# Variational formulation for models of shear shallow water flows and ideal turbulence

## TL;DR

This paper develops a unified variational framework for shear shallow water flows and ideal turbulence, modeling Reynolds stress evolution as non-holonomic constraints, revealing deep mathematical similarities between these fluid dynamics systems.

## Contribution

It introduces a novel variational formulation that treats shear effects and turbulence models within a common mathematical structure.

## Key findings

- Unified variational formulation for shear shallow water and turbulence models
- Reynolds stress evolution equations as non-holonomic constraints
- Mathematical analogy between shear flows and ideal turbulence

## Abstract

The shallow water equations without shear effects are similar to the gas dynamics equations with a polytropic equation of state. When the shear effects are taken into account, the equations contain additional evolution equations mathematically analogous to those of the Reynolds stresses in turbulent flows of compressible fluids when the source terms are neglected (ideal turbulence). We show that the non-dissipative model of shear shallow water flows and the model of ideal turbulence admit a similar variational formulation where, in the both cases, the equations for the Reynolds stress tensor evolution are considered as non-holonomic constraints.