Semi-infinite Eady model

A uniformly stratified quasi-geostrophic approximation fluid with constant vertical shear occupies $z>0$, has a flat rigid lower wall, and has decaying perturbations aloft. On an f-plane, the basic interior potential vorticity is constant. Its smooth edge-wave modes have zero interior anomaly and propagate along the lower-boundary buoyancy gradient. The absence of an upper wall removes the pair of interacting edge waves that makes the finite-depth Eady model unstable.

A growing complex phase speed cannot equal the real basic velocity. The linear interior potential-vorticity equation then forces zero anomaly everywhere, decay fixes one exponential, and the wall fixes the real speed $c=\Lambda /m$. Thus there is no growing smooth decaying edge-wave eigenmode. This excludes exponential normal-mode instability without claiming that every initial disturbance lacks transient amplification or a singular neutral component.

A lower-boundary Eady edge wave in the semi-infinite Eady model has a smooth exponentially decaying vertical structure and a real frequency. It is supported by the horizontal boundary buoyancy gradient while its interior potential vorticity anomaly vanishes. Its speed equals the basic velocity at one penetration depth.

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.

A smooth zero-interior-potential vorticity mode hasDecay for $z\to \infty$ gives $\hat{\psi }=C{e}^{-mz}$. Its penetration depth $1/m$ increases with horizontal wavelength and decreases with buoyancy frequency. Singular neutral interior sheets are a distinct continuous-spectrum class.