Eady model

The Eady model describes quasi-geostrophic baroclinic instability in a uniformly stratified layer with constant vertical shear, zero interior potential-vorticity gradient, and two rigid horizontal boundaries.

When both rigid boundaries of a constant-shear Eady model have the same small slope $\alpha$, define $a=N^2\alpha /(f_0\Lambda )-1$, $M=N|k|D/|f_0|$ and $C=c/(\Lambda D)$. Zero-interior-PV modes have vertical structure $A\cosh (Mz/D)+B\sinh (Mz/D)$. The boundary conditions $(z/D-C){\hat{\psi }}_{z/D}+a\hat{\psi }=0$ at both endpoints give the displayed dispersion relation. For $a\ge 0$ its square root is real and there is no exponential baroclinic instability. The same-sign boundary-pseudomomentum criterion is necessary, not sufficient.

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 $({\partial }_t+\Lambda z{\partial }_x)q^{\prime }=0$. In the zero-interior-PV subspace, integrating ${\psi }_x^{\prime }{q}^{\prime }$ 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 $(N^2/f_0^3)\int {\psi }_x^{\prime }{q}^{\prime }dV$ and need not vanish. The restriction to zero interior PV is automatic for a growing smooth mode with complex phase speed.

For constant shear $U=\Lambda z$ between impermeable horizontal walls at $0,D$, with constant buoyancy frequency and no interior potential-vorticity gradient, define $m=NK/|f_0|$. The boundary conditions yieldThe bracket is negative for $0<mD<2.399\dots$, giving baroclinic instability for nonzero zonal wavenumber. For large separation the boundary waves decouple; the lower-wave speed tends to $\Lambda /m$, matching the dispersion relation of a semi-infinite Eady edge wave.

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.

A gently sloping lower boundary modifies the lower Boundary Rossby wave of the Eady model, while a horizontal upper boundary retains the usual upper wave. With $M=N|k|D/|f_0|$, $a=\alpha N^2/(f_0\Lambda )$ and $C=c/(\Lambda D)$, the two-boundary dispersion relation isFor $a>1$ both speeds are real for every nonzero wavenumber. For every $a<1$ there is a band of exponentially growing modes; this includes negative slopes. The case $a=1$ has no exponential instability but can have a defective neutral mode at the crossing of the two uncoupled speeds.

The discriminant of the Eady model with a sloping lower boundary isFor any $a<1$, choose the unique $M>0$ satisfying $M\tanh M=2-a$. Then $\Delta <0$, proving existence of an unstable band even when $a<0$. For large negative $a$, resonance occurs at large $M$ and the exponentially weak boundary-wave coupling makes the unstable band narrow.

Eady instability arises when counterpropagating Rossby waves localized on the upper and lower boundaries phase-lock and amplify one another.

A boundary Rossby wave is supported by a buoyancy perturbation or potential-temperature anomaly on a rigid horizontal boundary. Its intrinsic propagation depends on the boundary orientation and basic buoyancy gradient; the upper and lower boundary waves of the Eady model propagate oppositely relative to their local background flows.

For a semi-infinite fluid above a rigid horizontal boundary, constant buoyancy frequency $N$, and thermal wind $U=\Lambda z$, a zero-interior-potential vorticity perturbation has vertical dependence $e^{-\mu z}$, where $\mu =NK/|f_0|$ and $K=(k^2+l^2{)}^{1/2}$. Material conservation of boundary buoyancy gives $c{\varphi }^{\prime }(0)+\Lambda \varphi (0)=0$. Thus its phase velocity relative to the boundary flow is $c=\Lambda /\mu$. The wave's time dependence resides in its boundary condition; the interior field follows by potential-vorticity inversion. MIT's Eady-edge-wave notes develop this boundary-wave interpretation.

With uniform thermal wind $U=\Lambda z$ above a plane boundary, write the zero-interior-potential vorticity response as ${\psi }^{\prime }=\mathrm{Re}\{A{e}^{-\mu z+i(kx+ly)}\}$, where $\mu =N\sqrt{k^2+l^2}/f$ and $f>0$. Boundary vertical velocity $ϵ\cos (kx+ly-{\omega }_{0}t)$ gives $C=N^2ϵ/(f\mu )$ and natural angular frequency $\omega =\Lambda k/\mu$. An initially zero response has $A=C(e^{-i{\omega }_{0}t}-e^{-i\omega t})/[i(\omega -{\omega }_0)]$. Its continuous resonant limit is $Ct{e}^{-i\omega t}$: forcing in phase with the boundary mode supplies energy every cycle, and linear undamped theory predicts secular growth.

For a semi-infinite stratified fluid with thermal wind $U=\Lambda z$ above a gently sloping boundary $z=\alpha y$, a zero-interior-potential vorticity perturbation decays as $e^{-\mu z}$, where $\mu =N|k|/|f_0|$. The boundary buoyancy perturbation obeys a single oscillatory equation with phase velocity $c=(\Lambda -\alpha N^2/f_0)/\mu$. The topographic and thermal-wind contributions cancel when the boundary follows a background isopycnal.

Two vertically unbounded regions with buoyancy frequencies $N_1,N_2$ and thermal wind shears ${\Lambda }_1,{\Lambda }_2$ support a localized wave if pressure and normal velocity are continuous at their common material interface. With decays ${\varphi }_1=A{e}^{{\mu }_{1}z}$ below and ${\varphi }_2=A{e}^{-{\mu }_{2}z}$ above, ${\mu }_j=N_{j}K/|f_0|$, matching givesWhen $N_1/N_2\to \infty$ with bounded shear, the lower layer becomes effectively rigid and the upper-layer Eady edge wave speed ${\Lambda }_2/{\mu }_2$ is recovered.

Adding linear damping to one boundary of the Eady model makes the two boundary-wave phase speeds complex and can destabilize modes that were neutral without damping.