Exact linear-drag cosine basin circulation ($r{\Psi }^{\prime \prime }+\beta {\Psi }^{\prime }-r{m}^2\Psi =F_0$)

For streamfunction $\psi =\Psi (x)\cos (my)$ with impermeable walls at $x=0,L$, let $A=-F_0/(r{m}^2)$ and positive exponents satisfy $\lambda -\gamma =\beta /r$, $\lambda \gamma =m^2$. The exact profile is $\Psi =A[1-C_W{e}^{-\lambda x}-C_E{e}^{\gamma (x-L)}]$, where $C_W=(1-e^{-\gamma L})/(1-e^{-(\lambda +\gamma )L})$ and $C_E=(1-e^{-\lambda L})/(1-e^{-(\lambda +\gamma )L})$. Substitution proves the ordinary differential equation and both boundary conditions. The usual narrow western boundary current limit requires $\lambda L\gg 1$ and $\gamma L\ll 1$, with $\lambda \simeq \beta /r$ and $\gamma \simeq r{m}^2/\beta$; the opposite first inequality does not describe weak drag.