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Baroclinic instability in the presence of a stronghorizontal shearE. Knobloch a; H. C. Spruit ba Department of Physics, University of California, Berkeley, CA, U.S.Ab Max-Planck-Institut für Physik und Astrophysik, W. Germany

Online Publication Date: 01 July 1985To cite this Article: Knobloch, E. and Spruit, H. C. (1985) 'Baroclinic instability in thepresence of a strong horizontal shear', Geophysical & Astrophysical Fluid Dynamics,32:3, 197 - 216To link to this article: DOI: 10.1080/03091928508208785

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Geophys. Astrophys. Fluid Dynamics, 1985, Val. 32, pp. 197-216 0309- 1929/85/3204-0 197 $18.50/0 Q 1985 Gordon and Breach, Science Publishers, fnc. and OPA Ltd. Printed in Great Britain

Baroclinic Instability in the Presence of a Strong Horizontal Shear

E. KNOBLOCH Department of Physics, University of California, Berkeley, CA 94720, U. S. A.

and

H. C. SPRUIT

Max- Planck-lnstitut fur Physik und Astrophysik, Karl-Schwarzschild-Str. 1, 8046 Garching bei Munchen, W. Germany

(Receiued December 12, 1984)

The theory of baroclinic instability is extended to include strong horizontal shears. The $plane and anelastic approximations are used, but the assumption that the Rossby number is small is relaxed. Although this problem is nongeostrophic, much of the standard theory, including necessary conditions for instability and a semicircle theorem, generalize readily for disturbances of low zonal wavenumber.

1. INTRODUCTION

Baroclinic instability is of fundamental importance in geophysical and astrophysical fluid dynamics (e.g. Pedlosky, 1979), and its theory is now well developed. In the earth’s atmosphere the conditions are such that an appropriate scaling of the equations introduces a small parameter, the Rossby number, into the basic equations, with the result that at lowest order the instability is hydrostatic in the vertical

197

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198 E. KNOBLOCH AND H. C. SPRUIT

and satisfies geostrophic balance in the horizontal. This provides a useful simplification to the equations of motion, and facilitate the development of a fairly complete theory ( e g Pedlosky, 1979). In astrophysical fluid dynamics, on the other hand, the conditions tend to be rather different, and the assumption of hydrostatic balance and “geostrophy” may be inappropriate. Specifically, there may be large horizontal shears present (as, for example, in accretion disks), or the buoyancy frequency may be small (as for example, near convection zones, near the centre of a star or at the midplane of an accretion disk). Under these conditions the geostrophic approximation and the hydrostatic approximation respectively will fail. The present paper is concerned with the extension of the theory of baroclinic instability to conditions not covered by the usual theory.

Extensions of the theory of baroclinic instability have been considered before. Thus Stone (1966) considered analytically the effects of relaxing the geostrophic approximation in the context of the Eady model of baroclinic instability, and in a later paper (Stone, 197 1) relaxed the hydrostatic approximation as well. Stone found that for Richardson numbers less than about 1 nongeostrophic effects were dominant, and that these modes were also the ones most affected by non-hydrostatic effects, the growth rate of the instability being reduced under nonhydrostatic conditions. This conclusion was subsequently confirmed by the more detailed results of Hyun and Peskin (1976).

These authors did not include in their analysis the effect of horizontal (latitudinal) shear. This shear is not of particular importance in geophysical fluid dynamics, being generally smaller than the vertical shear, but in certain astrophysical applications (for example, in accretion disks) it dominates the vertical shear, and its inclusion then becomes of paramount importance. The resulting problem is harder than the’ standard problem, because the shear flow is now fundamentally “nonseparable” (cf. McIntyre, 1970).

The present paper is concerned with an extension of the analysis of Stone and Hyun and Peskin to cover the nonseparable, nonhydrostatic, nongeostrophic baroclinic problem. We find that the general theory in its essential aspects generalizes to this more complicated case. We are able to derive a generalization of the geostrophic approximation for baroclinic flows with strong horizontal shear, valid in the limit of small azimuthal (zonal) wave-

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BAROCLINIC INSTABILITY 199

numbers. In this limit the hydrostatic approximation also holds. Thus the long zonal wavelength of the instability is responsible for the hydrostatic approximation, even though the Rossby number (based on the latitudinal shear) need not be small. We make this distinction because in a strongly horizontally sheared baroclinic flow the instability becomes highly anistropic in the horizontal.

The most complete results can be obtained in thef-plane approximation, in which changes in the coriolis parameter with latitude are neglected. In the small zonal wavenumber limit the standard theory generalizes completely, and necessary conditions for the instability can be found, including the analogue for the present problem of the Charney-Stern theorem and Howard’s semicircle theorem.

The organization of the text is as follows. In Section 2 the basic nature of the problem is studied for the baroclinic circular vortex. In Section 3 the basic stability results are derived. The question to what extent sphericity effects can be introduced into the theory is studied in Section 4. It turns out from this analysis that a consistent reduction of the geometrical effects is possible only in the $plane model. The usual b-term can be introduced only together with the other geometrical terms, unless the horizontal Rossby number is assumed to be small. In that case, however, the theory reduces to the standard theory as summarized in, for example, Pedlosky (1979). A brief summary of the principal conclusions of the paper is given in Section 5.

2. THE BASIC PROBLEM

We begin this paper with a general formulation of the stability problem for the baroclinic circular vortex, that is the case where gravity is parallel to the rotation axis. In the anelastic approximation, the equations for adiabatic, inviscid motion are then

p(dtu + u - V U ) = - dxp + p2vR + pnzx , (2.la)

p(d,u + u VU) = - a y p - p2uR + pRZy, (2.lb)

p(8,w+u - Vw)= -d,p-pg, (2 .k)

(2.1 d,e) 8@+u * Vd=O, V * pu=O.

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Here u = (u, u, w ) are the velocity components in Cartesian coordinates (x, y, z) in a reference frame rotating with constant angular velocity R about the z-axis, p is the pressure, and 8 the potential temperature given by In 0 = y - In p - In p . The gravity g = (0, 0, -g) is taken to act downwards. As usual (e.g. Pedlosky, 1979, Ch. 6), we first linearize the equations with respect to a “reference” stratification which is time-independent and a function of z only. Denoting the reference state by a subscript s, we have

&Ps + P g = 0,

ps(d,u + u - VU) = - a x p + ps2uQ + psiz2x,

ps(d,u + u * VU) = - d y p - p,2uR + psR2y,

ps(atw+u - V w ) = -d,p-pg,

d,+U - vo =o, v (psu)=0,

where p , p and 8 are related by

e / e s = ( P / Y P s ) - b / P s l (In 8, = y - l In ps - In p,) .

With (2.4), the right hand side of (2.3~) can be written as

- P s ~ , ( P / P , ) + Pd, In 0, + o(P,/e,)g.

(2.2)

(2.3a)

(2.3b)

(2.3~)

(2.3d, e)

(2.4)

The second term in this expression is usually considered offending and is removed by constructing the reference stratification such that d,ln8,=O (cf. Pedlosky, 1979, equation 6.5.7). The result is a set of equations which is equivalent to the Boussinesq approximation, except that the anelastic continuity equation (2.3e) is used instead of an incompressible one. Thus, (2.3~) now reads

d,w + u * v w = - &(p/p , ) +(B/O,)g. (2.5)

The baroclinic basic state we study is a steady solution of equations (2.3), (2.5), and is an azimuthal (zonal) circulation uo = [uo(y, z),O,O]. This circulation must satisfy the thermal wind relation, found by

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BAROCLINIC INSTABILITY

eliminating p / p , from (2.3b) and (2.5):

20 1

2naZu0 +gos- d,80 = 0, (2.6)

where 8,(y,z) is the temperature field of the basic state. This relation describes the basic balance in a baroclinic state: a vertically sheared zonal flow is maintained by a horizontal temperature gradient, or equivalently by the inclination of constant entropy surfaces relative to the equipotentials.

The baroclinic instability extracts energy from the vertical shear of the zonal flow by means of motion with directions within the small angle between the constant temperature surfaces and equipotentials. Such motions reduce this angle, and hence the baroclinicity of the basic state. The instability may be viewed as ,a form of shear instability of vertically sheared flows, or equivalently, as a form of horizontal convection, driven by horizontal temperature gradients.

In the following we shall study the linear stability of the baroclinic zonal wind uo. We first nondimensionalize the equations by expressing time in units of (2R)-l, and distances in units of a length L (to be chosen later). We define a dimensionless pressure p” and potential temperature e’ through

P l P S ( 4 = p”4Q2L2, 8/8,(z) = 84R2 L/g, (2.798)

and the dimensionless velocities by

ii = u/( 2RL). (2.9)

The linearized forms of equations (2.3), (2.5) then become, after dropping the on all scaled quantities:

a,u + u0axu + EW = - d,p + yv, (2.10a)

atv + uOaxV = - ayp - u, (2.10b)

a,W + uoaxw = - aZp + 0, (2.10c)

ate+ u0aX8 --EU + N ~ W = 0, (2.10d)

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202

where

E. KNOBLOCH AND H. C. SPRUIT

6(y, z ) = d y U O ( =+sz-la,*uO*), (2.1 la)

y(y,z)=1-J, (2.1 1 b)

F ( Y , z ) = B,u,( =g- ldz*uo*), (2.1 lc)

In (2.10d) the thermal wind relation (2.6) has been used. The expressions in brackets in (2.11) define the quantities 6, E, and N 2 in terms of the physical (dimensional) quantities and coordinates. In what follows we shall refer to E as the baroclinicity of the basic state, 6 is the Rossby number of the flow, and N 2 the dimensionless buoyancy frequency. Equations (2.10) are the basic equations deter- mining the linear stability of a baroclinic state under nongeostrophic and nonhydrostatic conditions.

3. BASIC RESULTS

3.1 Derivation of the eigenvalue equations

Since equations (2.10) are separable in both x and t, we look for solutions of the form

where c is the complex eigenvalue whose real part determines the phase speed of the baroclinic wave, and whose imaginary part determines its growth. With (3.1) equations (2.10) become

im(u, -c)u + EW = - i m p + yu, (3.2a)

im(u, - c)u = ~ dyp - u, (3.2b)

im(u, - c)w = - a z p + 8, (3 .2~)

im(u, - C ) d - & U f N2W =o, (3.2d)

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(3.2e) im psu + d,(p,u) + d,(p,w) = 0.

The stability problem is most conveniently described in terms of an equation for the pressure perturbation p . The relation of this method to that employed by Stone (1971) is explained in Appendix A. Solving equations (3.2) for u, u, w we find:

ti= D- [ -rim(r2 + P ) p - [ ( r2 + i '?2)y-~2]d , ,p+r2~d,p] , (3.3a)

u = ~ - ' [ i r n ( r ~ + P ) p - r ( r 2 + ~ ~ ) d , p - r s a , p ] , (3.3b)

w = D- '[simp - srd,p - r(y + r2)d,p], (3.3c)

where

r = irn(u, - c), D = ( r2 + ~ ' ) ( y + r 2 ) - s2. (3.495)

Substituting these expressions into equation (3.2e) we obtain an equation for p. After some simplification, it can be written in the form

where

Since we are interested in studying unstable solutions (ci # 0) the quantity uo-c never vanishes, and hence we are to solve the problem

where L is a second order self-adjoint operator. Thus, the general form of the problem persists even when the usual geostrophic and hydrostatic approximations are relaxed. The self-adjointness of L is

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due to the thermal wind relation, which makes the coefficient of v in (3.2d) equal but of opposite sign to the coefficient of w in (3.2a). This happens more generally in rotating systems, e.g. that studied by Braginskiy (1967).

Equation (3.8) is an eigenvalue problem for c, and is to be solved subject to some boundary conditions. For simplicity we take the boundary conditions to be

p + o as zz+y2-+m. (3.9)

If fixed boundaries at some finite distance are assumed it is well known that the stability problem changes significantly, and that special forms of instability are possible [cf. the discussion in Pedlosky (1979), Ch. 71. Although the case of fixed horizontal boundaries is of particular relevance in the geophysical context, it need not be considered separately, since such boundaries can be simulated by an appropriate choice of p,(z). Thus the boundary conditions (3.9) are sufficiently general for the present discussion.

Necessary conditions for instability may be derived by multiplying equation (3.8) by p*, the complex conjugate of p , and integrating over y ,z . Integrating by parts, and using (3.9), we obtain

Since J ; g and h are complex when c,#O, we cannot derive useful necessary conditions for instability from a formulation that is this general. However, an important result can be obtained in either of the two limiting cases m-0 or N’-m. The case m+O, corresponding to the limit of long azimuthal wavelengths, is considered in detail below.

3.2 General results for the case m+O

The baroclinic instability is essentially a long wavelength nonaxi- symmetric instability. In contrast to buoyancy oscillations it is characterized by a growth rate that vanishes as the azimuthal wavenumber vanishes, a fact that is captured by the assumed

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BAROCLlNlC INSTABILITY 205

decomposition (3.1). To demonstrate instability it is then sufficient to study the case m+O. This limit can be taken directly in (3.10), but in order to discuss the physics involved we return briefly to equations (3.2), and rescale the perturbations as follows:

u = ii, u = mi7, w =m3, 8= g , p =F. (3.11)

- (Note that the meaning of the in m equations (3.2) become:

has been redefined.) To lowest order

quo -c)C +&3 = -ip"+yv", (3.12a)

o = -a,c-ii, o = -a,fi+& (3.12b, c)

i(uo - c)B- &i7 + N 2 3 = 0, (3.12d)

ip,ii + ay(psq + &(ps3) = 0. (3.12e)

Thus, owing to their long timescale the perturbations are in hydrostatic balance [equation (3.12c)], while the long azimuthal wavelength produces a geostrophic balance in the latitudinal direction [equation (3.12b)], but not in the azimuthal direction. This is in contrast to most geophysical cases studied before, where the small- Rossby number approximation usually employed assures geostrophic balance in both horizontal directions. In our case the strong horizontal shear (with an associated Rossby number which is not assumed small) has prevented us from using the same scales for u and u [cf: equation (3.11)]. This partial geostrophy is a characteristic of the strongly sheared problem.

One can proceed as in Section 3.1 to derive a single equation for p , or alternatively one may take the limit m-tO in (3.8). One obtains

ho = P B Y P O , Do = E ~ ( N ~ ~ & - ~ - 1). (3.14c, d)

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Proceeding as in Section 3.1, we obtain from (3.13), with boundary condition (3.9):

Since fo,go, ho are now real, the left-hand side of (3.15) is real, and the imaginary part of (3.15) therefore yields

A necessary condition for instability (ci # 0) is therefore that

this quantity plays the role of the potential vorticity gradient. For instability it is necessary that

n,=0 (3.19)

somewhere in the flow. This analogue of Rayleigh’s inflection point criterion can be generalized in the manner suggested by Fjnrtoft: from the real part of (3.15) one obtains

It is easy to show that the right-hand side is negative definite if both

N2>0 and foho-gi>O. (3.2 1 a, b)

The former condition ensures stability against convection, the latter condition yields

~ N ’ / E ~ > 1. (3.22)

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In terms of the dimensional quantities, denoted temporarily by an index *, this can be written [cf. (2.1 l)]

(1 -(2Q)- a y * U O * ) N ~ / ( a z * U O * ) ’ > 1. (3.23)

This is just a special case of the Solberg-Herjland condition for stability against axisymmetric instabilities [e.g. Eliassen and Kleinschmidt (1957); see also Stone (1970), Section 41. If the Richardson number N ’ / E ~ is large, for example, (3.23) reduces to Rayleigh’s condition for axisymmetric stability in a differentially rotating fluid. It follows that (3.21) is equivalent to the condition that all axisymmetric modes are stable. Assuming that this is the case the right-hand side of (3.20) is negative, so that it is necessary for instability that

somewhere in the flow. This is the analog of Fjerrtoft’s condition for the instability of an inflection point shear profile, except that in the present two-dimensional problem we cannot make the stronger statement that (3.24) must hold everywhere in the flow (cf. Drazin and Reid, 1980).

We can prove stronger results for the present problem along the lines of Howard (1961) and Pedlosky (1979). If we let

4 = P / ( U o - c) (3.25)

and multiply (3.13) by q* and integrate over all space, the result may be written, after integration by parts, in the form

G.A.F.D. B

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208 E. KNOBLOCH AND H. C. SPRUTT

As before, a considerable simplification is obtained upon a further integration by parts of the terms on the right-hand side:

J-J- (uo - c) 2A d y dz - J"j- (uo - c)d, 14) 2 d y dz - JJ" 6 1412 d y dz = 0, (3.27)

[cf. equation (3.21)]. A final integration by parts of the second term yields the remarkably simple result

JJ (uo - C)2A d y dz = 0. (3.29)

From the real and imaginary parts of equation (3.29), we find that

j j [(.o - c r y - c f ] A d y dz = 0, (3.30)

and

c, JJ A dy dz = JJ uoA d y dz. (3.31)

From these equations we obtain finally the result

(c? +cl")Jj A dy dz =JJ u; A dy dz. (3.32)

It follows from (3.31) that if uo varies between urnin and urnax in { - c o < y , z < c o ) , then

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This is Howard's semicircle theorem for the nonseparable baroclinic instability problem with a strong horizontal shear. Note that if um,,=umin, i.e. if the flow is uniform, then ci=O, and the flow is necessarily stable.

3.3 The limit Ri>>l

The familiar condition for baroclinic instability (Charney and Stern, 1962) can be recovered from condition (3.19) by taking the large Richardson number limit. Taking E ~ / N ~ < < ~ , we get with the defi- nitions of y and E

a,,u,+ Ps1Y2a,CPs(YNZ) - l & u o l =o. (3.36)

In the usual formulation of the problem, the Rossby number a,u, is small, so that we must take y= 1, which yields

somewhere in the flow. This is identical to the usual condition on the potential vorticity [Pedlosky, 1979, equations (7.2.1 l), (7.3.38)], when applied near the pole.

4. INSTABILITY ON A SPHERE

In this section we investigate to what extent the results of the previous section, which are valid for a plane rotating layer with rotation and gravity parallel, can be generalized to an arbitrary latitude on a sphere.

We use spherical coordinates (4 ,A,r) , where 4 is the longitude (positive eastward), A the latitude (positive northward) and r the radius. The equations of motion for inviscid flow are then (e.g., Pedlosky, 1979, Ch. 6):

duldt + u w r - l -uurpl tan A - 2Q sin A v + 2R cos A w = ( p r cos A)- ' a + p

(4.la)

(4.lb)

(4.1~)

du/dt + w v r - ' + u2rp tan A + 2R sin A u = - ( p r ) - ' d,,p

dw/dt -(u2 + u2)rp ' -2R cos A u = -p- ' d,p -g.

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Here (u,u,w) are the ( 4 , A , r ) components of the velocity and the centrifugal acceleration has been absorbed into g.

As in Section 3, we first linearize the equations with respect to a “reference” stratification, i.e.

P = P s + P1, P = Ps + P1, (4.2)

where us=v,=w,=O, and ps,ps are functions of r only, independent of time, and satisfying pS-’drps= -g. With the definition of the potential temperature we have, as before

The right-hand side of (4 .1~) can then be written as

As before, we construct the reference stratification such that

a, In 8, = 0. (4.4)

Dropping the subscript 1 on all quantities, equations (4.1) then become

du/dt + u w r - l - uvr tan A - 2R sin A v + 2Q cos A w

= - ( r cos 4 - ag(P/P,), (4.54

dv/dt + v w r - l +u2rp’ tanA+2R sin A u= - r - l ~ , ( p / p , ) , (4.5b)

dw/dt-(u2+v2)rp1-2RcosAu= -d,(p/p,)+gQ/Q,. (4 .5~)

The continuity equation yields:

dtp + 2p,wrp + d,(p,w) + ( r cos A) -‘d,(p,v cos A) +

The (adiabatic) heat equation becomes:

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d(1n 0Jdt + d(0/OS)/dt = 0.

211

(4.7)

The complications of the full spherical geometry can be avoided only if the relevant horizontal scale of the process is sufficiently small. Since the characteristic wavelength of baroclinic instability is of the order r f /N, where f is the coriolis parameter and N the buoyancy frequency, we must for consistency, assume that

We choose a horizontal scale L, such that

Since the vertical scale of baroclinic instability is of the order L f /N [see, for example, McIntyre (1972)], we choose a vertical scale

H = aL. (4.10)

In agreement with these assumed scales, we introduce local, dimensionless, Cartesian coordinates (x, y, z ) such that

( r cos A)-’ 8, = L- lax; r-’aA = L-ld,; a,= (EL) - a,. (4.1 1)

The coriolis parameter is

f = 2Q sin Ao, (4.12)

where A. is our reference latitude. We introduce dimensionless velocities, and a timescale, as follows:

( ~ , ~ ) = L f ( u ” , q , w = L f a G , d,= fa, (4.13)<4.15)

From (4.13) it is seen that a Rossby number of order unity is implied. The pressure and the potential temperature are non- dimensionalized as follows:

0 = e,f’r/g8, ?J=f 2L2 P P S - (4.16a, b)

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With these scalings, we expand equations (4.5)-(4.7) in a, keeping terms up to first order. We drop the on the dimensionless quantities. This yields

-

8,u + udxu + vd,u + wdZu - cluu tan A, - v( 1 + By) + uw cot A, = - d g ,

(4.17a)

d,v + udxv + od,u + wd,v + uu2 tan A, + u( 1 + By) = - d,p, (4.17b)

--CI cot AOU= - d,p + 8. (4.17c)

where B=acotA,. With (4.16a,b) and (4.3b) the density p can be written as

P = P S X C L 2 / ( Y 4 P - 81, (4.18)

where

x = f 2rlg> h- = 18, In psi. (4.19,20)

The scale height h can be of order L2/r. Assuming that the centrifugal acceleration is small compared to gravity, we have

X<<1. (4.21)

The term drp in the continuity equation then vanishes, so that the motions are anelastic. The scaled form of the continuity equation then becomes

p;' d,(psw)+dyu-a tanA,v+d,u=O. (4.22)

The heat equation (4.7) becomes, using (4.4):

Consider now the linear stability of a purely zonal flow u,(y,z), with a corresponding temperature field 8,. The thermal wind relation becomes, to order a:

d,u0(2clu, tan A, + 1 + B y ) = - ay80, (4.24)

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and the pertubed equations are:

213

(a, + u,d,)u + v d p , + wd,u0 -v(au0 tan A. + 1 +by) + CIW cot A,

= -a,p, (4.25a)

(4.25b)

(4.25~)

(4.25d)

(at + u,d,)v + u(2au0 tan A, + 1 +by) = - d,p,

- a cot Aou = - d,p + 0,

(a, + U,a,)8 + uayoo + wd,8, = 0,

where u, v, w, p , 0 now stand for the perturbed quantities.

4.1 The long wavelength limit

As in Section 3, we now assume that the zonal wavelength is long, since only in this limit a useful stability condition is obtained. Due to the spherical geometry, the longest possible wavelength is of the order r. With this wavelength assumed, we can look for solutions of the form

F ( t , x, y, z ) = F(y , (4.26)

As in Section (3.2), it is useful to rescale v and w:

V = afi; W = a3. (4.27)

To lowest order in a, the linearized equations (4.22), (4.25) then yield

= - ayp, o= -a,p+o, (4.28b, c)

quo - c)o + fiayoo + 3a,o, = 0, (4.28d)

while the thermal wind relation becomes

azuO = - a,e,. (4.30)

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These equations are identical to equations (3.12), the only difference being that the quantity 2R has been replaced by the coriolis parameter f. We conclude that the results of Section 3.2 are also applicable to instability on a sphere, in the f-plane approximation.

It would seem natural to seek a generalization of the procedure to a fi-plane approximation, but this does not seem possible without introducing certain inconsistencies. The reason is that, due to the assumption that the Rossby number is of order one, the fi-terms in equations (4.25) have no special role compared with the other sphericity effects, i.e. the other terms of order a. It is possible to rescale equations (4.25) by assuming that the Rossby number is in fact small (of order a, say). To lowest order in a, equations (4.25) then yield a purely geostrophic, hydrostatic balance, and the dynamics has to be inferred from the next order in CI. This procedure, however, is identical to that outlined in Pedlosky (1979, Ch. 6), so that no new results are found.

We conclude that an f-plane approximation is the best spherical model for which the results of Section 3 can be used.

5. SUMMARY AND DISCUSSION

In the present work we have extended the theory of baroclinic instablity to include strong horizontal shear. The motivation for this work comes from astrophysics rather than geophysics, and specifically the physics of accretion disks, in which the strong horizontal Kepler shear dominates the vertical shear produced by the baroclinicity of the basic state. The Kepler shear is of fundamental importance, and its effects cannot be neglected when discussing baroclinic instability in accretion disks. However, the results of the paper are of more general interest, for they show that the small Rossby number assumption of the standard theory can be relaxed without losing the powerful results that the theory is based upon. We have shown that for long-wavelength disturbances the hydrostatic balance in the vertical continues to hold, although the geostrophic balance is modified. We have found that an expansion in the azimuthal wavenumber of the disturbance is a powerful method for deriving the eigenvalue equation (3.13) under these more general conditions. From this equation necessary conditions for instability including the semicircle theorem could bc obtained by standard

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BAROCLINIC INSTABILITY 21 5

methods. These results were found to hold in general for anf-plane model. However, we were unable to generalize the results to the B- plane model, because when the Rossby number (based on the latitudinal scale) of the wave is of order unity the sphericity effects can no longer be lumped into one parameter.

Acknowledgement

The work of E.K. was supported by the California Space Institute under grants CS 13-83 and CS 14-84, and the Institute of Geophysics and Planetary Physics (IGPP) at the University of California. This work was begun while E.K. was visiting the Max-Planck-Institute fur Astrophysik, and the continuing hospitality of the Institute is gratefully acknowledged. We thank an anonymous referee for pointing out the work of Braginskiy.

Appendix A In Section 3 we have found it convenient to use the pressure perturbation p as our dependent variable. In this appendix we discuss the four consequences of this choice, and show that the present method can be used to recover the results obtained by Stone (1971) in his study of the nonhydrostatic Eady model.

If we set 6 = O (no horizontal shear) and E = 1 (linear vertical shear), we obtain from (3.12) the equation

pozz + 2ikp0, - k2 N 2 p o = 0.

where k is the horizontal (latitudinal) wavenumber and we assumed that the Brunt-Vaisala frequency is constant. Note that because of our scaling N 2 = R i , where Ri is the Richardson number as defined by Stone (1971).

The problem (Al) is degenerate, and no condition on the frequency is obtained by imposing boundary conditions in z. It can be shown that this property of the problem persists to all orders in the azimuthal wavenumber m. However, the basic state may still be unstable if one imposes boundary conditions on the vertical velocity perturbation w as done by Stone (1971). The boundary conditions used by Stone, w =O on z=O, 1 are somewhat artificial, particularly in the astrophysical context, and suggest that the instability described by Stone might be due only to the boundary conditions he chose.

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In terms of the velocity perturbation w l .

w1 = - i[po + ( z - c)(3 ik p o +poz)1/(N2 - l),

we find instead the eigenvalue problem

wlZz + [ik- ~ ( z - c ) - ’ ] w l Z - [ $ N 2 k 2 + ik(z-c) ~ ‘ 1 w1= 0. (A3)

In the limit k+O, this problem reduces to

WlZZ - 2(z - c) - ’ WlZ = 0. (A41

Solving this equation subject to the boundary conditions w=O on z=O, 1, gives (Stone 1971)

w1 =(z-c)3+c3, 645) with c given by

~ = $ ( 1 + 3 - ~ ’ ~ i ) .

The state is therefore unstable, the instability being due to the boundary condition at z = 1.

References

Braginskiy, S. I., “Magnetic waves in the earth’s core,” Geomagn. & Aeron. 7, 851

Charney, J. G. and Stern, M., “On the stability of internal baroclinic jets in a rotating

Drazin, P. and Reid, W., Hydrodynamic Stability, Cambridge University Press,

Eliassen, A. and Kleinschmidt, E., “Dynamical Meteorology” in Handbuch der Pkysik

Howard, L. N., “Note on a paper by John Miles,” J . Fluid Mech. 10, 509 (1961). Hyun, J. M. and Peskin, R. L., “Baroclinic instabilities of a deep fluid,” J . Atmos. Sci.

McIntyre, M. E., “On the non-separable baroclinic flow instability problem,” J . Fluid

McIntyre, M. E., “Baroclinic instability of an idealized model of the polar night jet.”

Pedlosky, J., Geophysicalfluid dynamics, Springer Verlag, New York (1979). Stone, P. H., “On nongeostrophic baroclinic stability,” J . Atmos. Sci. 23, 390 (1966). Stone, P. H., “Baroclinic stability under non-hydrostatic conditions,” J . Fluid Mech.

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Q.J.R. Meteorol. Soc. 98, 165 (1972).

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