Energetics of a fluid under the Boussinesq approximation

TL;DR

This paper develops a comprehensive energy budget theory for fluids under the Boussinesq approximation, clarifying energy conversions and conservation laws, including extensions for oceanographic applications involving salinity effects.

Contribution

It introduces a consistent energy budget framework for the Boussinesq approximation and extends it to include salinity effects in oceanography, clarifying energy conversions and mass conservation constraints.

Findings

No potential energy is available under the Boussinesq approximation.

Buoyancy work due to temperature changes converts kinetic and internal energy.

Salinity-related buoyancy work converts kinetic and potential energy under the extended approximation.

Abstract

This paper presents a theory describing the energy budget of a fluid under the Boussinesq approximation: the theory is developed in a manner consistent with the conservation law of mass. It shows that no potential energy is available under the Boussinesq approximation, and also reveals that the work done by the buoyancy force due to changes in temperature corresponds to the conversion between kinetic and internal energy. This energy conversion, however, makes only an ignorable contribution to the distribution of temperature under the approximation. The Boussinesq approximation is, in physical oceanography, extended so that the motion of seawater can be studied. This paper considers this extended approximation as well. Under the extended approximation, the work done by the buoyancy force due to changes in salinity corresponds to the conversion between kinetic and potential energy. It also turns out that the conservation law of mass does not allow the condition $\nabla\cdot\boldsymbol{u}=0$ on the fluid velocity $\boldsymbol{u}$ to be imposed under the extended approximation; the condition to be imposed instead is presented.