 Solution

Write the vertical complex amplitude as $\Psi (z)=a(z)e^{i\varphi (z)}$. A constant streamfunction phase has $kx-k{c}_{r}t+\varphi (z)=\mathrm{const}$, hence $dx/dz=-{\varphi }_z/k$. For $k>0$, westward tilt with height implies ${\varphi }_z>0$. With $v^{\prime }={\psi }_x^{\prime }$ and $b^{\prime }=f_0{\psi }_z^{\prime }$, a zonal average over the wave givesfor 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 $-\overline{v^{\prime }{b}^{\prime }}{\bar{b}}_y/N^2>0$. Correlated rising warm anomalies and sinking cold anomalies have positive $\overline{w^{\prime }{b}^{\prime }}$, 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.