The Stability, Modulation and Long Wave Resonance of a Planetary Wave in a Rotating, Two-Layer Fluid on a Channel Beta-Planet

Transcription

1 160 Journal of the Meteorological Society of Japan Vol. 58, No. 3 The Stability, Modulation and Long Wave Resonance of a Planetary Wave in a Rotating, Two-Layer Fluid on a Channel Beta-Planet By Toshio Yamagata Research Institute for Applied Mechanics, Kyushu University, Hakozaki , Higashi-ku, Fukuoka 812, Japan (Manuscript received 28 December 1979, in revised form 14 March 1980) Abstract The stability of planetary waves in a two-layer fluid on a channel beta-plane is discussed. The lowest mode of baroclinic waves is shown to be always unstable in the sense of triad resonance instability. The lowest mode of barotropic waves, however, is stable if the zonal wavenumber is smaller than the critical value *c. The transition corresponds to the occurrence of resonating phenomenon between the group velocity of the primary wave and the phase speed of long waves. This phenomenon is important as a mechanism generating relatively strong zonal flow. The coupled evolution equations which govern the phenomenon are derived. Also derived is the evolution equation which governs the modulation for a barotropic wave whose zonal wavenumber is smaller than *c. These evolution equations have exact solutions. The stability of exact plane wave solutions is examined and related to the long wave resonance instability and the sideband instability reported by Plumb (1977). The exact solutions of a solitary type, which seem to be final states after above instabilities, are also obtained and presented as planetary solitons. 1. Introduction In recent years many authors have discussed the nonlinear properties of planetary waves. Especially, much attention has been given to the stability of planetary waves concerning the predictability in numerical atmospheric models (Lorenz, 1972; Hoskins, 1973; Lilly, 1973; Baines, 1976). Gill (1974) extended the problem posed by Lorenz to that of propagating plane planetary waves, thus first suggesting the application to the observed intense meso-scale motion in the ocean. Since then the stability problem, which is intimately connected with the basic processes of energy and enstrophy transfer among quasi-geostrophic motions, has been investigated mainly in the oceanic context by several authors (Yamagata, 1976, 1977; Gavrilin and Zhmur. 1977; Jones, 1977, 1979; Kim, 1978). The stability of planetary waves on an "infinite" beta-plane depends on (i) the orientation of the wavenumber vector, (ii) the non-dimensional parameter M(= The brief outline of this article is available in Japanese. See Yamagata (1979). U/(*L2) U is the velocity amplitude and L is the reciprocal of the wave number), (iii) the non-dimensional parameter F(=(L/ LR)2 LR is the Rossby's radius of deformation) and (iv) the depth ratio * if we consider the two-layer fluid. For small values of M, the instability reduces to the triad resonance instability. Due to Fj*rtoft (1953)'s criterion concerning red cascade of energy in a conservative two-dimensional fluid motion, the above instability of a barotropic wave is prohibited in some cases on a "finite" beta -plane. For instance, Plumb (1977) showed that the barotropic planetary wave in a straight channel is stable for k/l<*c (=*-3+2*3) k is the zonal wavenumber and l is the meridional wavenumber. However, the wave becomes unstable to weaker side-band disturbances in the same region of k/1. This instability just contributes to the narrow spectral broadening of the wave in contrast to the triad instability. He also showed that the long wave resonance instability occurs when k/l=*c, mediating between the triad instability and the weaker side-band instability. In order to assess the effect of stratification

2 June 1980 T. Yamagata 161 on the stability of planetary waves in a channel 2. The interaction equation beta-plane, we first extend the analysis by Plumb Let us consider a channel of infinite east-west to a two-layer system with arbitrary layer thicknesses and derive the interaction equation to extent and north-south width L on a beta-plane. The non-dimensional potential vorticity equations argue the triad instability in a channel betaplane. It is shown that the baroclinic planetary layer fluid system are for an inviscid, adiabatic, quasi-geostrophic two- wave becomes always unstable since the stratification relaxes the Fj*rtoft's criterion and it is concluded that the higher order modulation of the baroclinic planetary wave packets is of no significance. Thus, in subsequent sections, we focus our interests on the modulation of the the subscripts 1 and 2 refer to the upper barotropic planetary wave packets for k/l<*c and lower layers respectively and *1=1, *2=-1. and derive the nonlinear Schrodinger equation and y are the non-dimensional * longitude and describing the long-time evolution of the envelope*. The stability of the exact plane wave dimensional time (scaled with *-1L-1). *n latitude coordinates (scaled with L). t is the non- solution is investigated and related to the sideband instability discussed by Plumb (1977). The (n=1, 2) are the non-dimensional geostrophic pressures (scaled with UL, U being the characteristic horizontal velocity). *2 is the two-dimen- exact envelope soliton solution which is known to be a final state of the side-band instability sional Laplacian operator. The non-dimensional (Hasimoto and Ono, 1972) is shown and proposed parameters which appear in (2.1) are defined as as another type of a planetary solitary wave which is able to exist without a mean lateral shear flow. The coupled two partial differential the internal rotational Froude number, equations governing the long wave resonance are the local Rossby number, also derived and the stability of the plane wave *1 and *2 are the fluid densities, D1 and solution is discussed. The exact envelope soliton D2 are the mean layer thicknesses and f0+ *Ly solution which is accompanied with the solitary the Coriolis parameter. long wave (nearly zonal flow) is exhibited. The Introducing the barotropic and baroclinic long wave resonance may be important since it vertical modes such as is capable of generating a relatively strong zonal flow in a form of a quasi-geostrophic planetary solitary wave**. The similar discussion concerning the wave depth ratio * is defined as D2(D1+D2)-1, envelope of topographic planetary waves in a (2.1) is rewritten as continuously stratified fluid is possible and will be reported else. See Yamagata (1980a). Grimshaw (1977a, b) derived the * nonlinear Schrodinger equation for continental shelf waves and internal waves, discussing the modulational instability. He also discussed the long wave resonance for these waves. Since planetary waves on a beta-plane have relatively simple structures compared with other geophysical waves, it is one of our motivations to show, making use of the planetary waves, the compact structure about the relevant phenomena. After the ** work was finished, we became aware of the similar work by Prof. Taniuti (Nagoya University) concerning drift waves in a nonuniform, magnetized plasma. Since planetary waves are analogous to drift waves as shown by Hasegawa et al. (1979), the work here can be translated by use of the terminology in plasma physics. (See also Mima and Lee, 1980.) F is defined as (D1+D2)-1 and J denotes the Jacobian operator. Hereafter we assume M is small; we focus our interests on a weakly non-linear problem of planetary waves.*** Under the boundary condi- *** Strictly speaking, (2.1) is obtained by assuming M>0(1) in mid-latitude. So we should regard (2.1) as the model equation hereafter.

3 162 Journal of the Meteorological Society of Japan Vol. 58, No. 3 tions on y=0 and *, the solution is sought of the form ln is the positive integer. In (2.4) we have introduced two time scales t and T which correspond to the linear wave propagation time scale of O(1) and the time scale of O(M-1) describing the triad interaction, respectively. We have also introduced two space scales * and X which correspond to the wavelength scale of (1) and the wave packet scale of O(M-1) respectively. O Hence (2.3) is modified by the trans- formation 3. Triad instability In this section we discuss the stability of a planetary wave to order M. Therefore it is sufficient to consider a triad for which the following relations hold. Substituting (2.4) and (2.5) into (2.3) and using Kronecker's **, we obtain the interaction equation To leading order the interaction equation (2.6) gives the dispersion relation of a barotropic mode and that of a baroclinic mode At O(M) we find the equations which govern the long time behavior of the amplitude such as group velocities with

4 June 1980 T. Yamagata 163 are introduced. Here k*=(k,-l) when k= (k,l) and b* denotes the complex conjugate of b. As is well-known, a single barotropic or baroclinic planetary wave is a solution of (3.3) and we study the stability of the solution. Now let us consider the case for which a barotropic wave of amplitude CI is a primary one. By adding the barotropic perturbations bpi=zpi and bqi= ZqI, keeping the inequality we obtain the equations In a manner similar to the barotropic disturbances, we obtain the expression for * similar to (3.8). Then the necessary condition for instability is reduced to Assuming the solution of the form we have the expression for * such as Here if the imaginary part of * is negative, the perturbation grows. The necessary condition for instability is (3.13) shows the Fj*rtoft's result modified by the two-layer structure. It is easily shown that other combinations of perturbations do not exist for the primary barotropic wave. Unstable modes which satisfy (3.10) or (3.13) correspond respectively to the mode AI and mode AII discussed by Yamagata (1977). Next let us examine the stability of a primary baroclinic wave of amplitude CII. By adding mixed baroclinic and barotropic perturbations bq=zpii, bq=zpi, we obtain the equations Explicitly, this condition can be rewritten as which means the well-known Fj*rtoft's result that energy transfer to or from higher wavenumbers must be accompanied by the transfer to or from lower wavenumbers in a two-dimensional conservative system. In a channel model 1 must be integer. Thus, appealing to the resonance condition (3.1), we can easily show that the barotropic planetary wave is stable for as discussed by Plumb (1977). This is a direct consequence of the above Fj*rtoft's result. Retaining the primary barotropic wave and adding baroclinic perturbations ZqII, we obtain the equations bp=zqii and bq= suffices (p, q) are interchangeable, corresponding to Mode BI and Mode BII in Yamagata (1977). Assuming the solution similar to (3.7), we obtain the expression for * and the necessary condition for instability becomes When *2-1 (unequal layer thicknesses), the other mode of bp=zpii, bq=zqii is possible.

5 164 Journal of the Meteorological Society of Japan Vol. 58, No. 3 This mode does not appear in Yamagata (1977) as pointed out by Jones (1979) since *=2-1 there. The equations which govern the mode are Hereafter we call this mode Mode BIII. Assuming the solution of the form (3.7), we obtain the necessary condition for instability as Fig. 2(a), (b) As Figure 1, but for the growing baroclinic perturbation. Three cases: F=0.5, 1.0 and 2.0 are shown. Here * is fixed at 0.5. It should be noted that (3.18) has the same form as (3.10). The resonance condition (3.1) and the maximum growth rate for K=0 and *=2-1 are calculated for some cases and shown in Figs. Fig. 3(a), (b) As Figure 1, but for the growing mixed barotropic (p) and baroclinic (q) perturbation to the primary baroclinic wave (n). Two cases: F=0.5 for *=0.5. and 1.0 are shown Fig. 1 (a) Zonal wavenumbers kp, kq of the growing barotropic perturbation to the primary barotropic wave with kn=(kn,1). Crosschannel mode is the lowest one (lp=-2, 1q=1). (b) Growth rate of the disturbance for K=0 and CI =1. Note that the disturbance grows for kn>*c. 1*4. As to the primary baroclinic wave, we see, using the explicit expression of *, that the depth ratio * appears in the growth rate formula in two different ways. One form is coupled with F and the other appears only as a positive multiplication factor of the growth rate. As shown in Fig. 4, the larger the value of F, the larger the growth rate. Thus, we may conclude that one of perturbed modes for the primary baroclinic waves always becomes unstable for all

6 June 1980 T. Yamagata 165 nonlinearity becomes important for the barotropic wave. Hereafter we treat the homogeneous system since the modulation of barotropic planetary wave in a layered system is exactly the same as that in the homogeneous system. The discussion is quite similar to Grimshaw (1977a) but the treatment is much simpler than his. Then the basic potential vorticity equation is L is the linear operator and Q is the non-linear term As the primary wave we choose Fig. 4(a), (b) As Figure 3, but for the interchanged cross-channel mode (lp=1, 1q=-2). Three cases: F=0.05, 0.5 and 1.0 are shown for =0.5. A is the amplitude and *n satisfies the * equation kn/ In (1n=1) in the sense of triad instability. The other mode shown in Fig. 3, however, becomes stable for kn/ln (ln=1) smaller than a with the boundary condition. critical value. At the critical point, the barowave. As shown in section 5, the critical point For the purpose of describing the wave modulation, corresponds to the resonating phenomenon betropic we assume that A is the function of long perturbation (kp) reduces to the long time and length scales tween the primary wave and the associated long wave. When the long wave is generated by the self-interaction of the primary wave, it must have a barotropic structure in the vertical direction Here VnI is the group velocity of the n-th mode. and have a first harmonic form across the It is the main purpose of this section to derive channel in order to meet the triad resonance the evolution equation of A. We seek the condition in our two-layer model. The mode solution of (4.1) of the form shown in Fig. 3 can satisfy the above requirement for 6*(1-*)>F but the one in Fig. 4 can not. This supplements the reason why the baroclinic wave has the mode which is unstable for all values of kn/ln (ln=1). The primary barotropic waves, however, becomes always stable for and kn/ln< *c(ln=1) concerning triad interaction (Figs. 1 and 2). To leading order, (*1ei+-1e-i*) is given by 4. Modulation of a barotropic planetary wave (4.4). Then, *0, *2, *-2 are anticipated to be We have seen that the baroclinic primary (M) at most and *m( m >2) are anticipated Oto wave is unstable in the sense of triad instability be OMm-1) from (4.1). Substituting (4.8) into but that the barotropic planetary wave which (4. 1), we obtain has kn/ln(1n=1) smaller than is always stable. Thus, the modulation of the primary wave which occurs at higher order of

7 166 Journal of the Meteorological Society of Japan Vol. 58, No. 3 We also expand the non-linear term Q such as Here Q0, Q2, Q-2 are anticipated to be O(M) at most and Q1, Q-1, Q3, Q-3 are anticipated to be O(M2). Then it follows that First we consider the case m=1. The necessary and sufficient condition for the case of (4.12) to have a solution is Next we must evaluate the right hand side of (4.18), especially Q1. Since we need Q1 correct to O(M2)), it is necessary to consider the nonlinear interactions between *-1 and *2 and also between *1 and *0. However it is sufficient to consider the interaction between *-1 and *0 because *2 is zero to O(M2). This situation makes the calculation fairly simple compared with problems concerning other waves. Now let us derive *0: a wave-associated zonal mean flow. From (4.1) and (4.4) we find the governing equation of *0 *n is the eigenfunction of the complete set which satisfies (4.5) and (4.6) with the eigenvalue cn. Explicitly, *n is written as Neglecting the term M(*/*T), we have, to leading order, *0 satisfies the equation with the eigenvalue (phase velocity) In connection with the result in 3, we are concerned with the case of n=1 (corresponding to n=1). I f we define l and the boundary condition The inhomogeneous solution is easily obtained and given by we can rewrite (4.13) as provided that Expanding the left hand side of (4.17) in Taylor series and utilizing the relation D(*n,k)=0, we obtain Because of the degeneracy concerning the long time T, it is arbitrary to add any zonal flow of the same order. In this section, however, we are only concerned with the zonal flows associated with the primary wave itself. Thus we remove the arbitrariness. We note that the condition (4.26) is necessary to avoid the resonance among the group velocity of the n-th mode and the phase speed of the long wave (zonal flow). If the resonance occurs, we must change the scaling of time and space coordinates to describe the long time behaviour. This case is discussed in the next section. Using (4.4), (4.22) and (4.25) to evaluate Q1 and substituting Q1 into (4.18), we finally find From (4.16), it follows that

8 June 1980 T. Yamagata 167 (which corresponds to k/l<*c) and the maximum growth rate is and when (4.27) is known as a non-linear Schrodinger equation. One of the exact solutions of (4.27) is the plane wave solution These results are identical with Eqs. (40)(42) in Plumb (1977). Another well-known exact solution of (4.27) is the envelope soliton solution R is the amplitude and real, * denotes the phase and ** is an arbitrary real constant. The stability of the above solution was discussed by Hasimoto and Ono (1972). Plumb (1977) showed that a monochromatic finite amplitude barotropic planetary wave in a channel is unstable to small side-band perturbation when kn/ln is smaller than *c. Here we mediate the stability problem of (4.28) and Plumb's result. We perturb (4.28), writing which exists only when *r<0. Since (4.27) is invariable to the transformation the general form of the solution becomes * is a small parameter, * and * are the amplitude perturbation and phase perturbation respectively. Substituting (4.29) into (4.27) and linearizing for small perturbations, we obtain from the imaginary part and from the real part. We consider the plane harmonic disturbance Substituting (4.32) into (4.30) and (4.31), we obtain the dispersion relation It is known that the solution (4.38) behaves like an elementary particle keeping its identity. As pointed out by Hasimoto and Ono (1972), it should be noted that the condition for existence of the envelope soliton is identical with the necessary condition for the plane wave solution to be unstable. It should also be noted that the preferred scale of the instability is nearly equal to the length scale of the envelope soliton. Although the envelope soliton of a barotropic planetary wave can exist in the limited parameter range, it may be listed as a planetary soliton on a beta-plane. 5. Long wave resonance In the preceding section, we cannot estimate when VnI=-(4n2)-1. This condition *0 is explicitly written as Hence the disturbance can grow if *<0**** which is solved as **** The tentative extrapolation of our result in the long wave limit (k*0) predicts modulational instability for long waves. However we must be careful to take the limit since our basic assumption of quasi-geostrophy breaks down in the limit. Grimshaw (1977a) showed the modulational stability for long continental shelf waves. (See Yamagata, 1980b.) As compared with the result in section 3, this resonance condition is identical with the marginal one for the triad instability. In this section we discuss the stability behaviour near the resonance. Adopting the procedure due to Grimshaw (l977a), we introduce tentatively the long wave

9 168 Journal of the Meteorological Society of Japan Vol. 58, No. 3 and the primary wave Using the above complete set, the coefficient a, is determined as * is an unknown constant to be determined later. Assuming that *-1 is the long length scale and *-2 is the long time scale, we obtain the modulation equation From (5.3) and (5.9), we see Here Q1(0) is the nonlinear term evaluated by use of (5.3) and (5.4), the order of which is Thus, if we choose the rela- O (M*+l) at most. tion and We introduce a new parameter * which is a measure of resonance as we can balance the left hand side of (5.5) (linear dispersive term) with the right hand side (nonlinear term). The equation which governs *0 is similar to (4.21) but now we have the homogeneous equation at O(M*) since the right hand side drops out. Requiring that the both sides should balance, we obtain Substituting (5.19) into (5.10) and integrating from 0 to * after multiplying (5.10) by we obtain for r=s=2n. Here * is defined as From (5.7) and (5.8), it follows that Substitution of (5.3) and (5.4) into (5.5) yields Thus the equation which determines *0 becomes Then (5.5) becomes Let us seek a solution of (5.10) in the form *r(0) is the cigenfunction of the complete set which satisfies the equation Thus, we finally obtain coupled equations (5.20) and (5.24) which govern the long wave resonance. These equations have plane wave solutions subject to the boundary condition C is real and * denotes the phase. If we perturb these solutions as Explicitly, it follows that with the eigenvalue (phase speed of a long wave) and substitute (5.26) into (5.20) and (5.24), we obtain

10 June 1980 T. Yamagata 169 When * is large in this case, the disturbance grows if i.e. after linearization. Here let us consider the plane harmonic disturbance since * is positive. (5.37) corresponds to i.e. the domain of the modulational instability. When * is large with the condition */K is (1), the instability occurs if Substituting (5.28) into (5.27), we obtain the dispersion relation with the condition O Since in * is positive, (5.39) reduces to *>0, which corresponds to in the limit *0, the disturbance grows if and only if (5.30) is identical with that obtained by Grimshaw (1977). When * is large, (5.30) reduces to i.e. the necessary condition for the triad instability. Thus we see that the equations (5.20) and (5.24) which govern the long wave resonance mediate implicitly the modulational instability and the resonance instability. The equations (5.20) and (5.24) have the exact solutions since c2n(0) is negative. The above condition is equivalent to since *~-*/* in the limit. Notice that (5.32) is identical with the necessary condition for the modulational instability of the plane harmonic solution of the nonlinear Schrodinger equation. However, if we keep K/* *O(1) when * becomes large, the disturbance grows if or with the condition (5.34) is equivalent to (3.10): the necessary condition for triad instability. On the other hand, in the limit *0, we recover the result obtained by Plumb (1977) as In the limit * *0 (~O(M-2/3)), ** tends to and * tends to O(M-2/3). Then we can find * (4.38) again in the limit. Thus it is anticipated that (5.42) will be the final state of the instability discussed above. (5.42) is composed of the envelope soliton and the long wave soliton. The amplitude of the long wave soliton is O(M4/3) This may be significant because the relatively strong (O(M4/3)) zonal flow can be generated through the long wave resonance by the primary barotropic planetary wave of O(M) on a channel beta-plane.

11 170 Journal of the Meteorological Society of Japan Vol. 58, No. 3 The long wave resonance can generate relatively strong zonal flows of O(M4/3) on the time 6. Summary and discussion We have discussed the stability of planetary scale of O(M-4/3). This aspect will be important waves in a two-layer fluid on a channel betaplane. We have shown that a pure baroclinic final state of the plane wave solution is anti- but it does not seem to be familiar. Further, the planetary wave becomes unstable for all values cipated to be composed of the envelope soliton of kn/ln (ln=1) in the sense of triad instability. and the long wave soliton. Although long wave Therefore the self-modulation occurring at the resonance can occur within the limited parameter higher order of nonlinearity is of no significance space, it can be listed as one of the possible for a baroclinic planetary wave since the wave mechanisms for generating planetary solitons. breaks down via triad resonance instability and loses its identity promptly. However, Hide (1958), Acknowledgements Hide and Mason (1975) and Hide, Mason and I express my thanks to Dr. R. A. Plumb for Plumb (1977) showed experimentally that the helpful comments on the manuscript. I must "regular" baroclinic wave exists for kn/l n*1.4 thank anonymous reviewers for their laborious in a differentially heated rotating annulus. Loesch task of checking the typescript. (1974) investigated triad resonant interactions in a two-layer baroclinic system with the betaeffect and showed that the energy transfer among References three modes changes drastically at F=10.5. For Baines, P. G., 1976: The stability of planetary F*10.5, a strong energy transfer occurs to both waves on a sphere. J. Fluid Mech., 73, neutral waves so that the unstable mode loses Gavrilin, B. L, and V. V. Zhmur, 1977: Stability of its identity. The transition occurs kn/ln= Rossby waves in a baroclinic ocean. Oceanology, 0.71, corresponding to the long wave resonance 1.6, as pointed out by Hide, Mason and Plumb Gill, A. E., 1974: The stability of planetary waves (1977). Recently, Yoden (1979) showed, using on an infinite beta-plane. Geophys. Fluid Dynam., the extended two-layer quasi-geostrophic spectral model of the kind of Lorenz (1960), that the Grimshaw, R., 1977a: The stability of continental mode which has kn/ln equal to 1 breaks down shelf waves. I: Side band instability and long and becomes irregular if the dissipation is weak. wave resonance. J. Austral. Math. Soc., 20(B), This is not inconsistent with the result of Loesch , 1977b: The modulation of an internal (1974). gravity-wave packet and the resonance with the Unifying these results, we suggest that the mean motion. Stud. Appl. Math., 56, incorporation of the zonal mean shear into our Hasegawa, A., C. G. Maclennan, and Y. Kodama, model will produce a kn/ln domain in which the 1979: Nonlinear behavior and turbulence spectra baroclinically unstable wave becomes stable in of drift waves and Rossby waves. Phys. Fluids, the sense of triad interactions. The reader is 22, referred to Pedlosky (1972) in this context. Hasimoto, H, and H. Ono, 1972: Nonlinear modulation of gravity waves. J. Phys. Soc. Japan, 33, In this article, however, we have not pursued the subject; we have focussed our interests on Hide, R., 1958: An experimental study of thermal describing the modulational instability and the convection in a rotating liquid. Phil. Trans., 250, long wave resonance of a barotropic wave by use of the nonlinear evolution equations. We - and P. J. Mason, 1975: Sloping convection in a rotating fluid. Adv. in Phys., 24, 47- have shown that the amplitude evolution for kn/ln*c can be described by the well-known 100. nonlinear Schrodinger equation. The exact plane -, - and R. A. Plumb, 1977: Thermal convection in a rotating fluid subject to wave solution is modulationally unstable and the envelope soliton solution evolves. The transition at kn/ln=*c corresponds to the long wave temporal characteristics of fully developed baro- horizontal temperature gradient: spatial and clinic waves. J. Atmos. Sci., 34, resonance. We have shown that the phenomenon Hoskins, B. J., 1973: Stability of the Rossbycan be described by the coupled nonlinear Haurwitz wave. Quart. J. Roy. Met. Soc., 99, Schrodinger and Korteweg-deVries equations. The instability of the exact solution mediates the Jones, S., 1979: Rossby wave interactions and modulational instability and the triad instability. instabilities in a rotating, two-layer fluid on a

12 June 1980 T. Yamagata 171 beta-plane. Part II. Geophys. and Astrophys. Fluid Dynam., 12, Kim, K., 1978: Instability of baroclinic Rossby waves; energetics in a two-layer ocean Deep- Sea Res., 25, Lilly, D. K., 1973: A note on barotropic instability and predictability. J. Atmos. Sci., 30, Loesch, A. Z., 1974: Resonant interactions between unstable and neutral baroclinic waves. Part I and II. J. Atmos. Sci., 31, Lorenz, E., 1960: Energy and numerical weather prediction. Tellus, 12, : Barotropic -, instability of Rossby wave motion. J. Atmos. Sci., 29, Mima, K, and Y. C. Lee, 1980: Modulational instability of strongly dispersive drift waves and formation of convective cells. Phys. Fluids, 23, Pedlosky, J., 1972: Finite-amplitude baroclinic wave packets. J. Atmos. Sci., 29, Plumb, R. A., 1977: The stability of small amplitude Rossby waves in a channel. J. Fluid Mech., 80, Yamagata, T., 1976: Stability of planetary waves in a two-layer system. J. Oceanogr. Soc. Japan, 32, , 1977: Stability of planetary waves in a two-layer system (small M limit). J. Meteor. Soc. Japan, 55, : Stability -, of planetary waves. Marine Science, 11, (in Japanese). 1980a: On the nonlinear -, modulation of planetary topographic waves in a rotating stratified ocean. Geophys. Astrophys. Fluid Dynam., in press. -, 1980b: On long planetary waves. Part I: a pathology of (Q-G.) P.V.E. approach and Part II. Yoden, S., 1979: Some dynamical properties of nonlinear baroclinic waves in a quasi-geostrophic model. J. Meteor. Soc. Japan, 57,