domain: sde spi nlin

text:
M A Manna

S Noubissié

J Touboul

B Simon

R A Kraenkel

The role of constant vorticity on weakly nonlinear surface gravity waves

This manuscript describes the derivation of systems of equations for weakly nonlinear gravity waves in shallow water in the presence of constant vorticity. The derivation is based on a multi-layer generalization of the traditional columnar Ansatz. A perturbative development in a nonlinear parameter and a dispersive parameter allow us to obtain sets of equations, for the horizontal fluid velocity and the free surface, able to describe propagation of weakly nonlinear and dispersive surface waves moving in water with some prescribed initial constant vorticity. We have shown that vorticity plays a central role on the dispersive properties of the system. When it is weak, it acts as a correction in linear and nonlinear dispersive terms. When stronger, it can also influence the nondispersive behavior of the system. Explicit steady solutions of the system corresponding to zero, weak, normal or strong vorticity are obtained. They correspond to solitary waves. Evolution of the soliton celerity, amplitude and width for these four cases are discussed.

Introduction

When propagating in coastal waters, surface waves often encounter currents. These currents, with intensities ranging from weak to very strong are generated through various mechanisms, such as oceanic circulation, tides, wind action or wave breaking. Given these generation processes, the currents are often observed to vary with depth, and result in an underlying vorticity. Such background vorticity, included within strong tidal currents [START_REF] Soulsby | Tidal current boundary layers[END_REF] or in wind driven currents [START_REF] Johnson | A problem in the classical theory of water waves: weakly nonlinear waves in the presence of vorticity[END_REF] could be important and should be taken into account in modeling the propagation of water waves [START_REF] Touboul | Extended mild-slope equation for surface waves interacting with a vertically sheared current[END_REF]. This underlying vorticity is especially observed in shallow water environments. For instance, strong currents, linearly sheared, were observed in the surf zone, in strong rip currents, in situ [START_REF] Macmahan | Ripex: observations of a rip current system[END_REF] or in laboratory experiments [START_REF] Haas | Laboratory measurements of the vertical structure of rip currents[END_REF]. More recently, a similar vertical structure of the current was observed over coral reefs [START_REF] Sous | Circulation patterns in a channel reef-lagoon system, Ouano lagoon[END_REF].

From a theoretical point of view the role played by constant or variable vorticity constitutes a vast and classical subject in fluid mechanics. Da Silva and Peregrine have studied [START_REF] Da Silva | Steep steady surface waves on water of finite depth with constant vorticity[END_REF] steady surface waves in water of finite depth with constant vorticity. In Refs. [START_REF] Constantin | Symmetry of steady periodic gravity water waves with vorticity[END_REF][START_REF] Constantin | Symmetry of steady deep water water waves with vorticity[END_REF][START_REF] Constantin | Exact steady periodic water waves with vorticity[END_REF][START_REF] Constantin | Rotational steady water waves near stagnation[END_REF][START_REF] Constantin | Steady periodic water waves with constant vorticity: Regularity and local bifurcation[END_REF], Constantin et al. have developed a complete study of exact steady periodic water waves with vorticity. Castro and Lannes [START_REF] Castro | Fully nonlinear long-wave models in the presence of vorticity[END_REF] investigated fully nonlinear long-waves under the action of vorticity with a Green-Naghdi equation. Numerical studies of this problem can be found in references Ko and Strauss [14,[START_REF] Ko | Effect of vorticity on steady water waves[END_REF] and Milewski, Vanden-Broeck and Z. Wang [START_REF] Milewski | Dynamics of steep two-dimensional gravity-capillary solitary waves[END_REF] However very few works have concentrated to modeling nonlinear dynamics of long surface waves in finite depth under the actions of a sheared current. The pioneering work in this domain was that of Benjamin [START_REF] Benjamin | The solitary wave on a stream with arbitrary distribution of vorticity[END_REF] who generalized the classical solitary wave theory to surface waves under the action of vorticity. Freeman and Johnson [START_REF] Freeman | Shallow water waves in shear flows[END_REF] derived a Korteweg-de Vries equation in a flow of arbitrary vorticity. Choi has derived the Green-Naghdi equation and the associated Boussinesq equation for small amplitude wave. In Ref. [START_REF] Johnson | A problem in the classical theory of water waves: weakly nonlinear waves in the presence of vorticity[END_REF] Johnson studied the problem of two-way propagation as described by the Boussinesq equation and derived a new Boussinesq-type equation valid for constant vorticity. The solitary-wave solution for this new equation is exhibited. The Camassa-Holm equation for water waves with constant vorticity and its solitary-wave solution were described. Kharif, Abid and Touboul have studied the action of vorticity on the rogue waves dynamics [START_REF] Kharif | Rogue waves in shallow water in the presence of a vertically sheared current[END_REF]. More recently Kharif and Abid [START_REF] Kharif | Nonlinear water waves in shallow water in the presence of constant vorticity: A whitham approach[END_REF] have studied the role played by constant vorticity on surface waves using a generalized Whitham equation [START_REF] Whitham | Linear and Nonlinear Waves[END_REF].

The main objective of this work is to derive higher order evolution equations for weakly nonlinear long surface gravity waves in shallow water in the presence of constant vorticity beyond all orders in nonlinearity, dispersion and nonlinear dispersion. The employed method is based on the Johnson's work [START_REF] Johnson | Camassa-Holm, Korteweg-de Vries and related models for water waves[END_REF]. A generalized columnar pattern Ansatz is introduced, together with the two perturbative parameters : the nonlinear parameter α = a/h < 1 and the dispersive parameter δ = kh < 1, where h refers to the constant depth of the fluid while a and k are a typical amplitude and wavenumber of the wave being studied. This generalization is based on two main hypothesis. Firstly, the horizontal velocity of the fluid is represented by means of an infinite series in powers of z 2 δ 2 . Secondly, the conservation of vorticity is enforced, which provides a consistent closure of the problem. This results in a generalized Serre-Green-Naghdi model with vorticity equivalent to a multi-layer model in place of the single-layer model (see Kim et al. [23,[START_REF] Kim | A strongly-nonlinear model for water waves in water of variable depth -The irrotational Green-Naghdi model[END_REF]). An interesting advantage of this approach is that it provides an inter-comparison of the terms describing frequency dispersion, and nonlinear dispersion, even when they involve vorticity. The main result in relation to previous studies is that Ω plays a fundamental role on the dispersive properties of the system. Furthermore, when it is of stronger order of magnitude, vorticity also acts as a corrective term in the non dispersive terms of the system. The role of vorticity can thus be discussed, considering nil, weak, normal or strong vorticity. Furthermore, solitary wave solutions are derived in these four cases, and the role of vorticity on the solitonic properties are fully investigated.

The combined effect of vorticity, dispersion, and nonlinearity on long water waves

General equations of the problem

We consider the water particles to be located by a to a two-dimensional Cartesian coordinate system with origin 0 and axes x, z, z being orientated upward in the vertical direction. The governing equations are the Euler equations and the equation of mass conservation in x, z and t (time) completed with appropriate surface and bottom boundary conditions.

Before perturbations the fluid lies between the impermeable bottom at z = 0 and the still water free surface at z = h with h constant. The undisturbed initial state is incompressible, inviscid, with zero surface tension and with a superimposed current profile in the x direction and depending only linearly on z i.e ⃗ U = Ωz⃗ e x with ⃗ e x the unitary vector in the x direction. The perturbations to the free surface, horizontal and vertical velocities and pressure (relative to the hydrostatic pressure in the undisturbed initial state) are η(x, t), u(x, z, t), w(x, z, t) and p(x, z, t) respectively. The governing equations are made nondimensional as follow (primes mean dimensionless variables):

z ′ = z/h, x ′ = kx, t ′ = kt √ gh, η ′ = η/h, with
k an average or a typical wavenumber, a the amplitude and g the gravity. This brings to

u ′ = u √ gh , w ′ = w kh √ gh , p ′ = p ρgh , Ω ′ = Ω k √ gh ,
We now introduce the two small parameters α = a/h ≪ 1 and δ = kh ≪ 1, which are used as perturbative parameters, as it is classically done in small perturbative methods for studying surface water waves problems. The final form of the governing equations require to scale u ′ , w ′ , p ′ and η ′ with α i.e.:

(u ′ , w ′ , p ′ , η ′ ) → α(u ′ , w ′ , p ′ , η ′ ).
As a result we obtain for 0 ≤ z ≤ 1 + αη and dropping the primes

u x + w z = 0, u t + (δΩz + αu) u x + w(δΩ + αu z ) + p x = 0, w t + (δΩz + αu) w x + αww z + αp z + 1 αδ 2 = 0 (1) p = 0 and η t + [δΩ(1 + αη) + αu] η x -w = 0, on z = 1 + αη; and w = 0, on z = 0. (2)
A classical treatment of the system (1)-(2) relies on its linearization, which admits solutions for the horizontal and vertical velocities u L and w L as: u L (x, z, t) = 2aiδ exp{iθ (x, t)} cosh δz, w L (x, z, t) = 2a exp i{θ (x, t)} sinh δz with exp{iθ (x, t)} = exp{i(xσ t)} and σ the linear frequency of expression

σ = δΩ - Ω 2 tanh δ ± { Ω 2 tanh 2 δ 4 + tanh δ 2 } 1 2
.

The Johnson derivation of the single-layer model equations (in Ref. [START_REF] Johnson | Camassa-Holm, Korteweg-de Vries and related models for water waves[END_REF]) means, from a physical mathematical point of view, the lower order in δ of the linear horizontal velocity solution u L proportional to exp iθ (x, t) is replaced by an unknown function u(x, t) which is supposed to bring an approximate solution of the entire nonlinear problem. Our approach can be viewed as a generalization of this columnar hypothesis where each one of the orders δ 0 z 0 exp{iθ (x, t)}, δ 2 z 2 exp{iθ (x, t)}, δ 4 z 4 exp{iθ (x, t)}, . . . is replaced by

u 0 (x, t), δ 2 z 2 u 2 (x, t), δ 4 z 4 u 4 (x, t), . . . so, u(x, z, t) = ∞ ∑ n=0 u 2n (x, t)δ 2n z 2n , ∀n (3)
In these two developments, only even terms are retained in the decomposition of u, while uneven terms are kept for the decomposition in w. This assumption, classical in Boussinesq type equations, is justified by the Taylor's expansions of cosh(δz) and sinh(δz) respectively. From the continuity equation and the boundary condition w = 0 on z = 0 it follows that

w(x, z, t) = - ∞ ∑ n=0 u 2n,x δ 2n z 2n+1 2n + 1 . (4)
Substituting u(x, z, t) and w(x, z, t) in the equation for w t integrating for z ∈ [z, 1 + αη] and using p = 0 at z = 1 + αη we obtain p(x, z, t)

p = 1 + αη -z - δ 2 2 [ (1 + αη) 2 -z 2 ] [ u 0,xt + αu 0 u 0,xx -αu 2 0,x ] - δ 3 3 [ (1 + αη) 3 -z 3 ] Ωu 0,xx + O ( δ 4
) .

(

) 5
Using the expression of p in u t and integrating for z ∈ [0, 1 + αη] lead to

u 0,t + αu 0 u 0,x + η x = δ 2 3(1 + αη) {[ (1 + αη) 3 ( u 0,xt + αu 0 u 0,xx -αu 2 0,x )] x -(1 + αη) 3 ( u 2,t + α(u 0 u 2,x -u 2 u 0,x ) )} + δ 3 Ω { (1 + αη) 3 ( u 2,x 6 - u 0,xxx 4 -αu 0,xx η x (1 + αη) -1 )} + O ( Ωδ 5 , δ 4 ) . (6)
Finally equation for η(x, t) using u(x, z, t) and w(x, z, t) at z = 1 + αη gives

η t + (1 + αη) δΩη x + [u 0 (1 + αη)] x + δ 2 3 [ u 2 (1 + αη) 3 ] x + O ( δ 4 ) = 0. (7)
Expressions ( 6) and (7) are a system of two coupled equations in three fields (and theirs derivatives): u 0 , u 2 and η. In order to reduce this system to only two fields u 0 (and derivatives) and η we use the conservation of vorticity equation

D ⃗ ω Dt = 0
We obtain

ω t + (δΩz + αu)ω x + αwω z = 0, with ω = | ⃗ ω|, α ⃗ ω = ( δΩ + αu z -αδ 2 w x ) ⃗ e y , (8)
with ⃗ e y the unitary vector in the y direction. This yields the following expression

u 2n+2 = - u 2n,xx (2n + 1)(2n + 2) , n = 0, 1, 2, . . . (9)
This recurrence allows us to reduce ( 6) and [START_REF] Da Silva | Steep steady surface waves on water of finite depth with constant vorticity[END_REF], at any degree of approximation, in a system in u 0 , η and their derivatives.

Coming back to Eqs. ( 6) and [START_REF] Da Silva | Steep steady surface waves on water of finite depth with constant vorticity[END_REF], we might substitute u 2 (x, t) with u 0 (x, t) by using the recurrence [START_REF] Constantin | Symmetry of steady deep water water waves with vorticity[END_REF]. Then, if dropping terms of greater order than δ 3 and rearranging, we obtain

u 0,t + αu 0 u 0,x + η x - δ 2 2 { (1 + αη) 2 ( u 0,xt + αu 0 u 0,xx -αu 2 0,x )} x -δ 3 Ω { 1 3 (1 + αη) 3 u 0,xxx + α(1 + αη) 2 u 0,xx η x } + O ( δ 4 , Ωδ 4 ) = 0, (10)
η t + (1 + αη)δΩη x + [u 0 (1 + αη)] x - δ 2 6 [ u 0,xx (1 + αη) 3 ] x + O ( δ 4 ) = 0. ( 11
)
The system of Eqs. ( 10)-( 11) is obtained for small values of the parameter α and δ. However, it is emphasized that no assumption was performed on the order of magnitude of Ω. The purpose of this work is to analyze the role played by the vorticity on this system. Following this purpose in the next section, a further assumption is to be introduced, the order of magnitude of the vorticity being compared to the dispersive parameter of the system δ. Indeed, three cases will be distinguished, which are weak, normal or strong vorticity, assuming respectively that

Ω = Ω 0 δ = O (δ), Ω = Ω 0 = O (1), and Ω = Ω 0 /δ = O ( δ -1 )
. In this further decomposition, Ω 0 is, of course, of order 0 in δ, setting the value of the constant vorticity.

Long wave evolution equations 2.2.1. The irrotational case Ω = 0

We shall start by considering the zero vorticity case, with Ω = Ω 0 = 0. Starting from Eqs. ( 10) and ( 11), and introducing this further assumption, we obtain

u 0,t + αu 0 u 0,x + η x - δ 2 2 { u 0,xt + αu 0 u 0,xx -αu 2 0,x + 2αηu 0,xt } x = 0, (12)
η t + u 0,x + α(u 0 η) x - δ 2 6 u 0,xxx - αδ 2 2 (u 0,xx η) x = 0. ( 13
)
These equations are different from the classical Green-Naghdi system because we have used the conservation of the vorticity theorem. En passant, we note that if D ⃗ ω/Dt = 0 had been used independently from the order in δ, the Green-Naghdi system would be reduced (as it can easily be shown) to the Shallow Water Equation.

We now make a further assumption, by searching for traveling waves. Though, we might introduce the change in variables r = xct, c being the speed of propagation. We thus exclude waves traveling to the (-x) direction. Inserting this new variable in the system above, and integrating equation ( 12), we can eliminate the unknown δ from the system. We thus obtain the equation

( 1 -c 2 ) u 0 + 3α 2 cu 2 0 + δ 2 ( c 2 2 - 1 6
)

u 0,rr + αδ 2 2 [( 2c 3 -3c ) u 0 u 0,rr + cu 2 0,r ] = 0. ( 14
)
2.2.2. The weak vorticity case, Ω = Ω 0 δ Now, we investigate the weak vorticity case, by introducing the relation Ω = Ω 0 δ = O(δ) in Eqs. ( 10) and [START_REF] Constantin | Rotational steady water waves near stagnation[END_REF].

To introduce this assumption in the system, we proceed as done in Section 2.2.1. By replacing the unknown u 2 with its expression in u 0 , introducing the traveling wave hypothesis, integrating the system, and eliminating the unknown η, we obtain the equation on u 0 , which reads

(1 -c 2 )u 0 + 3α 2 cu 2 0 + δ 2 [( c 2 2 - 1 6
)

u 0,rr + Ω 0 cu 0 ] + αδ 2 2 [( 2c 3 -3c ) u 0 u 0,rr +cu 2 0,r + (c 2 -1)Ω 0 u 2 0 ] = 0. ( 15
)
2.2.3. The rotational case, Ω = Ω 0 ̸ = 0 Again, we shall start from Eqs. ( 10) and [START_REF] Constantin | Rotational steady water waves near stagnation[END_REF], and introduce the assumption Ω = Ω 0 = O(1). We shall now follow the procedure described in Section 2.2.1, and obtain the equation

(1 -c 2 )u 0 + 3α 2 cu 2 0 + δΩ 0 cu 0 - αδ 2 (1 -c 2 )Ω 0 u 2 0 +δ 2 ( c 2 2 - 1 6 ) u 0,rr + αδ 2 2 [( 2c 3 -3c ) u 0 u 0,rr + cu 2 0,r ] = 0. ( 16
)

The strong vorticity case

, Ω = Ω 0 δ -1
Finally, in this section, we examine the hypothesis where the vorticity is of great order of magnitude with respect to the dispersive properties. This assumption writes Ω = Ω 0 /δ = O(δ -1 ). Again, the procedure applied in the previous subsections, with the further assumption of traveling waves, leads to the following equation

(1 -c 2 + Ω 0 c)u 0 + α 2 ( 3c + Ω 0 c 2 -Ω 0 ) u 2 0 + δ 2 [ c 2 2 - 1 6 - Ω 0 6 (2Ω 0 -c) ] u 0,rr + αδ 2 2 [( 2c 3 -3c + Ω 0 + Ω 0 c (4Ω 0 -15c) 3 ) u 0 u 0,rr + (c -Ω 0 )u 2 0,r ] = 0. ( 17
)
Yet, it is emphasized that the terms of order O(Ωδ 4 ), which were dropped in the system (10)- [START_REF] Constantin | Rotational steady water waves near stagnation[END_REF] for sake of brevity were kept in this derivation, to insure the consistency of Eq. ( 17).

The combined effect of vorticity, dispersion, and nonlinearity on long water waves

The four Eqs. ( 14)-( 17) correspond to four long wave evolution equations, involving the influence of the vorticity, for various order of magnitude. The differences between the four equations are presented in Table 1, which collects each and every term, as a function of their order of magnitude in α, δ and their combinations.

In this table, the line denoted Ω = 0 corresponds to our reference case, in the absence of vorticity. Collecting the terms of Eq. ( 14), as it is classically done in Boussinesq like models, we identify the contribution of each and every term on the dispersive properties of the system. The term in O(1) corresponds to the non-dispersive contribution, the term of

Ω = 0 (1 -c 2 )u 0 3 2 cu 2 0 ( c 2 2 - 1 6 ) u 0,rr ( 2c 3 -3c ) u 0 u 0,rr + cu 2 0,r Ω = O(δ) (1 -c 2 )u 0 3 2 cu 2 0 ( c 2 2 - 1 6
)

u 0,rr +Ω 0 cu 0 ( 2c 3 -3c ) u 0 u 0,rr +cu 2 0,r + (c 2 -1)Ω 0 u 2 0 Ω = O(1) (1 -c 2 )u 0 3 2 cu 2 0 Ω 0 cu 0 - 1 2 (1 -c 2 )Ω 0 u 2 0 ( c 2 2 - 1 6
) u 0,rr

( 2c 3 -3c ) u 0 u 0,rr + cu 2 0,r Ω = O(δ -1 ) (1 -c 2 )u 0 + Ω 0 cu 0 3c 2 u 2 0 -Ω 0 (c 2 -1)u 2 0 ( c 2 2 - 1 6
)

u 0,rr - Ω 0 6 (2Ω 0 -c)u 0,rr ( 2c 3 -3c ) u 0 u 0,rr + Ω 0 + Ω 0 c(4Ω 0 -15c) 3 u 0 u 0,rr +(c -Ω 0 )u 2 0,r order O(α)
is the nonlinear correction, also known as shoaling effect, and the terms in O(δ 2 ) and O(αδ 2 ) are respectively known as the linear dispersion (or frequency dispersion) and nonlinear dispersion. More details about this classification can be found in [START_REF] Dingenmans | Water Wave Propagation Over Uneven Bottoms: Part 2. Non-Linear Wave Propagation[END_REF]. Table 1 provides a direct comparison of these terms, when vorticity is involved at various order,

namely Ω = O(δ), O(δ O ) and O(δ -1 ).
When weak vorticity is considered (Ω = O(δ)), the behaviour of the system is almost unmodified. The vorticity enters the equation as a direct correction in terms O(δ 2 ) and O(αδ 2 ), but the structure of the equation remains unchanged. This is understood to be a modification of the dispersive properties of long waves, both linear and nonlinear.

On the other hand, the cases of normal and strong vorticity are more complicated. For the first case, if Ω = O(δ 0 ), two new terms appear in the evolution equation, in the sense these terms are expressed at an order of magnitude which was not involved precedently. These two terms are respectively of order O(δ) and O(αδ), and are respectively proportional to u 0 and u 2 0 . This is very similar to the first terms observed in Eqs. ( 14) and ( 15), expressed at orders O(1) and O(α). It is thus understood that vorticity, when involved to this order of magnitude, acts as a non dispersive correction, at lower order, though (O(δ)). It also enters as a correction at lower in δ in the nonlinear term. In the meantime, it is emphasized that the dispersive behaviour of the equation (the terms in O(δ 2 ) and O(αδ 2 )) are unaffected by the vorticity.

Finally, the strong vorticity case, involving Ω = O(δ -1 ), presents the largest complexity. In this case, the vorticity is expressed in each and every term. First, it plays a role on non dispersive effects and nonlinear effects, at the leading order. In the meantime, vorticity is also involved in the dispersive terms of order O(δ 2 ) and O(αδ 2 ), also at the leading order. It is thus understood that when vorticity is strong, it changes completely the nature of the evolution equation, playing a significant role on non-dispersive and nonlinear terms, but also on the dispersive properties of the system, both in frequency and nonlinear dispersion.

Effect of the vorticity on traveling waves solutions

Solitonic solutions of the evolution equations

The four long wave evolution equations obtained in Section 2.2 exhibit an imbalance between linear dispersion and nonlinear dispersion terms. Thus, we expect these four equations to admit solutions describing solitary waves. Finding these solutions is the purpose of this subsection.

The irrotational case Ω = 0

Starting from Eq. ( 14), we now seek for solitary waves solutions. This might be done by assuming the existence of solutions in the form u 0 (r) = Asech 2 (pr) + αBsech 4 (pr), [START_REF] Freeman | Shallow water waves in shear flows[END_REF] with c a supercritical velocity defined by c = c 0 + αV = 1 + αV , and where V , p and B are constants in r, yet functions of A. Substituting u 0 (r) in [START_REF] Ko | Large amplitude steady rotational water waves[END_REF], we obtain the unknown V , p, B,

which read V = A 2 , p = √ 3αA 2δ [ 1 - 3 4 αA ] , B = 3A 2 20
.

Thus, the soliton solution for u 0 (x, t) reads

u 0 (x, t) = Asech 2 [p(x -ct)] + α 3A 2 20 sech 4 [p(x -ct)] , c = c 0 + α A 2 .
Finally, the soliton solution in terms of η is obtained by introducing the expression of u 0 (r) (and its derivatives) in Eq. ( 13).

We obtain

η(x, t) = A(1 -αA)sech 2 [p(x -ct)] + α 19 10 A 2 sech 4 [p(x -ct)]. ( 19
)
Furthermore, the parameter α = a/h is, by definition, the dimensionless amplitude of η. The maximum of η being observed at xct = 0, the relation α = η(0) has to be fulfilled. Thus, the dependence of A with respect to α is necessarily the positive solution of the quadratic polynomial

A 2 ( 19 20 -1 ) α + A -α = 0.
(20)

The weak vorticity case, Ω = Ω 0 δ

We now seek for solitary solutions for the cases involving weak vorticity, as described by Eq. ( 15). This is achieved by reproducing the previous, and seeking for solitary solutions in the form of [START_REF] Freeman | Shallow water waves in shear flows[END_REF]. Doing so, we obtain

c 0 = 1 + 1 2 δ 2 Ω 0 , V = A [ 1 2 + 1 4 δ 2 Ω 0 ] , p = √ 3αA 2δ [ 1 + 1 4 δ 2 Ω 0 ] [ 1 - 3 4 αA ( 1 + 1 2 δ 2 Ω 0 )] , B = 3A 2 20
, the solitary wave, in terms of velocity, being given by

u 0 (x, t) = Asech 2 [p(x -ct)] + α 3A 2 20 sech 4 [p(x -ct)], c = c 0 + α A 2 [ 1 + 1 2 δ 2 Ω 0 ] ,
and in terms of elevation,

η(x, t) = A ( c 0 -αA [ 1 + 5 4 δ 2 Ω 0 ]) sech 2 [p(x -ct)] +αA 2 [ 19 10
+ 93 40 δ 2 Ω 0 ] sech 4 [p(x -ct)]. (21)
Finally, the wave elevation at the crest has to be α, which compels the value of A. Indeed, the latter is obtained as the positive root of the quadratic polynomial

A 2 ( - 1 20 + 179 120 δ 2 Ω 2 0 ) α + c 0 A -α = 0. ( 22
)
3.1.3. The rotational case, Ω = Ω 0 ̸ = 0 Again, we reproduce the procedure described in Section 3.1.1. We seek for solutions of Eq. ( 16) in the form [START_REF] Freeman | Shallow water waves in shear flows[END_REF]. In this equation, the critical velocity satisfies: 1 + δΩ 0 c 0 -c 2 0 = 0, which admits the positive solution

c 0 = 1 + 1 2 δΩ 0 + 1 8 δ 2 Ω 2 0 .
Thus, if u 0 (r) is a solitary wave of the form (18) with a supercritical velocity c = c 0 + αV , we find

V = A [ 1 2 + 1 4 δΩ 0 + 1 6 δ 2 Ω 2 0 ] , p = √ 3αA 2δ [ 1 + 1 4 δΩ 0 + 19 96 δ 2 Ω 2 0 ] [ 1 - 3 4 αA ( 1 + 1 2 δΩ 0 + 1 3 δ 2 Ω 2 0 )] , B = 3A 2 20
.

Hence, examining the case of constant vorticity Ω of order O(δ 0 ), we obtain the expression

u 0 (x, t) = Asech 2 [p(x -ct)] + α 3A 2 20 sech 4 [p(x -ct)], c = c 0 + α A 2 [ 1 + 1 2 δΩ 0 + 1 3 δ 2 Ω 2 0 ] ,
for u 0 (x, t), and for η(x, t),

η(x, t) = Asech 2 [p(x -ct)] ( c 0 -αA [ 1 + 5 6 δΩ 0 + 13 12 δ 2 Ω 2 0 ]) + αA 2 [ 19 10
+ 93 40 δΩ 0 + 303 160 δ 2 Ω 2 0 ] sech 4 [p(x -ct)], (23)
Finally, the crest amplitude of the solitary wave having an elevation of α, the value of A is provided by the positive root of the polynomial

A 2
( -

1 20 + 179 120 δΩ 0 + 389 480 δ 2 Ω 2 0 ) α + c 0 A -α = 0. ( 24
)
3.1.4. The strong vorticity case, Ω = Ω 0 δ -1 Finally, we examine the case of strong vorticity, when Ω = O(δ -1 ). Starting from a solution in the form u 0 (r) = Asech 2 (pr) + αBsech 4 (pr), we obtain the expression of the constants

c 0 = Ω 0 2 + √ 1 + Ω 2 0 4 V = Ac 0 3 + Ω 2 0 3(2c 0 -Ω 0 ) , p = 1 2δ √ αAc 0 (3 + Ω 2 0 ) Ω 2 0 -c 2 0 + 2 [ 1 + αAc 0 (3 + Ω 2 0 ) 2(2c 0 -Ω 0 )(Ω 2 0 -c 2 0 + 2) ( 5Ω 0 6 -c 0 ) ] , B = 3A 2 20 [ Ω 2 0 c 0 + 6Ω 0 + c 0 Ω 2 0 -c 2 0 + 2
] .

Following the procedure described previously, we obtain the expression of the solitary wave, in terms of velocity u 0 (x, t),

u 0 (x, t) = Asech 2 [p(x -ct)] + α 3A 2 20 [ Ω 2 0 c 0 + 6Ω 0 + c 0 Ω 2 0 -c 2 0 + 2 ] sech 4 [p(x -ct)], c = c 0 + αAc 0 (3 + Ω 2 0 ) 3(2c 0 -Ω 0 ) ,
and in terms of elevation η(x, t),

η(x, t) = c 0 Asech 2 [p(x -ct)]+c 0 (3 + Ω 2 0 ) ( 1 3(2c 0 -Ω 0 ) + 2Ω 0 -3c 0 6(Ω 2 0 -c 2 0 + 2) ) αA 2 sech 2 [p(x -ct)] + 3c 0 Ω 2 0 -c 2 0 + 2 ( Ω 2 0 c 0 + 6Ω 0 + c 0 20 + (3 + Ω 2 0 )(3c 0 -2Ω 0 ) 12 ) αA 2 sech 4 [p(x -ct)], (25)
A being determined as the positive root of the polynomial

A 2 ( c 0 (3 + Ω 2 0 ) 3(2c 0 -Ω 0 ) - c 0 (3 + Ω 2 0 )(2Ω 0 -3c 0 ) 12(Ω 2 0 -c 2 0 + 2) + 3c 0 (Ω 2 0 c 0 + 6Ω 0 + c 0 ) 20(Ω 2 0 -c 2 0 + 2) ) α + c 0 A -α = 0.
(26)

Characteristics of the solitary waves

The quasi-steady solutions obtained in the previous section present various behavior, depending on the order of magnitude considered for the vorticity. The purpose of this section is to describe the influence of vorticity magnitude on the properties of solitary waves. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Effect of the vorticity on the soliton properties

The purpose of this section is to describe the influence of vorticity magnitude on the properties of solitary waves. Fig. 1 presents the main characteristic of the solitary waves obtained for all cases of vorticity. Fig. 1(a) corresponds to the water elevation η, while Fig. 1(b) illustrates the distribution of the mean horizontal velocity u 0 induced by the perturbation. In both subfigures, non dimensional parameters are chosen such as α = 0.6, δ = 0.5. In every case considered the reference vorticity Ω 0 is chosen to be unity. In these figures red lines describe the results obtained when no vorticity is considered.

Green lines correspond to the weak vorticity case, the blue lines are the result obtained with the normal vorticity, while the yellow lines describe the results obtained when the vorticity is strong. Since the value of α is prescribed, the four curves describing water elevation (Fig. 1(a)) present the same maximum values. Differences appearing only concern the waves width, and thus the local slopes. The four curves presented in Fig. 1(a) show a similar tendency. Solitary waves are narrower in the presence of vorticity. Indeed, if the curves obtained with no vorticity, and with weak vorticity are almost perfectly superimposed, one can notice that the model involving a unit vorticity describes a slightly narrower wave. The effect of vorticity when considering these two vortical conditions is almost insignificant, in terms of wave width. On the other hand, when considering the model assuming strong vorticity, the wave obtained is clearly narrower than the previous ones. The trend observed previously emphasized, and is more visible here.

When considering the average velocity u 0 , differences are much more significant induced by the solitary wave. These velocities are presented in Fig. 1(b). From this figure, it clearly appears that the maximum velocity induced by the free surface disturbance is strongly affected by the magnitude of the vorticity. Namely, the stronger is the vorticity, the weaker is the fluid disturbance. It appears that the maximum of the velocity disturbance can be reduced up to 40% when the vorticity becomes strong. From Eq. ( 17), we observed that strong vorticity adds a dispersive term and a nonlinear dispersive term. It is responsible for the trend leading to the weakest solitonic profile downstream propagating for strong vorticity. The effects on u 0 (x, t) of normal and weak vorticities occur only through η(x, t). For strong vorticity u 0 (x, t) directly depends on Ω through the new dispersive and nonlinear dispersive term.

The evolution of the fluid velocity, observed in previous figures has to be related with the kinematic properties of the solitary wave. The evolution of its velocity and width are depicted in Fig. 2. On Fig. 2(a), the evolution of the wave celerity c of the solitary wave is plotted as a function of the vorticity Ω, for the four models. Here again, the nonlinear and dispersive parameters (α, δ) are set equal to (0.6, 0.5), while Ω varies. Since the models do not cover the same range of values for Ω, four colors are used, in order to distinguish the area described by each model. Obviously, the model without vorticity corresponds to a single point in this diagram, while the green line correspond to the model with weak vorticity, the blue line is the model of vorticity of order unity, and the yellow line is the strong vorticity assumption. The first result observed is an opposite trend to the behaviour shown by the fluid disturbance velocity u 0 . Namely, the celerity of the wave increases with vorticity, while the components of the fluid disturbance decreases with vorticity. For the parameters considered, the celerity is increased by about 85%. Another interesting remark, here, is to notice the relative continuity exhibited by the four models. Indeed, they have been derived independently, under various asymptotic assumptions, but present very compatible trends, and values. In Fig. 2(b), the evolution of the width of the wave, λ, defined as λ = 1/p, is presented versus vorticity Ω for the four models. This figure confirms the trend forebode on Fig. 1(a). Here again, the four models are juxtaposed, where the model without vorticity corresponds to a single point, a red square, in this diagram, the green line correspond to the model with weak vorticity, the blue line is the model of vorticity of order unity, and the yellow line is the strong vorticity assumption. The three models involving vorticity present a similar trend, the width decreasing when vorticity increases.

Effect of the vorticity on the korteweg-de Vries soliton

The behavior of λ(Ω) is quite difficult to analyze from the soliton expressions of u 0 (x, t) and η(x, t) of ( 10)- [START_REF] Constantin | Rotational steady water waves near stagnation[END_REF].

However this becomes possible with soliton solutions of the Korteweg-de Vries equations associated to the system (10)- [START_REF] Constantin | Rotational steady water waves near stagnation[END_REF]. Let us have a look at λ(Ω) of the soliton solutions of KdV equations for u 0 (x, t) in the two extreme cases Ω = 0 and Ω = Ω 0 δ -1 . Using asymptotic methods (see [START_REF] Whitham | Linear and Nonlinear Waves[END_REF]) in ( 14) and ( 17) we obtain u 0,t for these two extreme cases. They read

u 0,t + 3 2 αu 0 u 0,x + δ 2 6 u 0,xxx = 0, Ω = 0, (27)
(Ω 2 0 + 4) 1 2 u 0,t + c 0 (Ω 2 0 + 3)αu 0 u 0,x + δ 2 6 (c 2 0 -Ω 2 0 )u 0,xxx = 0, Ω = Ω 0 /δ, c 0 = 1 2 [Ω 0 + (Ω 2 0 + 4) 1/2
].

(

) 28
The soliton solutions of ( 27) and (28) read

u 0 = A s sech 2 [ 1 2δ (3αA s ) 1 2 (x - A s α 2 t)], u 0 = A s sech 2 { 1 2δ (3αA s ) 1 2 [ c 0 (3 + Ω 2 0 ) 3(c 2 0 -Ω 2 0 ) ] 1 2 [x -( αA s c 0 (3 + Ω 2 0 ) 3(4 + Ω 2 0 ) )t]}, (29)
with A s an arbitrary constant amplitude. The associated soliton widths read

λ(Ω = 0) = 2δ (3αA s ) 1 2 , λ(Ω = Ω 0 δ ) = 2δ (3αA s ) 1 2 [ c 2 0 -Ω 2 0 c 0 (1 + Ω 2 0 3 ) ] 1 2 . ( 30
)
From expressions (30) (taken Ω 0 = 1) we obtain

λ(Ω= 1 δ )
λ(Ω=0) ∼ 0, 5 < 1, and (29) in the case Ω = Ω 0 δ -1 is a downstream wave narrower than in the case Ω = 0. So for downstream propagation the stronger the vorticity the narrower the width. This is a very well known result not only for KdV but also for the Boussinesq equation and the Camassa-Holm equation (see [START_REF] Johnson | Camassa-Holm, Korteweg-de Vries and related models for water waves[END_REF]).

Effect of the vorticity on the inner flow

Another interesting point with this approach is to take into account the evolution of the velocity field within the fluid layer. Indeed, accordingly to Eqs. ( 3) and ( 4), the velocity can be reconstructed as a power series of δ 2 z 2 . If closed with the recurrence relation [START_REF] Constantin | Symmetry of steady deep water water waves with vorticity[END_REF], the velocity in the total fluid layer might be reconstructed, as

u(x, z, t) = u 0 (x, t) - 1 2 u 0xx (x, t)δ 2 z 2 + O(δ 4 z 4 ), (31) w
(x, z, t) = -u 0x (x, t)z + 1 6 u 0xxx (x, t)δ 2 z 3 + O(δ 4 z 4 ). ( 32
)
These are the fields shown in Fig. 3, for a solitary wave of nonlinear parameter α = 0.6 and dispersive parameter δ = 0.5, where Ω 0 is taken equal to unity. In this figure, the horizontal components of the velocities within the fluid layer are presented on the left column, while the vertical components of the velocity are shown on the right column. Each line corresponds to one of the models. Thus, sub-figures (a-e) present the two components of the velocity obtained in the absence of vorticity, sub-figures (b-f) are the fields obtained under the assumption of weak vorticity, sub-figures (c-g) describe the velocity fields obtained with the assumption of unit vorticity, while sub-figures (d-h) correspond to the fields obtained under the assumption of a strong vorticity. From these figures, two main results are observed. First, the magnitude of the velocity field decreases when vorticity increases. This result was obtained when considering the mean velocity u 0 . It is confirmed here within the entire fluid layer, concerning both components (u, w) of the velocity. Secondly, these figures allow to compare the evolution of the velocity fields with depth. It appears that the velocity disturbance vanishes faster with depth when vorticity increases. This comes from the fact that in the presence of vorticity, for waves of a given height α, the velocity disturbance diminishes when vorticity increases. Thus, the second term in the decomposition (31),( 32) is relatively smaller with respect to the first term, than it is when considering no vorticity.

Effect of α and δ on the soliton properties

Another interesting aspect of this approach is to allow comparisons of the influence of the parameters α and δ on the wave properties. Following that purpose, Fig. 4 describes the evolution of the wave width λ in the parameter map (α, δ). First, it has to be emphasized that these figures present very different color scales, confirming the trends observed in Fig. 2. In the meantime, it is striking to notice that the qualitative dependence on both the nonlinear and the dispersive parameters are very similar. Oppositely, Fig. 5 illustrates the evolution of the wave celerity in the parameter map (α, δ).

Sub-figure (a) corresponds to the behavior of a solitary wave propagating in the absence of vorticity, sub-figure (b) is the weak vorticity case, sub-figure (c) is the unit vorticity model, while sub-figure (d) is the strong vorticity configuration. It is interesting to notice that, in the absence of vorticity, the traveling celerity does not depend on the dispersive parameter δ, which confirms classical behavior of traveling solitary waves in the absence of vorticity. Noteworthily, this result does not hold when vorticity is involved. Indeed, it is interesting to notice that, when weak or unit vorticity are considered, the wave celerity exhibits a strong dependence on α and δ. However, the respective influence of α and δ on this quantity is not completely similar. In the weak vorticity model, the dispersive parameter δ seems to play a less significant role than in the unit vorticity approach. Finally, when considering the strong vorticity case, the celerity becomes constant again with respect to δ. This asymptotic case exhibits characteristics very similar to those observed in the absence of vorticity.

Conclusions and perspectives

We have introduced a new hierarchy of weakly nonlinear equations for surface gravity in finite depth in the presence of constant vorticity Ω. The members of the hierarchy are systems of equations for the z-integrate horizontal velocity u(x, z, t) and the wave amplitude η(x, t). They involve weak nonlinearities, dispersion and nonlinear dispersion and are characterized by the powers of two perturbative parameters α and δ (nonlinearity and dispersion) and their products (nonlinear dispersion). The vorticity Ω appears in the equations of motion in factors Ωδ s (s = 1, 3, . . . ), so we have scaled Ω as Ω = Ω 0 δ q with Ω 0 ∼ O(δ 0 ) and q integer. The negatives values of q must be chosen such that the neglected term O(Ωδ 5 , δ 4 ) in Eq. ( 6) remains negligible. A remarkable finding is that the physical origin of dispersion and nonlinear dispersion for u 0 (x, t) can be associated or not with the vorticity Ω. In the η(x, t) dynamics, vorticity is never directly associated with nonlinear dispersion. In order to analyze some physical-mathematical consequences while remaining relatively simple in formalism, we have only discussed equations at orders α, δ n and αδ n with n = 1, 2. Non zero cases clearly shown that vorticity changes linear and nonlinear dispersive behavior of the system with respect to the zero vorticity case. Normal vorticity adds a weak correction to the non-dispersive and nonlinear terms. For strong vorticity the same modification is observed, but at the leading order. Furthermore, a modification in the linear and nonlinear dispersion terms is present simultaneously.

Furthermore, the solitonic solutions of these long wave evolution equations have been obtained analytically when involving nil, weak, normal and strong vorticity. For each case we have numerically analyzed these solitonic solutions, and their properties (celerity, amplitude and wave width).

CRediT authorship contribution statement

Fig. 1 .

 1 Fig. 1. Comparison of the free surface elevation η (a) and the mean velocity u 0 (b) obtained with the four model equations. The red lines correspond to the case without any vorticity (Ω = 0), the green lines are the model with weak vorticity (Ω = Ω 0 δ), the blue lines describe the model with unit vorticity (Ω = Ω 0 ), and the yellow lines correspond to the strong vorticity case (Ω = Ω 0 /δ). The parameters used are α = 0.6, and δ = 0.5.

Fig. 2 .

 2 Fig. 2. Evolution of the wave velocity (a), and the wave width (b) as a function of the vorticity. The red squares correspond to the case without any vorticity (Ω = 0), the green lines are the model with weak vorticity (Ω = Ω 0 δ), the blue lines describe the model with unit vorticity (Ω = Ω 0 ), and the yellow lines correspond to the strong vorticity case (Ω = Ω 0 δ -1 ). The parameters used are α = 0.6, and δ = 0.5. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 3 .

 3 Fig. 3. Distribution of the horizontal (left column) and vertical (right column) components of the velocity within the fluid layer. (a-e) Ω = 0, (b-f) Ω = Ω 0 δ, (c-g) Ω = Ω 0 ,(d-h) Ω = Ω 0 δ -1 . The parameters used are α = 0.6, and δ = 0.5.

  Sub-figure (a) corresponds to the behavior of a solitary wave propagating in the absence of vorticity, sub-figure (b) is the weak vorticity case, sub-figure (c) is the unit vorticity model, while sub-figure (d) is the strong vorticity configuration.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Evolution of the width of the wave as a function of the dispersive parameter δ and the nonlinear parameter α. (a) Ω = 0, (b) Ω = Ω 0 δ, (c) Ω = Ω 0 , (d) Ω = Ω 0 δ -1 .

M

  .A. Manna: Conception and design of study, Writing -original draft, Writing -review & editing. S. Noubissié: Conception and design of study, Acquisition of data, Writing -original draft. J. Touboul: Conception and design of study, Acquisition of data, Analysis and/or interpretation of data, Writing -original draft, Writing -review & editing. B. Simon: Acquisition of data, Analysis and/or interpretation of data, Writing -original draft. R.A. Kraenkel: Conception and design of study, Analysis and/or interpretation of data, Writing -original draft.

Table 1

 1 Modifications of the long wave equations in the presence of weak, normal or strong vorticity.

	O(1)	O(α)	O(δ)	O(αδ)	O(δ 2 )	O(αδ 2 )

Acknowledgments

The authors would like to thank FAPESP (São Paulo Research Foundation) and Key Initiative Sea & Coast of the Montpellier University of Excellence (MUSE) for their financial support. The French DGA and ANR are also deeply acknowledged for their financial support through the grant 13-ASTR-0007.

Approval of the version of the manuscript to be published (the names of all authors must be listed): M.A. Manna, S. Noubissié, J. Touboul, B. Simon R. A. Kraenkel.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


E Didenkulova

E Pelinovsky

J Touboul

Benjamin -Bona-Mahony

Long-wave approximations in the description of bottom pressure

Keywords:

The role of various long-wave approximations in the description of the wave field and bottom pressure caused by surface waves, and their relation to evolution equations are being considered. In the framework of the linear theory, these approximations are being tested on the well-known exact solution for the wave spectral amplitudes and pressure variations. The famous Whitham, Korteweg-de Vries (KdV) and Benjamin-Bona-Mahony (BBM) equations have been used as evolutionary equations. It has been shown that if the wave is long, though steep enough, the BBM approximation gives better results than the KdV approximation, and they are quite close to the exact results. The same applies to the description of rogue waves, though formed from smooth relatively long waves, are often short and steep, then they may be invisible in variations of the bottom pressure. Another advantage of the BBM approximation for calculating the bottom pressure is the ability to analyze noisy series without preliminary filtering, which is necessary when using the KdV approximation.

Introduction

Measuring sea waves using buoys in the coastal zone is a very difficult task for many reasons. The coastal buoy is heavily influenced by tipping waves and coastal currents. Many of them disappear due to vandalism of passing ships crews. That is why, the installation of bottom pressure sensors, which are easier to fix at the bottom and hide from vandals, has become a popular technology in measurements. For example, the DART (Deep-ocean Assessment and Reporting of Tsunamis) system is well-known for tsunami registration in the open ocean, which allows pre-registering a tsunami wave. In this case, however, an important practical problem of determining the elements of surface waves from variations in bottom pressure arises. This problem in the mathematical formulation belongs to the class of ill-posed problems, and although there are certain practical achievements, it is not considered to have been solved so far, as can be seen from engineering publications [START_REF] Cavaleri | Wave measurement using pressure transducer[END_REF][START_REF] Wang | Time series surface-wave recovery from pressure gage[END_REF][START_REF] Bishop | Measuring waves with pressure transducers[END_REF][START_REF] Baqueizo | Transfer function between wave height and wave pressure for progressive waves[END_REF][START_REF] Zaslavsky | About recovery of surface wave spectrum from pressure sensor[END_REF][START_REF] Bonneton | A nonlinear weakly dispersive method for recovering the elevation of irrotational surface waves from pressure measurements[END_REF][START_REF] Martins | Non-hydrostatic, non-linear processes in the surf zone[END_REF][START_REF] Tsai | On the recovery of surface wave by pressure transfer function[END_REF] and mathematical ones [START_REF] Esher | On the recovery of the free surface from the pressure within periodic travelling water waves[END_REF][START_REF] Deconink | Relating the bottom pressure and surface elevation in the water wave problem[END_REF][START_REF] Constantin | On the recovery of solitary wave profiles from pressure measurements[END_REF][START_REF] Oliveras | Recovering surface elevation from pressure data[END_REF][START_REF] Clamond | Recovery of steady periodic wave profiles from pressure measurements at the bed[END_REF][START_REF] Hsu | Recovering surface profiles of solitary waves on a uniform stream from pressure measurements[END_REF][START_REF] Vasan | Water-wave profiles from pressure measurements: Extensions[END_REF][START_REF] Vasan | A method to recover water-wave profiles from pressure measurements[END_REF][START_REF] Bonneton | Recovering water wave elevation from pressure measurements[END_REF].

In our opinion, even in the direct correct problem of determining the bottom pressure under sea waves, there is still no complete clarity. Analytical formulas have been derived for a certain class of wave motions such as traveling waves in a basin of constant depth (such formulas can be found in the papers cited above), but they describe difficult integral expressions which are hard to make practical algorithms. The pressure formulas are explicit only in the long-wave approximation in the framework of the highly nonlinear and weakly dispersive Green-Naghdi model [START_REF] Green | A derivation of equations for wave propagation in water of variable depth[END_REF][START_REF] Barthelemy | Nonlinear shallow water theories for coastal waves[END_REF]. They are used, for example, in the analytical calculation of bottom pressure under a solitary wave [START_REF] Pelinovsky | Bottom pressure caused by passage of a solitary wave within the strongly nonlinear green-naghdi model[END_REF]. Numerical calculations of pressure under solitary waves are carried out in the framework of various models that demonstrate a certain difference in the results; see, for example, [START_REF] Touboul | On the use of linear theory for measuring surface waves using bottom pressure distribution[END_REF][START_REF] Touboul | Bottom pressure distribution under a solitonic wave reflecting on a vertical wall[END_REF][START_REF] Slunyaev | The pressure field beneath intense surface water wave groups[END_REF].

In this paper, we evaluate the role of long-wave approximations in estimating the characteristics of bottom pressure for modeling unsteady wave processes leading to the formation of rogue waves. All results are obtained analytically allowing investigating definition parameters of process. Evolution equations like Whitham, Korteweg-de Vries and Benjamin-Bona-Mahony equations are briefly discussed in Section 2. Long-wave approximations associated with these equations for the bottom pressure are derived in Section 3. Analytical calculations of bottom pressure induced by wave train focusing into the rogue wave are given in Section 4. The influence of the wave shape on the variation of bottom pressure is analyzed in Section 5. Bottom pressure induced by the focusing of frequency modulated (chirped) wave train is considered in Section 6 using the stationary-phase method. Obtained results are summarized in Conclusion.

The evolution equations for small amplitude surface waves

In the case of small-amplitude waves in a basin of constant depth, the original Euler equations can be exactly reduced to the Whitham equation in the linear approximation [START_REF] Whitham | Linear and Nonlinear Waves[END_REF] ∂η ∂t

+ c 0 ∫ +∞ -∞ D(x -y) ∂η(y, t) ∂y dy = 0, (1)
D(ξ ) = 1 2π ∫ +∞ -∞ dk √ tanh(kh) kh exp(ikξ ). (2)
Here η(x, t) is the elevation of the free surface; h is the basin depth assumed to be constant; g is the acceleration of gravity; c 0 = √ gh is the speed of long waves; t is time, and x is the coordinate. This integral equation, actually, reflects the fact that the frequency of the monochromatic wave ω is related to the wave number k by the dispersion relation for surface gravity waves

ω(k) = √ gk tanh(kh). (3)
In fact, the general solution of Eq. ( 1), as well as the original Euler equations, can be represented by the Fourier integral

η(x, t) = 1 2π Re ∫ +∞ -∞ η(k) exp {i [ω(k)t -kx]} dk, (4)
where η(k) is the Fourier spectrum of the wave determined by the initial condition

η(k) = ∫ +∞ -∞ η(x, 0) exp(ikx)dx. (5)
In linear theory, of course, there is no difference between the application of evolution equations and the Fourier transform, but in the nonlinear problem, their difference becomes fundamental. Since our interest will be related to the calculation of bottom pressure, we will briefly first discuss the evolution equation (1) in various approximations. We will focus on the relatively long waves that ''reach'' the bottom and can be detected by bottom pressure sensors.

In the case of very long waves (kh ≪ 1), the dispersion relation becomes very simple

ω(k) = c 0 k, (6)
and the Whitham equation is reduced to the transport equation

∂η ∂t + c 0 ∂η ∂x = 0, ( 7
)
its solution is trivial

η(x, t) = η 0 (x -c 0 t). ( 8
)
This is a well-known solution of linear shallow water equations that we will not discuss further. Accounting for the next correction in kh leads to the modification of the dispersion relation ( 6)

ω = c 0 k ( 1 - k 2 h 2 6
) ,

and the transformation of the Whitham equation into a differential equation This is a linear approximation of the well-known Korteweg-de Vries (KdV) equation, which played a crucial role in the nonlinear theory of water waves, in particular for describing solitons.

∂η ∂t + c 0 ∂η ∂x + c 0 h 2 6 ∂ 3 η ∂x 3 = 0. ( 10
)
With the same accuracy (like 1 1+ε ≈ 1 -ε, ε ≪ 1), we can propose another approximation of the dispersion relation

ω = c 0 k 1 + k 2 h 2 6 , (11)
leading to the next transformation of the Whitham equation

∂η ∂t + c 0 ∂η ∂x - h 2 6 ∂ 3 η ∂t∂x 3 = 0, (12)
which is known as the linear part of the Benjamin-Bona-Mahony -BBM equation [START_REF] Benjamin | Model equations for long waves in nonlinear dispersive systems[END_REF]. It is not integrable as the Korteweg-de Vries equation, and is not actually used in the practice of calculating water waves.

The accuracy of various long-wavelength dispersion approximations is illustrated in Fig. 1.

Here Ω = ω √ h/g is the dimensionless frequency. As we see in the range kh ∼ 1, the long-wave approximations coincide well with the exact curve. In the short-wave region, the differences are more significant, especially in the case of the KdV approximation. This difference can affect long waves due to nonlinear interactions; in the linear approximation the difference affect the convergence of solutions. In principle, it is possible to find the long-wave approximation of the dispersion relation using the Padé decomposition, which can expand the coincidence region [START_REF] Madsen | Boussinesq-Type formulations for fully nonlinear and extremely dispersive water waves: derivation and analysis[END_REF]. In the simple variant, it is a combination of Eqs. ( 10) and [START_REF] Oliveras | Recovering surface elevation from pressure data[END_REF]; it has been considered, for example, in [START_REF] Francius | Wave dynamics in nonlinear media with two dispersionless limits for long and short waves[END_REF]. However, we restrict here to the analysis of the long-wave approximation in the framework of the KdV and BBM equations.

Bottom pressure caused by surface waves -long-wave approximations

It is convenient to measure the bottom pressure through the so-called displacement of the mercury column:

p = ρg(h + ξ ), ( 13
)
where ρ is the density of sea water. Then from the linearized equations of hydrodynamics we obtain the following relationship between the spectra of bottom pressure and displacement of the water surface [START_REF] Mei | Applied Ocean Surface Waves[END_REF] ξ

(k) = η(k) cosh(kh) . ( 14
)
Using the direct Fourier transform, we can obtain the integral relationship between the bottom pressure and the displacement of the water surface. However, in practice it is easier to use direct and inverse Fourier transforms. The function 1/cosh(kh) is a transfer function, or rather its spectrum. Similar transfer function is appeared in the problems of tsunami generation by the instantaneous vertical motion of the sea bottom [START_REF] Liu | Analytical solutions of tsunamis generated by underwater earthquakes[END_REF]. Formula ( 14) implies the well-known fact that short waves do not reach the seabed and cannot lead to changes in bottom pressure. Therefore, the main attention is paid to waves of large and moderate lengths (compared to the depth of the basin. In this case, by analogy with the results of the previous section, we can expand the transfer function in a Taylor series, limiting by two terms, rewrite [START_REF] Hsu | Recovering surface profiles of solitary waves on a uniform stream from pressure measurements[END_REF] approximately in the form

ξ KdV (k) ≈ [ 1 - k 2 h 2 2 ] η(k). (15)
Here we have used the abbreviation KdV, since the same decomposition is used to derive the Korteweg-de Vries equation [START_REF] Esher | On the recovery of the free surface from the pressure within periodic travelling water waves[END_REF]. In this case, the direct Fourier transform leads to an explicit formula for bottom pressure variations

ξ (x) = η(x) + h 2 2 d 2 η dx 2 . ( 16
)
In this approximation, there are no residual variations in bottom pressure after the passage of the sea wave. It is important to note, it is the formula ( 16) that constitutes the linear part of the bottom pressure in the framework of the weakly dispersed strongly nonlinear Green-Naghdi model [START_REF] Bonneton | A nonlinear weakly dispersive method for recovering the elevation of irrotational surface waves from pressure measurements[END_REF][START_REF] Touboul | Bottom pressure distribution under a solitonic wave reflecting on a vertical wall[END_REF], which is actively used to calculate sea waves in the coastal zone. Essentially, our conclusion shows where such a term comes from. With the same accuracy, the transfer function in the region of long waves can be represented in the form:

ξ BBM (k) ≈ η(k) 1 + k 2 h 2 2 , (17)
where we use the abbreviation BBM, since such an expansion is used in [START_REF] Constantin | On the recovery of solitary wave profiles from pressure measurements[END_REF]. Applying the direct Fourier transform to [START_REF] Bonneton | Recovering water wave elevation from pressure measurements[END_REF], we obtain a differential equation for the spatial distribution of bottom pressure:

ξ (x) - h 2 2 d 2 ξ dx 2 = η(x). ( 18
)
This equation is easily solved:

ξ (x) = √ 2 h ∫ x 0 η(y) sin h ( √ 2 y -x h ) dy ( 19
)
and it is convenient for numerical integration. Thus, there are two long-wavelength approximations of the transfer function, and their accuracy is illustrated in Fig. 2.

As we see, in the region of very long waves (kh ≪ 1) the curves coincide. The BBM approximation ( 6) is close to the exact curve to approximately kh ∼ 2, and qualitatively coincides with it in the entire range of wave numbers. The KdV approximation [START_REF] Esher | On the recovery of the free surface from the pressure within periodic travelling water waves[END_REF] seems to be the worst, since the transfer function in the region kh > 1.42 becomes negative and grows strongly for short waves, contrary to the well-known fact that short waves do not reach the bottom. It underlines the modern Green-Naghdi models for describing long water waves. 21)) and dashed blue line -BBM (Eq. ( 22)).

Rogue wave image in bottom pressure variations

We will demonstrate the difference in the predictions of various models using the example of an ''ideal'' rogue wave in the form of a delta-function

η(x) = A 0 δ(x/l), (20)
where the amplitude and length parameters have been introduced only for dimension. Such a wave satisfies all the criteria of a rogue wave, since it significantly exceeds all possible surrounding waves in amplitude [START_REF] Ch | Rogue Waves in the Ocean[END_REF]. Its spectrum is A 0 l 0 and the direct Fourier transform using the exact transfer function ( 14) gives an answer [31, Eq. 3.981.3]:

ξ (x) = πA 0 l h sech ( πx 2h ) . (21)
As can be seen, the ''bottom rogue wave'' looks like a short smooth pulse (its length is of the order of depth), and the amplitude depends on the ratio between the ''initial'' length and depth. Similar calculations with the BBM approximation of the transfer function lead to [31, Eq. 3.723.2]

ξ (x) = π √ 2A 0 l h exp ( - √ 2 |x| h ) . (22)
We do not use the KdV approximation [START_REF] Vasan | Water-wave profiles from pressure measurements: Extensions[END_REF], since the corresponding integrals and functions diverge. A comparison of formulas ( 21) and ( 22) is shown in Fig. 3 (normalized amplitude). As expected, the BBM approximation overestimates the amplitude of the bottom pressure and does not prevent the formation of a singularity in the profile (jump in the derivative), the value of which decreases with the increasing depth of the basin.

Single wave image in bottom pressure variations

Obviously, an ''ideal'' wave is short, and it is natural to expect that long-wave approximations would not work. Nevertheless, as we see from Fig. 3, the BBM approximation describes well the form of pressure variations, although it greatly overestimates the value of bottom pressure. Let us therefore consider a more realistic case of a long wave of a rectangular shape (this choice is due to the receipt of analytical expressions)

η 0 (x) = A 0 { 1 -l < x < l 0 out of interval (23)
Then an exact expression for the bottom pressure is in the form [31, Eq. 4.122.1]

ξ (x) = 2A 0 π atan [ sinh(π l/2h) cosh(π x/2h)
] .

(

) 24
Similar calculations using the BBM approximation lead to the following expression [31, Eq. 3.725.3] These dependences are presented in Fig. 4 for the case of a long wave l/h = 15 (the amplitude is normalized to A 0 ). As expected, the pressure in the long wave is hydrostatic, but on the wave slopes it decreases sharply. In this case, the BBM approximation comes up to the form of pressure variations very well, so it can be used even if there are sections with a large steepness in the wave field. The KdV approximation cannot describe an impulse with sharp slopes, and then endless surges appear in pressure.

ξ BBM (x) = A 0 { 1 -cosh( √ 2x/h) exp(- √ 2l/h) |x| < l sinh( √ 2l/h) exp(- √ 2x/h) |x| > l (25)
Even if the wave is not very long (l/h = 2), the BBM approximation still works well (Fig. 5), although the difference in pressure from hydrostatics becomes noticeable and the shape of the pressure variations differs significantly from a rectangular one. The last analytical example in this section is a smooth pulse on the surface with small oscillations at the edges

η(x) = A 0 [ 1 + ( h l ) 2 - 2h 2 x 2 l 4 ] exp [ - ( x l ) 2 ] . ( 26
)
In this case, calculations of the bottom pressure in the long-wave approximation result in the following expressions:

ξ BBM (x) = A 0 exp [ - ( x l ) 2 ] , (27)
ξ KdV (x) = A 0 [ 1 - 3h 4 l 4 + 12h 4 x 2 l 6 - 4h 4 x 4 l 8 ] exp [ - ( x l ) 2 ] . ( 28
)
Fig. 6. The response of a long wave of a smooth shape in variations of the bottom pressure (solid red curve is a surface displacement, dashed blue curve is the BBM approximation, and dashed dotted black curve is the KdV approximation), l/h = 5.

Fig. 7.

The response of a long wave of a smooth shape in variations of the bottom pressure (solid red curve is a surface displacement, dashed blue curve is the BBM approximation, and dashed dotted black curve is the KdV approximation), l/h = 2.

If the wave is very long (l/h = 5), then the difference in predictions of the BBM and KdV models is insignificant (Fig. 6), which could be expected for waves whose entire spectrum is concentrated in long waves. However, for shorter waves (l/h = 2), the difference becomes noticeable (Fig. 7).

Dispersion focusing of frequency-modulated wave packets

The dispersive compression of frequency-modulated packets is one of the mechanisms of rogue wave formation [START_REF] Ch | Rogue Waves in the Ocean[END_REF] and is often used in laboratory tanks to generate large-amplitude pulses. For this, there must be an optimal distribution of spectral components in space, described by a self-similar solution to the kinematic equation [START_REF] Whitham | Linear and Nonlinear Waves[END_REF][START_REF] Ch | Rogue Waves in the Ocean[END_REF]:

c gr [k(x, t)] = dω dk = x f -x t f -t , (29)
or close to it. Here x f is the position of the focal point at which all wave packets converge at time t f . Having the initial distribution of the wave number at t = 0 in accordance with [START_REF] Liu | Analytical solutions of tsunamis generated by underwater earthquakes[END_REF], one may be sure that the rogue wave will be the largest.

The problem, however, is that having ''organized'' the optimal focusing within the framework of the exact equations of hydrodynamics, we will not be able to reproduce it in the framework of long-wavelength models. Indeed, the group velocity for surface waves is always positive; while in the KdV and BBM models it is negative for short waves. It is natural, therefore, to be limited to the values of kh ∼ 1, when the differences in the approximations of the dispersion relation

have not yet become crucial. It is quite difficult to solve the problem of focusing a frequency-modulated (chirped) wave packet, however, you can use the invariance property of the hydrodynamic equations with respect to the change of signs of time and coordinate, and solve the problem of transforming a rogue wave into a frequency-modulated packet, and then invert it in space. This technique has been used in a number of works [START_REF] Ch | Rogue Waves in the Ocean[END_REF][START_REF] Chabchoub | Time-reversal generation of rogue waves[END_REF]. Choosing, for example, a rogue wave in the form of a Gaussian pulse:

η 0 (x) = A 0 exp ( - x 2 l 2 ) , (30)
it is easy to calculate the structure of the wave packet at large times using the stationary phase method [START_REF] Wong | Asymptotic Approximations of Integrals[END_REF] η(x,

t) = A[x, t; k * (x, t)] cos [ω(k * )t -k * x -π /4] , (31)
where the envelope of the wave packet is

A[x, t; k * (x, t)] = A 0 l √ 2 t|d 2 ω/dk 2 | exp ( - k 2 * l 2 4
) ,

and

c gr [k * (x, t)] = dω dk = x t . ( 33
)
The nature of the amplitude distribution over the spectrum is determined by the function (the normalizations are the same as in Fig. 1, and K = kh):

F (K ) = 1 √ |d 2 Ω/dK 2 | exp [ - K 2 4 ( l h ) 2 ] . (34)
In particular, in the KdV approximation spectral amplitude is

F KdV (K ) = 1 √ K exp [ - K 2 4 ( l h ) 2 ] . (35)
In BBM approximation the amplitude function is

F BBM (K ) = (1 + K 2 /6) 3/2 √ K ⏐ ⏐ 1 -K 2 /18 ⏐ ⏐ exp [ - K 2 4 ( l h ) 2 ] . (36)
Functions ( 35) and (36) are shown in Fig. 8 for the case l/h = 2. As we see, the curves differ from each other, but not too much for kh < 2. The spectral amplitudes of bottom pressure variations have been obtained from (34) using the transfer function in various approximations. In particular, the bottom pressure spectral amplitudes are

P KdV (K ) = F KdV (K ) ⏐ ⏐ ⏐ ⏐ 1 - K 2 2 ⏐ ⏐ ⏐ ⏐ , (37)
P BBM (K ) = F BBM (K ) 1 + K 2 2 . ( 38
)
These functions are presented in Fig. 9. Here the difference between KdV and BBM approximations is more visible. As it is known, the stationary phase method is not valid in the vicinity of the leading wave where the wavenumber values are small. This problem has been analyzed in [START_REF] Yu | About surface waves in the ocean generated by underwater earthquakes[END_REF] with use of the exact dispersion relation and it was shown that leading wave amplitude is bounded. In the KdV approximation, the leading wave is described by the Airy function and this solution is given in many books on nonlinear waves. The difference in results is small on large times. The same is expected for BBM approximation of the dispersion relation. Inversing the wave field in space we obtain the focusing of weak chirped wave train in the rogue wave of Gaussian shape [START_REF] Ch | Rogue Waves in the Ocean[END_REF]. If its amplitude is large and length is small (almost ''ideal'' rogue wave), it will be manifested in the bottom pressure variations as a small signal, see Section 4.

Conclusion

The applicability of various long-wave approximations has been investigated in the description of the wave field and bottom pressure caused by surface waves. Within the framework of linear theory, an exact formula that relates the spectral amplitudes of a wave and pressure variations, which allows testing long-wave approximations is known. We show how these formulas are related to the well-known evolution equations used in the theory of water waves. Exactly propagating waves are described by the Whitham equation in the linear approximation, while the Korteweg-de Vries (KdV) and Benjamin-Bona-Mahony (BBM) equations are used in the long-wave approximation. In the framework of these approximations, the bottom pressure can be calculated using simple differentiation and integration operations without using the Fourier transform, which requires a sufficiently long record. The obvious fact is demonstrated that if the wave is long and sufficiently smooth (kh ∼ 1), both long-wave approximations provide sufficient accuracy and there is no particular difference in the use of one or another approximation. However, if the wave is long, though steep enough, the BBM approximation gives better results than the KdV one, and is quite close to the exact results. The same relates to the description of rogue waves, though formed from smooth relatively long waves, are often short and steep. Another advantage of the BBM approximation for calculating the bottom pressure is the ability to analyze noisy series without preliminary filtering, which is necessary when using the KdV approximation. It is important to note that in the existing highly nonlinear models of wave movements in the coastal zone, the KdV approximation (in the linear approximation) is used to calculate the pressure; so the question of the ''correct'' formula for pressure is not solved yet. Finally, the important conclusion that follows from our analysis is that, although the BBM approximation works well in almost the entire range of wave numbers, it results in small bottom pressures if the rogue wave is relatively short and steep (the same conclusion follows from the exact solution). Therefore, it is still unclear how correctly to distinguish rogue waves from bottom pressure records.

Fig. 1 .

 1 Fig. 1. Different approximations of the dispersion relation for surface gravity waves.

Fig. 2 .

 2 Fig. 2. Various approximations of the transfer function.

Fig. 3 .

 3 Fig. 3. The profile of the bottom ''ideal'' rogue wave. Solid red line -exact (Eq. (21)) and dashed blue line -BBM (Eq. (22)).

Fig. 4 .

 4 Fig. 4. The response of a long wave of a rectangular shape in variations of the bottom pressure (the solid red curve is the exact solution, and the dashed blue line is the BBM approximation), l/h = 15.

Fig. 5 .

 5 Fig. 5. The response of a long wave of a rectangular shape in variations of the bottom pressure (the solid red curve is the exact solution, and the dashed blue line is the BBM approximation), l/h = 2.

Fig. 8 .

 8 Fig. 8. Normalized spectral amplitudes of the wave field at l/h = 2. The solid red line is the KdV and the dashed blue line is the BBM approximation.

Fig. 9 .

 9 Fig. 9. The normalized spectral amplitudes of pressure variations at l/h = 2. The solid red line is the KdV and the dashed blue line is the BBM approximation.

Acknowledgments

This work is done within the joint Russian-French grant (RFBR No 19-55-15005). Particular support from the RFBR grants (20-05-00162 and 18-05-80019) and the grant for state support of scientific research of leading scientific schools of the RF (NSh-2485.2020.5) is also acknowledged. The CNRS is greatly acknowledged for the French part of the Russian-French grant.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.