Pag 10a18 Geofisical Fluid Dinam 2

8/13/2019 Pag 10a18 Geofisical Fluid Dinam 2

1/9

528 Geophysical Fluid Dynamicsfirst step is to simplify the vorticity equation for quasi-geostrophic motions,a ssuming tha t the uelocity is geostrophic lo the lowest order The small departuresfrom geostrophy, however, a re important, sinc e the y de termine the euolutionof the ftow with time .

We s ta rt wi th the sha llow wa te r poten tial vor tici ty equat ion

+ / ) =0DI hwhich can be written as

D Dh +1 -+ 1 -=0DI DIWe now expand the material deriva tive and subs ti tu te h = H r , where Histhe uni fo rm undisturbed depth ofthe laye r, and r is the surface displacement.This gives

a a a ) a r a r a r )H+T) -+u-+v-+f:lv - +fo) -+u-+v- =0 114)a l sx a y a l l x a yHere we have used DII Dt =v dfl dy =f3 v We have a lso replaced I by io inthe sec ond term, since the ,B-plane a pproximation neglects the variation ofIexcept when i t involve s df/ dy . For small perturbations we can neglect thequa dratic nonlinear terms in 114), obtaining

a a rH-+Hf3v-fr. -=0a l o iJI 115 )This is the linearized form of the potential vortrcity equation. Its quasi-geost rophic ver sion is obtained i f we subs ti tu te the approx imate geos trophicexpressions for velocity:

. g a T Julo a yg a rvfo a x

From this the vorticity is found asg a 2 T J a 2r ) ta x 2 a y 2

so that the vorticity e quation 115) become s

I16)

gH a 2T i r ) gHf:l a r _ lo iJT=OI al a x 2 a y 2 io a x iJlDenoting e =. .JgH , this becomes

i J2r a 2r _i ) f3 iJT= OiJl a x 2 al e2 T J iJx 117 )

1

15 Rossby Wave 529

This is the quasi -geos trophic forrn of the l inea rized vor tici ty equation, whichgove rns the flow of large-scale motions. The ratio el io is re cognized as theRossby radius. Note that we have not set a r /i J t = o in 115) during theder iva tion of 117), a lthough a s tr ic t val id ity of the geost roph ic re la tions 116)would requi re tha t the hor izontal d ive rgence , and hence a T ) f a l , be zero . Thisi s be cause the departure from stric t geostrophy determines the evolution oft he flow, desc ribed by 117). We can therefore use the geostrophic relationsfor velocity everywhere except in the hor izontal divergenc e term in the vorticityequation.Dispersin RelationAssume solutions of the form

= j e(k: +I)-wt)We shall regard w as positive ; the signs of k and then determine the directionof phase propagat ion. A subst itut ion into the vor tici ty equat ion 117 gives

11 8)3 kw = e+/2+ f fe2

This is the dispersion relation for Rossby waves The asymmetry of the disper-sion relation with respect to k and signifies that the wave motion is notisotropic in the horizontal, which is expected because ofthe f:l-effect. Althoughwe have derived it for a single homogeneous layer, it is equally applicable tostratified ows if is replaced by the corresponding infernal value , which isc=yg H for reduced gravity model see Section 7.17) and c= NHln7T forthe nth rnode of a continuously stratified model. For the ba rotropic mode e isvery large, and t J / c 2 is usually negligible in the denominator of 118).

The dispersion relation w k, 1 in 118) can be displayed as a surface,taking k ano 1 along the horizontal axes and w along the vertical axis. Thesection of thi s surfa ce along 1=0 is indicated in the upper panel of Figure13.29, and se ctions of the surface for three values of w are indicated in thebot tom panel. The contours of constant w are circles, since the dispersionrelation 118) can be written as k + L ) 2 + 1 2 = L ) 2 _ I 42w 2w eThe def ini tion of group ve loc ity

a w a we =-- ak al

shows that the group velocity vector is the gradient of w in the wavenumberspace. The direction of cg is therefore perpendicular to the w contours, asindica ted in the lower panel of F igure 13.29. For 1= O, the maximum frequencyand zero group spee d a re a ttained at kc/ fa = 1 corresponding to wma,fo/f3c=0.5. The maximum frequency ismuch srna ller than the Coriol is f requency. For


2/9

530 Geophysicat Huid Dy namics

O

5

wfo c

nondi spersi regio n

-3 -2 1 okCl fo

lefo2

-5fo

Fig.I3.29 Dispersion relation w k, 1 for a Rossby wave. The upper panel shows w versus k for O. Regions of positive and negative group velocity e.x are indicated. The lower panel showsa plan view of the surface wk, 1), showing contours of constant w on a klplane. The values ofwfo f3 e for the three cireles are 0.2, 0.3, and 0.4. Arrows perpendicular ro w contours indicatedirections of group velocity vector eg [Adapted from Gil 1982).)

example, in he ocean the ratio cuma. fo =0.5f3 c/ f is of arder 0.1 for thebarotropic mode, and of order 0.001 for a baroclinic mode, taking a typicalmid-Jatitude vaJue of lo -4SI a barotropic gravity wave speed ofe20a m/s, and a baroclinic gravity wave speed of e 2 m/s. The shortest periodof mid-latitude barocJinic Rossby waves in the ocean can therefore be morethan ayear.

1l

15. Rossby Wave

The eastward phase speed isw f3 I19)e+{2+ f /e2

The negative sign shows that he phase propaga/ion is a{ways westward. Thephase speed reaches a maximum when {2-,> O, corresponding to very largewavelengths represented by the region near the origin of Figure 13.29. In thisregion the waves are nearly nondispersive and have an eastward phase speed

f3e e =x f

With f3=2 X 10- m - s - a typical baroclinic value of e - 2 m i s, and a mid-latitude value of fo - 10-4S-I this gives e x _10-2 m/s. At these slow speedsthe Rossby waves would take years to eross the width of the ocean at mid-latitudes. The Rossby waves in the ocean are therefore more important atlower latitudes, where they propagate faster. [The dispersion relation 118),however, is not valid within a latitude band of 3 from the equator, for thenthe assump tion o f a near geostrophic balance breaks down. A difIerent analysisis needed in the trapics. A discussion of the wave dynamics of the tropies isgiven in Gill 1982 and in the review paper by McCreary 1985 .] In theatmosphere e is much larger, and consequently the Rossby waves propagatefaster. A typical large atmospheric disturbance can propagate as a Rossbywave at a speed of several meters per second.

Frequently the Rossby waves are superposed on a strong eastward meancurrent, such as the atmospheric Jet Str eam. If U is the speed of th is eastwardcurrent, then the observed eastward phase speed is

{3e x = U k2 2f/ c2

Stationary Rossby waves can therefore form when the eastward current cancelsthe westward phase speed, giving e=0. This is how stationary waves areformed downstream of the topographic step in Figure 13.21. A simpleexpression for the wavelength results if we assume =O and the flow isbarot ropic, so th atf1 e2 is negligible in 120 . This gives U =f31 k2 for station-ary solutions, so that the wavelength is 2 1T, j Uf3.

Lastly, note tbat we have be en rather cavalier in deriving the quasi-geostrophic vorticity equation in this section, in the sense that we have sub-stituted the approximate geostrophic expressions for velocity without a formalordering of the scales. Gill 1982 has given a mor e p recise derivation, expand-ing in terms of a small para meter. Another way to justify the dispersion relation118) is to obtain it from the general d ispersion relation 76) derived in Section10:

120)

w3 e2wk2 [2) - fJw - c2 {3k =O 121For cuj the first term is negligible compared to the third, reducing 121) to118).

531


3/9

532 Geophysical Fluid Dynamics

6 Barotropic Instability Section 11 .9 we discussed the inviscid stability of a shear ftow Uy) in anonrotating system, and demonstrated that a necessary condition for its insta-bility is that d2 U1 dy2 must change sign somewhere in the ftow. This was caIledRayleigh s point of inflection crilerion. In terms of vorticity [= +d U 1dy thecriterion says that d D dy must change sign somewhere in the ftow. We shallnow show that, on a rotating earth, the criterion requires that d[ f)1 dymust change sign somewhere within the flow.

Consider a horizontal current U y) in a medium of uniform density. the absenc e of horizontal density gradients only the barotropicmode is allowed,and U y does not vary with depth. The vorticity equation ist u . v f 122)This is identical lo the potential vorticity equation DI Dt [ f)1 h] O,withthe added simplification that the layer depth is constant because w O. Lethe total flow be decomposed into background ftow plus a disturbance:

u Uy)+uv

The total vorticity is then- . dU av a u ) dU = + l=- dy ax - ay dy +\7

where we have defined the perturbation strearn functionu= _ a l

ayv,=af

axSubstituting into 122) and linearizing, we get the perturbation vorticityequation

a a d 2U)a,_ \72,)+U-\72f)+ 13--- =a l ax dy2 axSince the coefficients of 123) are independent of x and 1, there can be solutionsof the form

123)

f J y) ekX-CThe phase speed e is complex and solutions are unstable ir its imaginary partc>O.The perturbation vorticity equation 123) then becomes

[ d2 2 J A [ d 2 U ] AU-e) --k + 13-- =0dy2 dy2Comparing this with Equation 11.76), derived without Coriolis forces, it isseen that the effect of planetary rotation is the replacement of _d Uy? by{3- d Uy2). The analysis of that section therefore carries over to the present

17. Baroelinic Instability

y

J

u

Eq uator

Fig.13.30 Profiles of velocity and vorticity of a westward tropical wind. The velocity distributionis barotropically unstable, since d[ f)1 dy changes sign within the Row . [After Houghton 1986).]

case, resulting in the foIlowing criterion: A neeessary condition [or the inviscidinstability of a b arotropic current Uy) is that the gradient ofthe absolute uorncity

d _ d2U-+f)={3--dy dy? 124)must change sign somewhere in the flow. This result was first derived by Kuo1949).

Barotropic instability quite possibly plays an important role in the nstabil-ity of currents in the atmosphere and in the ocean. The instability has nopreference for any latitude, because the criterio n involves 13nd notf However,the mechanism presumably dominates in the tropies, because mid-latitudedisturbances prefer the baroclinic instability mechanism discussed in the follow-ing section. An unstable distribution of westward tropical wind is shown inFigure 13.30.

7 Baroclinic InstabilityThe weather maps at rnid-Iatitudes invariably show the presence of wave-likehorizontal excursions of temperature and pressure contours, superposed oneastward mean flows such as the Jet Stream. Similar undulations are aIsofound in the ocean on eastward currents such as the Gulf Stream in the northAtlantic. A typical wavelength of these disturbances is observed to be of theorder of the internal Rossby radius, that is about 4000 km in the atmosphereand 100 km in the ocean. They seem to be propagating as Rossby waves, buttheir erratic and unexpected appearance suggests that they are not forced byany external agency, but are due to an inherent instabilit y of mid-Iatitudeeastward ftows. otherwords, the eastward flows have a spontaneous tendencyto develop wave-like disturbances. this section we shall investigate the

533


4/9

534 Geophysicat Huid D namc5

instability mechanism that is responsible for the spon taneous relaxation ofeastward je ts i n t o a meanderi ng s ta te .

The pol ewa rd decr ea se of the sola r i rradiation results in a pol ewarddecrease of the ternperature and a consequent increase of t he density. Anidea liz ed distribution of the atmospheric density in the nor thern hemisphereis shown in Figure 13.31. The density incr eases northward due to the lowertempera tu re s nea r the pol es and dec reas es upward because of static stability.According t o the therma l wind relation (15), an eastward flow (such as theJet Stream in the atmosphere or the GulfStream in the Atlantic) in equilibriumwith such a density structure must have a vel oc it y that i nc re a s es wi th hei gh t.A systern with inclin cd density surfaces, such as the one i n F igure 13. 31 , hasmore potential energy than a system with horizontal density surfaces, just asa syst em wi th an incl ined f re e surface has more poren tial energy than a systemwith a horizontal free surface. It is therefore potentially unstable, for it canrelease the s to red pot en ti al energy by means of an instability that wou ld causet he density surfaces to flatten out. In the process , vertical shear of the rneanfiow U(z) would decrease, and per turbat ions would ga in k ineti c ene rgy.

Jnstability of baroclinic jets that release potential energy by laueni ng outthe density sur fa ce s i s c al led t he baroclinic instabitity. Our analysis would showthat the preferred scale of the unstab e waves is indeed of the order of theRossby r adius, as observed for the mid-Iatitudc weather dis turbances. Thetheory of barocIinic instability was developed in the 1940s by Bjerkes,Holmboe, Eady, and Charney and is considered on e of the major t riumphso f geophysical fluid m echanics. Our presenration is esse ntial ly based on thereview article by PedJosk y (1971).Considcr a basic state in which the density is s tably s trat if ied in the verticalwith a uniform buoyancy frequeney N, and increases northward at a constanrate D /5 / a y Aceording to the thermal wind relation, the constaney of ii pj a yrequir es t hat t he ver tical shear of the basic eastward flow U(z) also be constant.The j3-eflect is neglected, since it is not an essential requirement of theinstability. [The j3.effect do es modify the insrability, however.] This is borneout by th.e spontaneous appearance of undulations in laboratory experimentsin a ro rating ann ulus, in which the inner wall is maintained at a highertempcra t u r e t han the outer wal . The f:J-efTect is absenr in such a n experiment.

. I.>- ,:o v < "'-/5"

zO

w H

ynonh Equator

Fig 3 3 Lines of co nstan t density in thc northcrn h emispheric atmos pb ere . The lines are nearlyhorizontal, tbc slopcs being greatly ex agg eraf ed in Ihe figure The velocity U{ z) is into t he planoof pa per.

7 Baroclnic lnstability 535

Perturbation Vorticity EquationThe equati ons f or t ot al t l.owa r e

au au au ap- u- v--1v= ---a ilx a y Po a xav ilv clv 1 i'ip- u- v-u=---a l ilx ay P o a y

i 'ip= pgaz 125ilu dV ilw =0ilx y az

ilp ilp-u-v-w-=Oa l a x a y a zwhere P o is a constant reference clcnsity. We assurne that rhe total flow iscomposed of a basic eastward jet U(z) in geostrophic equilibrium with thebasic density structure (y, z ) shown in Figure 13.31, plus perturbations. That15

U(z)+ u(x, y, z )v v x, y, z

w=w x,y,Z) 126p p y, Z)+ p x, y, z p jy, z);' p x, y, z)

The bas ic flow is in geostrophic and hydrostatic balance:IV _. E

Po Jy 127p O= pg8zElmnating the pressure, we get the therrnal wind relat on

dU g a j5=dz lpo a y (28)which says that the eastward ow rnust incre ase with height because aj 5/a y > For s ir npl i ci ty we assume that apl a)' is constant, and that U at the surfacez 0, so hat the background flow is

U= Uo zH

where U; is the velocity at the IOp of (he layer at z H.


5/9

536 Geopnycot Huid Dynamics

\Ve first form a vorticity equation by cross-differentiating the horizontalequations of motion in (125), obtaining

a ; e; e c aw-+u-+v--( ;+ f) -=0al ax ay dZThis is identical lo (92), except for the exclusion of the ,B-effect here ; thealgebraic steps are therefore not rcpeated. Substituting the decoroposition(126), and noting that =1 ; because the basic ftow U =Uoz/ H has no verticalcornponent of vorticity, (129) becomes

(129)

elr a ? ; a , --I-U--J-=Oal dX dZ (130)where rh e nonlinear terrns have been neglect ed. This is the per turbation vorticityequation, whieh we shall now write in terrns of p,

Assume that the perturbations are large-scale and slow, so that the veloeityis nearly geostrophic:

apu=---PoJ ayv,=_I_iJp'

Po ax (131)from which ihe perturbation vorticity is found as

1r=-v p' 13 2)PoWe now express w in (130) in terrns of p', The density equation gives

e a a a- (IH p')+( U -1-u') - (+p')+ v'- -I- p')+w (+p') =Oal ax ay azLinearizing, we gel

ap 1 U+ u ap _PoN'w' =0al ax ay g

where N' = gp; '(elP dZ l, The perturbation density p ean be written in t errnsof p by using the hydrostatic balance in (125), an d subtracting the basic state(J 27). This gives

(133)

dP' ,0= ---p g

(134)which says that (he perturbations are hydrostatic. Equation

1 [ ( a a ) a p dU ap] = poN' ,+ U ax a ;- dz dX(133) then gives

(135)where we have writren ap la ) in terms of the therma wind dU / dz. Using (132)and (135), the perturbation vorticity equation (130) becomes

( a a ) [ J' a p ]u- p-- =0al iJx H N'a z' (136)

I

11I

7 Baroclinc lnsmbiliry 537

This is the equation t hat governs the quasi-geostrophic perturba tions on aneastward current U( z).Wave Solution\Ve assume t hat the flow is confined be t ween two horizontal planes a t z=Oan d z =H a n d t hat it is unbou nde d in X a nd y. Real flows are likely to bebounded in (he y dircction, especially in a labo raro ry situation of flow in anannular region, where the walls set boun dary conditions parallel O the f1ow.The boundedness in however, simply seis up normal modes in rhe fonnsin(nrryl L), where L is the width 01' the channel. Each of ihese modes can bereplaced by a periodicity in y. Accordingly, we assurne wave-like solutio ns

P =p( ) e +'-''''The perturbation vorricity cquai io n (136) rhen gives

dfi ,_ -(lp=Od

( 137)

( 138)where

N 2 , : n-= (k'+/')r 139The solution 01 (138) can be written as

p = A cosh e x ( , - ) Bsinh e x (z - i) ( 1401Boundary conditions h ave 10 be imposed on sol ut ion (140) in order to derivean in st abilit y criterion.Boundary ConditionsThe conditions are

, = O at z = O , HThe corresponding conditions on p cm be Iound rom (135) ami U= Voz /N,We ge t

a O p U"Z;'P' U"iJp'- -- - - -- + - --- =Oal a z H dX az H Jx at z = 0, Hwhr wetherefore

have also use d U = Uoz The \VO boundary conditions area O p U ( l p ---- _.__= al iJz H dX at z = O

a U e l a ,-.- . -.._--"L+uo-P_=o l z H elx s x l e al z =H


6/9

538 Geophysical ui DynamicsInstabiliry CriteronUs ing (137 ) and (140 ), the f or egoing boundary condi rions r equir e

[ aH o; aH]A acsinh---cosh-2 H 2

[aH Va aH]B -accosh-+-sinh- =02 H 2

[ aH UD aH]A a U -c)sinh---cosh-o 2 H 2

[ aH Ua aH]+B a U,,-c cosh---sinh- =02 H 2

where c= wik. i s t he eastward phase velocity.This is a pair of homoge neous equations for the constants and B Fornontrivial solut io ns exi st, the determinant o f the coefficients must vanish.Thisgives, after some str ai gh tf orwa rd a lgeb ra , the pha se vel oc ity

-u, Un aH aH n aHe 2 aH J(z-tanh-2- Z-:-COlh-2- (141)Whe ther t he sol ut i on g rows with time dep end s on the sign of the radicandoThe behavior of the Iunctio ns under the radical sign is sketche d in Figure13.32. It is apparent that the first factor in the radicand is positive since

aH /2> tanh(aH /2) Ior a values of H. However, the second factor isncgat ive f or small values of aH for wh ich aH/2


7/9

540 Geopnysical Fluid ynamics

However, the wavelength A 2.61\ does not grow al [he fastest rateo lt canbe shown from (141) that the wavelength with the largest growth rate is

I A ,=3.91\ IThis is therefore the wavele ngth that i s observed when the instability develops.Typical values for J N, and suggest that A, , - 4000 km in the atmosphereand 200 km in the oeean, which agree with observations. Waves much smallerthan the Rossby radius do no grow, and the ones much larger than the Rossbyradius grow very slowly.EnergeticsThe foregoing analysis suggests ihat the ex.istence of weather waves is dueto the fact that small perturbations can grow spontaneously when superposedon an east ward current maintained by the sloping densiry surfaces (figure13.31). Although the basic current does have a vertical shear, the pert urbationsdo not grow by extracting energy from the vertical shear field. Instead, theyextraer the ir energy from the poiential energy storcd in the syst ern of slopingdensity surfaces. The energetics of t he baroclinic instability is the refo re quitediferenr than thai of the Kelvin- H elmholtz instability (which 3150 has a verticalshear of the rne an flow), where the perturbati onReynolds stress UIV inieractswith the vertical sh ea r and exrracts energy from the mean shear flo w. Thebarociinic instability is not a shear flow insrability; the Reynolds stresses aretoo small beca use of the small IV in quasi-ge ostrophic large-scale flows.

The energetics ofthe baroclinic instability can be understood by examiningthe equation for the perturbation kin etic cnergy. Sueh an equation can bederived by multiplying the cquations for a L a l and avlal by u and vrespectively, adding rhe two , and integrating over the region of f low . Because1 the assumed periodicity in x a nd J, the ex ten t of the region of integrationis chosen to be one wavelength in either direction. During this inregration, theboundary eondition of zero normal flow on the walls and periodiciry in x andy are used repeatedly. Th e procedure is similar to that for the derivation ofEquation n1.83) and is not re peate d here. Thc result is

J = g IVpdxdvdzdt -where K is the global perturbation kineric energy

K f (,,2 + 0.2 dx dy d:In unsiable flows we rnust have dK] dI> O, which rcqui res t hal the volumeintegral of wo mJst be negative. Let us denote the volume average of wpby wp A negative IVp means that on the average rhe Jighter uid rises andrhe heavier uid sink s. y such an interchange the cerner of gr avity of thesysrcm, and thercfore its potential energy, is lowered. The interesti ng point isthat this can not happen in a stably stratified system with horizont al density

8 Geostrophic Turbulence 541

J o y

;g ;

y

fig 13.33 Wedge of instabitiry shaded in a baroclinc inst ability. The wedge is bounded yconsrant dens ity lines and the horizontal. Unstable waves have a pa ni de rrajectory that Iallswirhin {h wedge

surfuces; in that case an exchange offluid particles raises the pot enrial energy.Mor eover, a basic state with inelinee density surfaces (Figure 13.31) cannothave lVp O if the particJe excursions are vertical, If, however, the particleexcursions a ll wi thi n the wedge formed by the constant density lines and thehorizontal (Figure 13.33), then an exchange of fluid particles takes lighterparticles upward (and northward) and denser particles downwar d (and south-ward). Such an interchange would tend to rnake the density surfaces morehorizontal, releasing potential energy from th e mea n density field with aconscquent growth of the perturbation energy. This type of convection is c alJedsloping coniection According lO figure 13.33 the exchange of fluid partic1eswithin the wedge of insiability results in a net poleward transport of heat fromthe tropies, which servcs to redistribute the largcr solar heat received by thetropies.

ln su mmary, baroc1inic instability draws energy frorn the potential energyof the rncan derisity field. The resulting eddy rnoticn has particJe trajectoriesthat are oriented at a small angle with the horizontal, so thar rhe resulting heattransfer has a poleward co mponent. The preferred seale of the disturbance isthe Ro ssby radius.

8 eostrophic TurlrulenceTwo corn mon modes of instability of a Jarge-scale curre nt system were pre sen-ted in lhe preceding sections. When the flow is strong enough, such instabiJitiescan make a flow chaotic or turbulent. A peculiarity of large-scale turbulencein the atrnosphere or the ocean is that it is esse ntially two-dimensional innature. The existence of the Coriolis force, stratification, and small thicknessof geophysical media severely restricts the vertical velocity in large-scale flows,which tend 10 be quasi-ge ostrophic, with the Coriolis force balancing the


8/9

542 Ceophyscat Fluid Dynamcs

horizontal pressure gradient to the lowest order. Since vortex stretching, a keymechanisrn by which ordnary three-dmensional turbuleru ows transferenergy from la rge to small scales, is absent in two-dimensional ft ow, oneexpects that the dynamics of geostrophic turbuleoce are lkely lO be funda-mentally diflerent from that of ihree-dirnensional laboratory-scale turbulencediscussed in Chapter 12. However, we can stil\ eall the motion tu rbulent ,because it is unpredietable and diffusive.

A key result on the subject was discovered by the meteorologist Fjortoft(1953), and since then Kraichnan, Leith, Batehelor, and others have con tributedto various aspects of the problem. A good discussion is given in Pedlosky(1987), to which the reader is referred for a fuller treatme nt. Here we shallonly point out a few important results.

An irnporta nt variable in the discussion Q[lwo-dimensional turbulence isenstrophy; which is the mean square vorticity t In an iso tr op ic t urbulent fieldwe can define an energy spectrum S(K), a funcrion of the magnitudc of thewavenumber K as

; ? = [5(K)dKlt can be shown that the enstrophy spectrurn is K 2 S(K), that is

[2= r K2S(K) sxwhih mak es sense because vorticity involves the sparial gradient of velocity.We consider a freely evolving iurbuleru field, in which the shape of thevelocity spectrum changes with time. The large sea les are esse ntial ly inviscid,so that both energy and enstrophy are nearly conserved:

J f . . . S(K)dK=Od od f KS(K)dK=Odt o

wher e terms proportional to the rnolecular viscosiiy v have be en neglected onthe right sides of the equations. The enstrophy conservatio n is unique IOtwo-dirnensional turbulence because of the abse nce of vortex stretching.

Suppose th at the energy spectrurn initially contains all its energy alwavenumber Ko. Nonlinear inter actions transfer this energy to other wavenum-bers, so that t he sharp spectral peak srnears out. For the sak e of argument,suppose t hat all of the initial energy goes ro I\VO neighboring wavenurnbersK and K with K,


9/9

544 Geoph sical Fluid n arnio

region. I r energy is injected into the system al a rate E, then the energy spectrumin t he energy caseade region has the form S( K) e x E2Jl K -53; the argument isessentially the same as in the case of ihe Kolmogorov spectrum in three-dimensional turbulence (Section 12.9), except that the transfer is backwards.A dimensional argument also shows that the energy spectrum in the enstrophycascade region isofthe form S K) e x af3 K -, where a is the forward enstrophyf lux to h igher wavenurnbers. There is negligible energy flux in the enstrophycascade region.

As the eddies grow in sizc, they become increasingly immune 10 viscousdissipation, arid the inviscid assumption implied in (143) becomesincreasingly applicable. (This would not be rhe case in three-dimensionalturbulence, in which the eddies eontinue 10 decrease in s ize until viseous eflectsdrain energy out of the system.) In contrast, the corresponding assumption inthe enstrophy conservation equation (144) becomes less and less valid asenstrophy goes to smaller sea les, where viscous dissipation drains enstrophyOUI of the system. At later stages in the evolution, then, (144) may nOI be agood assumption. However, it can be shown (see Pedlosky, 1987) thar ihedissipation of enstrophy actually intensifies the process of energy transfer tolarger scales, so that the red cascade (that is, transfer la larger scales) ofenergy is a general result of two-dimensional turbulence.

The eddies, however, do not grow in size indefinitely. They becomeinereasingly slower as their length scale incr eases, while their velocity scaleti remains constant. The s lower dynamies rnak es thern increasingly wave-Iik e,and the eddies transforrn into Rossby wave packets as heir length sea lebecomes of order (Rhines, 1975)ti,1- -{ (Rhines Ierigt h)where f3 dj] dy and u is th e rms fluctuating speed. The Rossby wave propaga-tion results in an anisotropic elon gation ofthe eddies in the ea st-west ( zonal )direction, while the eddy size in the north-south direction stops growing al U f 7 3 FinaIJy the velocity.field consists of zonally directed jets whose north-south extent is of order - f U 7 7 This has been suggested as an ex planaticn forthe existence of zonal jets in the atmosphere of the pla net Jupiter (Williams,1979). The inverse energycascade regime rnay not occur in the ea rths atmos-phere and the ocean at m id-latitudes, beca use ihe Rhines length (about 1000 kmin the atmosphere and 100 km in ihe ocean) is of the order of the internalRossby radius, where t he energy is inje cted by baroclinie insiability. (For theinverse cascade to occur, - f U 7 7 needs to be larger th an rhe sea le al whichenergy is injected.)

Eventually, however, the kin eric energy has 10 be d issipate d by mnleculareffects at the Kolmogor ov microscale 7 , which is of the order of a fewmillimeters in the oeean and the atmosphere. A fair hypothesis is that processessuch as internal waves drain energy out of the mesoscale eddies, and breakinginternal waves generare three-dimensional turbulence that finally cascadesenergy to molecular scales.

Exercises 545

Exercisesl. The Gulf Strearn flows northward along the east coast of the Unired

States with a surface current ofaverage magnitude 2 m/s.lfthe ow is assumedto be in geostrophic balance, find the ave rage s lope of the sea surface acrossthe current at a latitude of 45CN. Answer: 2.1cm per km]

2. A plate containing water ( 10-6 m2/s) above it rotates at arate of]0 revolutions per m inut e. Find the depth of the Ekman layer, assuming thatthe flow is laminar.

3. Assume ihat t he atmospheric Ekman layer ove r the earths surface ata latitude 1 45N can be approximated by an eddy visc os ity of v; = IO m 2/s.ff t he geostrophic veloci ty above the Ekman layer is Om/ s, whar is the Ekmantranspon across isobars? Allslrer: 2203 n/Is]

4. Find rhe axis rario of a hodograph plot for a semi-diurnal tide in th emiddJe of the ocean at a latitude of 45N. Assume t hat the mid-oeean tidesare rotational surface gravity waves of long wavelength and are unaffected byproximity of coasral boundaries. Ir t he depth of the o cean is 4 km, find th ewave lengt h, rhe phase velocity, an d the group velocity. Note, howe vcr, thatthe wavelength is comparable to the width of the oce an, 5 that the ncgle ctof coast al boundaries is not very realistic.

5. An internal Kelvin wave on rhe ther mocline of the ocean propagaresalong the west coasr of Australia. Thc rhermocline has a depth 50 11 1 an dhas a nearly discorninuous dcnsity change of 2 kg/ m across it. The luycrbelow the thermocline is deep. At a latitude of 30S, find the direci ion andmagnitudc of the propagation speed ancl the decay scale perpendicular 10 thecoast.

6. Using the dispersion relation m=kN2-1,j)/ ,,/-f) for interna:waves , show ihar the group ve locity vector is given by

N -e) ,(c . c,J = m+ k )-1 m 2/2 + k N) /,l m, -kJ

H ill /: Difterentiare the dispcrsion relation partially wirh respecr to an d mShow that cg and e are perpendicular and ha ve oppositely directed verticalcomponents. Verify that e is par allel 1 u.

7.. Supposethe atrnos phere at a latitude of45cN is idealized by a uniformlystrarified layer of height 10 km. across which rhe potential temperatureincreases by 50C.

(i) What is the value of the buoyancy frequeney N?(ii) Find the speed of a long gravity wave corresponding to the 11 = 1

barcclinic mode.(iii) For t he /1 J mode, find the westwar d speed of nondispersive i hat

is, very large wavelcngrh) Rossby waves. A/1sIVer: N=O.01279s-; C=40.71 mis; c,=-3.J2m/s]

.tt

Pag 10a18 Geofisical Fluid Dinam 2

Documents