Vorticity and divergence are scalar quantities that 272...

272 Atmospheric Dynamics

the velocity in the direction transverse to the flow.Curvature is the rate of change of the direction of theflow in the downstream direction. Shear and curva-ture are labeled as cyclonic (anticyclonic) and areassigned a positive (negative) algebraic sign if theyare in the sense as to cause an object in the flow torotate in the same (opposite) sense as the Earth’srotation , as viewed looking down on the pole. Inother words, cyclonic means counterclockwise in thenorthern hemisphere and clockwise in the southernhemisphere. Diffluenceconfluence is the rate ofchange of the direction of the flow in the directiontransverse to the motion, defined as positive if thestreamlines are spreading apart in the downstreamdirection. Stretchingcontraction relates to the rate ofchange of the speed of the flow in the downstreamdirection, with stretching defined as positive.

7.1.2 Vorticity and Divergence

Vorticity and divergence are scalar quantities thatcan be defined not only in natural coordinates, butalso in Cartesian coordinates (x, y) and for the hori-zontal wind vector V. Vorticity is the sum of the shearand the curvature, taking into account their algebraicsigns, and divergence is the sum of the diffluence andthe stretching.

It can be shown that vorticity is given by

2 (7.1)

where is the rate of spin of an imaginary puckmoving with the flow. This relationship is illustratedin the following exercise for the special case of solidbody rotation.

Exercise 7.1 Derive an expression for the vorticitydistribution within a flow characterized by counter-clockwise solid body rotation with angular velocity (see Fig. 7.2b).

Solution: Vorticity is the sum of the shear and thecurvature. Using the definitions in Table 7.1, theshear contribution is Vn Vr, where r isradius in polar coordinates, centered on the axis ofrotation. In one circuit rotates through angle 2.Therefore, the curvature contribution Vs

V(22r) Vr. For solid body rotation with angular

Table 7.1 Definitions of properties of the horizontal flow

Vectorial Natural coords. Cartesian coords.

Shear

Curvature

Diffluence

Stretching

Vorticity k V

Divergence DivHV V

Deformation ux

vy

; vx

uy

ux

vy

V

n

Vs

x

uy

V

s

Vn

Vs

V

n

V

s

Vn

ψ

ny

P

s

x

Fig. 7.1 Natural coordinates (s, n) defined at point P in ahorizontal wind field. Curved arrows represent streamlines.

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7.1 Kinematics of the Large-Scale Horizontal Flow 273

velocity the contribution from the shear is there-fore d(r)dr and the contribution from the cur-vature is rr . The shear and curvaturecontributions are in the same sense and thereforeadditive. Hence, 2.

Divergence V (or DivHV) is related to the timerate of change of area. Consider a block of fluid of areaA, moving downstream in the flow. In Cartesian (x, y)coordinates, the Lagrangian time rate of change is

If the dimensions of the block are very small com-pared to the space scale of the velocity fluctuations,we may write

where the partial derivative x in this expressionindicates that the derivative is taken at constant y.The time rate of change of y can be expressed in ananalogous form. Substituting for the time derivativesin the expression for dAdt, we obtain

Dividing both sides by xy yields

(7.2)

where the right-hand side may be recognized as theCartesian form of the divergence in Table 7.1. Hence,divergence is the logarithmic rate of expansion of thearea enclosed by a marked set of parcels moving withthe flow. Negative divergence is referred to as con-vergence.

Some of the relationships between the variouskinematic properties of the horizontal wind field areillustrated by the idealized flows shown in Fig. 7.2.

(a) Sheared flow without curvature, diffluence,stretching, or divergence. From a northernhemisphere perspective, shear and vorticity arecyclonic in the top half of the domain andanticyclonic in the bottom half.

1A

dAdt

ux

vy

dAdt

xy ux

xy vy

ddt

(x) u ux

x

dAdt

ddt

xy y ddt

x x ddt

y

(b) Solid body rotation with cyclonic shear andcyclonic curvature (and hence, cyclonicvorticity) throughout the domain, but withoutdiffluence or stretching, and hence withoutdivergence.

(c) Radial flow with velocity directlyproportional to radius. This flow exhibitsdiffluence and stretching and hencedivergence, but no curvature or shear andhence no vorticity.

(d) Hyperbolic flow that exhibits both diffluenceand stretching, but is nondivergent because thetwo terms exactly cancel. Hyperbolic flow alsoexhibits both shear and curvature, but isirrotational (i.e., vorticity-free) because thetwo terms exactly cancel.

Vorticity is related to the line integral of the flowalong closed loops. From Stokes’ theorem, it followsthat

(7.3)

where Vs is the component of the velocity along thearc ds (defined as positive when circulating aroundthe loop in the same sense as the Earth’s rotation),and C is referred to as the circulation around theloop. The term on the right hand side can be rewrittenas where is the spatial average of the vorticitywithin the loop and A is the area of the loop. The

A,

C Vsds dA

(a) (b)

(c) (d)

Fig. 7.2 Idealized horizontal flow configurations. See textfor explanation.

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analogous relationship for the component of thevelocity Vn outward across the curve is

(7.4)

which follows from Gauss’s theorem.

Exercise 7.2 At the 300-hPa (10 km) level along40 °N during winter the zonally averaged zonal wind[u] is eastward at 20 m s1 and the zonally averagedmeridional wind component [v] is southward at30 cm s1. Estimate the vorticity and divergenceaveraged over the polar cap region poleward of 40 °N.

Solution: Based on (7.3), the vorticity averagedover the polar cap region is given by

In a similar manner, the divergence over the polarcap region is given by

7.1.3 Deformation

Deformation, defined in the bottom line of Table 7.1,is the sum of the confluence and stretching terms. Ifthe deformation is positive, grid squares orientedalong the (s, n) axes will tend to be deformed intorectangles, elongated in the s direction. Conversely,if the deformation is negative, the squares will bedeformed into rectangles elongated in the directiontransverse to the flow. In Cartesian coordinates, thedeformation tensor is made up of two components:the first relating to the stretching and squashing ofgrid squares aligned with the x and y axes, and thesecond with grid squares aligned at an angle of 45°with respect to the x and y axes. Figure 7.2(d) shows

1.01 107 s1

[v]RE

cos 40

(1 sin 40)

DivHV [v]40 N ds

40 N dA

[u]RE

cos 40

(1 sin 40) 6.74 106 s1

[u]40 N ds

40 N dA

2RE[u] cos 40

2R2E 90

40 cos d

Vnds DivH VdA

an example of a horizontal wind pattern consistingof pure deformation in the first component. Herethe x axis corresponds to the axis of dilatation (orstretching) and the y axis corresponds to the axis ofcontraction. If this wind pattern were rotated by 45°relative to the x and y axes, the deformation wouldbe manifested in the second component: rectangleswould be deformed into rhomboids.

Figure 7.3 illustrates how even a relatively simplelarge-scale motion field can distort a field of passivetracers, initially configured as a rectangular grid,into an elongated configuration in which the indi-vidual grid squares are stretched and squashedbeyond recognition. Some squares that were ini-tially far apart end up close together, and vice versa.

Deformation can sharpen preexisting horizontalgradients of temperature, moisture, and other scalar

L

A

D

E

B

C

Fig. 7.3 (Top) A grid of air parcels embedded in a steadystate horizontal wind field indicated by the arrows. Thestrength of the wind at any point is inversely proportional tothe spacing between the contours at that point. (A–E) Howthe grid is deformed by the flow as the tagged particles movedownstream; those in the upper right corner of the grid mov-ing eastward and those in the lower left corner moving south-ward and then eastward around the closed circulation. [FromTellus, 7, 141–156 (1955).]

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7.1 Kinematics of the Large-Scale Horizontal Flow 275

variables, creating features referred to as frontalzones. To illustrate how this sharpening occurs, con-sider the distribution of a hypothetical passive tracer,whose concentration (x, y) is conserved as it is car-ried around (or advected) in a divergence-free hori-zontal flow. Because ddt 0 in (1.3), it followsthat the time rate of change at a fixed point in spaceis given by the horizontal advection; that is

(7.5a)

or, in Cartesian coordinates,

(7.5b)

The time rate of change of at a fixed point (x, y) ispositive if increases in the upstream direction inwhich case V at (x, y) must have a componentdirected down the gradient of ; hence the minussigns in (7.5a,b).

Suppose that initially exhibits a uniform horizon-tal gradient, say from north to south. Such a gradientwill tend to be sharpened within regions of negativevy. In the pattern of pure deformation in Fig. 7.2d,vy is negative throughout the domain. Such a flowwill tend to sharpen any preexisting north–south tem-perature gradient, creating an east–west orientedfrontal zone, as shown in Fig. 7.4a. Frontal zones canalso be twisted and sharpened by the presence of awind pattern with shear, as shown in Fig. 7.4b.

7.1.4 Streamlines versus Trajectories

If the horizontal wind field V is changing with time,the streamlines of the instantaneous horizontal windfield considered in this section are not the same asthe horizontal trajectories of air parcels. Consider, forexample, the case of a sinusoidal wave that is propa-gating eastward with phase speed c, superimposed ona uniform westerly flow of speed U, as depicted inFig. 7.5. The solid lines represent horizontal stream-lines at time t, and the dashed lines represent thehorizontal streamlines at time t t, after the wavehas moved some distance eastward. The trajectoriesoriginate at point A, which lies in the trough of thewave at time t. If the westerly flow matches the rateof eastward propagation of the wave, the parcel willremain in the wave trough as it moves eastward, asindicated by the straight trajectory AC. If the west-

t u

x v

y

t V

(b)

(a)

Fig. 7.4 Frontal zones created by horizontal flow patternsadvecting a passive tracer with concentrations indicated bythe colored shading. In (a) the gradient is being sharpened bydeformation, whereas in (b) it is being twisted and sharpenedby shear. See text and Exercise 7.11 for further explanation.

Fig. 7.5 Streamlines and trajectories for parcels in a wavemoving eastward with phase velocity c embedded in a westerlyflow with uniform speed U. Solid black arrows denote initialstreamlines and dashed black arrows denote later stream-lines. Blue arrows denote air trajectories starting from point Afor three different values of U. AB is the trajectory for U c;AC for U c, and AD for U c.

A

B

CD

erly flow through the wave is faster than the rate ofpropagation of the wave (i.e., if U c), the air parcelwill overtake the region of southwesterly flow aheadof the trough and drift northward, as indicated by the

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Ω2RAR

A

Ω

gg*

P

Equator

Ω

Oc´

c

C

7.2 Dynamics of Horizontal Flow 277

horizontal motion V can be written in vectorialform as

(7.7)

where f, the so-called Coriolis parameter, is equal to2 sin and k is the local vertical unit vector, definedas positive upward. The sin term in f appropriatelyscales the Coriolis force to account for the fact that thelocal vertical unit vector k is parallel to the axis of rota-tion only at the poles. Accordingly, the Coriolis force

C f k V

in the horizontal equation of motion increases with lat-itude from zero on the equator to 2V at the poles,where V is the (scalar) horizontal wind speed. TheCoriolis force is directed toward the right of the hori-zontal velocity vector in the northern hemisphere andto the left of it in the southern hemisphere. On Earth

where day in this context refers to the sidereal day,3

which is 23 h 56 min in length.

2 rad day1 7.292 105 s1

3 The time interval between successive transits of a star over a meridian.

The role of the Coriolis force in a rotating coor-dinate system can be demonstrated in laboratoryexperiments. Here we describe an experiment thatmakes use of a special apparatus in which the cen-trifugal force is incorporated into the verticalforce called gravity, as it is on Earth. The appara-tus consists of a shallow dish, rotating about itsaxis of symmetry as shown in Fig. 7.7. The rotationrate is tuned to the concavity of the dish suchthat at any given radius the outward-directed cen-trifugal force exactly balances the inward-directedcomponent of gravity along the sloping surface ofthe dish; that is

where z is the height of the surface above somearbitrary reference level, r is the radius, and isthe rotation rate of the dish. Integrating from thecenter out to radius r yields the parabolic surface

The constant of integration is chosen to make z

0 the level of the center of the dish.Consider the horizontal trajectory of an ideal-

ized frictionless marble rolling around in the dish,as represented in both a fixed (inertial) frame of

z 2r2

2 constant

dzdr

2r

reference and in a frame of reference rotatingwith the dish. (To view the motion in the rotatingframe of reference, the video camera is mountedon the turntable.)

In the fixed frame of reference the differentialequation governing the horizontal motion of themarble is

It follows from the form of this differential equa-tion that the marble will execute elliptical trajec-tories, symmetric about the axis of rotation, with

d2rdt2

2r

7.1 Experiment in a Dish

Continued on next page

Axis of rotation

Camera

Parabolic surface

Rotating turntable

Fig. 7.7 Setup for the rotating dish experiment. Radius r isdistance from the axis of rotation. Angular velocity is therotation rate of the dish. See text for further explanation.

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period 2, which exactly matches the period ofrotation of the dish. The shape and orientation ofthese trajectories will depend on the initial posi-tion and velocity of the marble. In this example,the marble is released at radius r r0 with no ini-tial velocity. After the marble is released it rollsback and forth, like the tip of a pendulum, alongthe straight line pictured in Fig. 7.7, with a periodequal to 2, the same as the period of rotationof the dish. This oscillatory solution is representedby the equation

where t is time and radius r is defined as positiveon the side of the dish from which the marble isreleased and negative on the other side. The veloc-ity of the marble along its pendulum-like trajectory

is largest at the times when it passes through thecenter of the dish at t 2, 32 . . . , and themarble is motionless for an instant at t , 2 . . .when it reverses direction at the outer edge of itstrajectory.

In the rotating frame of reference the onlyforce in the horizontal equation of motion is theCoriolis force, so the governing equation is

where c is the velocity of the marble and k is thevertical (normal to the surface of the dish) unitvector. Because dcdt is perpendicular to c, it fol-lows that c, the speed of the marble as it movesalong its trajectory in the rotating frame of refer-ence, must be constant. The direction of the for-ward motion of the marble is changing with time atthe uniform rate 2, which is exactly twice the rateof rotation of the dish. Hence, the marble executesa circular orbit called an inertia circle, with period22 (i.e., half the period of rotation ofthe dish), with circumference c (), and radiusc ()2 c2. Because dcdt is to the right

dcdt

2k c

drdt

r0 sin t

r r0 cos t

of c, it follows that the marble rolls clockwise,i.e., in the opposite sense as the rotation of the dish.

As in the fixed frame of reference, the trajec-tory of the marble depends on its initial positionand relative motion. Because the marble isreleased at radius r0 with no initial motion in thefixed frame of reference, it follows that its speedin the rotating frame of reference is c r0, andthus the radius of the inertia circle is r02

r02. It follows that the marble passes through thecenter of the dish at the midpoint of its trajectoryaround the inertia circle.

The motion of the marble in the fixed and rotat-ing frames of reference is shown in Fig. 7.8. Therotating dish is represented by the large circle. Thepoint of release of the marble is labeled 0 andappears at the top of the diagram rather than onthe left side of the dish as in Fig. 7.7. The pendu-lum-like trajectory of the marble in the fixedframe of reference is represented by the straight

7.1 Continued

12

3

4

5

6

7

8

9

101112

8

7

9

11

0

10

112

23

34

45

56

7

8

9

1011

Fig. 7.8 Trajectories of a frictionless marble in fixed(black) and rotating (blue) frames of reference. Numberedpoints correspond to positions of the marble at varioustimes after it is released at point 0. One complete rotationof the dish corresponds to one swing back and forth alongthe straight vertical black line in the fixed frame of referenceand two complete circuits of the marble around the blueinertia circle in the rotating frame of reference. The lightlines are reference lines. See text for further explanation.

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In (7.13a) the horizontal wind field is defined on sur-faces of constant geopotential so that 0, whereasin (7.14) it is defined on constant pressure surfaces sothat p 0. However, pressure surfaces are suffi-ciently flat that the V fields on a geopotential surfaceand a nearby pressure surface are very similar.

7.2.4 The Geostrophic Wind

In large-scale wind systems such as baroclinic wavesand extratropical cyclones, typical horizontal velocitiesare on the order of 10 m s1 and the timescale overwhich individual air parcels experience significantchanges in velocity is on the order of a day or so(105 s). Thus a typical parcel acceleration dVdt is10 m s1 per 105 s or 104 m s2. In middle latitudes,where f 104 s1, an air parcel moving at a speed of10 m s1 experiences a Coriolis force per unit mass C103 m s2, about an order of magnitude larger thanthe typical horizontal accelerations of air parcels.

In the free atmosphere, where the frictional forceis usually very small, the only term that is capable ofbalancing the Coriolis force C is the pressure gradi-ent force P. Thus, to within about 10%, in middleand high latitudes, the horizontal equation of motion(7.14) is closely approximated by

Making use of the vector identity

it follows that

For any given horizontal distribution of pressure ongeopotential surfaces (or geopotential height on pres-sure surfaces) it is possible to define a geostrophic5

wind field Vg for which this relationship is exactlysatisfied:

(7.15a)

or, in component form,

V 1f (k )

V 1f (k )

k (k V) V

f k V

(7.15b)

or, in natural coordinates,

(7.15c)

where V is the scalar geostrophic wind speed and nis the direction normal to the isobars (or geopoten-tial height contours), pointing toward higher values.

The balance of horizontal forces implicit in thedefinition of the geostrophic wind (for a location inthe northern hemisphere) is illustrated in Fig. 7.9. Inorder for the Coriolis force and the pressure gradientforce to balance, the geostrophic wind must blowparallel to the isobars, leaving low pressure to theleft. In either hemisphere, the geostrophic wind fieldcirculates cyclonically around a center of low pres-sure and vice versa, as in Fig. 1.14, justifying the iden-tification of local pressure minima with cyclones andlocal pressure maxima with anticyclones. The tighterthe spacing of the isobars or geopotential height con-tours, the stronger the Coriolis force required to bal-ance the pressure gradient force and hence, thehigher the speed of the geostrophic wind.

7.2.5 The Effect of Friction

The three-way balance of forces required for flow inwhich dVdt 0 in the northern hemisphere in thepresence of friction at the Earth’s surface is illus-trated in Fig. 7.10. As in Fig. 7.9, P is directed normal

V 1f

n

u 1f

y, v

1f

x

5 From the Greek: geo (Earth) and strophen (to turn)

Fig. 7.9 The geostrophic wind V and its relationship to thehorizontal pressure gradient force P and the Coriolis force Cin the northern hemisphere.

C

P

Vg

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to the isobars, C is directed to the right of the hori-zontal velocity vector Vs, and, consistent with (7.11),Fs is directed opposite to Vs. The angle between Vs

and V is determined by the requirement that thecomponent of P in the forward direction of Vs mustbe equal to the magnitude of the drag Fs, and thewind speed Vs is determined by the requirement thatC be just large enough to balance the component ofP in the direction normal to Vs; i.e.,

It follows that sCssPs and, hence, the scalar windspeed Vs sCsf must be smaller than V sPsf.The stronger the frictional drag force Fs, the largerthe angle between V and Vs and the more sub-geostrophic the surface wind speed Vs. The cross-isobar flow toward lower pressure, referred to as theEkman6 drift, is clearly evident on surface charts, par-ticularly over rough land surfaces. That the winds

f Vs &P&cos

usually blow nearly parallel to the isobars in the freeatmosphere indicates that the significance of thefrictional drag force is largely restricted to theboundary layer, where small-scale turbulent motionsare present.

The same shear stress that acts as a drag force onthe surface winds exerts a forward pull on the surfacewaters of the ocean, giving rise to wind-driven cur-rents. If the ocean surface coincided with a surface ofconstant geopotential, the balance of forces justbelow the surface would consist of a two-way bal-ance between the forward pull of the surface windand the backward pull of the Coriolis force inducedby the Ekman drift, as shown in Fig. 7.11. Althoughthe large scale surface currents depicted in Fig. 2.4tend to be in geostrophic balance and orientedroughly parallel to the mean surface winds, theEkman drift, which is directed normal to the surfacewinds, has a pronounced effect on horizontal trans-port of near-surface water and sea-ice, and it largelycontrols the distribution of upwelling, as discussed inSection 7.3.4. Ekman drift is largely confined to thetopmost 50 m of the oceans.

7.2.6 The Gradient Wind

The centripetal accelerations observed in associationwith the curvature of the trajectories of air parcelstend to be much larger than those associated with

Fig. 7.10 The three-way balance of forces required forsteady surface winds in the presence of the frictional dragforce F in the northern hemisphere. Solid lines represent iso-bars or geopotential height contours on a weather chart.

P

Fs

Vsψ

ψ

C

Vg

6 V. Walfrid Ekman (1874–1954) Swedish oceanographer. Ekman was introduced to the problem of wind-driven ocean circulationwhen he was a student working under the direction of Professor Vilhelm Bjerknes.7 Fridtjof Nansen had approached Bjerknes with aremarkable set of observations of winds and ice motions taken during the voyage of the Fram, for which he sought an explanation.Nansen’s observations and Ekman’s mathematical analysis are the foundations of the theory of the wind-driven ocean circulation.

7 Vilhelm Bjerknes (1862–1951) Norwegian physicist and one of the founders of the science of meteorology. Held academic positionsat the universities of Stockholm, Bergen, Leipzig, and Kristiania (renamed Oslo). Proposed in 1904 that weather prediction be regardedas an initial value problem that could be solved by integrating the governing equations forward in time, starting from an initial statedetermined by current weather observations. Best known for his work at Bergen (1917–1926) where he assembled a small group of dedi-cated and talented young researchers, including his son Jakob. The most widely recognized achievement of this so-called “Bergen School”was a conceptual framework for interpreting the structure and evolution of extratropical cyclones and fronts that has endured until thepresent day.

Fig. 7.11 The force balance associated with Ekman drift inthe northern hemisphere oceans. The frictional force F is inthe direction of the surface wind vector. In the southern hemi-sphere (not shown) the Ekman drift is to the left of the sur-face wind vector.

C F

VEkman

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the speeding up or slowing down of air parcels asthey move downstream. Hence, when dVdt is large,its scalar magnitude can be approximated by thecentripetal acceleration V2RT, where RT is the localradius of curvature of the air trajectories.8 Hence,the horizontal equation of motion reduces to thebalance of forces in the direction transverse to theflow, i.e.,

(7.16)

The signs of the terms in this three-way balancedepend on whether the curvature of the trajectoriesis cyclonic or anticyclonic, as illustrated in Fig. 7.12.In the cyclonic case, the outward centrifugal force(the mirror image of the centripetal acceleration)reinforces the Coriolis force so that a balance can beachieved with a wind speed smaller than would berequired if the Coriolis force were acting alone. Inflow through sharp troughs, where the curvature ofthe trajectories is cyclonic, the observed wind speedsat the jet stream level are often smaller, by a factor oftwo or more, than the geostrophic wind speedimplied by the spacing of the isobars. For the anticy-clonically curved trajectory on the right in Fig. 7.12the centrifugal force opposes the Coriolis force,necessitating a supergeostrophic wind speed in orderto achieve a balance.

The wind associated with a three-way balancebetween the pressure gradient and Coriolis and

n V2

RT f k V

centrifugal forces is called the gradient wind. Thesolution of (7.16), which yields the speed of the gra-dient wind can be written in the form

(7.17)

and solved using the quadratic formula. FromFig. 7.12, it can be inferred that RT should be speci-fied as positive if the curvature is cyclonic and nega-tive if the curvature is anticyclonic. For the case ofanticyclonic curvature, a solution exists when

7.2.7 The Thermal Wind

Just as the geostrophic wind bears a simple relation-ship to , the vertical shear of the geostrophic windbears a simple relationship to T. Writing thegeostrophic equation (7.15a) for two different pres-sure surfaces and subtracting, we obtain an expres-sion for the vertical wind shear in the interveninglayer

(7.18)

In terms of geopotential height

(7.19a)

or in component form,

(7.19b)

This expression, known as the thermal wind equation,states that the vertically averaged vertical shear of thegeostrophic wind within the layer between any twopressure surfaces is related to the horizontal gradientof thickness of the layer in the same manner in whichgeostrophic wind is related to geopotential height. Forexample, in the northern hemisphere the thermal wind

(v)2 (v)1 0

f (Z2 Z1)

x

(u)2 (u)1 0

f (Z2 Z1)

y,

(V)2 (V)1 0

f k (Z2 Z1)

(V)2 (V)1 1f k (2 1)

&& f 2 RT

4

Vr 1f &&

V2r

RT

8 In estimating the radius of curvature in the gradient wind equation it is important to keep in mind the distinction between streamlinesand trajectories, as explained in Section 7.1.4.

Fig. 7.12 The three-way balance involving the horizontal pres-sure gradient force P, the Coriolis force C, and the centrifugalforce sVs2RT in flow along curved trajectories in the northernhemisphere. (Left) Cyclonic flow. (Right) Anticyclonic flow.

V

2

RT

V

2

RT

Vgr

C P

VgrC

P

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(namely the vertical shear of the geostrophic wind)“blows” parallel to the thickness contours, leaving lowthickness to the left. Incorporating the linear propor-tionality between temperature and thickness in thehypsometric equation (3.29), the thermal wind equa-tion can also be expressed as a linear relationshipbetween the vertical shear of the geostrophic wind andthe horizontal temperature gradient.

(7.20)

where is the vertically averaged temperaturewithin the layer.

To explore the implications of the thermal windequation, consider first the special case of an atmos-phere that is characterized by a total absence of hori-zontal temperature (thickness) gradients. In such abarotropic9 atmosphere T 0 on constant pressuresurfaces. Because the thickness of the layer betweenany pair of pressure surfaces is horizontally uniform,it follows that the geopotential height contours onvarious pressure surfaces can be neatly stacked ontop of one another, like a set of matched dinnerplates. It follows that the direction and speed of thegeostrophic wind must be independent of height.

Now let us consider an atmosphere with horizontaltemperature gradients subject to the constraint thatthe thickness contours be everywhere parallel to thegeopotential height contours. For historical reasons,such a flow configuration is referred to as equivalentbarotropic. It follows from the thermal wind equa-tion that the vertical wind shear in an equivalentbarotropic atmosphere must be parallel to the winditself so the direction of the geostrophic wind doesnot change with height, just as in the case of a purebarotropic atmosphere. However, the slope of thepressure surfaces and hence the speed of thegeostrophic wind may vary from level to level inassociation with thickness variations in the directionnormal to height contours, as illustrated in Fig. 7.13.

In an equivalent barotropic atmosphere, the iso-bars and isotherms on horizontal maps have thesame shape. If the highs are warm and the lows arecold, the amplitude of features in the pressure fieldand the speed of the geostrophic wind increase withheight. If the highs are cold and the lows are warm,

T

(V)2 (V)1 Rf

ln p1

p2k (T )

the situation is just the opposite; features in the pres-sure and geostrophic wind field tend to weaken withheight and the wind may even reverse direction if thetemperature anomalies extend through a deepenough layer, as in Fig. 7.13b. The zonally averagedzonal wind and temperature cross sections shown inFig. 1.11 are related in a manner consistent with Fig.7.13. Wherever temperature decreases with increas-ing latitude, the zonal wind becomes (relatively)more westerly with increasing height, and vice versa.

Exercise 7.3 During winter in the troposphere 30°latitude, the zonally averaged temperature gradient is0.75 K per degree of latitude (see Fig. 1.11) and thezonally averaged component of the geostrophic windat the Earth’s surface is close to zero. Estimate themean zonal wind at the jet stream level, 250 hPa.

Solution: Taking the zonal component of (7.20)and averaging around a latitude circle yields

[u]250 [u]1000 R

2sin [T ]y

ln 1000250

9 The term barotropic is derived from the Greek baro, relating to pressure, and tropic, changing in a specific manner: that is, in such away that surfaces of constant pressure are coincident with surfaces of constant temperature or density.

Fig. 7.13 The change of the geostrophic wind with height inan equivalent barotropic flow in the northern hemisphere:(a) V increasing with height within the layer and (b) V revers-ing direction. The temperature gradient within the layer is indi-cated by the shading: blue (cold) coincides with low thicknessand tan (warm) with high thickness.

p = p2

p = p2

p = p1

p = p1

(a)

(b)

(Vg )

2

(Vg )

1

(Vg )

2(V

g )1

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Noting that [ug]1000 0 and R Rd

in close agreement with Fig. 1.11.

In a fully baroclinic atmosphere, the height andthickness contours intersect one another so that thegeostrophic wind exhibits a component normal to theisotherms (or thickness contours). This geostrophicflow across the isotherms is associated with geostrophictemperature advection. Cold advection denotes flowacross the isotherms from a colder to a warmer region,and vice versa.

Typical situations corresponding to cold and warmadvection in the northern hemisphere are illustratedin Fig. 7.14. On the pressure level at the bottom ofthe layer the geostrophic wind is from the west sothe height contours are oriented from west to east,with lower heights toward the north. Higher thick-ness lies toward the east in the cold advection case(Fig. 7.14a) and toward the west in the warm advec-tion case (Fig. 7.14b).10

The upper level geostrophic wind vector V2 blowsparallel to these upper-level contours. Just as theupper level geopotential height is the algebraic sumof the lower level geopotential height Z1 plus thethickness ZT, the upper level geostrophic wind vectorV 2 is the vectorial sum of the lower level geostrophicwind Vg1 plus the thermal wind VT. Hence, thermalwind is to thickness as geostrophic wind is to geopo-tential height.

From Fig. 7.14 it is apparent that cold advectionis characterized by backing (cyclonic rotation) ofthe geostrophic wind vector with height and warmadvection is characterized by veering (anticyclonicrotation). By experimenting with other configura-tions of height and thickness contours, it is readilyverified that this relationship holds, regardless ofthe direction of the geostrophic wind at the bottom

0.75 K

1.11 105 m 36.8 m s1,

[u]250 287 J deg1 kg1

2 7.29 105 s1 sin 30 ln 4

of the layer or the orientation of the isotherms, andit holds for the southern hemisphere as well.

Making use of the thermal wind equation it is pos-sible to completely define the geostrophic wind fieldon the basis of knowledge of the distribution ofT(x, y, p), together with boundary conditions foreither p(x, y) or V(x, y) at the Earth’s surface or atsome other “reference level.” Thus, for example, a setof sea-level pressure observations together with anarray of closely spaced temperature soundings fromsatellite-borne sensors constitutes an observing sys-tem suitable for determining the three-dimensionaldistribution of V.

10 Given the distribution of geopotential height Z1 at the lower level and thickness ZT, it is possible to infer the geopotential height Z2

at the upper level by simple addition. For example, the height Z2 at point O at the center of the diagram is equal to Z1 ZT, and at point Pit is Z1 (ZT Z), and so forth. The upper-level height contours (solid black lines) are drawn by connecting intersections with equal val-ues of this sum. If the fields of Z1, Z2, and ZT are all displayed using the same contour interval (say, 60 m), then all intersections betweencountours will be three-way intersections. In a similar manner, given a knowledge of the geopotential height field at the lower and upperlevels, it is possible to infer the thickness field ZT by subtracting Z1 from Z2.

Fig. 7.14 Relationships among isotherms, geopotentialheight contours, and geostrophic wind in layers with (a) coldand (b) warm advection. Solid blue lines denote the geopo-tential height contours at the bottom of the layer and solidblack lines denote the geopotential height contours at the topof the layer. Red lines represent the isotherms or thicknesscontours within the layer.

(Vg)1

(Vg)1

(Vg)2

(Vg)2

O

Z1 +

ZT +

δ Z

ZT

+ δ Z

Z1

Z1 +

ZT

Z 1 +

Z T +

δ Z

Z 1 +

Z T

OP

P

VT

VT

Z1

Z1+ δ ZZ T

+ δ Z

Z T +

δ Z

Z T

Z T

(a)

(b)

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parameter f, which can be interpreted as the plane-tary vorticity that exists by virtue of the Earth’s rota-tion: 2 (as inferred from Exercise 7.1) times thecosine of the angle between the Earth’s axis of rota-tion and the local vertical (i.e., the colatitude).Hence (f ) is the absolute vorticity: the sum of theplanetary vorticity and the relative vorticity of thehorizontal wind field. In extratropical latitudes,where f 104 s1 and 105 s1, the absolute vor-ticity is dominated by the planetary vorticity. In trop-ical motions the relative and planetary vorticityterms are of comparable magnitude.

In fast-moving extratropical weather systems thatare not rapidly amplifying or decaying, the diver-gence term in (7.21) is relatively small so that

(7.22a)

or, in Lagrangian form,

(7.22b)

In this simplified form of the vorticity equation,absolute vorticity ( f) behaves as a conservativetracer: i.e., a property whose numerical values areconserved by air parcels as they move along with(i.e., are advected by) the horizontal wind field.

This nondivergent form of the vorticity equation wasused as a basis for the earliest quantitative weatherprediction models that date back to World War II.13

The forecasts, which were based on the 500-hPa geopo-tential height field, involved a four-step process:

1. The geostrophic vorticity

(7.23)

is calculated for an array of grid points, using asimple finite difference algorithm,14 and addedto f to obtain the absolute vorticity.

2. The advection term in (7.22a) is estimated toobtain the geostrophic vorticity tendency gtat each grid point.

v

x

u

y

1f 2

x2 2

y2

ddt

( f ) 0

t V ( f )

3. The corresponding geopotential height tendency t at each grid-point is found by invertingand solving the time derivative of Eq. (7.23).

4. The field is multiplied by a small time step tto obtain the incremental height change at eachgrid point and this change is added to the initial500-hPa height field to obtain the forecast of the500-hPa height field.

Let us consider the implications of applying thisforecast model to an idealized geostrophic wind pat-tern with a sinusoidal wave with zonal wavelength Lsuperimposed on a uniform westerly flow with veloc-ity U. The geostrophic vorticity is positive in thewave troughs where the curvature of the flow iscyclonic, zero at the inflection points where the flowis straight, and negative in the ridges where the cur-vature is anticyclonic. The vorticity perturbationsassociated with the waves tend to be advected east-ward by the westerly background flow: positive vor-ticity tendencies prevail in the southerly flow nearthe inflection points downstream of the troughs ofthe waves, and negative tendencies in the northerlyflow downstream of the ridges (points A and B,respectively, in Fig. 7.15). The positive vorticity ten-dencies downstream of the troughs induce heightfalls, causing the troughs to propagate eastward, andsimilarly for the ridges. Were it not for the advectionof planetary vorticity, the wave would propagateeastward at exactly the same speed as the “steeringflow” U.

In addition to the vorticity tendency resulting fromthe advection of relative vorticity , equatorwardflow induces a cyclonic vorticity tendency, as air fromhigher latitudes with higher planetary vorticity is

13 The barotropic model was originally implemented with graphical techniques using a light table to overlay various fields. Later it wasprogrammed on the first primitive computers.

14 See J. R. Holton, An Introduction to Dynamic Meteorology, Academic Press, pp. 452–453 (2004).

Fig. 7.15 Patterns of vorticity advection induced by the hor-izontal advection of absolute vorticity in a wavy westerly flowin the northern hemisphere.

f

f – ∆f

A B

f + ∆f

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will be positive and will increase without limit as

grows.17 This asymmetry in the rate of growth of rela-tive vorticity perturbations is often invoked to explainwhy virtually all intense closed circulations arecyclones, rather than anticyclones, and why sharpfrontal zones and shear lines nearly always exhibitcyclonic, rather than anticyclonic vorticity.

7.2.10 Potential Vorticity

It is possible to derive a conservation law analogousto (7.22) that takes into account the effect of thedivergence of the horizontal motion. Consider alayer consisting of an incompressible fluid of depthH(x, y) moving with velocity V(x, y). From the con-servation of mass, we can write

(7.26)

where A is the area of an imaginary block of fluid,but it may also be interpreted as the area enclosed bya tagged set of fluid parcels as they move with thehorizontal flow. Making use of (7.2), we can write

Hence the layer thickens wherever the horizontalflow converges and thins where the flow diverges.Using this expression to substitute for the divergencein (7.21b) yields

In Exercise 7.35 the student is invited to verify thatthis expression is equivalent to the simple conserva-tion equation

(7.27)

The conserved quantity in this expression is calledthe barotropic potential vorticity.

From (7.27) it is apparent that vertical stretchingof air columns and the associated convergence of the

ddt

f

0

ddt

( f ) (f )

H dHdt

1H

dHdt

1A

dAdt

V

ddt

(HA) 0

horizontal flow increase the absolute vorticity (f )in inverse proportion to the increase of H. The “spinup” of absolute vorticity that occurs when columnsare vertically stretched and horizontally compressedis analogous to that experienced by an ice skatergoing into a spin by drawing hisher arms and legsinward as close as possible to the axis of rotation.The concept of potential vorticity also allows for con-versions between relative and planetary vorticity, asdescribed in the context of (7.22).

Based on an inspection of the form of (7.27) it isevident that a uniform horizontal gradient in thedepth of a layer of fluid plays a role in vorticitydynamics analogous to the beta effect. Hence,Rossby waves can be defined more generally aswaves propagating horizontally in the presence of agradient of potential vorticity.

An analogous expression can be derived for theconservation of potential vorticity in large-scale, adi-abatic atmospheric motions.18 If the flow is adiabatic,air parcels cannot pass through isentropic surfaces(i.e., surfaces of constant potential temperature). Itfollows that the mass in the column bounded by twonearby isentropic surfaces is conserved following theflow; i.e.,

(7.28)

where p is the pressure difference between twonearby isentropic surfaces separated by a fixedpotential temperature increment , as depicted inFig. 7.16. The pressure difference between the top

Ap

constant

17 Divergence tends to be inhibited by the presence of both planetary vorticity and relative vorticity. In this sense, prescribing thedivergence to be constant while the relative vorticity is allowed to increase without limit is unrealistic.

18 The conditions under which the assumption of adiabatic flow is justified are discussed in Section 7.3.3.

Fig. 7.16 A cylindrical column of air moving adiabatically,conserving potential vorticity. [Reprinted from Introductionto Dynamic Meteorology, 4th Edition, J. R. Holton, p. 96,Copyright 2004, with permission from Elsevier.]

θ + δθ

δp

θ

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7.3 Primitive Equations 293

flattening out the isentropes. In time, the isentropicsurfaces would become completely flat, were it notfor the meridional heating gradient, which is continu-ally tending to lift them at high latitudes and depressthem at low latitudes. The role of the horizontaladvection term in producing rapid local temperaturechanges observed in association with extratropicalcyclones is illustrated in the following exercise.

Exercise 7.4 During the time that a frontal zonepasses over a station the temperature falls at a rateof 2 °C per hour. The wind is blowing from the northat 40 km h1 and temperature is decreasing with lati-tude at a rate of 10 °C per 100 km. Estimate theterms in (7.37), neglecting diabatic heating.

Solution: Tt 2 °C h1 and

The temperature at the station is dropping only halfas fast as the rate of horizontal temperature advec-tion so large-scale subsidence must be warming theair at a rate of 2 °C h1 as it moves southward. It fol-lows that the meridional slopes of the air trajectoriesmust be half as large as the meridional slopes of theisentropes.

Within the tropical troposphere the relative mag-nitude of the terms in (7.37) is altogether differentfrom that in the extratropics. Horizontal temperaturegradients are much weaker than in the extratropics,so the horizontal advection term is unimportant andtemperatures at fixed points vary little from one dayto the next. In contrast, diabatic heating rates inregions of tropical convection are larger than thosetypically observed in the extratropics. Ascent tendsto be concentrated in narrow rain belts such as theITCZ, where warming due to the release of latentheat of condensation is almost exactly compensatedby the cooling induced by the vertical velocity termin (7.37). The prevailing lapse rate throughout thetropical troposphere all the way from the top of theboundary layer up to around the 200-hPa level isnearly moist adiabatic so that the lifting of saturatedair does not result in large temperature changes, evenwhen the rate of ascent is very large. Within themuch larger regions of slow subsidence, warming dueto adiabatic compression is balanced by weak radia-tive cooling.

4 C h1

V T vTy

40 km h1 10 C100 km

7.3.4 Inference of the Vertical Motion Field

The vertical motion field cannot be predicted on thebasis of Newton’s second law, but it can be inferredfrom the horizontal wind field on the basis of thecontinuity equation, which is an expression of theconservation of mass.

Air parcels expand and contract in response to pres-sure changes. In general, these changes in volume areof two types: those associated with sound waves, inwhich the momentum of the air plays an essential role,and those that occur in association with hydrostaticpressure changes.When the equation for the continuityof mass is formulated in (x, y, p) coordinates, only thehydrostatic volume changes are taken into account.

Consider an air parcel shaped like a block withdimensions x, y, and p, as indicated in Fig. 7.17. Ifthe atmosphere is in hydrostatic balance, the mass ofthe block is given by

Because the mass of the block is not changing withtime,

(7.38)

or, in expanded form,

Expanding the time rates of change of x, y, and p interms of partial derivatives of the velocity components,as in the derivation of (7.2), it is readily shown that

(7.39a)ux

vy

p 0

y p ddt

x x p ddt

y x y ddt

p 0.

ddt

(x y p) 0

" x y z x y p

ν + δν

u + δuω

δx

ω + δ ω νδp

δyu

Fig. 7.17 Relationships used in the derivation of thecontinuity equation.

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or, in vectorial form,

(7.39b)

where V is the divergence of the horizontal windfield. Hence, horizontal divergence ( V 0) isaccompanied by vertical squashing (p 0), andhorizontal convergence is accompanied by verticalstretching, as illustrated schematically in Fig. 7.18.

Within the atmospheric boundary layer air parcelstend to flow across the isobars toward lower pressurein response to the frictional drag force. Hence, thelow-level flow tends to converge into regions of lowpressure and diverge out of regions of high pressure,as illustrated in Fig. 7.19. The frictional convergencewithin low-pressure areas induces ascent, while thedivergence within high-pressure areas induces subsi-dence. In the oceans, where the Ekman drift is in theopposite sense, cyclonic, wind-driven gyres are charac-terized by upwelling, and anticyclonic gyres by down-welling. In a similar manner, regions of Ekman driftin the offshore direction are characterized by coastal

p V

upwelling, and the easterly surface winds along theequator induce a shallow Ekman drift away from theequator, accompanied by equatorial upwelling.

The vertical velocity at any given point (x, y) can beinferred diagnostically by integrating the continuityequation (7.39b) from some reference level p* tolevel p,

(7.40)

In this form, the continuity equation can be usedto deduce the vertical velocity field from a knowl-edge of the horizontal wind field. A convenientreference level is the top of the atmosphere, wherep* 0 and 0. Integrating (7.40) downwardfrom the top down to the Earth’s surface, wherep ps we obtain

Incorporating this result and (7.34) into (7.32) andrearranging, we obtain Margules’22 pressure tendencyequation

(7.41)

which serves as the bottom boundary condition forthe pressure field in the primitive equations.

At the Earth’s surface, the pressure tendency termpst is on the order of 10 hPa per day and theadvection terms Vs ps and ws(pz) are usu-ally even smaller. It follows that

where V is the mass-weighted, vertically-averaged divergence. Because and1 day 105 s, it follows that

V 107 s1

ps 1000 hPa

ps

0 ( V) dp ps V 10 hPa day1

ps

t Vs ps ws

pz

ps

0 ( V) dp

s ps

0( V) dp

(p) (p*) p

p*( V) dp

ω > 0 ω < 0

• V > 0 ∆

• V < 0 ∆

Fig. 7.18 Algebraic signs of in the midtroposphere asso-ciated with convergence and divergence in the lowertroposphere.

Fig. 7.19 Cross-isobar flow at the Earth’s surface inducedby frictional drag. Solid lines represent isobars.

L

H

22 Max Margules (1856–1920) Meteorologist, physicist, and chemist, born in the Ukraine. Worked in intellectual isolation on atmos-pheric dynamics from 1882 to 1906, during which time he made many fundamental contributions to the subject. Thereafter, he returned tohis first love, chemistry. Died of starvation while trying to survive on a government pension equivalent to $2 a month in the austere post-World War I period. Many of the current ideas concerning the kinetic energy cycle in the atmosphere (see Section 7.4) stem fromMargules’ work.

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7.3 Primitive Equations 295

That the vertically averaged divergence is typicallyabout an order of magnitude smaller than typicalmagnitudes of the divergence observed at specificlevels in the atmosphere reflects the tendency (firstpointed out by Dines23) for compensation betweenlower tropospheric convergence and upper tropos-pheric divergence and vice versa. Vertical velocity ,which is the vertical integral of the divergence, tendsto be strongest in the mid-troposphere, where thedivergence changes sign.24

It is instructive to consider the vertical profile ofatmospheric divergence in light of the idealized flowpatterns for the two-dimensional flow in two differ-ent kinds of waves in a stably stratified liquid inwhich density decreases with height. Fig. 7.20a showsan idealized external wave in which divergence isindependent of height and vertical velocity increaseslinearly with height from zero at the bottom to amaximum at the free surface of the liquid. Fig. 7.20bshows an internal wave in a liquid that is bounded bya rigid upper lid so that w 0 at both top and bot-tom, requiring perfect compensation between lowlevel convergence and upper level divergence andvice versa. In the Earth’s atmosphere, motions thatresemble the internal wave contain several orders ofmagnitude more kinetic energy than those thatresemble the external wave. In some respects, thestratosphere plays the role of a “lid” over the tropo-

sphere, as evidenced by the fact that the geometricvertical velocity w is typically roughly an order ofmagnitude smaller in the lower stratosphere than inthe midtroposphere.

7.3.5 Solution of the Primitive Equations

The primitive equations, in the simplified form thatwe have derived them, consist of the horizontal equa-tion of motion

(7.14)

the hypsometric equation

(3.23)

the thermodynamic energy equation

(7.36)

and the continuity equation

(7.39b)

together with the bottom boundary condition

(7.41)

Bearing in mind that the horizontal equation ofmotion is made up of two components, the system ofprimitive equations, as presented here, consist of fiveequations in five dependent variables: u, v, , , andT. The fields of diabatic heating J and friction F needto be prescribed or parameterized (i.e., expressed asfunctions of the dependent variables). The horizontalequation(s) of motion, the thermodynamic energyequation, and the equation for pressure on thebottom boundary all contain time derivatives and are

ps

t Vs p ws

pz

ps

0 ( V) dp

p V

dTdt

!Tp

Jcp

p

RTp

dVdt

f k V F

23 Willam Henry Dines (1855–1927) British meteorologist. First to invent a device for measuring both the direction and the speed ofthe wind (the Dines’ pressure-tube anemometer). Early user of kites and ballons to study upper atmosphere.

24 The midtroposphere minimum in the amplitude of V justifies the use of (7.22) in diagnosing the vorticity balance at that level.Alternatively (7.22) can be viewed as relating to the vertically averaged tropospheric wind field, whose divergence is two orders of magni-tude smaller than its vorticity.

(a) (b)

Fig. 7.20 Motion field in a vertical cross section through atwo-dimensional wave propagating from left to right in a sta-bly stratified liquid. The contours represent surfaces of con-stant density. (a) An external wave in which the maximumvertical motions occur at the free surface of the liquid, and(b) an internal wave in which the vertical velocity vanishes atthe top because of the presence of a rigid lid.

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7.4 The Atmospheric General Circulation 297

The flow becomes progressively more zonal witheach successive time step, until the meridional com-ponent of the Coriolis force comes into geostrophicbalance with the pressure gradient force, as illustratedin Fig. 7.22. As required by thermal wind balance, thevertical wind shear between the low level easterliesand the upper level westerlies increases in proportionto the strengthening equator-to-pole temperaturegradient forced by the meridional gradient of the dia-batic heating. The Coriolis force acting on the rela-tively weak meridional cross-isobar flow isinstrumental in building up the vertical shear.Frictional drag limits the strength of the surface east-erlies, but the westerlies aloft become stronger witheach successive time step. Will these runaway wester-lies continue to increase until they become supersonic

or will some as yet to be revealed process interveneto bring them under control? The answer is reservedfor the following section.

7.4 The Atmospheric GeneralCirculationDating back to the pioneering studies of Halley26 in1676 and Hadley in 1735, scientists have sought toexplain why the surface winds blow from the east atsubtropical latitudes and from the west in middle lat-itudes, and why the trade winds are so steady fromone day to the next compared to the westerlies inmidlatitudes. Such questions are fundamental to anunderstanding of the atmospheric general circulation;i.e., the statistical properties of large-scale atmos-pheric motions, as viewed in a global context.

Two important scientific breakthroughs thatoccurred around the middle of the 20th century pavedthe way for a fundamental understanding of the gen-eral circulation. The first was the simultaneous discov-ery of baroclinic instability (the mechanism that givesrise to baroclinic waves and their attendant extratropi-cal cyclones) by Eady27 and Charney.28 The second wasthe advent of general circulation models: numerical

Hadleycells

NorthPole

Tropospheric jet streamJ

J

L L

H H H

L L

H H H

Ferrel cell

Ferrel cell

NorthPole

NorthPole

NorthPole

S.P. S.P.

S.P. S.P.

(d)

(b)(a)

(c)

low pressure

high pressure

high pressurecooling

heating

cooling

Fig. 7.21 Schematic depiction of the general circulation asit develops from a state of rest in a climate model for equinoxconditions in the absence of land–sea contrasts. See text forfurther explanation.

Fig. 7.22 Trajectories of air parcels in the middle latitudesof the northern hemisphere near the Earth’s surface (left) andin the upper troposphere (right) during the first few timesteps of the primitive equation model integration described inFig. 7.21. The solid lines represent geopotential height con-tours and the curved arrows represent air trajectories.

(b)(a)

VgVg

26 Edmund Halley (1656–1742) English astronomer and meteorologist. Best known for the comet named after him. Determined thatthe force required to keep the planets in their orbits varies as the inverse square of their respective distances from the sun. (It remainedfor Newton to prove that the inverse square law yields elliptical paths, as observed.) Halley undertook the business and printing (“at hisown charge”) of Newton’s masterpiece the Principia. First to derive a formula relating air pressure to altitude.

27 E. T. Eady (1915–1966) English mathematician and meteorologist. Worked virtually alone in developing a theory of baroclinic insta-bility while an officer on the Royal Air Force during World War II. One of us (P. V. H.) is grateful to have had him as a tutor.

28 Jule G. Charney (1917–1981) Made major contributions to the theory of baroclinic waves, planetary-waves, tropical cyclones, and anumber of other atmospheric and oceanic phenomena. While working at the Institute for Advanced Studies at Princeton University hehelped lay the groundwork for numerical weather prediction. Later served as professor at Massachusetts Institute of Technology whereone of us (J. M. W.) is privileged to have had the opportunity to learn from him. Played a prominent role in fostering large internationalprograms designed to advance the science of weather prediction.

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7.4 The Atmospheric General Circulation 299

motion is replaced by the hydrostatic equation,which has no vertical acceleration term. In large-scale motions, kinetic energy is imparted directly tothe horizontal wind field. When warm air rises andcold air sinks, the potential energy that is releaseddoes work on the horizontal wind field by forcing itto flow across the isobars from higher toward lowerpressure. In the equation for the time rate of changeof kinetic energy,

(7.42)

the cross-isobar flow V is the one and onlysource term. Flow across the isobars toward lowerpressure is prevalent close to the Earth’s surface,where the dissipation term F V is most intense.Frictional drag is continually acting to reduce thewind speed, causing it to be subgeostrophic, so thatthe Coriolis force is never quite strong enough tobalance the pressure gradient force. The imbalancedrives a cross-isobar flow toward lower pressure(Fig. 7.19), which maintains the kinetic energy andthe surface wind speed in the presence of frictionaldissipation. This process is represented by the secondterm on the right-hand side of (7.42). It can be shownthat, in the integral over the entire mass of theatmosphere, the generation of kinetic energy by theV term in (7.42) is equal to the release ofpotential energy associated with the rising of warmair and sinking of cold air.

V dVdt

ddt

V2

2 V F V

The trade winds in the lower branch of the Hadleycell are directed down the pressure gradient, out ofthe subtropical high pressure belt and into the belt oflow pressure at equatorial latitudes. That the winds inthe upper troposphere blow from west to eastimplies that decreases with latitude at that level,and hence that the poleward flow in the upperbranch of the Hadley cell is also directed down thepressure gradient.

Closed circulations like the Hadley cell, which arecharacterized by the rising of warmer, lighter airand the sinking of colder, denser air and the preva-lence of cross-isobar horizontal flow toward lowerpressure, release potential energy and convert itto the kinetic energy of the horizontal flow.Circulations with these characteristics are referredto as thermally direct because they operate in thesame sense as the global kinetic energy cycle. Otherexamples of thermally direct circulations in theEarth’s atmosphere are the large-scale overturningcells in baroclinic waves, monsoons and tropicalcyclones. Circulations such as the Ferrel cell, thatoperate in the opposite sense, with rising of colderair and sinking of warmer air are referred to as ther-mally indirect.

Because thermally direct circulations are continu-ally depleting the atmosphere’s reservoir of potentialenergy, something must be acting to restore it.Heating of the atmosphere by radiative transfer andthe release of the latent heat of condensation of watervapor in clouds is continually replenishing the poten-tial energy in two ways: by warming the atmospherein the tropics and cooling it at higher latitudes and byheating the air at lower levels and cooling the air athigher levels. The former acts to maintain the equa-tor-to-pole temperature contrast on pressure surfaces,and the latter expands the air in the lower tropo-sphere and compresses the air in the upper tropo-sphere, thereby lifting the air at intermediate levels,maintaining the height of the atmosphere’s center ofmass against the lowering produced by thermallydirect circulations. Hence, the maintenance of large-scale atmospheric motions requires both horizontaland vertical heating gradients analogous to those inthe laboratory experiment depicted in Fig. 7.23. Incontrast, the maintenance of convection requires onlyvertical heating gradients.

The most important heat source in the troposphereis the release of latent heat of condensation thatoccurs in association with precipitation. Latent heat

Heatsink

Heat sink

Heat source

Heatsource

Fig. 7.23 Vertical cross section through a steady-state circu-lation in the laboratory, driven by the distribution of heatsources and heat sinks as indicated. The colored shading indi-cates the distribution of temperature and density, with cooler,denser fluid represented by blue. The sloping black lines repre-sent pressure surfaces. Note that the flow is directed down thehorizontal pressure gradient at both upper and lower levels.

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release tends to occur preferentially in rising air,which becomes saturated as it cools adiabatically.This preferential heating of the rising air tends tomaintain and enhance the horizontal temperaturegradients that drive thermally direct circulations, ren-dering the motions more vigorous than they wouldbe in a dry atmosphere. Condensation heating playsa supporting role as an energy source for extratropi-cal cyclones and it plays a starring role in the ener-getics of tropical cyclones.

The kinetic energy cycle can be summarized interms of the flowchart presented in Fig. 7.24. Potentialenergy generated by heating gradients is converted tokinetic energy of both large-scale and smaller scaleconvective motions. Kinetic energy is drained fromthe large-scale reservoir by shear instability and flowover rough surfaces, both of which generate small-scale turbulence and wave motions, as discussed inChapter 9. Kinetic energy in the small-scale reservoiris transferred to smaller and smaller scales until itbecomes indistinguishable from random molecularmotions and thus becomes incorporated into theatmosphere’s reservoir of internal energy. This finalstep is represented in Fig. 7.24 as “a drop in thebucket” to emphasize that it is a small contribution tothe atmosphere’s vast internal energy reservoir: onlylocally in regions of very strong winds such as thosenear the centers of tropical cyclones is frictional dissi-pation strong enough to affect the temperature.

7.4.2 The Atmosphere as a Heat Engine

The primary tropospheric heat sources tend to be con-centrated within the planetary boundary layer, as dis-cussed in the previous chapter, and in regions of heavyprecipitation. The primary heat sink, radiative coolingin the infrared, is more diffuse. In a gross statisticalsense, the center of mass of the heating is located at aslightly lower latitude and at a slightly lower altitudethan the center of mass of the cooling, and is thereforeat a somewhat higher temperature. The atmosphericgeneral circulation can be viewed as a heat engine thatis continually receiving heat at a rate QH at thathigher temperature TH and rejecting heat at a rate QC

at the lower temperature TC. By analogy with (3.78)for the Carnot cycle, we can write an expression forthe thermal efficiency of the atmospheric heat engine

(7.43)

where W is the rate at which work is done in generat-ing kinetic energy in thermally direct circulations. Theintegral of the kinetic energy generation termV over the mass of the atmosphere is estimatedto be 1–2 W m2. QH, the net incoming solar radia-tion averaged over the surface of the Earth, is 240W m2. Hence, based on this definition, the thermalefficiency of the atmospheric heat engine is less than1%. In contrast to the situation in the Carnot cycle,much of the heat transfer that takes place within theatmosphere is irreversible. For example, cold, polarair masses that penetrate into the tropics are rapidlymodified by the heat fluxes from the underlying sur-faces and radiative transfer. It follows that relation-ships derived from the reversible Carnot cycle are notstrictly applicable to the atmospheric heat engine.

7.5 Numerical Weather PredictionUntil the 1950s, day-to-day weather forecasting waslargely based on subjective interpretation of synopticcharts. From the time of the earliest synoptic networks,it has been possible to make forecasts by extrapolatingthe past movement of the major features on the charts.As forecasters began to acquire experience from abacklog of past weather situations, they were able torefine their forecasts somewhat by making use of his-torical analogs of the current weather situation, with-out reference to the underlying dynamical principlesintroduced in this chapter. However, it soon became

# W

QH

Thermally direct circulations

ShearinstabilityBoundary

effects

Kinetic energy of random

molecular motions

Kinetic energy of large-scale motions

Convection

Horizontalheating gradients

Low-level heatingHigh-level cooling

Available potential energy

Energycascade

Kinetic energy of small-scale

motions

Fig. 7.24 Schematic flow diagram for the kinetic energycycle in the atmospheric general circulation.

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L

H

H

ANALYSISANALYSISANALYSIS

L

DAY 1

8m 99.7%

L

DAY 3

36m 95.3%

H L

DAY 5

53m 89.8%

HDAY 7

81m 76.1%

DAY 10

109m 56.8%

H

H

H

L

L

L

L

L

L

L

H

H H

L

L

L

L

L

L

L

H

H

H

H

H

L

L

L

L

L

H

H H

L

LL

L

L

L

L

H

H H

L

L

LL

L

H

H

H

H

L

L

L

L

L

L

Fig. 7.25 Forecasts and verifying analysis of the northern hemisphere 500-hPa height field for 00 UTC February 24, 2005. Thenumber printed in the lower left corner is the root mean squared error and the percentage shown in the lower right corner of eachpanel is the anomaly correlation, a measure of the hemispherically averaged skill of the forecast. [Courtesy of Adrian J. Simmons,European Centre for Medium Range Weather Forecasts (ECMWF).]

Fig. 7.26 Three sets of ensemble forecasts performed using the highly simplified three-variable representation of the atmospheredescribed in Box 1.1. The time-dependent solution of the governing governing equations traces out the three-dimensional geo-metric pattern, referred to as the Lorenz attractor. Each panel represents a numerical experiment in which the initial conditions arethe ensemble of points that lie along the purple ellipse and the forecasts at successive forecast times are represented by the“down-stream” black ellipses. Each point on the ellipse of initial conditions is uniquely identified with a point on each of the suc-cessive forecast ellipses. [Courtesy of T. N. Palmer, ECMWF.]

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7.5 Numerical Weather Prediction 303

proceeds, as evidenced by the decreasing size of suc-cessive forecast ellipses. In the second experiment(middle panel), successive forecast ellipses widenmarkedly as the points representing the individualforecasts that make up the ensemble approach thebottom of the attractor and spread out into the twoseparate loops. In the third experiment (right panel)the rapid increase in the uncertainty of the forecastsbegins earlier in the forecast interval, and the pointsalong the bottom of the attractor spread rapidlyaround their respective loops as the forecast proceeds.Hence the predictability of this simple model is highlydependent on the initial state.

Ensemble forecasts carried out with the full set ofgoverning equations for large-scale atmosphericmotions provide a means of estimating the uncertain-ties inherent in weather forecasts at the time whenthey are issued. The results are not as easy to interpretas those for the idealized model based on the Lorenzattractor, but they are nonetheless informative. As inthe idealized experiments, the ensemble forecasts alsoprovide an indication of the range of atmosphericstates that could develop out of the observed initial

conditions. In current operational practice, differentforecast models, as well as perturbed initial conditions,are used to generate different members of the ensem-ble. At times when the entire hemispheric circulationis relatively predictable, members of the ensemble donot diverge noticeably from one another until rela-tively far into the forecast. Often the errors grow mostrapidly over one particular sector of the hemispheredue to the presence of local instability in the hemi-spheric flow pattern. The rate of divergence of theindividual members of the ensemble provides a meas-ure of the credibility of the forecasts in various sectorsof the hemisphere and the length of the time intervalover which the forecasts can be trusted.

Figure 7.27 shows 7-day ensemble forecasts for atypical winter day. The mean is considerablysmoother than its counterpart in Fig. 7.25 because itrepresents an average over many individual forecasts.Some of the individual forecasts, like the one in thelower left panel, capture the features in the verifyinganalysis with remarkable fidelity. Unfortunately, thereis no way of identifying these highly skillful forecastsat the time that the ensemble forecast is made.

ANALYSIS DAY 7

MEAN

HDAY 7

BEST

DAY 7

WORST

L

L

L

L

L

L

L

H

H

H

HL

H

H

H

H

L

L

L

L H

H

H

L

L

L

L

L

L

L

H

H

L

L

L

Fig. 7.27 As in Fig. 7.25 but for the 7-day forecasts generated by the ensemble forecasting system in current use at ECMWF.Mean is the average of the 50 members of the ensemble Best and Worst forecasts are selected based on anomaly correlations withthe verifying analysis. [Courtesy of Adrian J. Simmons, ECMWF.]

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Solution: From the results of Exercise 7.2,

Making use of (7.39), we have

To express the vertical velocity in height coordi-nates we make use of the approximate relation(7.33)

Note that this is the 300-hPa vertical velocityaveraged over the area of the anvil. The verticalvelocity within the updraft may be much larger.In contrast, divergences and vertical motionsobserved in large-scale weather systems are muchsmaller than those considered in this exercise.

7.42 Figure 7.31 shows the pressure and horizontalwind fields in an atmospheric Kelvin wave that

1.5 m s1

7 km

300 hPa 6.66 102 hPa s1

w300 300

RTp

300 Hp

300

6.66 102 hPa s1

0 3.33 104 s1 200 hPa

300 100 ( V) (300 100) hPa

V 1A

dAdt

0.20600 s

3.33 104 s1

propagates zonally along the equator. Pressureand zonal wind oscillate sinusoidally withlongitude and time, while v 0 everywhere.Such waves are observed in the stratospherewhere frictional drag is negligible in comparisonwith the other terms in the horizontal equationsof motion. (a) Prove that the waves propagateeastward. (b) Prove that the zonal windcomponent is in geostrophic equilibrium withthe pressure field.

7.43 Averaged over the mass of the atmosphere, theroot mean squared velocity of fluid motions is17 m s1. By how much would the center ofgravity of the atmosphere have to drop in orderto generate the equivalent amount of kineticenergy?

7.44 Suppose that a parcel of air initially at rest onthe equator is carried poleward to 30° latitudein the upper branch of the Hadley cell,conserving angular momentum as it moves.In what direction and at what speed will it bemoving when it reaches 30°?

7.45 On average over the globe, kinetic energy isbeing generated by the cross-isobar flow at arate of 2 W m2. At this rate, how long wouldit take to “spin up” the general circulation,starting from a state of rest?

7.46 The following laboratory experiment provides alaboratory analog for the kinetic energy cycle inthe atmospheric general circulation. The studentis invited to verify the steps along the way and toprovide physical interpretations, as appropriate.

A tank with vertical walls is filled with a homo-geneous (constant density) incompressible fluid.The height of the undulating free surface of thefluid is Z(x, y).

(a) Show that the horizontal component of thepressure gradient force (per unit mass) isindependent of height and given by

(b) Show that the continuity equation takes theform

where V is the horizontal velocity vectorand w is the vertical velocity.

V wz

0

P Z

Fig. 7.30 Physical situation in Exercise 7.41.

100 hPa

300 hPa

Fig. 7.31 Distribution of wind and geopotential height on apressure surface in an equatorial Kelvin wave in a coordinatesystem moving with the wave.

HL EQ

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and the four remaining terms in the two toplines as

With these substitutions, we obtain

Rearranging slightly (noting that ft 0)and reintroducing vector notation yields theconservation law

or, in Lagrangian form,

These equations appear in the text as Eq.(7.21a,b).

7.33 Consider a sinusoidal wave along latitude with wavelength L and amplitude in themeridional wind component. The wave isembedded in a uniform westerly flow withspeed U. (a) Show that the amplitude of thegeopotential height perturbations associatedwith the wave is fL2g where f is theCoriolis parameter and is the gravitationalacceleration. (b) Show that the amplitude ofthe associated vorticity perturbations is(2L)v. Show that the maximum values ofthe advection of planetary and relativevorticity are and (2L)2Uv, respectively,and that they are coincident and of opposingsign. (c) Show that the advection terms exactlycancel for waves with wavelength

LS is referred to as the wavelength of a stationaryRossby wave.

Ls 2√Ub

.

ddt

( f ) ( f ) ( V)

t (f ) V (f ) (f )( V)

t u

x v

y ( f )u

x

vy v

fy

u

x v

y

ux

vy

7.34 Suppose that the wave in the previous exercisepropagating is along 45° latitude and has awavelength of 4000 km. The amplitude of themeridional wind perturbations associated withthe wave is 10 m s1 and the background flowU 20 m s1. Assume that the velocity field isindependent of latitude. Using the results of theprevious exercise, estimate (a) the amplitude ofthe geopotential height and vorticityperturbations in the waves, (b) the amplitude ofthe advection of planetary and relative vorticity,and (c) the wavelength of a stationary Rossbywave embedded in a westerly flow with a speedof 20 m s1 at 45° latitude.

7.35 Verify the validity of the conservation ofbarotropic potential vorticity (f )H, asexpressed in Eq. (7.27). [Hint: You mightwish to verify and make use of the identitydyy dxx, where y 1x.]

7.36 Consider barotropic ocean eddies propagatingmeridionally along a sloping continentalshelf, with depth increasing toward the east,as pictured in Fig. 7.28, conserving barotropicpotential vorticity in accordance with (7.27).There is no background flow. In whichdirection will the eddies propagate? [Hint:Consider the vorticity tendency at points Aand B.]

7.37 During winter in middle latitudes, themeridional temperature gradient is typicallyon the order of 1° per degree of latitude, whilepotential temperature increases with height ata rate of roughly 5 °C km1. What is a typicalslope of the potential temperature surfaces inthe meridional plane? Compare this result

B

?

A

Fig. 7.28 Barotropic ocean eddies propagating along acontinental shelf, as envisioned in Exercise 7.36. Lighter blueshading indicates shallower water.

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Exercises 309

with the slope of the 500-hPa surface inExercise 7.17.

7.38 Making use of the approximate relationshipshow that the vertical motion

term

in Eq. (7.37) is approximately equal to

where %d is the dry adiabatic lapse rate and % isthe observed lapse rate.

7.39 Figure 7.29 shows an idealized trapezoidalvertical velocity profile in a certain rain area inthe tropics. The horizontal convergence intothe rain area in the 1000 to 800-hPa layer is105 s1, and the average water vapor contentof this converging air is 16 g kg1. (a)Calculate the divergence in the 200 to 100-hPalayer. (b) Estimate the rainfall rate using theassumption that all the water vapor iscondensed out during the ascent of the air.

Solution: (a) Let ( V)L and ( V)H referto the divergences in the lower and upper layers,respectively. Continuity of mass requires that

200

100( V)H dp 1000

800( V)L dp

w(%d %)

!Tp

Tpv

w,

Hence,

(b)

Making use of the relationship w, thevertical mass flux w can be obtained by dividing (in SI units) by . Therefore, the rate at whichliquid water is being condensed out is

We can express the rainfall rate in more conven-tional terms by noting that 1 kg in2 of liquidwater is equivalent to a depth of 1 mm. Thus therainfall rate is 3.27 104 mm s1 or

which is a typical rate for moderate rain.

7.40 In middle-latitude winter storms, rainfall(or melted snowfall) rates on the order of 20 mmday1 are not uncommon. Most ofthe convergence into these storms takesplace within the lowest 1–2 km of the atmosphere(say, below 850 hPa) where the mixing ratios areon the order of 5 g kg1. Estimate the magnitudeof the convergence into such storms.

7.41 The area of a large cumulonimbus anvil inFig. 7.30 is observed to increase by 20% overa 10-min period. Assuming that this increase inarea is representative of the average divergencewithin the 300 to 100-hPa layer and that thevertical velocity at 100 hPa is zero, calculate thevertical velocity at the 300-hPa level.

28.3 mm day1

3.27 101 mm s1 8.64 104 s day1

3.27 106 kg m2 s1

2 101 Pa s1

(9.8 m s2) 0.016

2 103 hPa s1

(105 s1) (200 hPa) 0

( V)L (1000 hPa 800 hPa) v1000

800 800

1000( V)L dp 1000

2 105 s1

( V)H ( V)L 1000 800200 100

Fig. 7.29 Vertical velocity profile for Exercise 7.39.

0

100

200

400

600

800

Pre

ssur

e (h

Pa)

Vertical Velocity (ω)

UpwardMotion

DownwardMotion

+–

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Solution: From the results of Exercise 7.2,

Making use of (7.39), we have

To express the vertical velocity in height coordi-nates we make use of the approximate relation(7.33)

Note that this is the 300-hPa vertical velocityaveraged over the area of the anvil. The verticalvelocity within the updraft may be much larger.In contrast, divergences and vertical motionsobserved in large-scale weather systems are muchsmaller than those considered in this exercise.

7.42 Figure 7.31 shows the pressure and horizontalwind fields in an atmospheric Kelvin wave that

1.5 m s1

7 km

300 hPa 6.66 102 hPa s1

w300 300

RTp

300 Hp

300

6.66 102 hPa s1

0 3.33 104 s1 200 hPa

300 100 ( V) (300 100) hPa

V 1A

dAdt

0.20600 s

3.33 104 s1

propagates zonally along the equator. Pressureand zonal wind oscillate sinusoidally withlongitude and time, while v 0 everywhere.Such waves are observed in the stratospherewhere frictional drag is negligible in comparisonwith the other terms in the horizontal equationsof motion. (a) Prove that the waves propagateeastward. (b) Prove that the zonal windcomponent is in geostrophic equilibrium withthe pressure field.

7.43 Averaged over the mass of the atmosphere, theroot mean squared velocity of fluid motions is17 m s1. By how much would the center ofgravity of the atmosphere have to drop in orderto generate the equivalent amount of kineticenergy?

7.44 Suppose that a parcel of air initially at rest onthe equator is carried poleward to 30° latitudein the upper branch of the Hadley cell,conserving angular momentum as it moves.In what direction and at what speed will it bemoving when it reaches 30°?

7.45 On average over the globe, kinetic energy isbeing generated by the cross-isobar flow at arate of 2 W m2. At this rate, how long wouldit take to “spin up” the general circulation,starting from a state of rest?

7.46 The following laboratory experiment provides alaboratory analog for the kinetic energy cycle inthe atmospheric general circulation. The studentis invited to verify the steps along the way and toprovide physical interpretations, as appropriate.

A tank with vertical walls is filled with a homo-geneous (constant density) incompressible fluid.The height of the undulating free surface of thefluid is Z(x, y).

(a) Show that the horizontal component of thepressure gradient force (per unit mass) isindependent of height and given by

(b) Show that the continuity equation takes theform

where V is the horizontal velocity vectorand w is the vertical velocity.

V wz

0

P Z

Fig. 7.30 Physical situation in Exercise 7.41.

100 hPa

300 hPa

Fig. 7.31 Distribution of wind and geopotential height on apressure surface in an equatorial Kelvin wave in a coordinatesystem moving with the wave.

HL EQ

P732951-Ch07.qxd 9/12/05 7:45 PM 0

Copyright © 2006 Elsevier Inc. All rights reserved.

Additional Credits Chapter 7 Figure 7.16 [Reprinted from Introduction to Dynamic Meteorology, 4th Edition, J. R. Holton, p. 96, Copyright 2004, with permission from Elsevier.]

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