= Solution
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> is
$$
\boxed{\beta\bar v_1=W,\qquad \beta\bar v_2=0.}
$$
For 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 $W=(\nabla_h\times\boldsymbol\tau)_z/(\rho_0h)$, so $\beta h(\bar v_1+\bar v_2)=(\nabla_h\times\boldsymbol\tau)_z/\rho_0$.
For $\beta>0$ and $W<0$, the upper-layer interior transport is \b[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:
$$
\beta v_n+\frac{D_n}{Dt}\nabla_h^2\psi_n-F\frac{D_n}{Dt}(\psi_n-\psi_m)=\begin{cases}W,&n=1,\\0,&n=2,\end{cases}\qquad m\ne n.
$$
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 interior>
$$
\boxed{\bar\psi_1=Vx,\qquad \bar\psi_2=0,\qquad V=\frac W\beta.}
$$
Unforced 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.
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