Emil Novruzov

On the Propagation of Equatorial Waves Interacting With a Non-Uniform Current

Keywords: Wave-current interactions, Hamiltonian formulation, Mathematics Subject Classification (2010): 35Q35, 37N10, 37K05 ]ocnmp

We consider the propagation of equatorial waves of small amplitude, in a flow with an underlying non-uniform current. Without making the too restrictive rigid-lid approximation, by exploiting the available Hamiltonian structure of the problem, we derive the dispersion relation for the propagation of coupled long-waves: a surface wave and an internal wave. Also, we investigate the above-mentioned model of wave-current interactions in the general case with arbitrary vorticities.

Introduction

The propagation of ocean waves in equatorial regions presents great physical relevance and offers many mathematical challenges. For example, the equatorial region of the Pacific extends over 13000 km (about one third of its total length) and it is responsible for one of the most important climate phenomena: the El Nino events, starting every few years and having a global impact several months afterwards (see the discussion in [START_REF] Kessler | Oceanic equatorial waves and the 1991-1993 El Niño[END_REF]). Moreover, the ocean flow in this region is characterized by underlying depth-varying currents (westward at the surface and eastward at about 100-200 m depth, while at great depths the water is almost still -see [START_REF] Boyd | Dynamics of the Equatorial Ocean[END_REF]). A further complication is the fact that equatorial ocean regions present the strongest stratification, with a thermocline that is quite well-defined, separating two layers of practically constant density. The study of wave-current interactions in equatorial regions is of great current interest (see [START_REF] Constantin | Equatorial wave-current interactions[END_REF], [START_REF] Constantin | The dynamics of waves interacting with the Equatorial Undercurrent[END_REF], [START_REF] Constantin | An exact, steady, purely azimuthal equatorial flow with a free surface[END_REF], [START_REF] Ionescu-Kruse | Local stability for an exact steady purely azimuthal equatorial flow[END_REF] and references therein). As we noted above the El Nino events is one of the most important climate phenomena. In the present paper we consider the model of wave-current interactions (see [START_REF] Constantin | Equatorial wave-current interactions[END_REF]) with parameters which correspond to El Nino case. Also, we investigate the general case with arbitrary vorticities (the case of only non-zero constant vorticity in the near-surface homogeneous layer have been considered in [START_REF] Novruzov | On equatorial wave-current interactions[END_REF]).

Preliminaries

Near the Equator the Coriolis force acts like a waveguide, inducing the azimuthal propagation of waves and currents, and we can therefore investigate two-dimensional flows in the f -plane approximation. Following the setting of the recent paper [START_REF] Constantin | Equatorial wave-current interactions[END_REF], we consider equations of motion of the fluid in the domain bounded by flat bed z = -h from below and with the upper boundary z = h 1 + η 1 (x, t) which is a free surface of elevation (where h ≈ 8 and h 1 ≈ 0.24 are some non-dimensional constants, which reflect the fact that the near-surface layer is about 100-200 m deep, the average depth of the Pacific ocean near the Equator being about 4km). The thermocline z = η(x, t) separates the two layers of different constant densities in the domain under consideration. Thus, the thermocline divides the domain into two parts: the deep (colder) layer

D(t) = {(x, z) : -h < z < η(x, t)}
and the relatively shallow near-surface layer

D 1 (t) = {(x, z) : η(x, t) < z < h 1 + η 1 (x, t)} .
We consider an inviscid setting, in which the wind effects are captured by the winddrift near-surface current of constant vorticity (see the data in [START_REF] Ewing | Wind, wave and current data for the design of ships and offshore structures[END_REF] for the reasonable assumption that wind-generated currents are appropiately described by the assumption of constant vorticity, and note that non-zero vorticity means non-uniform currents -see the considerations in [START_REF] Constantin | Global bifurcation of steady gravity water waves with critical layers[END_REF]). Therefore, above the thermocline, in the region D 1 (t) the equations of motion are Euler's equations

u 1,t + u 1 u 1,x + 1 u 1,z + ω 1 = -p x 1,t + u 1 1,x + 1 1,z -ωu 1 = -p z -g,
where (u 1 , 1 ) is the velocity field in upper region, p is the pressure, ω = 0, 15 is the non-dimensional constant that captures the Coriolis effect due to the Earth's rotation and g ≈ 2 × 10 4 is the non-dimensional gravity. Taking into account the equations of mass conservation (for constant density) and the vorticity distribution typical of mixing in the near-surface layer, we have

u 1,x + 1,z = 0 u 1,z -1,x = γ 1 ,
where γ 1 < 0 is the vorticity (constant in the near-surface layer, the sign corresponding to the westward trade winds in the equatorial Pacific, for an eastward orientation of the horizontal axis, as in the paper [START_REF] Constantin | Equatorial wave-current interactions[END_REF]).

Denoting with (u, ) the velocity field in lower region D(t) the equations of motion are again Euler's equations

u t + uu x + u z + ω = - 1 1 + r p x , t + u x + z -ωu = -
with r 1 a constant that takes into account the stable density stratification (the less dense, warmer, fluid overlying the abyssal colder fluid). Below the thermocline, in the region D(t), the equations of mass conservation and vorticity distribution can be written as follows:

u x + z = 0, u z -x = γ,
where γ > is the vorticity, constant in the region below the thermocline (thus capturing the eastward orientation of the Equatorial Undercurrent, which occurs practically from about 100-200 m depth to 500m depth: see the data provided in [START_REF] Constantin | The dynamics of waves interacting with the Equatorial Undercurrent[END_REF] and [START_REF] Constantin | An exact, steady, purely azimuthal equatorial flow with a free surface[END_REF].

We now present the relevant boundary conditions. Firstly, at the free surface z = h 1 + η 1 (x, t) the pressure is the constant atmospheric pressure, so that we complement the above-metioned equations with the following the dynamic and kinematic boundary conditions

1 = η 1,t + u 1 η 1,x on z = h 1 + η 1 (x, t), 1 = η t + u 1 η x on z = η(x, t), = η t + uη x on z = η(x, t), = 0 on z = -h.
Besides, as noted in [START_REF] Constantin | Equatorial wave-current interactions[END_REF], the available field data for equatorial flows suggests a continuous transition between the two layers (velocity discontinuities across the thermocline are not detected in measurements), so that that it is required that a tangential velocity balance holds:

1 η x + u 1 = η x + u on z = η(x, t) ,
in addition to the condition that the pressure is continuous across the thermocline z = η(x, t).

The complicated nature of the governing equations described above is to some degree compensated by the fact that they have a Hamiltonian structure (see the discussion in [START_REF] Constantin | Equatorial wave-current interactions[END_REF]). We recall that Zakharov's discovery [START_REF] Zakharov | Stability of periodic waves of finite amplitude on the surface of a deep fluid[END_REF] of the Hamiltonian formulation for irrotational deep-water gravity waves represented an important advance, enabling in-depth studies (see the discussion in [START_REF] Craig | Mathematical aspects of surface waves on water. (Russian)[END_REF]). Subsequently Hamiltonian formulations for irrotational internal waves were provided in [START_REF] Craig | Hamiltonian long-wave expansions for free surfaces and interfaces[END_REF], for waves with constant vorticity [START_REF] Wahlén | A Hamiltonian formulation of water waves with constant vorticity[END_REF] and for wave-current interactions in stratified rotational flows with piecewise constant vorticity (see [START_REF] Constantin | Hamiltonian formulation for wave-current interactions in stratified rotational flows[END_REF] and [START_REF] Constantin | Equatorial wave-current interactions[END_REF]). In all these results, the key idea is that for harmonic functions one reduce the twodimensions to one (on the boundary) by means of Dirichlet-Neumann operators (scalar with no stratification, but if there is a thermocline, one needs a coupling of a scalar and a matrix Dirichlet-Neumann operator -see the discussion in [START_REF] Craig | Hamiltonian long-wave expansions for free surfaces and interfaces[END_REF], [START_REF] Constantin | Hamiltonian formulation for wave-current interactions in stratified rotational flows[END_REF] and [START_REF] Constantin | Equatorial wave-current interactions[END_REF]). For this to be analytically tractable, one needs the assumption of irrotational flow (as in [START_REF] Craig | Hamiltonian long-wave expansions for free surfaces and interfaces[END_REF]), or of constant vorticity (as in [START_REF] Constantin | Hamiltonian formulation for wave-current interactions in stratified rotational flows[END_REF] and [START_REF] Constantin | Equatorial wave-current interactions[END_REF], where a nonlinear separation result was proved, showing that, at the level of the nonlinear governing equations, wave-current interactions ]ocnmp[ E Novruzov for flows with constant vorticity correspond to a harmonic wave on a non-uniform purecurrent background state). To take advantage of this feature, we define

Φ(x, t) = ϕ(x, η(x, t), t), Φ 1 (x, t) = ϕ 1 (x, η(x, t), t) , and Φ 2 (x, t) = ϕ 1 (x, h 1 + η 1 (x, t), t) ,
where ϕ in D(t) and ϕ 1 in D 1 (t) are harmonic perturbed velocity potentials which are defined as follows:

u = ϕ x + γ (z + h) and = ϕ z in D(t), u 1 = ϕ 1,x + γ 1 z + γh and 1 = ϕ 1,z in D(t). Then, let u = (η, η 1 , ξ, ξ 1 ) T ,
where the superscript denotes the transpose and

ξ = (1 + r) Φ -Φ 1 , ξ 1 = Φ 2 ,
with a positive constant r (defined above; typically r = 10 -3 -see [START_REF] Kessler | Oceanic equatorial waves and the 1991-1993 El Niño[END_REF]).

Using the Fourier transform

∧ f = R f (x) e -ikx dx
for f in the Schwartz class S(R) in each component of u above-mentioned linearised equations of motion are transformed for any fixed k ∈ R into the linear autonomous system of ordinary differential equations (see [START_REF] Constantin | Equatorial wave-current interactions[END_REF]):

∂ t ∧ u (k, t) = M (k) ∧ u (k, t) , (1)
where

M (k) =   -iγhk + iµkΘ (k) iµ 1 k sec h (h 1 k) Θ (k) k 2 Θ (k) k 2 sec h (h 1 k) Θ (k) iµk sec h (h 1 k) Θ (k) -iΓ 1 k + iµ 1 k Θ 1 (k) k 2 sec h (h 1 k) Θ (k) k 2 Θ 1 (k) Γ -µ 2 Θ (k) -µµ 1 sec h (h 1 k) Θ (k) -iγhk + iµkΘ (k) iµk sec h (h 1 k) Θ (k) -µµ 1 sec h (h 1 k) Θ (k) ωΓ 1 -g -µ 2 1 Θ 1 (k) iµ 1 k sec h (h 1 k) Θ (k) -iΓ 1 k + iµ 1 kΘ 1 (k)   and µ = (1+r)γ-γ 1 +rω 2 , µ 1 = γ 1 +ω 2 , ω = 0, 15 , Γ = -r ∧ g , ∧ g = g -ωhγ, Θ (k) = tanh(hk) k[1+r+tanh(hk) tanh(h 1 k)] , Θ 1 (k) = tanh(hk)+(1+r) tanh(h 1 k) k[1+r+tanh(hk) tanh(h 1 k)] .
Here a fixed nondimensional value of k corresponds to a harmonic oscillation of the free surface and/or of the thermocline of wavelength 1000π |k| m. Therefore we have that the unique solution to [START_REF] Boyd | Dynamics of the Equatorial Ocean[END_REF] with initial data

∧ u 0 (k) = R u 0 (x) e ikx dx, is ∧ u (k, t) = e M (k)t ∧ u 0 (k)
for t ≥ 0. This solution corresponds, by means of the inverse Fourier transform, to the solution

u (x, t) = 1 2π R e M (k)t ∧ u 0 (k) e ikx dk
of the linearized problem under consideration, with initial data u 0 ∈ S (R). Note that the Schwartz class is good to describe localized waves, which arise as perturbations of a pure-current background state with a flat free surface and a flat thermocline.

An eigenvalue with non-zero real part leads to instability due to growth in time (see the discussion in [START_REF] Constantin | Equatorial wave-current interactions[END_REF]). On the other hand, to a purely imaginary eigenvalue with frequency |k| 2π , propagating at the constant speed c and the general solution of ( 1) is a linear combination of solutions of the form t n e Λ(k)t U , where n is a nonnegative integer, U is a constant vector and Λ(k) is an eigenvalue of M (k) (see the discussion in [START_REF] Constantin | Equatorial wave-current interactions[END_REF]). This way, the linear wave propagation is reduced to the study of the eigenvalues of the matrix M (k).

Λ(k) = -ikc
Finding accurate estimate for the eigenvalues of M (k) turns out to be quite a challenge (see [START_REF] Constantin | Equatorial wave-current interactions[END_REF] and [START_REF] Novruzov | On propagation of linear waves for model of equatorial wave-current interactions[END_REF]), but this is at the core of understanding the evolution of wave packets at the surface and along the thermocline. In this context, it is worth to note that, Λ(k) ∈ C is an eigenvalue of M (k) with eigenvector (v 1 , v 2 , v 3 , v 4 ) T if and only if

λ(k) = iΛ(k) k
is an eigenvalue with corresponding eigenvector (v 1 , v 2 , iv 3 , iv 4 ) T of the real matrix

R (k) =     γh -µΘ (k) -µ 1 sec h (h 1 k) Θ (k) Θ (k) sec h (h 1 k) Θ (k) -µ sec h (h 1 k) Θ (k) Γ 1 -µ 1 Θ 1 (k) sec h (h 1 k) Θ (k) Θ 1 (k) -Γ + µ 2 Θ (k) µµ 1 sec h (h 1 k) Θ (k) γh -µΘ (k) -µ sec h (h 1 k) Θ (k) µµ 1 sec h (h 1 k) Θ (k) -ωΓ 1 + g + µ 2 1 Θ 1 (k) -µ 1 sec h (h 1 k) Θ (k) Γ 1 -µ 1 Θ 1 (k)     .

The propagation of linear waves in the El Nino setting

A careful consideration of the field data reveals that h 1 /h 1 typically. Moreover, during an El Nino event the trade winds in the equatorial mid-Pacific loose considerably in intensity and, as a consequence, the depth of the thermocline diminishes a great deal throughout the Pacific. These physical consideration motivate us to investigate the linear problem in the limiting case h 1 → 0.

The following lemma holds.

]ocnmp[

Lemma 1. In the limiting case h 1 → 0 we have that the matrix R(k) has only real eigenvalues, given explicitly by:

λ 1,2 = γh, λ 3,4 = γh -(µ 1 Θ + µΘ) ± Θ 2 (µ 1 + µ) 2 + Θ ∧ g (1 + r).
Proof. To investigate roots of the characteristic polynomial

p(λ) = det(R -λI 4 ) ,
we first perform two sets of operations to simplify its structure. Add the first row multiplied by µ to the third row, and the second row multiplied by µ 1 to the fourth row, and in the outcome add the third column multiplied by µ to the first column and the fourth column multiplied by µ 1 to the second column to obtain a determinant expressed in terms of λ, which corresponds to the wave speed relative to the maximum speed of the current, given by

Λ p (λ) = -X 0 Θ Θ 0 γ 1 h 1 -X Θ Θ -Γ -2µX 0 -X 0 0 ∧ g -2µ 1 X 0 γ 1 h 1 -X , where h 1 = 0, X = λ -γh, ∧ g = g -ωhγ, Γ = -r ∧ g.
Therefore, we have

Λ p (λ) = -X 0 Θ Θ 0 -X Θ Θ r ∧ g -2µX 0 -X 0 0 ∧ g -2µ 1 X 0 -X = X 4 + (2µ 1 Θ + 2µΘ) X 3 + -Θ ∧ g -r ∧ gΘ X 2 .
Now, it is easy to see that all the roots of the equation Λ p (λ) = 0 are real. Indeed, we have

Λ p (λ) = X 4 + (2µ 1 Θ + 2µΘ) X 3 + -Θ ∧ g -r ∧ gΘ X 2 = 0 or X 2 + (2µ 1 Θ + 2µΘ) X + -Θ ∧ g -r ∧ gΘ X 2 = 0. Consequently, λ -γh -(µ 1 Θ + µΘ) + Θ 2 (µ 1 + µ) 2 + Θ ∧ g (1 + r) × λ -γh -(µ 1 Θ + µΘ) -Θ 2 (µ 1 + µ) 2 + Θ ∧ g (1 + r) (λ -γh) 2 = 0. Therefore, λ 1,2 = γh -(µ 1 Θ + µΘ) ± Θ 2 (µ 1 + µ) 2 + Θ ∧ g (1 + r), λ 3,4 = γh.
Taking into account that Θ = tanh(hk) k(1+r) , ∧ g = g -ωhγ (where g >> γhω) and µ 1 + µ = (1+r)(γ+ω) 2

we get

λ 1,2 = γh - tanh (hk) 2k (γ + ω) ± tanh 2 (hk) k 2 (γ + ω) 2 + tanh (hk) k (g -ωhγ), (2)
λ 3,4 = γh. (3)
Therefore, the matrix M (k) has the following purely imaginary eigenvalues

Λ 1,2 = - ik 2 γh - tanh (hk) 2k (γ + ω) ±ik tanh 2 (hk) k 2 (γ + ω) 2 + tanh (hk) k (g -ωhγ)
.

Λ 3 = -ikγh.
which corresponding to oscillations with speeds given by formulas ( 2) -(3). Thus, the previous considerations prove the main result.

Theorem 1. In an El Nino setting, the propagation of linear waves generated by equation ( 1) is characterized by the fundamental oscillation modes propagating at the constant speeds given by the formulas ( 2) -(3).

4 The propagation of linear waves for a case of non-zero constant vorticities.

In [START_REF] Novruzov | On propagation of linear waves for model of equatorial wave-current interactions[END_REF] the model of wave-current interactions (see [START_REF] Constantin | Equatorial wave-current interactions[END_REF]) ) in the case of the absence of vorticity has been considered. However, this permits a more detailed analysis, and then the model with non-zero vorticities can be regarded as a perturbation of the the irrotational setting. In this respect, the above-mentioned paper [START_REF] Novruzov | On propagation of linear waves for model of equatorial wave-current interactions[END_REF] and [START_REF] Novruzov | On equatorial wave-current interactions[END_REF] where γ = 0 and γ 1 = 0, are closely connected with general case and reasoning from here can be adapted to the case with nonzero γ 1 and γ if we replace λ with X = λ -γh (see Lemma 1 in [START_REF] Novruzov | On propagation of linear waves for model of equatorial wave-current interactions[END_REF]), taking into account some technical details.

Thus, we can investigate the model with non-zero vorticities based on general scheme of proofs from [START_REF] Novruzov | On propagation of linear waves for model of equatorial wave-current interactions[END_REF][START_REF] Novruzov | On equatorial wave-current interactions[END_REF] with necessary references to the above-mentioned papers.

As we noted in Intoduction, Λ(k

) ∈ C is an eigenvalue of M (k) with eigenvector (v 1 , v 2 , v 3 , v 4 ) T if and only if λ(k) = iΛ(k)
k is an eigenvalue with corresponding eigenvector (v 1 , v 2 , iv 3 , iv 4 ) T of the real matrix R (k). Moreover, the following leema holds (see [START_REF] Constantin | Equatorial wave-current interactions[END_REF], page 28) Lemma 2. The matrix M (k) has four distinct purely imaginary eigenvalues for 0 < |k| < 64.

Thus, we can investigate the model in the physically relevant regime 0 < |k| < 64, with non-zero vorticities based on general scheme of proofs from [START_REF] Novruzov | On propagation of linear waves for model of equatorial wave-current interactions[END_REF][START_REF] Novruzov | On equatorial wave-current interactions[END_REF] with necessary references to the above-mentioned papers.

]ocnmp[

E Novruzov

Lemma 3. The eigenvalues of R(k) for 0 < |k| < 64 satifies the following estimates:

|λ i | ≤ ( 16 k 2 tanh 2 (k (h + h 1 )) + 2γ 2 1 h 2 1 -8γ 1 h 1 µ 1 Θ 1 + 2Θ 1 γ 2 1 h 1 + 4 k (g -ωγh) tanh k (h + h 1 ) -8µµ 1 Θ tanh (h 1 k) k ) 1/2 + γh. (4)
Proof. To investigate roots of the characteristic polynomial p(λ) = det(R -λI 4 ), we first perform two sets of operations to simplify its structure. Add the first row multiplied by µ to the third row, and the second row multiplied by µ 1 to the fourth row, and in the outcome add the third column multiplied by µ to the first column and the fourth column multiplied by µ 1 to the second column to obtain a determinant edxpressed in terms of X = λ -γh, corresponding to the wave speed relative to the maximum speed of the

Λ p (λ) = -X 0 Θ sΘ 0 γ 1 h 1 -X sΘ Θ 1 -Γ -2µλ 0 -X 0 0 ∧ g -2µ 1 X 0 γ 1 h 1 -X , where s = sech (h 1 k), Γ = rωγh -rg, ∧ g = g + γ 2 1 h 1 -ωγh. We obtain: X 4 + (-2γ 1 h 1 + 2µ 1 Θ 1 + 2µΘ) X 3 + (γ 2 1 h 2 1 -Θ 1 γ 2 1 h 1 -Θ 1 (g -ωγh) -4µΘγ 1 h 1 +4µµ 1 ΘΘ 1 -s 2 Θ 2 -rΘ (g -ωγh))X 2 + (2γ 1 h 1 rΘ (g -ωγh) + 2µΘγ 2 1 h 2 1 -2 rgµ 1 + µg + µγ 2 1 h 1 -rωγhµ 1 ΘΘ 1 -s 2 Θ 2 )X- γ 2 1 h 2 1 rΘ (g -ωγh) + (rg 2 + rγ 2 1 h 1 (g -ωγh) -rωγhg) ΘΘ 1 -s 2 Θ 2 = 0. (5)
In view of above-mentioned Lemma 2 (see [START_REF] Constantin | Equatorial wave-current interactions[END_REF] ) all the roots of this equation are real. Taking into account coefficients of equation ( 5) we have following equalities:

X 1 + X 2 + X 3 + X 4 = -2γ 1 h 1 + 2µ 1 Θ 1 + 2µΘ, (6)
X 1 (X 2 + X 3 + X 4 ) + X 2 (X 3 + X 4 ) + X 3 X 4 = γ 2 1 h 2 1 -Θ 1 γ 2 1 h 1 -Θ 1 (g -ωγh) -4µΘγ 1 h 1 +4µµ 1 Θ tanh (h 1 k) k -rΘ (g -ωγh) , (7)
where X i are roots of the equation [START_REF] Constantin | An exact, steady, purely azimuthal equatorial flow with a free surface[END_REF]. Thus, by squaring the right-hand and left-hand sides of equation ( 6) and using equation (7) multiplied by 2, carrying same manipulations as in papers [START_REF] Novruzov | On propagation of linear waves for model of equatorial wave-current interactions[END_REF][START_REF] Novruzov | On equatorial wave-current interactions[END_REF] we obtain the validity of Lemma 3 and since the further proof of this fact repeats almost word for word the corresponding proof from the above-mentioned papers, we omit the further details.

Theorem 2. The propogation of linear waves generated by equation ( 1) is characterized by the the fundamental oscillation modes propagating at the constant speeds given by the formulas ( 12) -(13). Moreover, the following estimates hold:

|λ i | ≤ ( 16 k 2 tanh 2 (k (h + h 1 )) + 2γ 2 1 h 2 1 -8γ 1 h 1 µ 1 Θ 1 + 2Θ 1 γ 2 1 h 1 + 4 k (g -ωγh) tanh k (h + h 1 ) -8µµ 1 Θ tanh (h 1 k) k ) 1/2 + γh.

  with c ∈ R\{0} of the 4 × 4 matrix M (k), with corresponding eigenvector b(k) = 0, we can associate the oscillatory mode solution e M (k)t b(k) = e -ikx b(k) .Also, a purely imaginary eigenvalue Λ(k) = -ikc of M (k) corresponds to the fundamental oscillation mode e ik(x-ct) b(k)

c The author(s). Distributed under a Creative Commons Attribution 4.0 International License

Remark. From physical point of view γ 1 is a negative constant (see [START_REF] Constantin | Equatorial wave-current interactions[END_REF], page 11) greater than 1 ( approximately equal to -12.5) and since r < 1 (and therefore µ 1 = γ 1 +r 2 < 0 ) we get that term 8γ 1 h 1 µ 1 Θ 1 is a positive. Thus, formula (4) acquires a more simpler form:

Besides, to obtain roots of the equation we also can use the following standard substitution

and the problem boils down to the consideration of the following equation

where

Following a procedure from the paper [START_REF] Novruzov | On propagation of linear waves for model of equatorial wave-current interactions[END_REF] under notations

2 we consider the following three cases generated by equation y 3 -3y = d which is related to resolvent cubic of the above-mentioned quartic equation ( 9): i) Q < 0 (which corresponds to the case |d| < 2); ii) Q = 0 (which corresponds to the case |d| = 2); iii) Q > 0 (which corresponds to the case |d| > 2). Then, by the similar manipulations as in [START_REF] Novruzov | On propagation of linear waves for model of equatorial wave-current interactions[END_REF] under condition p 1 = -1 12 p 2 -l < 0, we obtain that M (k) has the four disjoint purely imaginary eigenvalues:

where z = y 1 36 p 2 + 1 3 l -1 3 p and y are some constants depending on d as in formulas below (according to cases (i-iii)

.

In other words, by formulas ( 10) -(11) we get four types of oscillations with speeds given by formulas ( 12) -(13) below

where 1 12 p 2 + l > 0 (note that since g is a dominated term, this condition is not very restrictive. Besides for long-waves, in our case µ 1 and µ are sufficiently close to zero, thus this condition holds directly).

The previous considerations prove the following result.