= Solution
Write the vertical complex amplitude as $\Psi(z)=a(z)e^{i\phi(z)}$. A constant <streamfunction> phase has $kx-kc_rt+\phi(z)=\mathrm{const}$, hence $dx/dz=-\phi_z/k$. For $k>0$, westward tilt with height implies $\phi_z>0$. With $v'=\psi'_x$ and $b'=f_0\psi'_z$, a zonal average over the wave gives
$$
\boxed{\overline{v'b'}=\frac{f_0k}{2}a^2\phi_z\cos^2(ly)e^{2kc_it}>0}
$$
for the northern-hemisphere conventions used here. This is the <westward tilt and poleward eddy heat flux> relation. Warm anomalies move northward and cold anomalies southward, transporting heat down the mean gradient $\bar b_y=-f_0\Lambda<0$. The tilt describes the relative phases needed for a nonzero flux, not the literal trajectories of fluid particles along a tilted <streamfunction> crest.
The flux reduces the mean meridional <temperature> contrast and converts mean <available potential energy> into eddy <available potential energy>. In the perturbation buoyancy-variance budget its production term is $-\overline{v'b'}\bar b_y/N^2>0$. Correlated rising warm anomalies and sinking cold anomalies have positive $\overline{w'b'}$, converting that eddy potential <energy> into <kinetic energy>. The flow is therefore energized by the baroclinic reservoir associated with tilted <density>/potential-temperature surfaces. It is not simply a claim that the disturbance amplifies while leaving the basic state's thermal structure unchanged.
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