Geofisical Fluid Dinam 1d1

8/13/2019 Geofisical Fluid Dinam 1d1

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480 Geophy ical Fluid Dynamics

4. Approrimate Equations lo r a Thin Layeron a Rotat ing Sphere 481

au iJw7 xz = 7 z., = pl Iv PlJH. az aX

au. iJU;j = pl I -- ----IiJxj iJx

where j is the v isco us stress tenso r. The s tr es s i n a laminar flow is caused bythe molecular exchanges of momentum. From Equation 4 .36 ), th e v isco usstress tensor in an isotropic incompressible medium in laminar flow is givenby 6)

W HU L

4 Approximate Equationsfor a Thin Layer on a Rotatill.gSphere

The atmosphere and the ocean are very thin layers in which the d ep th scaleof flo w is a f ew k il om et er s, w he re as t he h or iz on ta l s ca le is of the o rd er ofhundreds, or even thousands, of kilometers. The trajectories of fluid elementsarevery shalIow, and theverticalvelocitiesare muchsmallerthan the horizontalvelocities. In fact, the continuityequation suggests that the scale of the verticalvelocity W is related to that of the horizontal velocity U by

W it h t he a ss um ed f or m f or t he t ur bu le nt s tr es s, t he components of thefrictional force F = a ij aX become

Estimates of th e edd y coefficients v ary g reatly . Typ ical su gg ested v alues arevv lO m2/s and l JH - 105 m2/s for the lo wer atmosphere, and l v 1 m2/sand VH - 100 rn 2/s for the upper ocean. In comparison, the molecular valuesare lJ = 1.5 X 10- 5 m2/s for air and lJ = 10- 6 m2/s for water.

5av aw7,_z = 7 z,. = p l Iv PlJH az ay au av7 xy = 7yx = PVH - ay ax

In large-scale g eo ph ysical flo ws, h owev er, the frictio nal forces are p rov idedb y tu rb u len t mix in g, and the molecular ex ch ang es are n eg ligible. Th e complexity of turbulent behavior makes it i mp os si bl e t o r el at e t he s tr es s t o t hev elocity field in a simp le way . To proceed, then, we adopt the eddy viscosityhypothesis, assuming that the turbulent stress is proportional to the v elocitygradient field.

Geophysical media are in the form of sh allow stratified layers, in whichthe vertical velocities are much smallerthan horizontal velocilies. This meansthat the exchange of momentum across a h o rizon tal su rface is much weakerthan that a cr os s a v er ti ca l s ur fa ce . We e xp ec t then that t he v er ti ca l e dd yviscosity l v is much smaller than the h orizo n tal ed dy v isco sity VH, and wea ss um e t ha t t he turbulent stress co mp o nen ts h av e th e form

The d ifficu lty with set 5 ) is that the expressions for T and T,.z depend o n t hefluid rotation i n t he v er ti ca l p lane and not jus t th e d efo rmatio n . In Sectio n4.9 we saw t ha t a requirement for a con stitutive equation is that the stressesshould be independent of flu id ro tatio n and should depend only on thedeformation. Therefore, Txz should only depend on the combination au/az aw/ax , whereas the express ion in 5) depends on both deformation androtation. A tensorially correct geophysical treatment of th e frictio nal terms isd iscussed , for ex amp le, in Kamenkovich I96 7). Ho wever, th eassumed form 5 ) lead s to a s im pl e formulation for v isco us effects, as we shall see sh ortly .Since the eddy viscosity assumption is of questionable validity [which Pedlosky1 9 71 ) d escrib es as a rather disreputable and desperate altempt ), there doesno t seem to be any purpose in fo rmu latin g the stress-strain relatio n in morecomplicated ways, merely t o obe y the requirement of invariance with respectto rotation.

iJw7 Zl = 2p lJaz

;

where H is t he d ep th s ca le and L is the horizontal length scale. Stratificationan d C oriolis effects u su alIy con strain the v ertical v elocity to b e even smalIerthan UH L.Large-scale geophysical f1 w problems should b e solv ed u sin g sp hericalpolar co ord in ates. If, h owever, the h orizo ntal leng th scales are smaller thanth e radiu s of th e earth =6371 k m), th en the curvature of t he e ar ih c an beignored, and th e motion can b e stud ied b y adopting a local Cartesian systemon a tangent plane Figure 1 3.3). On this plane we t ak e a n xy z coordinates ys tem, w it h x i nc re as in g e as tw ar d, y northward, and z upward. Thecorresponding velocity components are u eastward), v northward) and wupward).

The earth rotates at araten = T rad/day = 0.73 x 10- 4 S

around the polar axis, in an anticlockwise sense looking from aboye the northpoleo From Figure 13.3, the components of angular velocity of the earth in th e


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482 Geophysical Fluid Dynamics 4. Approximate Equationslor a Thin Layer on a RotatingSphere 483

9)

n

Fig. 13.3 Local Carte,ian coordinales. The x - ax i s i s i nl o l he p l an e of Ehe paper.

veJocity, f i s c al le d t he p/anelary vorlicity. More c ommonly, f is re fe rre d toas the Corio/is parameler, or the Coriolis frequency. It is positive in thenorthern hemisphere an d n eg at iv e i n t he southern hemisphere, va rying fram 1.45x 10-4 s- at th e p ol es t o zera at the equator. This m ak es s en se , s in ce ap e rs o n s t an di n g a t the n o rt h p o le spins ar ound himself in a n a ntic loc kwisesense at arate n , w he re as a person standing at the equator d oes n ot s pi naround himself but simply translates. The quantity

Ti =2Tr/fis c aB ed t he inertial period, for reasons t ha t will b e c le ar i n S ec ti on 1i.

Th e vertical component of t he C or io li s f or ce , namely 2 nu cos e isgeneraBy negligible compared to the dominant terms in the vertical equationof motian, namely gp lpo an d po (ap laz). Using 6) an d 7), the equationsof motian 2) reduce to

Du 1 ap (a2u a2u) a2 u f v= v H 2 + V y - , o ax ax 2 ay az-Dv l ap (a 2v a v) a2v f u= vH V yo ay ax 2 ai az 2

=2n[i w cos e- vsin {)+ju sin {)-ku cos )]In the term multiplied by i we can use the condition w cos 8 v sin 8, beca usethe thin sheet appr oximation requires that w v. T he thr ee components of theCoriolis forc e a re there fore

Plane ModelTh e variation with latitude can be approximately representedby expandingf in a Taylor series about the c e ntral la ti tude ea:

Dw=_.. ..ap_gp+v (iw+a w) + v a2w o az o H ax a l Yaz

These a re t he equations of moton f or a t hi n s he ll on a r o ta ti ng earth. Notet h at o n ly the vertical component of the e a rth s angular velocity appears as acansequence of the flatness of th e fluid trajectories.f Plane ModelThe Coriolis parameter f = 2n sin 8 va ries with la ti tude Howe ver, we shallsee later that this va ria tion is important only for phenomena ha ving ve ry longtimescales several weeks) or very longlengthscales thousands of kilometers).Fo r many purposes we can assumef to be a c onstant, say fo = 2n sin 80 , where80 is the c entra l la ti tude of the re gion under s tu dy . A m od el u si ng a constantCariolis parameter is called an f-p/ane modelo

10)= fo+{3ywhere we de fined

7) 20 x u)x = - 2 n sin 8 v = fv 2 0 x u ) y = 2n sin 8 u = fu 2 0 x u lz = - 2n cos 8 u

n x = on). =n cos n z = n sin e

where ) is th e latitude. The Coriolis forc e is there fore

u v wj k

2f i x u = O 2n cos 2n sin

local Cartesian system are

where w e h av e d ef in ed I f=2n s in 8 I 8) ( df df de . 2 n cos eaf dy 90 = d8 dy 90 RHere w e h av e u se d f=2D sin 8 a nd d8/dy= l/R, where the radius of the earthis nearlyt o b e t wi ce th e verlical component of 0_ Since vorticity is twice the angular R=6371km


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Geophysical Fluid Dynamics

Fig.13.5 Thermal wind, indicated by heavyarrows poioting inlo lhe plane of papero Isobars areindicated by salid lines, and contours of constantdensity are indicared by dashed lines.

5 Geostrophic Flow 487

(18)

(17)

(16)2.l1v = _ app x ap2 l1u=1 apO= gp z

Taylor-Proudman TheoremA striking phenomenon occurs in the geostrophic flow of a homogeneous fluid ..I t c an only be observed in a laboratory experiment, since stratification effectscanno t be avoided in natu ra l flows. Consider then a laboratory experiment inwhich a tank of fluid is steadily rotated at a high angular speed n and a solidbody is moved slowly along the bot tom of the tank. The purpose of makingl1 large and the movement of the solid bodyslow is t o make the Coriolis forcemuch larger than the accelera tion terms , which must be made neg ligible forgeostrophic equilibrium. Away f rom the frictional effects of boundaries, thebalance is therefore geostrophic in the horizontal and hydrostatic in the vertical:

; 1JQx

---2

486

resuI ting in a positive op ox whose magnitude increases with height. Since thegeostrophic wind is to the right of the horizontal pressure force ( in the northernhemisphere), it follows that tl)e geostrophic velocity is into the plane of the paper,and its magnitude increases with height.

T hi s is easy to demonst rat e f rom an analysis of t he geos troph ic andhydrostatic balance

Elimination of by cross differentiation between the horizontal momentumequations gives v u2 l =0y xUsing the continuity equation, this gives

apfv Po x (12)=az (19)

(20)Equations (19) and (20) show that

Also, differentiating (16) and (17) with respect to z and using (18), w e getav au= =0az az

fa13)

(14)

apfu - Po apO g paz

(15) showng that the velocity vector cannot vary in the direction of fi . otherwords, steady slow motions in a rotating, homogeneous, inviscid f lu id a retwo-dimensional. This is the Tay/or-Proudman theorem, first derived by Proudman in 1916 and demonstrated experimentally by G. 1. Taylor soon afterwards.

In Taylor s experiment, a tan k w as made to rotate as a sol id body, and asmall cylinder was slowly dragged along the bottom of t he tank (Figure 13.6).Oye was introduced from a point A aboye the cylinder and directly ahead of

Eliminating p between (12) and (14), and also between (13) and (14), we get ,respectively,av = _-L apz o f xau=-L apz of

Meteorologists cal these the therma/ wind equations, because they give thevertical variation of wind f rom measurements of horizontal temperaturegradients. The thermal wind is a baroclinic phenomenon, because the surfacesof constant p and d o n ot coincide (Figure 13.5).

(2 j)


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488 Geophysical Fluid Dynamics Ekman Layer af a Free Surjace 489Rodies moving either parallel o r perpendicu lar to the axis of rotat ion carryalong their motion a so-called Taylor column of fluid, oriented parallel t o t h e .axis. The phenomenon is analogous to th e horizontal blocking caused by asolid body s ay amountain) in a strongly stratified system, shown in Figure 7.33.

6. Ekman Laye . al a Free Surface

26) 25) 24)

27

22)

duz=Ov v r aldz

dv z=O= atdzu v 7 O as z7

Multiplying 2 3) b y i = R and adding 2 2) , we getd 2 V Vdz I v

where we have defined the complex veIocityVE u iv

d2 uf v VY 2dzd 2vfu = Vy dZ2 23

Taking the z-axis vertically upward from the surface ofthe ocean, the boundaryconditions are

In the preceding section,we discussed a steady linear inviscid motion expectedto be valid away from frictional boundary layers. We shall now examine themotion with in f ric tiona l laye rs over horizontal surfaces. In visc ous f lowsunaffected by Coriolis forces and pressure grad ient s, the only terro which canbalance the viscous force is either the t ime derivative auj al oI the advectionu . Vu The balance of aujal and t he viscous force g iv es r is e t o a v is co us layerwhose thickness increases with t ime, a s in the suddenly accelerated platediscussed in Section 9.7. The balance of u V u and the visc ous f or ce give s r iseto a visc ous layer whose thickness increases in the direct ion of f low, as i n t heboundary Iayer over a semi-infinite pIate discussed in Section lOA In a rotatingflow, however, we can have a balance between the CorioIis and the viscousforces, and the thickness of the viscous Iayer can be invarant in time andspace. Two examples of such layers ar e given in th is and the following sections.

Consider first t he cas e of a f r ic tional l ayer near the free surface of theocean, which is a ct ed o n b y a wind stress r in the x direction. We shall no tconsider how the flow adjusts to the s teady s ta te bu t only examine the steadysolution. We shall assume that the horizontal pressure gradients are zero andthat t he f iel d i s horizontally homogeneous. From (9), the horizontal equationsof motion are

1 II II II1 Taylor columnA Si B I ,I I cm I II II IT 11I1

2.5 cm I C

Siue yiew

--TOflyiew

Fig. 13.6 G. Taylor s experiment in a strongly rotating flow of a homogeneous fluid.

it. In a nonrotating fluid t he wat er ~ o u l pass o ve r t he t op of the movingcylinder. In the rotat ing experiment, however, the dye divides at a point S, a sif it h ad b een b Io cked by an upward extension of the cy linder, and 1I0wsaround this imaginary cylinder, caIled the Taylor column Oye released froma point B within the Taylor co lumn remained there and moved with thecylinder. The conclus ion was that the f low outs ide the upward extension ofthe cy linder is t he s ame as if the cylinder ext ended acros s t he ent ir e wat erdepth and that a column of water directly aboye th e cylinder moves w ith it.The motion is two-dimensional, although th e solid bodydoes not extend acrossthe ent ire water depth. Taylor did a second experiment, in which he draggeda solid body paraUel to th e axis of Jotation. In accordance with awjaz = O heobserved that a column of f lu id i s pushed ahead . The l at er al velocity components u and v werezero. In both ofthese experiments, there are shear layersat lhe edge of the Taylor column.

ln summary Taylor s experiment established th e folIowing striking faetfor steady inviscid motion of homogeneous f lu id i n a s t r o ~ g l y rotating system:

Geofisical Fluid Dinam 1d1

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