744 JOURNAL OF THE ATMOSPHERIC SCIENCESInertial Taylor Columns and Jupiter's Great Red Spot1 ANDREW P. INGERSOLLDivision of Geological Sciences, California Institute of Technology, Pasadena(Manuscript received 18 July 1968, in revised form 4 February 1969)ABSTRACT A homogeneous fluid is bounded above and below by horizontal plane surfaces in rapid rotation about avertical axis. An obstacle is attached to one of the surfaces, and at large distances from the obstacle therelative velocity is steady and horizontal. Solutions are obtained as power series expansions in the Rossbynumber, uniformly valid as the Taylor number approaches infinity. If the height of the obstacle is greater than the Rossby number times the depth, a stagnant region (Taylorcolumn) forms over the obstacle. Outside this region there is a net circulation in a direction opposite therotation. The shape of the stagnant region and the circulation are uniquely determined as part of thesolution. Possible geophysical applications are discussed, and it is shown that stratification renders Taylor columnsunlikely on earth, but that the Great Red Spot of Jupiter may be an example of this phenomenon, as Hidehas suggested.1. Introduction Taylor (1917) and Proudman (1916) first showed thatsteady, weak currents in a rotating homogeneous fluidare two-dimensional: the fluid moves about in colunmswhose axes are parallel to the rotation vector. Moreover,since the currents tend to be deflected around solidobstacles, those columns of fluid which intersect anobstacle are isolated from the rest of the flow. Theseisolated regions are called Taylor columns. They werefirst observed in the laboratory by Taylor (1923) andmore recently by Hide and Ibbetson (1966) and Hideet at. (1968). They are not observed in the earth's atmosphere, but Hide (1961, 1963) has suggested that theGreat Red Spot is t~ manifestation of a Taylor columnin the atmosphere of Jupiter. However, there is no entirely satisfactory theory ofTaylor columns to compare with observation, either inthe laboratory or in planetary atmospheres. If anviscid theoretical model is used, the flow depends on theinitial conditions for all time (e.g., Stewartson, 1953).A steady, invisdd solution is obtained if one assumesthat the fluid is initially in a state of uniform rotation,but this solution is not unique. In addition, the laboratory experiments show that the fluid within the Taylorcolumn in stagnant, but the inviscid models with themost obvious initial conditions do not give this result. Jacobs (1964) obtained a unique solution, valid onlyfor slow, viscous flow in which accelerations relative tothe rotating reference frame are negligible. In his model,a homogeneous fluid is contained between planes perpendicular to the rotation vector, with an axially ~ Contribution No. 1550 of the Division of Geological Sciences,California Institute of Technology, Pasadena.symmetric obstacle attached to one plane. The fluidwithin the Taylor column is found to be stagnant,and the streamlines are symmetric to the left and rightof the obstacle, as well as upstream and downstreamfrom it. Jacobs' solution emphasizes the importance ofdissipation in determining unique steady solutions, butthe relative accelerations are not negligible for mostlaboratory and geophysical flows. In fact, the streamlines observed by Hide and Ibbetson are decidedlyasymmetric, as are the markings around Jupiter's GreatRed Spot. In this paper, we consider steady solutions which arevalid when the relative accelerations predominate overthe viscous accelerations. The model resembles theearlier invlscid models, but the lack of uniqueness hasbeen removed by formulating the problem with viscosityincluded, and then letting the viscosity tend to zero. The general nonlinear problem is discussed in thenext section for a homogeneous fluid bounded by horizontal planes which are perpendicular to the rotationvector. The fluid is in rapid rotation, so the flow isquasi-horizontal and quasi-two-dimensional. The equation is derived for the lowest order interior stream function, which describes the flow outside the Ekmanboundary layers in the limit as the Rossby number approaches zero and the Taylor number approachesinfinity. Anticipating that Taylor columns will appearas regions of closed streamlines above the obstacle, weshow that the fluid must be motionless within any suchregion in the steady state. This follows since the Ekmanlayer transport is always to the left of the interiorvelocity at both the top and the bottom surfaces. Theassumption is made that the edge of the Taylor columnis not a high vorticity layer. The assumption is based onJuLY 1969 ANDREW P. INGERSOLL 745a heuristic argument, since the appropriate nonlinearboundary layer equation has not been solved. From thiscondition, and the condition of no motion within theTaylor column, we derive an important free surfaceboundary condition which is applied on the closed portion of the critical streamline. No attempt is made to solve the general problem,when the effects of inertia and friction are of equalimportance. Solutions are obtained only in the inviscidlimit, although the effects of friction are retained in thefree surface boundary condition. However, the inviscidproblem is still nonlinear, even when the solutions areexpanded as power series in the Rossby number. Thisis because the location of the free surface at the edge ofthe Taylor column is not known a priori. The differential equation becomes linear when the flow is uniformat infinity away from the obstacle, but the nonlinearitydue to the free surface boundary condition remains.Therefore, the inviscid problem with uniform flow atinfinity is solved only in two special cases, but theseprovide insight into the solutions for more generalobstacle sizes and shapes. This paper concludes with a brief discussion of the effects of stratification, baroclinicity and the fi-effect as they apply to the formation of Taylor columns in planetary atmospheres. Stratification and baroclinicity are important in the earth's atmosphere, which explains why Taylor columns are not observed on earth. How ever, the importance of these effects varies inversely as the horizontal scale of the phenomenon, so they may not be important in the dynamics of Jupiter's Great Red Spot. The fi-effect probably is important in the flow around the Great Red Spot, but this fact is not in consistent with Hide's hypothesis that the Spot is a Taylor column.2. General formulation Consider a homogeneous, incompressible fluid contained between two horizontal planes. The systemrotates counterclockwise about a vertical axis, andvelocity is measured relative to the state of uniformrotation. The obstacle is a slight irregularity in one ofthe boundaries, and at large distances from the obstacle the flow is uniform and horizontal. All components of the velocity must vanish at solid boundaries,and the flow is steady in the rotating reference frame. Let z be a dimensionless vertical coordinate, scaledby the distance H between the planes. Let x and y behorizontal coordinates, also scaled by//. Let u, v, w bethe x, y, z components of velocity, scaled by V, acharacteristic velocity. Then q= (u,v,w) is the dimensionless velocity vector. The dimensionless quantity pis the actual pressure minus the hydrostatic pressure(due to the gravitational and centrifugal accelerations),scaled by the quantity 2~Vttp, where [2 is the rotationrate and p the density. Let z= 1 be the equation of theupper boundary, and z = h (x,y) that of the lower boundary. We assume that h-~ 0 as x~+y~-~ o~, so h is theheight of the obstacle relative to the depth H. Then ~= V/(mR), (~)is the Rossby number and R=~R2/~, (2)is the Taylor number, where p is the kinematic viscosityof the fluid. In this section we consider the limit e--* 0, eR~= O(1), h/~=O(1), (3)corresponding to weak flow in a rapidly rotating system.The quantity eR- is the ratio of the time constant fordecay of vorticity by Ekman layer suction [the spin-uptime (Greenspan and Howard, 1963)-] to the advectiontime H/V. And the quantity hie is the ratio of thetopographically-induced vorticity to the vorticity V/It. In this notation, the continuity equation and NavierStokes equations are u,+v~+w,=O, (4) 1~q.vu-v = -p~-k--V'-'u, (5a) 2R 1~q.Vv+u = - p.~d---V2v, (Sb) 2R 1eq.Vw = --p,+--V2w. (5c) 2RFrom these we derive the z component of the vorticityequation: 1 ~q'V~'-w~(lnt-e~')q-~(w~v~-wuu~) =--V2~, (6) 2Rwhere ~= v~--uu is the z component of vorticity. Finally,we derive the Bernoulli equation 1q.V (p+eE) =--(uV2u+vWv+wWw), (7) 2Rwhere E=-(uXd-v~+w2) is the dimensionless kineticenergy of the fluid. In accordance with (3), we introduce an ordinarypower series expansion in ~, i.e., u = u(-~+ ~u<~+ ~u(~)+ .... (8)It is consistent to assume that ur, v(~) and w(n) arequantities of order unity whose derivatives with respectto z are large only in the vicinity of the top and bottomsurfaces. Henceforth, we shall refer only to the interiorflow; modifications due to the Ekman boundary layerswill be introduced as boundary conditions to be appliedat z = 1 and z = h. From (5) we have, for the interior flow ,<0)=p?), u(0)=_p?), O=p?); (9)746 JOURNAL OF THE ATMOSPHERIC. SCIENCES VOLUME 26whence from (4) . u, (0) = v~<-~ = w~ (0) = 0. (10)Thus, to lowest order the interior flow is two-dimensional with p<0) as stream function. Most theories of Ekman boundary layers are derivedunder the assumption ~RI<<I (e.g., Jacobs, 1964). When~Ri is not ~mall, care must be exercised to ensure thatthe expansion in ~ is uniformly valid over the entirerange of the boundary layer coordinate. Nevertheless,in all cases which have been investigated, the conditionon the interior velocity, to lower order in e, is the sameas in-, the strictly linear theories_ (e.g., Benney, 1965;Greenspan and Weinbaum, 1965). Thus, from Jacobs'analysis, the vertical velocities at the interior edges ofthe Ekman boundary layers at z= 1 and z=h are --R-,~'(0)/2 and u(-)hx+v(-)h~-+-R-;'~'<~)72, (11)respectively, where we have used the fact that h and allits derivatives are 0(-). To lowest order in ~, theseboundary conditions may be applied at z= 1 and z= 0.From (3), these vertical velocities are O(~) at most, sofrom (10) we conclude that wr = 0 in the interior of thefluid. The differential equation for the interior stream function pC0) is derived from (6). To lowest order in ~ wehave w?)= 0, which is satisfied automatically by (10).To first order in e, Eq. (6) becomes q(0) .v~-(o) _ V2~r =w,% (12) 2eRwhere q(0) and ~<o) may be expressed in terms bf thestream function p(0) by (9). The left side of (12) isindependent of z, so w(~) must be linear in z. Th~n with(11) we obtain [ h~ i i " q(0) .V ~(0) + .... ~.o) +_~V~(0). (13) ~Ri 2~REq. (13) is the required'nonlinear equation for theinterior stream function p(0)(x,y). The first term on theright side represents the .rate of destruction of relativevorticity by Ekman layer suction. The Second termrepresents the laterial diffusion of vo~ticity. The secondterm will be comparable to the fi?st in. vei;ical shearlayers of thickness R-I. Outside of vertical shear layers,the second term will be negligible. If the flow is uniform far from the obstacle, there arestreamlines p(0> (x,y) which terminate at infinity. Letus assu~e that one of these intersects itself, enclosing aregion of closed streamlines. We now show that thefluid within such a region is stagnant. Integrat~ (13)with respect to x and y over the region bounded by aclosed streamline. The left side is the divergence of avector [~'(-)q-h/~q(-), which vanishes on integrationbecause the component of q(0> normal to the closedstreamline is zero. Let us tentatively assume that thereare no high-vorticity layers outside the Ekman boundary layers. Then the second term on the right will beO(1) only in layers of vorticity discontinuity of thickness R-i, so the integral of the second term will beO (R-i). The integral of the first term on the right canbe expressed as the circulation round the closed streamline divided by -Rt. Thus, (13) implies f[q dr]= (R-t) (14) (0). Oaround any closed streamline, provided the streamlinedoes not pass through a high-vorticity region. Since thevelocity q(0) cannot change sign except on the criticalstreamline, where there are one or more stagnationpoints, (14) implies that the velocity within the regionenclosed by the critical streamline must be O(R-I). The above conclusion rests on the assumption thatthe vorticity ~-(0) is O (1). We might hope to analyze (13)using boundary layer methods in order to decidewhether free shear layers can exist, but the boundarylayer equation is nonlinear, and solutions have not beenobtained. Instead, we present a heuristic argument toshow that vorticity must be O(1) everywhere in thefluid. Vorticity 'is produced at the rate _q<0>.Vh/e,which is O(1), by (3). Ekman layer suction resultssimply in the destruction of vOrticity, by (13). Thisleaves lateral diffusion, important only in verticalshear layers. Let ns assume that such a shear layerexists, outside of which vorticity is O (1). Then on thestreamline where vorticity is at its maximum, (13)implies that the downstream derivative of 1~'<-)I isnegative. Th~s; the source of high vorticity within theshear layer must lie upstream, which is impossiblesince streamlines are either closed, or terminate atinfinity, where the flow is uniform. Vorticity ~-<0> must be O(1), although there may belayer~ across which it changes abruptly. If the layerthickness is O (R-i), as implied by (13), only the O (R-i)component' of velocity may exhibit boundary layerstructure; the O (1) component of velocity is continuous.Thus, we may assume that the O(1) velocity can beobtained from continuous solutions of (13) without thesecsnd term on the right. As part of this assumption,the free surface boundary condition is : q(0) =kxvp(0)=0, [p (0) = ~?) ], (15)'~vhere k is the vertical unit Vector. This.condition isapplied on the closedSportion of the critical streamlinep(0) = p?), and serves to fix its locktion.3. The inviscid limit In describing the flow outside :the Critical streamline,we may neglect both terms on the right in' (13) provided --~0, -Rt-~oo, h/-=O(1). (16)JuLY 1969 ANDREW P. INGERSOLL 747The free surface boundary condition (15) was derivedfrom (13) under the assumption that the Ekman layersuction term was O(1), and therefore much larger thanthe neglected terms of order ~, e2, etc. However, we shallshow that the neglected terms do not change the freesurface boundary condition (15), even' when they arelarger than the Ekman layer suction term. Thus, in theinviscid limit, the differential equation is (13) with rightside equal to zero, and the free surface boundary condition is (15), valid uniformly as e-~ 0 and The condition (15) arises because the existence ofclosed curves on which p(0) is constant implies a contradiction unless the circulation round the curve iszero. i~irculation implies a net vorticity within theregion',' which implies a net influx of fluid: from theEkman layers, by (11). For eR~=O(1), the contradiction arises because the lateral boundary of the region isa streamline, and so there cannot be an effiux of fluid tobalance the Ekman layer influx. In generalizing (15) tothe case eR--~ o,, we must show to all orders in ~ thatthere cannot be an efflux from a region within a curveon which p(0) is constant. To do this, we consider thecirculation theorem. From (5), it follows that1where the integral is taken around a closed contourwhich moves with the fluid. Moreover, if the flow issteady, and if there are closed particle paths, then thefirst term on the left is zero when the integral is takenaround such a path. Assume that there are closed curves in the x--y planeon which p(0) is constant. Let So be the cylindrical surface formed ffom one of these curves with generatorsparallel to the z axis. Let us assume that we have provedthe theorem through order n; in other words, that thenet flux through So is O (e"+~). Then there exists a surface S~ on which the velocity .~ q(O)+~q(~)q_ . q-e"q(") (18)is everywhere tangent to the surface. Moreover, S,~ maybe made almost to coincide with So, the error in positionbeing O(e). For the moment we neglect the influx offluid at the top and bottom due to Ekman layer suction,as well as the term on the right in (17). Then particlepaths will be nearly closed contours lying almost entirely in S~, the error being O(e~+~). The first term onthe left in (17) will be O(e'~+~), if the integral is takenaround a particle path in S~. Therefore the second term, X dr +f~k X ~+~q(~+~)3. dr, (19)taken around the same path, must alan he O (e~+e). However, the path dement dr is parallel to the nth orderapproximation to the velocity (18), so the first term in(19) is zero. This leavesf[k Xq('~+~)~- dr--O(e),(20)where the integral is taken around a particle path in S~.This path coincides, to O (~), with the lowest order particle paths in So, so (20) is valid on the surface So aswell. However, the vector normal to So is horizontal, so(20) is equivalent to the statement/q (~+~) - dAo = O (t),(21)where dA0 is an element of area on So. From (21) itfollows that the net effiux throuth So is O(e~+2), and sowe have proved the theorem to one higher power of e. In the preceding discussion the effects of Ekmanlayer suction and lateral diffusion were neglected. Nowassume that R-~=O(e"+~), forsome non-negative integer n. To lowest order in ~, the influx of fluid from theEkman layers is still given by (11), and is thereforeO(~+~). Therefore, particle paths will still be closed onthe surface S~, so the proof still holds through ordernq-1, provided the term on the right in (17) is negligible. In this case the effiux through So is O(e~+~), whichimplies that the Ekman layer influx must also beO(e~+~). However, by (11) this influx will be O(R-~)= O(e~+~) unless the O(1) circulation about the closedstreamline is zero. Thus, we are led again to (15) as the boundary condition to be applied on the closed portion of the criticalstreamline, provided the right side of (17) can beneglected. In proving this, we assume that the vorticity~(0) is O(1) as before, since the appropriate equation isstill (13), but without the Ekman lay.? suction term.The thickness of layers across which ~(0~ may changeabruptly is now (2eR)-, which is simply (Re)-~, whereRe--VH/~, is the Reynolds nnmber. Thus, the rightside of (17) is O(e~R-~)=O(e~+a~), which is small compared to e~+~, and is therefore negligible. In the inviscid limit (16), the equation to be solved is(13) with right side equal to zero, subject to the boundary condition at infinity, and to the free surface condition (15). The latter has a simple physical interpretation in terms of the pressure. If there is no motionwithin the Taylor column, the pressure p must be constant around its edge. Thus, at the edge of the Taylorcolumn we haveq-Vp=0, (22)which is satisfied automatically to lowest order in ~,since to this order the edge of the Taylor column is astreamline p(-)=constant. The terms in (22) proportional to ~ may be expressed in terms of p(o) by means of748 JOURNAL OF THE ATMOSPHERIC SCIENCES VoLu~ag 26 2,0 ~.5 t.0 0.5 0.0-0.5-~.0 -1.5-2.0 -z.o -t.5 -1.0 -0.5 0.0 o.$ t.0 t.5 2.0 F~o. 1. Streamlines over a flat, cylindrical obstacle of heighth0=e. The flow is from right to left if the rotation is counter~clockwise, left to right if the rotation is clockwise, and the separation of streamlines at infinity is 0.4.Here we have introduced circular coordinates r= (x~+y2)-, O= tan-~ y/x, and have arbitrarily set ~k= 0on the critical streamline. Note that both ~ and itsnormal derivative must vanish on the closed curve atthe edge of the Taylor column.a. Flat, cylindrical obstacle We set h(r)=ho=constant for r< i, and h(r)=0 forr> 1. This example is somewhat artificial, because ofthe discontinuity in depth at r = 1. However, a formalsolution of (25)-(26) exists, as first pointed out byVenezian (private communication), and so the exampleis a useful one. When the height of the obstade h0 is lessthan 2e, the streamlines are distorted in the vicinity ofthe obstacle, but the), do not cross (Fig. 1). In this casethe solution is I h0 ~e Inr+r sinr-- 1 (r> 1),~=L h0 --~(r2--1)-tTr sinO--1 (r~l).(27)the Bernoulli equation (7). Thus, we have q'Vp= -- ~qC-~'VE<-> =0, Ep(-> =p?>], (23)to first order in ,. Thus, E(-> must be constant alongthe dosed portion of the critical streamline, and sinceit is zero at the stagnation point, we havewhich is the same as (15).4. Two examples The differential equation in the inviscid limit issimplified considerably if the flow is uniform at infinity. Eq. (13) with right side equal to zero implies thatthe quantity I-r<-)+h/,-] is constant along streamlines,and is therefore zero everywhere outside the criticalstreamline. Thus, we have r r + ~/~ = W~+ h/~ = O, (25)where the lowest order pressure p(0) has been replacedby the symbol ~(x,y). Henceforth, we shall measurehorizontal distafices in uints of L, a characteristiclength which may be different from the depth//. Thestream function is measured in units of LV, and theheight of the obstacle is measured in units of H. Then(25) is unchanged provided we use the Rossby number~= V/29L based on the horizontal scale. We let thevelocity at infinity be unity in nondimensional notation, directed along the negative x axis. Then (25) mustbe solved subject to the boundary conditions ~u -~ 1, ~p~--~ 0 (r--~ o~), .(26a) V~k= 0 (~= 0, closed portion). (26b)The boundary condition (26a) is sufficient to determine the flow everywhere in the x--y plane. Thereis a clockwise circulation ~rho/e around the body,and an associated force on the body ~rh0 in the negativey direction. (In the present notation, this is exactly theCoriolis force which would act on an equal volume offluid flowing in the positive x direction with the freestream velocity.) When the height h0> 2e, the solution (27) gives closedstreamlines, but the condition (26b) is not satisfied onthe critical streamline, and there is flow within theTaylor column. Eq. (27) is the only solution of (25) and(26a) which is regular everywhere in the x--y plane,and is the solution usually offered in inviscid theories.However, (25) is not necessarily valid on closed streamlines, since these do not extend to r--~ o,; as we haveshown, this fact is properly taken into account byapplying (26b) on the closed portion of the criticalstreamline. The correct solution for ho> 2e is ' ho K r' ---- lnr+-- ln---Cr sinr--I (r> 1), 2e 2~r r~~ =- ho K r' (28) ----(r2-1)---- ln--+r sinr -- 1 4e 2,r r, (r< 1, r'>r,).The Taylor column is a circle centered at (x=0,y=2e/ho), circumscribed within the obstacle and tangent to it at (x=0,y=l). The radius of the Taylorcolumn is r,= 1--2e/ho, and r' is the distance from itscenter to the point (x=r cosO, y=r sinO). Finally, theJULY 1969 ANDREW P. INGERSOLL 749constant K=~rhorc2/e is the additional circulationnecessary to satisfy (26b). The net clockwise circulationabout the obstacle is 4*r(1--e/h0), corresponding to aforce 4~re(1-e/h0) acting in the negative y direction.Examples of this solution are plotted in Figs. 2-3, forvarious values of hole. For a high obstacle (h0/e>>l),the Taylor column coincides with the obstacle itself,and the force on the obstacle is 4re in the negative ydirection. (This is 4e times the Coriolis force on avolume of fluid equivalent to the obstacle plus Taylorcolumn.) Recently, T. Maxworthy (private communication)has observed Taylor columns in the laboratory for thecase l>>e>>R--. He finds that stagnation first occursabove a lens-shaped body whose height relative to thecontainer depth is roughly 3-4 times the Rossby number. Taking hie =C(1 --r2), we find from (25) and (26a),that stagnation is expected to occur for C>=3(3/2)-~3.68, in excellent agreement with observation. Maxworthy also finds that the sense of the asymmetry isthe same as in Figs. 1-3 of this paper, with the circulation in a direction opposite the rotation, and thestagnant region of the right-looking downstream, forcounterclockwise rotation?b. General symmetric obstacle Exact solutions of (25)-(26) have only been obtainedfor the flat cylinder. However, it is possible to obtainapproximate solutions valid for high obstacles (relativeto e) which terminate abruptly at r= 1. Consider thecase h(r)=0 for r> 1, and h(r)=ho'(1--r)q-ho"(1--r)2/2-t-. -., for r< 1, where the slope h0' is large compared to~. Because the slope is large, we assume that the boundary of the Taylor column almost coincides with the edge~,00.50.0-0.5-~.0-%5- 2.0 ~ -2.0 I I ~ , ~ I I .-~.5 -t.0 -o.5 0.O 0.5 ~.0 *.5 2.0Fro. 2. Same as Fig. 1 except that h0=2~.Note added in proof. 0 .,~ III \\~-2.0 ~ -2.0 -4.5 -I.0 -0.5 0.0 0.5 t.0 q.5 2.0Fro. 3. Same as Fig. 1 except that/~0 = 4~.of the obstacle at r= 1. Thus, an approximate solutionis ~=&lnrq-(r--l-h sinO (r>l), (29a) 2r \ rl ho' C - 1~b=--~ee(1--r)aq-~lnrq-(r--7)sinE)(0< 1--r<-1), (29b)where C is the circulation around the obstacle, to bedetermined. Note that ~ and its normal derivative arecontinuous at r= 1, and that the critical streamline- = 0 coincides with the circle r= 1. However, (26b) is not satisfied exactly at r= 1. Infact, we may use (26b) to determine the position of theTaylor column, and ultimately the circulation C. Firstnote that no portion of the Taylor column may extendto r> 1. In this region, ~ satisfies LaPlace's equation,and the function and its normal derivative cannot beequal to zero on a finite curve. Therefore, from (29a)we have -- 2 <= C/2~r <__ 2. (30) We now use (29b) to derive an approximate expression for the curve r~(O) on which (26b) is satisfied.Making use of the fact that ho'/~ is large, we find (r~--l)~ =-2-~-e r---C-t-2 sinr1. (31) ho'L2~'But if h0~>0, (30) and (31) are consistent only when F4e ]~ ,/O)=l-Lc0,(1-sinOj. (32)750 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUMe26The solution for r> 1 is the same as that obtained forthe high flat cylinder (h0>>e). The clockwise circulationis 4~r, and the force on the obstacle is 4,re in the negativey direction. It is convenient at this point to discuss the forces onobstacles. In the previous examples, where the flow is inthe negative x direction, the force on a stationary obstacle is in the negative y direction. To interpret thisphysically, it is simpler to let the fluid be stationary atr-ore, and have the body moving in the positive xdirection. Derivatives of Ca now give the velocity of thefluid relative to that of the body, and the pressurep(0) is now qa--y. The force on the obstacle computedfrom the stream function ~k, as in the examples, is nowthe net force, including forces exerted by the fluid andthe Coriolis force due to the motion of the body. Forthe flat cylinder with h0< 2e (no Taylor column), thisnet force is simply the Coriolis force *rh0 in the negativey direction; the fluid exerts no force in this case. Forh0> 2e, the net force is less than the Coriolis force whichwould act on a volnme of fluid equal to the body plusTaylor column. For hc>>e, the net force is O(e) timesthe Coriolis force; the fluid almost balances the Coriolisforce due to the motion of the obstacle. This behaviorwas observed by Taylor (1923) in his experiments ontowed spheres and cylinders. 5. Geophysical applications Phillips (1963) gives equations appropriate to a stratified, inviscid ideal gas on a rotating sphere. These equations are based on an dxpansion in the Rossby number similar to that presented in this paper. The motion is still horizontal and nondivergent, to lowest order in the Rossby number, but now the lowest order stream function depends On the vertical coordinate. This simply reflects the fact that the Taylor-Proudman theorem (10) does not hold for a stratified fluid, and so Taylor columns will not form when stratification is important. We now describe the relevant nondimen sional parameters which determine the importance of stratification, and of the spherical geometr3(. We let x and y be horizontal coordinates eastwardand poleward, respectively, scaled by a length L. Theequations then hold over a limited portion of the spherecentered at latitude ~0, for which the ratio Z/a< 1,where a is the radius of the sphere. The Rossby numberis now based on the vertical component of the rotationvector at latitude ~0, e = V/~ 29L sin~01. The height ofthe obstacle h is measured relative to the scale heightH=RT/g, evaluated at the surface, where'T is the- temperature, g the acceleration of gravity, and R thegas constant of the atmosphere. Finally, we let z bethe vertical coordinate, normal to the. surface of thesphere, and cp the specific heat of the gas at constantpressure. Then the effects of stratification and sphericalgeometry (8 effect) depend on the magnitude of theparameters B and b, where gtt~ l (dT B= (2ill sin,I,0)2 ~[x~.z+~-v]' b=L/(ae). (33)In deriving the basic equations, Phillips assumes that Band b are O(1), with tt/L, L/a, and e all small. To thisapproximation, the coordinates x, y, z define a Cartesiansystem; the only effect of the spherical geometry is thatthe vorticity ~' is replaced by ,;'+by, where y is the poleward coordinate measured from the mean latitude ~I'0.This is known as the tS-plane approximation. The effects of stratification and spherical geometryare small when B and b are small, but Phillips does notpresent a scheme for proceeding to the limit as B and btend to zero. In what follows, we shall simply assumethat stratification and spherical geometry can beneglected when B and b are small. This assumptionshould be verified; the case B --~ 0 is especially troublesome because of difficulties with the boundary conditionat z -~ o,. It is possible that the effects of stratificationare never negligible as B--~ 0, at sufficient heights inthe atmosphere. The effects of vertical shear due to horizontal temperature differences (baroclinicity) can be estimated bythe relation AV/V ~ (B/e) (ATn//XT,), (34)where /xV is a typical difference in horizontal velocityover one scale height//, and/XTn and AT~ are typicalhorizontal and vertical temperature differences overdistances L and H, respectively. We shall assume thatbaroclinicity is unimportant when B/e<l, sinceATa/zXT~ is typically O(L/a), a small quantity., We now discuss the magnitude of the constants e, Band b for the atmospheres of the earth and Jupiter. Forthe earth, taking V = 15 m sec-~, L = llY km,//= 8 km,- 0= 30- and dT/dz=--6.5C km-~, we obtain e~0.2, B~l.5, b~0.8. (35)Thus, the expansion in e is only a rough approximationand the effects of stratification, baroclinicity and spherical geometry (fi effect) are all important. Because ofstratification, the flow will not be two-dimensional, andthe Taylor-Proudman theorem (10) will not hold. Thisfact probably explains why Taylor columns are notobserved on the earth. The main difference between the earth's atmosphereand Jupiter's is the immense horizontal scale of.phenomena on Jupiter. The Great Red Spot coverssome 10- of latitude and 30- of longitude, correspondingto Lml0~ km. And taking V=50 m sec-~ (Reese andSmith, 1968), H--8 km (0pik, 1962), dT/dz=O (extreme stratification), and c~/R=3.5 (diatomic gas),we obtain ~0.04, B~0.03, b~4. (36)JULY 1969 ANDREW P. INGERSOLL 751These are conservative estimates; if the velocity islower (Hide, 1963), and the lapse rate dT/dz is morenearly adiabatic, e and B will be smaller, and b will belarger. Thus, stratification and baroclinicity are notlikely to be important for large scale phenomena suchas the Great Red Spot. We therefore assume that thestream function ~b(x,y) is independent of the ,,erticalcoordinate, and obeys the equation q. V (~'q- by-+-h/e) = 0, (37)where u= --~bu, v=-x, and ~'= V2~b, as before. With thesedefinitions, (37) is valid in both the Northern andSouthern Hemispheres, provided x is always the eastward (prograde) coordinate, and y is always the poleward coordinate. Note that (37) is identical to (13) inthe inviscid limit, with an additional term by due tothe spherical geometry. We are assuming that there is a rigid surface belowJupiter's atmosphere, the equation of which is z = h (x,y).We also assume that this surface exerts a small frictional drag on the overlying fluid, whence the boundarycondition at the edge of the Taylor column (15). Inshort, we are investigating the implications of Hide'shypothesis that the Spot is a Taylor column formedas a result of interaction between the atmosphere andthe surface of Jupiter. Problems concerning the natureand motion of this surface have been discussed by Hide,and will not be mentioned here. We let the stream function at infinity be Uy, corresponding to uniform flow in the negative x direction(retrograde), if U is positive. Then (37) reduces to v~-- b(~- Uy)/U+h/~ = 0, (38) 2.0 ~.5 ~0 0.5 0.~-0.5-~.0-L-2.0 - ~:.0 t ~ I I I I i -~.5 -~.0 -0.5 0,0 0,5 t.0 ~.5 2.0 Fro. 4. Streamlines on a/g plane for retrograde ttow over anobstacle of height h (r)= (L/4a)(1--r~). The flow is from east towest (right to left in the Northern Hemisphere, left to right in theSouthern Hemisphere), and the separation of streamlines atinfinity is 0.4. A smooth transition at r=l takes place in aninertial boundary layer of thickness (ea/L)t2.0]'5t0.5I I [ ] I I I 0.0-0,5 -1.0-t.5 --a.o . , I I I I I I -~.0 -q .5 -~ .0 -0.5 0.0 0.5 t .0 t.5 ~.0 F~o. 5. Same as Fig. 4 except for h(r)= (L/a)(1--r2). Inertialboundary layers form at r=l and at the edge of the stagnantregion.which must be solved subject to the boundary conditions ~pu-~ U, ~k~--~ O (r-~,~), (39a) V~b= 0 (~b'= 0, closed portion). (39b)Eq. (38), like (25), is linear in ~ with constant coefficients, but the free surface boundary condition (39b),like (26b), introduces a critical nonlinearity into theproblem. The only simple case which can be analyzedoccurs when b/U is large and positive (retrograde flow).An approximate solution of (38) and (39a) is then ~= Uy+hU/(b4. (40)There is no circulation about the object at r-~ o,, andstreamlines ~b~=constant are deflected toward theequator by ~n object with h> 0 (Figs. 4-5). For retrograde flow, the asymmetry to the left and right of theobstacle is the same as for solutions of (25). Closedstreamlines will appear for h= O(eb)= O(L/a), where his the height of the obstacle relative to the scale heighttt= RT/g. For the Great Red Spot, this critical heightis about 2 km. However, the solution (40) does not satisfy (39b) onthe closed portion of the critical streamline. In fact, aninertial boundary layer of thickness (U/b)~ devdops inthe vicinity of this streamline. Within the inertialboundary layer, complementary solutions appear whichsatisfy the equation W-~-- (*/U)~= O,and which vanish exponentially at the outer edge of thelayer. It is not difficult to show that the location of thestreamlines is adequately represented by (40), although752 JOURNAL OF THE ATMOSPHERIC SCIENCES VoLrJtalg 26the velocity components obtained from (40) are onlycorrect outside the inertial boundary layer. If the flow is prograde rdative to the obstacle (U< 0),solutions of (41) are no longer confined to boundarylayers, but exhibit wavelike behavior. A similar situation exists at the western boundaries of ocean basinswhere the flow in the open ocean is to the east (e.g.,Carrier and Robinson, 1962). Such regions are not wellunderstood, but it appears that simple relations of theform (40) are not valid everywhere in the interior of thefluid. Thus, when b/U is large and the flow is prograderelative to the obstacle, steady inviscid Taylor columnsmay not exist, or they may be very different from theexamples given here. The flow around Jupiter's Great Red Spot bears someresemblance to the solutions (40) for retrograde flow(Fig. 5). In the first place, the parameter b/U is large,as implied in (36). Second, the streamlines are clearlydeflected towards the equator, as implied by the existence of the Red Spot Hollow on the equatorward sideof the Spot. Third, Reese and Smith (1968) reportrepeated observations of counterclockwise circulationaround the edge of the Spot, which is only consistentwith the present theory if the flow is retrograde. Onthe other hand, the observed east-west vdocity at thelatitude of the Spot is more complicated. Equatorwardof the Spot, in the South Equatorial Bdt, extremeretrograde motion is observed with rotation periodsgreater than 9~58m (Peek, 1958; Reese and Smith,1968). Poleward of the Spot,' in the South TemperateBelt, extreme prograde motion is observed with rotation periods about 9a53'~. For comparison, the meanrotation period of the Spot during the last hundredyears is 9~55m38~. The fact that there is prograde flow at the polewardedge of the Spot means that simple solutions of theform (40) are not valid over the entire region. Furthertheoretical work is needed to understand the effectsof nonuniform, prograde flow at infinity, as well as toverify that stratification can be neglected for smallvalues of B. The motion of a floating body on a g planeshould also be investigated. In this regard, the fact thatthe Spot lies in a zone of extreme negative relittivevorticity (taking the planetary vorticity to be positive)may prove significant. Acknowledgments. The author wishes to thank Dr.Giulio Venezian for his many helpful suggestions, andfor the interest he has shown in this work. Financialsupport was provided partly by the National Aeronautics and Space Administration under Grant NGL05-002-003. REFERENCESBenney, D. J., 1965: The flow induced by a disk oscillating about a state of steady rotation. Quart. J. Mech. Appl. Math., 18, 333-345.Carrier, G. F., and A. R. Robinson, 1962: On the theory of thewind-driven ocean circulation. J. Fluid Mech., 12, 49-80.Greenspan, H. P., and L. N. Howard, 1963: On a time-dependentmotion of a rotating fluid. J. Fluid Mech., 17, 385-404.---, and S. Weinbaum, 1965: On non-linear spin-up of a rotating fluid. J. Math. Phys., 44, 66-85.Hide, R., 1961: Origin of Jupiter's Great Red Spot. Nature, 190, 895-896. , 1963: On the hydrodynamics of Jupiter's atmosphere. -a 2Physique des lPlanetes, Belgium, Institut d'Astrophysique, Cointe-Sclessin, 481-505.--, and A. Ibbetson, 1966: An experimental study of "Taylor columns." Icarus, 5, 279-290. and M. J. Lighthitl, 1968: On slow transverse flow past obstacles in a rapidly rotating fluid. J. Fluid Mech., 32, 251 272.Jacobs, S. J., 1964: The Taylor column problem. J. Fluid Mech., 20, 581-591.0pik, E. J., 1962: Jupiter: Chemical composition, structure, and origin of a giant planet. Icarus, 1, 200-257.Peek, B. M., 1958: The Planet Jupiter. London, Faber and Faber, 283 pp.Phillips, N. A., 1963: Geostrophic motion. Rev. Geophys., 1, 123 176.Proudman, J., 1916: On the motion of solids in a liquid possessing vorticity. Proc. Roy. Soc. (London), A92, 408-424.l~eese, E. J., and B. A. Smith, 1968: Evidence of vorticity in the Great Red Spot of Jupiter. Icarus, 9, 474-486.Stewartson, K., 1953: On the slow motion of an ellipsoid in a ro- tating fluid. Quart. J. Mech. A ppl. Math., 6, 141-162.Taylor, G. I., 1917: Motion of solids in fluids when the flow is not irrotational. ;Proc. Roy. Soc. (London), A93, 99-113. ,1923: Experiments on the motion of solid bodies in rotating fluids. Proc. Roy. $oc. (London), A104, 213-218.

## Abstract

A homogeneous fluid is bounded above and below by horizontal plane surfaces in rapid rotation about a vertical axis. An obstacle is attached to one of the surfaces, and at large distances from the obstacle the relative velocity is steady and horizontal. Solutions are obtained as power series expansions in the Rossby number, uniformly valid as the Taylor number approaches infinity.

If the height of the obstacle is greater than the Rossby number times the depth, a stagnant region (Taylor column) forms over the obstacle. Outside this region there is a net circulation in a direction opposite the rotation. The shape of the stagnant region and the circulation are uniquely determined as part of the solution.

Possible geophysical applications are discussed, and it is shown that stratification renders Taylor columns unlikely on earth, but that the Great Red Spot of Jupiter may be an example of this phenomenon, as Hide has suggested.