Eady-wave phase tilt and heat flux ($\overline{v^{\prime }{b}^{\prime }}=f_0{A}^{2}kK\alpha /2$)

For ${\psi }^{\prime }=A[\cosh (Kz)\cos (kx)-\alpha \sinh (Kz)\sin (kx)]$ with $k,K,\alpha >0$, its phase increases with height and its crests tilt westward. The derivative $b^{\prime }=f_0{\psi }_z^{\prime }$ has decreasing continuous phase and eastward-tilting crests. Multiplying $v^{\prime }={\psi }_x^{\prime }$ by $b^{\prime }$ and averaging over a zonal wavelength gives the displayed eddy buoyancy flux, using ${\cosh }^2(Kz)-{\sinh }^2(Kz)=1$. Its sign is poleward for the associated eastward-increasing thermal wind in either hemisphere. An untilted separable perturbation has zero zonally averaged flux under periodic endpoint conditions.