= Energy balance of the incompressible vertical-shear model
{title2=$\partial_t(K+\rho_0\Phi_t)+\nabla\cdot[(K+P+\rho_0\Phi_t)\mathbf u]=-2\rho_0\Omega^2qzu_x$}
For constant density $\rho_0$ and body force $\Omega^2(3x-2qz)\mathbf e_x$, dotting the <incompressible flow> momentum equation with $\rho_0\mathbf u$ gives the <kinetic energy> flux $(K+P)\mathbf u$ and source $\rho_0\Omega^2(3x-2qz)u_x$, where $K=\rho_0|\mathbf u|^2/2$. <Coriolis acceleration> does no work. Absorbing $\Phi_t=-3\Omega^2x^2/2$ gives the displayed balance. For nonzero $q$, the remaining force has <curl> $-2\Omega^2q\mathbf e_y$, so it cannot be incorporated into a scalar mechanical potential. The model can exchange energy with its maintained vertical-shear background; at $q=0$ ordinary <Jacobi energy in a shearing sheet> conservation is recovered under zero boundary flux.
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