Instability of some equatorially trapped waves

[1] A high-frequency asymptotics approach within the Lagrangian framework shows that some exact equatorially trapped three-dimensional waves are linearly unstable when their steepness exceeds a specific threshold. Citation: Constantin, A., and P. Germain (2013), Instability of some equatorially trapped waves, J. Geophys. Res. Oceans, 118, 2802–2810, doi:10.1002/jgrc.20219.


Introduction
[2] Exact solutions play an important role in the study of geophysical flows since many apparently intangible wave motions can often be viewed as perturbations thereof. By controlling the perturbations one can extract relevant information (qualitative as well as quantitative) about the dynamics of more complex flows. The successful implementation of this approach is contingent upon two aspects. First, it is essential to unveil as much as possible the detailed structure of the exact solution. Explicit solutions for ideal flows are known in the Lagrangian and in the Eulerian framework but only the simplest ones have a tractable form in both descriptions. Since in the Eulerian formalism the flow is described by the determination of the fluid velocity at fixed points in space as a function of time, whereas in the Lagrangian framework one specifies the motion of the individual particles, Lagrangian solutions present the great advantage that the fluid kinematics may be described explicitly [see, Bennett, 2006]. Once an exact solution is available, the stability issue becomes important. For stable flows small perturbations do not alter the main characteristics of the motion-all perturbations which are small initially remain small for all time. Instability occurs when the effect of some disturbance of the forces acting on the fluid grows as time progresses. The nature of the instability is important for understanding the factors that might trigger the transition from the large-scale coherent structure represented by the exact solution to a more chaotic fluid motion.
[3] The aim of the paper is to present a stability analysis of the three-dimensional geophysical flow that was recently derived by Constantin [2012]. The explicit Lagrangian so-lution to the governing equations in the -plane approximation describes the eastward propagation of equatorially trapped waves (see section 2), having only an implicit representation within the Eulerian description of fluid flows. The investigation of the stability of this wave pattern relies on the implementation of the theory of short-wavelength instabilities developed independently by Bayly [1987], Friedlander and Vishik [1991], and Lifschitz and Hameiri [1991] (see also the survey, Friedlander and Yudovich [1999]). In section 3.1, we present the short-wavelength instability approach for geophysical equatorial flows, while section 3.2 is devoted to applying the method to the specific study of the equatorially trapped waves. In particular, we identify perturbations in the meridional direction as a source of instabilities with an exponentially growing amplitude and we show that the growth rate of the instabilities depends on the steepness of the wave profile travelling eastward along the Equator.

Description of the Equatorially Trapped Wave
[4] Explicit solutions for gravity fluid flow within the Lagrangian framework, while very desirable from a kinematical viewpoint, are not that numerous. Moreover, those solutions describing flows with a free surface are all generalizations of the rotational deep water gravity water wave due to Gerstner [1809]-see, the discussions in Aleman and Constantin [2012] ; Constantin [2001]; Stuhlmeier [2011]; Weber [2011]. Recently, the approach pioneered by Gerstner was extended by Constantin [2012] to geophysical fluid flows. We now present the main features of this specific explicit equatorially trapped solution.
[5] The geophysical wave is symmetric about the Equator, being confined to a region within 5 latitude from the Equator (corresponding to an equatorial band of width about 1110 km, centered on the Equator). Approximating the shape of the Earth by a sphere of radius R ¼ 6378 km, consider a rotating framework with the origin at a point on the Earth's surface, with the x axis chosen horizontally due east, the y axis horizontally due north and the z axis upward (see Figure 1). Since the meridional distance from the Equator is moderate, the governing equations for geophysical ocean waves are the Euler equations in the -plane approximation, together with the condition of incompressibility [6] On the free surface the kinematic boundary condition for an impermeable free surface-a material boundary across which there is no flux of matter of the macroscopic scale (a particle initially on the boundary will remain on the boundary), and the dynamic boundary condition requiring the stress to be continuous across the free surface which separates the air and the water, must hold. In (4) P atm stands for the (constant) atmospheric pressure, thus decoupling the water flow from the motion of the air above it (see, the discussion in Constantin [2011]). We also require the velocity field to decay rapidly with depth, so that at great depth there is practically no motion. In (1) and (2), U ¼ U ; V ; W ð Þ is the fluid velocity, ¼ 7:29 Á 10 À5 rad s À1 is the (constant) rotational speed of the Earth round the polar axis toward the east, g ¼ 9:8ms À2 is the (constant) gravitational acceleration at the Earth's surface, is the constant water's density, and P is the pressure. The -plane effect y noticeable in (1), with ¼ 2 R ¼ 2:28Á 10 À11 m À1 s À1 , is the result of linearizing the Coriolis force in the tangent plane approximation: although the Earth was assumed to be spherical, since the spatial scale of motion is moderate, the region occupied by the fluid can be approximated by a tangent plane and the linear term of the Taylor expansion captures the -plane effect [see, the discussions in Cushman- Roisin and Beckers, 2011;Gallagher and Saint-Raymond, 2007].
[7] Let the Lagrangian positions X ¼ x; y; z ð Þ of fluid particles be given as functions of the labeling variables q; r; s ð Þand time t by [8] Here k is the wave number, is the wave speed, and captures the decay of particle oscillation in the meridional direction. The labeling variables q and s cover the real line, while r 2 À1; r 0 ð for some fixed r 0 < 0. Here ffiffiffiffiffiffiffi 2kc g q can be identified as the inverse equatorial Rossby radius R for this problem. Since 2 ( kg for all physically reasonable waves, R can be approximated by ffiffiffiffi c 2 q , which is the form given by Gill and Clarke [1974] in the barotropic shallowwater case. [9] The equation (5) define an equatorially trapped wave propagating eastward, the free surface z ¼ x À ct; y ð Þ at latitude y ¼ s being obtained by setting r ¼ r 0 s ð Þ, where r 0 s ð Þ r 0 is the unique solution to e 2k rÀf s ð Þ ½ 2k À r ¼ e 2kr 0 2k À r 0 . The fluid velocity field, the pressure and the free surface exhibit an (x,t) dependence of the form (x-ct), the flow is oriented eastward with a vanishing meridional velocity V, and all particles move in a vertical plane. Note that all particles move on circles (see Figure 2), a feature that is in contrast to the case of irrotational gravity water waves [discussed in Constantin, 2006;Constantin and Strauss, 2010;Henry, 2008]. The restriction of the flow pattern to a fixed latitude replicates a two-dimensional wave motion (see Figure 3), with the three-dimensional character of the flow captured by the decay in the meridional direction. The considerations in Constantin [2012] show that the explicit tractable form (5) of the flow in Lagrangian coordinates corresponds to an intricate implicit representation of the velocity field within the Eulerian framework. This feature is highlighted by the fact that at each fixed latitude y ¼ s, the free surface profile z ¼ x À ct; s ð Þ is trochoidal (see Figure 4). We emphasize Figure 1. The rotating frame of reference with the origin at a point on the Earth's surface: x corresponds to longitude, y to latitude, and z to the local vertical.
the nonlinear character of the flow pattern (5). The particle velocity can be computed from (5) as where we set ; cs e À cos ð Þ ð Þ : [11] Consequently, in the Euler equation (1) with V 0, both terms U t : UU x : WU z : 2W ð Þand W t : UW x : ð WW z : À2U Þ have orders of magnitude in a ratio 1 : e : e : 2 kc À Á since cos; sin; e 6cos are all of order 1 due to constraint e e kr 0 < 1. Note that along the Equator we have ¼ kr, so that by (5) the steepness of the equatorial wave profile, defined as the amplitude multiplied by the wave number, is precisely e kr 0 since the height (that is, the difference in elevation between the crest and the trough) of the wave propagating eastward along the Equator is 2e kr 0 =k. The above considerations show that an increasing steepness enhances the nonlinear character of the flow (5). A further issue of interest is the significance of the Coriolis terms 2W ; À2U ð Þ . For the wavelength L ¼ 63 m we have k % 10 À1 m À1 , so that (6) yields c % 10ms À1 . Choosing r 0 ¼ À12m, depths of about 53 m beneath the surface correspond to r % À65m. This yields ¼ kr % À6:5. Therefore at these specific depths along the Equator, e , being roughly e À6:5 % 15 Á 10 À4 , is comparable to 2 kc % 15 Á 10 À5 , thus confirming the geophysical character of the flow. Note that while at depths in excess of 88 m beneath the surface the water is practically still since for r À100m the height of the particle oscillation (5) is of the order 2 k e kr < 1mm , at about 50 m beneath the surface the motion induced by the surface wave is not negligible since the height of the particle oscillation (5) is of the order 2 k e kr % 3cm . Also, since the wave steepness of the wave profile propagating along the Equator is e kr 0 % 0:301, the linear and the weakly nonlinear regime are not appropriate and the wave motion is genuinely nonlinear. For this concrete example the maximal wave height of the trapped wave is 2 k e kr 0 % 6m, so that the wave can be classified among the relatively high waves [cf., Holden, 2012;Kinsman, 1965]. The corresponding equatorial radius of  . The motion at a fixed latitude : each particle beneath the surface wave describes a circle, and they are all in the same phase. The diameter of these circles decreases exponentially with the distance to the surface. In particular, at a depth of half a wavelength the diameter is reduced to about 4% of its free-surface value, illustrating the deep water wave character of the motion.
deformation R is approximately ffiffiffiffi c 2 q % 500km . At meridional distances R and s ! 1000km from the Equator the wave height reduces to 2 k e k r 0 Àf R ð Þ ½ % 4:5m and 2 k e k r 0 Àf s ð Þ ½ < 2m respectively. Note that for this concrete example the vorticity r Â U of the flow in a near-surface layer L centred on the Equator, less than 88 m deep, extending about 1000 km in the meridional direction and all across the whole length (about 13,000 km) of the ocean basin, is well approximated by 0; À2kce 2kr ; 0 À Á . Since at depths in excess of 88 m beneath the surface the motion is practically ground to a halt, the above formula for the vorticity is indicative of a near-surface westward flowing current. To elucidate this aspect, let us fix the latitude s. Since by (8) the average of the horizontal fluid velocity U over a wave period T ¼ L=c clearly vanishes, the mean Lagrangian velocity is zero: [12] Note that the crest/trough levels of the surface wave that propagates eastward at latitude s are z 6 s ð Þ ¼ r 0 s ð Þ Àr 0 6 1 k e k r 0 Àf s ð Þ ½ , while the mean water level is as one can see by using in the second step the outcome of the differentiation of the x component of (5) with respect to the q variable. A fixed depth z ¼ z 0 beneath the level z ¼ z À s ð Þ of the surface wave troughs is characterized in Lagrangian variables by means of the relation which yields a functional dependence [13] By differentiating (9) with respect to the q variable we obtain 0 ¼ q þ q e cos À e sin; so that [14] This permits us to compute the Eulerian mean velocity hU i E z 0 ; s ð Þ at the latitude s and at the depth z 0 due to (5), (8), (10), (11). Thus which shows that the Eulerian mean flow is westward. Since the Lagrangian mean flow is zero, the Stokes drift, defined by Longuet-Higgins [1969], as the difference between the Lagrangian and the Eulerian mean velocities, is eastward. Note that differentiating (9) with respect to z 0 yields 1 ¼ z0 þ z0 e cos; so that z0 ¼ 1 1 þ e cos > 0: [15] Combining this with (12), we get @ z0 hU i E s; z 0 ð Þ¼À 2kc L Z L 0 e 2 1 þ e cos dq < 0: [16] Therefore the Eulerian mean velocity hU i E s; z 0 ð Þ depends monotonically on the depth, with the westward flow of significance only in the near-surface region and barely noticeable at great depths. This is the hallmark of a nonuniform wave-induced westward current.
[17] We conclude our discussion of the flow (5) by investigating the mass transport. Recall that hU i L is sometimes called the mass-transport velocity, being the mean velocity of a marked particle [cf., Longuet-Higgins, 1969;Constantin, 2013]. Let us compute the mass flux past x ¼ x 0 at a fixed latitude s, given by and representing the instantaneous zonal transport across a fixed longitude. From (8) and differentiating with respect to r the third component of (5), evaluated at q ¼ r; t; s ð Þ determined by the constraint we get ce cos 1 þ e cos À r e sin ð Þ dr: ð14Þ [18] Differentiating (13) with respect to r yields 0 ¼ r À e sin À r e cos; so that t ¼ e sin 1 À e cos ; and (14) becomes e cos 1 À e 2 1 À e cos dr: [19] Note that the wave crests/troughs lie on the line x ¼ x 0 if and only if cos ¼ 61, in which case (15) shows that mass is carried forward/backward, respectively. Since hU i L ¼ 0, the average of the mass flux over a wave period T vanishes. This fact is encoded in (15): differentiating (13) with respect to t yields 0 ¼ t À t e cos þ ce cos; so that t ¼ À ce cos 1 À e cos ; and (15) takes on the form [20] The T-periodicity of the function t 7 ! r; t; s ð Þ confirms that the average of the mass flux over a period T vanishes.

Instability Analysis
[21] In this section, we first present the short-wavelength instability approach for a general geophysical equatorial flow. Subsequently, we study the specific case of the equatorially trapped waves.

Short-Wavelength Instability Approach
[22] Since the stability issue concerns the evolution of small perturbations with time [cf., Drazin, 2002;Yudovich, 1984], it is reasonable to pursue its investigation within a linear framework by neglecting nonlinear terms arising from products of the perturbed quantities. The stability analysis of a basic flow represented by the velocity field U of an inviscid incompressible fluid relies on the study of the growth of infinitesimal disturbances u. Within the short-wavelength instability approach one considers the evolution of a rapidly varying localized wave packet following a particle. Choose an initial disturbance in the form where the vector b 0 represents the normalized amplitude, 0 is the normalized wave vector subject to the transversality condition 0 Á b 0 ¼ 0, and the small parameters " and ensure that the small disturbance oscillates rapidly in space (see Figure 5). The choice of a function b 0 of small support Figure 5. The initial high-frequency disturbance, localized near X 0 , is moved by the flow. Both its amplitude "b 0 X ð Þ and its normalized wave vector 0 change throughout this process. If t7 !X X 0 ; t ð Þ is the Lagrangian trajectory of the particle with label X 0 , transported by the basic flow U, at a later time t the disturbance evolves to a wave packet localized near X X 0 ; t ð Þ, of amplitude "b X; t ð Þ and with normalized wave vector t ð Þ.
and sharply peaked at a point X 0 in the fluid localizes the initial disturbance near X 0 and specifies b 0 X 0 ; 0 ð Þ as its main direction. The orthogonality between the wave vector 0 and the wave amplitude b 0 is forced by the incompressibility constraint (see later) and confers to the perturbation the characteristic of transverse waves, thus inducing a particle displacement that is orthogonal to the direction of wave propagation. The initial disturbance u 0 is moved by the basic flow U, so that at time t the fluid particle at X 0 has moved to a point X X 0 ; t ð Þ. With this specific choice of initial disturbance, we represent, at leading order in powers of ", the subsequent evolution of the velocity and pressure perturbations in the form and respectively, where the scalar function d measures the amplitude of the pressure perturbation p. The initial conditions [23] Writing (1) and (2) for the perturbed flow leads to and r Á u ¼ 0; respectively, where we denoted [24] At highest order in the expansion of (21) in powers of , due to (17) while, as long as b 6 ¼ 0, (20) yields [25] Since (23) is equivalent to u Á rÈ ¼ 0, the above equation leads to the eikonal equation [26] Taking the gradient of (25) gives now the evolution equation for the field of wave vectors ¼ rÈ, where rU ð Þ T is the transpose of the basic velocity gradient tensor [27] On the other hand, at the highest order in the expansion of (20) in powers of ", we have if we recall (24). Taking the time derivative of (23), and taking advantage of (26) and (27) leads to (23). Solving for d, we obtain [28] The above expression can now be used in (27) [29] At leading order in powers of ", the path of the particle located initially at X 0 is determined by solving the ordinary differential equation _ X ¼ U X; t ð Þ with initial data X 0 ð Þ ¼ X 0 : the perturbation velocity field u does not contribute to the advection of the observation point X t ð Þ. Since the material derivative _ f ¼ f t þ U Á rf describes the rate of the change of the vector f as a particle moves due to the bulk flow (induced by the unperturbed velocity field U), taking into account (26) and (28), we deduce that the evolution in time of X, of the amplitude b and of the wave vector ¼ rÈ is governed, at leading order in an expansion in powers of " and , by the specific coupled system of ordinary differential equations [30] The initial conditions (at s ¼ 0) associated to (29) are with 0 Á b 0 ¼ 0. The growth rate of the vector b is analogous to the concept of a Lyapunov exponent for a dynamical system: if for some initial position X 0 we have then exponential stretching occurs in the flow -certain particles are separated at an exponential rate and this separation, visualized as stretching, is a hallmark of instability [cf., the discussion in Friedlander, 1999;Friedlander and Yudovich, 1999]. From the point of view of spectral analysis, if Ã X 0 ð Þ > 0 for some X 0 , then the unbounded operator associated to the linearization around the basic flow has a nonempty unstable continuous spectrum in the space of square integrable functions [cf., Friedlander and Yudovich, 1999]. Note that while in (29) the second and third equation are linear, the first equation is usually nonlinear but decouples from the other two. The first equation provides the particle trajectory of the basic (undisturbed) flow, while the second and third equation govern to leading order the evolution along this trajectory of the local wave vector and of the amplitude of the perturbation, respectively.

Instability of the Equatorially Trapped Wave
[31] To prove the instability of the geophysical fluid flow (5), it is not necessary to investigate the associated system (29) for all initial data. It suffices to exhibit a choice for the initial disturbance that results in an exponentially growing amplitude b. This is our aim.
[32] The class of disturbances characterized by 0 ¼ 010 ð Þ T is of particular interest since in this case the solution to the equation for in (29) can be computed explicitly: t ð Þ ¼ 010 ð Þ T for all t ! 0. Using this fact, we see that the [33] The dependence of on t prevents the above linear differential system from being autonomous. Thus the possibility of solving (31) explicitly seems remote. Fortunately, there is some hidden structure of the system. First of all, (31) is reduced to the planar nonautonomous linear differential equation [34] In the rotating frame of reference corresponding to the change of variables induced by the matrix the system (32) becomes autonomous and therefore tractable. Indeed, where D ¼ kce 1 À e 2 sin kq ð Þ À kce 1 À e 2 cos kq ð Þ À 2 þ kce 2 1 À e 2 À kc 2 À kce 1 À e 2 cos kq ð Þ þ 2 À kce 2 1 À e 2 þ kc 2 À kce 1 À e 2 sin kq ð Þ 0 B @ 1 C A: [36] The asymptotic behaviour of Q(t) as t ! 1 is determined by the eigenvalues of the matrix D. These are found by solving the quadratic equation [37] Consequently, exponential growth of the solution Q(t) occurs for t ! 1 if and only if (recall from section 2 that e < 1). Using (6), we can express (34) as [38] Since the matrix P(t) is time periodic and B t ð Þ ¼ P t ð ÞQ t ð Þ, the temporal growth of the shortwavelength disturbances b is exponential if the wave propagating eastward along the Equator is sufficiently steep. Indeed, recall from section 2 that the steepness of the equatorial wave profile is precisely ¼ e kr 0 . The exponential growth rate of the short-wavelength disturbances is given by the positive root of the equation (33). These considerations shows that the equatorially trapped wave (5) with wave number k is linearly unstable when the steepness of the wave propagating eastward along the Equator is strictly larger than the expression on the right-hand side of (35)-(35): an initial disturbance of the form u 0 x; y; z ð Þ¼" b 1 x; y; z ð Þ 0 b 3 x; y; z ð Þ 0 @ 1 A e iy=" , localized in the nearsurface layer, will grow exponentially fast at the rate ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 2 10k 2 c 2 þ 32 þ kc ð Þ ½ À 4 þ kc ð Þ 2 À 4 4 þ 3kc ð Þ 2 q 2 1 À 2 ð Þ ð36Þ given by the positive root of equation (33). For all physically realistic equatorially trapped waves we have that 2 ( kg. Consequently, while the right-hand side of (35) is always larger than 1/3, it is very close to 1/3. The previous considerations show that a steepness of the wave propagating eastward along the Equator in excess of 1/3 triggers instability.

Discussion
[39] The theory of short-wavelength perturbations has been applied to prove the linear instability of some recently derived equatorially trapped waves when the wave profile that propagates along the Equator is sufficiently steep. Despite the efficiency of the approach for steep waves, beneath the threshold specified in (34) and (35) this method appears to be inconclusive. Even in this regime stability is unlikely but analytic or numerical evidence is currently not available.
[40] By inspection one can see that the instability condition (34) simplifies to the requirement that the wave steepness exceeds 1/3 if one eliminates the small 2W ; À2U ð Þ terms in the equation (1). In particular, setting ¼ ¼ 0 in (5), we obtain Gerstner's gravity water wave and we recover the results obtained by Leblanc [2004], investigation that appears to be the first stability analysis of a Lagrangian flow that is not explicitly available in the Eulerian representation and was a source of inspiration for the present paper.
[41] It is of interest to compare our results with the studies of the effect of zonal currents on surface equatorial Kelvin waves -waves that are usually associated with anomalies in surface wind stress and penetrate the entire depth of the ocean [cf., McPhaden and Ripa, 1990;Wang, 2003]. Like (5), these equatorially trapped waves propagate eastward and their meridional velocity vanishes everywhere. However, unlike (5), the equatorial Kelvin waves are not exact solutions to the governing equations (1)-(4) but approximate linear solutions in the shallow water regime: neglecting the vertical motion and with a vanishing meridional velocity, the linearized shallow-water equations in the equatorial -plane take the form [42] [cf., Gill, 1982]. In (37) the constant H is the mean depth of the fluid and is the perturbation of the flat free surface z ¼ 0. Note that the first and third equation in (37) yield the linear wave equation tt À gH xx ¼ 0. Seeking solutions in the form U ; ½ ¼Ũ y ð Þ; h y ð Þ Â Ã e i kxÀ!t ð Þ , the dispersion relation ! 2 ¼ gHk 2 , relating the frequency ! and the wave number k, emerges. The third equation in (37) forces !h ¼ kHŨ , so that the second equation in (37) becomes [43] Since the solution with exponential growth has to be ruled out on physical grounds, the only realistic solution is the equatorial Kelvin wave propagating eastward with speed c ¼ ffiffiffiffiffiffi ffi gH p . The surface equatorial Kelvin waves are destabilized by a weak crossequatorial shear of the type " 0; y; 0 ð Þ, modeling a mean equatorial current with meridional shear: numerical studies [see, Boyd and Christidis, 1982;Boyd and Natarov, 2002;Boyd, 2005;Natarov and Boyd, 2001] show that the imaginary part of the complex phase speed (i.e., the growth rate) is an exponentially small function of the strength " of the cross-equatorial shear, being of order exp À1=" 2 ð Þ. In contrast to this, the instability mechanism for steep equatorially trapped waves of type (5) predicts an exponential growth rate of the instabilities that depends on the steepness of the wave profile traveling eastward along the Equator.