On a new type of solving procedure for Laplace tidal equation

TL;DR

This paper introduces a novel method for solving Laplace tidal equations by reducing the problem to a system of nonlinear ODEs, providing analytical insights and demonstrating limitations in cases with nonzero pressure and potentials.

Contribution

A new ansatz for solving LTE is proposed, reducing the momentum equation to a system of nonlinear ODEs and exploring the analytical solvability of the system.

Findings

Reduced LTE to 3 nonlinear ODEs for velocity components

Established the non-existence of analytical solutions with nonzero pressure and potentials

Obtained a proper partial solution using invariant dependence

Abstract

In this paper, we present a new approach for solving Laplace tidal equations (LTE) which was formulated first in [S.V.Ershkov, A Riccati-type solution of Euler-Poisson equations of rigid body rotation over the fixed point, Acta Mechanica, 228(7), 2719 (2017)] for solving Poisson equations: a new type of the solving procedure for Euler-Poisson equations (rigid body rotation over the fixed point) is implemented here for solving momentum equation of LTE, Laplace tidal equations. Meanwhile, the system of Laplace tidal equations (including continuity equation) has been successfully explored with respect to the existence of analytical way for presentation of the solution. As the main result, the new ansatz is suggested here for solving LTE: solving momentum equation is reduced to solving system of 3 nonlinear ordinary differential equations of 1-st order in regard to 3 components of the flow velocity (depending on time t), along with the continuity equation which determines the spatial part of solution. Nevertheless, the proper elegant partial solution has been obtained due to invariant dependence between temporary components of the solution. In addition to this, it is proved here that the system of Laplace tidal equations has not the analytical presentation of solution (in quadratures) in case of nonzero fluid pressure in the Ocean, as well as nonzero total gravitational potential and the centrifugal potential (due to planetary rotation).