Semiclassical and spectral analysis of oceanic waves
Christophe Cheverry, Isabelle Gallagher, Thierry Paul, Laure Saint-Raymond
aa r X i v : . [ m a t h . A P ] J un SEMICLASSICAL AND SPECTRAL ANALYSIS OF OCEANIC WAVES
CHRISTOPHE CHEVERRY, ISABELLE GALLAGHER, THIERRY PAUL,AND LAURE SAINT-RAYMOND
Abstract.
In this work we prove that the shallow water flow, subject to strong wind forcingand linearized around an adequate stationary profile, develops for large times closed trajec-tories due to the propagation of Rossby waves, while Poincar´e waves are shown to disperse.The methods used in this paper involve semi-classical analysis and dynamical systems forthe study of Rossby waves, while some refined spectral analysis is required for the study ofPoincar´e waves, due to the large time scale involved which is of diffractive type. Introduction
The problem we consider is motivated by large-scale oceanography: the main physical phe-nomenon leading this study is the existence of persistent oceanic eddies, which are coherentstructures of vortex type, spreading over dozens of kilometers and propagating slowly overperiods from one year to one decade. These structures have been observed long past byphysicists [16, 17, 19, 26, 27] who gave heuristic arguments (reproduced below) to explaintheir formation due both to wind forcing and to convection by a macroscopic zonal current.Giving a (much less precise) mathematical counterpart of those arguments, even at a linearlevel, requires careful multiscale analysis and rather sophisticated tools of semiclassical andmicrolocal analysis. In this paper we simplify the model by considering particular macro-scopic currents, which are stationary solutions of the forced equations. This allows to exhibittrapped Rossby waves, by solving the dynamics associated with an appropriate integrableHamiltonian system. We prove also that the other waves produced by the dynamics, namelyPoincar´e waves, disperse on the same time scales (which turn out to be of diffractive type).1.1.
Physical observations.
Simple observations show that large-scale ocean dynamics can be decomposed as the sum of the solid-body rotation together with the Earth, convectionby macroscopic currents (such as the Gulf Stream in the North Atlantic, the Kuroshio in theNorth Pacific, Equatorial or Circumpolar currents), and motion on smaller geographical zones,due for instance to the fluctuations of the wind and more generally to the coupling with theatmosphere. While the spatial extent of macroscopic currents is of the order of a hundred toa thousand kilometers, those fluctuations are typically on dozens of kilometers. We thereforeexpect eddies to be particular forms of those fluctuations. The point is to understand whythey are quasi-stationary, or in other words why they do not disperse as other waves. At thisstage we have to describe briefly the different kinds of waves that can be found in theocean as linear responses to exterior forcing. They are usually classified into two families,
Mathematics Subject Classification.
Key words and phrases.
Semiclassical analysis; microlocal analysis; integrable systems; Bohrn-Sommerfeldquantization; Geophysical flows. depending on their typical period and on their dynamical structure. The exact dispersionrelation of all these waves can be computed explicitly [4, 10, 13, 14, 24] in simplified cases (noconvection, linear approximation of the Coriolis parameter). • Poincar´e waves , the period of which is of the order of a day, are fast dispersivewaves. They are due to the Coriolis force, that is to the rotation of the Earth ; • Rossby waves propagate much slower, since the departure from geostrophy (thatis equilibrium between pressure and Coriolis force) is very small. They are actuallyrelated to the variations of the Coriolis parameter with latitude. In particular, theypropagate only eastwards.The heuristic argument leading to the existence of quasi-stationary coherent structures isthen as follows (as suggested by physicists): the wind forcing produces waves, in particularRossby waves which would propagate, in the absence of convection, with a speed comparableto the bulk velocity of the fluid ¯ v ∼
10 ms − ; the convection by zonal flow may then stopthe propagation, creating ventilation zones which are not influenced by external dynamics, inparticular by continental recirculation. We are then led to studying wave propagation underthe coupled effects of the pressure, the Coriolis force and zonal convection, that is to studyinga system of linear PDEs with non constant coefficients.1.2. The model.
The system we will consider is actually a toy model insofar as many physicalphenomena are neglected. Our aim here is only to get a qualitative mechanism to explainthe trapping of Rossby waves. More precisely, we consider the ocean as an incompressible,inviscid fluid with free surface submitted to gravitation and wind forcing, and furthermake the following classical assumptions : the density of the fluid is homogeneous ρ = ρ =constant ; the pressure law is given by the hydrostatic approximation p = ρ gz ; the motion isessentially horizontal and does not depend on the vertical coordinate, leading to the so-called shallow water approximation . For the sake of simplicity, we shall not discuss the effectsof the interaction with the boundaries, describing neither the vertical boundary layers, knownas Ekman layers, nor the lateral boundary layers, known as Munk and Stommel layers.We consider a purely horizontal model, and assume an infinite domain for the longitude(omitting the stopping conditions on the continents) as well as for the latitude (this maybe heuristically justified using the exponential decay of the equatorial waves to neglect theboundary). The evolution of the water height h and velocity v is then governed by the Saint-Venant equations with Coriolis force (1.1) ∂ t ( ρ h ) + ∇ · ( ρ hv ) = 0 ∂ t ( ρ hv ) + ∇ · ( ρ hv ⊗ v ) + ω ( ρ hv ) ⊥ + ρ gh ∇ h = ρ hτ where ω denotes the vertical component of the Earth rotation vector Ω, v ⊥ := ( − v , v ), g isthe gravity and τ is the - stationary - forcing responsible for the macroscopic flow. It dependsin particular on time averages of the wind forcing, temperature gradients and topography. Theequations are written in cartesian coordinates ( x , x ), where x corresponds to the longitude,and x to the latitude (both will be chosen in R ). The vertical component of the Earth rotationis therefore Ω sin( x /R ), where R is the radius of the Earth, but it is classical in the physicalliterature to consider the linearization of ω (known as the betaplane approximation) ω ( x ) =Ω x /R ; most of our results will actually hold for more general functions ω , but in somesituations we shall particularize the betaplane case in order to improve on the results. In EMICLASSICAL AND SPECTRAL ANALYSIS OF OCEANIC WAVES 3 order to analyze the influence of the macroscopic convection on the trapping of Rossbywaves, we will consider small fluctuations around the stationary solution h = ¯ h, ∇ · (¯ v ⊗ ¯ v ) + ω ¯ v ⊥ = τ, div ¯ v = 0 , where ¯ h is a constant. Physical observations show that the nonlinear convection term isessentially negligible compared to the Coriolis term, so that the previous equation is nothingelse than the Sverdrup relation (see [27]).1.3. Orders of magnitude and scaling.
Let us introduce the observation length, time andvelocity scales l (of the size of the radius of the Earth R ), t and v , and the nondimensionalvariables ˜ x = x/l , ˜ t = t/t , and u = ( v − ¯ v ) /v . We also define the typical height variation δh and the corresponding dimensionless variable η = ( h − ¯ h ) /δh . We denote by v c the typicalvalue of the velocity of the macroscopic current: ¯ u = (¯ u ,
0) = ¯ v/v c . The length scale l andthe convection velocity v c are fixed by the macroscopic flow: typical values for the Gulf Streamare l ∼ km and v c ∼
10 ms − . As we are interested in structures persisting during manymonths, a relevant choice for the observation time scale is t = 10 s ( ∼ ,
38 months) . Theassociated
Rossby number is then Ro := 1 / ( t | Ω | ) = 0 .
01, recalling that | Ω | = 7 . × − s − .The variations of water height which can be observed are typically of the order δh ∼ h ∼ m . The influence of gravity (through hydrostatic pressure) is measuredby the
Froude number Fr := ( v l ) / ( t gδh ) ∼ . , considering namely fluctuations oforder v ∼ . − . Defining ε := Fr and dropping the tildas (note that as often in Physics, ε is not really a very small number), we therefore end up with the following scaled system(1.2) ∂ t η + 1 ε ∇ · u + ¯ u · ∇ η + ε ∇ · ( ηu ) = 0 ,∂ t u + 1 ε bu ⊥ + 1 ε ∇ η + ¯ u · ∇ u + u · ∇ ¯ u + ε u · ∇ u = 0 , where b := ω/ | Ω | . We shall compute the response to the wind forcing, assuming that the windinduces a pulse at time t = 0: since the wind undergoes oscillations on small spatial scales,the initial data is further assumed to depend both on x and x/ε . Typically(1.3) ( η ε , u ε ) | t =0 = ( η k ( x ) , u k ( x )) exp (cid:18) i k · xε (cid:19) , for some k ∈ Z . More generally we shall consider initial data which are microlocalized (inthe sense of Appendix B) in some compact set of T ∗ R .1.4. Local well-posedness.
The local existence of a solution to the scaled Saint-VenantCoriolis system (1.2) supplemented with initial data in the form (1.3) comes from the generaltheory of hyperbolic quasilinear symmetrizable systems. Defining the sound speed u by η = (cid:2) (1 + ε u / − (cid:3) /ε , we indeed obtain that (1.2) is equivalent to(1.4) ε ∂ t U + A ( x , εD ) U + ε Q ( U ) = 0 , U = ( u , u , u )where A ( x , εD x ) is the linear propagator(1.5) A ( x , εD ) := ε ¯ u · ε ∇ ε∂ ε∂ ε∂ ε ¯ u · ε ∇ − b ( x ) + ε ¯ u ′ ε∂ b ( x ) ε ¯ u · ε ∇ , CH. CHEVERRY, I. GALLAGHER, T. PAUL, AND L. SAINT-RAYMOND and Q ( U ) := S ( U ) ε∂ U + S ( U ) ε∂ U with(1.6) S ( U ) := u u u u
00 0 u and S ( U ) := u u u u u . Because of the specific form of the initial data, involving fast oscillations with respect to x ,we introduce semi-classical Sobolev spaces H sε = { U ∈ L / k U k H sε < + ∞} with k U k H sε = X | k |≤ s k ( ε ∇ ) k U k L . We shall also need in the following to define weighted semi-classical Sobolev spaces (in thespirit of [10]), adapted to the linear propagator as explained in Section 7:(1.7) W sε := n f ∈ L ( R ) / (1 − ε ∂ ) s (1 − ε ∂ + b ( x )) s f ∈ L ( R ) o . A classical result based on the Sobolev embedding (see Section 7 for related results) k ε ∇ U k L ∞ ≤ Cε k∇ U k H sε for any s > , implies that (1.2) has a unique local solution U ε ∈ L ∞ ([0 , T ε ) , H s +1 ε ). Note that the lifespan of U ε depends a priori on ε . One of the goals of this article is to show existence onan ε -independent time interval.2. Main results and strategy of the proofs
Most of this paper is concerned with the analysis of the solution to the linear equation(2.1) ε ∂ t V + A ( x , εD ) V = 0 , V = ( v , v , v )which is expected to dominate the dynamics since we consider small fluctuations. The de-scription of the linear dynamics is provided in Theorem 1 below. The comparison betweenlinear and nonlinear solutions is postponed to the final section of the paper (see Theorem 2).For technical reasons we shall restrict our attention in this paper to the case of a shear flow,in the sense that ¯ u ( x ) = (¯ u ( x ) , , where ¯ u is a smooth, compactly supported function. Weshall further assume for simplicity that the zeros of ¯ u , in the interior of its support, are oforder one . We shall also suppose throughout the paper that b is a smooth function with asymbol-like behaviour:(2.2) ∀ α ∈ N , ∃ C α , ∀ y ∈ R , | b ( α ) ( y ) | ≤ C α (1 + b ( y )) , and we shall further assume that lim y →∞ b ( y ) = ∞ , and that b has at most a finite number ofcritical points (that is to say points where b ′ vanishes). Without such an assumption one couldnot construct Rossby waves. We shall also suppose that the initial data is microlocalized (see Appendix B) in some compact set of T ∗ R (which we shall identify to R in the following),denoted C and satisfying(2.3) C ∩ { ξ = 0 } = ∅ . EMICLASSICAL AND SPECTRAL ANALYSIS OF OCEANIC WAVES 5
Thanks to this assumption, which is propagated by the linear flow, one can diagonalize thesystem into Rossby and Poincar´e modes. Finally in order to avoid pathological trapped Rossbytrajectories we shall also require that(2.4)
C ∩
Σ = ∅ , where Σ is a codimension 1 subset of R defined in Proposition 4.6.2.1. Statement of the main results.
In this paragraph we shall state the two main theo-rems proved in this paper. The first result deals with the linear system (2.1).
Theorem 1 (The linear case) . There is a submanifold Λ of R , invariant under translationsin the x − direction, such that the following properties hold.Let U ε, be ε − microlocalized in a compact set C satisfying Assumptions (2.3)-(2.4). For anyparameter ε > , denote by V ε the associated solution to (2.1). Then for all t ≥ one canwrite V ε ( t ) as the sum of a “Rossby” vector field and a “Poincar´e” vector field: V ε ( t ) = V Rε ( t ) + V Pε ( t ) , satisfying the following properties: (1) There is a compact set K of R such that ∀ t ≥ , k V Rε ( t ) k L ( K ) = O ( ε ∞ ) if and only if the ε -frequency set of V Rε (0) intersects Λ . (2) Suppose that b has only one non degenerate critical value (meaning that ( b ) ′ onlyvanishes at one point, where ( b ) ′′ does not vanish). Then for any compact set Ω in R , one has ∀ t > , k V Pε ( t ) k L (Ω) = O ( ε ∞ ) . In particular supposing that b has only one non degenerate critical value, then there is acompact set K of R such that ∀ t > , k V ε ( t ) k L ( K ) = O ( ε ∞ ) if and only if the ε -frequency set of V Rε (0) intersects Λ . Remark 2.1.
Actually Λ corresponds to the set of initial positions and frequencies in thephase space giving rise to trapped trajectories for the Rossby hamiltonian. This will be mademore precise in Section 4, where we shall prove that under some additional (non restrictive)assumptions on ¯ u , Λ is of codimension one. In particular it will be shown that some of thosetrapped trajectories actually exhibit a singular behaviour in large times, in the sense that theyconverge in physical space towards a point, while the ξ frequency goes to infinity. This couldbe interpreted like the creation of some sort of oceanic eddies.The result (2) is related to dispersive properties of the Poincar´e hamiltonian on diffractivetype times (of the type O (1 /ε ) ), which requires some spectral analysis. Due to the assumptionon b one can write a rather simple proof; more general conditions could be treated, but theBohr Sommerfeld quantization would require to decompose the phase space into various zonesaccording to the geometry of the level sets of the Hamiltonian, which is much more technicaland beyond the scope of this article. Actually in [11] we propose a different approach, basedon Mourre estimates, which allows to relax very much the assumptions on b and on ¯ u . CH. CHEVERRY, I. GALLAGHER, T. PAUL, AND L. SAINT-RAYMOND
The final section of this paper is devoted to the proof of the following theorem, which statesthat the very weak coupling chosen in this paper implies that nonlinear dynamics are governedby the linear equation. We also consider more generally the following weakly nonlinear system(with the notation (1.5) and (1.6)):(2.5) ε ∂ t U + A ( x, εD x ) U + ε η S ( U ) ε∂ U + ε η S ( U ) ε∂ U = 0 , η ≥ . The case η = 0 corresponds of course to the original system (1.4) presented in the introduction. Theorem 2 (The nonlinear case) . Let U ε, be any initial data bounded in W ε . Then thefollowing results hold. (1) The case η = 0 : (a) There exists some T ∗ > such that the initial value problem (2.5) with η = 0 hasa unique solution U ε on [0 , T ∗ [ for any ε > . (b) Assume that the solution V ε to the linear equation (2.1) satisfies k εV ε k L ([0 ,T ∗ [; L ∞ ) → as ε → . Then the solution U ε to (2.5) with η = 0 satisfies k U ε − V ε k L → uniformly on [0 , T ∗ [ as ε → . (2) The case η > : Le T > be fixed. Then there is ε > such that for any ε ≤ ε , theequation (2.5) has a unique solution U ε on [0 , T ] . Moreover, k U ε − V ε k L → uniformly on [0 , T ] as ε → . Remark 2.2.
Result (1b), joint with Theorem 1, implies in particular that as soon as b hasonly one non degenerate critical value, then for positive times the energy of U ε on any fixedcompact subset is carried only by Rossby waves. The refined L ∞ estimate on the linear solutionrequired in result (1b) should be proved by using WKB tools. For the sake of simplicity, weshall not consider such technical estimates here, all the less that we do not expect them to beenough to get an optimal result regarding the nonlinear problem (see Remark 7.1). That is thereason why we consider, in result (2), a weaker coupling still. That result implies in particularthat the L norm of U ε on any fixed compact subset may remain bounded from below only ifthere are trapped Rossby waves, i.e. only if the ε -frequency set of the initial data does intersect Λ (with the notation of Theorem 1). Some related studies.
This work follows a long tradition of mathematical studies offast rotating fluids, following [28] and [18]; we refer for instance to [4] and [14] for a numberof references. The present study concerns the case when the penalization matrix does nothave constant coefficients. A first study in this type of situation may be found in [12], wherea rather general penalization matrix was considered. Due to the generality of the situation,explicit computations were ruled out and no study of waves was carried out. In order tocompute explicitly the modes created by the penalization matrix, various authors (see [8], [9],[10] as well as [13]) studied the betaplane approximation, in which the rotation vector dependslinearly on the latitude. In that case explicit calculations may again be carried out (or someexplicit commuting vector fields may be computed) and hence again one may derive envelopeequations. In this paper we choose again to work with a more general rotation vector, thischoice being made possible by a semi-classical setting (see the next paragraph); in particularthat setting enables us diagonalize the system approximately, therefore to compute waves;note that a related study is performed by two of the authors in [5] via a purely geometric
EMICLASSICAL AND SPECTRAL ANALYSIS OF OCEANIC WAVES 7 optics approach, where no explicit diagonalization is performed (actually the initial data isstrongly polarized so that only Rossby modes are present, including in the non linear setting).Another feature of our study is that it is a multi-scale problem, in the sense that the oscillationfrequency is much bigger than the variation of the coefficients of the system. This is dealtwith by using semi-classical analysis (which, compared to the previous paragraph, enablesus to compute almost commuting vector fields although the penalization matrix no longerdepends only linearly on the latitude). Such techniques are classical in geometrical optics,but in our case the additional difficulty is that the propagators are linked to different timescales: this is due to the fact that the system has eigenvalues at different scales (one isactually a subsymbol). In particular we are mostly interested in the role of the subsymbolin the dynamics, as this subsymbol is responsible for the trapping phenomenon we wantto exhibit: this implies, by semi-classical analysis, the need to study the dynamical systeminduced by that subsymbol. On the other hand this also means that the dynamics linked toother eigenvalues must be analyzed on diffractive-type time scales, therefore much longer thanthat allowed by semi-classical analysis. We are able to show the dispersion of those waves byusing spectral analysis and Bohr-Sommerfeld quantization.2.3.
Organization of the paper.
The proof of Theorems 1 and 2 requires a number of stepswhich are described in this paragraph.2.3.1.
Reduction to scalar propagators.
Persistent structures are related to the propagationof Rossby waves. Our first task is therefore to transform the original linear system (2.1) intothree scalar equations. One is polarized on Rossby waves while the two others are polarizedon Poincar´e waves. This is done in Section 3 by proving some necessary conditions for theexistence of those propagators. The general strategy is the following:(1) Consider the system A ( x , εD ) U = iτ U . Take the Fourier transform in x , which ispossible since the equation is translation invariant in x . Then extract from this system(by linear combinations and substitutions) a linear equation on one component u k of U ,of the type h ( x , εD ; ξ , ε, τ ) u k = 0.(2) The symbolic equation corresponding to the PDE writes h ( x , ξ ; ξ , ε, τ ) = 0. It hasthree roots (with respect to τ ) , τ ± ( x , ξ ; ε ) (Poincar´e roots) and τ R ( x , ξ ; ε ) (Rossbyroot). We find τ ± ( x , ξ ; ε ) = τ ± ( x , ξ ) + O ( ε ) and τ R ( x , ξ ; ε ) = ε e τ R ( x , ξ ) + O ( ε ).(3) Those roots are not necessarily symbols. To guarantee these are indeed symbols oneneeds a microlocalization. Given a compact set and a truncation on that compactset χ , one can construct three operators T χj (via a general theorem, stated and provedin an abstract way in Theorem 3 of Appendix A) whose principal symbols are preciselythe Rossby and Poincar´e symbols τ ± and τ R .2.3.2. Trapping of Rossby waves.
Section 4 is devoted to the study of the Rossby propaga-tor T , and in particular to the proof of result (1) in Theorem 1. For the time scale considered,it is easy to see that the energy propagates according to the trajectories of the semiclassicalRossby hamiltonian e τ R . The first step of the analysis therefore consists in studying the dy-namical system giving rise to those trajectories. It turns out that the trajectories are alwaysbounded in the x direction. One is therefore reduced to studying the trajectories in the x variable and in identifying the set Λ of initial data in the cotangent space giving rise to trap-ping in x . One then checks that Λ is of codimension one under some additional assumptionson ¯ u , and the last step of the study consists in studying more precisely the trajectories insome specific situations, in particular in the case of the betaplane approximation.2.3.3. Dispersion of Poincar´e waves.
The next step of our analysis of wave propagation con-sists in proving, in Section 5, that Poincar´e waves propagate so fast that they exit from any(bounded) domain of observation on the time scale that we consider, which proves result (2)in Theorem 1. Note that, because of the very long time scaling, usual tools of semiclassicalanalysis cannot be applied for the Poincar´e waves : we actually need deeper arguments suchas the Bohr-Sommerfeld quantization to conclude.2.3.4.
A diagonalization result.
Once the Rossby and Poincar´e propagators have been well un-derstood, we can retrace the steps followed in Section 3 to prove that the necessary conditionson the scalar propagators are sufficient. The difficulty is that the operators Π enabling one togo from the original system to the scalar equations and back (computing an approximate leftinverse Q of Π at the order O ( ε ∞ )) are only continuous on microlocalized functions; moreoverthe scalar propagators T χj are themselves only defined on microlocalized functions. So thatrequires understanding the persistence of the microlocalization of the solutions to the scalarequations. That is achieved in the two previous sections, where it is proved that if the initialdata is conveniently microlocalized, then for any time t ≥ K and one can construct T χj as in Section 3, so that the solution to the scalar equations withpropagators T χj is microlocalized in K (actually for Poincar´e modes the microlocalization is inthe variables ( x , ξ , ξ ) only, which is enough for our purpose). This enables us in Section 6to conclude rather easily by computing explicitly the matrix principal symbols of Π and Q .2.3.5. The analysis of the nonlinear equation.
Section 7 is devoted to the proof that thesolution of the nonlinear equation remains close to that of the linear equation. The method ofproof consists first in proving the wellposedness of the nonlinear equation on a uniform timeinterval, by using semi-classical weighted Sobolev type spaces, whose additional feature is to bewell adapted to the penalization operator A ( x , εD ): one therefore constructs a matrix-valuedpseudo-differential operator which approximately commutes with A ( x , εD ). The convergenceof U ε − V ε to zero relies on a standard L energy estimate on U ε − V ε , and Gronwall’s lemma.2.3.6. Two appendixes.
In Appendix A one can find the statement and the proof of a generaltheorem, used in Section 3, allowing to associate to a linear evolution PDE a number of opera-tors describing the dynamics of the equation; those operators are constructed by writing downthe symbolic equation associated to the PDE and in quantizing the roots of that polynomial(in the time derivative). Appendix B finally collects a number of prerequisites on microlocaland semiclassical analysis, that are used throughout the paper.
EMICLASSICAL AND SPECTRAL ANALYSIS OF OCEANIC WAVES 9 Reduction to scalar propagators
Introduction.
Let us first recall that the propagator A ( x, εD ) = ε ¯ u · ε ∇ ε∂ ε∂ ε∂ ε ¯ u · ε ∇ − b + ε ¯ u ′ ε∂ b ε ¯ u · ε ∇ can, in the particular case when ¯ u ≡ b ( x ) = βx , be diagonalized without any errorterm (in particular for any finite ε ), using a Fourier basis (exp( iε x ξ )) ξ ∈ R in x and a Hermitebasis ( ψ εn ( x )) n ∈ N in x . Precisely, the following statement is proved in [13]. Proposition 3.1 (Gallagher & Saint-Raymond,[13]) . For all ( ξ , n, j ) ∈ R × N × {− , , } ,denote by τ ( ξ , n, j ) the three roots (in increasing order in j ) of (3.1) τ − ( ξ + βε (2 n + 1)) τ + εβξ = 0 . Then there exists a complete family of L ( R × R , R ) of pseudo-eigenvectors (Ψ εξ ,n,j ) of theoperator A ( x, εD ) (where ¯ u ≡ and b ( x ) = βx ): (3.2) ∀ ( ξ , n, j ) ∈ R × N × {− , , } , A ( x, εD )Ψ εξ ,n,j = iτ ( ξ , n, j )Ψ εξ ,n,j where Ψ εξ ,n,j can be computed in terms of the n -th Hermite function ψ εn ( x ) and its derivatives. In other words, the three scalar propagators (numbered by j ) can be obtained from thesymbolic equation (3.1) remarking that βε (2 n + 1) is the quantization of the harmonic oscil-lator − ε ∂ + β x . It is proved in [13] that as ξ and n go to infinity τ ( ξ , n, ± ) ∼ ± q ξ + βε (2 n + 1) , and τ ( ξ , n, ∼ εβξ ξ + βε (2 n + 1) · We are interested here in deriving a symbolic equation similar to (3.1) for a general zonalcurrent ¯ u = (¯ u ( x ) ,
0) and Coriolis parameter b = b ( x ). The difficulty comes from the factthat the propagation of waves is governed by a matrix of differential operators with non-constant coefficients, the diagonalization of which is not a standard computation. Of course,in the semiclassical limit ε →
0, we expect to get a good approximation of the propagation atleading order by considering the matrix of principal symbols(3.3) A ( x, ξ ) := iξ iξ iξ − b ( x ) iξ b ( x ) 0 and by computing the scalar propagators associated to each eigenvalue(3.4) τ ± ( x , ξ , ξ ) := ± q ξ + ξ + b ( x )and 0. The eigenvalue 0 corresponds to the Rossby modes, whereas the two O (1) eigenval-ues ± p ξ + ξ + b ( x ) are the Poincar´e modes. Nevertheless, this approximation is relevantonly for times of order O ( ε ), and we are interested here in much longer times, of order O ( ε ).This means that we need to compute the next order of the expansion of the eigenvalue 0. Oncethat is done, we need to quantify these symbol eigenvalues to deduce scalar propagators. The Rossby modes.
Finding precisely the Rossby modes (up to an O ( ε ) error) requiresmore intricate calculations than merely diagonalizing the matrix of principal symbols A ( x, ξ )given in (3.3). So let iτ be an eigenvalue of the propagator, assumed to be of the form τ = ε e τ R + O ( ε ). Explicit computations lead to the subsystem (cid:18) ε ¯ u ξ − τ ξ ξ ε ¯ u ξ − τ (cid:19) (cid:18) ρu (cid:19) = (cid:18) iε∂ u − i ( b − ε ¯ u ′ ) u (cid:19) and defining α τ ( x , ξ ) := ε ¯ u ξ − τ and p R ( x , ξ ) = − ξ + α τ to the scalar equation(3.5) (cid:0) ε∂ p − R (cid:0) iα τ ε∂ + iξ ( b − ε ¯ u ′ ) (cid:1) − bp − R ( iξ ε∂ + iα τ ( b − ε ¯ u ′ )) + iα τ (cid:1) u = 0where ξ is the Fourier variable corresponding to x /ε . From now on we assume that ξ isfixed, and is bounded away from zero (recalling Assumption (2.3)). Note that equation (3.5)makes sense because we are assuming here that τ = ε e τ R + O ( ε ), so p − R is well defined. Thatwould not be the case for the Poincar´e modes (where τ = ± p | ξ | + b ( x ) + O ( ε )) so we shalluse another subsystem in the next paragraph to deal with the Poincar´e operators.In order to derive a symbolic equation associated with the differential equation (3.5), we shallproceed by transforming (3.5) into a differential equation which is the left quantization (in thesense recalled in Appendix B) of a symbol, polynomial in ε and τ . This leads to a differentialequation of the type(3.6) P ( x ; ε, τ )( ε∂ ) u + P ( x ; ε, τ ) ε∂ u + P ( x ; ε, τ ) u = 0 , where each P j ( x ; ε, τ ) is a smooth function in x , and has polynomial dependence in ε andin τ (precisely of degree at most 5 in ε and τ ). This generalizes (3.1); one can compute inparticular, using the fact that ε∂ α τ = O ( ε ), that P ( x ; ε, τ ) = ip R α τ + O ( ε ) , P ( x ; ε, τ ) = O ( ε ) , and P ( x ; ε, τ ) = ip R (cid:0) ξ εb ′ − ( b + ξ ) α τ (cid:1) + O ( ε ) . The differential operator appearing on the left-hand side of (3.6) is the left quantization ofthe following symbol:(3.7) h ( x , ξ ; ε, τ ) := −P ( x ; ε, τ ) ξ + i P ( x ; ε, τ ) ξ + P ( x ; ε, τ )which belongs for each τ to S ( g τ ) for some function g τ of the type g τ ( x , ξ ) = (1 + τ ) (cid:0) b ( x ) + ξ (cid:1) recalling that ¯ u and ¯ u ′ are bounded from above, as well as Assumption (2.2).Now we recall that h ( x , ξ ; ε, τ ) is a polynomial of degree 5 in τ hence has five roots, amongwhich 2 are actually spurious: these are of the form ± ξ + O ( ε ), and they appear because wehave multiplied the equation by a polynomial in τ which cancels at the point ± ξ at first orderin ε . An easy computation allows to obtain the two other O (1) roots, which are precisely thePoincar´e roots ± p | ξ | + b ( x ) + O ( ε ), and an asymptotic expansion allows also easily toderive the Rossby O ( ε ) root: one finds τ R ( x , ξ , ξ ; ε ) := ε e τ R ( x , ξ , ξ ; ε ) + O ( ε ) , where e τ R ( x , ξ , ξ ) := b ′ ( x ) ξ ξ + ξ + b ( x ) + ξ ¯ u ( x ) . (3.8)Now that the root τ R has been computed, our next task is to prove the existence of the Rossbypropagator T R = ε e T R whose principal symbol is precisely ε e τ R . Actually this result is a direct EMICLASSICAL AND SPECTRAL ANALYSIS OF OCEANIC WAVES 11 consequence of Theorem 3 stated and proved in Appendix A: with the notation of Theorem 3,one has ν = 1 and ∂ τ h ( x, ξ,
0) = − iξ ( ξ + ξ + b ( x )) . In the following statement, the timevariable s is defined as s = t/ε . Proposition 3.2 (The Rossby propagator) . Let e τ R be the symbol defined in (3.8). Thenfor any compact set K satisfying Assumptions (2.3,2.4) there exists a formally self-adjointpseudo-differential operator e T R of principal symbol e τ R such that if ϕ R is microlocalized in K and solves (3.9) ∂ s ϕ R = iε e T R ϕ R , then P ( x ; ε, ∂ s )( ε∂ ) ϕ R + P ( x ; ε, ∂ s )( ε∂ ) ϕ R + P ( x ; ε, ∂ s ) ϕ R = O ( ε ∞ ) . Definition 3.3 (The Rossby operator) . We shall call Π R the Rossby operator defined by Π R := P − R (cid:0) i ( − iε ¯ u ∂ − ε e T R ) ε∂ + ε∂ ( b − ε ¯ u ′ ) (cid:1) − P − R (cid:0) − ε ∂ + i ( − iε ¯ u ∂ − ε e T R )( b − ε ¯ u ′ ) (cid:1) Id , where P R := ε ∂ − ( iε ¯ u ∂ + ε e T R ) . Remark 3.4. (1)
Notice that Π R is well defined since the principal symbol of P R isbounded from below (see Appendix B). (2) Proposition 4.6 shows that if ϕ R | t =0 is microlocalized in a compact set K satisfy-ing (2.4), then the solution to (3.9) is microlocalized for all t ≥ in a compact set K t . (3) The above computations allow to formally recover the original shallow-water equation,up to O ( ε ∞ ) . Indeed retracing the steps which enabled us above to derive equation (3.6)shows that if ϕ is a smooth function conveniently microlocalized, and if ϕ solves ε∂ t ϕ = i e T R ϕ, ϕ | t =0 = ϕ , then the vector field U := Π R ϕ satisfies (2.1) up to O ( ε ∞ ) . This property will be maderigorous in Section 6. The Poincar´e modes.
In this paragraph we shall follow the method used above inthe case of Rossby modes to infer Poincar´e propagators T ± and operators Π ± . Actuallyone cannot use precisely the same method since the symbol ( ε ¯ u ξ − τ ) − ξ may vanishwhen τ = τ ± + O ( ε ). So we shall instead consider the subsystem (cid:18) ε ¯ u ξ − τ − iε∂ − iε∂ ε ¯ u ξ − τ (cid:19) (cid:18) ρu (cid:19) = (cid:18) − ξ u ibu (cid:19) and the scalar equation(3.10) (cid:16) ξ p − P ( − α τ ξ − ε∂ b ) + α τ + ( b − ε ¯ u ′ ) p − P ( ε∂ − α τ b ) (cid:17) u = 0 , where as before α τ ( x , ξ ) := ε ¯ u ξ − τ , and p P ( x , ξ ) := ε ∂ + α τ . We notice here that α − τ is well defined when τ = τ ± since τ ± is bounded away from zero by the assumption on ξ ,and the same goes for τ ± − ξ so p − P is also well defined. Then it remains to follow the stepsof Paragraph 3.2 to obtain a new scalar PDE of the same type as (3.6), as well as a symbolequation of the type (3.7):(3.11) ˜ h ( x , ξ ; ε, τ ) := − ˜ P ( x ; ε, τ ) ξ + i ˜ P ( x ; ε, τ ) ξ + ˜ P ( x ; ε, τ ) . Of course τ ± are roots of that equation up to O ( ε ), and ετ R is a root up to O ( ε ). Then theapplication of Theorem 3 implies a similar result to Proposition 3.2, noticing that with thenotation of Theorem 3, ν = 0 and ∂ τ h ( x, ξ, τ ± ) = − ξ + ξ + b ( x )) . Proposition 3.5 (The Poincar´e propagator) . Let τ ± be the symbol defined in (3.4). Considera compact set K P ⊂ R ∗ × T ∗ R . Then there exists formally self-adjoint pseudo-differentialoperators T ± of principal symbols τ ± such that if ϕ ± solves ∂ s ϕ ± = iT ± ϕ ± , and ϕ ± is mi-crolocalized in R × K P , then ˜ P ( x ; ε, ∂ s )( ε∂ ) ϕ ± + ˜ P ( x ; ε, ∂ s )( ε∂ ) ϕ ± + ˜ P ( x ; ε, ∂ s ) ϕ ± = O ( ε ∞ ) . Definition 3.6 (The Poincar´e operator) . We shall call Π ± the Poincar´e operator defined by Π ± := P − ± ( i ( − iε ¯ u ∂ − T ± ) ε∂ + ε∂ b )Id P − ± ( i ( − iε ¯ u ∂ − T ± ) b − ε∂ ε∂ ) , where P ± := ( iε ¯ u ∂ + T ± ) + ε ∂ . Remark 3.7.
As in Remark 3.4, one sees formally that if ϕ is a smooth function convenientlymicrolocalized, and if ϕ solves ε ∂ t ϕ = iT ± ϕ, ϕ | t =0 = ϕ , then the vector field U := Π ± ϕ satisfies (2.1) up to O ( ε ∞ ) . This property will be made rigorousin Section 6. Study of the Rossby waves
The dynamical system.
For the time scale considered here, the propagation of energyby Rossby waves is given by the transport equation (see Appendix B) ∂ t f + { e τ R , f } = 0where e τ R is the principal symbol of the Rossby mode computed in (3.8):(4.1) e τ R ( ξ , x , ξ ) = b ′ ( x ) ξ ξ + ξ + b ( x ) + ¯ u ( x ) ξ . As e τ R is a smooth function of ( x , ξ , ξ ), the energy is propagated along the bicharacteristics,i.e. along the integral curves of the following system of ODEs: (cid:26) ˙ x t = ∇ ξ e τ R ( ξ t , x t , ξ t ) , x = ( x , x )˙ ξ t = −∇ x e τ R ( ξ t , x t , ξ t ) , ξ = ( ξ , ξ ) . Since the condition (2.3) avoids the set { ξ = 0 } , we can suppose that ξ = 0. Moreoversince e τ R does not depend on x , we find ξ t ≡ ξ . The ODE to be studied is therefore(4.2) ˙ x t = ¯ u ( x t ) + b ′ ( x t ) ( − ξ + ξ t + b ( x t ))( ξ + ξ t + b ( x t )) ˙ x t = − b ′ ( x t ) ξ ξ t ( ξ + ξ t + b ( x t )) ˙ ξ t = − ¯ u ′ ( x t ) ξ + 2 b ( x t ) b ′ ( x t ) ξ ( ξ + ξ t + b ( x t )) − b ′′ ( x t ) ξ ξ + ξ t + b ( x t ) · Due to the assumptions on ¯ u and on b , the map ( x, ξ ) ( ∇ ξ e τ R , −∇ x e τ R )( x, ξ ) is bounded,so the integral curves are globally defined in time. The strategy to study their qualitative EMICLASSICAL AND SPECTRAL ANALYSIS OF OCEANIC WAVES 13 behaviours is to first (in Section 4.2) consider the motion in the reduced phase space ( x , ξ ) ∈ R and then (in Section 4.3) to study the motion in the x direction.4.2. Trajectories in the reduced ( x , ξ ) phase space. In this section we study the tra-jectories in the reduced ( x , ξ ) phase space. We shall denote ξ := ξ .4.2.1. Energy surfaces.
Since the Hamiltonian e τ R and ξ are conserved along any trajectory,trajectories are submanifolds of E τ,ξ := (cid:8) ( x , ξ ) ∈ R ; e τ R ( ξ , x , ξ ) = τ (cid:9) . In the following we shall note for any energy τ and any ξ ∈ R ∗ V τ,ξ ( x ) := b ′ ( x ) ξ τ − ¯ u ( x ) ξ − ξ − b ( x ) , so that if D := (cid:8) x (cid:14) V τ,ξ ( x ) ≥ (cid:9) , then E τ,ξ = n(cid:0) x , ± q V τ,ξ ( x ) (cid:1) , x ∈ D o . Notethat V τ,ξ ( x t ) becomes singular if x t reaches a point x such that τ = ¯ u ( x ) ξ . Proposition 4.1.
The projection of E τ,ξ on the x -axis is bounded.Proof. We recall that on E τ,ξ we have b ′ ( x ) ξ ξ + ξ + b ( x ) + ¯ u ( x ) ξ = τ. Suppose the trajectory in x is not bounded, then in particular it escapes the support of ¯ u .In the case when τ = 0, letting | x | go to infinity yields a contradiction due to the assumptionson b . In the case τ = 0, for x out of the support of ¯ u we have b ′ ( x ) ξ ξ + ξ + b ( x ) = 0and the only possibility is for x to be fixed on a zero point of b ′ (hence in particular does notgo to infinity). (cid:3) Proposition 4.1 shows that to prove that some trajectories are trapped in physical space, itsuffices to study their behaviour in the x direction. However before doing so, let us preparethat study by classifying the trajectories in the reduced phase space. Up to a change ofparameter, namely expressing time t as a function of x dt = ± b ′ ( x ) ξ ( τ − ξ ¯ u ( x )) p V τ,ξ ( x ) dx , which is justified locally, and will give the convenient global behaviour by suitable gluing, weare brought back to the study of the hamiltonian system ξ − V τ,ξ ( x ) describing the motionof a particle in the potential − V τ,ξ .For smooth potentials V such that V ( x ) → −∞ as | x | → ∞ , the possible behaviours ofsuch a system are well-known and the trajectories are usually classified as follows (see forinstance [1, 2, 3, 20, 22]): periodic orbits, fixed points, homoclinic and heteroclinic orbitsconnecting unstable fixed points. Here the situation is more complex insofar as V τ,ξ admitssingularities. We shall classify the trajectories according to their motion in the x variable. Periodic trajectories.
These correspond to the case when there exists [ x min , x max ] in R with x min = x max , containing x such that • V τ,ξ has no singularity and does not vanish on ] x min , x max [; • V τ,ξ ( x min ) = V τ,ξ ( x max ) = 0; • the points x min and x max are reached in finite time.The extremal points x min and x max are then turning points, meaning that the motion isperiodic. The fact that x min and x max are reached in finite time is equivalent to V ′ τ,ξ ( x min ) = 0and V ′ τ,ξ ( x max ) = 0. Indeed if V ′ τ,ξ ( x max ) = 0 (resp. V ′ τ,ξ ( x min ) = 0), then ( x max ,
0) (resp( x min , V ′ τ,ξ ( x max ) = 0 (resp. V ′ τ,ξ ( x min ) = 0), an asymptotic expansionin the vicinity of x max (resp. x min ) shows that the extremal point is reached in finite time. Definition 4.2.
We will denote by P the subset of the phase space T ∗ R consisting of initialdata corresponding to periodic motions along x . A rather simple continuity argument allows to prove that P is an open subset of R × R ∗ × R .Denote indeed by (˜ x , ˜ ξ , ˜ x , ˜ ξ ) any point of P , and by ˜ x min and ˜ x max the extremal pointsof the corresponding (periodic) trajectory along x . As V ′ ˜ τ, ˜ ξ (˜ x max ) = 0 and V τ,ξ is a smoothfunction of τ and ξ outside from the closed subset of singularity points, the implicit functiontheorem gives the existence of a neighborhood of ( ˜ ξ , ˜ τ ) such that there exists a unique x max which satisfies V τ,ξ ( x max ) = 0. Furthermore x max depends continuously on τ and ξ , inparticular V ′ τ,ξ ( x max ) = 0 . Using the same arguments to build a suitable x min , we finallyobtain that there exists a neighborhood of (˜ x , ˜ ξ , ˜ x , ˜ ξ ) for which the motion along x is anon degenerate periodic motion. Moreover, x min , x max and also ξ min , ξ max and the period T depend continuously on the initial data. Fixed points correspond to the degenerate case when x min = x max = x , which impliesthat either ξ = 0 or b ′ ( x ) = 0. The latter case is completely characterized by the condi-tion b ′ ( x ) = 0, so let us focus on the case when b ′ ( x ) = 0. Fixed points correspond then tolocal extrema of V τ,ξ . They can be either stable or unstable depending on the sign of V ′′ τ,ξ .Stable fixed points are obtained as a limit of periodic orbits when the period T →
0, whereasunstable fixed points are obtained in the limit T → ∞ as explained below. Stopping trajectories belong to the same energy surfaces as unstable fixed points and reachsome unstable fixed point in infinite time : they correspond to the case when there exists aninterval [ x min , x max ] of R containing x such that • V τ,ξ has no singularity and does not vanish on ] x min , x max [; • x min and x max are either zeros or singularities of V τ,ξ ; • as t → ∞ , x t → x ∞ such that V ′ τ,ξ ( x ∞ ) = 0 or b ′ ( x ∞ ) = 0.They are in some sense also a degenerate version of periodic trajectories since for arbitrarilyclose initial data, one can obtain periodic orbits. Definition 4.3.
We will denote by δ P the subset of the phase space T ∗ R consisting in initialdata corresponding to fixed points and stopping motions along x . EMICLASSICAL AND SPECTRAL ANALYSIS OF OCEANIC WAVES 15
Using the characterization of the energy surfaces which carry such pathological motions, wecan prove that δ P is a codimension 1 subset of the phase space. We first consider the energysurfaces containing a fixed point ˜ x such that b ′ (˜ x ) = 0. We then have τ = ξ ¯ u (˜ x ). Thecorresponding set of initial data { ( x , ξ , x , ξ ) / e τ R ( ξ , x , ξ ) = ξ ¯ u (˜ x ) } is of codimension 1.As we have assumed that b has only a finite number of critical points, the union of thesesets is still of codimension 1. We then consider the energy surfaces containing a fixed point(˜ x , ˜ ξ , ˜ x ,
0) such that V ˜ τ, ˜ ξ (˜ x ) = V ′ ˜ τ, ˜ ξ (˜ x ) = 0. We then have − ¯ u ′ (˜ x ) + 2 b (˜ x ) b ′ (˜ x )( ˜ ξ + b (˜ x )) − b ′′ (˜ x )˜ ξ + b (˜ x ) = 0 . For each ˜ x , there are at most two values of ˜ ξ such that the previous quantity vanishes. Wetherefore deduce that n ( x , ξ , x , ξ ) / e τ R ( ξ , x , ξ ) = b ′ (˜ x ) ξ ξ + b (˜ x ) + ¯ u (˜ x ) ξ and − ¯ u ′ (˜ x ) + 2 b (˜ x ) b ′ (˜ x )( ˜ ξ + b (˜ x )) − b ′′ (˜ x )˜ ξ + b (˜ x ) = 0 o is of codimension 1.It consists indeed of at most eight manifolds, each one of them parametrized by the realparameter ˜ x . In the sequel, we shall avoid these pathological motions assuming that theinitial data is microlocalized outside δ P .4.2.3. Asymptotic trajectories.
These correspond to the case when there exists an inter-val [ x min , x max ] of R containing x such that • V τ,ξ has no singularity and does not vanish on ] x min , x max [; • x min and x max are either zeros or singularities of V τ,ξ ; • as t → ∞ , x t → x ∞ where x ∞ ∈ { x min , x max } is a pole of multiplicity 1 of V τ,ξ . Forthe sake of simplicity, we further impose that b ′ ( x ∞ ) = 0.This situation therefore corresponds to a motion which is not periodic.Depending on the sign of ξ , an asymptotic trajectory will either encounter a turning pointand then converge asymptotically to the singular point, or converge monotonically to thelimiting point. As x ∞ is such that ¯ u ( x ∞ ) = τ /ξ , one has • either τ = 0 and x ∞ belongs to the support of ¯ u , • or τ = 0 and ¯ u ( x ) = − b ′ ( x ) / ( ξ + ξ + b ( x )) = 0, meaning that x belongs tothe support of ¯ u . Therefore, x ∞ is either min { y > x / ¯ u ( y ) = 0 } or max { y We will denote by A the subset of the phase space consisting of initial datacorresponding to asymptotic motions along x . The same kind of arguments as in the previous paragraph allow to prove that A is an opensubset of the phase space R × R ∗ × R . Consider indeed some (˜ x , ˜ ξ , ˜ x , ˜ ξ ) ∈ A , and thecorresponding asymptotic point ˜ x ∞ . As x ∞ is a pole of multiplicity 1 of V ˜ τ, ˜ ξ , the implicit function theorem shows that V − τ,ξ admits locally a unique zero, which depends continuouslyon τ and ξ . Using further the continuity of the possible turning point, we get that A containsa neighborhood of (˜ x , ˜ ξ , ˜ x , ˜ ξ ). Moreover, we can obtain bounds on the expansion (withrespect to time) of any compact subset of A . Here we will focus on the growth of ξ t , andproves that it depends continuously on the initial data in A . Without loss of generality, wecan consider the case when the asymptotic point is x max . Then we recall thatlim t → + ∞ ξ t = ∞ and lim t → + ∞ x t = x ∞ , with ξ ¯ u ( x ∞ ) = τ. As x tends to x ∞ , we have (recalling that b ′ ( x ∞ ) = 0) V τ ( x ) ∼ − b ′ ( x ∞ )¯ u ′ ( x ∞ ) ( x − x ∞ ) − . This implies that | ξ t | ∼ − b ′ ( x ∞ )¯ u ′ ( x ∞ ) ( x − x ∞ ) − and ˙ x t ∼ − b ′ ( x ∞ ) ξ ξ t | ξ t | ∼ | ¯ u ′ ( x ∞ ) | / | ξ || b ′ ( x ∞ | / ( x ∞ − x ) / . By integration, we get x t ∼ x ∞ + C t − , ξ t ∼ C t where C and C depend continuously on x ∞ , and consequently on the initial data. Singular trajectories are a degenerate version of the asymptotic trajectories above: they cor-respond to the case when there exists an interval [ x min , x max ] of R containing x such that • V τ,ξ has no singularity and does not vanish on ] x min , x max [; • x min and x max are either zeros or singularities of V τ ; • as t → ∞ , x t → x ∞ where x ∞ ∈ { x min , x max } is either a singularity of order greaterthan 1 of V τ,ξ , or a singularity which is also a zero of b ′ .As previously, it is easy to check that x ∞ is necessarily in the support of ¯ u . Furthermore,we have ¯ u ′ ( x ∞ ) = 0 . Indeed, in the vicinity of x ∞ , we have V τ,ξ ( y ) ∼ b ′ ( x ∞ ) ξ ¯ u ( x ∞ ) − ¯ u ( y )which has a singularity of order greater than 1, or a singularity which is also a zero of b ′ ifand only if ¯ u ′ ( x ∞ ) = 0 . Definition 4.5. We will denote by δ A the subset of the phase space consisting of initial datacorresponding to singular motions along x . The previous condition ¯ u ′ ( x ∞ ) = 0 shows that δ A is included in the union of energy surfaces E τ,ξ with τ = ¯ u ( y ) for some y ∈ supp (¯ u ) such that ¯ u ′ ( y ) = 0 . As ¯ u ′ has only a finite number of zeros in the support of ¯ u , this implies that δ A is acodimension 1 subset of the phase space.Gathering all the previous results together, we obtain the following EMICLASSICAL AND SPECTRAL ANALYSIS OF OCEANIC WAVES 17 Proposition 4.6. The phase space R × R ∗ × R admits the following partition R × R ∗ × R = P ∪ A ∪ Σ where P and A are the open sets of initial data giving rise respectively to periodic motionsand asymptotic motions along x , and Σ := δ P ∪ δ A is the codimension 1 set of initial datagiving rise to pathological motions along x .For any compact set K ⊂ A ∪ P , and for any time T > , we further have a uniform boundon the image of K by the flow up to time T .Proof. The first statement just tells us that all trajectories belong to one of the four categoriesdescribed above. Indeed, for any initial data ( x , ξ , x , ξ ), one has V τ,ξ ( x ) = ( ξ ) ≥ , and V τ,ξ ( x ) → −∞ as | x | → ∞ , so that there exists an interval [ x min , x max ] of R containing x such that V τ,ξ has no singularityand does not vanish on ] x min , x max [, and x min and x max are either zeros or singularitiesof V τ,ξ . The second statement is then a simple corollary of the continuity results establishedon P and A . (cid:3) Analysis of the trajectories in the x direction: trapping phenomenon. Propo-sition 4.1 states that the trajectories are always bounded in the x variable, so it remains tostudy the x variable. Our aim is to find a set Λ ⊂ R × R ∗ × R such that any initialdata ( x , ξ , x , ξ ) in Λ gives rise to a trapped trajectory, meaning that Z t ˙ x s ds is uniformly bounded for t ∈ R + . The criterion of capture. Let us prove the following result. Proposition 4.7. A necessary and sufficient condition for a trajectory with initial data in A∪P to be trapped is lim t → T t Z t ˙ x s ds = 0 , where T denotes the (finite) period of the motion along x in the periodic case, and T = + ∞ in the asymptotic case.Proof. We will study separately the different situations described in the previous section,namely the case of periodic and asymptotic trajectories in ( x , ξ ). • In the case of a periodic motion in ( x , ξ ) of period T > 0, the function ˙ x t is also periodic,with the same period. Writing x t = x + Z t (cid:18) ˙ x s − T Z T ˙ x s ′ ds ′ (cid:19) ds + tT Z T ˙ x s ds we see that depending on the average of ˙ x t over [0 , T ], x t is either a periodic function, orthe sum of a periodic function and a linear function. It follows that trapped trajectoriesare characterized by the criterion R T ˙ x t dt = 0. Note that, depending on the period T , thetrajectory can explore a domain in x the size of which may be very large. Nevertheless the continuity statement in Proposition 4.6 shows that we have a uniform bound on this size onany compact subset of P . • For asymptotic motions, we need to check that˙ x t − ¯ u ( x ∞ ) t is integrable at infinity.We have indeed ˙ x t = ¯ u ( x t ) + b ′ ( x t )( − ξ + ξ t + b ( x t ))( ξ + ξ t + b ( x t )) , which, together with the asymptotic expansions of x t and ξ t obtained in the previous section,implies that ˙ x t = ¯ u ( x ∞ ) + O ( t − ) . It is then clear that the trajectory is trapped if and only if¯ u ( x ∞ ) = lim t →∞ t Z t ˙ x s ds = 0 . Using again the continuity statement in Proposition 4.6, we also get a uniform bound on thesize of the time evolution of any compact subset of A . (cid:3) Remark 4.8. Note that, in the case of a singular asymptotic motion along x , the criterionof capture is equivalent to ¯ u ( x ∞ ) = τ = 0 . In the case of periodic motions along x , thiscriterion - even more complicated - can also be expressed in terms of the initial data. Themap x t is indeed a smooth bijection from a time interval ] t , t + T [ to ] x min , x max [ , so Z T ˙ x t dt = 2 Z t + T/ t ˙ x t dt = 2 Z x max x min ˙ x t | x t = x ( ˙ x t ) − | x t = x dx . The trajectory is therefore trapped if and only if (4.3) Z x max x min h ξ − ( τ /ξ ) b ′ ( y )2 (cid:0) ( τ /ξ ) − ¯ u ( y ) (cid:1) i V τ,ξ ( y ) − / dy = 0 (notice that since we are only looking for a criterion for the function to vanish, we can re-place ξ by p V τ,ξ without discussing the sign). Such a formula would be useful to investigatenumerically the initial data giving rise to trapped trajectories. We shall also use it wheninvestigating in more detail the case of the betaplane approximation, in Paragraph 4.4. Exhibiting a subset of Λ of codimension 1. Let us prove the following proposition. Proposition 4.9. Let ¯ u ∈ C ∞ c ( R ) be a function which is not identically positive, with somezero of finite multiplicity. Then the set Λ sing consisting of data in Λ giving rise to singularand trapped trajectories, is nonempty. It contains a submanifold of R × R ∗ × R which is ofcodimension .Proof. Without loss of generality, we can assume that there exist y < y where ¯ u vanishes,with ¯ u ′ ( y ) > u ( y ) < y , y [. As we want to study trapped asymptotic trajecto-ries, we will restrict our attention to the case when τ = 0, which is a necessary condition forasymptotic trajectories to be trapped. Extremal points of the trajectories are then defined interms of the function V ,ξ ( y ) = − b ′ ( y )¯ u ( y ) − b ( y ) − ξ . EMICLASSICAL AND SPECTRAL ANALYSIS OF OCEANIC WAVES 19 More precisely, if we introduce the auxiliary function ̺ ( y ) := − b ′ ( y )¯ u ( y ) − b ( y ) , y ∈ ] y , y [ , we obtain turning (or stopping) points y s if ρ ( y s ) = ξ , and singular points y s if lim y → y s ρ ( y ) =+ ∞ or equivalently ¯ u ( y s ) = 0. By definition of y and y , one haslim y → y + ̺ ( y ) = + ∞ and lim y → y − ̺ ( y ) = + ∞ . Let us define N := max (cid:0) y ∈ ] y ,y [ ̺ ( y ) (cid:1) ∈ R + . For ξ such that ξ ≥ N , we then define h ( ξ ) := sup (cid:8) y ∈ ] − ∞ , y [ ; ̺ ( y ) ≤ ξ (cid:9) ∈ ] y , y [ . We therefore have that ∀ y ∈ ] h ( ξ ) , y [ , y is neither a turning point nor a singular point.As h is a decreasing function on ] − ∞ , −√ N ], all ξ ∈ ] − ∞ , −√ N ] except a countable numberare continuity points. Choose then some ˜ ξ to be a continuity point of h and ˜ x ∈ ] h ( ˜ ξ ) , y [.By continuity of h , there exists a neighborhood ˜ V of (˜ x , ˜ ξ ) such that ∀ ( x , ξ ) ∈ ˜ V , x − h ( ξ ) > . The set { (cid:0) x , ξ , x , ( ̺ ( x ) − ξ ) (cid:1) ; ( x , ξ , x ) ∈ R × ˜ V (cid:9) is a submanifold of R × R ∗ × R having codimension 1. Furthermore, for any initial data in this set, we have ˙ x | t =0 > x t is an increasing function of time. In particular x t → y as t → ∞ . This proves Proposition 4.9 (cid:3) Some examples in the betaplane approximation. We are concerned here with thebetaplane approximation, that is when b ( x ) ≡ βx . One has therefore e τ R ( ξ , x , ξ ) = ¯ u ( x ) ξ + βξ ξ + ξ + β x and with the notation of Paragraph 4.2, V τ,ξ ( x ) = βξ τ − ξ − β x . • In the absence of convection, one can characterize exactly the set Λ of initial data givingrise to trapped trajectories. One can notice that for τ such that τ ξ − ∈ ]0 , β ξ − ], the energysurface E τ,ξ is simply the ellipse (cid:8) ( x , ξ ) / ξ + β x = β ξ τ − − ξ (cid:9) . Let us go through the previous analysis and study the trajectories in this situation. Onenotices that fixed points correspond to x ( t ) = ξ ( t ) = 0, with x ( t ) = x . There are no asymptotic trajectories (singular trajectories would correspond to τ = 0, which is not possiblehere since ξ = 0). Finally let us consider periodic trajectories . In order to get trapping onemust check that if the energy level τ and the frequency ξ are fixed, we have Z x max x min β (cid:16) β ξ τ − ξ (cid:17)q βξ τ − ξ − β x dx = 0 . Let us choose for instance any ( x , ξ ) ∈ R × R ∗ and define the energy level τ = β/ (2 ξ ).Then the integral is identically equal to zero hence the corresponding trajectory is trapped.It corresponds to x t = x , and to x t , ξ t satisfying for all times ξ t + β x t = ξ . Such an explicit characterization cannot be obtained if there is some convection, in particu-lar in physically relevant situations. Nevertheless, we are able to prove that, under suitableassumptions, Λ is not empty, and more precisely that it contains both singular trapped tra-jectories and periodic trapped trajectories. • Proposition 4.9 shows that under the assumption that ¯ u ∈ C ∞ c ( R ) is a function which is notidentically positive, with some zero of finite multiplicity, Λ contains a subset of codimension 1of initial data giving rise to singular trapped trajectories. • Finally let us construct periodic trajectories. We suppose to simplify the notation that β = 1,and that ¯ u has a local maximum at zero, with, say0 < ¯ u (0) < / u ′′ (0) < − . Then we shall prove that the set Λ per is nonempty and contains a submanifold of R × R ∗ × R which is of codimension 1. Let τ be an energy level. We notice that for η small enoughand y ∈ ] − η, η [ ¯ u ( y ) < ¯ u (0) < τξ so there is no singular point for V τ,ξ in the interval ] − η, η [. Now let us show that there aretwo turning points inside ] − η, η [ for η small enough; this will imply that there is a periodictrajectory of energy τ . Define the function H ξ ( y ) := τξ − ξ + y − ¯ u ( y ) . Then H ′ ξ (0) = 0 and an easy computation shows that if η is chosen small enough, then forall ( y, ξ ) ∈ ] − η, η [ × [1 , + ∞ [, H ′′ ξ ( y ) = 2 ξ − y ( ξ + y ) − ¯ u ′′ ( y ) ≥ / > . Now let us choose η such that sup ] − η,η [ ¯ u ′′ ( y ) < − , and let us define the number δ := 1 − η ξ > τ ≡ τ ( ξ ) := ¯ u (0) ξ + δξ so that H ξ (0) = − − δξ < H ξ is decreasing on ] − η, 0[ and increasing on the interval ]0 , η [; moreover H ξ ( ± η ) ≥ H ξ (0) + η − (1 − δ ) + η ≥ η > . It follows that, for all ξ ∈ [1 , + ∞ [, there are two points x min ( ξ ) (in ] − η, x max ( ξ )(in ]0 , η [) satisfying the requirements of periodic trajectories, in the sense of Section 4.2.2.According to the criterion (4.3), we define the following function on [1 , + ∞ [: G ( ξ ) := Z x max ( ξ ) x min ( ξ ) h ξ − ( τ /ξ )2 (cid:0) ( τ /ξ ) − ¯ u ( y ) (cid:1) i V τ,ξ ( y ) − / dy, EMICLASSICAL AND SPECTRAL ANALYSIS OF OCEANIC WAVES 21 and let us prove it vanishes. We notice that recalling that ¯ u (0) < / G (1) ≥ Z x max (1) x min (1) h − (cid:16) ¯ u (0) + δ (cid:17) δ i V τ,ξ ( y ) − / dy ≥ δ − − δ δ Z x max (1) x min (1) V τ,ξ ( y ) − / dy > η ∈ ]0 , ξ goes to + ∞ , we find that x min ( ξ ) → x max ( ξ ) → τ ( ξ ) = ¯ u (0) ξ − δ/ξ , G ( ξ ) ∼ − ¯ u (0) ξ δ Z x max ( ξ ) x min ( ξ ) V τ,ξ ( y ) − / dy, so lim ξ → + ∞ G ( ξ ) < . By construction, the function G is smooth so there is some ˜ ξ belong-ing to ]1 , + ∞ [ such that G ( ˜ ξ ) = 0. We have generically (in τ ) G ′ ( ˜ ξ ) = 0. This impliesthat Λ per = ∅ , where Λ per is the subset of Λ giving rise to periodic trajectories, and that Λ per contains a submanifold of R × R ∗ × R which is of codimension 1.5. Study of the Poincar´e waves The strategy. In this section we want to prove the dispersion property of the Poincar´epolarisation, namely result (2) of Theorem 1. We recall that the principal symbols τ ± areof order one, so one needs to study diffractive-type propagation, on a time scale of the or-der 1 /ε . Computing the bicharacterstics, as in the previous section, is therefore not enoughto understand the classical flow. We shall instead rely on a spectral argument to prove thatPoincar´e waves do disperse, and escape from any compact set in the physical space. The firststep consists in taking the Fourier transform in x (recalling that the problem is invariant bytranslations in the x direction). We also recall that the data is microlocalized on a compactset C such that ξ is bounded away from zero. Since τ ± = ± p ξ + b ( x ) + ξ , functionalcalculus implies that one can find classical pseudo-differential operators H ± ( ξ ) of principalsymbols ξ + b ( x ) such that T ± = ± p H ± ( ξ ) + ξ . Let us now call λ k ± ( ξ ) and ϕ k ± ( ξ ; x )the eigenvalues and eigenfunctions of H ± ( ξ ). The following proposition will be proved in thenext paragraph. Proposition 5.1. Let φ be an eigenfunction of H ± ( ξ ) , microlocalized on an energy surfacewhich interstects C . Then φ and its associate eigenvalue λ are C ∞ functions of ξ . More-over ε ∂ ξ λ is bounded on compact sets in ξ . Proof of Theorem 1(2) assuming Proposition 5.1. Let us now carry out this pro-gram. We consider an initial data denoted ϕ , microlocalized in C . One can take the Fouriertransform in x which gives ϕ ( x ) = 1 √ πε Z ˆ ϕ ( ξ , x ) e − i x ξ ε dξ . Now let us consider a coherent state (in Fourier variables) at ( q, p ) (see Appendix B), that is:(5.1) ϕ qp ( ξ ) := 1( πε ) e i ξ qε e − ( ξ − p )22 ε . After decomposition onto coherent states we get ϕ ( x ) = 1 √ πε πε ) Z ˜ ϕ ( q, p, x ) e i ξ q − x ε e − ( ξ − p )22 ε dqdpdξ where ˜ ϕ ( q, p, x ) := (cid:0) ϕ qp | ˆ ϕ ( · , x ) (cid:1) L . We notice that the integral over p and q is, mod-ulo O ( ε ∞ ), on a compact domain due to the microlocalization assumption on ϕ . Finallydecomposing onto the eigenfunctions ϕ k ± ( ξ ; x ) gives ϕ ( x ) = 1 √ πε πε ) X k Z ϕ ( q, p ; k, ξ , x ) ϕ k ± ( ξ ; x ) e i ξ q − x ε e − ( ξ − p )22 ε dqdpdξ where ϕ ( q, p ; k, ξ , x ) := (cid:16) ϕ k ± ( ξ ; · ) | ˜ ϕ ( q, p, · ) (cid:17) L . Note that the dependence of ϕ on ξ isonly through the eigenfunction ϕ k ± , so ϕ depends smoothly on ξ , as stated in Proposition 5.1.The sum over k contains O ( ε − ) terms, due to the fact that λ k ± ( ξ ) remains in a finite interval(this will be made more precise in the next section, see Remark 5.3). Now it remains topropagate at time t/ε this initial data, which gives rise to the following expression:1 √ πε πε ) X k Z ϕ ( q, p ; k, ξ , x ) ϕ k ± ( ξ ; x ) e i ξ q − x ε e − ( ξ − p )22 ε e ± i ( λ k ± ( ξ )+ ξ ) tε dqdpdξ The stationary phase lemma then gives that this integral is O ( ε ∞ ) except if there exists astationary point, given by the conditions: ξ = p and ε ( x − q ) ± ξ + ∂ ξ λ k ± ( ξ )2 q λ k ± ( ξ ) + ξ t = 0 . The second condition gives2 q λ k ± ( ξ ) + ξ ( x − q ) = ∓ ε (cid:16) p + ∂ ξ λ k ± ( ξ ) (cid:17) t. Therefore, since p = 0 and the λ k ± ’s are bounded, with ∂ ξ λ k ± ( ξ ) = O ( ε ), there is no criticalpoint for x in a compact set. Proposition 5.1 therefore allows to apply the stationary phaselemma and to conclude the proof of result (2) of Theorem 1. Notice that the (fixed) lossesin ε (namely the negative powers of ε appearing in the integrals and the number of k ’s in thesum) are compensated by the fact that the result is O ( ε ∞ ); it is important at this point thatas noticed above, the function ϕ depends smoothly on ξ .5.3. Proof of Proposition 5.1. The first step of the proof consists in using the theoryof normal forms in order to reduce the problem to the study of functions of the harmonicoscillator (as in Bohr-Sommerfeld quantization, paying special attention to the dependenceon ξ ). The second step then consists in checking that the eigenvectors and eigenvalues havethe required dependence on ξ . In the following we shall only deal with H + to simplify, andwe shall write H := H + . The first step relies on the following lemma. Lemma 5.2. Suppose that ξ lies in a compact set away from zero. There is an ellipticFourier Integral Operator U : D ( R ) → D ( R ) , independent of ξ , and a pseudodifferentialoperator V ( ξ ) with C ∞ symbol in ξ such that microlocally in any compact set K ⊂ T ∗ R onehas ( V U ) ∗ = ( V U ) − and V U H ( V U ) − = f ( − ε ∂ + x ; ξ ; ε ) + O ( ε ∞ ) EMICLASSICAL AND SPECTRAL ANALYSIS OF OCEANIC WAVES 23 where f ( I ; ξ ; ε ) = f ( I ) + X ε j f j ( I ; ξ ; ε ) while f is a smooth bijection on R + and the f j ’s are C ∞ functions of I and ξ .Proof. The proof consists in using techniques linked to the isochore Morse lemma (see [6],[30]). Actually we introduce a canonical change of variables (the corresponding operatorbeing the FIO U ) allowing to pass from the variables ( x , ξ ) to action-angle variables (see [1]for instance). Let us make this first step more precise : we recall that the action variables aregiven by I := I ξ dx , where the integral is taken on a constant energy curve H = h . The angle variables are thengiven by solving θ = ∂ I S ( I, ξ ), where K is the hamiltonian in the new variables, which onlydepends on I , and where S is defined by dS I = constant = ξ dx . Note that this is a global change of variables. It is now well known (see [7] for instance) thatsuch a canonical change of variables is associated with an FIO U (independent of ξ since theprincipal symbol of H does not depend on ξ ) such that(5.2) U HU − = f ( − ε ∂ + x ) + εF where f is a smooth, global bijection on R + . In this case we can actually write (see [7]) thefollowing formula for U : for any L function ϕU ϕ ( x ) = 1(2 πε ) Z e i S ( x ,ξ − yξ ε a ( x , ξ , ε ) u ( y ) dydξ where a is constructed so that U is unitary (up to O ( ε )). Using those new coordinates andthe formula giving U as well as the formula for the principal symbol of the adjoint (whichis here the inverse) given in Appendix B, it is not difficult to show that (5.2) holds. Oncethe function f is obtained we proceed by induction: we first look for a symbol q suchthat Q := Op ε ( q ) satisfies e iQ U HU − e − iQ = f , ( − ε ∂ + x ) + ε F . In order to compute q we notice that e iQ U HU − e − iQ = U HU − + Z t ∂ t H t dt where H t := e itQ U HU − e − itQ . One sees easily that ∂ t H t = i [ Q , H t ]= e itQ [ Q , f ( − ε ∂ + x ) + εF ] e − itQ so the principal symbol of ∂ t H t is therefore the Poisson bracket ε { q , f ( x + ξ ) } . One thenremarks that the equation { q, f ( x + ξ ) } = g has a solution if and only if g has zero mean value (in action-angle variables): one has indeednecessarily ∂ θ q = Ig∂ I f · Note that if f and g are smooth, then so is q since ∂ I f > q simply by solving { q , f ( x + ξ ) } = f − f where f is the average of f f ( I ) = 12 π Z π f ( R θ ( x , ξ ); ξ ) dθ where R θ denotes the rotation of angle θ . This implies that with this choice of q andwriting Q := Op ε ( q ) one has e iQ U HU − e − iQ = ( f + εf )( − ε ∂ + x ) + ε F . One proceeds similary at all orders. (cid:3) The next step consists in using that lemma to check that the eigenvalues and eigenfunc-tions enjoy the expected smoothness properties. Actually this is rather straightforward sinceif ( ψ, λ ) satisfy Hψ = λψ + O ( ε ∞ )then e ψ := V U ψ satisfies V U H ( V U ) − e ψ = λ e ψ + O ( ε ∞ )hence f ( − ε ∂ + x ) e ψ = λ e ψ + O ( ε ∞ ) . This implies that λ = λ n = f (cid:18) ε (cid:0) n + 12 (cid:1) ; ξ ; ε (cid:19) and e ψ = e ψ n = 1 ε h n (cid:18) x √ ε (cid:19) where h n is the n -th Hermite function. The conclusion follows recalling that f ( I ; ξ ; ε ) = f ( I ) + P ε j f j ( I ; ξ ; ε ) (where f does not depend on ξ ) and that each eigenfunction ismicrolocalized in a compact set. Remark 5.3. Since we have a complete spectral description of T ± , with discrete spectrumfor each given ξ , it is obvious that if the initial data is microlocalized in ( ξ , x , ξ ) , then thesolution to the equation ε ∂ t ϕ = iT ± ϕ remains microlocalized in the set of energy surfacescontaining ( ξ , x , ξ ) . Notice also that there are O (1 /ε ) eigenvalues in a compact energysurface. Diagonalization In this section we shall prove that the scalar propagators defined in Section 3 correspondindeed to a diagonalization of the original linear system (2.1). Let us prove the followingproposition. Proposition 6.1. Consider a compact set K ⊂ R ∗ × T ∗ R . With the notation of Defini-tions 3.3 and 3.6, the operator Π := (Π − Π R Π + ) maps continuously H ∞ ε functions, microlo-calized in R × K and satisfying (2.3), onto H ∞ ε . Moreover it is left-invertible modulo ε ∞ , andits left inverse Q (modulo ε ∞ ) maps continuously those functions onto H ∞ ε . EMICLASSICAL AND SPECTRAL ANALYSIS OF OCEANIC WAVES 25 Remark 6.2. This proposition, along with Proposition 4.6 and Remark 5.3 showing the prop-agation of the microlocal support (in ( ξ , x , ξ ) ) of Rossby and Poincar´e modes, prove the firstpart of Theorem 1.Proof. The main step consists in showing that Π does have a left inverse, and in computingits principal symbol. The construction of a left inverse can be done symbolically as follows.We first compute the matrix-principal symbol P of Π. One gets P = − ξ p ξ + b − iξ bξ + b − ibξ ξ p ξ + b − iξ bξ + b − ξ ξ ξ ξ + ib p ξ + b ξ + b ξ ξ − ib p ξ + b ξ + b . This shows that Π maps microlocalized functions in R × K onto H ∞ ε . A simple computationshows that | det P | = 2( ξ + b ) ( ξ + b ) | ξ | ≥ , therefore Q = P − exists. Let us call Q the matrix obtained by Weyl quantization (termby term) of Q . By symbolic calculus we have that: Q Π = Id + εI Let us call Q = −I Q − , where I is the (matrix) principal symbol of I . Again by symboliccalculus we have that ( Q + εQ ) Π = Id + ε I where Q has principal symbol Q . Defining now Q = −I Q − we get (cid:0) Q + εQ + ε Q (cid:1) Π = Id + ε I where Q has principal symbol Q , and so on. This allows to invert the matrix Π up to O ( ε ∞ ),and the principal symbol of the (approximate) inverse matrix Q is given by Q : we have Q = 12( ξ + b ) ibξ − ξ p ξ + b ξ + b − ib p ξ + b + ξ ξ ibξ − ξ ξ ξ ibξ + ξ p ξ + b ξ + b ib p ξ + b + ξ ξ . Note that each term of the expansion of the symbol of Q is a polynomial of increasing orderin ξ , but that is not a problem due to the microlocalization assumption. The operator Q therefore clearly maps continuously H ∞ ε onto itself. The proposition is proved. (cid:3) Control of the nonlinear terms for a weak coupling We are now interested in describing the behaviour of our initial nonlinear system, whichincludes the effect of the convection by the unknown ( u , u ). We shall prove Theorem 2 inthis section. We recall that the system reads as follows:(7.1) ε ∂ t U + A ( x, εD x ) U + ε η S ( U ) ε∂ U + ε η S ( U ) ε∂ U = 0 , η ≥ with(7.2) S ( U ) = u u u u 00 0 u , S ( U ) = u u u u u and U = ( u , u , u ). The usual theory of symmetric hyperbolic systems provides the localexistence of a solution to (7.1) in H s ( R ) for s > 2, on a time interval depending on ε a priori.Because of the semiclassical framework (fixed by the form of the initial data), it is actuallynatural to rather consider ε -derivatives. Moreover, as the derivative with respect to x doesnot commute with the singular perturbation, we expect even the semiclassical Sobolev normsto grow like exp (cid:0) Ctε (cid:1) , and therefore the life span of the solutions to (7.1) to be non uniformwith respect to ε . That is the reason why the W sε spaces were introduced in (1.7).7.1. Propagation of regularity for the linear singular perturbation problem. Letus first remark that derivatives with respect to x do commute with A ( x, εD x ), so we canpropagate as much regularity in x as needed. Extending a result by Dutrifoy, Majda andSchochet [10] obtained in the particular case when b ( x ) = βx , we will actually prove thatthere is an operator of principal symbol ( ξ + b )Id which “almost commutes” with A ( x, εD x )in the semiclassical regime. • The first step, as in [10], is to perform the following orthogonal change of variable˜ U := (cid:18) u + u √ , u − u √ , u (cid:19) in order to produce the generalized creation and annihilation operators L ± := 1 √ (cid:0) ε∂ ∓ b (cid:1) . The system (7.1) can indeed be rewritten ε ∂ t ˜ U + ˜ A ( x, εD x ) ˜ U + ε ˜ S ( ˜ U ) ε∂ ˜ U + ε ˜ S ( ˜ U ) ε∂ ˜ U = 0with ˜ A ( x, εD x ) := ε ¯ u ε∂ + ε∂ L + + ε √ ¯ u ′ ε ¯ u ε∂ − ε∂ L − − ε √ ¯ u ′ L − L + ε ¯ u ε∂ , and˜ S ( ˜ U ) := u − ˜ u √ u − u √ u − ˜ u √ , ˜ S ( ˜ U ) := ˜ u u + ˜ u 40 ˜ u ˜ u + ˜ u u + ˜ u u + ˜ u u . • Next, remarking that [ ε ∂ − b , ε∂ ± b ] = ± εb ′ ( ε∂ ± b ) ± ε b ′′ , we introduce the operator D ε := ε ∂ − b + 2 εb ′ ε ∂ − b − εb ′ 00 0 ε ∂ − b . EMICLASSICAL AND SPECTRAL ANALYSIS OF OCEANIC WAVES 27 We notice that D ε is a scalar operator at leading order. Moreover one can compute thecommutator [ D ε , ˜ A ( x, εD x )]: we find[ D ε , ˜ A ( x, εD x )] = [ ε ∂ , ε ¯ u ] ε∂ ε √ (cid:0) [ ε ∂ , ¯ u ′ ] + 2 εb ′ ¯ u ′ − b ′′ (cid:1) ε ∂ , ε ¯ u ] ε∂ − ε √ (cid:0) [ ε ∂ , ¯ u ′ ] − εb ′ ¯ u ′ − b ′′ (cid:1) − ε √ b ′′ ε √ b ′′ [ ε ∂ , ε ¯ u ] ε∂ • At this stage, we have proved that(7.3) [ D ε , ˜ A ( x, εD x )] = O ( ε (Id − ε ∂ − D ε ))meaning that the commutator [ D ε , ˜ A ( x, εD x )] is of order O ( ε ) with respect to the ellipticoperator Id − ε ∂ − D ε . That implies that the regularity of the solution to the linear equation ε ∂ t V + ˜ A ( x, εD x ) V = 0can be controlled by an application of Gronwall’s lemma: one has ε k ( Id − ε ∂ − D ε ) V ( t ) k L ≤ ε k ( Id − ε ∂ − D ε ) V k L + Cε Z t k ( Id − ε ∂ − D ε ) V ( s ) k L ds , where C depends on the W , ∞ norms of ¯ u and b , so k ( Id − ε ∂ − D ε ) V ( t ) k L ≤ C k ( Id − ε ∂ − D ε ) V k L e Ct . Uniform a priori estimates for the nonlinear equation. Since the extended har-monic oscillator controls two derivatives in x , we get a control on the Lipschitz norm of U of the type(7.4) k ε∂ j U k L ∞ ≤ Cε ( k D ε U k L + k ε ∂ U k L + k U k L ) . As D ε is a scalar differential operator at leading order in ε , the antisymmetry of the higherorder nonlinear term is preserved. More precisely, we have, using the Leibniz formula, ε ∂ t D ε ˜ U + ˜ A ( x, εD x ) D ε ˜ U + ε ˜ S ( ˜ U ) ε∂ D ε ˜ U + ε ˜ S ( ˜ U ) ε∂ D ε ˜ U = − [ D ε , ˜ A ( x, εD x )] ˜ U − ε ˜ S ( ˜ U )[ D ε , ε∂ ] ˜ U − ε [ D ε , ˜ S j ( ˜ U )] ε∂ j ˜ U as well as ε ∂ t D ε ˜ U + ˜ A ( x, εD x ) D ε ˜ U + ε ˜ S ( ˜ U ) ε∂ D ε ˜ U + ε ˜ S ( ˜ U ) ε∂ D ε ˜ U = − [ D ε , ˜ A ( x, εD x )] ˜ U − ε ˜ S ( ˜ U )[ D ε , ε∂ ] D ε ˜ U − ε [ D ε , ˜ S j ( ˜ U )] ε∂ j D ε ˜ U + D ε (cid:16) − ε ˜ S ( ˜ U )[ D ε , ε∂ ] ˜ U − ε [ D ε , ˜ S j ( ˜ U )] ε∂ j ˜ U (cid:17) . and in the same way, for 1 ≤ ℓ ≤ ε ∂ t ( ε∂ ) ℓ ˜ U + ˜ A ( x, εD x )( ε∂ ) ℓ ˜ U + ε ˜ S ( ˜ U )( ε∂ ) ℓ +1 ˜ U + ε ˜ S ( ˜ U ) ε∂ ( ε∂ ) ℓ ˜ U = − ε ℓ X k =1 C ℓ ( ε∂ ) ℓ ˜ S j ( ˜ U ) ε∂ j ( ε∂ ) ℓ − k ˜ U . In all cases, the terms of higher order disappear by integration in x and the other terms arecontrolled with the following trilinear estimate (writing generically ˜ Q ( ˜ U ) for all the nonlin-earities): for all 0 ≤ k ≤ ≤ ℓ ≤ | < D kε ˜ U | D kε ˜ Q ( ˜ U ) > | + | < ( ε∂ ) ℓ ˜ U | ( ε∂ ) ℓ ˜ Q ( ˜ U ) > |≤ C k ˜ U k W , ∞ ε ( k D ε ˜ U k L + k ( ε∂ ) ˜ U k L + k ˜ U k L ) ≤ Cε (cid:16) k D ε ˜ U k L + k ( ε∂ ) ˜ U k L + k ˜ U k L (cid:17) . Remark 7.1. Note that because of the bad embedding inequality k∇ U k L ∞ ≤ ε k U k W ε , welose one power of ε , which seems not to be optimal considering for instance the fast oscillatingfunctions x exp (cid:18) ik x ε (cid:19) . A challenging question in order to apply semiclassical methodsto nonlinear problems is to determine appropriate functional spaces (in the spririt of [23] )which measures on the one hand the Sobolev regularity of the amplitudes, and on the otherhand the oscillation frequency. We are finally able to obtain a uniform life span for the weakly nonlinear system, thusproving result (1a) of Theorem 2. Indeed combining the trilinear estimate (7.5) and thecommutator estimate (7.3), we obtain the following Gronwall inequality ε ddt (cid:16) k D ε ˜ U k L + k ( ε∂ ) ˜ U k L + k ˜ U k L (cid:17) ≤ Cε (cid:16) k D ε ˜ U k L + k ( ε∂ ) ˜ U k L + k ˜ U k L (cid:17) from which we deduce the uniform a priori estimate k D ε ˜ U k L + k ( ε∂ ) ˜ U k L + k ˜ U k L ≤ ( C − Ct ) − where C depends only on the initial data. Such an estimate shows that the life span of thesolutions to (7.1) is at least T ∗ = C /C .7.3. Approximation by the linear dynamics. In this paragraph we shall prove re-sults (1b) and (2) of Theorem 2. The proof of both results relies on standard energy estimates.We have ε ∂ t ( U ε − V ε ) + A ( x, εD x )( U ε − V ε ) + ε η ˜ S ( U ε ) ε∂ U ε + ε η ˜ S ( U ε ) ε∂ U ε = 0 • If η = 0 and εV ε → L ∞ , we use the decomposition ε ∂ t ( U ε − V ε ) + A ( x, εD x )( U ε − V ε ) + ε ( ˜ S j ( U ε ) − ˜ S j ( V ε )) ε∂ j U ε + ε ˜ S j ( V ε ) ε∂ j U ε = 0and obtain the following L estimate ε ddt k U ε − V ε k L ≤ ε k ε∂ j U ε k L ∞ k U ε − V ε k L + 3 ε k V ε k L ∞ k ε∂ j U ε k L k U ε − V ε k L ≤ Cε ( ε k ε∂ j U ε k L ∞ + k ε∂ j U ε k L ) k U ε − V ε k L + Cε ( ε k V ε k L ∞ ) from which we conclude by Gronwall’s lemma k U ε − V ε k L ≤ C Z t ( ε k V ε ( s ) k L ∞ ) exp C (cid:18)Z ts ( ε k ε∂ j U ε k L ∞ + k ε∂ j U ε k L ) dσ (cid:19) ds on [0 , T ∗ [, and that proves result (1b). EMICLASSICAL AND SPECTRAL ANALYSIS OF OCEANIC WAVES 29 • If η > 0, the same arguments show that the life span of the solutions to (2.5) tends toinfinity as ε → T ε ≥ Cε − η , and that these solutions are uniformly bounded in W ε onany finite time interval. Furthermore, on any finite time interval [0 , T ], the previous energyestimate gives ε ddt k U ε − V ε k L ≤ Cε η k ε∂ j U ε k L ∞ k U ε k L , from which we deduce k U ε − V ε k L ≤ Cε η Z t ε k ε∂ j U ε ( s ) k L ∞ k U ε ( s ) k L ds . Result (2) of Theorem 2 is proved. Appendix A. A diagonalization theorem In this appendix we shall state and prove the crucial theorem allowing to diagonalize semi-classically the matrix of pseudo-differential operators A ( x , εD ). The construction of Rossbyand Poincar´e modes (see Propositions 3.2 and 3.5 in Section 3) are direct corollaries of thattheorem. We refer to Appendix B for the notation and results of semi-classical analysis usedin this paragraph. We consider an order function g on R d , and a symbol h ( X ; ε, τ ) in S d ( g ),depending polynomially on ε and τ , where we have defined X = ( x, ξ ) ∈ R d . The statementis the following. Theorem 3. Let g be an order function on R d , and let h ( X ; ε, τ ) be a classical symbolin S d ( g ) , depending polynomially on ε and τ : there is an integer N such that (A.1) ∀ X ∈ R d , h ( X ; ε, τ ) = N X j,ℓ =0 h j,ℓ ( X ) ε ℓ τ j , where the symbols h j,ℓ ( X ) belong to S d ( g ) . Let ˜ τ ε = ˜ τ ε ( X ) be a root of the polynomial h ( X ; ε, · ) which can be written for some ν ∈ N , ˜ τ ε ( X ) = ∞ X k =0 ε ν + k τ k ( X ) + O ( ε ∞ ) , with τ = 0 andwhere τ is a symbol. Finally let h ( X, τ ) be the principal symbol of h , satisfying the followingassumption: (A.2) ∃ C > , ∀ X ∈ R d , ∀ ε ∈ ]0 , , | ∂ τ h ( X ; ε ν τ ) | ≥ C. Let K be a compact subset of R d . Then there is a pseudo-differential operator T of principalsymbol ε ν τ such that if ψ ∈ S ′ ( R d ) is microlocalized in K and satisfies i∂ s ψ = T ψ then (A.3) Op ε ( h ( ε, i∂ s )) ψ = O ( ε ∞ ) in L ( R d ) . Proof. The idea of the proof of the theorem is the following. Let us define a smooth function χ ,compactly supported in R d , identically equal to one on K . Then for any integer N , we shallcompute recursively the coefficients of the symbol τ Nε := ε ν τ + P Nk =1 ε ν + k τ − k so T χ := Op ε ( τ χε )satisfies the required property, where τ χε , unique up to O ( ε ∞ ), is given by τ χε := ε ν τ + ∞ X k =1 ε ν + k χτ − k + O ( ε ∞ ) . This allows to replace the root e τ ε by an actual symbol. The above strategy will be achievedin the following way. We notice that if i∂ s ψ = T χ ψ then of course i∂ s ψ = Op ε ( τ χ,Nε ) ψ + O ( ε N + ν +1 ) , where τ χ,Nε := ε ν τ + N X k =1 ε ν + k χτ − k . If moreover ψ is microlocalized in K , then by definition of χ one has using (B.3-B.4) and thefact that χ is identically equal to one over K i∂ s ψ = Op ε ( τ Nε ) ψ + O ( ε N + ν +1 ) . This meansone can (and shall) compute recursively ( τ − k ) ≤ k ≤ N so thatOp ε ( h ( ε, i∂ s )) ψ = O ( ε N +1 ) , when i∂ s ψ = Op ε ( τ Nε ) ψ + O ( ε N +1 ) . Note that it is convenient in the computations to compute T χ as the “right”-quantization ofthe symbol τ χε . Now let us carry out the algebraic computations allowing to achieve the result.We shall start by dealing with the case when ν = 0, as the computations can be carried out inan easier way, and then we shall discuss the case when ν = 0. Recalling that h ( X ; ε, ˜ τ ε ( X )) = 0we infer that h ,ℓ ≡ ℓ < ν and(A.4) h ,ν ( x, ξ ) + h ( x, ξ ) τ ( x, ξ ) = 0 . Now we recall that for any ψ in S ′ ( R d ), one hasOp ε ( h ( ε, i∂ s )) ψ ( x ) = (2 πε ) − d Z R d e i ( x − y ) · ξε h ( x, ξ ; ε, i∂ s ) ψ ( y ) dydξ. The above integral, as all the ones appearing in this proof, is to be understood in the distri-butional sense. So if τ ε and ψ are such that Op ε ( τ χε ) ψ = i∂ s ψ with ψ microlocalized in K ,Op ε ( h ( ε, i∂ s )) ψ ( x ) = (2 πε ) − d Z R d e i ( x − y ) · ξε h ( x, ξ ; ε, Op ε ( τ χε )) ψ ( y ) dydξ = (2 πε ) − d Z R d e i ( x − y ) · ξε h ( x, ξ ; ε, Op ε ( τ Nε )) ψ ( y ) dydξ + O ( ε N +1 ) . (A.5)Now we need to compute h ( x, ξ ; ε, Op ε ( τ Nε )). Using the fact that h is polynomial in τ , h ( x, ξ ; ε, Op ε ( τ Nε )) = N X j,ℓ =0 h j,ℓ ( x, ξ ) ε ℓ (cid:0) Op ε ( τ Nε ) (cid:1) j =: Op ε (˜ h ( x, ξ, y, η ; ε ))where using compositions rules recalled in Appendix B, the symbol ˜ h ( x, ξ, y, η ; ε ) (in the ( y, η )variable) can be expanded as˜ h ( x, ξ, y, η ; ε ) = ε ν ∞ X k =0 ˜ h k ( x, ξ, y, η ) ε k + O ( ε ∞ )where the principal symbol of ˜ h ( x, ξ, y, η ; ε ) is(A.6) ε ν ˜ h ( x, ξ, y, η ) = ε ν (cid:0) h ,ν ( x, ξ ) + h , ( x, ξ ) τ ( y, η ) (cid:1) . We recall indeed that according to Appendix B, one has (cid:0) Op ε ( τ Nε ) (cid:1) j =: Op ε ( m j ( y, η ; ε )) , where m ( y, η ; ε ) = τ Nε ( y, η ) and m j ( y, η ; ε ) = X k ≥ ( iε ) k k ! ∂ ky m j − ( y, η ; ε ) ∂ kη τ Nε ( y, η ) + O ( ε ∞ ) . EMICLASSICAL AND SPECTRAL ANALYSIS OF OCEANIC WAVES 31 In particular the principal symbol of (cid:0) Op ε ( τ ε ) (cid:1) j is ε νj τ j ( y, η ), which yields (A.6). We noticethat due to (A.4), (A.6) implies that(A.7) ˜ h ( x, ξ, x, ξ ) = 0 , so that ˜ h ( x, ξ, x, ξ ; ε ) = O ( ε ν +1 ). More generally, plugging the expansion of m j into the for-mula defining ˜ h and noticing that h , ( x, ξ ) = ∂ τ h | τ =0 ( x, ξ ), one finds that there is h k ( x, ξ, y, η ),depending on the symbol coefficients of h ( x, ξ ), and in the ( y, η ) variables on the sym-bols τ ( y, η ) , τ − ( y, η ) , . . . , τ − k − ( y, η ) only, such that(A.8) ∀ k ≥ , ˜ h k ( x, ξ, y, η ) = h k ( x, ξ, y, η ) + ∂ τ h | τ =0 ( x, ξ ) τ − k ( y, η ) . Finally going back to (A.5) we find thatOp ε ( h ( ε, λ )) ψ ( x ) = (2 πε ) − d Z R d e i ( x − y ) · ξε e i ( y − y ′ ) · ηε ˜ h ( x, ξ, y ′ , η ; ε ) ψ ( y ′ ) dydy ′ dξdη + O ( ε N ) . We can first perform the integration in the y variable, which creates a Dirac mass at ξ − η ,and therefore we have(A.9) Op ε ( h ( ε, λ )) ψ ( x ) = (2 πε ) − d Z R d e i ( x − y ′ ) · ξε ˜ h ( x, ξ, y ′ , ξ ; ε ) ψ ( y ′ ) dy ′ dξ + O ( ε N ) . Now we shall construct τ − ensuring that the order of Op ε ( h ( ε, λ )) ψ ( x ) is O ( ε ν +1 ) insteadof O ( ε ν ). The argument will easily be adaptable by induction to Op ε ( h ( ε, λ )) ψ ( x ) = O ( ε N )for any N ∈ N , by a convenient choice of τ − j , for j ≤ N − 1. We notice that in (A.9),the quantity ˜ h ( x, ξ, y ′ , ξ ; ε ) can easily be replaced by ˜ h ( x, ξ, x, ξ ; ε ) by Taylor’s formula: moreprecisely we write(A.10) ˜ h ( x, ξ, y ′ , ξ ; ε ) = ˜ h ( x, ξ, x, ξ ; ε ) + ( y ′ − x ) · ( ∇ y ′ ˜ h )( x, ξ, x, ξ ; ε ) + O ( | y ′ − x | )which gives after integrations by partsOp ε ( h ( ε, λ )) ψ ( x ) = Z R d e i ( x − y ′ ) · ξε ˜ h ( x, ξ, x, ξ ; ε ) ψ ( y ′ ) dy ′ dξ (2 πε ) d − iε Z R d ∇ ξ e i ( x − y ′ ) · ξε · ( ∇ y ′ ˜ h )( x, ξ, x, ξ ; ε ) ψ ( y ′ ) dy ′ dξ (2 πε ) d + O ( ε ν +2 )= Z R d e i ( x − y ′ ) · ξε ˜ h ( x, ξ, x, ξ ; ε ) ψ ( y ′ ) dy ′ dξ (2 πε ) d + iε Z R d e i ( x − y ′ ) · ξε ∇ ξ · ( ∇ y ′ ˜ h )( x, ξ, x, ξ ; ε ) ψ ( y ′ ) dy ′ dξ (2 πε ) d + O ( ε ν +2 ) . Now using (A.7), it remains to choose τ − so that Z R d e i ( x − y ′ ) · ξε (cid:0) ˜ h ( x, ξ, x, ξ ; ε ) + iε ∇ ξ · ( ∇ y ′ ˜ h )( x, ξ, x, ξ ; ε ) (cid:1) ψ ( y ′ ) dy ′ dξ (2 πε ) d = O ( ε ν +2 ) . That is possible simply by looking at formula (A.8) and choosing τ − ( x, ξ ) = − ˜ h ( x, ξ, x, ξ ) + i ∇ ξ · ( ∇ y ˜ h )( x, ξ, x, ξ ) ∂ τ h | τ =0 ( x, ξ ) · Note that Assumption (A.2) guarantees that τ − is well defined. The argument may be pursuedat the next order simply replacing (A.10) by˜ h ( x, ξ, y ′ , ξ ; ε ) = ˜ h ( x, ξ, x, ξ ; ε ) + ( y ′ − x ) · ( ∇ y ′ ˜ h )( x, ξ, x, ξ ; ε )+ 12 ( y ′ − x ) ⊗ ( y ′ − x ) : ( ∇ y ′ ˜ h )( x, ξ, x, ξ ; ε ) + O ( | y ′ − x | )and using integrations by parts again. Then the choice τ − ( x, ξ ) = − ˜ h ( x, ξ, x, ξ ) + i ∇ ξ · ( ∇ y ˜ h )( x, ξ, x, ξ ) − ∇ ξ : ( ∇ y ˜ h )( x, ξ, x, ξ ) ∂ τ h | τ =0 ( x, ξ )gives that Op ε ( h ( ε, λ )) ψ ( x ) = O ( ε ν +3 ). We leave the rest of the induction argument to thereader. To end the proof of Theorem 3 we need to consider the case ν = 0. The argument issimilar to the case ν = 0 treated above, though the formulas are slightly more complicated.We shall use the same notation as in the previous case. We start by noticing that by definitionof ˜ τ ε we have in particular for all ( x, ξ ) ∈ R d , N X j =0 h j, ( x, ξ ) τ j ( x, ξ ) = 0 . Then as before letus write h ( x, ξ ; ε, Op ε ( τ Nε )) =: Op ε (˜ h ( x, ξ, y, η ; ε )) . One computes easily that the principalsymbol of ˜ h ( x, ξ, y, η ; ε ) is (unlike the case ν = 0)(A.11) ˜ h ( x, ξ, y, η ) = N X j =0 h j, ( x, ξ ) τ j ( y, η ) . Note that as above one has ˜ h ( x, ξ, x, ξ ) = 0 , so that ˜ h ( x, ξ, x, ξ ; ε ) = O ( ε ). One can alsocompute the next orders, and as in the case ν = 0 they can be written in the following form: ∀ k ≥ , ˜ h k ( x, ξ, y, η ) = h k ( x, ξ, y, η ) + τ − k ( y, η ) N X j =1 h j, ( x, ξ ) jτ j − ( y, η )where h k ( x, ξ, y, η ) depends on the symbol coefficients of h ( x, ξ ), and in the ( y, η ) variableson τ ( y, η ) , τ − ( y, η ) , . . . , τ − k − ( y, η ) only. In particular we notice that(A.12) ∀ k ≥ , ˜ h k ( x, ξ, x, ξ ) = h k ( x, ξ, x, ξ ) + τ − k ( y, η ) ∂ τ h | τ = τ . Now that these formulas have been established, it remains to go through exactly the samecomputations as in the case ν = 0, and we find τ − ( x, ξ ) = − ˜ h ( x, ξ, x, ξ ) + i ∇ ξ · ( ∇ y ˜ h )( x, ξ, x, ξ ) ∂ τ h | τ = τ ( x, ξ )which is well defined thanks to Assumption (A.2). The other orders are obtained exactly asin the case ν = 0. This ends the proof of Theorem 3. (cid:3) Appendix B. Some well-known facts in semi-classical analysis In this section we recollect some well-known facts in semi-classical analysis, which have beenused throughout the paper. Most of the material is taken from [7], [21], [25], [30] and [29].B.1. Semi-classical symbols and operators. EMICLASSICAL AND SPECTRAL ANALYSIS OF OCEANIC WAVES 33 B.1.1. Definitions. We recall that an order function is any function g ∈ C ∞ ( R d ; R + \ { } )such that there is a constant C satisfying ∀ X ∈ R d , ∀ α ∈ N d , | ∂ α g ( X ) | ≤ Cg ( X ) . For instance g ( x, ξ ) = (1 + | ξ | ) =: h ξ i is an order function. Note that the variable X usuallyrefers to a point ( x, ξ ) in the cotangent space T ∗ R n ≡ R n , or to a point of the type ( x, y, ξ )with y ∈ R n . A semi-classical symbol in the class S d ( g ) is then a function a = a ( X ; ε )defined on R d × ]0 , ε ] for some ε > 0, which depends smoothly on X and such that forany α ∈ N d , there is a constant C such that | ∂ α a ( X, ε ) | ≤ Cg ( X ) for any ( X, ε ) ∈ R d × ]0 , ε ].If ( a j ) j ∈ N is a family of semi-classical symbols in the class S d ( g ), we write that a = ∞ X j =0 ε j a j + O ( ε ∞ )if for any N ∈ N and for any α ∈ N d , there are ε and C such that ∀ X ∈ R d , ∀ ε ∈ ]0 , ε ] (cid:12)(cid:12)(cid:12) ∂ α (cid:16) a ( X, ε ) − N X j =0 ε j a j ( X, ε ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ Cε N g ( X ) . Conversely for any sequence ( a j ) j ∈ N of symbols in S d ( g ), there is a ∈ S d ( g ) (unique upto O ( ε ∞ )) such that a = ∞ X j =0 ε j a j + O ( ε ∞ ) . An ε -pseudodifferential operator is defined asfollows: if a belongs to S n ( g ), and u is in D ( R n ), then (cid:16) Op ε ( a ) (cid:17) u ( x ) := 1(2 πε ) n Z e i ( x − y ) · ξ/ε a ( x, y, ξ ) u ( y ) dydξ. B.1.2. Changes of quantization. If a ∈ S n ( g ) and t ∈ [0 , 1] then a t ( x, y, ξ ) := a ((1 − t ) x + ty, ξ )belongs to S n ( g ), and one defines Op tε ( a ) := Op ε ( a t ). When t = 0 this corresponds to the classical , or “ left ” quantization, when t = 1 / Weyl quantization (and is usually denoted by Op Wε ( a ) = Op ε ( a )), while when t = 1 one refers to the “ right ”quantization. Furthermore, if a belongs to S n ( h ξ i m ) for some integer m (or more generallyif a belongs to S n ( g ) where g is a H¨ormander metric [21]), then there is a unique symbol a t belonging to S n ( h ξ i m ) (resp. a t ∈ S n ( g )) such that Op tε ( a t ) = Op ε ( a ), and one has a t ( x, ξ ) = X α ( iε ) α α ! ∂ αξ ∂ αη a ( x + tη, x − (1 − t ) η, ξ ) | η =0 + O ( ε ∞ ) . A classical symbol is a symbol a in S n ( h ξ i m ) such that a ( x, ξ ; ε ) = ∞ X j =0 ε j a j ( x, ξ ) + O ( ε ∞ )with a not identically zero, and a j ∈ S n ( h ξ i m ) independent of ε . For any real number ν , theterm ε ν a is the principal symbol of the classical pseudo-differential operator A = ε ν Op tε ( a )(and this does not depend on the quantization). On the other hand ε ν +1 a is the subprincipal symbol of A = ε ν Op Wε ( a ) (in the Weyl quantization only). In the following we shall denoteby σ t ( A ) the symbol of an operator A = Op tε ( a ) (in other words a = σ t ( A )), and by σ P ( A )its principal symbol. B.1.3. Microlocal support and ε -oscillation. If u is an ε -dependent function in a ball of L ( R n ),its ε -frequency set (or microlocal support ) is the complement in R n of the points ( x , ξ )such that there is a function χ ∈ S n (1) equal to one at ( x , ξ ), satisfying k Op Wε ( χ u ) k L ( R n ) = O ( ε ∞ ) . We say that an ε -dependent function f ε bounded in L ( R n ) is ε -oscillatory if for everycontinuous, compactly supported function ϕ on R n ,(B.1) lim sup ε → Z | ξ |≥ R/ε | ϕ b f ε ( ξ ) | dξ → R → ∞ . An ε -dependent function f ε bounded in L ( R n ) is said to be compact at infinity if(B.2) lim sup ε → Z x ≥ R | f ε ( x ) | dx → R → ∞ . B.1.4. Adjoint and composition. Let a be a symbol in S n ( g ), where g is a H¨ormander met-ric [21], and define a ∗ ( x, y, ξ ) := a ( y, x, ξ ) . Then the operator (Op ε ( a )) ∗ := Op ε ( a ∗ ) satisfiesfor all u, v in S ( R n ), (cid:16) (Op ε ( a )) ∗ u, v (cid:17) L = (cid:16) u, (Op ε ( a )) ∗ v (cid:17) L and is therefore called the formal adjoint of Op ε ( a ) . In particular Op ε ( a ) is formally self-adjoint if a is real. Let a and b be two symbols in S n ( g ) and S n ( g ) respectively, where g j are H¨ormander metrics. For all t ∈ [0 , c t in S n ( g g ) which allowsto obtain Op tε ( a ) ◦ Op tε ( b ) = Op tε ( c t ). Moreover one has(B.3) c t ( x, ξ ; ε ) = e iε [ ∂ u ∂ ξ − ∂ η ∂ v ] ( a ((1 − t ) x + tu, η ) b ((1 − t ) v + tx, ξ )) | u = v = xη = ξ =: a t b. This can be also written c t ( x, ξ ; ε ) = X k ≥ ε k i k k ! ( ∂ η ∂ v − ∂ ξ ∂ u ) k ( a ((1 − t ) x + tu, η ) b ((1 − t ) v + tx, ξ )) | u = v = xη = ξ + O ( ε ∞ ) . In particular one has σ t ( A ◦ B ) = σ t ( A ) σ t ( B ) + O ( ε ) . For example in the case when t = 0then Op ε ( a ) ◦ Op ε ( b ) = Op ε ( c ), with(B.4) c ( x, ξ ) = a b = X α ε α i α α ! ∂ αξ a ( x, ξ ) ∂ αx b ( x, ξ ) + O ( ε ∞ ) . In particular if a and b are two classical symbols in the sense described above, and if onedefines A := Op tε ( a ) and B = Op tε ( b ), then the principal symbols satisfy (if σ P ( A ) σ P ( B doesnot vanish identically) σ P ( AB ) = σ P ( A ) σ P ( B ) . B.2. Semiclassical operators, Wigner transforms and propagation of energy. Oneof the main interests of the semiclassical setting is that it allows a precise description of thepropagation of the energy, on times of the order of O ( ε ). We refer for instance to [15] (Section6) for the proof of the following property (actually in the more general setting of matrix-valuedoperators): consider a scalar symbol τ ε ( x, ξ ) defined on R n , belonging to the class S n ( h ξ i σ )for some σ ∈ R (or more generally to S n ( g ) where g is a H¨ormander metric). We assume EMICLASSICAL AND SPECTRAL ANALYSIS OF OCEANIC WAVES 35 moreover that Op Wε ( τ ε ) is essentially skew-self-adjoint on L ( R n ). Then consider f ε an ε -oscillatory initial data in the sense of (B.1), bounded in L ( R n ) and compact at infinity inthe sense of (B.2), and the PDE ε∂ t f ε + Op Wε ( τ ε ) f ε = 0 , f ε | t =0 = f ε . Then the Wigner transform W ε ( t, x, ξ ) of f ε ( t ) defined by W ε ( t, x, ξ ) := (2 π ) − n Z R n e iv · ξ f ε ( x − ε v ) ¯ f ε ( x + ε v ) dv converges, locally uniformly in t , to the solution W of ∂ t W + { τ , W } = 0 where τ is theprincipal symbol of τ ε , and where the Poisson bracket is given by { τ , W } := ∇ ξ τ · ∇ x W − ∇ x τ · ∇ ξ W. The interest of Wigner transforms lies in particular in the fact that under the assumptionsmade on f ε , for any compact set K ⊂ R n one has R K | f ε ( t, x ) | dx = W ε ( t, K × R n ) due tothe fact that | f ε ( t, x ) | = R R n W ε ( t, x, ξ ) dξ .B.3. Coherent states. A coherent state is Φ p,q ( y ) := ( πε ) − n e i ( y − q ) · pε e − ( y − q )22 ε . Any tem-pered distribution u defined on R n may be written u ( y ) = (2 πε ) − n Z T u ( p, q )Φ p,q ( y ) dpdq, where T is the F BI (for Fourier-Bros-Iagolnitzer) transform T u ( p, q ) := 2 − n ( πε ) − n Z e i ( q − y ) · pε e − ( y − q )22 ε u ( y ) dy. This transformation maps isometrically L ( R n ) to L ( R n ). The above formula simply trans-lates the fact that u = T ∗ T u .B.4. Fourier Integral Operators. A Fourier Integral Operator (FIO) is an operator U which can be written, for any f ∈ L ( R n ) U f ( x ) = 1(2 πε ) n Z R n e i Φ( x,y,τ ) /ε a ( x, y, τ ) f ( y ) dydτ where a is a symbol of order 0, compactly supported in x and y , Φ is real valued and homoge-neous of degree 1 in τ , smooth for τ = 0. One requires also a non degeneracy condition on thephase (see [29], Chap. 9, Par. 6.11) on the support of a . Then U is continuous over L ( R n ). Acknowledgements. The authors are grateful to B. Texier for having answered manyquestions concerning geometric optics and semi-classical analysis. They also thank P. G´erardfor interesting discussions around the study of Poincar´e waves in Section 5. Finally theythank the anonymous referees for a very careful reading of the manuscript. I. Gallagher andL. Saint-Raymond are partially supported by the French Ministry of Research grant ANR-08-BLAN-0301-01. References [1] V.I. Arnold. Mathematical Methods of Classical Mechanics . Springer-Verlag, New York, 1978.[2] D.K. Arrowsmith, C.M. Place. An introduction to dynamical systems , Cambridge University Press, Cam-bridge, 1990.[3] G. Birkhoff & G.C. Rota. Ordinary Differential Equations , Ginn, Boston , 1962.[4] J.-Y. Chemin, B. Desjardins, I. Gallagher & E. Grenier, Basics of Mathematical Geophysics , OxfordUniversity Press , 2006, xii+250 pages.[5] C. Cheverry & T. Paul, On some geometry of propagation in diffractive times, accepted for publication, DCDS-A .[6] Y. Colin de Verdi`ere and J. Vey, Le lemme de Morse isochore, Topology (1979), p. 283-293.[7] M. Dimassi & S. Sj¨ostrand, Spectral Asymptotics in the Semi-Classical Limit ; Cambridge University Press,London Mathematical Society Lecture Note Series , 1999.[8] A. Dutrifoy & A. J. Majda, Fast Wave Averaging for the Equatorial Shallow Water Equations, Comm.PDE (2007), Vol. 32, Issue 10, pp. 1617 – 1642.[9] A. Dutrifoy & A. J. Majda, The Dynamics of Equatorial Long Waves: A Singular Limit with Fast VariableCoefficients,” Comm. Math. Sci., Vol. 4, No. 2, pp. 375 – 397, 2006.[10] A. Dutrifoy, A. J. Majda & S. Schochet, A Simple Justification of the Singular Limit for EquatorialShallow-Water Dynamics, in Communications on Pure and Applied Math. LXI (2008) 0002-0012.[11] I. Gallagher, T. Paul and L. Saint-Raymond, On the propagation of oceanic waves driven by a strongmacroscopic flow, arXiv:1011.4435 , submitted .[12] I. Gallagher and L. Saint-Raymond, Weak convergence results for inhomogeneous rotating fluid equations, Journal d’Analyse Math´ematique , (2006), 1-34.[13] I. Gallagher & L. Saint-Raymond, Mathematical study of the betaplane model: equatorial waves andconvergence results. M´em. Soc. Math. Fr. (N.S.). 107 (2006), v+116 pp.[14] I. Gallagher & L. Saint-Raymond, On the influence of the Earth’s rotation on geophysical flows, Handbookof Mathematical Fluid Dynamics , S. Friedlander and D. Serre Editors Vol 4, Chapter 5, 201-329, 2007.[15] P. G´erard, P. Markowich, N. Mauser & F. Poupaud, Homogenization limits and Wigner transforms, Communications on Pure and Applied Mathematics (1997) Vol. L, 323-379.[16] A. E. Gill, Atmosphere-Ocean Dynamics, International Geophysics Series , Vol. 30 , 1982.[17] A. E. Gill & M. S. Longuet-Higgins, Resonant interactions between planetary waves, Proc. Roy. Soc.London , A 299 (1967), pages 120–140.[18] E. Grenier: Oscillatory Perturbations of the Navier–Stokes Equations, Journal de Math´ematiques Pureset Appliqu´ees (1997), , pages 477-498.[19] H.P. Greenspan, The theory of rotating fluids , Cambridge monographs on mechanics and applied math-ematics, 1969.[20] J. Hale & H. Ko¸cak. Dynamics and Bifurcations . Springer-Verlag, New-York, 1991.[21] L. H¨ormander, The Analysis of Linear Operators, Vol III., Springer-Verlag 1985.[22] J.H. Hubbard & B.H.West. Differential equations: a dynamical system approach II , Texts in appliedmathematics, Springer, New-York, 1995.[23] J.-L. Joly, G. M´etivier & J. Rauch, Coherent nonlinear waves and the Wiener algebra, Ann. Inst. Fourier(Grenoble) (1994), no. 1, 167–196.[24] A. Majda, Introduction to PDEs and waves for the atmosphere and ocean . Courant Lecture Notes inMathematics, 9. New York University, Courant Institute of Mathematical Sciences, New York; AmericanMathematical Society, Providence, RI, 2003.[25] A. Martinez, An introduction to semiclassical and microlocal analysis, Springer (2002)[26] J. Pedlosky, Geophysical fluid dynamics, Springer (1979).[27] J. Pedlosky, Ocean Circulation Theory, Springer (1996).[28] S. Schochet: Fast Singular Limits of Hyperbolic PDEs, Journal of Differential Equations (1994) , ,pages 476-512.[29] E. Stein, Harmonic Analysis , Princeton University Press, 1993.[30] S. V˜u Ngoc, Syst`emes int´egrables semi-classiques : du local au global, Panoramas et Synth`eses , 2006. EMICLASSICAL AND SPECTRAL ANALYSIS OF OCEANIC WAVES 37 (Ch. Cheverry) Institut Math´ematique de Rennes, Campus de Beaulieu, 263 avenue du G´en´eralLeclerc CS 74205 35042 Rennes Cedex, FRANCE E-mail address : [email protected] (I. Gallagher) Institut de Math´ematiques UMR 7586, Universit´e Paris VII, 175, rue du Chevaleret,75013 Paris, FRANCE E-mail address : [email protected] (T. Paul) CNRS and Centre de Math´ematiques Laurent Schwartz, ´Ecole Polytechnique, 91128Palaiseau Cedex, France E-mail address : [email protected] (L. Saint-Raymond) Universit´e Paris VI and DMA ´Ecole Normale Sup´erieure, 45 rue d’Ulm, 75230Paris Cedex 05, FRANCE E-mail address ::