= 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> $\pi$, give
$$
f_0\mathbf e_z\mathbin\times\mathbf U_h=-\nabla_h\pi,
\qquad \pi_z=b.
$$
Differentiate the first equation in $z$ to obtain <thermal-wind balance>:
$$
f_0\Gamma\mathbf e_z\mathbin\times\mathbf u_h=-\nabla_h b.
$$
Taking the <dot product> with $\mathbf U_h$ proves
$$
\boxed{\mathbf U_h\cdot\nabla_h b=0.}
$$
Thus 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> gives
$$
b=f_0\Gamma\Psi+b_0(z),\qquad
v=\Gamma z\Psi_x,
$$
and hence
$$
vb=\partial_x\left[\Gamma z\left(\frac{f_0\Gamma}{2}\Psi^2+b_0\Psi\right)\right].
$$
For a zonally <periodic function> $\Psi$, or vanishing endpoint contributions, the zonal integral is zero. Likewise, the anomalous <eddy buoyancy flux> is
$$
\boxed{\overline{v'b'}=f_0\Gamma^2z\,\overline{\Psi'_x\Psi'}=0.}
$$
The required correlation between meridional <velocity> and thermal anomaly is absent. A nonzero vertical <phase> tilt, as in the next part, permits that correlation.
\b[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 $b=f_0\Gamma x+N^2z$. These satisfy the displayed <thermal-wind balance>, but $vb=\Gamma z(f_0\Gamma x+N^2z)$ 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.
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