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For purely meridional wavevectors, $k=0$, $l\ne 0$, the growth rate of a uniform meridional upper-layer flow $V$ over a resting lower layer isGrowth requires $V\ne 0$ and $0<l^2<2F$. The perturbation has no northward velocity, so planetary vorticity advection does not explicitly stabilize it. At fixed normalized wind stress curl, $V=W/\beta$ means stronger $|\beta |$ nevertheless reduces the shear and growth rate.

Use the quasi-geostrophic streamfunction convention $u_n=-{\psi }_{n,y}$, $v_n={\psi }_{n,x}$, and write $F=k_I^2$, $f=f_0+\beta y$. In a steady, large-scale basin interior, neglect the material derivatives of relative vorticity and interfacial stretching compared with advection of planetary vorticity. With small Rossby number, weak nonlinear eddy terms, no significant interior friction or topographic forcing, and the specified forcing confined to layer 1, Sverdrup balance isFor equal depths $h$, the depth-integrated meridional transport is $h({\bar{v}}_1+{\bar{v}}_2)=hW/\beta$. Here $W$ is the normalized potential vorticity source appearing in the evolution equation. If the dimensional wind stress curl is used, its usual layer forcing is $W=({\nabla }_h\times \mathbf{\tau }{)}_z/({\rho }_{0}h)$, so $\beta h({\bar{v}}_1+{\bar{v}}_2)=({\nabla }_h\times \mathbf{\tau }{)}_z/{\rho }_0$.

For $\beta >0$ and $W<0$, the upper-layer interior transport is southward. In a closed subtropical basin, negative wind stress curl also corresponds to downwelling Ekman pumping for $f_0>0$ and an anticyclonic gyre. A northward return transport is needed to close the circulation; its narrow western boundary current requires processes outside the frictionless Sverdrup balance. The local interior equations by themselves do not specify the detailed boundary-current structure.

Expanding the layer equation shows the physical budget:The first term is the change of planetary vorticity as a fluid parcel moves north or south. The second is the change of relative vorticity. The last is vortex stretching in layered quasi-geostrophic flow: displacement of the interface changes layer thickness and thus the stretching contribution to potential vorticity. Its opposite signs in the two layer definitions express their thickness changes in opposite directions. The wind stress curl supplies or removes upper-layer potential vorticity. Each $D_n/Dt$ follows its own layer velocity, rather than a common velocity for both layers.

For uniform $W$, choose the local two-layer Sverdrup interiorUnforced background zonal currents and arbitrary additive interface offsets have been set to zero. This choice is also an exact uniform-flow solution of the stated forced equations, not just a leading balance: ${\bar{q}}_1=f_0+\beta y-FVx$, ${\bar{q}}_2=f_0+\beta y+FVx$, and ${\bar{v}}_1{\bar{q}}_{1,y}=V\beta =W$. The local interface slope can be nonzero even though its stretching contribution is constant along each basic-state trajectory. A streamfunction linear in $x$ is an interior-patch description, not a complete globally bounded basin solution.

For a uniform normalized upper-layer wind stress curl $W$ and no lower-layer forcing, a uniform local Sverdrup balance isThe upper layer flows meridionally and the lower layer rests. Although the relative vorticity vanishes, the potential vorticity has the zonal gradients ${\bar{q}}_{1,x}=-FV$, ${\bar{q}}_{2,x}=FV$. These are essential for the perturbation stability and do not vanish merely because the background velocity is uniform.