A Method for Finding Exact Solutions to the 2D and 3D Euler-Boussinesq Equations in Lagrangian Coordinates

TL;DR

This paper introduces a general method for deriving exact solutions to the 2D and 3D Euler-Boussinesq equations in Lagrangian coordinates, expanding the set of known explicit fluid flow solutions.

Contribution

It extends previous methods for homogeneous Euler equations to include the Euler-Boussinesq system, enabling the construction of a wide range of exact solutions.

Findings

Derived conditions for Lagrangian fluid maps in Boussinesq approximation

Presented a general method to find exact solutions in 2D and 3D

Showcased a Gerstner type solution for 2D Euler-Boussinesq equations

Abstract

We study the Boussinesq approximation for the incompressible Euler equations using Lagrangian description. The conditions for the Lagrangian fluid map are derived in this setting, and a general method is presented to find exact fluid flows in both the two-dimensional and the three-dimensional case. There is a vast amount of solutions obtainable with this method and we can only showcase a handful of interesting examples here, including a Gerstner type solution to the two-dimensional Euler-Boussinesq equations. In two earlier papers we used the same method to find exact Lagrangian solutions to the homogeneous Euler equations, and this paper serves as an example of how these same ideas can be extended to provide solutions also to related, more involved models.