Variational formulations of steady rotational equatorial waves

Jifeng Chu; Joachim Escher

doi:10.1515/anona-2020-0146

Article Open Access

Variational formulations of steady rotational equatorial waves

Jifeng Chu and Joachim Escher

Published/Copyright: August 22, 2020

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From the journal Advances in Nonlinear Analysis Volume 10 Issue 1

Abstract

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.

Keywords: Steady periodic water waves; Equatorial flows; Vorticity; Variational formulations

MSC 2010: Primary 76B15; 35J60; 47J15; 76B03

1 Introduction

The mathematical study of geophysical flows 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 fluid dynamics is the study of fluid motion where the Earth’s rotation plays a significant role, the Coriolis forces are incorporated into the governing Euler equations, and applies to a wide range of oceanic and atmospheric flows [10, 20, 29, 44]. Because geophysical fluid 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 flows [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 stratification 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 flat, 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, 12, 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 flows, 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 flows. 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 flows, we refer the reader to [18] for the constant vorticity, [17] for the piecewise constant vorticity allowing for stratification, [15, 16, 37] for the extension of the Hamiltonian formulation to various scenarios pertaining to equatorial water flows. 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 Korteweg-de 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 effects - the pressure is given as the constant atmospheric pressure. Of course the Earth’s rotation plays a significant 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 significant 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 different period from the steady wave. Bridges and Mielke [2] studied the existence and linear stability for the Stokes periodic wavetrain on fluids of finite depth, by the Hamiltonian structure of the water-wave problem. Zakharov [46] and Mackay and Saffman [42] discussed the linear stability for the Hamiltonian system that arises with the use of the velocity potential in the irrotational case.

2 Preliminaries

The vanishing of the Coriolis parameter along the Equator confers the flows in this part of the ocean a two-dimensional character. The vorticity equation plays an appreciable role in proving the two-dimensionality of gravity wave trains over flow 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

ut+uux+wuz+2Ωw=−1ρPx,wt+uwx+wwz−2Ωu=−1ρPz−g, (2.1)

together with the equation of mass conservation

ux+wz=0, (2.2)

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

w=−ηt−uηxonz=−η(x,t), (2.3)

and

w=0on z=−d. (2.4)

Beneath the thermocline, the motionless colder water has a slightly higher density ρ+Δρ (for example, for the equatorial Pacific the typical value of Δρ/ρ is 0.006, see the discussion in [30]). For this reason, the dynamic boundary condition

P=P0−g(ρ+Δρ)zon z=−η(x,t). (2.5)

See [9] and [4] for the details on the above equations (2.1)-(2.5).

Given c > 0, 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 flow, that is,

u<cfor −η(x)≤z≤−d, (2.6)

throughout the fluid. Due to (2.2), we can define the stream function ψ(x, z) by

ψx=−w,ψz=u,for −η(x)<z<−d.

Thus

−Δψ=ω=wx−uz,

where ω is the vorticity.

Throughout this paper, let 𝓡 := {0 < x < L, –η(x) < z < –d} and S := {(x, –η(x)), 0 < x < L} be the thermocline, B := {(x, –d), 0 < x < L} be the upper boundary of the centre layer. Since on S the function ψ – cz is constant, we can choose ψ – cz = 0 on S. Thus on B, ψ – cz = m, where^[1]

m:=∫−η(x)−d(u(x,z)−c)dz<0

is the relative mass flux. It is not difficult to verify that the equations of motion (2.1)-(2.5) are expressed as

(ψz−c)ψxz−ψxψzz−2Ωψx=−1ρPx,for −η(x)<z<−d,−(ψz−c)ψxx+ψxψxz−2Ωψz=−1ρPz−g,for −η(x)<z<−d,ψx=(ψz−c)ηx,on z=−η(x),ψ−cz=0,on z=−η(x),ψ−cz=m,on z=−d. (2.7)

As was shown in [4], the condition (2.6) ensures that there exists a C¹ vorticity function y such that

−Δψ=ω=y(ψ−cz).

From the first two equations in (2.7) we obtain in analogy with Bernoulli’s law for gravity water waves [7], which states that the expression

E:=ψx2+(ψz−c)22−2Ωψ+gz+Pρ−Γ(cz−ψ)

is constant throughout 𝓡, where

Γ(p):=∫0py(−s)ds,0≤p≤−m.

As shown in [4], the governing equations (2.1)-(2.5) are equivalent to problem

Δψ=−y(ψ−cz),for −η(x)<z<−d,|∇(ψ−cz)|2−2(g~+2Ωc)z=Q,on z=−η(x),ψ−cz=0,on z=−η(x),ψ−cz=m,on z=−d, (2.8)

where Q:=2(E−P0ρ) is the physical constant and

g~:=gΔρρ

is the reduced gravity [30].

3 Main results

3.1 Invariants

Let 𝓡(t) := {(x, z) ∈ ℝ² : 0 < x < L, –η(x, t) < z < –d} be a periodic cell of the fluid domain. We will obtain several invariants for the equatorial flow, which are in analogy to the well-known results in [38] for the classical gravity water waves. Two of them have to be modified due to the Earth’s rotation and the non-constant density, cf. the dynamic boundary condition (2.5). First the fluid mass

M:=∬R(t)dzdx

is invariant. Secondly, for an arbitrary C¹–function F, the integral

F:=∬R(t)F(ω)dzdx

is invariant^[2]. In fact, as done in [24], to prove that 𝓕 is invariant, we only need to show

DωDt=ωt+uωx+wωz=0,

which is indeed a fact following from the equations (2.1). Now we consider the third invariant given as

E:=∬R(t)[u2+w22−g~z]dzdx.

In fact, let C be the boundary of 𝓡(t), using Green’s identity, the conditions (2.5) and (2.2), we obtain

dEdt=∬R(t)DDt(u2+w22)dzdx−g~∬R(t)DDt(z)dzdx=−1ρ∬R(t)(uPx+wPz)dzdx−g∬R(t)wdzdx−g~∬R(t)wdzdx=−1ρ∫CP(wdx−udz)−g∬R(t)wdzdx−g~∬R(t)wdzdx=−P0ρ∫C(wdx−udz)+g(ρ+Δρ)ρ∫Cz(wdx−udz)−(g~+g)∬R(t)wdzdx=−P0ρ∫C(wdx−udz)+g(ρ+Δρ)ρ∬R(t)wdzdx−(g~+g)∬R(t)wdzdx=−P0ρ∫C(wdx−udz)=−cP0ρ∫0Lηx(x,t)dx=0.

Finally we consider the fourth invariant defined as

U:=∬R(t)(u(x,z,t)+2Ωz)dzdx.

To see that 𝓤 is invariant, by the fact ∫_Czdz = ∫_Sη(x, t)η_x(x, t)dx = 0, we compute

dUdt=∬R(t)DDt(u)dzdx+2Ω∬R(t)DDt(z)dzdx=∬R(t)(ut+uux+wuz)dzdx+2Ω∬R(t)wdzdx=∬R(t)(−1ρPx−2Ωw)dzdx+2Ω∬R(t)wdzdx=−1ρ∬R(t)Pxdzdx=1ρ∫C(P0−g(ρ+Δρ)z)dz=1ρ∫CP0dz.

3.2 Variational formulation

Now we will write the above functionals in terms of ψ and η, which are defined on the function space

F:={(ψ,−η)∈Cper2(R×(−∞,−d])×Cper1(R)×R:ψz<c},

where c is the travelling speed. We will restrict perturbations (ψ₁, –η₁) of (ψ, –η) to the subspace

D:={(ψ,−η)∈F:∫Bψzdx=0}.

We assume that F : ℝ → ℝ is a C²–function for which F″ vanishes nowhere, that is, F is either strictly convex or strictly concave. Let us define the C¹–function y by y = (F′)^–1. Obviously, the function y is monotone.

We say that (ψ, –η) ∈ 𝔽 is a steady periodic equatorial water wave with the vorticity function y if there exist k, Q ∈ ℝ such that

Δψ=−y(ψ−cz−k),for −η(x)<z<−d,|∇(ψ−cz)|2−2(g~+2Ωc)z=Q,on z=−η(x),ψ−cz=k,on z=−η(x),ψ−cz=m+k,on z=−d, (3.1)

where the constants g̃, Q are described above and

m:=∫−η(0)−d(ψz(0,z)−c)dz.

Obviously, (ψ – k, –η) solves the equations (2.8) if (ψ, –η) is a steady periodic equatorial water wave with the vorticity function y.

We remark that the stream function ψ, determined up to a constant by (3.1) and the free surface profile η completely determine the steady flow.

Theorem 3.1

Any critical point in 𝔽 of 𝓔 – 𝓕, subject to the constraints of 𝓜 and 𝓤, is a steady periodic equatorial water wave with the vorticity function y.

Proof

Let (ψ, –η) ∈ 𝔽 be a critical point of 𝓔 – 𝓕. Then (ψ, –η) satisfies the Euler-Lagrange equation

δ(E−F)=λδU+μδM, (3.2)

where λ and μ are Lagrange multipliers. Let (ψ₁, –η₁) ∈ 𝔻 denote a perturbation of (ψ, –η) and set ω := –Δψ and ω₁ := –Δψ₁. By direct computations, we obtain

δM(ψ,−η)(ψ1,−η1)=limε→0M(ψ+εψ1,−η−εη1)−M(ψ,−η)ε=∫Sη1dx,δU(ψ,−η)(ψ1,−η1)=∬Ru1dzdx+∫S(u+2Ωz)η1dx,δF(ψ,−η)(ψ1,−η1)=∬RF′(ω)ω1dzdx+∫SF(ω)η1dx,

and by Green’s formula

δE(ψ,−η)(ψ1,−η1) = ∬R(uu1+ww1)dzdx+∫S(u2+w22−g~z)η1dx=∬R∇ψ⋅∇ψ1dzdx+∫S(|∇ψ|22−g~z)η1dx=∫S{(|∇ψ|22−g~z)η1+ψ(ψ1z+ηxψ1x)}dx−∬RψΔψ1dzdx−∫Bψψ1zdx.

By definition, any critical point (ψ, –η) satisfies (3.2), so that

∫S{|∇ψ|22−(g~+2Ωλ)z−F(ω)−μ−λψz}η1dx−∫Bψψ1zdx+∫Sψ(ψ1z+ηxψ1x)dx−∬R{ψΔψ1+F′(ω)ω1+λψ1z}dzdx=0,

which can be rewritten in the following equivalent form

∫S{|∇(ψ−λz)|22−λ22−(g~+2Ωλ)z−F(ω)−μ}η1dx−∫B(ψ−λz)ψ1zdx+∫S(ψ−λz)(ψ1z+ηxψ1x)dx−∬R{(ψ−λz)Δψ1+F′(ω)ω1}dzdx−λ∬R(ψ1z+zΔψ1)dzdx−λ∫Bzψ1zdx+λ∫Sz(ψ1z+ηxψ1x)dx=0.

Let n be the unit outer normal on the surface S and dl be the measure of arclength. It is easy to see that

∬R(zΔψ1+ψ1z)dzdx−∫Sz∂ψ1∂ndl+∫Bzψ1zdx=0.

Therefore we obtain that

∫S{|∇(ψ−λz)|22−λ22−(g~+2Ωλ)z−F(ω)−μ}η1dx−∫B(ψ−λz)ψ1zdx+∫S(ψ−λz)∂ψ1∂ndl−∬R{(ψ−λz)Δψ1+F′(ω)ω1}dzdx=0. (3.3)

Let us choose four different types of perturbation functions.

Firstly, we take η₁ = 0 and aim ψ₁ to be a solution of the elliptic problem

Δψ1=−ω1,in R,ψ1z=0,on B∂ψ1∂n=0,on S.

Then (3.3) reduces to

∬R{ψ−λz−F′(ω)}ω1dzdx=0.

The latter is valid for all smooth functions ω₁ with ∬_𝓡ω₁dzdx = 0, implying that

ψ−λz=F′(ω)+kinR, (3.4)

for some constant k. Therefore ω = y(ψ – λz – k). By taking λ = c, we obtain the first equation in (3.1).

Plugging (3.4) into (3.3), we obtain

∫S{|∇(ψ−λz)|22−λ22−(g~+2Ωλ)z−F(ω)−μ}η1dx−∫B(ψ−λz−k)ψ1zdx+∫S(ψ−λz−k)∂ψ1∂ndl=0. (3.5)
Secondly, we choose η₁ = 0 and ψ₁ is a solution of the elliptic problem

Δψ1=0,in R,ψ1z=0,on B∂ψ1∂n=f,on S,

where f is an arbitrary smooth function defined on S. Now (3.5) reduces to

∫S(ψ−λz−k)fdl=0.

Thus ψ – λz – k = 0 on S. Therefore, we get the third identity in (3.1) by taking λ = c.
Next we take η₁ = 0 and ψ₁ is a solution of the elliptic problem

Δψ1=0,in R,ψ1z=f,on B∂ψ1∂n=0,on S,

where f is an arbitrary smooth function defined on B with ∫_Bfdx = 0. Now (3.3) reduces to

∫B(ψ−λz−k)fdx=0.

Thus ψ – λz – k = k_B on B with a constant k_B. Therefore, we get the fourth identity in (3.1) by taking λ = c. Note that k_B = m when λ = c.
Finally, we take ψ₁ = 0 throughout 𝓡 with η₁ arbitrary, we obtain

∫S{|∇(ψ−λz)|22−λ22−(g~+2Ωλ)z−F(ω)−μ}η1dx=0

for all smooth functions η₁. Therefore,

|∇(ψ−λz)|22−λ22−(g~+2Ωλ)z−F(ω)−μ=0,onS.

It follows from (3.4) and (ii) that F′(ω) = 0 on S, and since F′ is strictly monotone, this is only possible, if ω is constant on S, e.g., ω = ω₀ on S. Hence letting λ = c we find that

|∇(ψ−cz)|22−(g~+2Ωc)z=c22+F(ω0)+μ,onS.

It remains to choose Q = c²+ 2μ + 2F(ω₀) in order to fullfil also the second equation in (3.1).□

Theorem 3.2

If (ψ, –η) ∈ 𝔽 is a solution of the equations (2.8) then

δH(ψ,−η)(ψ1,−η1)=0, (3.6)

for all (ψ₁, –η₁) ∈ 𝔻, where the functional 𝓗 is given as

H(ψ,−η)=∬R[|∇(ψ−cz)|22−(g~+2Ωc)z−Q2−F(−Δψ)]dzdx

so that the variation of 𝓗 at (ψ, –η) is given by δ𝓗 = δ𝓗₁ + δ𝓗₂ + δ𝓗₃, where

δH1=∬R{(ψ−cz)ω1−F′(ω)ω1}dzdx,δH2=∫S{|∇(ψ−cz)|22−(g~+2Ωc)z−Q2−F(ω)}η1dx,δH3=∫S(ψ−cz)∂ψ1∂ndl−∫B(ψ−cz)ψ1zdx.
Conversely, assume that (ψ, –η) ∈ 𝔽 satisfies (3.6) for all (ψ₁, –η₁) ∈ 𝔻. Then (ψ, –η) is a steady periodic water wave with the vorticity function y.

Proof

We first show that (3.6) holds for any (ψ₁, –η₁) ∈ 𝔻 if (ψ, –η) satisfies the equations (2.8). Note that –Δψ = ω = y(ψ – cz) implies F′(ω) = ψ – cz, so that δ𝓗₁ = 0. By the later two boundary conditions of (2.8) and the fact ∫_Bψ_zdx = 0, we know that

∫S(ψ−cz)∂ψ1∂ndl=0

and

∫B(ψ−cz)ψ1zdx=m∫Bψ1zdx=0,

and therefore δ𝓗₃ = 0. By the second equation of (2.8), we obtain

δH2=−∫SF(ω)η1dx=0

by letting F(y(0)) = 0 and using the fact ω = y(0) on S. The choice of F satisfying F(y(0)) = 0 does not change the vorticity function y since y = (F′)^–1.
Let (ψ, –η) ∈ 𝔽 be given and assume that for any (ψ₁, –η₁) ∈ 𝔻 (3.6) holds. As in the proof of Theorem 3.1 it follows that

ψ−cz=F′(ω)+kinR

for some constant k. Plugging this into (3.6), we have

0=∫S{|∇(ψ−cz)|22−(g~+2Ωc)z−Q2−F(ω)}η1dx+∫S(ψ−cz−k)∂ψ1∂ndl−∫B(ψ−cz−k)ψ1zdx. (3.7)

By choosing η₁ = 0 and

ψ1(x,z)=zχ(z+dϵ)f(x)

with a cut-off function χ ∈ C0∞ (ℝ) satisfying χ(z) = 1 for |z| ≤ 1 and f ∈ Cper1 (ℝ) with ∫_Bfdx = 0. Taking ϵ > 0 small enough, we obtain

∫B(ψ−cz−k)fdx=0

for any f ∈ Cper1 (ℝ) with ∫_Bfdx = 0, so that

ψ−cz−k=C,onB (3.8)

for some constant C. Moreover, for any f ∈ Cper1 (ℝ), we can construct ψ₁ satisfying ∂ψ1∂n=f on S and ψ₁ = 0 outside of a small neighborhood of S, so that one has

ψ−cz−k=0,onS. (3.9)

Now let us take ψ₁ = 0 throughout 𝓡 with η₁ arbitrary, we obtain

∫S{|∇(ψ−cz)|22−(g~+2Ωc)z−Q2−F(ω)}η1dx=0.

Thus

|∇(ψ−cz)|22−(g~+2Ωc)z−Q2−F(ω)=0,onS. (3.10)

Since F′ is strictly monotone and F′(ω) = ψ – cz – k = 0 on S, we know that ω is constant on S, e.g., ω = ω₀ on S. Hence

|∇(ψ−cz)|22−(g~+2Ωc)z−Q2−F(ω0)=0,onS.

Thus we obtain

Δψ=−(F′)−1(ψ−cz−k),for −η(x)<z<−d,|∇(ψ−cz)|2−2(g~+2Ωc)z=Q+2F(ω0),on z=−η(x),ψ−cz−k=C,on z=−η(x),ψ−cz−k=0,on z=−d. (3.11)

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 < 0. Thus we obtain the equations (3.1) by taking m = C and F(ω₀) = 0, and also obtain the equations (2.8) by adding the constant k to ψ. The choice of F satisfying F(ω₀) = 0 does not change the vorticity function y since y = (F′)^–1.□

Remark 3.3

It is worthwhile to note that the choice of the spaces 𝔽 and 𝔻 is important to ensure the efficiency of Theorem 3.2. To explain this point, we first note that the restriction ψ_z < c in the definition 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.
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 obtain^[3]

ψ−cz=k,onB,

and consequently also (3.9). Thus we get the equations (3.11) with C = 0, which is essentially the same as (2.8) with m = 0, and this contradicts the assumption (2.6).

3.3 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 (ψ₁, –η₁) ∈ 𝔻 and (ψ₂, –η₂) ∈ 𝔻. We further let ω₂ = –Δψ₂.

Theorem 3.4

Let (ψ, –η) ∈ 𝔽 be a solution of the equations (2.8). Then the second variation of 𝓗 is

δ2H=∫S(∂ψ∂z−c){∂ψ2∂nη1+∂ψ1∂nη2}dl+∬R{∇ψ2⋅∇ψ1−F″(ω)ω1ω2}dzdx+∫S(g~+2Ωc)−12∂∂z[|∇(ψ−cz)|2]η1η2dx.

Proof

We start from the formulas given in Theorem 3.2 and calculate further variation of each term. First,

δ2H1=∬R{ψ2ω1−F″(ω)ω2ω1}dzdx+∫S{(ψ−cz)ω1−F′(ω)ω1}η2dx=∬R{ψ2ω1−F″(ω)ω2ω1}dzdx=∬R{∇ψ2⋅∇ψ1−F″(ω)ω2ω1}dzdx−∫Sψ2∂ψ1∂ndl+∫Bψ2∂ψ1∂zdx,

where we used the fact ψ – cz = F′(ω) = 0 on S.

We can compute the remaining two terms as follows

δ2H2=∫S{ψxψ2x+(ψz−c)ψ2z−F′(ω)ω2}η1dx−∫S{ψxψxz+(ψz−c)ψzz−(g~+2Ωc)−F′(ω)ωz}η2η1dx=∫S{ψxψ2x+(ψz−c)ψ2z}η1dx−∫S{ψxψxz+(ψz−c)ψzz−(g~+2Ωc)}η2η1dx=∫S(ψz−c)∂ψ2∂nη1dl+∫S(g~+2Ωc)−12|∇(ψ−cz)|2zη2η1dx,

and

δ2H3=∫S{ψ2+(ψz−c)η2}∂ψ1∂ndl−∫Bψ2∂ψ1∂zdx.

Combining all the terms, we obtain

δ2H=∬R{∇ψ2⋅∇ψ1−F″(ω)ω2ω1}dzdx+∫S(ψz−c)∂ψ2∂nη1dl+∫S(g~+2Ωc)−12|∇(ψ−cz)|2zη2η1dx+∫S{(ψz−c)η2}∂ψ1∂ndl.

Now we obtain the desired equality and the proof is finished.□

The second variation of the functional 𝓗 is related to the stability properties of steady periodic waves. In order to explain this, let us take ψ₁ = ψ₂ and η₁ = η₂ in Theorem 3.4, to obtain the quadratic form

δ2H=∬R{|∇ψ2|2−F″(ω)|ω2|2}dzdx+2∫S(ψz−c)∂ψ2∂nη2dl+∫S(g~+2Ωc)−12|∇(ψ−cz)|2z|η2|2dx. (3.12)

Definition 3.5

The traveling wave (ψ, –η) is linear stable if for any (ψ₂, –η₂) ∈ 𝔻, the quadratic form δ²𝓗 is nonnegative.

Therefore, when we try to obtain the stability results, we need to find 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 ω_z > 0. Then a classical travelling wave is linearly stable if the surface is unperturbed.^[4]

Proof

The hypothesis ω_z > 0 implies that F″ < 0. Therefore the first integral in (3.12) is nonnegative. When the surface is unperturbed, we have η₂ = 0, and thus all the rest terms in (3.12) are zero. Therefore in this case, δ²𝓗 ≥ 0.□

In order to analyze the cases of perturbed surface, we first prove the following result.

Lemma 3.7

Assume that

y(m)(c−u|B)<g~+2Ωc. (3.13)

Then we have

(g~+2Ωc)−12|∇(ψ−cz)|2z≥0

on the thermocline S.

Proof

We calculate

{12|∇(ψ−cz)|2}z−(g~+2Ωc)= ψxψxz+(ψz−c)ψzz−(g~+2Ωc)= {−Pρ+2Ω(ψ−cz)+Γ(cz−ψ)−(g+g~)z}z.

It follows from (2.7) that

ΔPρ=2Ω(ψxx+ψzz)+2ψxxψzz−2ψxz2.

Since Δψ = –y(ψ – cz), we obtain

ΔΓ(cz−ψ)=−y′(ψ−cz)|∇(ψ−cz)|2−y(ψ−cz)(ψxx+ψzz)=−y′(ψ−cz)|∇(ψ−cz)|2+(Δψ)2.

Therefore,

Δ{−Pρ+2Ω(ψ−cz)+Γ(cz−ψ)−(g+g~)z}= −2ψxxψzz+2ψxz2−y′(ψ−cz)|∇(ψ−cz)|2+(Δψ)2= 2ψxz2+ψxx2+ψzz2−y′(ψ−cz)|∇(ψ−cz)|2≥0

because y′ < 0. Thus

Pρ−2Ω(ψ−cz)−Γ(cz−ψ)+(g+g~)z

is superharmonic and by the maximum principle [31], its minimum can only be attained on the thermocline S or on the upper boundary B of the centre layer unless it is a constant.

On the upper boundary z = –d of the centre layer, we have ψ_x = w = 0 and ψ – cz = m, thus

{−Pρ+2Ω(ψ−cz)+Γ(cz−ψ)−(g+g~)z}z= ψxψxz−(ψz−c)[ψxx+y(ψ−cz)]−(g~+2Ωc)= −(ψz−c)y(ψ−cz)−(g~+2Ωc)= y(m)(c−u|B)−(g~+2Ωc)<0

by using the condition (3.13). Thus the minimum of

Pρ−2Ω(ψ−cz)−Γ(cz−ψ)+(g+g~)z

must be attained on the thermocline S.

However, on S we know that the function is constant because

Pρ−2Ω(ψ−cz)−Γ(cz−ψ)+(g+g~)z=P0ρ−2Ω(ψ−cz)−Γ(cz−ψ)=P0ρ,

by the condition (2.5). Thus it is minimized at every point of S. Therefore

0<{Pρ−2Ω(ψ−cz)−Γ(cz−ψ)+(g+g~)z}z|S=−12|∇(ψ−cz)|2z+(g~+2Ωc)

by the Hopf maximum principle [31].□

Theorem 3.8

Assume that ω_z > 0 and that (3.13) is satisfied. Then a classical travelling wave is linearly stable if the surface is perturbed only normally.

Proof

The hypothesis ω_z > 0 is equivalent to F″ < 0. By Lemma 3.7 the first 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 ∂ψ₂/∂n = 0 on S. Therefore the second term in (3.12) vanishes. Therefore, δ²𝓗 ≥ 0 and the proof is finished.□

Acknowledgments

The authors are grateful to the anonymous referee for the careful reading, for pointing out interesting references, and for several suggestions which improved the first version of our contribution.

Jifeng Chu was supported by the Alexander von Humboldt-Stiftung of Germany, and the National Natural Science Foundation of China (Grants No. 11671118 and No. 11871273).

The publication of this article was funded by the Open Access Fund of Leibniz Universität Hannover.

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Received: 2020-05-25

Accepted: 2020-07-07

Published Online: 2020-08-22

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/anona-2020-0146

Keywords for this article

Steady periodic water waves; Equatorial flows; Vorticity; Variational formulations

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