I Fundamentals - University of California, San...

I

Fundamentals

This first chapter reviews the fundamental principles of fluid mechanics,emphasizing the relationship between the underlying microscopic description ofthe fluid as a swarm of molecules, and the much more useful (but less genuine)macroscopic description of the fluid as a set of continuous fields. Although it iscertainly possible to study fluids without recognizing their true particulate nature,such an approach avoids important ideas about averaging that are needed later,especially in the study of turbulence. It is best to encounter these ideas at theearliest opportunity.

Moreover, even if one adopts a strictly macroscopic viewpoint (as we even-tually shall), one still has the choice between Eulerian and Lagrangian fieldtheories. The Eulerian theory is the more useful and succinct, and most textbooksemploy it exclusively. However, the Lagrangian theory, which regards the fluidas a continuous field of particles, is the more complete and illuminating, andit represents a natural extension of the ideas associated with the underlyingmolecular dynamics to the macroscopic level of description.

All fluids are composed of molecules. We shall regard these molecules as pointmasses that exactly obey Newton's laws of motion. This assumption is not pre-cisely correct; the molecular motions are really governed by quantum mechanics.However, quantum effects are frequently unimportant, and the main ideaswe want to develop are independent of the precise nature of the underlyingmolecular dynamics. It only matters that there be an exact underlying moleculardynamics, so that one could in principle predict the behavior of the whole fluidby solving the equations governing all of its molecules.

3Salmon, Rick. Lectures on Geophysical Fluid Dynamics, Oxford University Press, Incorporated, 1998. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/ucsd/detail.action?docID=271803.Created from ucsd on 2019-03-14 13:03:38.

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In the ocean, L is typically several millimeters or more, whereas A is only about10 8cm. Thus seawater easily satisfies the necessary condition (0.2) for (0.1) tomake sense. We shall see that condition (0.2) recurs frequently as we attempt toderive equations for \(x,y,z,i) and the other macroscopic fields.

There are two general methods for deriving the equations governingmacroscopic variables like v(x,y,z,t) . The first, which we shall call the averagingmethod, is by direct averaging of the equations governing the point masses.Unfortunately, this method leads to equations that arc exact but mathema-tically unclosed until further assumptions are invoked. These further assump-tions require the methods of kinetic theory and non-equilibrium statisticalmechanics.

The second general method is the more traditional. This method treats thefluid as if it really were a continuous distribution of mass in space, and it derivesthe macroscopic equations by analogy with the equations for point masses. Thesuccess of this second general method can be gauged by the fact that fluidmechanics was a highly developed subject long before the existence of atoms andmolecules was generally accepted, at the beginning of the twentieth century. Forthe most part, we shall follow the traditional method, although the analogy withparticle mechanics will be even closer than in the usual presentation. However,we shall also follow the averaging method for a while, in order to appreciate thefundamental difficulties that arise when an exact but complicated dynamicsis "simplified" by averaging. We shall encounter these same fundamental dif-ficulties in our subsequent study of turbulence, where they cannot be so easilycircumvented.

4 Lectures on Geophysical Fluid Dynamics

It is of course utterly impractical to follow the motion of every molecule,because even the smallest volume of fluid contains an immense number ofmolecules. We are immediately forced to consider dynamical quantities thatrepresent averages over many molecules. For example, the velocity v(x,y,z,t) atlocation (x,y,z) and time t is defined as a mass-weighted average,

in which the sums run over all the molecules in a small volume 5V centered on(x,y,z). Here, ra, is the mass of the ith molecule, and v,(^)is its velocity. However,the definition (0.1) makes sense only if the volume 8V is neither too large nor toosmall. If 8V is too small, then it contains too few molecules for a meaningfulaverage. If, on the other hand, 5V is too large, then the average (0.1) smoothesout physically interesting features of the velocity field. In fact, the definition (0.1)makes sense only if v(x,y,z,t) is independent of the size of 5V for a considerablerange of sizes, that is, only if the smallest scale L over which the velocity fieldvaries appreciably is much larger than the average separation /I betweenmolecules,

Salmon, Rick. Lectures on Geophysical Fluid Dynamics, Oxford University Press, Incorporated, 1998. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/ucsd/detail.action?docID=271803.Created from ucsd on 2019-03-14 13:03:38.

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Fundamentals 5

I. Eulerian and Lagrangian Descriptions

For the moment, then, we regard our fluid as a continuum—a continuous distri-bution of mass in space. There arc two common descriptions of continuummotion. In the Eulerian description, the independent variables are the spacecoordinates x = (x,y,z) and the time t. The dependent variables include thevelocity v (x ,y ,z , t ) , the mass density p(x,y,z, t) , and the pressure p(x,y,z,t).

In the Lagrangian description, the independent variables are a set of particlelabels a = (a,b,c), and the time i=t. The dependent variables are the coordinates

at time i, of the fluid particle identified by (a,b,c). The particle labels varycontinuously throughout the fluid, but the values of (a,b,c) on each fluid particleremain fixed as the fluid particle moves from place to place. By fluid particle, wenow mean a tiny piece of the imaginary continuum—not a molecule!

In the Lagrangian description, we use the special symbol rfor time so that weshall know that d/dt means that (a,b,c) are held fixed, (d/dt means that (x,y,z) areheld fixed.) Thus dF/fais the rate of change in (arbitrary quantity) F measured byan observer following a fluid particle. In other words,

is the usual substantial derivative.We can think of a label space with coordinates (a,b,c) and a location space with

coordinates (x,y,z). Then the fluid motion (1.1) is a time-dependent mappingbetween these two spaces. Alternatively, we can think of the label variables(a,b,c) as curvilinear coordinates in location space. Then the fluid motion dragsthese curvilinear coordinates through location space.

The label variables (a,b,c) can be arbitrarily assigned. Commonly, the a aredefined to be the x-location of the corresponding fluid particle at a reference timeT= 0. We will give a different, more convenient definition. But no matter how thelabels are defined, each fluid particle keeps the same values of (a,b,c) for all time.

The derivatives with respect to Eulerian and Lagrangian coordinates arerelated by the chain rule. For example,

for any quantity F that can be regarded as a function of (x,y,z,i) or (a,b,c,T). Butthe substantial derivatives of (x,y,z) are, by definition, the components of thevelocity,

and thus (1.3) becomes


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Equation (1.5) is the familiar formula for the substantial derivative.

2. Mass Conservation

We can assign the labeling coordinates (a,b,c) so that

where dVahl. = dadbdc is an infinitesimal volume in a-space, and d(mass) is theenclosed mass.1 Since the labeling coordinates move with the fluid, (2.1) holds atall subsequent times. By definition,

where p is the mass density, and dVxyz = dxdydz. is the volume in x-spacecorresponding to dVahc. Thus, remembering that

we find that

Equations (1.4) and (2.4) define the Eulerian dependent variables v(x,f) andp(x,f) in terms of the Lagrangian dependent variables x(a,r). The correspondingdefinition of.p(x,t) will emerge later.

Next, define the specific volume

The substantial derivative of (2.5) is


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Thus the usual equation for mass conservation results from our requirement thatfixed volumes in (a,£>,c)-space always contain the same mass.

To complete our continuum equations, we need an equation for the con-servation of momentum. For this, we adopt the somewhat unusual approach ofinvoking Hamilton's principle of least action. This approach has several advan-tages over the more standard derivation. Here we mention only two. First, theHamiltonian derivation demonstrates the strength of the analogy between con-tinuum mechanics and particle mechanics. Second, mechanics and thermo-dynamics enter the Hamiltonian formulation on the same footing: In both cases,we prescribe the dynamics by specifying how energy depends on variables thatdefine the state of the system. The resulting forces—both mechanical andthermodynamical—are then the derivatives of the energy with respect to thesevariables. Only entropy and the concept of thermodynamical equilibriumseparate thermodynamics from mechanics.

The following two sections offer a very brief review of Hamiltonian particlemechanics.2

Fundamentals 7

That is,

We can rewrite (2.7) in the more familiar form,

3. Functionals and Variational Principles

A functional is a number that depends on all the values taken by a function with-in some range of its argument. For example, let x(t) be any function defined onf, < t < t2. Then

is a functional that depends on x(t).The calculus of variations addresses the following question: For what functions

x(t) is the functional F[x(t)] stationary? More precisely, for what x(t) is F un-changed by small, arbitrary changes in x(t) that vanish at t = tl and t2?

Suppose that x(t) is changed to x(t) + Sx(t), where &(t) is everywhere small,and &x(t1) = 8x(t2) = 0. Then F changes from (3.1) to


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where terms of order (Sx)2 have been neglected. The change in F is therefore

after integration by parts. For Flo be stationary, 5F must be zero. But since 5x(i)is arbitrary, its coefficient in the integrand of (3.3) must vanish. Therefore,

and only those x ( t ) satisfying (3.4) correspond to stationary values of F. Thedifferential equation (3.4) is said to be equivalent to the variatlonal principle8F = 0.

4. Hamilton's Principle for Point Masses

Now consider a system composed of N point-particles with masses m, (i = 1 to N)and locations x,(0- Let V(x,,. . . , x,v) be the potential energy of the system. If, forexample, the N point masses all have equal electric charge q, then

Whatever V, Newton's law of motion is

Hamilton's principle is the variational principle equivalent to Newton's law. Itstates that the action

is stationary, where the Lagrangian

is the difference between the kinetic energy '/' and the potential energy V of thesystem. For the collection of point masses,

where the summation runs over all the molecules. Hamilton's principle thusstates that

for arbitrary, independent variations {&c,-(f),(5y,-(i),&,•(/)} that vanish at /t and f2.Since <Sx;(O = <5x,(?2) = 0, (4.6) implies that


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Fundamentals 9

for all j. Newton's law (4.2) then follows by the arbitrariness of S\j(t).

5. Hamilton's Principle for a Barotropic Fluid

The only difference between the system of point masses and the fluid continuumis that, in the continuum, the mass is distributed continuously in space. Therefore,we must replace

by

and (4.5) becomes

In the fluid, the potential energy arises from external and interparticleforces that depend only on the particle locations x(a,b,c,T). The simplest assump-tion is that these particle locations enter the potential energy V in the specialform

where

is the specific volume, and the specific internal energy E(a) and external potential0(x(a,r)) are prescribed functions of their respective arguments. That is, theinternal energy E depends only on the amount by which the fluid has beencompressed (as measured by the Jacobian (5.5)), and the external potential 0(typically representing gravity) depends only on the fluid-particle locations x(a,r)(and not, say, on their derivatives). The precise form of E(d) depends on the typeof fluid being considered. We shall show that Hamilton's principle and thehypothesis (5.4) with given E(a) and 0(x) yield the momentum equation for anideal (barotropic) fluid.

According to Hamilton's principle, the action

must be stationary with respect to arbitrary variations 5\(a,b,c,T) in the locationsof the fluid particles. The integral in (5.6) is over the whole mass of the fluid, a


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fixed volume in a-space. We suppose that the fluid has two kinds of boundaries:rigid boundaries, with outward unit normal n, at which the fluid velocity in thenormal direction must vanish (d\/dT-n = 0) and at which the fluid-particle varia-tions arc correspondingly constrained (<5x-n = 0); and free-surface boundaries, atwhich 8x is wholly unconstrained.

By direct calculation, the change in (5.6) produced by the variation 5x(a,b,c,T)in the fluid-particle locations, is

But for any quantity F,

By the divergence theorem, this is

where the second term is an integration over the boundary of the fluid. Note,however, that the rigid portion of the boundary makes no contribution tothis integral, because 8x-n = 0 at rigid boundaries. Setting F= -dE/da and using(5.8-9) in (5.7), we see that Hamilton's principle implies that

where

Then, recalling that 8x is arbitrary, we conclude from (5.10) that


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and that p = 0 at the free-surface boundaries. From (5.12) we see that p must bethe pressure; then (5.11) agrees with a familiar equation from thermodynamics.According to (5.11), the pressure is positive if the fluid resists compression, thatis, if energy must be supplied to reduce the volume.

yields the same momentum equation (5.12) as before, but with p defined as thepartial derivative (6.3), with entropy held constant. Equations (6.3) and (6.4)

Fundamentals I !

6. Nonhomentropic Flow

The assumption that the internal energy E increases only through compression istoo restrictive. We know that the addition of heat can also increase the internalenergy. The simplest conceivable generalization of E = E(a) is

where r\ is an additional parameter, which we call the specific entropy. Thedifferential of (6.1) is

where

and

is the temperature. We recognize the term -p da in (6.2) as the change in energy(per unit mass) caused by mechanical compression. It follows that T dr\ must bethe change in energy caused by heating. If no heat is added to the fluid ortransferred between fluid particles, then dr\ = 0 and the entropy of each fluidnarticle does not change:

We can then regard the entropy T](a,b,c) as a prescribed function of the fluid-particle identity, determined by the initial conditions.

Since the entropy r/(a,b,c) is unaffected by changes in the dependence of x on(a,r), Hamilton's principle,


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I 2 Lectures on Geophysical Fluid Dynamics

agree with the usual thermodynamic definitions of pressure and temperature,respectively. However, it remains to be shown that the temperature defined by(6.4) has the familiar properties associated with temperature.

The existence of a. fundamental relation (6.1), at thermodyamic equilibrium,among E, a, and r\ is one of the basic postulates in the formulation of thermody-namics developed by Gibbs, Tisza, and Callen.3 (We introduce the other postu-lates when we discuss molecular transports.) Once again, the form of thefundamental relation (6.1) depends on the type of fluid under consideration andcannot be determined from thermodynamics alone. By using the fundamentalrelation in the Lagrangian for a moving fluid, we are assuming local thermody-namic equilibrium, that is, that the variables E, a, and 77, although changing inspace and time, are locally related as they would be in exact thermodynamicequilibrium. This assumption can be true only if the time required for molecularprocesses to bring the system into local equilibrium is much shorter than the timescale for macroscopic changes.

7. Variable Composition

The assumption that E depends only on a and 77 is true only for a fluid of fixedchemical composition. Seawater is a dilute solution of ionic salts whose composi-tion is specified by the concentrations of each of its ionic components. Let n, bethe mass fraction of component i, defined as the number of grams of i per gramof seawater. Since nt is generally different for different seawater particles, thefundamental relation for seawater takes the general form

where N is the number of (important) components. The most important compo-nents of seawater are Cl , Na+*, SO4 , and Mg++.

Measurements show that the mass fractions of the dissolved ions in seawateralways have approximately the same ratio to one another. This is because theionic concentrations change mainly by evaporation and precipitation at the seasurface, and these processes affect all the dissolved ions in the same way. Thus itis possible to specify the composition of seawater by a single parameter, thesalinity,

and each fluid particle conserves its salinity. Then, since S(a,b,c), like rj(a,b,c), isunaffected by variations in the dependence of x on (a,fo,c,r), Hamilton's principlegives the same momentum equation (5.12) as before, but now with

The fundamental relation can therefore be written

If there is no molecular diffusion of salt,


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obtained by eliminating the entropy 77 between (7.5) and (7.7), is also often calledthe equation of stale. All the equations (7.5-9) follow from the fundamentalrelation (7.3), which completely defines the thermal equilibrium state of the fluid.

8. Equations of Motion for an Ideal Fluid

Let us now collect our results. Our equations of motion are, apart from defini-tions, the result of applying Hamilton's principle to a hypothetical macroscopiccontinuum, whose internal energy (representing the energy stored in the other-wise ignored microscopic degrees of freedom) is assumed to be a prescribedfunction E(a,r\,S) of the specific volume, entropy, and salinity. The entropy andsalinity are conserved on each moving fluid particle. In conventional notation,our equations are

Fundamentals 13

Instead of (6.2), we now have

where

is the temperature, and

is the chemical potential of salt in scawater. In (7.6), the term u dS represents theinternal energy change caused by a change dS in salinity, assuming no change inthe specific volume or entropy.

Equations (7.5), (7.7), and (7.8) arc called equations of state. The equation


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form a complete set of equations, with c(p,p,S) a prescribed function. The func-tion c(p,p,S)—like all other thermodynamic functions—is uniquely determinedby the fundamental relation E(a,r],S). The reader should verify, by linearizing(8.5), that c is indeed the speed of sound.

Another formulation uses the temperature T as one of the dependent vari-ables. This formulation is probably preferable if molecular transports are to beintroduced subsequently. Eliminating the specific volume a between (7.5) and(7.7), we obtain


Equations (8.1), in which the molecular transports (i.e., diffusion) of momentum,heat, and salt are entirely missing, are the equations for an ideal fluid. If thefundamental relation E = E(a,r],S) is known (from laboratory measurements,say), then (8.1) forms a complete set of seven (scalar) equations in the sevendependent variables p = I/a, v, p, r], and S. In this section, we rewrite (8.1) inseveral useful equivalent forms.

We can transform (8.1) into a more convenient form in several ways. By onemethod, we rearrange (8.1c) into the form

Then, taking the substantial derivative of (8.2) and using (8.1d-e), we obtain

where

The quantity c turns out to be the speed of sound. The equations

It follows that


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Fundamentals 15

To put (8.7) in neater form, we must express its two thermodynamic partialderivatives in terms of tabulated quantities. Remembering that T dr\ representsthe increase in the internal energy per unit mass caused by heating, we see that

where Cp is the heat capacity at constant pressure (and salinity).The other partial derivative in (8.7) can be simplified as follows:

where ft is the coefficient of thermal expansion. The first equality in (8.9) is one ofMaxwell's thermodynamic relations. To prove it in the standard way, we rewrite(7.6) as

and require

Substituting (8.8) and (8.9) into (8.7), we obtain another complete set ofequations,

where (8.12c) is the equation of state, (7.9).We often neglect the Dp/Dt term in (8.12d). Let <5Tand Sp be typical tempera-

ture and pressure differences in the flow of interest. Then the right-hand side of(8.12d) is negligible if

For the typical oceanic values,


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this neglect is justified if

However, condition (8.14) is certainly violated when seawater particles undergopressure changes of hundreds of atmospheres with small temperature changes, asin the deep ocean. In fact, in the deep ocean, the most significant temperaturechanges are sometimes caused by pressure changes, via (8.12d).

A similar justification sometimes permits us to neglect Dp/Dt in (8.5c). Theresulting approximation,

bears a superficial resemblance to the mass conservation equation (8.5a), but it isnevertheless an approximation to (8.5c). Equation (8.15) can never be justified asan approximation to (8.5a).

9. The Method of Averaging

Let's begin again, and rederive the equations for macroscopic fluid motion usingthe first of the two general methods described at the beginning of the chapter:averaging over molecular motions. The resulting equations are exact (apart fromour assumption that the molecules obey Newton's law) but mathematicallyunclosed without further approximations. However, these exact equations auto-matically include the molecular transports of momentum, heat, and salt omittedpreviously.

Again we regard the fluid molecules as point masses, and let m, be the mass atlocation x,-(f). The governing molecular equations are

where V is the potential for intermolecular forces. The mass density at (x,f) canbe defined as an average over molecules, in the form

where the summation is an estimate of the mass per unit volume near x at time t.Here, R(r) is a sampling function that counts the molecules near x. The simplestdefinition of R is

where r0, the radius of the sampling sphere, is much larger than the averagemolecular separation A but much smaller than the smallest scale L on which p(x,?)varies significantly. However, it is more logical and convenient to define R(r) to


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Fundamentals 17

Figure I.I The sampling function used to define the fluidr density and velocity.

be an infinitely differentiable function with the general shape sketched in figure1.1. The only rigid requirement is the normalization condition,

Defining the macroscopic velocity by

(compare (0.1)) so that

which is therefore an exact consequence of our definitions of p and v.

we see that (9.5) is equivalent to the continuity equation,

The time derivative of (9.2) yields


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Suppose that there are two kinds of molecules: water molecules with mass mwand salt molecules with mass ms. These two kinds of molecules are separatelyconserved. By analogy with (9.2), we define the density of salt by

where

Thus (9.10) can also be written

where the sum is over salt molecules only. By the same steps as before, we findthat

defines the velocity v, of salt. By our previous definition of salinity,

Equation (9.13) is equivalent to

where

is the molecular flux of salt. A molecular salt flux occurs whenever the averagevelocity of salt molecules differs from the average velocity of all molecules. By(9.8), we can also write (9.14) as

If the molecular salt flux is negligible, (9.16) reduces to the ideal-fluid equationfor salinity, (8.1e).

Unfortunately, the exact derivation of (9.16) has introduced a new dependentvariable, vs (or Fs), without increasing the number of equations. As we shall see,the corresponding derivations of the equations for momentum and total energyalso introduce new variables. Thus, the method of averaging yields exact expres-sions for the molecular transports, which, however, render the resulting macro-scopic equations mathematically unclosed. This closure problem is an inevitableconsequence of the averaging. To close the equations, we must find approxima-tions for molecular transports like F, that involve only the original variables, p, S,


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Fundamentals 19

v, p, and T. The most commonly used approximation for the molecular saltflux is

for which, neglecting the possible spatial dependence of JJLS, (9.16) becomes

where KS = n,/p is the diffusion coefficient for salt. Approximation (9.17) can bejustified by an appeal to experimental results or by kinetic-theory argumentssimilar to those given in section 11.

Why does the equation (9.16) for salt contain a molecular transport term,whereas the corresponding equation (9.8) for all matter contains none? This isbecause our definition (9.6) of velocity is an average weighted by the total mass.Using a different convention would lead to differently placed molecular transportterms, but the closure problem would remain.

10. Momentum Equation by the Method of Averaging

We resume our program of rederiving the macroscopic equations using themethod of averaging over the molecular motions.4 Our next objective is anevolution equation for the macroscopic velocity v(x,t) defined by (9.7) and (9.2).Taking the time derivative of the x-component of (9.7), we obtain

where Ri = R(r,). As before,


We easily obtain the corresponding equations for d(pv)/dt and d(pw)/dt.Now we work on the right-hand side of (10.3). We want to extract the terms

that appear in the ideal-fluid momentum equation (8.1b). The remaining termswill represent the previously neglected molecular momentum flux. First, wedefine


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and note that v,'(x,/) is the difference between the velocity of the z'th molecule andthe continuum velocity at x, all at time t. Then the x-component of

The cross-terms cancel as a matter of definition. Treating the other two compo-nents of (10.5) similarly and substituting the results into (10.3), we obtain

Combining (10.7) with the corresponding equations for the momenta in the y-and z-directions and using the continuity equation to simplify the result, wefinally obtain

where F ( x , t ) is the symmetric tensor denned by

The last equivalence in (10.9) defines the angle-bracket term, the average prod-uct of molecular velocity components.

Now, everything so far is a matter of definition. To add the physics, we needNewton's law for molecules,

where the potential energy,

consists of one part, Vml, that arises from intermolecular forces and another part,Vea, that corresponds to external forces. In a uniform gravitational field (pointingin the minus z-direction),


is


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Fundamentals 21

where k is the unit vector in the vertical direction. We shall show that

for some tensor G, so that (10.13) can be written as

where

is the total momentum-flux tensor associated with the microscopic flow.The tensor G is sometimes negligibly small. In an ideal gas, for example, only

a tiny fraction of the molecules are interacting (colliding) at any particularinstant, and the noncolliding molecules make no contribution to Vml. Then Falone represents the momentum flux associated with the microscopic motion.

To understand F, suppose that Fxy > 0. This means that x-direction momentumflows toward positive y. And this agrees with definition (10.9), which states thatFxy > 0 if the molecules moving toward positive y (vmol > 0) have, on average, apositive Jt-direction momentum (umol > 0).

Now imagine that the fluid is enclosed by a rigid container and left undisturbedfor a long time. Intuition suggests that the macroscopic flow will eventually dieout, and that (away from boundaries) the microscopic, molecular motions willbecome statistically homogeneous and isotropic. This in turn implies that T isdiagonal, with all three diagonal components equal. These facts motivate thegeneral decomposition

where

is the dynamic pressure and T is the deviatoric stress tensor. The momentumequation (10.15) then becomes

To close (10.19), we need expressions for both p and t in terms of the macro-scopic variables v, p, and S.

In the hypothetical equilibrium situation described above, T = 0. If thedeviatoric stress tensor t is negligible, then (10.19) resembles the ideal-fluidmomentum equation (8.1b). However, in the ideal-fluid equations, p is the ther-modynamic-equilibrium pressure, determined by the fundamental relation ofthermodynamics. On the other hand, the/? in (10.19) is defined by (10.18). Withinthe context of this section, we should not even mention the fundamental relationunless we can derive it from the exact molecular dynamics. This turns out to be


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impossible—even in principle—without an additional hypothesis: the basic hy-pothesis of equilibrium statistical mechanics. We talk more about equilibriumstatistical mechanics in subsequent lectures. But here we note that even if it weresomehow possible to equate p to the pressure in (local) thermodynamic equilib-rium, we still lack a prescription for the deviatoric stess tensor t. And since Tevidently vanishes in thermal equilibrium, the expression for T requires ideasfrom nonequilibrium statistical mechanics. We illustrate the general flavor ofthese ideas in section 11, where we obtain an expression analogous to (9.17) forthe molecular flux of momentum.

Now we turn to the proof of (10.14). We assume that the interactions betweenmolecules are pairwise repulsions. Then

Now let S,7(x) be the unique solution to

where

and the prime denotes differentiation. The inequality (10.21) implies that themolecules repel one another, that is, that energy must be supplied to bring twomolecules closer to one another. The x-component of (10.14) is

Think about (10.23) in a coordinate system in which the molecules at x, and x, lieon one of the coordinate axes (the r-axis, say). Refer to figure 1.2. Then, if r is thedistance in the direction of x, - xr (10.23) is equivalent to

Thus Stj is nonzero (and negative) only in the cigar-shaped region betweenmolecules i and). Substituting (10.23) into (10.22), we obtain the x-component of(10.14) in the form

where


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Fundamentals 23

Figure 1.2 The scalar 5,7(x) is negative inside the cigar-shaped volume containing mol-ecules i and;', and zero outside. The sampling functions R, and Rt are nonzero only withinthe spheres of radius ra around molecules i and j, respectively.

is a symmetric tensor.To understand G, consider the contribution of two particular molecules, i and

j, to (10.26). With no loss in generality, we can assume that these two moleculeslie on the x-axis. Then these two molecules contribute only to the GM-componentof G, that is, to the flux of x-direction momentum in the x-direction. Since both<£'(>) and Sij are negative, this flux can never be negative, but it is nonzero onlywithin the region of nonvanishing Stj shown in figure 1.2.

In summary, F represents the microscopic momentum flux caused by mol-ecules moving from one location to another while conserving their momentum enroute. On the other hand, G represents the microscopic flux caused by intcrmo-lecular forces, and can be nonzero even if the molecules are not moving.

I I. An Example of Kinetic Theory

Using the averaging method, we have obtained equations for salinity and mo-mentum that contain exact expressions for the molecular transports. However, toclose the equations, we must find approximate expressions for these moleculartransports in terms of the macroscopic variables p, v, S, etc. Nonequilibriumstatistical mechanics based on kinetic theory provides the basis for this closure,but the full theory is far beyond our scope.5 Instead, we examine a very simpleversion of kinetic theory, which nonetheless illustrates the essential ideas.

Many authors define the p in (10.17) to be the pressure -dE/da given byequilibrium thermodynamics; then T is, by definition, the remainder. If one thenassumes that the components of T depend linearly on the first derivatives dvJdXjof the velocity components and that the relationship between these two tensors isisotropic, one eventually obtains the Navier-Stokes approximation,


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where K and u are undetermined scalars, and the subscripts denote directionalcomponents.6 Frequently,

where v = u /p is the kinematic viscosity.This conventional derivation of (11.1), though certainly very elegant, is also

somewhat empty: Why, after all, should the stress depend linearly on the strainrate? Furthermore, the conventional derivation leaves the coefficients u and K,and hence v, unspecified; we must then regard them as given functions, like thefundamental relation E(a,n). In contrast, kinetic theory offers both the physicalmotivation for a relation like (11.1) and a quantitative estimate of u and K, butaccurate numbers often demand impractically difficult computations.

To keep things simple, we suppose that all the molecules have the same mass,and consider the special situation in which the velocity v = (u (y) ,0 ,0) points onlyin the x-direction and depends only on y. We want to estimate the molecular fluxof x-direction momentum from y < 0 to y > 0, that is, through the dashed line infigure 1.3. We assume the following:

(a) The molecules conserve their momentum between collisions.(b) Before each collision, a molecule's momentum may differ from the average of

its neighbors, but after each collision this difference vanishes on average.(c) The molecular mean free path A is short compared to the scale L on which u(y)

varies.

By assumption (a), only F contributes to the total molecular momentum fluxT. Let s be the average speed of a molecule in any particular direction. That is, let

Figure 1.3 We seek the molecular flux of momentum across y = 0 in a flow with velocityu(y) in the x-direction.

and hence


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Fundamentals 25

Then, by (10.9), the diagonal components of T are

The macroscopic velocity u(y) has a negligible effect on these diagonal com-ponents, because u is so much smaller than s. By considering the momentumtransferred by elastic collisions to a plane surface immersed in the gas, we see that(11.5) is just the dynamic pressure p, the normal force per unit area of the gas onthe surface.

Now consider the (off-diagonal) flux of x-momentum in the x-direction,

at the location y = 0. If there were no macroscopic velocity, then (11.6) wouldvanish; thus Txy must depend on u(y). To estimate Txy' we divide the moleculescrossing y = 0 into two groups, and suppose that

where ( ) u p is the average (at y = 0) over up-going molecules and ()down is theaverage over down-going molecules. Now, on average, the down-going moleculescrossing y = 0 experienced their last collision at y = +A/2, whereas the up-goingmolecules crossing y = 0 experienced their last collision at y = -A/2. Therefore, byassumptions (a) and (b),

and

We use the Taylor-series expansion

to relate u(-A/2) to the velocity field at y = 0, the level at which we are calculatingthe flux. The third term in (11.10) is negligible provided that

that is, provided that

which is true by assumption (c). Similarly,Salmon, Rick. Lectures on Geophysical Fluid Dynamics, Oxford University Press, Incorporated, 1998. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/ucsd/detail.action?docID=271803.Created from ucsd on 2019-03-14 13:03:38.

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It is interesting to compare this prediction to the measured viscosity of theEarth's atmosphere. At 20°C, s = 5.0 x 104cm/sec and A = 6.5 x 10-6cm, so (11.15)pedicts that v~ .165cm2/sec. This agrees well with the measured value of .15cm2/sec. But don't be too impressed. There is no way to define the mean free pathprecisely. However, for any reasonable definition, (11.15) gives a reasonableestimate for the viscosity of an ideal gas.

Before leaving this subject, we briefly describe the more sophisticated methodactually used by statistical mechanicians to compute transport coefficients like v.One starts by deriving the Boltzmann equation, an equation describing theevolution of the probability distribution of molecular velocities. The derivationof the Boltzmann equation requires an assumption (Stosszahlansatz or "assump-tion of molecular disorder") that, like assumption (b), imposes a direction oftime. The Boltzmann equation has stationary solutions that correspond to ther-modynamic equilibrium. To estimate the transport coefficients, one expands theBoltzmann equation about one of these solutions, using a method pioneered byChapman and Enskog.7

Although we have emphasized that calculations of molecular transport coeffi-cients require a nonequilibrium theory based on the underlying molecular dy-namics, equilibrium thermodynamics actually places strong constraints on theforms these expressions can take. To explain these constraints, we turn, onceagain, to the strictly macroscopic point of view. First, however, we must completethe postulational basis of thermodynamics.

12. Thermodynamic Constraints on Molecular Diffusion

All of equilibrium thermodynamics can be deduced from the following fourfundamental postulates:

1. For any system at thermodynamic equilibrium, there exists a fundamental rela-tion n = n (E ,X 1 ,X 2 , . . . ,XN) among the entropy n, the energy E, and the othermacroscopic parameters X1,X2, . . . ,XN that describe the system.

2. At equilibrium, the entropy of an isolated system is a maximum.3. The entropy of a system is the sum of the entropies of its constituent subsystems.4. The entropy vanishes at zero temperature.

Substituting these results into (11.7), we obtain

Thus (11.3) holds, with molecular viscosity


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Fundamentats 27

Gibbs introduced the first three postulates. Nernst added the fourth, which isimportant only at temperatures so low that quantum effects become important.**

So far, we have used only the first postulate of thermodynamics, with X1 = aand X2 = S as the macroscopic parameters. That is, taking the system to be a gramof seawater, we have assumed the existence of a relation n(E,a,S) or, equiva-lently, E(a,S,n). From the standpoint of thermodynamics, these are given func-tions, which must be determined by experiment. However, equilibrium statisticalmechanics (discussed briefly in section 18) offers the means of actually calculat-ing n(E,a,S), removing (in principle!) the need for any experiments.

In this section, we discuss the constraints imposed by thermodynamics on theexchange of heat and salinity between seawater particles at a fixed pressure. Ourdiscussion requires the first three of the fundamental postulates listed previously.The system, shown schematically in figure 1.4, consists of two seawater particles(1 and 2), each of 1 gram, and an environment (3) of much greater size. The twoparticles exchange heat and salt with one another but not with the more distantenvironment. However, all three systems exchange mechanical work as the twoseawater particles adjust their volumes to keep their pressures the same as thepressure of the environment, pr. We imagine the whole system (1, 2, and 3) to beenclosed by a distant, rigid, adiabatic wall. In this way, we fulfill the requirementof postulate (2) that the whole system be "isolated."

By postulates (2) and (3), the total entropy

because the environment does not exchange heat with the two seawater particles.Thus postulate (2) reduces to the requirement that

But, by the conservation of total energy, volume, and salinity,

is a maximum in thermodynamic equilibrium. However,

be maximum in equilibrium. Now

Figure 1.4 Two seawater particles (1 and2) can exchange heat and salt with oneanother but not with the more distantenvironment (3).


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to leading order in the differences between the temperatures and potentials. Thetransfer coefficients kTT, kTu, kuT, and kw depend only on T1 (~ 7"2) and /i, (= fa).Furthermore, since (12.3) is a maximum at equilibrium, (12.8) must always bepositive. This puts a constraint on the transfer coefficients in (12.10).


Here, pr and V3 are the pressure and volume of the environment, and we haveagain used the fact that the environment exchanges only mechanical work withthe particles. We can treat p, as a constant because the environment is relativelylarge. Eliminating 5K3 between (12.5a,b), we find that

where

is the specific enthalpy. That is, the total enthalpy and salinity of the two seawaterparticles alone are conserved. By (12.4) and (12.6-7), we have

At equilibrium, (12.8) must be zero for arbitrary, infinitesimal Sh1 and SS1.Thus

at equilibrium. That is, for two seawater particles at equilibrium, the temperaturesand chemical potentials must be the same.

Next, suppose that our two-particle system is displaced away from equilibriumby a small but finite amount. Then dh1 and dS1 represent the (luxes (from fluidparticle 2 to particle 1) of enthalpy and salt that bring the system back intoequilibrium. What do we know about these fluxes? From the macroscopic view-point, these fluxes can depend only on the states of the two seawater particles.Since the pressure (pr) is uniform, these states are defined by T1, m1, T2, and fa(two variables for each particle). Furthermore, both fluxes must vanish whenT1 = T2 and fa = fa. If the differences between the temperatures and chemicalpotentials of the two seawater particles are small, it follows that the fluxes dependlinearly on their differences. That is,

d


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Fundamentals 29

Now suppose that the two seawater particles can exchange heat and mechani-cal work but not salinity (as if separated by a membrane). Then, for (12.8) to bepositive, the sign of(T2- T1) must be the same as that of dh1 In other words, heatflows from the particle with the higher temperature. Similarly, if the particlesexchange salt but not heat, then salt flows from the particle with the higher m/T.However, if, as is typically the case, our seawater particles can exchange bothheat and salt, then neither of these statements necessarily applies. Thermody-namics demands only that the total entropy production (12.8) be positive.

We easily extend these arguments to the fluid continuum. Consider a mass offluid, at a fixed pressure p,, that exchanges heat and salt between its constituentfluid particles but not with the surrounding fluid. The analog of (12.8) is

which must be positive. The analog of (12.6) is

where Fh and Fs are the fluxes of enthalpy and salt. Equation (12.12) requires thatthe enthalpy or salt lost by one fluid particle is gained by another. Substituting(12.12) into (12.11), integrating by parts, and using the fact that the normalcomponents of the fluxes vanish (by hypothesis) at the boundaries of the fluidmass, we obtain the analog of (12.8):

If the departure from thermodynamic equilibrium is slight, then the fluxes musttake the general forms

and

which are the analogs of (12.10). Again, this is true because the fluxes depend onT and n but must vanish when T and m are uniform. The flux laws (12.14-15) arethe first nonvanishing terms in an expansion about uniform T and m. Substituting(12.14-15) into (12.13) yields


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Even though we require laboratory measurements to determine the values of thediffusion coefficients in (12.14-15), we can be confident beforehand that theresults will satisfy (12.17).9

It is customary to rewrite (12.12) as equations for DT/Dt and DS/Dtthis requires the thermodynamic expression for h = h(T,S,pr), which maybe determined from the fundamental relation. Translating the flux law;(12.14-15) into flux laws for temperature and salinity, we obtain the finaequations:


which must be positive. Therefore, thermodynamics demands that

and

Because of the cross-terms in (12.18) and (12.19), a salt gradient generally drivesa temperature flux and vice versa. The K' term in (12.18) is negligible in theocean, but the K term in (12.19), called the Soret effect, can sometimes be as largeas the KS term. Nevertheless, both k' and K" arc customarily neglected.

The preceding discussion applies to two seawater particles at the same im-posed pressure. Since the pressures, temperatures, and salinity potentials of thetwo particles are equal in thermodynamic equilibrium, the two particles also havethe same equilibrium salinities. However, one can consider the more generalproblem of two seawater particles at different imposed pressures, or, more gen-erally, the thermodynamic equilibrium attained by an extensive mass of seawaterin a gravitational field. The analysis is nearly the same as the preceding, but onemust be careful to include the gravitational energy per unit mass (gz) as part ofthe energy. Again one finds that the temperature T and salinity potential n areuniform in thermodynamic equilibrium:

However, since T is uniform whereas p is not, it follows that the equilibriumsalinity in a gravitational field must be nonuniform; the change in S compensatesthe change in p in (12.20b). Fofonoff (1962) estimated that if the whole oceanwere in complete thermodynamic equilibrium, then its temperature would beuniform, but its salinity would decrease with depth at the rate of about 3 pptper kilometer. This is far larger than the observed midocean salinity gradient!Of course, this hypothetical equilibrium state has no practical relevance except,perhaps, as a reminder that however close the ocean may be to local ther-modynamic equilibrium, it is far, far from the state of global thermodynamicequilibrium.


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Fundamentals 3

13. Macroscopic Averages of the Equations of Motion

We have finally completed our derivation of the equations governing the motionof seawater. Subject to the approximations laid out in the preceding sections,these equations are:

Equations (13.1) apply to fields p, v, p, T, and S in which only the molecularfluctuations have been removed by averaging. However, we often wish to con-sider fields that represent a much more drastic averaging. Consider, for example,the typical wall map of ocean currents, in which the arrows represent the velocityv averaged over a very long time. This time-averaged v is the field of interest tooceanographers studying the general circulation, but it is the unaveraged v thatobeys (13.1). We can derive an equation for the time-averaged flow by time-averaging (13.1), but the resulting equations are—like the equations previouslyobtained by averaging over molecular motions—mathematically unclosed. And,unlike the previous case, there is no really justifiable way of closing them. Thisclosure problem is the central problem of oceanography and of fluid mechanics ingeneral.

To focus the discussion, we consider not the general equations (13.1), but thesimpler Navier-Stokes equations for constant-density flow,

In (13.2), the subscripts stand for vector components, repeated subscripts aresummed from 1 to 3, and the constant density has been absorbed into thepressure. Again, equations (13.2) hold only if the velocity and pressure aredefined, as before, as the averages of molecular quantities over sampling volumesbetween A3 and L3. That is, v and p include all the space and time scales of themacroscopic flow.

Let (F(x,f)) denote the macroscopic average of an arbitrary flow variableF(x,t). The commonly used types of averages include spatial averages,


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time averages,

and ensemble averages,

where P ( F ; x , t ) is the probability distribution of F(x,t) in an ensemble of flowrealizations. Although it is the space- or time-average that is usually computed,it is always much easier to think about the ensemble average, because onlyensemble averages have all of the following convenient mathematical properties:

The ensemble average of (13.2) is

where

is the Reynolds flux of i-direction average momentum in the j-direction. Thedifficulty with (13.6) is that the averaging has created new dependent variables,Rij, without adding any new equations. Of course, we can derive an equation fordRij/dt directly from (13.2), but this equation contains new variables like(Vi 'Vj 'v k ' ) , and so on. The tensor —Rij is sometimes called Reynolds stress.

The unclosed Reynolds equations (13.6-7) are closely analogous to the equa-tions previously obtained by averaging only over molecular motions. In the caseof constant density (and neglecting the momentum flux arising from intcrmolecu-lar forces), the latter could be written

Kinetic-theory assumptions led us to replace


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Fundamentals 33

Analogous arguments, called mixing-length theory, are frequently invoked toreplace

where

and ve is the coefficient of eddy viscosity. Then, if ve is approximately constant,(13.6) becomes

where dp has been absorbed into (p), and v into ve,. (Since 8p is much smaller thanp, and v is usually much smaller than ve, it is also logical to say that dp and vhavesimply been neglected.) Equations (13.12) are formally identical to (13.2), exceptthat the dependent variables now represent macroscopic averages, and themolecular-viscosity coefficient v has been replaced by the eddy-viscosity coeffi-cient vc.

Mixing-length theory resembles our simplified kinetic theory but with themolecules replaced by turbulent eddies that represent the fluctuating velocityfield v'. According to mixing-length theory, these eddies move an average dis-tance of Ae, the mixing length, before exchanging momentum with one another.This mixing of momentum between eddies is the macroscopic analogue of mol-ecular collisions, and thus Ae, is analogous to the mean free path A. If Se, is a speedcharacteristic of the eddies, then arguments analogous to those leading up to(11.15) predict that ve is of size Aese. However, none of the three assumptions(page 24) about molecular collisions really applies to the interactions betweeneddies. In particular, there is typically no scale separation between the macro-scopically averaged velocity (v) and the residual v'. Thus, although it may beconvenient and sometimes even accurate to replace the Reynolds stress by aneddy viscosity of type (13.10), the true justification for this step, if it exists, isevidently much more complex than mixing-length theory.

The problem of closure, so successfully addressed by kinetic theory in the caseof molecular averages, is the central problem of fluid mechanics. We can avoid itonly by considering completely unaveraged macroscopic flow fields that includethe full spectrum of space- and time-scales. In most oceanographic applications,this is either impossible or impractically difficult. To make progress, we adopteddy-viscosity-type closures. This amounts to regarding (13.1) as the equationsfor macroscopically averaged fields, with eddy coefficients v, KT, and KS manytimes larger than their molecular counterparts.


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The magnitude of these eddy coefficients depends on the precise definition ofthe average and on the particular flow under consideration. To appreciate this,consider the eddy diffusion coefficient in the equation for the average salinity (S),namely,

We can regard (13.13) as the definition of KS. (This definition makes sense only ifthe statistics are isotropic; otherwise, KS must be a tensor.) The definition (13.13)renders the equation for (S) exact, but that equation is useful only if we canindependently determine the value of Ks, which generally depends on locationand time. One way to determine KS is to measure the terms on the right-handside of (13.13), and then hope that KS is sufficiently uniform, in space and time,that the measured value will be useful in other situations for which there are nomeasurements.

The more inclusive the average, the larger the residuals v' and S', and thelarger the expected value of KS. However, even for a fixed type of average, KS alsodepends on the statistics of v and S, that is, on the particular flow under consid-eration. In this respect, it is completely unlike the molecular diffusion coefficient,which, according to thermodynamics, depends only on the local values of thethermodynamic state variables S, T, and p.

The dependence of the eddy diffusion coefficent on the statistics of the scalarbeing diffused (even when that scalar is, unlike salinity, dynamically passive) hasbeen insufficiently emphasized. There is no fundamental reason why the diffu-sion coefficient obtained, through (13.13), for a blob of scalar of characteristicsize, say, 10km, should be the same as that obtained for a blob of size 1000km,even if the velocity field and the definition of the average are the same.

14. Stirring and Mixing

So far, we have managed only to derive the equations of motion. Now, finally,we begin to study their solutions.10 Consider the general advection-diffusionequation,

with constant diffusion coefficient K. Several of the equations (13.1) fit this form.(In fact, if it weren't for the pressure-gradient term in (13.1b), even the velocitycomponents would obey equations like (14.1)). Hence, our conclusions about(14.1) will have general significance. It doesn't even matter whether 8 is anaverage over molecular fluctuations only (in which case k is the molecular diffu-sion coefficient) or a macroscopic average of the type considered in the previoussection (in which case K iS an eddy diffusion coefficient). Our results apply to bothcases.


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Fundamentals 35

In analyzing (14.1), we shall however assume that the velocity field V(x,y,z , t) isalready known, either from measurements or from the solution of other equa-tions. If the latter is the case, then 9 must be considered passive or we would haveneeded to know 9(x,y,z,t) to determine v. If v(x,y,z,t) is given, then (14.1) is alinear equation for 9 (x ,y , z , t ) , but the solutions of (14.1) are still surprisinglycomplicated for all but the simplest choices of V(x,y,z,i). However, our aim is tostate genera], qualitative properties of these solutions and to support our state-ments with a simple but explicit example.

First, consider a two-dimensional flow enclosed by rigid boundaries. For givenv(x,y,t), the scalar field Q(x,y,i) is determined by (14.1), the initial condition

and appropriate boundary conditions. The quantity

is a convenient measure of the spatial variability in 9 at time t. By directcalculation,

where n is the outward normal at the boundary of the fluid. If there is no flux of9 through the boundary, then V0-n is zero there, and, using (14.1), (14.4) reducesto

The last term in (14.5), which represents the effect of mixing, is always negative;mixing always reduces the variability C. The first term on the right-hand side of(14.5), which represents stirring, can have either sign. In fact, a sudden reversal ofthe velocity would change the sign of this term. However, this stirring term isusually positive, for the same reason that it is easier to stir things up than it is tounstir them. That is, stirring directly increases the variability C, on average.However, the eventual, indirect effect of stirring is to decrease the variability,because the increase in C directly caused by stirring makes the mixing moreefficient. These ideas are summarized in figure 1.5. Next, we consider a simplesolution of (14.1) that exhibits these behaviors explicitly.

Consider a spatially unbounded two-dimensional flow in which the velocityfield is a simple shear,

Equation (14.1) becomes


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Figure 1.5 The evolution of C(t), ameasure of scalar variability, when

stirring or mixing occurs alone, andwhen stirring and mixing are

simultaneously present.

Let (x0,y0) be the initial coordinates of the fluid particle located at (x,y) at time t.Then

arc the transformation equations from Eulerian coordinates (x,y,t) to Lagrangiancoordinates (x0,y0,t0). Thus,

and (14.7) transforms to

with initial condition

If K= 0, then the solution of (14.10-11) is

From (14.10), we see that the transformation to Lagrangian coordinates hassimplified the advection terms but made the diffusion terms much more compli-cated. However, if, as we assume next, the velocity u(y) depends linearly on y,


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Fundamentals 37

then (14.10) is an equation whose coefficients are independent of location, andwe can use a spatial Fourier transform to solve it. Suppose that

where a is a constant, so that

Then (14.10) takes the form

in which all the coefficients are independent of (x0,y0). The initial value problem(14.11, 14.15) can therefore be solved by Fourier analysis and superposition. Itsuffices to consider solutions of the form

where k and l are constants. The resulting ordinary differential equation

for A(t0) can be directly integrated. For simplicity, suppose / = 0, that is, that 9 isinitially independent of y. The resulting solution,

(where A0 is an arbitrary constant), is a sinusoid with x-direction wave number kand y-direction wave number

The amplitude of (14.18) is controlled by the final exp-factor.In the limit w —» 0, we obtain the solution corresponding to a steady shear,

u = ay, namely,


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If a - 0 (no shear), then the amplitude of (14.20) decreases like exp{-Kk2t}. Ifa ^ 0, then the initial decrease is similar, but at

This faster decrease is caused by the predominance of the y-direction wavenumber as the shear produces the pattern shown in figure 1.6.

If w=/ 0, then the shear reverses periodically, and, as anticipated by our generaldiscussion, the mixing is much less enhanced. For large t, the amplitude decreasesonly like

where

is the average, squared y-direction wave number.By (14.18), the variability C is proportional to

Figure 1.6 The evolution of isolines of scalar 9, in a simple shearing flow with velocity asshown on the left. The x-direction separation between isolines is constant, but the y-direction separation decreases with time.

the amplitude begins to decrease at a faster rate corresponding to


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Thus, as (w—> 0,

For small enough K, the stirring (a) causes an algebraic increase in C at small t,but the exponential decrease caused by mixing (K) eventually dominates. Be-cause of the t3-term in (14.26), this exponential decrease is ultimately faster withstirring than it would be without it. Thus (14.26) captures all of the behaviorshown in figure 1.5.

is called the hydrostatic relation.Equations (15.2-3) are two equations in the four dependent variables p(z),

S(z), T(z), and p(z)', thus there are many possible solutions. We could, forexample, choose S(z) and T(z) to be anything we please; then p(z) and p(z) aredetermined by (15.2-3). However, we shall see that only a subset of these solu-tions is stable with respect to tiny disturbances of the fluid.

Let p(z) be a density field satisfying (15.2-3), and suppose that a particle offluid is suddenly displaced a small vertical distance 8z from its position at z = 0 inthe state of rest. If the displacement occurs without exchange of heat or salt, thenthe change in particle density is

Fundamentals 3V

15. Static Stability

The simplest solutions of the equations of motion are states of rest. Suppose thatv = 0, and that all the other variables do not change with time. Taking 0 = gzand neglecting the diffusion of heat and salt, we reduce the equations of motion(13.1) to

and

The vertical component of (15.1), namely,

where po is the density of the fluid at z = 0. The displaced particle experiences abuoyancy force per unit volume of11


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where

Thus the fluid is stable if N2 > 0 and unstable if N2 < 0. If N2 > 0, an upwardlydisplaced water particle has a density greater than that of its surroundings, and ittends to sink. If N2 < 0, the converse is true. Note that the compressibility of thefluid—the g/c2-term in (15.8)—is always a destabilizing factor.

16. Potential Density and Potential Temperature

The term p0g/c2 in (15.6) is the rate at which density changes with depth solely asa result of the increasing pressure—the so-called adiabatic lapse rate. As we haveshown, stability depends on the difference between the actual lapse rate, -dp/dz,and the adiabatic lapse rate. As our calculation suggests, it is frequently conven-ient to remove the pressure effect on density.

Roughly speaking, potential density is density with the pressure-dependenceremoved. More precisely, the potential density peof a seawater particle is thedensity that the particle would have if moved adiabatically (that is, withoutexchanging heat or salt) to a prescribed reference pressure p0. Unless otherwisespecified, po is the average pressure at sea level. The potential temperature 9 issimilarly defined.

Neglecting terms of order 5z2, we obtain the equation of motion

for the displaced particle. More succinctly,

N is the Vaisala frequency. The solution to (15.7) is


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Fundamentals 41

Because the potential temperature and potential density are defined by hypo-thetical adiabatic displacements, they are closely related to the entropy. Regard-ing entropy as a function of salinity, temperature, and pressure,

we have, by definition,

Thus, for a given reference pressure po, the potential temperature depends onlyon the entropy and salinity. Since a particle moving adiabatically keeps the samevalue of 77 and S, it also conserves its potential temperature (no matter what thereference pressure p0). For a particle of fixed salinity,

If the equation of state is given in the form

then the potential density is, by definition,

Thus 77, S, 9, and p0 are all conserved on a particle moving adiabatically. For aseawater particle of fixed salinity,

where

As the calculation in section 15 suggests, the potential density gradient is moresimply related to the Vaisala frequency than is the density gradient itself. Again,the critical factor determining static stability is the difference between the densityof the adiabatically displaced fluid particle and the density of its surroundings.Suppose that the reference level for potential density is the level to which theparticle is displaced (figure 1.7). Its density there is, by definition, the potentialdensity at the level it originated, that is, pe(z). On the other hand, the ambientdensity at z + 8z is pa (z + &z). Thus the fluid is stable if

We find that


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Figure 1.7 A seawater particle is displaced from its initial elevation z to the reference levelfor the definition of potential density at z + &z.

However. (16.8-9) arc true only at the reference pressure,p0. Far from p0, (16.8-9) can be grossly inaccurate. For example, in the deep North Atlantic, dpg/dzactually becomes positive if p9 is defined with the reference pressure at the seasurface. We return to this point in section 17.

The potential temperature gradient d0/dz is not closely related to static stabil-ity. This is because (16.5) implies that

is a relation between thermodynamic variables, which, like all other equilibriumthermodynamic functions, is uniquely determined by the fundamental relationE = E(a,n,S). The choice of S, T, and p as the independent variables is merelyconventional. However, the true salinity, defined as the mass fraction of alldissolved solids (section 7), is very difficult to measure. Hence, since about 1978,occanographers have adopted practical salinity, a precisely defined function ofelectrical conductivity that agrees as closely as possible with true salinity. Thepractical salinity is much more easily measured, and, for most purposes, thedistinction between practical and true salinity is unimportant. The equation ofstate (17.1) is a complicated empirical formula involving practical salinity, tem-perature, and pressure. The most recent version was adopted in 1980.12

The best way to get a feel for the equation of state is to look at a temperature-salinity (7-5) diagram (figure 1.8), which has temperature and salinity axes andcontour lines of constant

42 Lectures on Geophysical F l u i d Dynamics

and the final term in (16.10) is often very important. Thus no particular signifi-cance attaches to the sign of dd/dz.

17. The Equation of State for Seawater

The equation of state


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Fundamentals 43

Figure 1.8 A temperature-salinity diagram, with lines of constant a. The pressure is thepressure at sea level.

at fixed pressure. Since p varies from about 1.02 to 1.07 gmcm-3 in the ocean, ovaries from about 20 to 70. There is a different T-S diagram for every fixedpressure. If one regards the temperature axis as potential temperature, and thefixed pressure as the reference pressure p0, then the contours represent potentialdensity. It has become conventional to denote by an the sigma corresponding topotential density at a reference pressure equivalent to a depth of n kilometers.Thus

where

where 92 is the potential temperature corresponding to a reference pressure of2000 decibars.

To assess the relative importance of salinity, temperature, and pressure varia-tions on density, we write the total differential of specific volume in the form


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The open-ocean ranges of temperature, salinity, and pressure, and the corre-sponding fractional changes in specific volume (or density) are

typical rangeda/a

T25 deg.005

S6ppt.005

10,000 dbar.04

Clearly, pressure is the most important determinant of the density. However, theeffect of pressure on density is not as important as the effects of temperature andsalinity, because the most significant density differences are those between fluidparticles at about the same depth. (This, again, is the motivation for the use ofpotential density.) Furthermore, the effects of salinity and temperature partlycancel, because the warmest waters are also usually the saltiest. Thus the densityvariation caused by temperature and salinity is nearer .3% than 1%. At thelargest scales of motion, the effect of temperature on density is usually moreimportant than that of salinity.

The equation of state for seawatcr has two interesting peculiarities. The first isthe concave-downward shape of isopycnals on the T-S diagram. Consider twoparticles of seawater (A and B) with the same density, but suppose that B has thehigher salinity (figure 1.9). A mixture of equal masses of A and B lies (approxi-mately) at the center of a straight line joining A and B on the T-S diagram. Thatis, the final salinity and temperature are approximately

However, because of the isopycnal curvature, the mixture at C is denser than thatat both A and B. This phenomenon (called cabbeling) may contribute to the

Figure 1.9 Two seawater particles (A and B)with the same density form a mixture (C)

with a greater density.

p


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Fundamentals 45

Figure 1 . 1 0 As pressure increases, lines of constant density rotate clockwise on the T-Sdiagram.

formation of deep water.13 Of course, equations (17.6) are not precisely correct.As shown in section 12, the correct equations are

where h is the specific enthalpy.14

The second peculiarity of the equation of state is the clockwise turning ofisopycnal lines on the T-S diagram as the pressure increases (figure 1.10). In otherwords, salty water is less compressible than freshwater. Thus, A in figure 1.10 canbe denser than B at sea level, but less dense than B at great depth. Something likethis actually occurs: North Atlantic Deep Water is denser than Antarctic BottomWater at pressures near sea level, but the converse is true when they finally meetat a depth of about 4km.15

18. Statistical Mechanics: The Ideal Gas

As previously emphasized, the equilibrium thermodynamics of a particular sub-stance is completely determined by its fundamental relation E = E(n, X1,. . . ,XN), where the Xi, are the macroscopic variables (besides the specific entropy 77)that determine the state of the substance. For seawater, N = 2, Xl = a, and X2 =S. From the standpoint of thermodynamics, the fundamental relation must begiven, that is, determined by laboratory measurements.

Statistical mechanics offers a means of calculating the fundamental relation ofthermodynamics, using Boltzmann's famous formula,

thereby completing the theoretical framework. In (18.1), k is Boltzmann's con-stant, and Q is the number of distinct microscopic states of the system thatcorrespond to the macroscopic values E,X1, . . . ,XN. For a gram of seawater, onewould need to count the number of microstates corresponding to energy E,


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volume a, and salinity S. To determine the fundamental relation, we would thensolve (18.1) for E.

Statistical mechanics offers precise rules for counting the number of states.However, the counting rules, though simple in principle, are quite difficult toapply to all but the simplest physical systems (which certainly excludesseawater!). Moreover, even when the counting rules can be applied, they oftenlead to results that are ambiguous or paradoxical, unless the true quantum natureof the microscopic states is taken into account. However, there is one type ofsystem—the ideal gas—in which the counting rules are both easy to apply andrelatively free from ambiguity.

Even in the case of an ideal gas, the general counting rule of statisticalmechanics requires a rather lengthy explanation. We defer that until chapter5, where we shall have further use for statistical mechanics. Here we proceedby means of a shortcut, the so-called equipartition theorem, which assertsthat, in thermodynamic equilibrium, each microscopic degree of freedomhas the average energy 1/2kT. Unlike the general counting rule of statisticalmechanics, the equipartition theorem applies only to systems in whichthe energy is a quadratic function; fortunately, the ideal gas is such asystem.

Consider a system of ideal gas in which (for simplicity) all the molecules areidentical; the results generalize easily. By the equipartition theorem, the totalenergy of the system is

where d is the number of degrees of freedom per molecule and NA isAvogadro's number. For a monatomic gas, d = 3, corresponding to location inthree-dimensional space. For a diatomic gas behaving like a rigid dumbell,d = 5, with the additional two degrees of freedom necessary to specify theorientation of the molecule. From (18.2), it follows that the specific internalenergy is

where m is the molecular weight, the numoer of grams per mole. Let

be the universal gas constant, and let

be the gas constant for the specific gas under consideration. Thenbecomes


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Fundamentals 47

Equation (18.6) does not, by itself, determine the fundamental relation. How-ever, we also know that, in an ideal gas, the pressure represents the momentumtransferred by collisions. Thus

Equations (18.6) and (18.10) are frequently taken as the definition of an idealgas. It can be shown, using (18.6,10) and recalling the definitions p = -dE/da andT = dE/dn, that

and (18.6) becomes

where const is a constant of integration. Thus (18.6,10) determine the fundamen-tal relation to within a constant factor. This undetermined factor is physicallyirrelevant (for the same reason that the entropy in classical thermodynamics isarbitrary to within a constant).

The choice of d, the number of degrees of freedom associated with a singlemolecule, is somewhat ambiguous. In a diatomic gas, for example, d = 5 if the twoatoms are rigidly connected, point masses; d = 6 if the atoms have a finite radiusand one takes the rotation about the axis of symmetry into account; and d = 7 ifthe distance between atoms can change. If the atoms also have internal degreesof freedom, then d must be further increased.

The ambiguity in d affects all the formulae in which d appears (in particular,(18.6) but not the equation of state (18.9) or (18.10)). However, since we caneliminate d in favor of the heat capacity, we can also say that the ambiguity affectsonly the predicted values of heat capacity. At constant volume, the change inenergy equals the heat added to the system, 8E = SQ; so, by (18.6), the heatcapacity C = SQ/ST is

where umol is the molecular velocity in, say, the x-direction. By the equipartitiontheorem,

Thus,

where n is the number of moles per unit volume, or,


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Boltzmann, noting that (18.17) with d = 3 and d = 5 yields predictions that agreeclosely with the measurements of monatomic and diatomic gases (respectively) atmoderate temperatures, argued that this proved that other degrees of freedomwere not excited by the collisions between molecules. Maxwell believed that thetheory was seriously flawed. Neither man lived to see quantum mechanics, whichresolves the paradox. For air (a predominantly diatomic gas), the measured ratioof heat capacities is so close to (18.17) with d = 5 that many textbooks simply statethe theoretical result.

To a very good approximation, the Earth's atmosphere behaves like an idealgas (except at very high altitudes, where the collisions between molecules are tooinfrequent to maintain statistical equilibrium).

Notes

1. Alternatively, one could regard (a,b,c) as the values of (x,y,z) at the initial time.Then the initial density p0(a,b,c) appears on the right-hand side of (2.4).

2. Readers with no previous experience with Hamilton's method should study the firstfew chapters of Goldstein (1980) or Lanczos (1970).

3. Throughout chapter 1, I use the postulational basis of thermodynamics developedby Gibbs (1878), Tisza (1966), and Callen (1985). This approach, with its emphasis onenergy, nicely parallels Hamilton's approach to mechanics. It is equivalent to, but muchhandier than, the thermodynamics of Clausius and Kelvin presented in most thermody-namics texts.

4. Section 10 is partly based on an unpublished manuscript by Carl Eckart (ca. 1963).5. For a thorough introduction to kinetic theory, see, for example, Balescu (1975).6. For a self-contained derivation of (11.1), see Owczarek (1968, pp. 274-285) .7. Once again, Balescu (1975) is a good source for details.8. For a complete development of thermodynamics based on these four postulates, see

Callen (1985).9. Actually, we know even more. Using a general argument based on the time-

reversibility of the microscopic dynamics, Onsager (1931a,b) showed that KTll = K U T .10. Section 14 is based on the article by Young, Rhines, and Garrett (1982).11. This is Archimedes' principle. The reader should be able to prove it from the

momentum equation for a hydrostatic fluid.


At constant pressure,

which by (18.13) and (18.10) is

The heat capacity at constant pressure is thus

and


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Fundamentals 49

12. For a more complete discussion of practical salinity, the equation of slate, andother thermodynamic functions, see Fofonoff (1985).

13. See Foster (1972) and McDougall (1987).14. See Fofonoff (1962) for an estimate of the error in (17.6).15. See Lynn and Reid (1968).


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