Variational formulations of steady rotational equatorial waves

: When the vorticity is monotone with depth, we present a variational formulation for steady periodic water waves of the equatorial flow in the f -plane approximation, and show that the governing equations for this motion can be obtained by studying variations of a suitable energy functional H in terms of the stream function and the thermocline. We also compute the second variation of the constrained energy functional, which is related to the linear stability of steady water waves


Introduction
The mathematical study of geophysical ows is currently of great interest since an in-depth understanding of the ongoing dynamics is essential in predicting features of these large-scale natural phenomena.Geophysical uid dynamics is the study of uid motion where the Earth's rotation plays a signi cant role, the Coriolis forces are incorporated into the governing Euler equations, and applies to a wide range of oceanic and atmospheric ows [10,20,29,44].Because geophysical uid dynamics is highly complex, one usually uses the f -plane approximation of Euler equations.This approximation has been applied widely in the study of the equatorial ows [3,4,9,34,35,41].
Because the Coriolis force vanishes along the equator, equatorial water waves exhibit particular dynamics.Besides, in this region the vertical strati cation of the ocean is greater than anywhere else.Both factors facilitate the propagation of geophysical waves that either raise or lower the equatorial thermocline, which is the sharp boundary between warm and deeper cold waters.The rigorous mathematical study of equatorial water waves was initiated by [9], in which Constantin presented the model of wave-current interactions in the f -plane approximation for underlying currents of positive constant vorticity.Starting with this pioneering paper, recently some essential results on equatorial water waves have been proved in the literature.See [5,6,9,10,14,19,20,32,36,45].In the model constructed by Constantin in [9], the upper boundary of the centre layer is assumed to be at, while the lower boundary is the thermocline near which the equatorial undercurrent resides.See [19,40] for analytical results concerning the dynamics of the thermocline in the equatorial region.In recent work [4], the authors continued to study such a model by considering the general vorticity, and we proved the existence of steady two-dimensional periodic waves in the f -plane approximation by an application of the Crandall-Rabinowitz bifurcation theory.We also derived the dispersion relations for various choices of vorticity, including the negative constant vorticity and non-constant vorticity.For the classical gravity water waves, we refer the reader to [11-13, 25, 27] for the existence of steady periodic waves and the related properties.The study of steady periodic water waves with vorticity has received much attention since the work [25] by Constantin and Strauss.See [7] for more detailed discussions.Vorticity plays the key role in describing oceanic ows, and this aspect was very recently emphasized in thorough analytical studies [21][22][23].
In this paper, we obtain the variational formulation for steady equatorial waves with vorticity of the model in [4,9].It has a long history to study the variational formulations for steady water waves for the irrotational ows.We refer the reader to [33,39] for a Lagrangian formulation and [33,43,46] for a Hamiltonian formulation.For the Hamiltonian formulation of the rotational ows, we refer the reader to [18] for the constant vorticity, [17] for the piecewise constant vorticity allowing for strati cation, [15,16,37] for the extension of the Hamiltonian formulation to various scenarios pertaining to equatorial water ows.There are many results on variational formulations of the various classical small-amplitude long-wave approximations to the governing equations-the shallow water equations, the Boussinesq, and the Kortewegde Vries equations all emerge from this process, see [28] and the references therein.For the steady water waves with vorticity, variational formulations have been given by Constantin, Sattinger and Strauss [24], in which they provide two variational formulations.When the vorticity varies monotonically with the depth, they provided a fundamental variational principle which can be expressed entirely in terms of the natural invariants (energy, mass, momentum and vorticity).One motivation of our research is to extend the results in [26] to the new setting presented here.Some new aspects originate from the dynamic boundary condition (2.5) below and the fact that the pressure on the thermocline is not a constant.Note that the latter is in contrast to the case of classical gravity water waves, where -in absence of surface tension e ects -the pressure is given as the constant atmospheric pressure.Of course the Earth's rotation plays a signi cant role in our analysis.
By computing and analyzing the second variation of the constrained energy functional, we prove linear stability results of steady water waves.Remarkable progress on the linear stability and nonlinear stability properties of steady water waves with vorticity was given by Constantin and Strauss in [26].In the literature, there are many works that deals with the stability of the full water wave equations (not their approximate models).Benjamin and Feir [1] presented a signi cant analysis for a small-amplitude approximation in the irrotational case, showing that there always is a sideband instability, which means that the perturbation has a di erent period from the steady wave.Bridges and Mielke [2] studied the existence and linear stability for the Stokes periodic wavetrain on uids of nite depth, by the Hamiltonian structure of the water-wave problem.Zakharov [46] and Mackay and Sa man [42] discussed the linear stability for the Hamiltonian system that arises with the use of the velocity potential in the irrotational case.

Preliminaries
The vanishing of the Coriolis parameter along the Equator confers the ows in this part of the ocean a twodimensional character.The vorticity equation plays an appreciable role in proving the two-dimensionality of gravity wave trains over ow with constant vorticity vector, and the boundary conditions are decisive in proving the two-dimensionality.See the rigorous analytical argument in [8].In a rotating framework, let the x-axis be chosen horizontally due east, the y-axis horizontally due north and the z-axis vertically upward.z = −d is the upper boundary of the centre layer and z = −η(x, t) is the thermocline.In the region −η(x, t) ≤ z ≤ −d, the full governing equations in the f -plane approximation near the equator are the Euler equations together with the equation of mass conservation where Ω is the rotation speed of the Earth, P is the pressure, g is the gravitational acceleration, ρ is the water's density.The kinematic boundary conditions are Beneath the thermocline, the motionless colder water has a slightly higher density ρ+∆ρ (for example, for the equatorial Paci c the typical value of ∆ρ/ρ is 0.006, see the discussion in [30]).For this reason, the dynamic boundary condition See [9] and [4] for the details on the above equations (2.1)-(2.5).
Given c > , we are looking for the periodic waves traveling at speed c, that is, u, w, P, η have the form (x − ct) and all of them are periodic with period L. In the new reference frame (x − ct, z) → (x, z), we assume that there are no stagnation points of the ow, that is, throughout the uid.Due to (2.2), we can de ne the stream function ψ(x, z) by where ω is the vorticity.Throughout this paper, let (2.7) As was shown in [4], the condition (2.6) ensures that there exists a C vorticity function γ such that From the rst two equations in (2.7) we obtain in analogy with Bernoulli's law for gravity water waves [7], which states that the expression As shown in [4], the governing equations (2.1)-(2.5)are equivalent to problem where Q := (E − P ρ ) is the physical constant and is the reduced gravity [30].

Main results
.

Invariants
Let R(t) := {(x, z) ∈ R : < x < L, −η(x, t) < z < −d} be a periodic cell of the uid domain.We will obtain several invariants for the equatorial ow, which are in analogy to the well-known results in [38] for the classical gravity water waves.Two of them have to be modi ed due to the Earth's rotation and the nonconstant density, cf. the dynamic boundary condition (2.5).First the uid mass dzdx is invariant.Secondly, for an arbitrary C −function F, the integral is invariant².In fact, as done in [24], to prove that F is invariant, we only need to show which is indeed a fact following from the equations (2.1).Now we consider the third invariant given as In fact, let C be the boundary of R(t), using Green's identity, the conditions (2.5) and (2.2), we obtain wdzdx The freedom of choosing F will be used later on.

= − ρ
wdzdx − (g + g) Finally we consider the fourth invariant de ned as To see that U is invariant, by the fact C zdz = S η(x, t)ηx(x, t)dx = , we compute .

Variational formulation
Now we will write the above functionals in terms of ψ and η, which are de ned on the function space where c is the travelling speed.We will restrict perturbations (ψ , −η ) of (ψ, −η) to the subspace We assume that F : R → R is a C −function for which F vanishes nowhere, that is, F is either strictly convex or strictly concave.Let us de ne the C −function γ by γ = (F ) − .Obviously, the function γ is monotone.
We say that (ψ, −η) ∈ F is a steady periodic equatorial water wave with the vorticity function where the constants g, Q are described above and Obviously, (ψ − k, −η) solves the equations (2.8) if (ψ, −η) is a steady periodic equatorial water wave with the vorticity function γ.
We remark that the stream function ψ, determined up to a constant by (3.1) and the free surface pro le η completely determine the steady ow.
which can be rewritten in the following equivalent form Let n be the unit outer normal on the surface S and dl be the measure of arclength.It is easy to see that Therefore we obtain that Let us choose four di erent types of perturbation functions.(i) Firstly, we take η = and aim ψ to be a solution of the elliptic problem on S.
The latter is valid for all smooth functions ω with R ω dzdx = , implying that for all (ψ , −η ) ∈ D, where the functional H is given as for some constant k.Plugging this into (3.6),we have By choosing η = and for some constant C.Moreover, for any f ∈ C per (R), we can construct ψ satisfying ∂ψ ∂n = f on S and ψ = outside of a small neighborhood of S, so that one has Now let us take ψ = throughout R with η arbitrary, we obtain Since F is strictly monotone and F (ω) = ψ − cz − k = on S, we know that ω is constant on S, e.g., ω = ω on S. Hence Thus we obtain To explain this point, we rst note that the restriction ψz < c in the de nition of the space F is quite natural in view of (2.6).However this restriction ensures that the constant C in the proof of Theorem 3.2 is negative.
(b) Next we show that the restriction in D is needed in Theorem 3.2.Indeed, if F is used as the space of perturbations, then, as in the in the proof of Theorem 3.2 we would obtain³ and consequently also (3.9).Thus we get the equations (3.11) with C = , which is essentially the same as (2.8) with m = , and this contradicts the assumption (2.6).

. Second variation
Next we calculate the second variation of H. Beginning with a critical point (ψ, −η) ∈ F, we denote a pair of variations of (ψ, −η) by (ψ , −η ) ∈ D and (ψ , −η ) ∈ D. We further let ω = −∆ψ .Proof.We start from the formulas given in Theorem 3.2 and calculate further variation of each term.First, Therefore, when we try to obtain the stability results, we need to nd suitable conditions under which the symmetric quadratic form (3.12) is nonnegative.First we state the following almost trivial stability result.
Theorem 3.6.Assume that ωz > .Then a classical travelling wave is linearly stable if the surface is unperturbed.⁴Proof.The hypothesis ωz > implies that F < .Therefore the rst integral in (3.12) is nonnegative.When the surface is unperturbed, we have η = , and thus all the rest terms in (3.12) are zero.Therefore in this case, δ H ≥ .
In order to analyze the cases of perturbed surface, we rst prove the following result.by the Hopf maximum principle [31].
Theorem 3.8.Assume that ωz > and that (3.13) is satis ed.Then a classical travelling wave is linearly stable if the surface is perturbed only normally.
Proof.The hypothesis ωz > is equivalent to F < .By Lemma 3.7 the rst and third integral in (3.12) are nonnegative.
For the velocity on the surface to be perturbed only normally, it means that the tangential component of the velocity perturbation vanishes.But this means that ∂ψ /∂n = on S. Therefore the second term in (3.12) vanishes.Therefore, δ H ≥ and the proof is nished.
be the thermocline, B := {(x, −d), < x < L} be the upper boundary of the centre layer.Since on S the function ψ − cz is constant, we can choose ψ − cz = on S. Thus on B, ψ − cz = m, where¹ m := −d −η(x) u(x, z) − c dz < is the relative mass ux.It is not di cult to verify that the equations of motion (2.1)-(2.5)are expressed as

(3. 11 )Remark 3 . 3 .
By the choice of the space F, we know that ψ − cz is strictly decreasing from the upper bound B to the surface S, and therefore C < .Thus we obtain the equations (3.1) by taking m = C and F(ω ) = , and also obtain the equations (2.8) by adding the constant k to ψ.The choice of F satisfying F(ω ) = does not change the vorticity function γ since γ = (F ) − .(a) It is worthwhile to note that the choice of the spaces F and D is important to ensure the e ciency of Theorem 3.2.