Wave front propagation with non-trivial bottom topography is studied within the formalism of hyperbolic long wave models. Evolution of non-smooth initial data is examined, and in particular the splitting of singular points and their short time behaviour is described. In the opposite limit of longer times, the local analysis of wavefronts is used to estimate the gradient catastrophe formation and how this is influenced by the topography. The limiting cases when the free surface intersects the bottom boundary, belonging to the so-called "physical" and "non-physical" vacuum classes, are examined. Solutions expressed by power series in the spatial variable lead to a hierarchy of ordinary differential equations for the time-dependent series coefficients, which are shown to reveal basic differences between the two vacuum cases: for non-physical vacuums, the equations of the hierarchy are recursive and linear past the first two pairs, while for physical vacuums the hierarchy is non-recursive, fully coupled and nonlinear. The former case may admit solutions that are free of singularities for nonzero time intervals, while the latter is shown to develop non-standard velocity shocks instantaneously. We show that truncation to finite dimensional systems and polynomial solutions is in general only possible for the case of a quadratic bottom profile. In this case the system's evolution can reduce to, and is completely described by, a low dimensional dynamical system for the time-dependent coefficients. This system encapsulates all the nonlinear properties of the solution for general power series initial data, and in particular governs the loss of regularity in finite times at the dry point. For the special case of parabolic bottom topographies, an exact, self-similar solution class is introduced and studied to illustrate via closed form expressions the general results.
The two-dimensional Green-Naghdi equations with uneven bottom topography are studied in this paper. The function defining the bottom topography can be dependent on time. Group classification of these equations with respect to the function describing the topography of the bottom is performed in the paper. The algebraic approach used for the analysis of the classifying equations.
We carry out the group classification of the class of two-dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The equivalence group of this class is found by the algebraic method. Using algebraic techniques, we construct additional point equivalences between some of the listed cases of Lie-symmetry extensions, which are inequivalent up to transformations from the equivalence group.
In this paper, we look for an operator that describes the relationship between small errors in representation of the bottom topography in a barotropic ocean model and the model's solution. The study shows that the model's solution is very sensitive to topography perturbations in regions where the flow is turbulent. On the other hand, the flow exhibits low sensitivity in laminar regions. The quantitative measure of sensitivity is influenced essentially by the error growing time. At short time scales, the sensitivity exhibits the polynomial dependence on the error growing time. And in the long t…
The Riemann problem of one dimensional shallow water equations with discontinuous topography has been constructed recently. The elementary waves include shock waves, rarefaction waves, and the stationary wave. The stationary wave appears when the water depth changes, especially when there exists a bottom step. In this paper, we are mainly concerned with the interaction between a stationary wave with either a shock wave or a rarefaction wave. By using the characteristic analysis methods, the evolution of waves is described during the interaction process. The solution in large time scale is also…
We classify zeroth-order conservation laws of systems from the class of two-dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The classification is carried out up to equivalence generated by the equivalence group of this class. We find additional point equivalences between some of the listed cases of extensions of the space of zeroth-order conservation laws, which are inequivalent up to transformations from the equivalence group. Hamiltonian structures of systems of shallow water equations are used for relating th…
The two-dimensional shallow water equations with a particular bottom and the Coriolis's force $f=f_{0}+Ωy$ are studied in this paper. The main goal of the paper is to describe all invariant solutions for which the reduced system is a system of ordinary differential equations. For solving the systems of ordinary differential equations we use the sixth-order Runge-Kutta method.
A basic shallow water system with variable topography is analyzed from the point of view of a Lagrangian derivation of momentum, energy, and pseudomomentum balances. A two-dimensional action and associated momentum equation are derived. The latter is further manipulated to derive additional equations for energy and pseudomomentum. This revealed structure emphasizes broken symmetries in space and a reference configuration, and preserved symmetry in time.
The Euler-Poincaré differential (EPDiff) equations and the shallow water (SW) equations share similar wave characteristics. Using the Hamiltonian structure of the SW equations with flat bottom topography, we establish a connection between the EPDiff equations and the SW equations in one and multi-dimensions. Additionally, we show that the EPDiff equations can be recast in a curl formulation.
We investigate the influence of the topography on the lake equations which describe the two-dimensional horizontal velocity of a three-dimensional incompressible flow. We show that the lake equations are structurally stable under Hausdorff approximations of the fluid domain and $L^p$ perturbations of the depth. As a byproduct, we obtain the existence of a weak solution to the lake equations in the case of singular domains and rough bottoms. Our result thus extends earlier works by Bresch and Métivier treating the lake equations with a fixed topography and by Gérard-Varet and Lacave treating th…
A Hamiltonian model for the propagation of internal water waves interacting with surface waves, a current and an uneven bottom is examined. Using the so-called Dirichlet-Neumann operators, the water wave system is expressed in the Hamiltonian form, and thus the motions of the internal waves and surface waves are determined by the Hamiltonian formulation. Choosing an appropriate scaling of the variables and employing the Hamiltonian perturbation theory from Hamiltonian formulation of the dynamics, we derive a KdV-type equation with variable coefficients depending on the bottom topography to des…
In this paper, we study the motion of the free surface of a body of fluid over a variable bottom, in a long wave asymptotic regime. We assume that the bottom of the fluid region can be described by a stationary random process $β(x, ω)$ whose variations take place on short length scales and which are decorrelated on the length scale of the long waves. This is a question of homogenization theory in the scaling regime for the Boussinesq and KdV equations. The analysis is performed from the point of view of perturbation theory for Hamiltonian PDEs with a small parameter, in the context of which we…
The one-dimensional shallow water equations in Eulerian and Lagrangian coordinates are considered. It is shown the relationship between symmetries and conservation laws in Lagrangian (potential) coordinates and symmetries and conservation laws in mass Lagrangian variables. For equations in Lagrangian coordinates with a flat bottom an invariant difference scheme is constructed which possesses all the difference analogues of the conservation laws: mass, momentum, energy, the law of center of mass motion. Some exact invariant solutions are constructed for the invariant scheme, while the scheme ad…
We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply-connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free su…
Holm (Proc. Roy. Soc 2015) introduced a variational framework for stochastically parametrising unresolved scales of hydrodynamic motion. This variational framework preserves fundamental features of fluid dynamics, such as Kelvin's circulation theorem, while also allowing for dispersive nonlinear wave propagation, both within a stratified fluid and at its free surface. The present paper combines asymptotic expansions and vertical averaging with the stochastic variational framework to formulate a new approach for developing stochastic parametrisation schemes for nonlinear wave fields. The approa…
