Harmonic maps and 2D Boussinesq equations

TL;DR

This paper introduces a novel method using harmonic mapping theory to derive explicit solutions for the 2D Boussinesq equations in Lagrangian coordinates, connecting classical solutions like Kirchhoff's vortex and Gerstner's wave.

Contribution

It develops a new approach employing harmonic maps, Schwarzian derivatives, and complex analysis to explicitly solve the 2D Boussinesq equations, unifying classical solutions.

Findings

Explicit solutions for 2D Boussinesq equations obtained.

Connection between harmonic mappings and classical fluid solutions established.

Method simplifies solving nonlinear differential systems in fluid dynamics.

Abstract

Within the framework of Lagrangian variables, we develop a method for deriving explicit solutions to the 2D Boussinesq equations using harmonic mapping theory. By reformulating the characterization of flow solutions described by harmonic functions, we reduce the problem to solving a particular nonlinear differential system in complex space. To solve this nonlinear differential system, we introduce the Schwarzian and pre-Schwarzian derivatives, and derive the properties of the sense-preserving harmonic mappings with equal Schwarzian and pre-Schwarzian derivatives. Our method yields explicit solutions in Lagrangian coordinates that contain two fundamental classes of classical solutions.: Kirchhoff's elliptical vortex (1876) and Gerstner's gravity wave (1809, rediscovered by Rankine in 1863).