= Boundary pseudomomentum of a sloping Eady layer
{title2=$\mathcal P=\int\left[\dfrac{\psi_z'^2(D)}{2f_0A_D}-\dfrac{\psi_z'^2(0)}{2f_0A_0}\right]dx\,dy$}
In a constant-shear <Eady model> with slopes $\alpha$ and $\gamma$, set $A_0=\Lambda-N^2\alpha/f_0$ and $A_D=\Lambda-N^2\gamma/f_0$. The interior <potential vorticity> obeys $(\partial_t+\Lambda z\partial_x)q'=0$. In the zero-interior-PV subspace, integrating $\psi'_xq'$ by parts and using the two <buoyancy> boundary equations proves conservation of the displayed boundary <quadratic form>. Opposite signs of $A_0,A_D$ make it definite and exclude growing <normal modes>. For arbitrary nonzero interior PV, its derivative is $(N^2/f_0^3)\int\psi'_xq'\,dV$ and need not vanish. The restriction to zero interior PV is automatic for a growing smooth mode with complex <phase speed>.
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