Experiments and analyses of upstream-advancing solitary waves generated by moving disturbances

In this joint theoretical, numerical and experimental study, we investigate the phenomenon of forced generation of nonlinear waves by disturbances moving steadily with a transcritical velocity through a layer of shallow water. The plane motion considered here is modelled by the generalized Boussinesq equations and the forced Korteweg-de Vries (fKdV) equation, both of which admit two types of forcing agencies in the form of an external surface pressure and a bottom topography. Numerical results are obtained using both theoretical models for the two types of forcings. These results illustrate that within a transcritical speed range, a succession of solitary waves are generated, periodically and indefinitely, to form a procession advancing upstream of the disturbance, while a train of weakly nonlinear and weakly dispersive waves develops downstream of an ever elongating stretch of a uniformly depressed water surface immediately behind the disturbance. This is a beautiful example showing that the response of a dynamic system to steady forcing need not asymptotically tend to a steady state, but can be conspicuously periodic, after an impulsive start, when the system is being forced at resonance. A series of laboratory experiments was conducted with a cambered bottom topography impulsively started from rest to a constant transcritical velocity U, the corresponding depth Froude number F = U/(gh[sub]0)^1/2 (g being the gravitational constant and h[sub]0 the original uniform water depth) being nearly the critical value of unity. For the two types of forcing, the generalized Boussinesq model indicates that the surface pressure can be more effective in generating the precursor solitary waves than the submerged topography of the same normalized spatial distribution. However, according to the fKdV model, these two types of forcing are entirely equivalent. Besides these and some other rather refined differences, a broad agreement is found between theory and experiment, both in respect of the amplitudes and phases of the waves generated, when the speed is nearly critical (0.9 F > 0.2, finally disappear at F ~= 0.2. In the other direction, as the Froude number is increased beyond F ~= 1.2, the precursor soliton phenomenon was found also to evanesce as no finite-amplitude solitary waves can outrun, nor can any two-dimensional waves continue to follow, the rapidly moving disturbance. In this supercritical range and for asymptotically large times, all the effects remain only local to the disturbance. Thus, the criterion of the fascinating phenomenon of the generation of precursor solitons is ascertained

Mechanical balance laws for two-dimensional Boussinesq systems

Most of the asymptotically derived Boussinesq systems of water wave theory for long waves of small amplitude fail to satisfy exact mechanical conservation laws for mass, momentum and energy. It is thus only fair to consider approximate conservation laws that hold in the context of these systems. Although such approximate mass, momentum and energy conservation laws can be derived, the question of a rigorous mathematical justification still remains unanswered. The aim of this paper is to justify the formally derived mechanical balance laws for weakly nonlinear and weakly dispersive water wave Boussinesq systems. In particular, two asymptotic expansions used for the formal and rigorous derivation of the Boussinesq systems and the same are employed for the derivation and rigorous justification of the balance laws. Numerical validation of the asymptotic orders of approximation is also presented

On the Galilean invariance of some dispersive wave equations

Surface water waves in ideal fluids have been typically modeled by asymptotic approximations of the full Euler equations. Some of these simplified models lose relevant properties of the full water wave problem. One of them is the Galilean symmetry, which is not present in important models such as the BBM equation and the Peregrine (Classical Boussinesq) system. In this paper we propose a mechanism to modify the above mentioned classical models and derive new, Galilean invariant models. We present some properties of the new equations, with special emphasis on the computation and interaction of their solitary-wave solutions. The comparison with full Euler solutions shows the relevance of the preservation of Galilean invariance for the description of water waves.Comment: 29 pages, 13 figures, 2 tables, 71 references. Other author papers can be downloaded at http://www.denys-dutykh.com

Zonal flow regimes in rotating anelastic spherical shells: an application to giant planets

The surface zonal winds observed in the giant planets form a complex jet pattern with alternating prograde and retrograde direction. While the main equatorial band is prograde on the gas giants, both ice giants have a pronounced retrograde equatorial jet. We use three-dimensional numerical models of compressible convection in rotating spherical shells to explore the properties of zonal flows in different regimes where either rotation or buoyancy dominates the force balance. We conduct a systematic parameter study to quantify the dependence of zonal flows on the background density stratification and the driving of convection. We find that the direction of the equatorial zonal wind is controlled by the ratio of buoyancy and Coriolis force. The prograde equatorial band maintained by Reynolds stresses is found in the rotation-dominated regime. In cases where buoyancy dominates Coriolis force, the angular momentum per unit mass is homogenised and the equatorial band is retrograde, reminiscent to those observed in the ice giants. In this regime, the amplitude of the zonal jets depends on the background density contrast with strongly stratified models producing stronger jets than comparable weakly stratified cases. Furthermore, our results can help to explain the transition between solar-like and "anti-solar" differential rotations found in anelastic models of stellar convection zones. In the strongly stratified cases, we find that the leading order force balance can significantly vary with depth (rotation-dominated inside and buoyancy-dominated in a thin surface layer). This so-called "transitional regime" has a visible signature in the main equatorial jet which shows a pronounced dimple where flow amplitudes notably decay towards the equator. A similar dimple is observed on Jupiter, which suggests that convection in the planet interior could possibly operate in this regime.Comment: 20 pages, 15 figures, 4 tables, accepted for publication in Icaru

Boussinesq and Anelastic Approximations Revisited: Potential Energy Release during Thermobaric Instability

Expressions are derived for the potential energy of a fluid whose density depends on three variables: temperature, pressure, and salinity. The thermal expansion coefficient is a function of depth, and the application is to thermobaric convection in the oceans. Energy conservation, with conversion between kinetic and potential energies during adiabatic, inviscid motion, exists for the Boussinesq and anelastic approximations but not for all approximate systems of equations. In the Boussinesq/anelastic system, which is a linearization of the thermodynamic variables, the expressions for potential energy involve thermodynamic potentials for salinity and potential temperature. Thermobaric instability can occur with warm salty water either above or below cold freshwater. In both cases the fluid may be unstable to large perturbations even though it is stable to small perturbations. The energy per mass of this finite-amplitude instability varies as the square of the layer thickness. With a 4-K temperature difference and a 0.6-psu salinity difference across a layer that is 4000 m thick, the stored potential energy is 0.3 m^2 s^−2, which is comparable to the kinetic energy of the major ocean currents. This potential could be released as kinetic energy in a single large event. Thermobaric effects cause parcels moving adiabatically to follow different neutral trajectories. A cold fresh parcel that is less dense than a warm salty parcel near the surface may be more dense at depth. Examples are given in which two isopycnal trajectories cross at one place and differ in depth by 1000 m or more at another