Calculation of Integrals of the Hugoniot–Maslov Chain for Singular Vortical Solutions of the Shallow-Water Equation

Abstract

We discuss the problems of the Hugoniot-Maslov chain integrability for singular vortical solutions of the shallow-water equations on the β plane. We show that the complex variables used to derive the chain automatically give most of the integrals of the complete and the truncated chains. We also study how some of these integrals are related to the Lagrangian invariant (potential vorticity) .W e discuss how to choose solutions of the chain that can be used to describe the actual trajectories of tropical cyclones. Here, the scalar "phase" S(x, t), the vector (or scalar) "background" f (x, t), and the "amplitude" g(x, t) are smooth (infinitely differentiable) functions. The scalar function F (τ ) smoothly depends on τ for τ � and has a singularity at τ = 0. The singularities of the function w(x, t) are thus determined by the zeros of the phase S. Hereafter, the left superscript t denotes transposition. These solutions also include shock waves; in this case, F =Θ (τ ), where Θ(τ ) is the Heaviside function (Θ =0f or τ< 0a nd Θ=1f orτ ≥ 0). These solutions also include infinitely narrow solitons F =S ol(τ ), where Sol(τ )=0f orτ � 0 and Sol(τ )=1f orτ = 0. In the one-dimensional problems, the phase S has the form S = x− X(t) in both cases. Another example is given by weak singular solutions whose singularity has the type of the square root of a quadratic form. In this case, F = √ τ , and in the first approximation, S is a nondegenerate positive quadratic form with respect to x centered at the points x = X(t )t hat determine the trajectory Γ = � x = X(t) � of the singularity motion. Some properties of singular solutions of such systems of shallow-water equations on the β plane are studied in this paper. In this case, x = t (x1 ,x 2) ∈ R 2 , the vectors w, f ,a ndg have three components, and we have the formulas   η u1 u2