On the Dynamics of Two-Dimensional Capillary-Gravity Solitary Waves with a Linear Shear Current

The numerical study of the dynamics of two-dimensional capillary-gravity solitary waves on a linear shear current is presented in this paper. The numerical method is based on the time-dependent conformal mapping. The stability of different kinds of solitary waves is considered. Both depression wave and large amplitude elevation wave are found to be stable, while small amplitude elevation wave is unstable to the small perturbation, and it finally evolves to be a depression wave with tails, which is similar to the irrotational capillary-gravity waves.


Introduction
A capillary-gravity wave on a fluid surface is influenced by both the effects of surface tension and gravity.The wavelength of capillary-gravity waves in water is typically less than a few centimeters; therefore, they are often referred to as ripples.On the open ocean, much larger surface water waves may result from coalescence of smaller wind-caused capillarygravity waves.In the past several decades, the irrotational capillary-gravity waves have been studied by many scientists.In the beginning stage of the wind-driven wave, shear layer is often generated by the wind where the capillary-gravity waves often run on.So it is more realistic to consider the whole system in the presence of vorticity, and the simplest way is to incorporate a linear background shear in the potential flow.
Steady gravity waves with constant vorticity have been considered numerically by applied mathematicians in 1980s and 1990s.Many new profiles due to the shear effect have been found (see [1][2][3][4] and references therein).In the recent ten years, pure mathematicians revisit this problem for both steady and unsteady cases.New Hamiltonian formulation for the modified potential system and model equations under different assumptions have been derived.For the steady problem, Constantin et al. considered quite general distribution of the vorticity in [5][6][7][8], while for the unsteady flow, most of the works focus on the constant vorticity, and the read is referred to the references [9][10][11].
In the numerical aspects of two-dimensional surface water waves, the boundary integral method is proven to be very accurate and efficient for finding permanent wave profiles.For the unsteady computation, a new method based on time-dependent conformal mapping was derived by Dyachenko et al. [12] and widely used by other scientists later on.Choi (2009) generalized this method to incorporate the background linear shear for gravity waves in [13].Most recently, the dynamics of capillary-gravity solitary waves without background shear are considered by Milewski et al. [14] which is based on the same technique.A survey of the numerical methods for time-dependent irrotational water wave is provided by Dias and Bridges (2006).
In [15], the authors studied the steady capillary-gravity solitary wave propagating at the surface of a fluid of finite depth with constant vorticity, using the asymptotic analysis for small amplitude waves and direct numerical computation for full potential equations.However, the dynamics of these waves have not been studied yet, and in this paper, we try to bridge the gap.The rest of the paper is organized as follows.In Section 2, the governing equation is transferred to be a potential equation under a simple variable substitution.In Section 3, we derive the numerical method for computing 2 Advances in Mathematical Physics the full potential surface water waves in the presence of constant vorticity.In Section 4, the main numerical results are presented, including the typical profiles of solitary waves, bifurcation diagrams, and stability analysis.

Mathematical Formulation
We consider 2D surface water waves and choose Cartesian coordinates so that the -axis points upwards and -axis is the direction of wave propagation.The governing equation of motion is incompressible Euler system as follows: where (, V) is the velocity field according to  and  direction; respectively, (, , ) denotes the pressure and  is the gravitational acceleration.The free boundary conditions for capillary-gravity waves are where the free surface displacement  = (, ) is supposed to be a graph of function, constant  is the fluid density,  > 0 denotes the coefficient of surface tension, and  0 is the constant atmospheric pressure.We suppose the bed is flat and given by  = − where  is the mean depth of the wave.Now the nonslip boundary condition V = 0 is applied to the bottom.Taking the curl of the momentum equations implies the conservation of vorticity along the particle path; that is, if we designate  = V  − V  the vorticity equation is in the following form: The mass equation (1) implies that there exists a stream function ψ such that φ = −V and φ = .Therefore, the vorticity can be expressed by the stream function by  = −Δ φ.
In this paper, we assume  is a constant initially.From the argument in (4), we know that if the initial voriticity is a constant then it remains constant all the time.Since we only consider solitary waves in this paper, we assume the uniform velocity on the surface as  → ±∞.If moving the frame with this velocity, we can simply assume zero horizontal velocity in the far flow field on the free surface.Then we can reduce the problem to one for Laplace equation by introducing a new stream function  = φ + (/2) 2 , that is, Δ = 0. Consequently, one can introduce its harmonic conjugate  satisfying Now we can reformulate the boundary conditions by making use of  and .The kinematic boundary conditions on the bottom and on the free surface take the following form: and the momentum equations now become Using ( 3), Bernoulli's Law, which is also referred to as the dynamical boundary condition, for the constant vorticity flow now becomes Finally, we nondimensionalize the whole system by the time scale (/ 3 ) 1/4 , the length scale (/) 1/2 , and the potential and steam scale ( 3 / 3 ) 1/4 ; therefore, with a little abuse of notations, the governing system for the constant vorticity flow takes the following form: Δ = 0 for −  <  <  (, ) ,   = 0 for  = −,   = −    +   +   for  =  (, ) , 3/2 = 0 for  =  (, ) . (9)

Numerical Method
Following the method presented in [12,14,16] and the notations in [14], where  =  2  +  2  .It follows from the kinematic boundary condition, the second equation of ( 6) and formulas (11), that Notice that   /  is an analytic function of  +  and it therefore follows that the real and imaginary parts of their boundary values are related by In the curvilinear coordinates, the kinematic boundary condition becomes (13), while the dynamic boundary condition is obtained by substituting the expressions of ( 11) and ( 13) into Bernoulli's equation (8).Consider Finally, ( 14) is a complete differential integral equation for updating  and Φ.

Results
Due to the small length scale of the capillary-gravity waves (∼1 cm), deep water approximation is an appropriate assumption in this paper.Therefore the surface Euler system There is a local minimum of the phase speed for each curve.
in the curvilinear coordinates takes the following form: where  is the famous Hilbert transform with the Fourier symbol  sgn().There are three well-known conservative quantities for the evolution system: conservation of mass, momentum, and energy.Consider 4 Advances in Mathematical Physics If we linearize the system (15), it is easy to obtain the dispersion relation between the wave number  and the frequency Ω which results in the following phase speed: The linear dispersion relation is presented in Figure 1, for different vorticity strengths.The plotting of the phase speed versus wave number shows that there is a local minimum for each curve, and it has been figured out in [16] that solitary waves exist only below this point, and two branches of solitary waves bifurcate from the minimum of the phase speed (see Figure 3).

Traveling Wave.
To study the dynamics of the capillarygravity solitary waves with linear background shear, it is the first step to seek traveling wave solution to the Euler equations ( 9) with wave propagating speed ; we assume all functions depend on  − ; then, the boundary conditions on the free surface read In the curvilinear coordinates, (18) can be rewritten as follows: Similarly, the dynamic boundary condition (19) can be simplified as follows: Equation ( 21) only depends on the variable  which is convenient for numerical simulation.As a special case, for irrotational capillary-gravity wave, (21) recovers (12) in [14].
If ignoring the surface tension, this equation for traveling wave is equivalent to the equation solved by Choi [13] under a different scaling.
The nonlinear integral-differential equation ( 21) is discretized in the Fourier space using 1024 modes and solved by Newton's method and continuation method.In Figure 2, bifurcation pictures are shown for different .For each , solitary waves are found below the minimum of the phase speed, and small  has small minimum phase speed.There are two branches of solitary waves-elevation wave with the center above zero and depression wave with the center below zero bifurcating from the minimum of the phase speed.For large depression solitary waves, limit profile of selfintersect configuration is numerically found for nonzero .This unphysical profile has already been found in [16] for the irrotational wave.Large amplitude depression solitary wave which is composed of two depression solitary waves is also found for the flow with shear background (see Figure 2).

Stability.
For irrotational capillary-gravity solitary waves, the stability analysis has been carried out by Calvo and Akylas (2002), and later by Milewski et al. [14].In this section, stability properties are examined by time-dependent numerical experiments for the capillary-gravity solitary waves when the linear background shear is added, and similar results to irrotational case are observed.Small and moderate depression solitary waves are proved to be stable in our numerical experiments.However elevation solitary waves show different stability behaviors (see Figure 4).Small amplitude elevation wave is unstable due to small perturbation, and it finally evolves into a stable depression solitary wave with some radiations.Large amplitude elevation we introduce the time-dependent conformal mapping (, , ) = (, , ) + (, , ) which maps the physical domain − <  < (, ) occupied by the fluid into a stripe − <  < 0. The following capital letters denote the values on the boundary  = 0:  (, ) =  (, 0, )  (, ) =  (, 0, ) , Φ (, ) =  (, 0, ) Ψ (, ) =  (, 0, ) .(10) Therefore the relations  =  − C[] and Φ = −C[Ψ] hold, where C is a linear operator with the Fourier symbol Ĉ =  coth().To derive the evolution equations for Φ and , the following equations from the chain rule of differentiation are useful:

2 Figure 1 :
Figure 1: Dispersion relations for different vorticity strengths .There is a local minimum of the phase speed for each curve.

Figure 2 :Figure 3 :
Figure 2: (a) Amplitude versus speed bifurcation.There are two branches for each  and bifurcate from the minimum of the phase speed.From left to right:  = −0.2,−0.05, 0.05, 0.2.(b) Energy versus speed curves for different .The upper branches are for elevation solitary wave, and the lower branches are for depression ones.From left to right:  = −0.2,−0.05, 0.05, 0.2, respectively.