have a flux of heat (or more importantly buoyancy) from the surface weintroduce another parameter into the system — the surface buoyancy flux:

B =g

θv0w′θ′v where [B] = m2s−3. (3.52)

To the extent that the buoyancy flux effects the flow we might expect the velocitygradient to be described by a function of three variables,

du

dz= f(z, u∗,B). (3.53)

In non-dimensional terms this parameter set has one-redundant dimension, whichcan be cast as a length-scale L:

L = − u3∗

kB(3.54)

where the additional factor of k is arbitrary but customary. The length-scale Ldefined in this manner is called the Obukhov length-scale, it is negative in re-gions of positive buoyancy fluxes. Physically it weights the relative contribution ofbuoyancy versus shear production of turbulent kinetic energy. This multiplicity oflength-scales suggests on dimensional grounds that the non-dimensional velocitygradient should scale as a function of ζ = z/L,(

kz

u∗

)du

dz= Φm(ζ). (3.55)

Here Φm is the non-dimensional velocity gradient. To reproduce the law of thewall Φm should approach unity as ζ → 0. If we expect that the flow will be less


stratified when buoyancy contributes to mixing, and more stratified when buoyancyresists mixing then we might expect that Φm > 1 for ζ > 0 and Φm < 1 otherwise.

In analogy one might expect that a similar function exists to describe the non-dimensional virtual potential temperature and specific humidity gradients,(

kz

θv∗

)dθv

dz= Φh(ζ), (3.56)(

kz

q∗

)dqvdz

= Φh(ζ), (3.57)

where qv is the water vapor specific humidity and θ∗ and q∗ are flux scales definedin analogy to u∗ such that:

u∗θv∗ = −w′θ′v and u∗q∗ = −w′q′. (3.58)

The fact that Φh appears in both (3.56) and (3.57) reflects the assumption that thenon-dimensional scalar gradient function is independent of the particular scalarbeing scaled.

1.51.00.5-0.5-1.0-1.5-2.0

1

2

3

4

5

6

-2.5 0

Φm=1+4.7ζ

Φm= (1-15ζ)-1/4

ζ

Φm

Figure 3.6: Non-dimensional ve-locity gradient, adapted fromBusinger, Fig. 2.1.

Just as the law of the wall produces a re-lationship between the profile of the wind andthe flux, (3.55)-(3.57) extend this relationshipto the profiles of heat and moisture and theirfluxes and in so doing so accounts for thebuoyancy driving of the flow through the non-dimensional length-scale ζ. These flux profilerelationships are known as surface-layer sim-ilarity theory, or Monin-Obukhov (or some-times just M-O) similarity theory in honor ofMonin and Obukhov who first suggested theabove scaling. Many experiments have beenconducted to experimentally evaluate the the-ory and within the range of expected validitythe theory can be considered to be a spectacu-lar success.

Ultimately one would like to be able to de-rive the similarity functions Φm and Φh.How-ever, in the absence of a better understand-ing of turbulent flows we are forced to esti-mate these functions empirically. One suchestimation, for the case of Φm is illustrated in


Fig. 3.6. It is derived from a series of mea-surements made over wheat stubble in Kansas.Many similar measurements have been made over a wide variety of surfaces withthe result being that

Φm ≈

1 + βmζ ζ > 0(1− γmζ)−1/4 ζ ≤ 0

(3.59)

Φh ≈

1 + βhζ ζ > 0(1− γhζ)−1/2 ζ ≤ 0

. (3.60)

Four our purposes it is sufficient to take βh = βm = 5 and γh = γm = 16.The actual values of these constants remains, however, a matter of some debatewith most studies finding that 5 < βm < 7, 5 < βh < 9, 16 < γm < 20,and 12 < γh < 16. These fits are accurate for |ζ| not too large, as for |ζ| 1little reliable data exists. Instead the general shape of the gradient functions isconstrained by asymptotic considerations.

The general form of (3.59) and (3.60) allows one to analytically integrate(3.55)-(3.57), thereby yielding non-dimensional profile functions:

u(z) =u∗k

[ln(z/z0)−Ψm(ζ) + Ψm(zo/L)] (3.61)

θ(z)− θ(z0) = θ∗ [ln(z/z0)−Ψh(ζ) + Ψh(z0/L)] (3.62)

where for ζ ≥ 0Ψm = −βmζ and Ψh = −βhζ, (3.63)

while for ζ < 0

Ψm = 2 ln(

1 + Φ−1m

2

)+ ln

(1 + Φ−2

m

2

)− 2 tan−1(Φ−1

m ) +π

2(3.64)

Ψh = 2 ln

(1 + Φ−1

h

2

). (3.65)

where we should note that θ(z0) is not necessarily the surface temperature, but forsimple surfaces should be close to its value, thus we will take it as such subse-quently. Actually estimating the flux, given θ(z) and u(z) requires one to solve theabove equations for u∗ and θ∗. Because L depends on u∗ and θ∗ this is not as triviala procedure as one might hope, nonetheless the dependencies are simple enoughin the stable case to yield closed form expressions. In the unstable case one mustresort to iteratively solving a system of implicit equations for the flux.


It is difficult to overstate the importance of these results. They provide a meansfor estimating the flux from the profiles and are undoubtedly the most importantresult from similarity theory in the atmospheric and oceanic sciences.

Lastly we note that the above relationships can be related to exchange coeffi-cients for heat and momentum, respectivelyKm andKh. If following Prandtl theseare defined as the ratio of the flux to the gradient then we find that

Km = kzu∗Φ−1m (3.66)

Kh = kzu∗Φ−1h . (3.67)

These indicate that in a mixing length theory of turbulent flowsKm andKh shoulddepend on the stability following the stability dependence of Φh and Φm. From theresult in previous chapters we do not expect turbulence for Richardson numbersgreater than 1/4. Thus one would expect the empirically derived values of Φm andΦh to become infinite in this limit, and indeed they do. Equations (3.66) and (3.67)also provide a basis for deriving a turbulent Prandtl number

Pr =Km

Kh=

Φh

Φm. (3.68)

Thus we see for stable through neutral coefficients (3.59) and (3.60) imply a turbu-lent Prandtl number of one. As the flow becomes increasingly driven by buoyancyPr decreases, indicating rather more efficient transport of heat relative to momen-tum.

3.3.2 Convective Boundary Layers

The last topic we would like to discuss in the concept of thermal boundary layers.For thermal boundary layers the distinguishing feature of the boundary is its ther-mal disequilibrium with the interior fluid. This results in a transport of heat into orout of the fluid, leading to boundary effects even in the absence of a mean flow.

To help one think about thermal boundary layers consider a semi-infinite uni-formly stratified fluid at rest subject to a constant heat flux at the surface. Such asituation is depicted in Fig. 3.7a. As time increases we expect the thermally mod-ified region of the flow to deepen. If we associate a boundary layer depth h withthis region of thermally modified flow, similarity suggests that

h = f(Q,Γ, κh, t), (3.69)

where Γ is the stratification of the initial flow, Q is the applied surface heat flux,and κh is the molecular diffusivity of heat. Given the various dimensions we can


look for solutions of the form

h =(Qt

Γ

)1/2

f(Π), where Π =κhΓQ

. (3.70)

Here the non-dimensional parameter of our system can be thought of as a non-dimensional diffusivity. It effectively measures the ratio of the diffusive timescaleto the surface heating timescale.

dΘ/dz = Γ

(a) (b)

w'θ'|z=0 = Q

h

hΓ

Qt

θ

z

Figure 3.7: Schematic of the development of a thermal boundary layer.

For the sake of argument, let us now assume that the fluid is such that its dif-fusivity is zero for dθ/dz > 0 and κh otherwise. In this case for Π large we mightexpect the thermal layer to be relatively uniform in θ, resulting in the situation de-picted schematically in Fig. 3.7 b. For such a situation energetic considerationsshow that complete similarity in Π should exist such that

h =(

2Qt

Γ

)1/2

. (3.71)

Thus we see a thermal boundary layer developing in the fluid and growing witht1/2 at a rate which decreases with increasing stratification of the outer fluid andincreases with the rate at which the fluid is heated.

The atmospheric analogy to the above problem is something called the convec-tive boundary layer. Such boundary layers are most pronounced during the daytimeover land, and can become very deep. In these boundary layers the heat from thesurface is not transported diffusively, but rather by turbulent eddies, plumes or ther-mals, that arise at the surface and penetrate through the depth of the layer, into thestratified fluid above. These plumes or thermals act to keep the boundary layerwell mixed. Because the plumes are buoyant they accelerate through the depthof the boundary layer, overshoot their level of neutral buoyancy, and mix warmerfree-atmospheric fluid into the boundary layer. The general evolution of such aboundary layer is illustrated in Fig. 3.8. The noteworthy features of this figure arethe seemingly self-similar growth of the profile; the degree to which turbulence


z [

m]

Θ [K]

t=0

t=10hrs

∆θ

Figure 3.8: Numerical simulation of self-similar evolution of a free convectivelayer.

acts to maintain a well-mixed boundary layer, with nearly uniform values of con-served quantities. At the base of the mixed layer there is a super-adiabatic surfacelayer and at the top of the well mixed layer we find a layer which is actually coolerthan it was initially. This cooler layer is often called an entrainment layer, as itresults from entrainment, the downward mixing of warmer air through the actionof turbulence. The top of the entrainment layer is bounded by a region of enhancedstratification whose strength at ten hours is denoted by ∆θ. One can think of thiscapping region as resulting from large eddies pushing up, into the stratificationaloft.

Thus the boundary layer deepens and warms at a rate faster than predicted by(3.71). Again by similarity we can propose that

h = A

(2Qt

Γ

)1/2

. (3.72)

where A − 1 represents an entrainment efficiency. Energetically it can be thoughtof as a measure of the extent to which turbulent kinetic energy generated by thesurface driving is used to do work on the outer layer, as opposed to being dissipatedin situ.

Although convective boundary layers often exist in the presence of a meanwind, the mean wind often plays a secondary effect, and many of the fundamentalconcepts can be thought of in the limit of vanishing mean wind. This limit is calledfree convection, it (like the neutral shear layer) is part of the foundation of our


understanding of the atmospheric boundary layer.Free convection gives rise to something called mixed-layer scaling. On some

level one could imagine that the important parameters for mixed layer scalingwould be just the set (Q,Γ, t). For practical reasons, and because the free atmo-sphere is rarely if ever uniformly stratified, the equivalent set

Q,∆θ, h, (3.73)

where ∆θ measures the jump in temperature at the top of the entrainment layer (seeFig. 3.8). In addition, if one is interested in turbulent quantities, the efficiency withwhich a heat flux produces a buoyancy flux g/Θ0 is also an important parameter.These parameters can be used to produce a turbulent velocity, temperature andtimescale:

w∗ =(g

θ0Qh

)1/3

, θ∗ = Q/w∗, and τ = h/w∗. (3.74)

As a result of this scaling a non-dimensional number defining the stability of thelayer

Ri = ∆θ/θ∗, (3.75)

and the non-dimensional height z/h emerge. One could posit that all turbulentquantities could thus be written as a function of these parameters, for instance

θ′2 = θ2∗f(z/h,Ri). (3.76)

This is the so-called free-convective, or mixed layer, scaling. Empirically it hasbeen found to well describe the convective boundary layer, with most results show-ing little dependence on Ri. Note that the mean wind is not a parameter in thisscaling.

h/L Regimeh/L > 0 stable boundary layerh/L ≈ 0 neutral (or Ekman when f 6= 0), layer−1 < h/L < 0 shear dominated, but weakly convective−10 < h/L < −1 convection with shearh/L < −10 free convection

Table 3.2: Boundary Layer Regimes.

Most boundary layers are neither purely convective, nor are they neutrally strat-ified shear layers. To quantify the various intermediate states often boundary layers


are classified according to h/L. Here h is non-dimensionalized by L because theoverall buoyancy production in the boundary layer scales with h, consequentlyh/L is a better measure of the overall stability of the boundary layer. Such a clas-sification scheme leads to the definition of different regimes as indicated in Table3.2. The regimes are often associated with varied phenomena, for instance for−10 < h/L < −1 convective rolls are often found int he boundary layer, whileplumes or thermals are found at more negative values of h/L.

3.3.3 Diurnal Evolution

These concepts should help us understand the diurnal march of the boundary layerover land. Through the course of its daily evolution the boundary layer transitsthrough most of the regimes mentioned above. During the night the cold groundstabilizes the lowest level of the atmosphere and effectively decouples the boundarylayer from the flow aloft. In this situation the atmospheric boundary layer existsas a thin, mechanically forced layer, often only tens of meters thick. As the sunrises and heats the surface buoyancy driving begins to dominate, and the boundarylayer grows, often to great depths. The buoyancy forcing generates turbulenceefficiently which mixes winds and scalars through the depth of the boundary. Thesetendencies are evident in Fig. 3.9 where the differences between the wind at 50mand at the surface are most pronounced at night, and almost negligible during theday.

Local Time [hrs]

Win

d S

pee

d [

m s

-1]

2

4

6

8

4 8 12 16 20

z = 50 m

z = 0.5 m

Figure 3.9: Diurnal march of winds at twoheights as observed during the Wangara ex-periment.

Often the repeated cycling ofthe boundary layer means thatthe growth of the boundary layeron any given day is through theremnants of the previous daysmixed layer. Because the strat-ification is so weak (Γ so smallin Eq. 3.72) the boundary layercan often grow explosively to theheight of the previous days mixedlayer, thereafter growing markedlymore slowly as it encounters lay-ers of the free atmosphere un-influenced by the previous daysboundary layer. This tendency ofthe previous layer to be left as aremnant or fossil layer above thenight time boundary layer has important implications for atmospheric chemistry,


as it separates the previous days pollutants from emissions at the surface during thenight. These are mixed when the boundary layer re-forms in the morning, and thiscan be a time of very interesting chemistry.

Exercises

1. Derive (3.19) and the associated boundary conditions starting from (3.14).

2. Ekman Problem . . . to be determined.

3. What is the similarity variable in M-O similarity theory

4. Often something called bulk aerodynamic formulae are used to calculatefluxes

w′u′ = Cdu2(z)

w′θ′ = Chu(z)∆θ

where ∆θ is the difference between the potential temperature at the height zand its surface (z0) value. Estimate Cd assuming that (a) z0 = 0.0001 andζ = −5; (b) z0 = 0.01 and ζ = −5; (c) z0 = 0.01 and ζ = 0.

5. Given the time-series data in class, which consists of temperatures and ve-locities at two different heights calculate the surface fluxes of heat and mo-mentum. Explain how you could use this information to estimate the surfaceroughness and surface temperature.

6. Show that the MO Similarity functions imply zero flux for the surface Richard-son number,

Ri =z g

θv0∆θv

u2(z)> β−1

h

where ∆θv > 0 is the difference between the θv(z) and its surface (z0) value.

7. Estimate the depth of the day-time Ekman layer over land, justifying anyassumptions you might make.

8. Estimate the Rayleigh number corresponding to the flow in Fig. 3.8 at 10hours. Hint, to do this associate θ∗ with Γ0H in Eq. ??.

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Contents · Contents 1 Atmos & Ocean Thermodynamics 5 ... These notes emerged from a graduate course which I used to teach at the Uni-versity of California Los Angeles. The course