 Solution

Let ${\mathbf{U}}_h=\Gamma z{\mathbf{u}}_h(x,y)$ denote the actual horizontal velocity field. With constant nonzero Coriolis parameter $f_0$, geostrophic balance and hydrostatic balance, using pressure per reference density $\pi$, giveDifferentiate the first equation in $z$ to obtain thermal-wind balance:Taking the dot product with ${\mathbf{U}}_h$ provesThus the horizontal velocity is tangent to the horizontal buoyancy contours at every height. For a fluid whose thermal equation of state makes horizontal buoyancy contours coincide with horizontal temperature contours, there is no horizontal advection across those contours. In a dry terrestrial atmosphere, use potential temperature as the materially conserved thermal variable. A salinity contribution to buoyancy would require separate treatment before interpreting the result as a statement about temperature.

The usual statement about net meridional heat transport also follows explicitly. Introduce a horizontal streamfunction $\Psi$ such that ${\mathbf{u}}_h=(-{\Psi }_y,{\Psi }_x)$. Integration of thermal-wind balance givesand henceFor a zonally periodic function $\Psi$, or vanishing endpoint contributions, the zonal integral is zero. Likewise, the anomalous eddy buoyancy flux isThe required correlation between meridional velocity and thermal anomaly is absent. A nonzero vertical phase tilt, as in the next part, permits that correlation.

Qualification of the wording. The result does not say that the local heat flux vector $b{\mathbf{U}}_h$ vanishes, or that an arbitrary open-domain integral vanishes without boundary conditions. For a concrete counterexample to that literal reading, take ${\mathbf{u}}_h=(0,1)$ and $b=f_0\Gamma x+N^{2}z$. These satisfy the displayed thermal-wind balance, but $vb=\Gamma z(f_0\Gamma x+N^{2}z)$ is generally nonzero. What the hypotheses alone guarantee is zero transport across horizontal buoyancy contours; the vanishing zonally integrated heat transport uses the stated endpoint condition.