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A barotropic mode has horizontal velocity independent of depth. It is the zeroth quasi-geostrophic vertical mode and usually has the largest gravity-wave speed and Rossby deformation radius.

The potential-vorticity gradients in a meridional two-layer current are important here: besides ${\bar{q}}_{n,y}=\beta$, one has ${\bar{q}}_{1,x}=-FV$ and ${\bar{q}}_{2,x}=FV$. They must be retained when linearizing, even though the basic relative vorticity vanishes.

Let $K^2=k^2+l^2$ and $A=K^2+F$. For a normal mode the two-layer quasi-geostrophic potential vorticity amplitudes areBecause the basic velocities are northward $(V,0)$ in the two layers, their advection frequencies are $\sigma -lV$ and $\sigma$. The uniform forcing has no perturbation, $W^{\prime }=0$. The linear equations are thereforeIn particular, a perturbation's zonal velocity advects the basic interface-induced zonal potential-vorticity gradient. Substitution of the plane wave givesA nonzero disturbance exists exactly when the determinant vanishes. The requested relation for linear stability of a meridional two-layer current isEquivalently, with $\mathcal{B}=K^2(K^2+2F)$,Its discriminant isFor $K>0$, exponential baroclinic instability occurs when this discriminant is negative; otherwise the two frequencies are real. This also displays the stabilizing contribution of the planetary vorticity gradient when $k\ne 0$. As a check, $V=0$ gives the uncoupled barotropic mode and baroclinic mode frequencies $-\beta k/K^2$ and $-\beta k/(K^2+2F)$.

The rigid lower boundary has $w=0$. For a time-dependent normal mode this givesThe free-surface condition from part i givesWriting $q=N/c$, the lower condition selectsThe upper condition therefore requiresEquivalently, with $\mu =NH/c$ and $ϵ=N^{2}H/g$,Each positive root ${\mu }_n$ determines the barotropic mode or a baroclinic mode through $c_n=NH/{\mu }_n$.

Both descriptions are depth-averaged hydrostatic shallow-layer models. In a one-layer free-surface shallow water equations, a homogeneous layer overlies a rigid bottom and its upper surface moves. Surface elevation changes give a restoring acceleration based on full gravity $g$, with external-wave speed $\sqrt{g{H}_0}$. Barotropic coastal storm surges, long surface gravity waves or a basin seiche are examples.

In the rigid-lid reduced gravity model, the movable boundary is a density interface beneath the active upper layer, while the much deeper lower layer remains at rest. The internal interface mode instead has speed $\sqrt{g^{\prime }{h}_0}$ and represents, for example, thermocline displacements and an upper-ocean baroclinic eddy. The mathematical shallow-water form is retained with $g$ replaced by $g^{\prime }$ and with the appropriate interface-depth convention. A freely moving two-layer surface would support both external and internal modes; the rigid lid filters out the fast external mode, and the resting abyss removes its independent dynamics.

For equal-depth layers coupled by an interface,The Laplacian terms are relative vorticity; the streamfunction differences encode opposite layer-thickness changes. Each material derivative follows that layer's own velocity. The sum has a barotropic mode without interface displacement, while the difference contains a baroclinic mode.