Steady and unsteady nonlinear internal waves incident on an interface

Transcription

1 Quarterly Journal of the Royal Meteorologial Soiety Q. J. R. Meteorol. So. 139: , Otober 013 B Steady and unsteady nonlinear internal waves inident on an interfae Roger Grimshaw a *and John MHugh b a Department of Mathematial Sienes, Loughborough University, UK b Department of Mehanial Engineering, University of New Hampshire, Durham, NH, USA *orrespondene to: R. Grimshaw, Mathematial Sienes, Loughborough University, Ashby Road, Loughborough LE11 3TU, UK. R.H.J.Grimshaw@lboro.a.uk Steady nonlinear internal waves are ommonly desribed by the Dubreil- Jaotin Long equation. This equation ontains unknown funtions of the stream funtion, representing the density and vortiity fields. Often these are determined by upstream onditions where the flow is assumed to be known. But for the ase when the waves are periodi in the horizontal diretion, these funtions need to be determined instead by onsideration of the soure of the waves, and in partiular by the wave-indued mean flow. Here we show that this situation is partiularly important for waves inident and refleted from an interfae, representing a sharp hange in the bakground density stratifiation, suh as that at the tropopause. The ombination of the inident and refleted wave-indued mean flows generates a sharp shear near the interfae. Key Words: DJL equation; tropopause; internal waves Reeived 16 July 01; Revised 7 November 01; Aepted 0 November 01; Published online in Wiley Online Library 17 January 013 itation: Grimshaw R, MHugh J Steady and unsteady nonlinear internal waves inident on an interfae. Q. J. R. Meteorol. So. 139: DOI:10.100/qj Introdution The Earth s atmosphere has a sudden inrease in stati stability, measured by the buoyany frequeny N(z), with inreasing altitude z at the tropopause, and again at the mesopause, with a sudden derease in stability at the stratopause. These sudden hanges in the stati stability are potential barriers for vertially propagating internal waves, with important impliations for weather, limate and aviation. An idealized but useful model of the dynamis at these altitudes is a two-layer flow with onstant buoyany frequeny in eah layer. For instane, this model represents the tropopause as an interfae where the density gradient is disontinuous while the density remains ontinuous. This interfaial model has been used extensively in the treatment of nonlinear steady mountain waves. Fully nonlinear steady two-dimensional flows are desribed by the Dubreil-Jaotin Long (DJL) equation (Dubreil-Jaotin, 1934; Long, 1953) whih has been extensively used in the literature for the study of mountain waves. For instane, Ikawa (1990) used the two-layer model in the DJL equation for weakly nonlinear mountain waves, while Durran (199) using the same model obtained fully nonlinear mountain waves with an iterative tehnique. An important feature of the DJL equation is that in general it ontains an arbitrary funtion of the stream funtion, whih for mountain waves is usually determined by assuming that all streamlines emanate from upstream, where the flow redues to a known basi state. Reently MHugh (009) onsidered steady monohromati periodi waves using the DJL equation for a two-layer flow, and found that, when a wave is inident on the interfae, the ombination of the inident and refleted wave generated a mean flow, proportional to the square of the wave amplitude. Beause, in this invisid theory, this mean flow is disontinuous aross the interfae, there is an impliation that this will lead to shear instability and turbulene near the interfae. However, for horizontally periodi waves, there are no upstream onditions that an be used to determine the arbitrary funtions in the DJL equation, and instead MHugh (009) determined them using the basi state when the waves are absent. In the Boussinesq approximation, the DJL equation is then a linear equation in eah layer, sine the buoyany frequeny is onstant in eah layer, and all nonlinearity resides in the interfae onditions, whih for steady flow are that the interfae is a streamline, aross whih the pressure 013 Royal Meteorologial Soiety

2 Nonlinear Internal Waves Inident on an Interfae 1991 is ontinuous. The main purpose of this artile is to restore wave transiene, and so reonsider this problem as the longtime steady outome of a well-posed initial value problem. It will transpire that the arbitrary funtions in the DJL equation an then be uniquely determined, but are different from those used in MHugh (009). Unsteady vertially propagating internal waves in a single layer have been onsidered by Grimshaw (1975), Shrira (1981), Voronovih (198), Sutherland (006) and Tabaei and Akylas (007) in theoretial studies, and Sutherland (001) performed diret numerial simulations. From the perspetive of this artile, the most important finding is that a vertially propagating wave paket generates a mean flow that is loalised to the wave paket. Reently, the unsteady development of internal waves in a two-layer invisid flow was onsidered in numerial simulations by MHugh (008) for horizontally periodi monohromati waves, with a modulated amplitude in the vertial, and by Sharman (pers. omm., 01) for mountain waves. These results show that the wave paket reates a mean flow that is strongest at the interfae. This generated mean flow may even be strong enough to form a ritial level, ausing further inident waves to break. Further ongoing work by MHugh (pers. omm., 01) treats the unsteady development of the waves assuming the wave amplitude is slowly varying in the vertial. The unsteady theory shows that the ombination of inident waves and refleted waves in the lower layer results in a brief period where both are reating a mean flow that is additive, reating a jet-like feature that is similar to the results from the numerial simulations. Furthermore, the mean flow is found to be disontinuous at the interfae. Here we reonsider the steady two-layer problem (as in MHugh, 009), but inlude the unsteady development of the waves from an initial value problem. Our attention is foused on a monohromati inident linear wave whih is periodi in the horizontal diretion, but in general has a wave paket in the vertial diretion. When the wave envelope is onstant, then the wave is also periodi in the vertial diretion, when it an be desribed by the DJL equation. More omplex ombinations of waves are ertainly possible and very likely our in the atmosphere and oean. One suh ombination is a resonant triad. The importane of wave triads was originally suggested by Phillips (1960), and were demonstrated to exist in the ontext of surfae gravity waves with surfae tension by MGoldrik (1965). Wave triads for internal waves in a uniformly stratified medium were treated by MEwen (1971), MEwen et al. (197), and MEwen and Robinson (1975). The inident wave paket treated here may indeed ombine with two other free wave pakets to form a resonant triad. How suh resonantly interation triads might then interat with a sharp interfae, suh as the tropopause, is a matter for further study. However, we note that, as the members of the triad have different group veloities, in general eah member of the triad will reah the interfae at a different time, and hene we expet that the interfae interation senario desribed here an be applied to eah wave, at least as a first approximation. One triad interation that may be of some speial importane here is when the inident and refleted waves ombine with a muh longer wave to form a triad. Sine the inident and refleted waves have the same horizontal wavenumber, then the long wave must have zero horizontal wavenumber, and hene the long wave is part of the mean flow. Thus this triad represents an interation between the waves and the wave-indued mean flow. Suh an interation was treated by Grimshaw (1977) for waves between horizontal rigid surfaes, and more reently by Tabaei and Akylas (007) for waves in a single layer with uniform stratifiation. For the ase onsidered here, this mean flow would have a vertial osillation with twie the vertial wavenumber of the inident waves, and the solution given below does indeed inlude suh a mean flow. It is possible that this interation might lead to instability, whih would result in a stronger mean flow than that given below. This senario is not investigated further here, but ould be a subjet for further study. Another important ombination of waves is a broad spetrum in both frequeny and wavenumber. Although how suh a wave system might interat with a sharp interfae is beyond the sope of the present study, we expet that the results obtained here will be useful as a guide to this more general situation. We note, however, that one partiular speial ase (when the waves all have the same frequeny and horizontal wavenumber, but have a vertial wavenumber spetrum) is treated here in the limit when that spetrum is narrow, leading to the present wave paket formation. We an then speulate that an inident wave with a broad vertial wavenumber spetrum would reate a refleted wave also with suh a broad spetrum. As refletion generally tends to favour higher-frequeny waves, this would skew the refleted spetrum. In the small-amplitude limit, members of the inident spetrum might then ombine with members of the refleted spetrum to form a ontribution to the mean flow, as desribed above. Overall, the inident wave spetrum beneath the interfae would be spread as a result of this mehanism. In ontrast, the spetrum above the interfae will be shifted to lower frequenies and beome more foused, as transmission favours lower frequenies. In setion we formulate the problem in the Boussinesq approximation. Then in setion 3 we briefly reall the linearized solution, desribe how the mean flow an be determined, and extend the solution to seond order in wave amplitude. Then in setion 4 we revisit the steady formulation using the DJL equation, but with a new determination of the arbitrary funtions using our previously determined mean flow results. We onlude in setion 5.. Formulation We onsider the two-dimensional flow of an invisid inompressible fluid. The full equations for the perturbation variables are, in standard notation, ρ 0 u t + p x = F 1 = ρ 0 (uu x + wu z ) ρ(u t + uu x + wu z ), (1) ρ 0 w t + p z + gρ = G 1 = ρ 0 (uw x + ww z ) ρ(w t + uw x + ww z ), () u x + w z = 0, (3) ζ t w = H 1 = uζ x wζ z, (4) gρ t wρ 0 N (z) = J 1 = ugρ x wgρ z, (5) Here p 0 (z), ρ 0 (z) are the bakground pressure and density, p 0z (z) = gρ 0 (z), and ρ 0 N = gρ 0z. In the sequel, N

3 199 R. Grimshaw and J. MHugh will have a disontinuity aross z = η, where the boundary onditions are ζ = η, at z = η, (6) [p + p 0 (z)] + = 0, atz = η. (7) The density equation (5) an be solved by ρ 0 (z) + ρ = ρ 0 (z ζ ), so that, to leading order in wave amplitude, ρ = ρ 0z ζ + ρ 0zzζ +. (8) The omitted terms are O(ζ 3 ). The boundary onditions (6),(7) an be expanded so that [p gρ 0 η + p z η + ρ 0N η ] + + = 0, ζ + ηζ z + =η, at z = 0, and then ombined to give [p gρ 0 ζ ] + = K 1 = [gρ 0 ζζ z + ρ 0N ζ ] + + at z = 0. (9) Thus the variables ρ, η are formally eliminated. Further, to leading order in wave amplitude, it is readily shown ζ z is ontinuous aross the interfae, and so (9) redues to [p gρ 0 ζ ] + = ρ 0ζ [ N ] + +, at z = 0. (10) In a Boussinesq fluid, we set ρ 0 = onstant, ρ = 0exept when multiplied by g. Thusdefinegρ = b (buoyany) and then (8) beomes gρ = b = ρ 0 N ζ (ρ 0N ) z ζ +. (11) 3. Unsteady theory 3.1. Linear theory Here F 1, G 1, H 1, J 1, K 1 = 0. Further, we make the Boussinesq approximation, and assume that N is onstant above and below the interfae, i.e. N = N 1, z <ηand N = N, z >η. Then assume an upward propagating wave paket, a downward refleted wave paket and a transmitted wave paket, ζ =E{A(T Z/ g1 )exp( im 1 z) + RA(T + Z/ g1 )exp(im 1 z)}+.., z < 0, (1) ζ = ESA(T Z/ g )exp( im z) +.., z > 0. (13) where E = exp (ikx iωt), Z = ɛz, T = ɛt, and ω = kn 1 (k + m = kn 1 )1/ (k + m. (14) )1/ Here 0 <ɛ 1 is a small parameter defining the sale of the wave paket. Without loss of generality, hoose k > 0 so that ω>0 and the waves propagate in the positive x diretion. Then, to ensure that the inident wave and transmitted waves propagate upwards, we must hoose m 1, > 0 as well. X = ɛx, T = ɛt (ɛ <<1) are slow variables defining the wave paket and g1, g are the vertial group veloities, given by g1 = ωm 1 (k + m 1 ), g = ωm (k + m ). (15) We suppose that A(ξ) 0, ξ so that there are no waves near the interfae when T. The refletion and transmission oeffiients are given by R = m 1 m, S = m 1. (16) m 1 + m m 1 + m Note that, while m 1 is required to be real-valued, m may be either real-valued or pure imaginary (possible if N < N 1 ). In the latter ase, let m = im, M > 0 and the waves in z > 0 are evanesent. The expression (1) for the inident and refleted waves still holds, and R =1. Also formally the expression (13) still holds, but g is omplex-valued. Hene A(T Z/ g ) is a funtion of a omplex-valued variable, and it is then required that A(ξ) be an analyti funtion of ξ. 3.. Mean flow Let denote an x-average, and set p =p, et. Then the nonlinear mean flow equations are obtained by averaging (1) (5). In the Boussinesq approximation, we get u t + uw z = 0, (17) p z + b + w z = 0, (18) w = 0, (19) ζ t + wζ z = 0, (0) b t + wb z = 0. (1) These are formally fully nonlinear, but it will be suffiient here to work to quadrati order in wave amplitude. Thus averaging (8) yields b= g ρ 0 (z ζ ) ρ 0 (z) = ρ 0 N ζ (ρ 0N ) z ζ +, () linking b and ζ. Using the linear expressions, (0) an be integrated to give ζ = 1 ζ ρ0 N ζ z, so that b =. (3) z Then (18) an be integrated to yield p = ρ 0N ζ + w. (4) Here an arbitrary funtion of z has been set to zero without loss of generality. In partiular, (3),(4) show that p gρ 0 ζ = gρ 0 ζζ z + ρ 0 N ζ + w. (5)

4 Nonlinear Internal Waves Inident on an Interfae 1993 Next, following Sutherland (006), note that the vortiity χ = u z w x satisfies the equation Averaging yields ρ 0 (χ t + uχ x + wχ z ) = b x. (6) χ t + wχ z = 0, χ = u z, (7) and then integration with respet to z yields u t + wχ =0. (8) Here an arbitrary funtion of z is set to zero, sine there is no mean flow when there are no waves, that is as z for upward propagating waves. This is equivalent to (17) sine it is easily shown that wχ = wu z = wu z.butitannow be shown that, orret to seond order in wave amplitude, on using the linear relation w ζ t, and so (17) beomes wu z = χζ t, (9) u t + χζ t = 0, or u = χζ. (30) 3.3. Nonlinear theory Formally, expand ζ = αζ (1) + α ζ () +, (31) where α<<1. Then ζ (1) is given by (1),(13). The seondorder term will ontain two parts, a seond harmoni term proportional to E and a mean term, independent of E, whih has been found in setion 3. To the required order α, the averaged interfae boundary ondition (10) is [ p gρ0 ζ ] + = ρ 0 ζ [ N ] +. (3) Noting that w, ζ z are ontinuous aross the interfae to O(α), we see that the mean boundary ondition (3) is automatially satisfied by the mean flow solution (5) generated on eah side of the interfae. Note that this result holds for arbitrary N(z) on eah side of the interfae. In the remainder of this setion, we put N = N 1, on eah side of the interfae. It remains to evaluate u from (8), and for this purpose, note that sine wave fields desribed by (1),(13), χ b/ρ 0 N ζ/, where = ω/k. Thus, we find that u = N 1, ζ. (33) Hene, when m is real-valued, and using (1),(13), u = N 1 {I + R + IR os (m 1 z)}, z < 0, (34) u = N T, z > 0, (35) where the inident, refleted and transmitted wave pakets are defined by I = A(T Z/ g1 ), R = RA(T + Z/ g1 ), T = SA(T + Z/ g ). (36) Note that here A, R, S and hene I, R, T are all real-valued. If instead m = im is pure imaginary, so that the waves are evanesent in z > 0, then the expression (33) still holds, but (34),(35) are replaed by u = N { 1 I + R +I R os (m 1 z+φ R ) }, (37) with φ R = arg R, z < 0, u = N T exp ( M z), z > 0. (38) We reall that I = A(T Z/ g1 ) is real-valued and R = RA(T + Z/ g1 ), where R is omplex-valued with R =1 and tan (φ R /) = M /m 1. Also T = SA(T Z/ g )where both S and g are omplex-valued. Note that in both ases the mean flow is disontinuous aross the interfae, [u] + = S A (T) ( N N1 ). (39) The origin of this disontinuity is the jump in the vortiity aross the interfae. There are two main ases of interest. First suppose that the inident wave paket is loalised, for instane A(ξ) = A 0 seh(ξ). (40) Then the expression (34), or (37), shows that there is a loalised mean flow assoiated with the inident wave paket, and another with the refleted wave paket, proportional to A (T Z/ g1 ) respetively, whih is a well-known feature of vertially propagating wave pakets. In addition, where these pakets overlap briefly near the interfae, there is an additional z-dependent mean flow, given by the last term in (34). In z > 0 there is an analogous loalised mean flow in the transmitted wave proportional to A (T Z/ g )in the propagating ase when m is real-valued, and a mean flow loalised near the interfae in the evanesent ase when m = im. Seond, suppose that the inident wave paket is a frontal wave, followed by a uniform wave train, for instane A(ξ) = A 0 {1 + tanh (ξ)}. (41) Then for large times, T, A A 0, and in the propagating ase when m is real-valued, the mean flow (34) beomes u N 1 A 0 {1+R +R os (m 1 z)}, z < 0, u = N A 0 S, z > 0. (4) In this ase, the refletion from the interfae has generated a z-dependent mean flow in z < 0, whereas the mean flow in z > 0 is a onstant. In the evanesent ase when m = im, the mean flow (38) beomes u 4N 1 A 0 {1+os (m 1 z+φ R )}, z < 0, (43) u= 4N A 0 {1+os (φ R )} exp ( M z), z > 0. Now there is again a z-dependent mean flow in z < 0, but also a z-dependent mean flow in z > 0 trapped near the interfae.

5 1994 R. Grimshaw and J. MHugh 4. Steady ase: DJL equation Here we re-examine the same problem regarded as steady in a frame of referene X = x t, moving with speed = + O(α ), where is the nonlinear wave speed. For the present purposes, we an use. It is neessary to assume here that after initiation the wave amplitude A is a onstant. It is onvenient here to use a stream funtion ψ suh that u = ψ z, w = ψ x. Then the steady-state solution of equations (4),(5) are ζ ψ = ζ ( ), b = gρ 0(z)+b( ), = ψ z. (44) The vortiity equation (6) redues to the DJL equation χ = ψ zz + ψ xx = b ( )ψ + F( ). (45) ρ 0 Here the funtions ζ ( ), b( ), F( ) are unknown. They are usually found from upstream onditions, for instane by assuming that ψ 0, ζ 0, b 0asx. In that ase ζ ( ) = 0, b( ) = gρ 0 ( and the DJL equation (45) redues to ), F( ) = 0, (46) ψ zz + ψ xx + N (z ψ/)ψ = 0. (47) However, for periodi waves, this proess annot be arried through. Instead it is not lear how these funtions an be determined. It seems that some further information is needed, suh as that from the initial onditions. Thus we an assume that the relation (8) holds in the steady state, that is b = gρ 0 (z ζ ) gρ 0 (z), and then Thus b( )= gρ 0 (z ζ ), z ζ = ζ ( ). (48) b ( ) ρ 0 = N (z ζ ) {1 + ζ ( )}. (49) Next for linear steady waves, ζ = ψ/, and hene ζ ( ) is seond order in wave amplitude. It follows that the DJL equation (45) an be redued to ψ zz + ψ xx + N (z ψ/)ψ = F( ), (50) orret to seond order in wave amplitude. It remains to determine F( ) whih is seond order in wave amplitude. Here we use the mean flow expression (33). First, using the DJL equation in the form (50), and orret to seond order in wave amplitude, u = N (z) ζ. (51) Then, seond, taking the average of the DJL equation (50) yields u z + N (z) ψ =F( z)+ (N ) z ζ, ψ z = u. (5) Sine u is given by (51), this determines the funtional form of F. Indeed, (5) redues to F( z) = N { ζ z + ψ}. (53) This analysis shows that the mean flow should not be determined using the steady DJL equation, and instead an only be used for the flutuating wave omponents. Instead the mean flow is determined from (33). For example, suppose that N is a onstant, and onsider a single vertially propagating wave of onstant amplitude A. Then ζ =A is a onstant, u = N A / is a onstant, and ψ = uz. It follows that then F( z) = N 4 A z/ 3, and so F( ) = N 4 A / 4 is a linear funtion of.but when there is also a refleted wave, the situation is more ompliated, as then u depends expliitly on z: (4) in the propagating ase when m is real-valued, and (43) in the evanesent ase when m = im. Thus, in the propagating ase, in z < 0, ζ = A 0 {1 + } R + R os (m 1 z)}, N1 u = A 0 {1 + (54) R + R os (m 1 z)}, z { ψ = u dz = N 1 A 0 (1+R )z+ R } sin (m 1 z), (55) m 1 F( z)= N4 1 A 0 (1+R )z 3 + N 1 A 0 R(k 3m 1 ) m 1 sin (m 1 z), (56) F( )= N4 1 A 0 (1 + R ) 4 N 1 A 0 R(k 3m 1 ) m 1 ( m1 sin ). (57) In z > 0, there is just a single vertially propagating wave, and so F( ) is a linear funtion of as desribed above, i.e. in z > 0, F( ) = N4 A 0 4. (58) Analogous expressions an be found in the evanesent ase, when m = im. Thus in z < 0, ζ =4A 0 {1 + os (m 1z + φ R )}, u = 4N 1 A 0 {1 + os (m 1z + φ R )}, (59) z { ψ = u dz = 4N 1 A 0 z+ 1 } sin (m 1 z+φ R ), (60) m 1

6 Nonlinear Internal Waves Inident on an Interfae 1995 F( z)= 4N4 1 A 0 z 3 + N 1 A 0 (k 3m 1 ) m 1 sin (m 1 z+φ R ), (61) F( )= 4N4 1 A 0 4 N 1 A 0 (k 3m 1 ) m 1 ( m1 sin ) φ R. (6) In z > 0, we have ζ =u = 4A 0 {1 + os (φ R)} exp ( M z), u = 4N A 0 {1 + os (φ R)} exp ( M z), (63) z ψ = u dz = N A 0 M {1+os (φ R)} exp ( M z), (64) F( z)= N4 A 0 (k +5M M )exp( M z), (65) ( ) F( )= N4 A 0 (k +5M M )exp M. (66) 5. onlusion In this artile we have re-examined the earlier work by MHugh (009) on the refletion of a nonlinear internal wave inident on an interfae separating two regions eah of onstant but different buoyany frequeny. That artile assumed a steady state and so, in the referene frame moving with the horizontal phase speed of the waves, the DJL equation (45) ould be employed. However for the situation as in MHugh (009) and here, when the wave field is periodi in the horizontal diretion, ertain unknown funtions in the DJL equation annot be determined unless some additional information is obtained about the soure of the waves. In this artile we show that some knowledge of the wave-indued mean flow allows for the determination of a unique DJL equation (57), (6). Speifially, we have reonsidered the steady two-layer problem of MHugh (009) by inluding the transient development of the waves into a final steady state. Then the wave front generates a horizontal mean flow, given by (33) to seond order in wave amplitude. This expliit expression for the mean flow is found by rearranging the usual meanflow equation (17) into (30), allowing integration in time. The resulting mean flow in eah layer is given by (34) and (35). To ahieve this same mean flow from the steady DJL equation, an additional assumption (48) must be made, whih is the requirement that there is no average mass flux through a horizontal surfae. The previous work (MHugh, 009) used a different DJL equation, in whih the arbitrary funtions were obtained with referene to a basi state without any waves. In effet that orresponds to a different soure of the waves than that used here. In MHugh (009), the dynami interfaial onditions were found to generate a ontribution to the mean flow due to an inhomogeneity at seond order in wave amplitude in the dynami interfaial boundary ondition. The present work demonstrates that the mean pressure field by itself balanes this inhomogeneity. u / A N ω/ N 1 Figure 1. Jump in the mean veloity u at the interfae versus wave frequeny. The mean flow in the final steady wave solution is shown here to be disontinuous aross the interfae. This disontinuity is present whether the waves in the upper layer are transmitted, as at the tropopause or the mesopause, or evanesent, as they may be at the stratopause. The strength of the jump in the mean flow is proportional to the jump in the square of the buoyany frequeny, and inversely proportional to the horizontal wave speed. The strength also depends on wave frequeny, as shown in Figure 1. The jump is strongest for low-frequeny waves and dereases monotonially as the frequeny inreases. This outome that the mean flow is disontinuous aross the interfae implies that it will be unstable to Kelvin Helmholtz instability. This instability will result in the unsteady growth of unstable modes, expeted finally to form a relatively thin turbulent layer entred around the interfae. The strong mixing assoiated with the turbulene would form a well-mixed uniform temperature field in this thin layer. Future inident waves would have to ontend with this thin well-mixed layer, rather than a sharp interfae. Inident waves with a long vertial wavenumber would behave as desribed here, but if the thikness of the layer approximates the vertial wavelength of the waves then the behaviour may be quite different. Overall, the onsequene of the results reported here is that internal waves, even after reahing a stationary state, will drive a steep shear layer at the tropopause. This fat implies that the tropopause will be more turbulent on average than other altitude loations. This onlusion also applies to other plaes where the buoyany frequeny hanges rapidly, suh as the stratopause and then the mesopause. This result is partiularly important for aviation weather, where turbulene avoidane is ritial to airraft safety and the tropopause has long been suspeted of being more turbulent than other altitudes (Partl, 196). Referenes Dubreil-Jaotin ML Sur la determination rigoureuse des ondes permanentes periodiques d amplitude finie. J. Math. Pure Appl. 13: Durran DR Two-layer solutions to Long s equation for vertially propagating mountain waves: how good is linear theory? Q. J. R. Meteorol. So. 118: Grimshaw R Nonlinear internal gravity waves and their interation with the mean wind. J. Atmos. Si. 3:

7 1996 R. Grimshaw and J. MHugh Grimshaw R The modulation of an internal gravity-wave paket, and the resonane with the mean motion. Studies Appl. Math. 56: Ikawa M Weakly nonlinear aspets of steady hydrostati mountain waves in a -layered stratified fluid of infinite depth over a - dimensional mountain.j. Meteorol. So. Japan 68: Long RR Some aspets of the flow of stratified fluid. I. A theoretial investigation. Tellus 5: MEwan AD Degeneration of resonantly-exited standing internal gravity waves. J. Fluid Meh. 50: MEwan AD, Robinson RM Parametri instability of internal gravity waves. J. Fluid Meh. 67: MEwan AD, Mander DW, Smith RK Fored resonant seondorder interation between damped internal waves. J. Fluid Meh. 55: MGoldrik LF Resonant interations among apillary-gravity waves. J. Fluid Meh. 1: MHugh J Mean flow generated by an internal wave paket impinging on the interfae between two layers of fluid with ontinuous density.theor. omput. Fluid Dyn. : MHugh J Internal waves at an interfae between two layers of differing stability. J. Atmos. Si. 66: Partl W lear air turbulene at the tropopause levels. Navigation 9: Phillips OM On the dynamis of unsteady gravity waves of finite amplitudes. Part I. The elementary interations. J. Fluid Meh. 9: Shrira VI On the propagation of a three-dimensional paket of weakly non-linear internal gravity waves. Int. J. Nonlin. Meh. 16: Sutherland BR Finite-amplitude internal wavepaket dispersion and breaking. J. Fluid Meh. 49: Sutherland BR Weakly nonlinear internal gravity wavepakets. J. Fluid Meh. 569: Tabaei A, Akylas TR Resonant long short wave interations in an unbounded rotating stratified fluid. Studies Appl. Math. 119: Voronovih AG On the propagation of a paket of weakly nonlinear internal waves in a medium with onstant Väisälä frequeny. Izv. Atmos. Oean Phys. 18: