To describe the evolution of a (in this example non-relativistic) fluid system, the evolution equations for all relevant variables, conservation laws, the second law of thermodynamics, and appropriate (a)n appropriate equation(s) of state have to be considered.

The evolution equation for momentum is the Navier-Stokes equation which in a geophysical context can be written as

\[\frac{d u}{d t} + (u\cdot\nabla)u = -\frac{\nabla p}{\rho} -\nabla\Phi +\frac{1}{\rho}\nabla S\]

$\Phi$ is the geopotential and $S$ is the stress tensor. Conservation of angular momentum is taken into account by imposing the contraint that the stress tensor

\[S = \rho \nu \{ \nabla \circ u + (\nabla \circ u)^T\} + \rho \eta I (\nabla \cdot u)\]

where $\nu$ and $\eta$ denote the dynamic and kinematic viscosity respectively, is symmetric.

The evolution equation for internal energy (or equivalently temperature) can be written as

\[\frac{d e}{d t} = \frac{p}{\rho^2}\frac{d\rho}{d t} + Q_{rad} + Q_{lat} -\frac{1}{\rho}\nabla J + \frac{1}{\rho}(S\nabla)\cdot u\]

The second law of thermodynamics is considered by demanding that the last term in the above equation

\[\epsilon = \frac{1}{\rho} (\nabla S)\cdot u\]

which describes the frictional heating or dissipation is positive definite.

Conservation of mass is considered by including the continuity equation

\[\frac{d\rho}{d t} = \frac{\partial\rho}{\partial t} +\nabla(\rho u)\]

into the relevant system of equations.

As you can see, this is a coupled system of equations. The kinetic energy dissipated due to the friction term in the Navier-Stokes equation reappears as dissipative heating $\epsilon$ in the internal energy (or temperature) equation, so the (kinetic) energy can not disappear.

A nice example of a study which includes the temperature equation in addition to the Navier-Stokes equation, is the stuy of Sukorianski et al., who investigate stochastically forced turbulent flows with stable stratification by making use of renormalization-group like methods. By deriving a coupled system of RG equations that describes the scale-dependence of the anisotropic diffusivities of velocitie as well as of (potential) temperature fluctuations, they are by assuming the presence of a Kolmogorov scale invariant subrange able to repoduce the correct kinetic energy cascade and by slightly extending their work, it should be possible to derive the corresponding scale-dependence of the spectrum of temperature fluctuations (or available potential energy) too.

If this answer is not exactly what you wanted, I hope that it helps a bit at least.