Conditional energy balance for the linear stratified Boussinesq model

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ for the Lebesgue measure and energy-integration convention. Let $I\subseteq \mathbb{R}$ be a nonempty open time interval, let ${\mathbb{T}}^3={\mathbb{R}}^3\mathrm{/}(2\pi \mathbb{Z}{)}^3$ have fundamental cell $Q_3=[0,2\pi {]}^3$ with Lebesgue measure ${\lambda }_3$, where $dx=d{\lambda }_3(x)$ and time-space integration means the iterated Lebesgue integral with the spatial integral first, and represent fields by $2\pi$-periodic lifts. Let $u:I\times {\mathbb{T}}^3\to {\mathbb{R}}^3$ be a velocity field and $b:I\times {\mathbb{T}}^3\to \mathbb{R}$ a scalar buoyancy-acceleration field, each with continuous first time and spatial derivatives and continuous second spatial derivatives; let $\pi :I\times {\mathbb{T}}^3\to \mathbb{R}$ be pressure per reference density with continuous first spatial derivatives. Let $f\in \mathbb{R}$, $N>0$, and $\nu ,\kappa \ge 0$ be constants, with SI units s${}^{-1}$, s${}^{-1}$, m${}^2$ s${}^{-1}$, and m${}^2$ s${}^{-1}$, respectively. Assume these fields satisfy the linear rotating constant-stratification Boussinesq equations adopted in the geophysical-model postulate,

$u_t+f{e}_3\times u+\mathrm{\nabla }\pi =b{e}_3+\nu \mathrm{\Delta }u,\mathrm{\nabla }\cdot u=0,b_t+N^2{u}_3=\kappa \mathrm{\Delta }b,$

where $e_3=(0,0,1)$, $e_3\times u=(-u_2,u_1,0)$, $u_3$ is the vertical velocity component, $\pi$ has units m${}^2$ s${}^{-2}$, $\mathrm{\nabla }q=({\mathrm{\partial }}_{x_1}q,{\mathrm{\partial }}_{x_2}q,{\mathrm{\partial }}_{x_3}q)$, $\mathrm{\nabla }\cdot w={\sum }_{j=1}^3{\mathrm{\partial }}_{x_j}{w}_j$, and $\mathrm{\Delta }q={\sum }_{j=1}^3{\mathrm{\partial }}_{x_j}^{2}q$, with $\mathrm{\Delta }$ applied componentwise to vector fields. Define the normalized energy density per unit constant reference density and its spatial integral by

$e(t,x):=\frac{1}{2}(\mathrm{\mid }u(t,x){\mathrm{\mid }}^2+\frac{b(t,x{)}^2}{N^2}),E(t):={\int }_{Q_3}e(t,x)dx,$

where $\mathrm{\mid }u{\mathrm{\mid }}^2={\sum }_{i=1}^3{u}_i^2$, $\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2={\sum }_{i,j=1}^3({\mathrm{\partial }}_{x_j}{u}_i{)}^2$, and $\mathrm{\mid }\mathrm{\nabla }b{\mathrm{\mid }}^2={\sum }_{j=1}^3({\mathrm{\partial }}_{x_j}b{)}^2$. Then, for every $t_0<t_1$ in $I$,

$E(t_1)-E(t_0)=-{\int }_{t_0}^{t_1}{\int }_{Q_3}(\nu \mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2+\frac{\kappa }{N^2}\mathrm{\mid }\mathrm{\nabla }b{\mathrm{\mid }}^2)dxdt\mathrm{.}$

Thus $E$ is conserved when $\nu =\kappa =0$ and is nonincreasing when $\nu ,\kappa \ge 0$. In SI units, $e$ has units m${}^2$ s${}^{-2}$ and $E$ has units m${}^5$ s${}^{-2}$; multiplication by the constant reference density in kg m${}^{-3}$ converts $E$ to joules. This is a conditional deduction from the adopted model and the stated smooth periodic hypotheses. It is not an approximation-error estimate, a nonlinear or compressible energy theorem, or evidence that the model describes an observed fluid.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): Countable Choice, used in the mathematical energy supplier's Lebesgue integration convention.

Geophysical approximation branches: adopted linear constant-stratification rotating Boussinesq branch and its physical scope.

Positive linear stratified energy and diffusion: exact mathematical energy identity for the same periodic pointwise equations and regularity hypotheses.

Geoffrey K. Vallis, Fluid Dynamics of the Atmosphere and Ocean, University of Exeter ECMM719 full lecture notes (Autumn 2018), §1.6.3, equations (1.74)–(1.78), printed pp. 15–17

Pavel Berloff, Lectures on Introduction to Geophysical Fluid Dynamics, Imperial College London full lecture notes, Boussinesq approximation and energy context, printed pp. 36–38