C Kharif
email: kharif@irphe.univ-mrs.fr

M Abid

J D Carter

H Kalisch

Stability of periodic progressive gravity wave solutions of the Whitham equation in the presence of vorticity

The modulational instability of two-dimensional nonlinear traveling-wave solutions of the Whitham equation in the presence of constant vorticity is considered. It is shown that vorticity has a significant effect on the growth rate of the perturbations and on the range of unstable wavenumbers. Waves with kh greater than a critical value, where k is the wavenumber of the solution and h is the fluid depth, are modulationally unstable. This critical value decreases as the vorticity increases. Additionally, it is found that waves with large enough amplitude are always unstable, regardless of wavelength, fluid depth, and strength of vorticity. Furthermore, these new results are in qualitative agreement with those obtained by considering fully nonlinear solutions of the water-wave equations.

Introduction

It is well known that small-amplitude, two-dimensional, periodic wave trains are stable with respect to the modulational instability when the dispersive parameter kh, where k is the wavenumber and h is the mean fluid depth, is less than the critical value 1.363. Nevertheless, McLean [START_REF] Mclean | Instabilities of finite-amplitude gravity waves on water of finite depth[END_REF] found that Stokes waves are modulationally unstable when kh = 1 and ak = 0.29, where a is the wave amplitude. We can conjecture that strongly nonlinear uniform wave trains are modulationally unstable with respect to infinitesimal perturbations in shallow water. To extend the results of McLean [START_REF] Mclean | Instabilities of finite-amplitude gravity waves on water of finite depth[END_REF] to shallower water, Francius and Kharif [START_REF] Francius | Three-dimensional instabilities of periodic gravity waves in shallow water[END_REF] investigated instabilities of periodic gravity waves in shallow water using the fully nonlinear potential Euler equations. For small values of ak, they found that the dominant instabilities are quasi-twodimensional whereas for moderate and large steepness, the dominant instabilities are three-dimensional.

Whitham [START_REF] Whitham | Linear and Nonlinear Waves[END_REF] proposed an extension of the KdV equation by using the full linear dispersion instead of its third-order truncated expression. Consequently, the Whitham equation presents an improvement over the KdV equation for short waves. In fact, Carter [START_REF] Carter | Bidirectional Whitham equations as models of waves on shallow water[END_REF] showed that the Whitham equation provides a more accurate model for experimental initial waves of depression than does the KdV equation. Similarly, Moldabayev, Kalisch and Du-tykh [START_REF] Moldabayev | The Whitham equation as a model for surface water waves[END_REF] showed that solutions of the Whitham equation stay close to solutions of the Euler equations, and Klein et al. [START_REF] Klein | On Whitham and related equations[END_REF] gave a mathematical proof that solutions of the Euler equations are well approximated by solutions of the Whitham equation on small to intermediate time scales.

Ehrnström and Kalisch [START_REF] Ehrnström | Traveling waves for the Whitham equation[END_REF][START_REF] Ehrnström | Global bifurcation for the Whitham equation[END_REF] proved rigorously that the Whitham equation admits small and large amplitude, periodic traveling-wave solutions and numerically computed traveling-wave solutions with a variety of amplitudes including those close to the highest wave. Later on, Kharif and Abid [START_REF] Kharif | Nonlinear water waves in shallow water in the presence of constant vorticity: a Whitham approach[END_REF] computed steadily propagating periodic waves in the presence of constant vorticity. The method of computing these solutions was developed by Ehrnström and Kalisch [START_REF] Ehrnström | Global bifurcation for the Whitham equation[END_REF] and is also found in Sanford et al. [START_REF] Sanford | Stability of traveling wave solutions to the Whitham equation[END_REF] and Kharif and Abid [START_REF] Kharif | Nonlinear water waves in shallow water in the presence of constant vorticity: a Whitham approach[END_REF]. Sanford et al. [START_REF] Sanford | Stability of traveling wave solutions to the Whitham equation[END_REF] studied the Whitham equation and found that two-dimensional, periodic wave trains with kh = 1 are stable when the wave steepness, ak, is less than approximately 0.142 and are unstable when the wave steepness is larger than this threshold. To a certain extent, this result is surprising because the Whitham equation is valid for weakly nonlinear water waves. The latter authors numerically corroborated the stability analysis of Hur and Johnson [START_REF] Hur | Modulational instability in the Whitham equation for water waves[END_REF] who found that small-amplitude waves with kh < 1.145 are stable and are unstable when kh > 1.145. Note that Benjamin and Feir [START_REF] Benjamin | The disintegration of wave trains on deep water. Part 1. Theory[END_REF] and Whitham [START_REF] Whitham | Non-linear dispersion of water waves[END_REF] showed that Stokes waves are unstable with respect to long wavelength perturbations if kh > 1.363. Later on, Hur and Johnson [START_REF] Hur | Modulational instability in the Whitham equation with surface tension and vorticity[END_REF] incorporated in the Whitham equation the effect of constant vorticity which modifies the threshold value of the dispersive parameter.

Following Whitham [START_REF] Whitham | Linear and Nonlinear Waves[END_REF], Kharif et al. [START_REF] Kharif | Rogue waves in shallow water in the presence of a vertically sheared current[END_REF] and Kharif and Abid [START_REF] Kharif | Nonlinear water waves in shallow water in the presence of constant vorticity: a Whitham approach[END_REF] proposed a new model derived from the Euler equations for fully nonlinear water waves propagating on a vertically sheared current of constant vorticity in shallow water that satisfies the unidirectional linear dispersion relation. From this model they derived, within the framework of weakly nonlinear waves, a generalization of the Whitham equation which they named the vor-Whitham equation. At the same time, Hur [START_REF] Hur | Shallow water wave models with constant vorticity[END_REF] and Bjørnestad and Kalisch [START_REF] Bjornestad | Shallow water dynamics on linear shear and plane beaches[END_REF] derived shallow water wave equations in the presence of constant vorticity.

In order to extend the previous studies of Sanford et al. [START_REF] Sanford | Stability of traveling wave solutions to the Whitham equation[END_REF] and Hur and Johnson [START_REF] Hur | Modulational instability in the Whitham equation with surface tension and vorticity[END_REF], we consider the spectral stability of twodimensional, periodic traveling-wave solutions of the Whitham equation in the presence of constant vorticity. Our study focuses on the modulational instability. The second aim is to show that the Whitham equation, which is an approximate equation that is easier to work with than the fully nonlinear water wave equations, may provide reliable stability results that are in qualitative agreement with those of the full equations.

In section 2, we present the vor-Whitham equation. In section 3, we describe how to compute the stationary solutions to this equation. Additionally, we present the stability of these solutions with respect to infinitesimal perturbations, the growth rates of instabilities, and the ranges of unstable Floquet parameters as functions of vorticity. Section 4 contains a conclusion of this work.

The vor-Whitham equation

We consider two-dimensional gravity waves that propagate on the surface of an inviscid, incompressible fluid with a shear current parallel to the direction of wave propagation that varies linearly in the vertical direction. We assume that the waves travel along the x-axis and that the z-axis is oriented upward with z = 0 representing the unperturbed free surface. In order to focus on the effects due to vorticity, we assume that the current velocity is zero at the free surface. In this situation, the current vorticity, , is constant and the vor-Whitham equation is given by

η t + c 1 ( )ηη x + K * η x = 0, (1)
where η(x, t) represents the free surface displacement and t represents time. The coefficient of the nonlinear term is

c 1 ( ) = 3gh + h 2 2 h 4gh + h 2 2
, where g is the gravitational constant of acceleration. The dispersive term is given by the convolution product, K * η x , which is the inverse Fourier transform of the product of the Fourier transforms of K (x) and η x (x, t). The integral kernel is

K (x) = 1 2π +∞ -∞ c(k)e ikx dk,
where the unidirectional dispersion of linear waves in the presence of vorticity is

c(k) = tanh(kh) 2k + g tanh(kh) k + 2 tanh 2 (kh) 4k 2
.

We consider traveling-wave solutions of the vor-Whitham equation of the form η(x, t) = η(xc 0 t) for a given phase velocity c 0 . Substituting this ansatz into equation ( 1) and integrating once leads to the equation that defines c and where B is the constant of integration. We choose B so that the solution η has zero mean.

η -c 0 η + c 1 ( ) η2 2 + K * η = B, (2)

Stability analysis

Steady waves

In the frame of reference moving with the traveling-wave solution, the vor-Whitham equation is given by

η τ -c 0 η X + c 1 ( )ηη X + K * η X = 0, (3)
where X = xc 0 t and τ = t. In this frame of reference, the traveling-wave solution, η(X), is stationary (independent of τ ) and satisfies equation (2). To compute these solutions, we use the numerical method of Ehrnström and Kalisch [START_REF] Ehrnström | Traveling waves for the Whitham equation[END_REF]. Details of the numerical method and its validation are found in Kharif and Abid [START_REF] Kharif | Nonlinear water waves in shallow water in the presence of constant vorticity: a Whitham approach[END_REF]. However, herein we add a supplementary equation that fixes the wave amplitude when following solutions using c 0 as the continuation parameter.

In order to put the equations in dimensionless form, h and h/g are chosen as the reference length and reference time, respectively. This choice corresponds to setting h = 1 and g = 1. We consider 2π -periodic traveling-wave solutions to the vor-Whitham equation. Consequently, the wavenumber of these solutions is k = 1.

Fig. 1 shows profiles of traveling-wave solutions to the vor-Whitham equation for two values of and four values of the wave steepness. These plots, along with others left out for brevity, demonstrate that: (i) solutions corresponding to different values of and L are qualitatively similar, (ii) for solutions with a given period and vorticity, increasing wave speed increases wave height and steepness, and (iii) there appears to be a solution of maximal height for all wave periods and values of vorticity.

Fig. 2 shows profiles of traveling-wave solutions to the vor-Whitham equation with wave height 0.34 and four values of . These plots, along with others left out for brevity, demonstrate that: (i) for solutions with a given period and wave height, increasing vorticity causes the width of the solution to increase and (ii) for solutions with a given period and wave height, increasing vorticity causes both the minima and maxima of η to decrease. values of the wave steepness, c 0 increases as increases. This feature was observed by Kharif and Abid [START_REF] Kharif | Nonlinear water waves in shallow water in the presence of constant vorticity: a Whitham approach[END_REF] within the framework of a fully nonlinear, generalized vor-Whitham equation, whereas the profiles shown in Fig. 1 are weakly nonlinear.

Stability of steady solutions

In order to study the stability of these solutions with respect to infinitesimal perturbations, let

η(X, τ ) = η(X) + η (X, τ ), |η | | η|, (4)
where η(X) and η (X, τ ) correspond to the 2π -periodic unperturbed steady solution and infinitesimal square integrable disturbance, respectively. Substituting equation (4) into equation ( 3) and linearizing gives the following equation which governs the (linear) evolution of the perturbations

η τ -c 0 η X + c 1 ( )( ηη X + ηX η ) + K * η X = 0. ( 5
)
The Fourier-Floquet-Hill method of Deconinck and Kutz [START_REF] Deconinck | Computing spectra of linear operators using the Fourier-Floquet-Hill method[END_REF] and Johnson [START_REF] Johnson | Stability of small periodic waves in fractional KdV-type equations[END_REF] establishes that all bounded solutions of this problem have the form

η (X, τ ) = exp(λτ ) exp(ip X) +∞ j=-∞ a j exp(ij X), ( 6
)
where p is a real number known as the Floquet parameter. Substituting equation ( 6) into equation [START_REF] Moldabayev | The Whitham equation as a model for surface water waves[END_REF] gives

+∞ -∞ (c 0 -c 1 ( ) η -c p+ j )i(p + j)a j exp(ij X) -c 1 ( ) ηX +∞ -∞ a j exp(ij X) = λ +∞ -∞ a j exp(ij X), (7)
where

c p+ j = tanh(p + j) 2(p + j) + tanh(p + j) (2(p + j) + 2 tanh 2 (p + j) 4(p + j) 2 .
Equation ( 7) is transformed into a generalized eigenvalue problem for λ, which after truncation at M Fourier modes can be written as follows Au = λBu, [START_REF] Ehrnström | Global bifurcation for the Whitham equation[END_REF] where u = (a -M , ..., a 0 , ..., a M ) T is the corresponding eigenvector. The 2M + 1 unknown coefficients (a -M , ..., a 0 , ..., a M ) are chosen to satisfy (7) at 2M + 1 collocation points equally distributed along one period of the unperturbed solution. We used M = 100 and checked that the results are the same within seven significant figures when doubling this value. The complex-valued matrices A and B depend on the unperturbed wave, η, the vorticity, , and the Floquet parameter, p. Once the unperturbed traveling-wave solution has been computed and p fixed, the generalized eigenvalue problem ( 8) is solved by using a standard numerical eigenvalue solver.

The eigenvalue spectrum corresponding to the flat surface ( η = 0) is

λ j = i(p + j)c 0 -i(p + j)c p+ j ,
where the phase velocity of the flat surface is

c 0 = tanh(1)/2 + tanh(1) + 2 tanh 2 (1)/4.
All of these eigenvalues lie on the imaginary axis. Therefore, the flat surface is spectrally stable. The corresponding eigenfunctions are η j = a j exp(λ j τ ) exp(i(p + j) X) which represent infinitesimal waves of frequency (p + j)c 0 -(p + j)c p+ j in the moving frame of reference and wavenumber p + j.

As the amplitude of the unperturbed wave increases from zero, the eigenvalues move on the imaginary axis and eigenvalue collisions occur. A necessary, but not sufficient, condition for instability is the collision of eigenvalues. The collision of eigenvalues can be expressed as λ j 1 (p) = λ j 2 (p), [START_REF] Kharif | Nonlinear water waves in shallow water in the presence of constant vorticity: a Whitham approach[END_REF] where the corresponding wavenumbers are k 1 = p + j 1 and k 2 = p + j 2 . McLean et al. [START_REF] Mclean | Three-dimensional instability of finite amplitude water waves[END_REF] divided the solutions of ( 9) into two classes. Depending upon whether j 1j 2 is even or odd, instabilities belong to class I or class II, respectively. Without loss of generality, it is convenient to assume that j 2 =j 1 for class I and j 2 =j 1 -1 for class II. Herein, we focus only on class I with j 1 = 1 which correspond to instability of modulational type. These assumptions allow equation ( 9) to be rewritten as

2c 0 = (1 + p)c 1+p + (1 -p)c 1-p , ( 10
) with 2k 0 = k 1 + k 2 , ( 11
)
where

k 0 = 1, k 1 = 1 + p and k 2 = 1 -p.
The subharmonic and superharmonic sidebands correspond to k 2 and k 1 , respectively. Equations ( 10) and ( 11) can be interpreted as the resonance of two infinitesimal waves (sidebands) with the basic wave, i.e. a resonant four-wave interaction.

Numerical results

As a check on our numerical approach, we considered the case of Sanford et al. [START_REF] Sanford | Stability of traveling wave solutions to the Whitham equation[END_REF] corresponding to the 2π -periodic solution shown in their figure 3(b) with c 0 = 0.8002. We found that the maximum rate of growth is 0.000356 and the frequency is 0.00751 corresponding to p = ±0.04056. These values obtained with = 0 are very close to those of Sanford et al. [START_REF] Sanford | Stability of traveling wave solutions to the Whitham equation[END_REF]. Note that we used their transformation η → 3η/4 in order to ensure that our Whitham equation and solution were the same as theirs. Additionally, Sanford et al. [START_REF] Sanford | Stability of traveling wave solutions to the Whitham equation[END_REF] showed that the traveling-wave solutions are stable to the modulational instability if their wave steepness is less than approximately 0.142 which corresponds to ak ≈ 0.19 in our scaling (see Table 1).

We carried out the stability computations for solutions with steepness ak = 0.05 and ak = 0.10 for a range of values. We found that they are both stable with and without vorticity. Consequently, we decided to examine the stability of traveling-wave solutions of wave steepness ak = 0.20, a value larger than the critical value in the = 0 case. Figs. 4 and 5 contain plots of Im(λ) and Re(λ) versus p, the wavenumber of the normal mode perturbation for three values of the vorticity and ak = 0.20. The upper plots in these figures show the collisions of two purely imaginary in the vicinity of the origin. These collisions give rise to instabilities corresponding to intervals of instability shown in the lower plots. These plots show that for a fixed value of the wave steepness both the rate of growth and the size of the interval of instability increase as decreases. These plots also show that there are no instabilities with the same wavenumber as the unperturbed solution (i.e. p = 0) for any of the values of vorticity we examined.

Fig. 6 shows the magnitudes of the coefficients a j for the most unstable perturbation corresponding to = 0.10 and ak = 0.20. The two dominant components, j = -1 and j = 1, correspond to subharmonic and superharmonic sidebands typical of modulational instability. Note that the wavenumbers of the subharmonic and superharmonic sidebands are 1p and 1 + p, respectively. Fig. 7 shows the amplitude spectrum of the unperturbed wave of wave steepness ak = 0.20 perturbed by its most unstable normal mode. The physical perturbation corresponds to the real part of the perturbation given in equation [START_REF] Klein | On Whitham and related equations[END_REF]. The amplitude of the modes has been normalized so that the fundamental mode, k = 1, has unit amplitude. The magnitude of the superharmonic mode of the perturbation, |a 1 |, is one tenth of the amplitude of the fundamental mode.

Fig. 8 contains plots of the maximum growth rate versus the Floquet parameter for the perturbation for three values of the vorticity. These plots show that the vorticity effect is twofold: (i) the growth rate increases as decreases except for very long unstable perturbations, (ii) the bandwidth of unstable wavenumbers increases as decreases. Note that these features were observed by Thomas et al. [START_REF] Thomas | A nonlinear Schrödinger equation for water waves on finite depth with constant vorticity[END_REF] and Francius and Kharif [START_REF] Francius | Two-dimensional stability of finite-amplitude gravity waves on water of finite depth with constant vorticity[END_REF] within the framework of the nonlinear Schrödinger equation and the fully nonlinear Euler equations, respectively. Hur and Johnson [START_REF] Hur | Modulational instability in the Whitham equation for water waves[END_REF] found that small-amplitude, 2π /k-periodic traveling-wave solutions of the Whitham equation are modulationally unstable if kh > kh crit ≈ 1.146. Later on, Hur and Johnson [START_REF] Hur | Modulational instability in the Whitham equation with surface tension and vorticity[END_REF] found a formula for the boundary in the ( , k)-plane that separates stable and unstable small-amplitude, periodic, travelingwave solutions to the vor-Whitham equation. A plot of this boundary is included in Fig. 9. We numerically corroborated this analytic result for a variety of and k values. For example, we considered solutions just below and just above the critical value of ( , k) ≈ (5, 0.96). We found that a small-amplitude (wave height of 4.3 * 10 -3 ) solution to the vor-Whitham equation with ( , k) = (5, 0.9) is spectrally stable, see Fig. 10(a) and that a smallamplitude (wave height of 3.3 * 10 -3 ) solution with ( , k) = (5, 1) is unstable, see Fig. 10(b). In summary, in the absence of vorticity we found a critical value in agreement with that of Hur and Johnson [START_REF] Hur | Modulational instability in the Whitham equation for water waves[END_REF] and Sanford et al. [START_REF] Sanford | Stability of traveling wave solutions to the Whitham equation[END_REF] and in the presence of vorticity, we found critical values in agreement with the finding of Hur and Johnson [START_REF] Hur | Modulational instability in the Whitham equation with surface tension and vorticity[END_REF]. The modulational instability of large-amplitude solutions in the presence of vorticity is a new finding.

Conclusion

The modulational instability of traveling-wave solutions of the Whitham equation in the presence of vorticity has been investigated numerically. The Whitham equation is an extension of the KdV equation which takes into account the full range of dispersion. We presented a sampling of our results which show the important qualitative results related to the modulational instability of periodic, traveling-wave solutions to the vor-Whitham equation. These results demonstrate that (i) the vorticity strongly modifies the growth rate of the modulational instability, (ii) a critical wave steepness above which the solutions become unstable exists whatever the value of the vorticity, and (iii) the critical depth under which the solutions restabilize depends on the vorticity.

Fig. 1 .

 1 Fig. 1. Plots of periodic traveling-wave solutions to the vor-Whitham equation with L = 2π , wave height, H (see captions), and (a) = -1.0, (b) = 1.0. Note that the vertical scales in the two plots are different.

Fig. 3

 3 Fig.3displays the phase velocity as a function of wave steepness for five values of the vorticity. These plots show that the phase velocity decreases as the wave steepness increases. For fixed

Fig. 2 .

 2 Fig. 2. Plots of periodic traveling-wave solutions to the vor-Whitham equation with wave height, H = 0.34, L = 2π , and four different values of .

Fig. 3 .

 3 Fig. 3. Phase velocity of the traveling-wave solutions as a function of the wave steepness for several values of the vorticity.= 0 (•), -0.1 ( * ), 0.1 ( ), -0.2 ( ), 0.2 ( ).

Fig. 4 .

 4 Fig. 4. Radian frequency (top) and rate of growth (bottom) of the normal mode perturbation against its wavenumber for = 0 and ak = 0.20.

Fig. 5 .

 5 Fig. 5. Radian frequency (top) and rate of growth (bottom) of the normal mode perturbation against its wavenumber for ak = 0.20 and (a) = -0.1 (b) = 0.1.

Fig. 6 .

 6 Fig. 6. Magnitude of the coefficient a j of the most unstable normal mode corresponding to = 0.1 and ak = 0.20.

Fig. 7 .

 7 Fig. 7. Amplitude spectrum of the unperturbed wave of wave steepness ak = 0.20 perturbed by its most unstable normal mode for = 0.1. The amplitudes of the Fourier components have been normalized by the amplitude of the fundamental mode k = 1 of the unperturbed wave.

Fig. 8 .Fig. 9 .

 89 Fig.8. Maximum growth rate against the wavenumber for = 0 (solid line), = -0.1 (dashed line), = 0.1 (dot-dashed line) and ak = 0.20.

Fig. 10 .

 10 Fig. 10. Stability spectra for (a) a small-amplitude solution to the vor-Whitham equation with ( , k) = (5, 0.9) and (b) a small-amplitude solution to the vor-Whitham equation with ( , k) = (5, 1). Note that the axes in the two plots are scaled differently. The spectrum in (a) lies purely on the imaginary axis and therefore the corresponding solution is spectrally stable. The spectrum in (b) has values with positive real parts and therefore the corresponding solution is unstable.

Table 1

 1 Growth rate of the most unstable perturbation as a function of the basic wave steepness without vorticity.

	ak	p	Re(λ) max
	0.20	0.101	1.67 × 10 -3
	0.19	0.005	0.811 × 10 -4
	0.18	0.003	4.19 × 10 -5
	0.17		O(10 -11 )
	0.15		O(10 -11 )
	0.10		O(10 -12 )
	0.05		O(10 -12 )