Unstably stratified geophysical fluid dynamics

of at~m~_~s and oceans ELSEVIER

Dynamics of Atmospheres and Oceans 25 (1997) 233-272

Unstably stratified geophysical fluid dynamics Jun-Ichi Yano a,*, JoE1 Sommeria h a NCAR, P.O. Box 3000, Boulder, CO 80307-3000, USA b Laboratoire de Physique (C.N.R.S. URA 1325), Ecole Normale Supdrieure de Lyon, 46, al. d'halie, Lyon Cedex 69364, France Received 17 September 1993; revised 8 June 1995; accepted 6 March 1996

Abstract

The linear dynamics of the unstably stratified geophysical flows is investigated with a two-layer formulation. A 'convective' deformation radius classifies the dynamics into three regimes: 1. the scales smaller than the deformation radius: the dynamics characterized by unstable inertial-gravity modes; 2. the scales larger than the deformation radius: a quasi-geostrophic regime; 3. the scales close to the deformation radius, where the dynamics transits from the inertial-gravity regime to the quasi-geostrophic regime. The Rossby wave can propagate eastward in the unstably stratified quasi-geostrophic regime. The baroclinic instabilities are basically realized as a larger-scale extent of the inertial-gravity instabilities, but the former can be isolated from the latter in a limit of small /3-effect, with a very deep lower layer. The results suggest that the convectively unstable Jovian atmospheric dynamics can be well described as a quasi-geostrophic system.

1. Introduction

The purpose of the present paper is to embark into a new frontier of geophysical fluid dynamics: the unstably stratified case. The standard geophysical fluid dynamics (e.g. Pedlosky, 1987) has been restricted to the case with the stable stratification. The rationale for this restriction is that the most familiar geophysical flows to us (namely, the terrestrial atmosphere and the oceans) satisfy this condition. However, the removal of this constraint is desirable for improving our understanding of the dynamics of various

* Corresponding author at: CRC-SHM, Monash University, Clayton, Victoria 3168, Australia. 0377-0265/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S 0 3 7 7 - 0 2 6 5 ( 9 6 ) 0 0 4 7 8 - 2

234

J.-l. Yano, J. Sommeria / Dynamics of Atmo~pheres and Oceans 25 (1997) 233-272

types of planetary atmospheres. For the atmospheres of the major planets, this constraint is no longer satisfied due to the existence of internal heat sources (e.g. Hanel et al., 1981a,b, 1983). A type of dynamics for this purpose has been devised by Busse (Busse, 1976, 1983) under a framework of a deep thermal convection theory. He assumes that the vertical scale of motions is no longer smaller than the horizontal scale, but the Coriolis force is still dominant compared with the non-linearities (i.e. the Rossby number is small). However, in a limit of weak viscosity, which is expected to be a relevant regime for the Jovian atmospheres, the most preferred scale of convection is asymptotically small. Hence, the implicit assumption of the dominance of the Coriolis force is no longer quite justified. Furthermore high Rayleigh number thermal convection (e.g. Siggia, 1994, as a review) efficiently mixes the interior, expelling the temperature gradients toward the boundaries, where the constraint of vanishing normal velocity prevents turbulent mixing. This is surely an important feature, observed in all laboratory experiments (e.g. Castaing et al., 1989; Zocchi et al., 1990). Such a thermal boundary layer also appears in models of the convective zone of the stars, using mixing length theory (e.g. Spiegel, 1971; Spruit et al., 1990). However, this is completely ignored in previous instability analysis, using the diffusive vertical temperature profile as the basic state. We agree with Busse (Busse, 1976, 1983; see also Ingersoll and Pollard, 1982: Yano, 1987a) that convective motions also exist in the interior, where heat is generated, but the dynamics of the thermal boundary layer must have a key influence. The present paper is a first step in modelling this dynamics, by considering a shallow fluid layer lying over a lighter deep fluid, assumed at rest. We study the different instabilities and propagating modes that can develop in such a layer. We therefore seek a dynamics confined to a shallow atmosphere, like Williams (1978, 1985), Yano (1987b), but the unstable stratification brings specific differences that we explore in this paper. A similar idea was recently developed in the oceanographic context (Jones and Marshall, 1993; Legg and Marshall, 1993). Some parts of the oceans (e.g. the Arctic Ocean, the Mediterranean Sea, especially in the Gulf of Lions) are strongly cooled at the top to form a shallow thermal boundary layer of order 200 m. According to the numerical experiment by Jones and Marshall with the Boussinesq approximation on the f-plane, a triggered convection can extend 2 km in depth. Legg and Marshall considered the interactions of these thermal plumes by heron formulation. It is interesting to note that Williams (Williams, 1978, 1985) has already proposed an oceanographical analogy to the Jovian atmosphere, where he has emphasized the neutral stratification of deep ocean, so that deep convection has been much less important for Jovian atmospheric circulation than the upper-level baroclinicity induced by the solar heating. We re-invoke this analogy in the present paper, but with view of the recently developed oceanographic thermal convection. We first study in Section 2 the Rayleigh-Taylor instability on the f-plane. It appears that the hydrostatic balance along the vertical is well verified, even almost at horizontal scales of the order of the layer thickness. We can then retain most of the assumptions of the standard geophysical fluid dynamics, besides the removal of the stable-stratification, and then apply this formalism to more complex situations. We consider a two-layer

J.-l. Yano, J. Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

235

system on the beta-effect in Section 3 and Section 4, and introduce the possibility of a vertical shear in Section 5. As a first approach to the problem, we restrict the analysis to the linear stability theory, with only qualitative discussions of the non-linear effects. We study the Rossby waves in Section 4, and already face a new aspect of the large-scale dynamics with unstable stratification. A first taste of the idea is obtained by simply writing down the dispersion relation of the Rossby wave in a two-layer system (see Eq. (24a) and Eq. (24b) below). A negative Froude number leads to a divergence of the frequency at a turning scale, which further implies a continuous modification of the Rossby wave into ah inertial-gravity wave in a phase space: it is extensively investigated in Section 4. We present a stability analysis under the presence of a vertical shear in Section 5, where an interesting possibility of baroclinic instability is found. Remind that the presence of the baroclinic instabilities offers a rationale to invoke a large-scale dynamics in a standard geophysical flow. In all cases, we find that the linearly most amplified modes are at small scales. However, these should quickly saturate as turbulence, so that they would not critically contribute to the dynamics of the large scales, apart from defining the mean stratification by a vertical mixing. A corresponding turbulent mixing of momentum is also expected, leading to some friction effects, and possibly more complex diabatic effects that could be modelled by mixing length theory. The analogous idea has been applied to the atmospheric dynamics of the Earth: though cumulus convection is of small scale of l0 km, it is usually assumed that a distinguished large-scale dynamics of 10 3 km exists separately. The development of the dynamical meteorology, and the geophysical fluid dynamics as its generalization have been based on this principle. The intention of the present paper is to extend this principle to the Jovian situation. Our analysis could find applications in various cases of thermo-convection in rotating systems. We discuss particularly on the Jovian case in Section 6.5. Our results depend on two parameters which are the layer thickness and relative density difference between the layer and the underlying fluid. Such quantities are still largely unknown, but could be estimated by a model of turbulent convection using some mixing length theory. The dynamics that we consider corresponds to the region just below the tropopause, where efficient cooling by infra-red radiation is possible (i.e. the atmosphere becomes optically thin). Above this layer, the stratification should be again stable, and we can represent this property by the second layer with lower density. The major coherent vortices (like the Great Red Spot) probably evolve in such a stably stratified atmosphere (e.g. Sommeria et al., 1991; Yano and Flierl, 1994). However, quite different structures are also observed, which on the contrary rapidly disperse from some localized perturbation, like incoming plumes. We indicate evidence that the evolution of such features is controlled by the dynamics of some deeper, unstably stratified layer.

2. Rayleigh-Taylor instability with rotation We consider a two-layer system (Fig. l(a)). The first layer of a constant density Pl with a mean depth HI lies over the second layer of another constant density P2 with a mean depth H 2. The top free surface fluctuation is designated by rh(x, y, t), while the

J.-I. Yano. J. Sommeria/ Dynamics of Atmospheres and Oceans 25 (1997) 233-272

236

(a)

(b) \\\\\\\\\\ Pl

_HzT

Htl

Pz

Pz

Hz \

7///////,

7

Fig. 1. The model configuration: (a) the standard case, (b) the free bottom surface case.

fluctuation of the interface of the two layers is given by rh(x, y, t). We introduce a length scale L and velocity scale U, and define a Froude number F: at each layer ( j = 1, 2), and a Rossby number ~j by

f~L 2

U

l~J- gHj(l - y ) "

~-- fo L

(1)

where f 0 ( > 0) is the Coriolis parameter, g the acceleration of gravity, and 3' = Pl/P2 the density ratio. We are interested in the unstably stratified case, 3' > 1, for which the Froude numbers are negative. For the moment, there is no imposed velocity scale, so it will be convenient to choose a velocity scale U = f0 L, so that ~ = 1. A natural length scale of this problem is the Rossby radius of deformation L R - L ~ ~/13~, which is imaginary in the unstably stratified case. We consider in this section the linear Rayleigh-Taylor instability problem on the f-plane. The case with two infinitely deep fluid layers is considered in Chandrasek_har (1961). Here, we extend this analysis to the case with a finite depth upper layer, but still keeping the lower layer infinitely deep ( H 2 ---> + ~). The basic state is at rest, and we do not assume the hydrostatic balance in this analysis. Hence, the perturbation equations are given by 0

1

,gtvj + ~ x vj -- - - - Vp~ - k~ Pj

(2a)

V.v: = 0

(2b)

at each layer ( j = 1, 2), where vj = (u j, v:, wj) is the three-dimensional velocity vector, and ~-= 1/(1 - "y)/~16 the non-dimensional gravity.

J.-I. Yano, J. Somraeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

237

We take a mean position of the interface of two layers at z = 0, so that the boundary conditions are WI=

a 7/1 0t ~

Pl=0

at

Z=6+g r/t

at z = 6 + - 3

wj = w 2

172

(3a) (3b)

772

at z = -

at

r/2 at z = ---;g

Pl =Pz w2 = 0

-z-r/l

at z ~ - ~

(3c) (3d) (3e)

where 6 = H J L is the aspect ratio of the upper layer (we write the interface displacements as r h / ~ and "q2/g to get non-dimensional values of order 1 for rb). As a standard procedure, we apply the operators k . V × and k . V x V x on Eq. (2a) to obtain O^ a --k. ( Vx vj) - --wj = 0 (4a) Ot 0

Oz

--V 2 0t

(j =

O^ wj+

(VX v j ) = 0

k-

(48)

1, 2). We assume a solution of the form w I = ( a e ~z + B e - ~ Z ) e i k X + i t y + i ,°, w 2 = ( a + B ) e az+ikx+ily+ic°t

so that the continuity of w over the interface (Eq. (3c)) is satisfied. Though R e ( a ) > 0 is formally required to satisfy the condition at far depth (Eq. (3e)), it turns out that this requirement is too restrictive to obtain a non-trivial solution when Re(a 2) < 0. We demand the condition R e ( s ) > 0, only when I m ( a ) = 0, anticipating that a wave component of the solution will take care of the condition Eq. (3e) at a very deep, fixed bottom, when I m ( a ) 4: 0. By taking a determinant of Eq. (4a) and Eq. (4b) we obtain the dispersion relation O~2 tO 2

K 2 _ °d 2

(5a)

with K 2 = k 2 + 12. T h e remaining boundary conditions (Eq. (3a), Eq. (3b) and Eq. (3d)) lead to another dispersion formula O~

to 2 =

(5b) Pta(-r + cotha6)

The combination of Eq. (5a) and Eq. (5b) determines a relation between the horizontal wavenumber K and the vertical decay rate a as x 2=

-

+ coth

(5c)

238

J.-I. Yano, J. S o m m e r i a / Dynamic,~ oJ A tmo~pheres and O c e a n s 25 (1997) 2 3 3 - 2 7 2

The mode structure is obtained from Eqs. (2a), (2b), (3a), (3b), (3c), (3d) and (3e), which yield 2 y - 1 + cothc~6 B= 1 - coth o~6 A (6) lg I ~

-- ioJu I

= i(o~/k)( lg 2 =

Ze ....

Be-":)e

(7a)

' k ' + .....

-- i ~u2

= i ( c e / k ) ( Z + B ) e . . . . . ~,+i .... (7b) where l = 0 is assumed for now without loss of generality. Examples of the dispersion curves are given in Fig. 2: Re(w2) is plotted as a function of K2 based on Eq. (5a) in Fig. 2(a), whereas K 2 is computed from a given vertical wavenumber o~2 by Eq. (5c) as shown in Fig. 2(b). An imaginary component of c~2 is added to obtain a real horizontal wavenumber, when Re( o~2) < 0. Note that the solution is vertically purely evanescent ( I m ( ~ ) = 0) for K 2 > _ t~, whereas we have a complex wavenumber for K 2 < _ / ~ . The former corresponds to the Rayleigh-Taylor instability, and the latter to the almost neutral inertial-gravity mode. There is a finite gap of a dispersion curve between these two regimes. The gap narrows with decrease of the aspect ratio ~5, whereas when the upper layer is sufficiently thick (6 > 0.8), no solution

8,)

1.4 1.2 1.0 .8 ~

,

0 -.4 -.6

-1.00

. ".~.. .'--...

' .~ ' .4 ' .6 ' .8 ' 1 . ' 0 ' 1 2 2 ' 1 2 4 ' 1 2 6 ' 1 2 8 ' 2 . 0 Kz

(b)

.

.

.

.

.

.

.

.

.

.

.

>2/

.8 .6

. _/.

-.2 ~

/"

/.z

-.4

/

/"

-.6 / i../ --,8 ./ --I

, 0

,

,

.2

.

.

.

.4

.

.

.

.6

.

.8

,

,

*

,

,

,

,

,

l

,

1.0 1.2 1.4 1.6 1.8 2.0 Kz

Fig. 2. The linear dispersion curve of the linear Raylcigh-Taylor system with /~t = - 1, 3' = 1.2, and 6 = 1 (solid), 0.5 (dash), 0. I (chain-dot): (a) Re( w 2 ), (b) Re( c~2) as a function o f K 2 The asymptotic formulas (Eq. (9), Eq. (10)) are represented as a short-dash line.

J.-l. Yano,J. Sommeria/ Dynamicsof Atmospheres and Oceans25 (1997) 233-272

239

satisfying Eq. (5c) is available at K 2 < _ Fl" It may be worthwhile to note that, as Eq. (4a) and Eq. (4b) implies, the geostrophic mode, which is a steady solution without vertical dependence in each layer, is available separately from the type of solution (the inertial mode) considered here. This mode is, however, excluded in Eq. (5c) due to its virtual singularity. It appears in Fig. 2 that the inertial-gravity wave (the inertial mode) joins to the geostrophic mode at K 2 = _ b~l, where its frequency vanishes. In the limit of large layer thickness 6, we recover the result of Chandrasekhar (1961)

O j 2 = ~ [ 1 - - ( 1 + 4 K 2 / F 2 ) 1/2] ol = p~ to 2 where we have defined a new Froude number/~= =/~l 6 ( y + 1), which remains finite. All the modes are purely amplified (to imaginary), althoush more slowly than in the absence of rotation. For large reduced wavenumbers K 2/p~, the growth rate tends to the value without rotation K/It~=]. We are rather interested here in the opposite limit of small thickness H t, more precisely ]c¢6I<< 1. Then the vertical velocity variations would be neglectable in the upper layer, and the vertical component of Eq. (2a) reduces to the hydrostatic balance 1 Op ----+~=0 ( j = 1,2) (8) pj Oz The dispersion relation Eq. (5b) reduces to w 2 = az/b~l, so that, combining with Eq. (5a), we get a2 = K2 + F l to 2 = 1 -1- K 2 / / ~ t

(9) (10)

Therefore modes with K 2 > IFll are purely amplified ( to imaginary), and are evanescent in the bottom medium ( a real positive). The vertical velocity at the interface is in phase with the displacement "q2, both purely growing like in the ordinary Rayleigh-Taylor instability (the effect of rotation is to deviate the divergent velocity u t, creating a transverse component vt). By contrast modes with K 2 < Ibetl are now stable (to real), corresponding to purely propagating inertial gravity modes. The vertical velocity and displacement r/2 are then in quadrature, as well as the two horizontal velocity components. The field in lower layer corresponds to an inertial wave propagating in the deep interior ( a imaginary), and its physical significance is made clearer by pushing the expansion of the dispersion relationships (Eqs. (5a), (5b) and (5c)) to next order in the parameter l a6 I. This yields a small imaginary component toi to the frequency given by TtK 2 tot = +

(11)

The sign of toi is such that the perturbation decays as it emits an inertial wave towards the bottom t i m ( a ) is of the same sign as Re(to)), or it grows by absorbing an incoming inertial wave t i m ( a ) and Re(to) are of opposite signs). This effect remains however

240

J.-l. Yano, J. Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

slow in the limit of a small thickness 6 << 1. When the frequency goes to zero (i.e. K 2 ~ [ / ~ ]), the horizontal velocity becomes purely transverse to the horizontal wavevector K, from Eq. (7a), and the departure from geostrophy - v ~ + i k p l / p ~ = - t o 2 v~ goes to zero. Therefore both the purely growing mode of the Rayleigh-Taylor instability and the inertial gravity mode smoothly tend to the geostrophic mode a s K 2 --->IFII, in agreement with the dispersion relationship. Notice however that as to --* 0, its imaginary part toi remains finite, indicating an effective coupling with internal inertial waves (with nearly horizontal wave-vectors and therefore low frequencies). We now discuss in more detail the condition of validity for the shallow layer approximation. The tendency to the asymptotic formulas (Eq. (9), Eq. (10): short-dash) is clearly seen in Fig. 2. It will be shown that Eq. (10) agrees with the dispersion relation obtained with the hydrostatic balance, so that the condition ]a6] << 1 will be required for the validity of the hydrostatic approximation. This condition can be restated, using Eq. (9), as

I(K

<< I

or dimensionally, defining a scale L equal to the inverse of the disturbance wavenumber, L2

L2

Hence the hydrostatic balance is satisfied for all scales L >> H~ as long as the radius of deformation L R is much greater than H L. (The equivalent condition has been obtained by Jones and Marshall, 1993, purely based on the scale analysis for the Boussinesq system.) In practice a scale separation by a factor of 2 or 3 is sufficient, since the convergence to the hydrostatic approximation is exponential. In all cases, the growth rate is maximum at the highest wavenumbers, where the influence of rotation vanishes, and the two-layer model loses physical relevance. However, small scale instabilities would quickly saturate by non-linear effects, and various arguments suggest that larger scales would eventually dominate the system. Even the purely propagating modes with K 2 < [F~[ could be excited by some initial perturbation. Although this mode does not grow in our analysis, it represents a coupling between waves guided at the interface and inertial waves propagating in the deep interior (see Eq. (11)). If this interior is not quite neutral, such waves will be weakly amplified in this medium and excite the vertically trapped modes. Non-linear interactions from smaller scales in the shallow layer could also excite the propagating modes. Similarly, long oceanic surface waves are non-linearly excited from shorter waves which are the most amplified by the wind shear. From a vortex point of view, merging processes efficiently transfer energy toward the larger scales. Such arguments, further developed in subsequent sections, indicate that a large scale dynamics could be relevant even in the unstably stratified case. Such dynamics will be now studied in the general frame of shallow water theory, making use of the hydrostatic approximation in the rest of the paper.

J.-I. Yano,J. Sommeria/ Dynamicsof Atmospheresand Oceans25 (1997)233-272

241

3. Two-layer system

Hereinafter, we restrict our consideration in the case under the hydrostatic balance (Eq. (8)), based on the analysis of the previous section, to focus our attention to the large-scale dynamics, so that we recover the standard formulation for the two-layer systems (e.g. Gill, 1982; Pedlosky, 1987) even with an unstable stratification (i.e. Pl > P2). By designating the eastward and the northward velocity components as u j, vj ( j = 1, 2), respectively, at each layer, the momentum equations are given by D , u _ (1 + f l y ) v 1 = -~t l

0 oxrll

(12a)

Dj 0 D---~v, + ( 1 + fly)ul = - -~y'ql

(12b)

for the upper layer, and by D2u _ (1 + fly)v2 = 0 Dtt 2 - ~ x ~/B

(13a)

D2 O D---~v2 + ( 1 + flY)U2 = - -~y ~/B

(13b)

for the lower layer, where Dj_ O O O -Dt - = Ot +^EUjo x - + ' e-v , - ,Oy

(j=l,2)

and riB = r/2 + y(rll -- rh) The mass conservation at each layer is given by

o2

n2) +

Dtt T/2+

(1 }(ou 0v,) )(o.2 0v2)

+?:rl2

+

- n2)

ax + 3y

-Ox -+

=0

Oy

=0

(14a)

(14b)

respectively. Here we use the external Froude numbers defined at each layer ( j = l, 2) from the internal Froude number Eq. (l) by Fj---(1-Y)4

(15)

A standard fl-plane approximation f ( y ) - f 0 ( 1 + fly~L) has been assumed by taking the y-coordinate northwards, whereas the time has been scaled by the Coriolis parameter f0. Also we reintroduce an arbitrary velocity scale U so that the Rossby number @ can be a small parameter in the following. A simple extension of the present formulation is to consider the case with a bottom free surface and a top rigid lid condition (Fig. l(b)). This configuration, which looks like

J.-l. Yano, J. Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

242

less realistic than the first case (Fig. l(a)), may offer a better idealization of Jovian atmospheres (P. Stone, personal communication, 1992): the tropopause acts like a rigid-lid boundary for the troposphere, while the lower unstably stratified layer may act like a free surface to a bottom of the tropospheric atmospheric layer. It turns out that this configuration (Fig. l(b)) is obtained by reversing the order of layers and also by reversing the sign of the (external) Froude numbers of both layers (or equivalently by taking the negative mean depth for two layers) in the first configuration (Fig. l(a)). The different possibilities are summarized in Table 1. In realistic cases the density ratio y should be close to 1. However, the ratio y can physically be any value with the second configuration, if the free surface is replaced by an interface with a very deep layer with a density P0. The original formulation is recovered in this case by replacing p~ and P2 by P~ - P0 and P2 - P0, respectively, and the gravity by a reduced gravity. Finally, the energy conservation of the system is given by

Fj

d

-~tKj=Wj

( j = 1,2)

d ~tP, = -w~-

(15a)

w3

(15b)

- w2 + w3

(15c)

d

~P2=

for the first configuration, where Kl=

+ ~(~, - ~2)

(u~+v~

(16a)

K~= (l (-~2+ ~2 )( u~+v~)

(16b)

are the kinetic energy of upper and lower layers, respectively (( * ) designates a domain integration), and

P2

~-

(,~

(17b)

Table 1 T h e classifications o f possible c o n f i g u r a t i o n s o f the m o d e l

T o p free surface B o t t o m free s u r f a c e

Stable-stratification Unstable-stratification Stable-stratification Unstable-stratification

El

F2

Pl

F2

3'

+ +

+ + -

+ +

+ +

< > > <

.

.

.

.

1 1 1 I

243

J.-l. Yano, J. Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

are the potential energy due to the top free surface and the interface fluctuations, respectively. The energy conversion rates are given by Wl = 'Y

IV* "~l + ~(Tll -- "I~2) l'I

(18a)

W2 = (nBV- ( ~---~+ ~:V2)V2)

(18b)

W3= y('O,V. (-~2 + ~'tl2)V2)

(18c)

The potential energy P2 produced by the interface deformation turns negative in an unstably stratified case. Note, there is a positive available potential energy at the basic state without any surface deformations. The kinetic energy of a perturbation can grow, compensated by the decreasing potential energy of the developing surface deformation. For the second configuration with a free bottom boundary condition, the signs of the energies in definitions Eq. (16a) and Eq. (16b) and Eq. (17a) and Eq. (17b) should be reversed. In this case, the potential energy Pl due to the free surface is always negative, while P2 due to the interface remains negative with the unstable stratification, as expected. ^ Byassumption of a slow time scale, i.e. O/Ot= ~a/OZand with a scaling @~/3( .~ 1), F~ ~ F 2 ~ 1, we obtain a generalized quasi-geostrophic (QG) system

[0-~r +J(Oj," )1

Qj = 0

(19)

with j = 1, 2, expressing the material conservation of the potential vorticity in each layer Qj= ~:[ A0j + ( - ) / ~ ( Y ( J - l ) 0 , -

02)] +/~Y

(20)

where the streamfunctions in the two layers are given by ~Oj = rh and q'2 = 7/B"

4. Waves and instabilities

4.1. Rossby waves The dispersion relationship of Rossby waves is obtained by linearizing Eq. (19), 3 c9--~[ A~, +/~l(~b2- ~])] +,By I = 0

(21a)

a o-7 [

+

-

+

= o

(21b)

244

J.-l. Yano,d. Sommeria/ Dynamicsof Atmospheresand Oceans25 (1997)233-272

By assuming a plane wave solution ~ exp(ioJr/~ + ikx + ily), we have

[o4K2 +/9,)-

=0

(22a)

wy/92~b,- [ w ( K 2 + / 9 2 ) - ~k]~b 2 = 0

(22b)

where K 2 = k 2 + [2 is the total wave number. By taking the determinant, we obtain a general expression for Rossby wave dispersions

Ck (/9, + f 2 + 2K2)+_

/9, + / 9 2 ) ~ - 4 ( I - 7 ) F , F21

K4+(I~,+~2)K2+(1-y)#,~

w=-2

2

(23)

A detailed analysis of this dispersion relationship Eq. (23) will follow subsequently in Section 4.3. The most remarkable is that even in the unstably stratified case (i.e. /~ < 0 and /~2 < 0) the Rossby waves (quasi-geostrophic linear motions) do not turn unstable. The shortest explanation is that because of a slow motion assumed for quasi-geostrophic formulation, the dynamics is strongly constrained by Taylor-Proudman theorem, so that no vertical overturning is possible. The closest analogy is found in a Rayleigh-Benard system with rotation (Chandrasekhar, 1961, Chapter liD, in which no steady marginal convective motion is possible in the inviscid case regardless of the temperature difference imposed between two plates due to this Taylor-Proudman constraint. By the same token, convective instability is effectively filtered out by the quasi-geostrophy in the present case. Such a neutral stability of the geostrophic mode has been already seen in Section 2. The physics is most easily understood in the limit y ~ 1 +, for which the upper surface behaves like a rigid lid, and Eq. (23) simplifies to w+

(24a)

fik w_= K---7

(24b)

corresponding to the baroclinic mode qq = -(He/HI)tO 2 and to the barotropic mode 0~ = qJ2, respectively. The reduced gravity case of Section 2 is recovered, by assuming a very deep lower layer (i.e. Y/92 ~ 0- ) with no motion in the lower layer (i.e. if/2 ~ 0 and T/B ~ 0), (A completely analogous argument can be made in the limit FI ~ 0-, ~0~ ~ 0, which is more relevant with the free-bottom configuration.) In this limit, the dispersion relation is given by setting b~2 = 0 in Eq. (24a). The baroclinic velocity field is like in a stably stratified case, as long as we stay in the quasi-geostrophic approximation, but the corresponding interface deformation r/2 = (02 - YqJl)/( 1 - 3') is reversed (since y > 1). However the most remarkable difference is in the dispersion relationship Eq. (24b), which behaves quite differently as /91 and /92 are negative. There is indeed a turning scale, where the frequency diverges to infinity,

J.-L Yano, J. Sommeria/ Dynamics of Atmospheres and Oceans 25 (1997) 233-272

245

and changes the direction of propagation by crossing. In either of the considered limits y ~ 1 ÷ or k~2 ~ 0-, the turning wavenumber K t is given by

(25 The Rossby wave propagates eastward in the scale larger than [ - ( F i +/~2)]-l/Z The change of the propagation direction for a larger scale can be understood in terms of potential vorticity (t~.v.) conservation, say, in the upper layer, most easily in the reduced gravity limit ( F 2 "* 0). When a vortex tube is moved, say, northward, there are mainly two ways to ensure the conservation of the potential vorticity q = (~" + f ) / H against the increase of the planetary vorticity f: (1) decrease the relative vorticity ~, or (2) increase the length H of the vortex tube. In a stably stratified case, a negative vorticity (anti-cyclone) corresponds to a long vortex tube (ridge) by the geostrophic balance. Therefore both possibilities (1) and (2) lead to the production of negative vorticity for a northward displacement, which consistently defines the propagating direction of a Rossby wave: a ridge, with negative vorticity, induces a northward velocity on its west side, producing there a negative vorticity by p.v. conservation, so that the whole disturbance propagates to the west. By contrast, in an unstably stratified case, a negative vorticity corresponds to a short vortex tube (a trough). Consequently, the two compensating mechanisms to conserve potential vorticity by following a latitudinal displacement of a vortex tube lead to the opposite tendencies for propagation. For a small scale, where the vortex conservation dominates, the Rossby wave propagates westward as in a stably stratified case. As the scale becomes larger, the vortex tube moving northward will be compressed as it receives negative relative vonicity, with opposite effects on its potential vorticity. Therefore a larger induced vorticity will be required to satisfy p.v. conservation under the change of planetary vorticity associated with the latitudinal displacement, resulting in a faster propagation. At the turning scale given by Eq. (25), the two competing tendencies completely balance, hence the propagating speed diverges. By crossing the turning scale, the vortex stretching dominates, and the Rossby wave propagates in opposite direction. Note, however, that the 'p.v.' way of thinking invoked here has a limited applicability under the unstable stratification (c.f. Section 6.2). An important implication of Rossby wave divergence is that the quasi-geostrophy is no longer a self-consistent scaling in the unstably stratified systems as in a traditional geo-physical system - a slow time scale compared with the rotation period (~f~0 1) is assumed in deriving the Rossby wave dispersion. However, we obtain a very fast propagation in the vicinity of the turning scale, so that the original assumption of slow dynamics is broken. A more general wave dispersion analysis is performed in the next subsection to elucidate this problem more. 4.2. The general reduced-gravity case

Because it is shown that the quasi-geostrophy is not always a self-consistent approximation, we return to the original equation system Eqs. (12a), (12b), (13a), (13b), (14a) and (14b) to derive a general dispersion relationship. The procedure is basically the same as the usual derivation of dispersion formula of inertial-gravity waves (e.g.

246

J.-l. Yano, J. Sommeria / Dynamics t~fAtmospheres and Oceans 25 (1997) 233-272

Gill, 1982, Chapter 8), apart from retaining the /3-term in the present case. We anticipate that a reduced-gravity wave is continuously connected to a Rossby wave. We set O/Ot = iw in the momentum Eqs. (12a), (12b), (13a) and (13b), which offer an expression for the velocity components uj, vj ( j = 1, 2) in terms of the surface displacements r/t and 7/B. By substitution of the expressions into the continuity Eq. (14a) and Eq. (14b), we obtain i~o

+~y]2_co2~,+F,(~B--rh) 2

+fl{(l+fly)

2-~o2}

^

0

[{(l+flY)2+~o2}~---~-2i~o(l+Cty)~y]rll=O

iw (l +~y):_w2rIB +l~2(Trh- rIB) +fl{(l+~y)

2-w?}

-2

^ c? [{(l+flY)2+~o2}~---~-2i~o(l+~y)-~y]rlB=O

The assumption of a fixed wavenumber k in the longitudinal direction (i.e. O/Ox = ik) leads to an eigenproblem to define the latitudinal structure of the waves. For the present purpose, we confine our interest to a local solution ( y ~ O, i.e. the latitudinal scale is small in comparison with the planetary radius), and set 1 + ~ y = 1, O/Oy = il in the above set of equations. As a result, we have ico -

1

+/~l(~/B-- r/,)

+

t~-'-

[ik(1 + w e ) + 2 w l ] r / , = 0

(1 - o9:) 2 (26a)

K2 i,o[ 1--~°20B+P2(' /~'--nB)1+(l--o)2) 2 [ ik( l + ~°2) + 2~°l]nB=O (26b) whose determinant leads to a general dispersion relation for both inertial-gravity waves and Rossby waves. In the reduced-gravity limit (/~2 ~ 0 - , r/B ~ 0), the determinant in Eq. (26a) and Eq. (26b) reduces to -[KZ+/~l(1-wz)](1-~o2)o~+/3[k(l

+oJ 2 ) - 2 i t o I ]

=0

(27)

With a scaling w ~ 1 >>/3, we obtain an inertial-gravity mode identical to Eq. (10), along with a purely inertial mode ~o2 = 1 (corresponding to K = 0). On the other hand, in a limit oJ ~ / 3 << 1 we recover the Rossby mode (Eq. (24a)) for the reduced-gravity system (/~2 = 0). The connection formula between Rossby and inertial-gravity modes is

J.-l. Yano, J. Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

247

obtained by a scaling K 2 + E l ~ j~2/3, to ~ ~ I/3 in the vicinity of the turning scale Eq. (25) and a wave faster than Rossby waves but slower than the inertial-gravity waves, from Eq. (27): K

to3_

--+1 /61

)

,,k to+fl--z-=0 F1

(28)

It can be shown that the connection formula Eq. (28) reduces to Eq. (10) in the limit to >>/~ ~ 0 and to Eq. (24a) in the limit to ~/3 << 1 and further connects these two formulae over the turning scale Eq. (25). In this respect, Eq. (28) can be considered as a generalization of both Eq. (24a) and Eq. (I0). The three types of dispersion curves derived from Eq. (24a), Eq. (10), and Eq. (28) are depicted in Fig. 3 ((a) real component of the frequency; (b) the growth rate) as dependence to the longitudinal wavenumber k with /3 = 0.1, E l = - 1, 1 = 0. Note that the case l =/=0 just changes the Rossby frequency by factor k / K , which is equivalent to change /~ with the same factor. (We will keep this convention l = 0 throughout the paper for the graphic representation of the results, if not otherwise mentioned.) The Rossby frequency Eq. (24a), presented by a long-clash curve, is positive (westward propagation) for k > 1 and increases with the decrease of the wavenumber k. It diverges at k = l, and turns the sign (eastward propagation) for the larger scale k < 1. On the other hand, the inertial-gravity modes Eq. (10), represented by short-dash curves, are either purely growing or damping for k > l, but turn neutral and oscillatory for k < 1. One of the generalized dispersion curves (Eq. (28): solid) corresponds to the westward-propagating Rossby wave (long-dash) in the limit of high wavenumbers. This curve gradually deviates from the Rossby-wave curve with decrease of the wavenumber, and gradually modulates into the westward-propagating inertial-gravity wave (short-dash) as crossing the turning wavenumber K t. This mode remains neutral for all wavenumbers. The corresponding velocity is purely transverse at high wavenumber, with negligible interface deformation, like in an ordinary barotropic Rossby wave. As the wavenumber approaches the turning wavenumber K t, the interface deformation becomes very large, corresponding to a significant horizontally divergent velocity component, in quadrature with the transverse velocity. As this divergent component becomes comparable to the transverse (rotational) one, an inertial gravity mode is obtained. The other two generalized dispersion curves closely follow those of two inertial-gravity modes in the limit of high wavenumbers: one is a growing and the other is a decaying mode. These modes are dominated at high wavenumbers by a horizontally divergent velocity, with a transverse deviation by the Coriolis force, increasing for lower wavenumbers. This transverse component is influenced by the background planetary vorticity, inducing a slow eastward propagation. The growth is neutralized at a critical wavenumber k c and remains neutral below this wavenumber. As crossing this critical wavenumber, these modes branch out into a slower and faster propagating modes. The former closely follows the dispersion curve for the Rossby mode, and the latter for the westward-propagating inertial-gravity mode in the limit of small wavenumbers. Fig. 3(b) shows that the fl-effect increases the range of the Rayleigh-Taylor instability by decreasing the critical wavenumber (the scale of the exchange of stability)

248

J.-I. Yano, J. Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

(a) 1.8 1.6 1.4 1.2 1,0 .8 .6 .4 .2

I

'

I

'

I

'

~

;

I

[

, ,~r--C~l

]

'

I

'

t

'

I

'

I

'

t

'

I

'\,\RT "'",~\ /. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0

2

,~'

- . 4 - . 6

8

1.0 -1.2 -1.4

2

i

.

,

4

i

.b

i

i

.8

1.0

,

~

,

1.2

i

,

I .4

i

,

~ .6

i

,

i

,

2.0

! .8

i

,

2.2

i

,

2.4

i

,

2,6

i

2,8

,

3.0

k

(b) .O

~

i

,

I

i

,

t T-I

,

i

,

i-,

i

,

i

,

i

,

i

,

i

,

i

,

i

, .

2.5

2.0 1.5 1.0 0 -.5

-1 .0 -1 .5 -2.0

IG

~

.

-2.5 -~.0

I

•

i

i

i

'2., .6 '8 ,!o'~ .

i .2

[

I

J

1.4

i 1.6

,

I 1

.8

~

i

2.0

,

t

2,2

,

i

2,4

+

2.5

~i~.o

k

Fig. 3. The dispersion relation in the reduced-gravity limit

(F~ --, 0-):

(a) frequency, (b) growth rate. The

solid curves are based on the connection formula, while the long-dash curves are quasi-geostrophic limit (baroclinic Rossby mode: R), the short-dash curves the inertial-gravity approximation (/3 = 0: IG). The barotropic Rossby mode (RT) is also added as the chain-dot curve for reference. The parameters are /~t = - 1, /3 = 0.1. (I = 0 in this and the following figures except Fig. 5.)

J.-I. Yano, J. Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

249

k c. In that respect the /3-effect more destabilizes the system (c.f. Eq. (37a) and Eq. (37b)). Also represented in Fig. 3(a) is the barotropic Rossby wave to = (with l = 0) by the chain-dot curve.

kfl/K 2

4.3. Thegeneraltwo-layercase The analysis of the general cases with a finite depth of lower layer (i.e. k~z =~0) is basically the same as the reduced-gravity case, but a little involved by a necessity to consider both baroclinic and barotropic modes simultaneously. With the scaling to ~ l >>/3 in Eq. (26a) and Eq. (26b), we obtain the general dispersion relation for the inertial-gravity mode:

(|-

]/)/~lF2(1- ¢.02)2"[- g2(F1 "Jr-F 2 ) ( l - 032) q'- K4-~-0

or by solving this algebraic equation

(,02= to~-~2 =

1-

2(-/-1)

~]-~ j 4(1 - 7) AF, F2

.

(29)

x 1+__ l-[x2( L +~2)+2(1_,)Lp212 where

A -(K2 + e,)(K2 + f~)- ~,f2

(30)

The frequencies tog+ and tog_ are either real or purely imaginary, and the two interfaces (given by Eq. (26a) and Eq. (26b)) move in phase or opposition, depending on wavenumber and stratification. On the other hand, the Rossby mode dispersion is obtained by the scaling to ~/3 as

~2k~-&(2K~ +~, + f~)o, + Ato~= o or to=to(~)-

~(

2KZ+F,+/~2)

4~

1_+ 1 - ( 2 K 2 + / ~ +F~) 2

(31)

which is just a reformulation of Eq. (23). It transpires that the turning wavenumber Kt, defined by the divergence of a Rossby mode, is given by the condition A -- 0:

K2

ffI+ff2([4(I-y)FIF%] I/2} •~" 1 + l--~

(32)

J.-l. Yam), J. Sommeria / Dynamics q]'Atmo,~pheres and Oceans 25 (1997) 233-272

250

which generalizes Eq. (25). In the vicinity of the turning point (A ~ 0), the frequencies of two Rossby modes are given by

fik(2K 2 + 1?, + 172) CO+ -

+ 0(1)

- -

w~a)=

A

2K2+I? I +17,

(33a)

+ O(A)

(33b)

respectively. It is seen that the baroclinic mode w~ ) diverges at the turning scale, while the barotropic mode toCR) remains finite at the turning scale. Similarly, in the limit A ~ 0, the two modes of the inertial-gravity waves reduce to

w~+=2 1 - 2 ( y - 1 )

F, fe

]

A w~_= K2(/~ +172)+2(1_y)171172 +O(A2)

(34b)

2 changes sign over The quantity w~+ 2 remains positive finite (with A > 1), while my_ the turning scale. The latter mode is expected to be continuously connected to the baroclinic Rossby mode. For this reason, we designate the modes we+ and we - in Eq. (34a) and Eq. (34b) barotropic and baroclinic, respectively. Finally, the general connection formula is obtained in similar manner as to obtain Eq. (28), with the scalings w ~ fil/3 and A ~ fi2/~:

+ 2(1 -

o. 2 + ( 1

-

)*,172.o +} = 0

(35)

Two examples of dispersion curves are shown in Fig. 4, for a lower layer ten times thicker than the upper one (a), and for two layers with equal thickness (b). Only the real components are shown, because the growth rate curves are much similar to Fig. 3(b) in these cases. The roots of Eq. (35) are represented by the solid curves, the baroclinic Rossby mode Eq. (33a) by a long dashed curve, the baroclinic inertial-gravity mode Eq. (34b) by short dashed curves, and the barotropic Rossby mode Eq. (33b) by a chain-dash curve. The barotropic inertial-gravity mode remains neutral with high frequencies, and is not shown for clarity of the figure. A major modification from the reduced-gravity case is the disappearance of a divergence for the barotropic Rossby frequency in the limit k --+ 0. It is shown that the maximum of the frequency oJ(m)+of barotropic Rossby mode is given at wavenumber

K 2= - 3 ( y - I ) ( I / / ~ '

+ 1/172)+ O ( y -

I) e

(36)

with l = O. Except for this difference, qualitative features of the dispersion diagram are quite similar to the reduced-gravity case. The inertial-gravity modes are similarly

J.-1. Yano, J. Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

251

(a) ]

I

I

'

I

I

I

I

I

I

~

I

[

I

I

I

I

I

I

I

,

18 '2 .I

I

I

I

I

[

I

I

I

I

I

u =_ g

-.2 --,4 --,6 --.8

-1.0 -1 .2 -1 .4

i

i

.2

1

i

.4

i

,

.6

i

i

.8

i

i

i

i

i

1.6 1.4 1.2 1.0 .8 .6 .4 .2

g

I

,

i

,

P

i

i

,

i

,

0 2.2 2.4 2.6 2.8

3.0

k

(b)

$

i

1.0 1.2 1.4 1.6 1

•

i

,

~

,

~

,

i

,

p

,

n

,

n I'

[

'

i

,

n

,

I

,

u

,

i

,

I I -,

/

o -.2 --.4

m

--.8

-1.0 -1.2 -1.4 -1.6 -1.8

J"

/

t i

.2

,

i

.4

.

r

,6

.

i

.8

.

i

1.0

,

i

1.2

,

II

,

1.4

I

1 ,b

,

I

1.8

,

I

2.0

,

i

2.2

,

i

2.4

,

~

2.6

,

i

2.8

,

3.0

k

Fig. 4. Same as Fig. 3(a) but with F2=-0.1, 7=1.2 (a), and with F 2 = - I , -y=1.2 (b). Note that the inertial-gravity and generalized dispersion curves are restricted to the baroclinic mode.

252

J.-I. Yano, J. Sommeria / Dynamics ¢q'Atmoapheres and Oceans 25 (1997) 233-272

destabilized by the/3-effect: the wavenumber K c for stability exchange shifts to a lower value given by (37a)

K 2 = K 2 - fi2/3A( K 2)

( k ) 2j-K,2

A(K~) = 3 -~

(37b)

from the turning wavenumber K t given by Eq. (32). An example of dispersion curves with the free-bottom configuration (i.e. 3' < 1 but Fj < 0) is shown in Fig. 5. A major modification is that the barotropic Rossby mode (chain-dash curve) diverges at the second turning wavenumber:

Over this second turning wavenumber Kt.2, the barotropic Rossby wave modulates into the barotropic inertial-gravity wave (the asymptotic curve not shown in figure). Below this turning scale, the barotropic inertial-gravity wave is destabilized in a similar manner as the baroclinic inertial-gravity wave is destabilized by crossing the first turning scale. Since the two layers have a strongly correlated velocity in the barotropic mode, they behave very much like a single layer.

1.8

.

,t.

,

,

,

,

,

.

,

. ~],

,

.

,

,

,

.

,

.

,

,

,

.

l

l

,

l

,

,

,

1.6 1.4

i R~r

1.2 1.0

t

__i

I R

.8 ,6 .4 .2

0 --.2

-I .0 -I .2 -I .4 -I .6 -I .8 -2.0

RT *

R

, i , i i 1 , 1 1

.2

.4

.6

.81.0

,

l

,

l

,

l

h

l

,

l

,

l

,

.21.41.61.82.02.22.42.62.B3.0 k

Fig. 5. Same as Fig. 4 but for a free bottom boundary case with 3' = 0.5, /~l = F2 = - 1, /3 = 0.1, l = 0.5. in contrast with Fig. 4, the generalized dispersion curves for the barotropic mode are also added as solid curves.

J.-L Yano, J. Sommeria/ Dynamics of Atmospheres and Oceans 25 (1997) 233-272

253

5. Effect of a vertical velocity shear and baroclinic instabilities

Since the baroclinic instabilities are a chief driving force of stably stratified geophysical flows, it is of much interest to consider the modification of the baroclinic instabilities under the unstable stratification. Since the Jovian atmospheres are dominated by the strong zonal flows (c.f. Ingersoll, 1990), the problem is of much interest in this point of view as well. It can be also considered as a generalization of dispersion analysis of the previous section with a vertical zonal shear. We consider the Phillips (1954) problem under the unstable stratification. We add a homogeneous zonal flow Ut, U2 to the upper, and the lower layers, respectively, as a basic state, with the corresponding interface tilts given by the geostrophic balance. A normal mode representation (a plane wave solution) of a perturbation equation against the basic state is obtained by the replacement of the frequency to by Doppler-shifted frequencies toj -- to + EkUi and fl by an effective beta /3j --/3 + ( - ) J - l~p)(yj-IUI __ U2) in each layer ( j = 1, 2) in the linear wave Eq. (26a) and Eq. (26b). We obtain the modification of the wave dispersion with inclusion of the basic zonal flows for the inertial-gravity modes and the Rossby waves by scaling toj ~ 1 and %. ~ fl, resfiectively; A generalized dispersion relation is also obtained by replacements of to, fl by toj, flj at each layer ( j = 1, 2) in Eq. (26a) and Eq. (26b) and by taking the determinant:

~1~2 k2- k[ tol ~2( K2 + El) + to2 ~l( g2"+ F2)] .~=tolto2[(K 2 ~t_pl)(K2 + F2)- '~/Pl f21 +toito2(-K2(to2/~', +6022/$2)+ ( 1 - T ) [ ( 1 - 0.)2)(1- to22) - 1]/$iJ62} = 0 (38) We first consider the case without/3-effect, because the expression for the dispersion relation is drastically simplified in this case. In addition, we choose Fz = F2( -= F), 3' ~ 1, and U l = - U2 -= AU, so that

K2+2I~([8I~(K2-2t~)

o)2= _ _ 4F

1+

4# 2 +(K2+2f

1. . . . . . . (~kAU) 2 ( K 2 + 2~) 2

2

] 1/2/

) (@kAU) 41

}

(39)

For a shear with large Rossby number [~kAUI>> 1, we recover the relationship for Kelvin-Helmoltz instability, t o = +i~kAU (although we have used the hydrostatic approximation). However, we are rather interested in regimes of small Rossby numbers I@kAUI"~ 1. The ( - ) - s i g n in Eq. (39) corresponds to the geostrophic mode in the large scale limit (K 2 + 2/~ ,~: -IZ~AUI)for which Eq. (39) expands as

032 =

(K

- 2P)(

( K 2 Jr 2F)

kav)

J.-l. Yano, J. Sommeria /Dynamics of Atmospheres and Oceans 25 (1997) 233-272

254

which is exactly the relationship directly obtained from the QG approximation (refer to e.g. Pedlosky, 1987, Section 7.11). The sign of oJ2 does not depend on the sign of F, so the stability property is independent of the sign of the stratification in the QG approximation: it is stable for K 2 > 2[/~1 and unstable for K 2 < -21b~l. In the unstably stratified case however, the QG approximation breaks down near the turning point K 2 = _ 2/~. The turning point approximation, valid for K 2 + 2 k~ ~ 0, yields

which connects to the (+)-sign mode in the small scale limit

-

-

+ 2F--* +zc:

(K2+213)(~:kAU) 2

K2 + 2 / ~ [ 0) 2 =

K 2

1

2K2F

( K 2 + 2/~) 2

This clearly is the Rayleigh-Taylor instability modified by a shear. Note that this branch of solution remains purely growing for all the wavenumbers. The other branch, corresponding to the inertial-gravity mode in the large scale limit, remains purely oscillatory for all the wavenumbers. An example of dispersion diagram is given by Fig. 6, where we represent generalized dispersion curves Eq. (38) by solid, while the baroclinic Rossby mode, the baroclinic inertial-gravity mode, and the barotropic Rossby mode by long-dash, short-dash, and a chain-dot curves, respectively. Some examples for a more general case are shown in Figs. 7-9. (Note that by inclusion of the vertical shear, the barotropic Rossby mode can diverge in the following cases (even with 7 > 1).) The baroclinic instability usually appears as an extension of Rayleigh-Taylor (inertial-gravity) instability to a larger scale. Typical examples are shown in Fig. 7 and Fig. 8: both of them are with equal depth of two layers (/~1 = / ~ 2 = - - 1), while Fig. 7 represents a case with a weak vertical shear (~U L= - ~ U 2 = 0.06) and Fig. 8 a case with a stronger vertical shear (~:U~ = - ~U2 = 0.1). The range of the baroclinic instability stretches with the increase of the vertical shear. As in the previous section, the further reduction of Eq. (38) by an approximation around the turning scale i.e. A - /~2/3, 091 - - O9 2 ~ / ~ 1 / 3 (note the latter further implies 0)1 ~ 0)2 ~ 0) with fi i/3 >> $) is useful to inquire the modification of the stability of the inertial-gravity waves by a mean vertical shear. In particular, the scale of the stability exchange in the turning-scale approximation is modified to

A(K2) ---3Kt

~

l+TKa+(y--~l)/~l/~

2

(4o)

from Eq. (37b), where

au+ -;b-u

(41)

J.-l. Yano, J. Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

255

and 0p.___~_*= 2K2(/~ ' _ F2) + [ 2 - y ( l

OAU

+ y ) ] ( K t 2 + b~,)/~

(42a)

3ix*

O----U= y ( 1 - T ) ( K 2 +/~,)F2

(428)

The parameter p.* defined by Eq. (41) represents the manner that both the vertical shear AU =- (U l - U2)/2 and the mean total flow U - (U l + U 2 ) / 2 enhance or suppress the Rayleigh-Taylor (inertial-gravity) instabilities. Particularly in the limit of Y --) 1 + ,

Consequently, when the lower layer is thicker F~ >/~2, the inertial-gravity waves are less destabilized by a positive vertical shear AU > 0, and vice versa. Worthwhile to note that a mean effective fi is given by fi, + fi2

2

fi + ¢ : [ 2 F , - y ( l + y) be2] AU + ~:T(I -

'y) P'~2U

in the same approximation. This implies that a larger effective fi tends to enhance the baroclinic-Rayleigh-Taylor instabilities (c.f. Fig. 3(b)). Note that with a larger AU the slope of the interface is enhanced, hence the topographic-/3 effect is also enhanced. However, note that the generality of the dispersion relation is no longer preserved by reduction of Eq. (38) by a turning-scale approximation, in contrast to the case without mean vertical zonal flows. In particular, the turning scale defined by Eq. (40) does not correctly predict the onset of baroclinic-Rayleigh-Taylor instability for smaller scales as represented by Fig. 7 and Fig. 8. The discrepancy between the ^turning^scale and the scale of onset of instability, enhances in the limits of F~ >>/~2 or F 2 >> F v The scale of onset of instability is better predicted by the quasi-geostrophic scalings. Furthermore, the baroclinic instabilities are not always realized as a larger-scale extension of RayleighTaylor (inertial-gravity) instabilities as indicated in the previous examples. An extreme example is shown in Fig. 9 with /3= 0.01, ~:Ul = -~:U 2 = 0.05, F I = - 1, F2 = - 0 . 1 , y =- 1: the range of the baroclinic instability appears independently from that of the Rayleigh-Taylor instability in the growth rate curve. Finally, the major change in the free-bottom surface case (Fig. l(b)) is that the barotropic Rossby wave can be also destabilized at a larger scale (K < 0.5), like in the absence of shear, because the whole two-layer system is residing over a light deep layer. The nature of the 'baroclinic instability' under the unstable stratification may be better understood by considering the energetics. The energy conversion cycle of the disturbance is presented in Appendix A. It turns out that the important energy transfer processes are those W ( P M, PE) from the mean potential energy PM tO the eddy potential energy PE, W ( P E , KE) from the eddy potential energy PE tO the eddy kinetic energy K E, and to a much weaker extend W ( P E, K)) from the eddy potential energy PE

256

J.-L Yano, J, Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

(a) beta=O,Fl=F2=-I, garn=l .ul =-u2=0.1 ~ , 5

,

i

i

i

,

i

,

t

,

i

,

i

,

L ,

(

,

i

,

~

,

i

,

i

,

i

i

t

2.0 1.5 1.0

0 --

,1 Z j~

.5

i

/'~-

-1 .o . . . . . . . . . . . . .

i!

-1 .5

l

-2 .0

t

-2 k

(b) beta=O.Fl:F2=-I ,gam=l ,ul =-u2:0.1

3.0 2.5 2.0 t.5 1.0 .5

i

o --.5

-1 .0

\1

-1.5 -2.0

t

-2.5 -3.0

! ,

l

.2

,

l

l

.4

l

,

.6

t

,

r

,

l

,

l

,

l

,

l

l

l

l

l

l

l

l

[

l

l

L

.81.01.21.41.bl.82.02.22.42.62.83.0 k

Fig. 6. Baroclinic instability (the case without /3-effect, /3 = 0) the case /~ = / ~ = - 1, 3~ = 1, ~:Uj = - ~:U2 = 0.1. The format is same as Fig. 3. Solid curves are based on general formula (Eq. (38)), long-dash curves the baroclinic Rossby, short-dash curves the baroclinic inertial-gravity, and the chain-dot curves the barotropic Rossby modes (approximations, respectively).

J.-I. Yano, J. Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

257

(a) be ta=O. I, F I =F2=-I, gem=1.2, u I =-u2=O. 06

1.4 , i , 1.2 1.0 .8 : ~ ' ~ .6 .4 .2 0

I

,

L ,

I

;

i

,

I

,

-.2 -.4

..,..

I

i

I

,

I

'

I

,

I

,

t

,

~

,

.4" jJ

-,8

-1 .0 -1.2 -1 .4

-1 ,6 -1.8 -2,0

I I I ' l l l ' i ' l

.2 .4 .6 .81G1.

2

' l l , l l l l l l l l l l l

1.41.61.82.02.22.42.62.83.0

k

(b) beta=O.I.F1=F2=-1.gam=1.2.u1=-u2=O.Ob .0

l

'

l

~

l

'

l

'

l

l

l

l

l

1

1

1

1

'

1.5

1

'

1

~

l

~

l

'

l

'

/../

1.0 .5

t

o --.5

-1.0 -1.5 - 2

0

•

I l l e l l l l l l l ¢ l l l l l l l l l l l l l l l

.2 .4 .b . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 2 . 0 2 . 2 2 . 4 2 . 6 2 . 8 3 . 0 k

Hg. 7. B ~ l m i c instability (wi~ ~ e ~ c t , wi~ a w e ~ she~: same as Hg. 6 but ~e c~e wi~ ~ = 0.1, #1=~=-1, y = 1 . 2 , ~Ul = - ~ = 0 . ~ .

to the interaction kinetic energy K r The wavenumber k dependencies of these energy conversion rates are plotted in Fig. 10. The case without /3-effect, corresponding to Fig. 6, is shown in Fig. 10(a). As the

J.-I. Yano, J. Sommeria / Dynamics of Atmospheres and Oceans 25 f 1997) 233-272

258

b e t a = O . 1, F1 = F 2 = - 1 . g a m = l . 2 , u l = - u 2 = O . 1

1.8 1.6 1.4 1.2 1.0 .8 .6 .4 ~- .2 0 .2 -

,4

Y -.8

-1 -1 -1 -1 -1

.0 .2 .4 .6 .8

! I ,

+

,

2

i

t

.4

i

i

.6

.8

,

i

,

,

i

,

i

i

t

,

t

i

i

t

i

,

l

+

i

l

1 .0 ! 2 I .4 I .6 t .8 2.0 2.2 2.4 2.6 2.B 3.0 k

(b) beta=O.1iFl=F2=-l,gam=1.2.ul=-u2=0.1 2 . 0

,

i

,

i

,

i

,

i

,

i

,

F

,

~

,

i

,

I

,

i

,

i

,

i

,

I

,

i

15 1.0

z z

~1, ,//

0

Iq'

.5

',,

1.0 -1 .5 -2 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . •

.2

.4

.6

.8

I .0

1.2

1 .4

I .b

1 .8

2.0

2+2

2.4

2.6

2.8

3.0

k

Fig. 8.

Baroclinic instability

( w i t h el-effect, w i t h a

strong shear):

s a m e as Fig. 7 b u t ~ U 1 = - g~U2 = 0.1.

Rayleigh-Taylor instability is replaced by a baroclinic instability with decrease of wavenumber k, a positive W ( P E, K E) (long-dash) is gradually replaced by a negative baroclinic conversion W ( P M, PE) (solid curve). The Rayleigh-Taylor instability is realized as a process such that the eddy potential energy PE is converted into the eddy

J.-I. Yano, J. Sommeria/ Dynamics of Atmospheres and Oceans 25 (1997) 233-272

259

bete=O.O1,Fl=-l,F2=-O.l,gm=l,ul=-u2=O.05

1.41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.2F

!

. o L ....

! ',

.6

0 L

F

,I

~

1. . . . ~ ti

-.2 -

'

4 6

..................

f./

/

/ i

/

i i

-1 -1 2 -1 4 0 '.~ '.~, .4 '.~,

I I l ,

i

i

.8

Ii

i

,

i

,

T

,

i

,

i

,

l

,

i

,

f

,

i

.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2 . 6 2.8 3.0 k

(b) beta=O.01 ,FI=-I ,F2=-O. I ,gm=1 ,u1=-u2=O.05

30

i

,

I

,

L

,

r

,

i

,

I

,

I

,

L

'

1

'

t

'

T

,

I

'

i

,

J

,

25 20 15 10 •

5

i

o

/7 i

-1.0

-1.5 -2.0 -2.5 -3.0

I

.2

,

I

.4

,

I

.6

,

I

.8

,

I

,

I

,

I

i

I

,

I

,

I

,

I

,

I

,

I

,

i

1 .0 1.2 1.4 1 .6 1.8 2.0 2.2 2.4 2.6 2 . 8 3.0 k

Fig. 9. Isolation of baroclinic instability with a weak /3-effect: the parameters are /~ = 0.01, @U~ = - ~:U2 = 0.05, /~1 = - 1, F2 = - 0 . 1 , 3' = 1.

kinetic energy K E. As a result, the magnitudes of both the eddy potential energy P E ( < O) and the eddy kinetic energy K E ( > O) increase with time (Fig. ll(a)). A much weaker energy transfer from the interaction kinetic K~ to the eddy potential energy P E is also noted.

260

J.-l. Yano, J. Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

(a)

2.5 2.0 1.5 ,~ 1 . 0 " .5

|

/ / /

/

/

0

,,ol J i-.5

-2.0 o 2,2

(b)

.~'.~'.8'1Jo'~22'114'126'128'2.o

.~

b

.

,

.

.

.

.

.

.

.

.

.

.

.

.

.

.

•

2° I 1.6

,

.

/

•: 1 . 8

/i

1.4

).~ .o "~1.21

/

.6 .4

.2

-.2

.4

.6

.8 1.0 1.2 1.4 1.6 1.8

2.0

k

.60

(c)

.

,

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

,55

,5o

/

~..4s

/

•~

/

~..15 "~ .lO

I

!

g~

,

o

~

- -L__L_a--~

~-.(~ - . 1 0

,

I

,

,

,

i

,

i

,1

.2

.3

.4

.2

.4

.6

,8

,

L

.5

,

J

.6

,

,

.7

,

,

,

i

,

.8

.9 1.0

1.0 1.2 1.4 1 . 6

1.8 2 . 0

.09

(d)

i

.08 .07

~..06

i ;

&

.05 04

~r,.- 0 3 .02

i .m 0

J.-L Yano, J. Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

(a) lt'<0: Rayleigh-Taylor Instability

(b) ~'<0: Baroclinic Instability

(e) I~>0: Baroclinic Instability

(d) Energy Conversion

KI ~ "

261

x,'Pu . > 0 P,)

Kz (

Pz >0 W(P,, Ks)

Fig. 11. Schematics of the energy cycles (a) Rayleigh-Taylor instability (inertial-gravity mode), (b) baroclinic instability with an unstable stratification, (c) baroclinic instability with a stable stratification, and (d) the definitions of the energy conversion terms (c.f. Appendix A).

By contrast the negative baroclinic conversion W(P M, P E ) < 0 at lower wavenumbers implies a reduction of the magnitude of the mean potential energy PM( < 0, and the corresponding latitudinal temperature-gradien0 with a compensating generation of a negative eddy potential energy PE (Fig. 1 l(b)). A slight energy conversion W(PE, K E) from the eddy potential to the eddy kinetic energies is dictated by a geostrophic adjustment of the system. Remarkably, the principal energy conversion term W(PM, PE) is the same for both stably and unstably stratified cases (c.f. Fig. 10(d)): both are accompanied by the identical end effect of reducing the magnitude of the mean potential energy PM, and henceforce the latitudinal temperature-gradient. However, the direction of the conversion reverses (c.f. Fig. 11(c)): this may be formally attributed to an opposite sign of the potential energies in two cases. More physically, it is a manifestation of the fact that the baroclinic available potential energy cannot be an energy source of the motion in an unstably stratified flow, as conjectured by Yano (1987a): the tilted interface has less potential energy than the horizontal one. In this respect, the baroclinic

Fig. 10. Energy conversion rates with change of the wavenumbcr k: W(P M, P#.) (solid curve), W ( P E, K E) (long-dash), W(P E, K I) (short-dash). The cases corresponding (a) to Fig. 6 ( f l = 0, Fi = F2 = -- 1, T = I, ~ U l = - - ~U2=0.1), (b) to Fig. 8 ( ~ = 0 . 1 , P~=ff2=-l, T = 1.2, ~ U , = - ~ U 2 = 0 . 1 ) , (c) to Fig. 9 ( f l = 0.01, ~U I = - ~U2 =0.05, Ft = - l, F2 = - 0 . 1 , T = l), and (d) a case with a stable-stratification ( ~ =0.1, Fi = ]$2 = 1, ~t = 0.8, ~U I = - ~U2 = 0.1). The amplitude of the disturbance is normalized by KE = 1.

262

J.-l. Yano. J. Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

instability under an unstable stratification is better understood as a 'convectively driven' baroclinic instability, in contrast to a 'baroclinically driven' standard baroclinic instability. When a /3-effect is introduced (Fig. 10(b): corresponding to the case of Fig. 8), a peak of the baroclinic instability is dictated by the peak of the baroclinic conversion rate W(PM, PE)" The virtual isolation of the baroclinic instability from the Rayleigh-Taylor instability seen in Fig. 9 can be substantialized on the basis of energetics Fig. 10(c)).

6. D i s c u s s i o n

6.1. Quasi-geostrophy Our linear analysis has shown that a shallow unstably stratified system submitted to rapid rotation shears many properties of usual stably stratified systems. In particular various quasi-geostrophic modes can propagate without amplification, though with qualitatively different wave dispersions. However, these slow modes must coexist with rapidly unstable inertial-gravity modes at small scales. These are conveniently approached with the shallow water model, making use of the vertical hydrostatic balance, but are clearly outside the scope of the quasi-geostrophic model, leading to diverging frequencies at the turning wavenumber K t (Eq. (32)). The next step would be to perform non-linear analysis or simulations. It would be then convenient to keep only slow motions by adopting the quasi-geostrophic approximation. Our anticipation is that such an approach would be still valid for the convectively unstable Jovian atmospheric layer, provided we restrict the analysis to wavenumbers smaller than K t. In this range of scales, the QG modes are indeed in competition with rapid inertial-gravity waves which remain neutral. Therefore the situation is quite similar to stably stratified systems, and the different arguments in favor of the quasi-geostrophic approximation can be readily adapted. We can use for instance an illuminating analysis proposed by Vallis (1992). He proved by a variational principle that a geostrophic balance 1 B v. × k = - -- V--

H

q

(44)

constitutes the energy minimum state in the shallow water system with a given potential vorticity q, where H is the depth, B =-gH + ~v~/2 the Bernoulli functional. His argument is that, since the system is expected to be equilibrated to an energy minimum state, this explains the predominance of the geostrophy in the Earth's atmosphere (say, instead of inertial oscillations). This proof is readily extended to the present two-layer model to show that the geostrophy Eq. (44) is still an energy extremum even with unstable stratifications (the first variation vanishes). Furthermore we can easily prove, by considering second variations, that this extremum is indeed a minimum in respect to any perturbation with wavenumber smaller than K,. By contrast, it is a maximum in respect to perturbations with wavenumbers larger than Kt, confirming that the QG

J.-L Yano, J. Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

263

approximation is probably not appropriate in this small scale range. The QG modes are then in competition with inertial-gravity modes with rapid growth, and it seems clear therefore that the latter should dominate the non-linear dynamics in this range of scales. Vallis' proof is further extended to a continuously stratified case under the constraint of conservation of Ertel's potential vorticity and the total energy (kinetic energy + any type of potential energy not depending on the velocity). The geostrophy as the energy extremum state is still maintained if the magnitude of vertical gradient of potential temperature is much larger than that of horizontal gradient, with a slightly modified definition for the geostrophy, A simple extension of the Vallis (1992) argument may be that, consequently, the large-scale motions are prevailed by the geostrophic balance even under an unstable continuous stratification. Such quasi-geostrophic structures should be only weakly perturbed by the presence of the motions at the turning scale, otherwise the quasi-geostrophy will lose its own independent status as a regime. This can be checked by a formal scale-analysis h la Pedlosky (1987), which proves that two dynamical regimes are independent to the leading order. We designate where ~0(G~ and ~(t) refer to the contribution from the geostrophic-scale (scale larger than the turning scale) and from the turning scale itself, respectively. We further divide each component into the parts driven by the slow time-scale ( ~ / 3 - ' , with the dependence to r) and the rapid time-scale (~/3-~/3, with the dependence to t): ~(c) = qt(c~(, ,'r) + tp(c)( ~0"~ = q,~( • ,t) + q,"~( *

*

,t)

,~)

and scale by gt~o)(*, ~-) ~ 1 and ~0~t)(*, t) ~ ,a¢. It is estimated that qjct)(, O')~ ~/,~' and qJ~c>(*,t)~ ~:~¢,//~./3; ~0
from the last order-estimate statement. However the aforementioned arguments expect that the geostrophy is prevailing for the geostrophic scale larger than the turning scale (i.e. K < K t) Hence ~O(c)( * ,t) ~ ~.ff'/f~ ,/3 << 1 or ~¢' .a~/~ ,/3/~: ~ ~:- 2/3. The last statement implies that the large scale modes are still in the geostrophic regime, even in the presence of turning scale modes with larger amplitude. The non-linear feedback of the turning-scale dynamics to the slow large-scale geostrophic dynamics is very weak as O(~:). Consequently, we expect that scales larger than the turning scale can undergo a free mode dynamics in the QC regime. We have seen in Section 5 that the baroclinic instability provides a natural source for such modes, like in usual atmospheric systems. The interaction with other modes would have also to be considered to provide sources

264

J.-l. Yano, J. Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

and sinks. The interaction with rapid convective motion (at scale of the order of the layer thickness H ) could be modelled as an average effect, using some mixing length formulation, which is also required to determine the thickness H and typical density ratio Y in the turbulent boundary layer. It would be also useful to extend the QG formulation to capture some of the dynamics at the turning scale. It is indeed expected to be an important source of excitation for the system due to the proximity of unstable modes. Since these modes are still slower than inertial-gravity waves, although more rapid than QG modes, some kind of balance approximation may be possible. 6.2. Potential vorticity invertibility The invertibility of the potential vorticity (p.v.) is well guaranteed in the conventional stably stratified geophysical flows (c.f. Hoskins et al., 1985). It is not only because the p.v. is defined by an elliptic operator on the streamfunction in the quasi-geostrophic limit, but furthermore because the geostrophic mode is well separated from the inertialgravity mode even with a fairly large Rossby number, so that a certain balance condition is applicable. These virtues of the p.v. invertibility, however, are not competent under the unstably stratification. The p.v. is no longer defined by the elliptic operator in the quasi-geostrophic limit. The breakdown of the p.v. invertibility is particularly manifested at the turning scale. Not only the quasi-geostrophic limit loses the validity, but the geostrophic mode (Rossby mode) is modified into the inertial-gravity mode over the turning scale. Hence, even no balance condition is available. In order to elucidate its dynamical significance, let us consider an isolated p.v. anomaly characterized by the horizontal and the vertical wavenumbers k, m. The p.v. Q is related to the streamfunction tp in the quasi-geostrophic limit by Q=-(k2+/~m2)~b with an appropriately defined Froude number/~. We anticipate that both the p.v. and the streamfunction remain in the same amplitude with time, i.e. Q ~ ~b~ const, from the conservation of the enstrophy Q 2 / 2 and the total energy [(A~b)2 + l~(&b/dz)2]/2 ~ [(k2~02 + 1~m2qJ2]/2. As a corollary, the effective total wavenumber of the p.v. anomaly remains the same order with time: k 2 + l~m 2 ~ const.

(45)

This means that with a stable stratification (i.e. /~ > 0), the anomaly approximately evolves around the circle defined by Eq. (45) in the wavenumber space: an increase of the horizontal scale is usually compensated by a decrease of the vertical scale and vice versa (Fig. 12(a)). As a result, an initial spherical p.v. blob tends to remain an ellipsoid (Fig. 13(a)). Hence, a p.v. anomaly tends to reserve its identity like a material entity under the stable stratification, apart from some mixings accompanied by filamentations and others (c.f. Mclntyre, 1990). However, this material-like property of the p.v. is lost under the unstable stratification. The same argument leads to Eq. (45) by invoking the same conservation laws, but with a very different implication: the increase of the horizontal scale is accompanied by an increase of the vertical scale, and a decrease of the vertical scale is accompanied by a

J.-L Yano, J. Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

265

^

(a)

F t/z m

(b)

(-r)~/

,

/

k z -~F)m

2 =const

Fig. 12. The tendency k 2 + Fm 2 - const, expected for a potential vorticity anomaly with a horizontal and a vertical wavenumbers k, m on a phase plane with (a) a stable stratification, and (b) an unstable stratification.

decrease of the horizontal scale (Fig. 12(b)). As a result, an initial spherical p.v. blob cannot remain a well-defined ellipsoid but evolves more erratically by following this tendency in the wavenumber space. It is physically interpreted that, when an initial A

( b ) 1~<0

Fig. 13. A schematic representation of possible deformation of an initial spherical p.v. anomaly (center) in a vertical cross-section: when it is horizontally stretched (to the right) and when it is horizontally squeezed (to the left): (a) with a stable stratification, (b) with an unstable stratification.

266

J.-I. Yano. J. Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

spherical p.v. anomaly is horizontally stretched, it is accompanied by a generation of small-scale thermal plumes, which have a larger vertical scale than an initial p.v. anomaly. Similarly, when the p.v. anomaly is horizontally squeezed, it is accompanied by a vertical train of thermal plumes having a smaller vertical scale than the initial p.v. anomaly (Fig. 13(b)). Consequently, the p.v. no longer preserves a clear material-like identity under the unstable stratification. This also constitutes the limitation of the applicability of the quasi-geostrophy discussed in the previous subsection. Such a breakdown of the p.v. anomaly into small thermal plumes must be properly parameterized in the quasi-geostrophic description with the unstable stratification. 6.3. Rossby-wave dispersion

The divergence of the Rossby wave dispersion that we obtain in the unstably stratified layer is also known in the continuously stably stratified case (T. Matsuno and S. Miyahara, personal communication, 1992). Note that Froude number is inverse proportional to the equivalent depth, h. (e.g. Andrews et al., 1987, Chapter 4). Hence a negative Froude number simply corresponds to a negative equivalent depth in a continuously stratified case. It is known that in some occasions an equivalent depth, which is defined as an eigenvalue for the vertical structure, can be negative in the Earth's atmosphere. In such cases, the usual westward-propagating Rossby waves modulate into the inertial-gravity waves with the decrease of the[ (qualitatively equivalent to decrease the horizontal wavenumber; Longuet-Higgins, 1968). Also the eastward-propagating Rossby wave and the eastward-propagating inertial-gravity wave coexist with a sufficiently small ]he[ and are connected by a single dispersion curve as in the present result (see Fig. 17-21 of Longuet-Higgins, 1968). Missing from the Longuet-Higgins' dispersion curves is an unstable eastward-propagating inertial-gravity mode, which is unique with the unstable stratification. 6.4. Relation to other thermal convection studies

Even our linear analysis with the hydrostatic approximation shows that convective instability (Rayleigh-Taylor) prefers the smallest scale for the fastest growth. For an understanding of these small-scale motions, we refer to the experimental results on thermo-convection at high Rayleigh and Taylor numbers. In the experiments of Fernando et al. (1991) with a constant heating at the bottom, well-organized helical vortex-plumes are generated from the bottom of the container under a strong constraint of Taylor-Proudman theorem (in a high Taylor number regime T ~ 101°). Jones and Marshall (1993) have performed numerical experiments, in contrast, with an initial strong cooling at the top of the system. They observed a generation of thermal-plumes from the top boundary. The helical vortex structure of the plumes may be interpreted as 'Taylor-plumes', which are under geostrophic balance from a top view. In both cases, a coherent vortex structure is seen on a horizontal plane - reminiscent of the two-dimensional turbulence result by McWilliams (1984). Jones and Marshall (1993) suggest that the convective Taylor-plumes are organized into a scale of the 'convective' deformation radius ILRI (the turning scale, non-dimensionally, ~ ( - ~ ) - 1 1 2 ) .

J.-l. Yano, J. Sommeria / Dynamics of Atmo,spheres and Oceans 25 (1997) 233-272

267

However, their studies were incapable of elucidating the large-scale behavior of Taylor-plumes because of the limited laboratory and numerical domain sizes. In contrast to their works, our attention is the dynamics of convectively unstable medium in the scales larger than the deformation radius. An attempt to investigate such a large-scale dynamics has been done by Legg and Marshall (1993). They modeled Taylor-plumes by pairs of discrete point vortices - hetons in a two-layer quasi-geostrophic system. Our approach is, in this respect, the extension of their work to a horizontally continuous vorticity fields. Our prediction of baroclinically unstable modes in a scale larger than the deformation radius, independent from the small scale inertial-gravity instability, is particularly remarked in this context. It supports the view by Jones and Marshall (1993) that Taylor plumes will be modified by the baroclinic instabilities in larger scales. 6.5. Application to the Jovian atmospheres

Finally, a few comments for the application of the results to the Jovian atmospheres are in order. Stone (1966, 1970, 1971) has carefully investigated the Eady (1949) model for baroclinic instability in a limit of weak stratifications (/~ ~ +~c) by intending the application to the Jovian atmosphere (Stone, 1967, 1976). The present study can be considered as an extension of his study to the unstably stratified case. However, it is not easy to see the direct connection of Stone's work to the present one due to the adoption of the severely restricted vertical resolution in the present study. Particularly, the possibility of the symmetric instability is automatically excluded in the present study by adoption of the two-layer formulation (Phillips, 1964; Stone, 1970). An only major conclusion obtained for the limit of the weak stratification (/~--* -~c) from the present study is that the critical scale for the onset of Rayleigh-Taylor instability (inertial-gravity mode: Section 2 becomes infinitely small ( k c ~ ( - l ~ ) - J / 2 ) . A work should still be done to connect the present study to Stone's classical work by investigating a wider possibility of the instabilities in the phase space by considering the vertical structure without discretization. This will be problem to be addressed in a separate work. However, remarkably, the present study shows that a baroclinic instability under an unstable stratification can be realized independently from the convective instabilities at scale larger than the 'convective' deformation radius tLR[, in particular the limit of /3 << F~,F 2. Even though the convective instability of the smaller scales has a much larger growth-rate, these small-scale disturbances are very inefficient to relax the baroclinicity of the atmosphere. The result indicates that such a convectively driven baroclinic instability can play an important role in the Jovian atmospheres. Our other major result is the various linear wave properties revealed, in particular the possibility of the eastward propagating Rossby waves. It will be worthwhile to carefully reinvestigate the propagating properties of various Jovian eddies in this point of view. Particularly mentioned is an analysis of the 'ribbon' planetary waves in Satumian atmosphere by Sromovsky et al. (1983). They fitted the analyzed dependence of the frequency to the longitudinal wavenumber to the barotropic Rossby wave dispersion (Eq. (24b)) by taking the latitudinal wavenumber l as a free parameter 1 *. The best fit was obtained with the latitudinal wavelength 2 ~ / 1 * = 30°, while the observed latitudi-

268

J.-l. Yano, J. Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

nal wavelength is 27r/1 = 13°. The simplest explanation of the discrepancy (an alternative explanation has been proposed by Polvani and Dritschel, 1993) is obtained by assuming a negative Froude number in the dispersion relation Eq. (24a) with f = / ~ + F2. By using the identity 1.2=12+/~

we obtain an estimate of 'convective' deformation radius iL R = 4000 km

Propagating disturbances detected with the infrared measurements by Magalh~es et al. (1989, 1990) may be interpreted in a similar manner. It is worthwhile to point out here that the eddies are strongly dispersive (and radiative) in a medium with a negative Froude number. A slight change of the wavenumber leads to a drastically different phase velocity in the vicinity of the turning scale. By this effect, an initial isolated eddy, which consists of a wide range of wavenumbers, disperses away by radiations. The compacting condition of the isolated vortex is totally not satisfied in this case (Yano and Flierl, 1994; see also Ingersoll and Cuong, 1981). For this reason, even the linear evolution of the eddies appears very 'turbulent' in this situation. Various disturbances observed in the Jovian atmospheres (e.g. Sanchez-Lavega et ai., 1991a, and the references therein) may be explained in this term. All these disturbances are characteristically initiated by an appearance of a few very bright spots, which subsequently trigger longitudinally propagating disturbances. A Saturnian storm, a Great White Spot generated at the end of September 1990, can be a very dramatic example (Sanchez-Lavega et al., 1991 b). These bright spots triggering the disturbances may be interpreted as convective Taylor-plumes generated from the deep interior and lifted to an almost convectively neutral atmospheric layer, where the Rossby wave dispersion initiates a secondary perturbation. Equally remarkable is a North-Tropical Disturbance event of Jupiter in February, 1990 reported by Sanchez-Lavega et al. (1991a). In this event, a westward-propagating disturbance is generated from a bright white plume. A large intrinsic phase velocity (c~ --- - 6 0 m s -~ ) of the disturbance along with its short characteristic wavelength ( ~ 5000 kin) indicates the turning-scale dynamics. (An alternative interpretation is an inertial-gravity wave characterized by a very shallow equivalent-depth (corresponding to the deformation radius of ~ 200 km: Asada et al., 1993), since, otherwise, the phase velocity is still too slow as an inertial-gravity wave.) These different observations point out to a large unstable radius of deformation ( ~ 4000 kin), which seems inconsistent with the estimated thickness and stratification of a turbulent boundary layer, according to the mixing length theory used in astrophysics. However, a recent model for the internal structure of Jupiter and Saturn, by Guillot et al. (1994) predicts a stably stratified layer at a depth about 500-1000 kin, associated with the presence of a radiative zone. Therefore the upper convective zone would be confined to this shallow depth, whose dynamics could be approached by our two-layer model with a radius of deformation 4000 kin. The major conclusion of the present paper is that a quasi-geostrophic formulation is still possible for unstably stratified geophysical flows, and it can be a useful tool to

J.-L Yano, J. Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

269

understand the various phenomena of Jovian atmospheres. Further studies in this direction are anticipated.

Acknowledgements JIY appreciated the personal communications with P.H. Stone, T. Matsuno, and S. Miyahara. The major part of the work was performed during October, 1991-January, 1992, when JIY was visiting E.N.S. de Lyon as a CNRS Research Fellow. JIY was supported by NOAA Climate and Global Change Program during the writing, and by the Australian Government cooperative Research Centres Program during the revision of the paper. We are indebted to the comments by three anonymous reviewers for the preparation of the final version of the paper.

Appendix A. Energetics for the disturbances The energetics for almost linear disturbances is derived analogically as in Orlanski (1968). We designate the mean variables by bars to make them distinct from the eddy variables in the following (so that fij = Uj, j = 1, 2). We define the mean kinetic and potential energies by

1

1 - ~,<~ > PM=2

<'~> +

2

(Ala)

(Alb)

the interaction kinetic energy by K I = ~:(7fil(rh - r/2)u I "~"fiZ'~2U2 )

(Ale)

the eddy kinetic and potential energies by

KE =

+ ~--~ <,,~>

2 P E = 2 - +

I - ~, 2

(Aid)

(Ale)

The energy cycle for the system is,subsequently,given by d ~ K M = -W(KM,K,) + W(P~,K~) d -~tPM = --W( PM ,KM) -- W( PM ,PE)

(A2a) (A2b)

J.-l. Yano,J. Sommeria/ Dynamicsof Atmospheres and Oceans25 (1997)233-272

270

d ~ t K , = W ( P E , K , ) + W( K M ,KI)

(A2c)

d

(A2d)

-~tKE= W( PE,KE) d

~ t P E : W(PM,PE) -- W ( PE,KE) - W( P E , K , )

(A2e)

where the energy conversion rates are defined by W(KM,K,)

-~ - F~ Y-Y--(?:'fi,u,17"v , ) - f--~2 1 ( ~-~: u217" v2 )

(A3a)

1

W ( P M , K M ) = --~-Y
W(PM,PE)

---=T ( e u l ( ' 9 1

3'

F~

:

- '92 ) v l) + ( ~ 2 ' 9 2 v 2 )

(A3c)

1

W( PE,KE) = "~1 <'91~7"UI> + ~-f2('gB~7" V2) W(Pe,K.)

(A3b)

:

(A3d)

= y ( ~ f i , ( ' r / , - ' 9 2 ) ( v , - O'9,/0x)) + (~fi2('92v2- O'gBIOX)) (A3e)

In defining the above energy conversion rates, the approximations fi << 1, ~]~jL << ~:bfi;[ << 1 ( j = 1, 2) have been made. Only the last three terms, Eqs. (A3c), (A3d) and (A3e) are considered in the main text. The conversion W ( K M, K~) of mean kinetic energy K M to interaction kinetic energy Kj is identically zero when l = 0 is assumed as in the main text. Also this term is expected to be unimportant as long as the horizontal shear of the zonal flow is absent. The energy conversion W ( P M, K M) between the mean energies are not only our concern here, but requires the second-order calculation to evaluate the residual mean meridional winds ~j ( j = I, 2). On the other hand, the distinction between the eddy and the interaction kinetic energies K E, K~, which are usually combined together, is made in this energetics. The phase structure of the unstable inertial-gravity modes implies that the interaction kinetic energy K~ has a negative value, which is realized by an energy transfer into the eddy kinetic energy K E. Note that the conversion term W ( P E, K l) vanishes in a quasi-geostrophic limit by the balances v~ ~ O'9~/Ox, v 2 ~-O'gB/OX , SO that this term is not important in standard baroclinic instabilities. An inspection of the phase structure shows that the Rayleigh-Taylor instability is typically realized by a negative correlation between "rh - ' 9 2 , "q2 and 17-v~, 17-v2, respectively, so that the anomaly of the depth of the layers is enhanced by the horizontal convergence of the flows,

J.-L Yano, J. Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

271

References Andrews, D.G., Holton, J.R. and Leovy, C.B., 1987. Middle Atmosphere Dynamics. Academic Press, New York. Asada, T., Gierasch, P.J. and Yamagata, T., 1993. Initial development of eddies in highspeed zonal flow: one interpretation for the NTB activity of Jupil~r. Icarus, 104: 60-68. Busse, F.H., 1976. A simple model of convection in the Jovian atmosphere. Icarus, 29: 155-160. Busse, F.H., 1983. A model of mean zonal flows in the major planets. Geophys. Astrophys. Fluid Dyn., 23: 153-174. Castaing, B., Gunaratne, G., Heslot, F., Kadanoff, L., Libchaber, A., Thomae, S., Wu, X.-Z., Zaleski, S. and Zanetti, G., 1989. Scaling of hard thermal turbulence in Rayleigh-Benard convection. J. Fluid Mech., 204:

1-30. Chandrasekhar, S., 1961. Hydrodynamic and Hydromagnetic Instability. Clarendon, Oxford. Eady, E.T., 1949. Long waves and cyclone waves. Tellus, 1: 33-52. Femando, H.J.S., Chen, R.-R. and Boyer, D.L., 1991. Effects of rotation on convective turbulence. J. Fluid Mech., 228: 513-547. Gill, A.E., 1982. Atmosphere-Ocean Dynamics. Academic Press, New York. Guillot, T., Chabrier G., Morel, P. and Gautier, D., 1994. Nonadiabatic models of Jupiter and Saturn. Icarus, 112: 354-367. Hanel, R.A., Conrath, B., Herath, L.W., Kunde, V.G. and Pirraglia, J.A., 1981a. Albedo, internal heat, and energy balance of Jupiter: preliminary results of the Voyager infrared investigation. J. Geophys. Res., 86: 8705- 8712. Hanel, R.A., Conrath, B., Flaser, M., Kunde, V., Maguire, W., Pearl, J., Pirraglia, J., Sammuelson, R., Herath, L., Allison, M., Cruikshank, D., Gautier, D., Gierasch, P., Horn, L., Koppany, R. and Ponnamperuma, C., 1981b. Infrared observations of the Saturnian system from Voyager 1. Science, 212: 192-200. Hanel, R.A., Conrath, B., Kunde, V.G., Pearl, J.C. and Pirraglia, J.A., 1983. Albedo, internal heat flux, and energy balance of Saturn. Icarus, 53: 262-285. Hoskins, B.J., Mclntyre, M.E., Robertson, A.W., 1985. On the use and significance of isentropics potential vorticity maps. Q.J.R. Meteor. Soc., I 11 : 877-946. Ingersoll, A.P., 1990. Atmospheric dynamics of the outer planets. Science, 248: 308-315. Ingersoll, A.P. and Cuong, P.G., 1981. Numerical model of long-lived jovian vortices. J. Atmos. Sci., 38: 2067-2076. Ingersoll, A.P. and Pollard, D., 1982. Motion in the interior and atmospheres of Jupiter and Saturn: Scale analysis, anelastic equations, barotropic stability criterion. Icarus, 52: 62-80. Jones, H. and Marshall, J., 1993. Convection with rotation in a neutral ocean: a study of open-ocean deep convection. J. Phys. Oceanogr., 23: 1009-1039. Legg, S. and Marshall, J., 1993. A heton model of the spreading phase of open-ocean deep convection. J. Phys. Oceanogr., 23: 1040-1056. Longuet-Higgins, M.S., 1968. The eigensolutions of Laplace's tidal equations over a sphere. Phil. Trans. R. Soc. London, A262:511-607. Magalh]es, J.A., Weir, A.L., Conrath, B.J., Gierasch, P.J. and Leroy, S.S., 1989. Slowly moving thermal features on Jupiter. Nature, 137: 444-447. Magalh]es, J.A., Weir, A.L., Conrath, B.J., Gierasch, P.J. and Leroy, S.S., 1990. Zonal motion and structure in Jupiter's upper troposphere from Voyager infrared and imaging observations. Icarus, 88: 39-72. Mclntyre, M.E., 1990. Middle atmospheric dynamics and transport: Some current challenges to our understanding. In: A. O'Neil (Editor), Dynamics, Transport and Photochemistry in the Middle Atmosphere of the Southern Hemisphere. Kluwer Academic Pub., pp. 1-18. McWilliams, J.C., 1984. The emergence of isolated, coherent vortices in turbulent flow. J. Fluid Mech., 146: 21-43. Orlanski, I., 1968. Instability of frontal waves. J. Atmos. Sci., 25: 178-200. Pedlosky, J., 1987. Geophysical Fluid Dynamics. 2nd edn., Springer-Verlag. Phillips, N.A., 1954. Energy transformations and meridional circulations associated with simple baroclinic waves in a two-level, quasi-geostrophic model. Tellus, 6: 273-286. Phillips, N.A., 1964. An overlooked aspect of the baroclinic stability problem. Tellus, 16: 268-270.

272

J.-l. Yano, J. Sommeria / Dynamics of Atmospheres and Oceans 25 (1997) 233-272

Polvani, L.M. and Dritschel, D.G., 1993. Wave and vortex dynamics on the surface of a sphere. J. Fluid Mech., 255: 35-64. Sanchez-Lavega, A., Miyazaki, 1., Parker, D., Laques, P. and Lecacheux, J., 1991a. A disturbance in Jupiter's high-speed north temperate jet during 1990. Icarus, 94:92 97. Sanchez-Lavega, A., Colas, F., Lecachuex, L., Laques, P., Miyazaki, 1. and Parker, D., 1991b. The Great White Spot and disturbances in Saturn's equatorial atmosphere during 1990. Nature, 353: 397-401. Siggia, E.D., 1994. High Rayleigh number convection. Ann. Rev. Fluid Mech., 26: 137-168. Sommeria, J., Meyers, S.D. and Swinney, H.L., 1991. Experiments on vortices and Rossby waves in eastward and westward jets. In: A.R. Osborne (Editor), Nonlinear Topics in Ocean Physics. North-Holland, Amsterdam, pp. 227-269. Spiegel, E.A., 1971. Convection in stars. Ann. Rev. Astron. Astrophys., 9: 323-352. Spruit, H.C., Nordlund, A. and Title, A.M., 1990. Solar convection. Ann. Rev. Astron. Astrophys,, 28: 263-301. Sromovsky, L.A., Revercomb, H.E., Krauss, R.J. and Suomi, V.E., 1983. Voyager 2 observations of Saturns northern mid-latitude cloud features: morphology, motions, and evolution. J. Geophys. Res., 88: 8650-8666. Stone, P.H., 1966. On non-geostrophic baroclinic stability. J. Atmos. Sci., 23: 390-400. Stone, P.H., 1967. An application of baroclinic stability theory to the dynamics of the Jovian atmosphere. J. Atmos. Sci., 24: 642-652. Stone, P.H., 1970. On non-geostrophic baroclinic stability: part 11. J. Atmos. Sci., 27: 721-726. Stone, P.H., 1971. Baroclinic stability under non-hydrostatic conditions. J. Atmos. Sci., 45: 659-671. Stone, P.H., 1976. The meteorology of the Jovian atmosphere. In: T. Gehrels (Editor), Jupiter. University of Arizona Press, pp. 586-618. Vallis, G.K., 1992. Mechanisms and parameterizations of geostrophic adjustment and a variational approach to balanced flow. J. Atmos. Sci., 49:1144-1160. Williams, G.P., 1978. Planetary circulations: 1. Barotropic representation of Jovian and terrestrial turbulence. J. Atmos. Sci., 35: 1339-1426. Williams, G.P., 1985. Jovian and comparative atmospheric modeling. Adv. Geophys., 28A: 381-429. Yano, J.l., 1987a. Rudimentary considerations of the dynamics of Jovian atmospheres. Part I: The depth of motions and energetics. J. Meteor. Soc. Jpn., 65: 313-327. Yano, J.l., 1987b. Rudimentary considerations of the dynamics of Jovian atmospheres. Part I1: Dynamics of the atmospheric layer. J. Meteor. Soc. Jpn., 65: 329-340. Yano, J.l. and Flierl, G.R., 1994. Jupiter's Great Red Spot: compacting conditions, stabilities. Ann. Geophysicae, 12: I - t 8 . Zocchi, G., Moses, E. and Libchaber, A., 1990. Coherent structures in turbulent convection, an experimental study. Physica A, 166: 387-407.