C H A P T E R

6The equations of fluid motion

6.1. Differentiation following the motion6.2. Equation of motion for a nonrotating fluid

6.2.1. Forces on a fluid parcel6.2.2. The equations of motion6.2.3. Hydrostatic balance

6.3. Conservation of mass6.3.1. Incompressible flow6.3.2. Compressible flow

6.4. Thermodynamic equation6.5. Integration, boundary conditions, and restrictions in application6.6. Equations of motion for a rotating fluid

6.6.1. GFD Lab III: Radial inflow6.6.2. Transformation into rotating coordinates6.6.3. The rotating equations of motion6.6.4. GFD Labs IV and V: Experiments with Coriolis forces on a parabolic rotating table6.6.5. Putting things on the sphere6.6.6. GFD Lab VI: An experiment on the Earth’s rotation

6.7. Further reading6.8. Problems

To proceed further with our discussion ofthe circulation of the atmosphere, and laterthe ocean, we must develop some of theunderlying theory governing the motion of afluid on the spinning Earth. A differentiallyheated, stratified fluid on a rotating planetcannot move in arbitrary paths. Indeed,there are strong constraints on its motionimparted by the angular momentum of thespinning Earth. These constraints are pro-foundly important in shaping the pattern ofatmosphere and ocean circulation and their

ability to transport properties around theglobe. The laws governing the evolutionof both fluids are the same and so ourtheoretical discussion will not be specific toeither atmosphere or ocean, but can and willbe applied to both. Because the propertiesof rotating fluids are often counterintuitiveand sometimes difficult to grasp, along-side our theoretical development we willdescribe and carry out laboratory experi-ments with a tank of water on a rotatingtable (Fig. 6.1). Many of the laboratory

81

82 6. THE EQUATIONS OF FLUID MOTION

FIGURE 6.1. Throughout our text, running in parallel with a theoretical development of the subject, we studythe constraints on a differentially heated, stratified fluid on a rotating planet (left), by using laboratory analoguesto illustrate the fundamental processes at work (right). A complete list of the laboratory experiments can be foundin Appendix A.4.

experiments we use are simplified versionsof ‘‘classics’’ of geophysical fluid dynamics.They are listed in Appendix A.4. Further-more we have chosen relatively simpleexperiments that, in the main, do not requiresophisticated apparatus. We encourage youto ‘‘have a go’’ or view the attendant movieloops that record the experiments carriedout in preparation of our text.

We now begin a more formal devel-opment of the equations that govern theevolution of a fluid. A brief summary ofthe associated mathematical concepts, def-initions, and notation we employ can befound in Appendix A.2.

6.1. DIFFERENTIATIONFOLLOWING THE MOTION

When we apply the laws of motion andthermodynamics to a fluid to derive theequations that govern its motion, we mustremember that these laws apply to materialelements of fluid that are usually mobile.We must learn, therefore, how to express

the rate of change of a property of a fluidelement, following that element as it movesalong, rather than at a fixed point in space.It is useful to consider the following simpleexample.

Consider again the situation sketched inFig. 4.13 in which a wind blows over ahill. The hill produces a pattern of wavesin its lee. If the air is sufficiently saturatedin water vapor, the vapor often condensesout to form a cloud at the ‘‘ridges’’ of thewaves as described in Section 4.4 and seenin Figs. 4.14 and 4.15.

Let us suppose that a steady state is set upso the pattern of cloud does not change intime. If C = C(x, y, z, t) is the cloud amount,where (x, y) are horizontal coordinates, z isthe vertical coordinate, and t is time, then

(∂C∂t

)fixed point

in space

= 0,

in which we keep at a fixed point in space,but at which, because the air is moving, thereare constantly changing fluid parcels. The

6.1. DIFFERENTIATION FOLLOWING THE MOTION 83

derivative(∂∂t

)fixed point

is called the Eulerian

derivative after Euler.1

But C is not constant following along a par-ticular parcel; as the parcel moves upwardsinto the ridges of the wave, it cools, watercondenses out, a cloud forms, and so Cincreases (recall GFD Lab 1, Section 1.3.3);as the parcel moves down into the troughsit warms, the water goes back in to thegaseous phase, the cloud disappears and Cdecreases. Thus

(∂C∂t

)fixed

particle

�= 0,

even though the wave-pattern is fixed inspace and constant in time.

So, how do we mathematically express‘‘differentiation following the motion’’? Tofollow particles in a continuum, a specialtype of differentiation is required. Arbitrar-ily small variations of C(x, y, z, t), a functionof position and time, are given to the firstorder by

δC =∂C∂t

δt +∂C∂x

δx +∂C∂y

δy +∂C∂z

δz,

where the partial derivatives ∂/∂t etc. areunderstood to imply that the other variablesare kept fixed during the differentiation. Thefluid velocity is the rate of change of positionof the fluid element, following that elementalong. The variation of a property C followingan element of fluid is thus derived by settingδx = uδt, δy = vδt, δz = wδt, where u is thespeed in the x-direction, v is the speed in

the y-direction, and w is the speed in thez-direction, thus

(δC)fixed

particle

=

(∂C∂t

+ u∂C∂x

+ v∂C∂y

+ w∂C∂z

)δt,

where (u, v, w) is the velocity of the materialelement, which by definition is the fluidvelocity. Dividing by δt and in the limit ofsmall variations we see that

(∂C∂t

)fixed

particle

=∂C∂t

+ u∂C∂x

+ v∂C∂y

+ w∂C∂z

=DCDt

,

in which we use the symbol DDt to identify

the rate of change following the motion

DDt

≡ ∂

∂t+ u

∂

∂x+ v

∂

∂y+ w

∂

∂z≡ ∂

∂t+ u.∇.

(6-1)

Here u = (u, v, w) is the velocity vector, and

∇ ≡(∂∂x

, ∂∂y

, ∂∂z

)is the gradient operator.

D/Dt is called the Lagrangian derivative(after Lagrange; 1736–1813) (it is also calledvariously the substantial, the total, or thematerial derivative). Its physical meaning istime rate of change of some characteristic of aparticular element of fluid (which in general ischanging its position). By contrast, as intro-duced above, the Eulerian derivative ∂/∂texpresses the rate of change of some char-acteristic at a fixed point in space (but withconstantly changing fluid element becausethe fluid is moving).

1 Leonhard Euler (1707–1783). Euler made vast contributions to mathematics in the areas ofanalytic geometry, trigonometry, calculus and number theory. He also studied continuummechanics, lunar theory, elasticity, acoustics, the wave theory of light, and hydraulics, andlaid the foundation of analytical mechanics. In the 1750s Euler published a number of majorworks setting up the main formulas of fluid mechanics, the continuity equation, and theEuler equations for the motion of an inviscid, incompressible fluid.

84 6. THE EQUATIONS OF FLUID MOTION

Some writers use the symbol d/dt forthe Lagrangian derivative, but this is bet-ter reserved for the ordinary derivative ofa function of one variable, the sense it isusually used in mathematics. Thus for exam-ple the rate of change of the radius of a raindrop would be written dr/dt, with the iden-tity of the drop understood to be fixed. In thesame context D/Dt could refer to the motionof individual particles of water circulatingwithin the drop itself. Another example isthe vertical velocity, defined as w = Dz/Dt;if one sits in an air parcel and follows itaround, w is the rate at which one’s heightchanges.2

The term u.∇ in Eq. 6-1 represents advec-tion and is the mathematical representationof the ability of a fluid to carry its proper-ties with it as it moves. For example, theeffects of advection are evident to us everyday. In the northern hemisphere, southerlywinds (from the south) tend to be warm andmoist because the air carries with it prop-erties typical of tropical latitudes; northerly

winds tend to be cold and dry because theyadvect properties typical of polar latitudes.

We will now use the Lagrangian deriva-tive to help us apply the laws of mechanicsand thermodynamics to a fluid.

6.2. EQUATION OF MOTION FOR ANONROTATING FLUID

The state of the atmosphere or ocean atany time is defined by five key variables:

u = (u, v, w); p and T,

(six if we include specific humidity in theatmosphere, or salinity in the ocean). Notethat by using the equation of state, Eq. 1-1,we can infer ρ from p and T. To ‘‘tie’’ thesevariables down we need five independentequations. They are:

1. the laws of motion applied to a fluidparcel, yielding three independent

1x 2 dx, y 2 dy, z 1 dz212

12

12

1x 2 dx, y 1 dy, z 2 dz212

12

12

1x 2 dx, y 2 dy, z 2 dz212

12

12 1x 1 dx, y 2 dy, z 2 dz21

212

12

FIGURE 6.2. An elementary fluid parcel, conveniently chosen to be a cube of sides δx, δy, δz, centered on (x, y, z).The parcel is moving with velocity u.

2Meteorologists like working in pressure coordinates in which p is used as a vertical coordinate rather than z. In thiscoordinate an equivalent definition of ‘‘vertical velocity’’ is:

ω =DpDt

,

the rate at which pressure changes as the air parcel moves around. Since pressure varies much more quickly inthe vertical than in the horizontal, this is still, for all practical purposes, a measure of vertical velocity, but expressed inunits of hPa s−1. Note also that upward motion has negative ω.

6.2. EQUATION OF MOTION FOR A NONROTATING FLUID 85

equations in each of the threeorthogonal directions

2. conservation of mass

3. the law of thermodynamics, a statementof the thermodynamic state in which themotion takes place.

These equations, five in all, together withappropriate boundary conditions, are suffi-cient to determine the evolution of the fluid.

6.2.1. Forces on a fluid parcel

We will now consider the forces onan elementary fluid parcel, of infinitesimaldimensions (δx, δy, δz) in the three coor-dinate directions, centered on (x, y, z) (seeFig. 6.2).

Since the mass of the parcel is δM =ρ δx δy δz, then, when subjected to a netforce F, Newton’s Law of Motion for theparcel is

ρ δx δy δzDu

Dt= F, (6-2)

where u is the parcel’s velocity. As discussedearlier we must apply Eq. 6-2 to the samematerial mass of fluid, which means wemust follow the same parcel around. There-fore, the time derivative in Eq. 6-2 is the totalderivative, defined in Eq. 6-1, which in thiscase is

Du

Dt=

∂u

∂t+ u

∂u

∂x+ v

∂u

∂y+ w

∂u

∂z

=∂u

∂t+ (u · ∇) u.

Gravity

The effect of gravity acting on the parcel inFig. 6.2 is straightforward: the gravitationalforce is g δM, and is directed downward,

Fgravity = −gρz δx δy δz, (6-3)

where z is the unit vector in the upwarddirection and g is assumed constant.

Pressure gradient

Another force acting on a fluid par-cel is the pressure force within the fluid.Consider Fig. 6.3. On each face of our par-cel there is a force (directed inward) actingon the parcel equal to the pressure on thatface multiplied by the area of the face. Onface A, for example, the force is

F(A) = p(x − δx2

, y, z) δy δz,

directed in the positive x-direction. Notethat we have used the value of p at themidpoint of the face, which is valid for smallδy, δz. On face B, there is an x-directed force

1x 2 dx, y 2 dy, z 1 dz212

12

12

1x 2 dx, y 1 dy, z 2 dz212

12

12

1x 2 dx, y 2 dy, z 2 dz212

12

12 1x 1 dx, y 2 dy, z 2 dz21

212

12

FIGURE 6.3. Pressure gradient forces acting on the fluid parcel. The pressure of the surrounding fluid applies aforce to the right on face A and to the left on face B.

86 6. THE EQUATIONS OF FLUID MOTION

F(B) = −p(x +δx2

, y, z) δy δz,

which is negative (toward the left). Sincethese are the only pressure forces acting inthe x-direction, the net x-component of thepressure force is

Fx =

[p(x − δx

2, y, z) − p(x +

δx2

, y, z)]

δy δz.

If we perform a Taylor expansion (seeAppendix A.2.1) about the midpoint of theparcel, we have

p(x +δx2

, y, z) = p(x, y, z) +δx2

(∂p∂x

),

p(x − δx2

, y, z) = p(x, y, z) − δx2

(∂p∂x

),

where the pressure gradient is evaluated atthe midpoint of the parcel, and where wehave neglected the small terms of O(δx2)and higher. Therefore the x-component ofthe pressure force is

Fx = −∂p∂x

δx δy δz.

It is straightforward to apply the sameprocedure to the faces perpendicular tothe y- and z-directions, to show that thesecomponents are

Fy = −∂p∂y

δx δy δz,

Fz = −∂p∂z

δx δy δz.

In total, therefore, the net pressure force isgiven by the vector

Fpressure =(Fx, Fy, Fz

)= −

(∂p∂x

,∂p∂y

,∂p∂z

)δx δy δz

= −∇p δx δy δz. (6-4)

Note that the net force depends only on thegradient of pressure, ∇p; clearly, a uniformpressure applied to all faces of the parcelwould not introduce any net force.

Friction

For typical atmospheric and oceanicflows, frictional effects are negligible exceptclose to boundaries where the fluid rubsover the Earth’s surface. The atmosphericboundary layer—which is typically a fewhundred meters to 1 km or so deep—isexceedingly complicated. For one thing, thesurface is not smooth; there are mountains,trees, and other irregularities that increasethe exchange of momentum between the airand the ground. (This is the main reasonwhy frictional effects are greater over landthan over ocean.) For another, the bound-ary layer is usually turbulent, containingmany small-scale and often vigorous eddies;these eddies can act somewhat like mobilemolecules and diffuse momentum moreeffectively than molecular viscosity. Thesame can be said of oceanic boundary layers,which are subject, for example, to the stir-ring by turbulence generated by the action ofthe wind, as will be discussed in Section 10.1.At this stage, we will not attempt to describesuch effects quantitatively but instead writethe consequent frictional force on a fluidparcel as

Ffric = ρ F δx δy δz, (6-5)

where, for convenience, F is the frictionalforce per unit mass. For the moment we willnot need a detailed theory of this term.Explicit forms for F will be discussed andemployed in Sections 7.4.2 and 10.1.

6.2.2. The equations of motion

Putting all this together, Eq. 6-2 gives us

ρ δx δy δzDu

Dt= Fgravity + Fpressure + Ffric.

Substituting from Eqs. 6-3, 6-4, and 6-5, andrearranging slightly, we obtain

Du

Dt+

1ρ∇p + gz = F . (6-6)

This is our equation of motion for a fluidparcel.

6.3. CONSERVATION OF MASS 87

Note that because of our use of vectornotation, Eq. 6-6 seems rather simple. How-ever, when written out in component form,as below, it becomes somewhat intimidat-ing, even in Cartesian coordinates:

∂u∂t

+ u∂u∂x

+ v∂u∂y

+ w∂u∂z

+1ρ

∂p∂x

= Fx (a)

∂v∂t

+ u∂v∂x

+ v∂v∂y

+ w∂v∂z

+1ρ

∂p∂y

= Fy (b)

∂w∂t

+ u∂w∂x

+ v∂w∂y

+ w∂w∂z

+1ρ

∂p∂z

+ g = Fz . (c)

(6-7)

Fortunately we will often be able to makea number of simplifications. One such sim-plification, for example, is that, as discussedin Section 3.2, large-scale flow in the atmo-sphere and ocean is almost always close tohydrostatic balance, allowing Eq. 6-7c to beradically simplified as follows.

6.2.3. Hydrostatic balance

From the vertical equation of motion,Eq. 6-7c, we can see that if friction and the

vertical acceleration Dw/Dt are negligible,we obtain

∂p∂z

= −ρg, (6-8)

thus recovering the equation of hydrostaticbalance, Eq. 3-3. For large-scale atmosphericand oceanic systems in which the verticalmotions are weak, the hydrostatic equationis almost always accurate, though it maybreak down in vigorous systems of smallerhorizontal scale such as convection.3

6.3. CONSERVATION OF MASS

In addition to Newton’s laws there isa further constraint on the fluid motion:conservation of mass. Consider a fixed fluidvolume as illustrated in Fig. 6.4. The volumehas dimensions

(δx, δy, δz

). The mass of the

fluid occupying this volume, ρ δx δy δz, maychange with time if ρ does so. However,mass continuity tells us that this can only

1x 2 dx, y 2 dy, z 1 dz212

12

12

1x 2 dx, y 1 dy, z 2 dz212

12

12

1x 2 dx, y 2 dy, z 2 dz212

12

12 1x 1 dx, y 2 dy, z 2 dz21

212

12

FIGURE 6.4. The mass of fluid contained in the fixed volume, ρδx δy δz, can be changed by fluxes of mass outof and into the volume, as marked by the arrows.

3It might appear from Eq. 6-7c that |Dw/Dt| << g is a sufficient condition for the neglect of the acceleration term. Thisindeed is almost always satisfied. However, for hydrostatic balance to hold to sufficient accuracy to be useful, thecondition is actually |Dw/Dt| << g�ρ/ρ, where �ρ is a typical density variation on a pressure surface. Even in quiteextreme conditions this more restrictive condition turns out to be very well satisfied.

88 6. THE EQUATIONS OF FLUID MOTION

occur if there is a flux of mass into (or outof) the volume, meaning that

∂

∂t(ρ δx δy δz

)=

∂ρ

∂tδx δy δz

=(net mass flux into the volume

).

Now the volume flux in the x-directionper unit time into the left face in Fig. 6.4 isu(x − 1/2 δ x, y, z

)δy δz, so the correspond-

ing mass flux is [ρu](x − 1/2 δx, y, z

)δy δz,

where [ρu] is evaluated at the left face.The flux out through the right face is[ρu]

(x + 1/2 δx, y, z

)δy δz; therefore the net

mass import in the x-direction into thevolume is (again employing a Taylorexpansion)

− ∂

∂x(ρu) δx δy δz.

Similarly the rate of net import of mass inthe y-direction is

− ∂

∂y(ρv) δx δy δz,

and in the z-direction is

− ∂

∂z(ρw) δx δy δz.

Therefore the net mass flux into the vol-ume is −∇ · (ρu) δx δy δz. Thus our equationof continuity becomes

∂ρ

∂t+ ∇ · (ρu) = 0. (6-9)

This has the general form of a physicalconservation law:

∂ Concentration∂t

+ ∇ ·(flux

)= 0

in the absence of sources and sinks.Using the total derivative D/Dt, Eq. 6-1,

and noting that ∇ · (ρu) = ρ∇ · u + u·∇ρ

(see the vector identities listed inAppendix A.2.2) we may therefore rewriteEq. 6-9 in the alternative, and often veryuseful, form:

Dρ

Dt+ ρ∇ · u = 0. (6-10)

6.3.1. Incompressible flow

For incompressible flow (e.g., for a liquidsuch as water in our laboratory tank or inthe ocean), the following simplified approxi-mate form of the continuity equation almostalways suffices:

∇ · u =∂u∂x

+∂v∂y

+∂w∂z

= 0. (6-11)

Indeed this is the definition of incompress-ible flow: it is nondivergent—no bubblesallowed! Note that in any real fluid, Eq. 6-11is never exactly obeyed. Moreover, despiteEq. 6-10, use of the incompressibility condi-tion should not be understood as implyingthat Dρ

Dt = 0. On the contrary, the densityof a parcel of water can be changed byinternal heating and/or conduction (see, forexample, Section 11.1). Although these den-sity changes may be large enough to affectthe buoyancy of the fluid parcel, they aretoo small to affect the mass budget. Forexample, the thermal expansion coefficientof water is typically 2 × 10−4 K−1, and so thevolume of a parcel of water changes by only0.02% per degree of temperature change.

6.3.2. Compressible flow

A compressible fluid, such as air, isnowhere close to being nondivergent—ρ

changes markedly as fluid parcels expandand contract. This is inconvenient in theanalysis of atmospheric dynamics. Howeverit turns out that, provided the hydrostaticassumption is valid (as it nearly always is),one can get around this inconvenience byadopting pressure coordinates. In pressurecoordinates,

(x, y, p

), the elemental fixed

‘‘volume’’ is δx δy δp. Since z = z(x, y, p

),

the vertical dimension of the elementalvolume (in geometric coordinates) is δz =∂z/∂p δp, and so its mass is δM given by

δM = ρ δx δy δz

= ρ

(∂p∂z

)−1

δx δy δp

=−1gδx δy δp,

6.5. INTEGRATION, BOUNDARY CONDITIONS, AND RESTRICTIONS IN APPLICATION 89

where we have used hydrostatic balance,Eq. 3-3. So the mass of an elementalfixed volume in pressure coordinates cannotchange! In effect, comparing the topand bottom line of the previous equa-tion, the equivalent of ‘‘density’’ inpressure coordinates—the mass per unit‘‘volume’’—is 1/g, a constant. Hence, in thepressure-coordinate version of the continu-ity equation, there is no term representingrate of change of density; it is simply

∇p · up =∂u∂x

+∂v∂y

+∂ω

∂p= 0, (6-12)

where the subscript p reminds us that we arein pressure coordinates. The greater simplic-ity of this form of the continuity equation,as compared to Eqs. 6-9 or 6-10, is one ofthe reasons why pressure coordinates arefavored in meteorology.

6.4. THERMODYNAMICEQUATION

The equation governing the evolution oftemperature can be derived from the firstlaw of thermodynamics applied to a movingparcel of fluid. Dividing Eq. 4-12 by δt andletting δt −→ 0 we find:

DQDt

= cpDTDt

− 1ρ

DpDt

. (6-13)

DQ/Dt is known as the diabatic heatingrate per unit mass. In the atmosphere, thisis mostly due to latent heating and cool-ing (from condensation and evaporationof H2O) and radiative heating and cool-ing (due to absorption and emission ofradiation). If the heating rate is zero thenDT/Dt = 1

ρcpDp/Dt, and, as discussed in

Section 4.3.1, the temperature of a par-cel will decrease in ascent (as it moves tolower pressure) and increase in descent (asit moves to higher pressure). Of course thisis why we introduced potential tempera-ture in Section 4.3.2; in adiabatic motion, θ

is conserved. Written in terms of θ, Eq. 6-13becomes

Dθ

Dt=

(pp0

)−κ ·Qcp

, (6-14)

where·

Q (with a dot over the top) is a short-hand for DQ

Dt . Here θ is given by Eq. 4-17,

the factor(

pp0

)−κconverts from T to θ,

and·

Qcp

is the diabatic heating in units

of K s−1. The analogous equations thatgovern the evolution of temperature andsalinity in the ocean will be discussed inChapter 11.

6.5. INTEGRATION, BOUNDARYCONDITIONS, AND

RESTRICTIONS IN APPLICATION

The three equations in 6–7, together with6–11 or 6–12, and 6–14 are our five equationsin five unknowns. Together with initial con-ditions and boundary conditions, they aresufficient to determine the evolution of theflow.

Before going on, we make some remarksabout restrictions in the application of ourgoverning equations. The equations them-selves apply very accurately to the detailedmotion. In practice, however, variables arealways averages over large volumes. Wecan only tentatively suppose that the equa-tions are applicable to the average motion,such as the wind integrated over a 100-kmsquare box. Indeed, the assumption that theequations do apply to average motion isoften incorrect. This fact is associated withthe representation of turbulent scales, bothsmall scale and large scale. The treatment ofturbulent motions remains one of the majorchallenges in dynamical meteorology andoceanography. Finally, our governing equa-tions have been derived relative to a ‘fixed’coordinate system. As we now go on to dis-cuss, this is not really a restriction, but isusually an inconvenience.