Dispersion relation of a semi-infinite Eady edge wave ($c=\Lambda |f_0|/(NK)$)

For basic shear $U(z)=\Lambda z$ with a resting lower boundary, the rigid-boundary buoyancy condition for quasi-geostrophic waves and the decaying vertical exponential giveThis is real. The edge wave propagates in the same laboratory direction as the basic flow above its resting lower wall. If the basic buoyancy is $M^{2}y+N^{2}z$, then $\Lambda =-M^2/f_0$.

The critical level of the regular edge wave $c=\Lambda /m$, $U=\Lambda z$, is $z_c=1/m$ for $\Lambda \ne 0$. At that height $U=c$, so mean advection and the phase term oppose in the wave frame. A literal condition $U=-c$ instead gives the unphysical height $-1/m$ below the lower wall.