Conservation Laws for One-Dimensional Shallow Water Models for One and Two-Layer Flows

Abstract

A full set of conservation laws for the two-layer shallow water equations is presented for the one-dimensional case. We prove that all the conservation laws are linear combination of the equations for the conservation of mass and velocity (in each layer), total momentum and total energy.This result generalizes that of Montgomery and Moodie that found the same conserved quantities by restricting their search to the multinomials expressions in the layer variables. Though the question of whether or not there are only a finite number of these quantities is left as an open question by the authors. Our work puts an end to this: in fact, no more conservation laws are admitted for the two-layer shallow water equations. The key mathematical ingredient of the method proposed leading to the result is the Frobenius problem. Moreover, we present a full set of conservation laws for the classical one-dimensional shallow water model with topography, by using the same techniques.

Multi-dimensional shear shallow water flows: Problems and solutions

“…Since the number of scalar unknowns is six (h, u and three independent components of P), the system is not in conservative form. The definition and computation of discontinuous solutions for non-conservative hyperbolic equations is a challenging problem (see examples of non-conservative systems in compressible turbulence [7,2], multi-layer shallow water flows [34,4,32,5,1,30], multi-phase fluid flows [3,24,40,41,42,19,14], solid-fluid systems [17,33]).…”

Hamiltonian Aspects of Three-Layer Stratified Fluids

“…Notable for our purposes are the results of Tsarev (1985Tsarev ( , 1991, showing that six quantities are guaranteed to be conserved by strong solutions: the Hamiltonian functional (3.10), +∞ −∞ η j dx, +∞ −∞ μ j dx, j = 1, 2 (these are Casimir functionals of the Poisson brackets), and the total horizontal momentum (x) = +∞ −∞ (η 1 μ 1 + η 2 μ 2 ) dx, this being the generator of the x-translational symmetry in the free-surface case. Furthermore, in Montgomery and Moodie (2001) and Barros (2006), the authors show that these conserved quantities are the only ones whose densities do not explicitly depend on x. Also notable in this regard are the results and conjectures in Ferapontov (1994), about the complete integrability of quasi-linear systems which are linearly degenerate (termed weakly nonlinear therein) but lack a complete set of Riemann invariants.…”

“…The free-surface case is then briefly addressed, in particular, for the sake of concreteness, when n = 2. For a comprehensive approach, see, e.g., Choi (2000), and Gavrilyuk et al (1998); Barros (2006) for a discussion of a variational approach. Extension of these results, considering dispersive terms (as in (2.6)), can be found in Barros et al (2007), Percival et al (2008).…”

Hamiltonian aspects of 3-layer stratified fluids

“…The free-surface case is then briefly addressed, in particular, for the sake of concreteness, when n = 2. For a comprehensive approach see, e.g., [15], and [29,1] for a discussion of a variational approach. Extension of these results, considering dispersive terms (as in (2.6) below), can be found in [3,35].…”