Unsteady Gerstner waves
Introduction
Gerstner waves are steady progressive waves on the surface of a liquid of infinite depth. They are described by the exact solution of the equations of a perfect incompressible fluid [1], [2]. The Gerstner solution was rediscovered by Froude [3], Rankine [4] and Reech [5]. This solution is remarkable as it is the only exact solution for gravity waves on deep water. In this sense, for gravitational waves on water studied in the framework of complete hydrodynamic equations the Gerstner waves are analogous to solitons in nonintegrable systems (Kivshar and Malomed [6], Grimshaw et al. [7], Zahibo et al. [8], Stepanyants [9]).
Key contributions to physical understanding and mathematical justification of the Gerstner wave solution were made by Dubreil-Jacotin [10], Lamb [2], Mollo-Christensen [11], [12], Constantin and Strauss [13], Constantin [14] and Henry [15], among others. Hydrodynamic stability of the Gerstner wave was investigated by Leblanc [16]. The Gerstner surface wave solution was extended to edge waves in fluids with a free surface and a plane sloping rigid boundary (Yih [17]; Constantin [18]). Gerstner surface and edge waves remain exact solutions in stratified incompressible fluids [10], [17] (see also Stuhlmeier [19], [20]), including fluids with density and flow velocity discontinuities [12]. Mollo–Christensen developed an analytical model of gravitational and geostrophic billows in the atmosphere by deriving an exact finite-amplitude solution for a wave on an interface of two fluids, with one fluid moving as in the Gerstner wave and the other fluid in a uniform motion at the speed of the wave [11]. Johnson used the Gerstner edge wave solution to construct asymptotic solutions valid for variable but small bottom slopes [21]. Within the f- plane approximation, extensions of Gerstner surface and edge waves to rotating fluids have been proposed by Pollard [22], Matioc [23], [24] and Weber [25]. Ionescu–Kruse applied the short-wavelength perturbation method to derive instability criteria for the three-dimensional nonlinear Pollard geophysical waves [26]. Mollo–Christensen obtained the exact solution describing nonlinear edge waves in a rotating fluid in the presence of a mean flow [27]. Constantin and Monismith studied the propagation of Gerstner waves in the presence of mean currents and rotation [28]. Gerstner-type solutions for equatorially trapped surface and internal waves in the ocean were obtained by Constantin [29], [30], [31] in the β-plane approximation. The exact solution for equatorially trapped surface waves was further extended by Henry by allowing for a uniform current in the direction of wave propagation [32]. Godin presented the solutions providing an extension of the Gerstner wave in an incompressible fluid with a free boundary to waves in compressible three-dimensionally inhomogeneous moving fluids [33].
At the same time, physical feasibility of Gerstner waves is still disputable. These waves possess vorticity, hence they cannot be generated from rest by conservative forces in an ideal fluid. Lamb pointed that such a wave motion may arise against the background of a shear flow having the same vorticity [2]. This idea was implemented in the laboratory experiments by Monismith et al. [34]. Liquid particles in a Gerstner wave travel around a circumference without drift flow (average over the period). The wave motion possessing this property was generated in a flume by Monismith's team. This means that Gerstner waves or their finite-depth relatives must have been observed.
Weber, in turn, hypothesized that a weakly nonlinear Gerstner solution may be realized taking into account the effects of liquid viscosity and surface films [35]. In a linear approximation with respect to the small parameter of wave steepness, liquid particles in a viscous fluid move around circumferences, the radii of which decrease exponentially with time. The wave vorticity is concentrated in a narrow near-surface boundary layer. In the quadratic approximation, the viscosity gives rise to mean drift, which Weber compared with the available experimental data and did not exclude that viscosity-modified Gerstner waves were observed in some wave tank experiments.
The mentioned above studies were performed assuming constant pressure on a free fluid surface. This approximation is justified in the absence of wind. The physical manifestations of wind are variable pressure on the fluid surface and a vortex character of the wave motion. Waseda and Tulin showed experimentally that the wind does not suppress the Benjamin–Feir instability [36]. The investigation of the self-consistent wind-wave interaction is a rather complicated problem. A possible way to simplify it is to choose a definite law of pressure variation on the free surface. In the works (Leblanc [37], Kharif et al. [38], Onorato and Proment [39], Chabchoub et al. [40], Brunetti et al. [41], Eeltink et al. [42]) where the dynamics of weakly nonlinear, narrow bandwidth trains of surface waves was studied, a variable external pressure was specified according to the Miles linear theory of wind wave excitation [43]. Yan and Ma explored the formation of strongly nonlinear waves in the framework of a complete system of equations of hydrodynamics in the presence of a uniform air flow and proposed a phenomenological model for air flow pressure distribution on a free surface [44].
In this paper we will investigate the dynamics of vortex surface waves under the action of time-harmonic pressure non-uniformly distributed on a free surface. The consideration is carried out in Lagrangian variables. The fluid motion is described by a class of exact solutions [45], generalizing the Gerstner solution. Like in the classical Gerstner wave, the trajectories of the fluid particles are circumferences. The particles on a free surface have the same radius of rotation but the centers of the circumferences lie on different horizons, thereby the wave has a variable profile. Such waves are called unsteady Gerstner waves. We will analyze a family of waves, in which variable pressure acts on a bounded section of the free surface. The law of external pressure distribution includes an arbitrary function and corresponds to a wide variety of boundary conditions. We will consider a solution describing the dynamics of a non-uniform packet of unsteady Gerstner waves. Wave and pressure profiles, as well as vorticity distribution will be addressed. A possibility of observing the studied waves in laboratory and in the real ocean will be discussed.
The rest of this paper is organized as follows. In Section 2 we introduce a class of Ptolemaic flows into consideration and show that Gerstner waves are their particular case. An exact solution for Gerstner-type waves with variable pressure on a free surface was obtained in Section 3. In the next section, the dynamics of the Gerstner wave packet is studied, when the variable pressure acts on a bounded interval of the free surface. In Section 5 we show that, in the case of a uniform traveling plane wave of external pressure, the Gerstner solution is valid but with a different form of the dispersion relation. Our findings are summarized in Section 6.
Section snippets
Ptolemaic flows and Gerstner wave
Consider gravity waves on the surface of a homogeneous liquid having density ρ. Neglecting viscosity, the equations of 2D hydrodynamics in Lagrangian variables are written in the form [2], [46]:where X, Y are the Cartesian coordinates of the liquid particle trajectory, a, b are its Lagrangian coordinates, p is pressure, g is acceleration of gravity, t is time, the subscript “0” denotes the value of the variable at the
Variable external pressure
Consider Ptolemaic flows of the form
This expression is a particular case of the relation (6). The function F in (6) is identical to that of the Gerstner wave, and the function G may vary. If G is a linear function, then the relation (10) coincides with Gerstner's solution.
The wave (10) has a number of common properties with the Gerstner wave:
- -
liquid particles move around circumferences with radius Aexp kb;
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liquid particles rotate around circumferences of the same
An example of exact solution
Let a variable external pressure act on a bounded section of a free surface. This requirement corresponds to the following asymptotic behavior of the function G:
On both infinities, the wave tends asymptotically to the Gerstner wave, where the pressure is constant. This allows us to assume that the dispersion relation is valid for the waves (10), (12), similarly to the Gerstner wave.
The function G is taken in the form
The parameters α, β have
Steady Gerstner wave generated by the running harmonic wave pressure
In the previous sections we assumed that the wind forces a bounded area of the free surface. Consequently, outside this interval the waves are regarded to be uniform and their frequency and wave number meet the dispersion relation for Gerstner waves. However, there may arise a situation when a variable external pressure acts along all the free surface. In this case, the wave packet parameters will be determined entirely by the form of pressure distribution.
We will restrict our consideration to
Conclusions
The class of exact solutions of two-dimensional hydrodynamics describing vortex gravity surface waves in the presence of an external time-harmonic pressure varying along the coordinate has been found and analyzed. The particles of the free surface in the wave rotate around circumferences of the same radii, like in an ordinary Gerstner wave. Therefore, it is suggested to call this class of waves unsteady Gerstner waves. The pressure distribution on the free surface includes a quite arbitrary
Acknowledgement
The publication was prepared within the framework of the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 2018–2019 (grant № 18-01-0006) and by the Russian Academic Excellence Project “5-100″.
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