= Solution
In a narrow western layer, $x$ derivatives dominate $y$ derivatives. Subtracting the forced interior balance leaves
$$
\frac{\beta}{f_0}\psi_{{\rm bl},x}
\simeq-\gamma\psi_{{\rm bl},xx}.
$$
If the layer has width $\delta$, then $\psi_x=O(\psi/\delta)$ and $\psi_{xx}=O(\psi/\delta^2)$. Balancing the beta effect with linear bottom drag gives
$$
\frac{\beta}{f_0\delta}
\sim\frac{\gamma}{\delta^2},
$$
and therefore the <Stommel boundary layer> width is
$$
\boxed{\delta=\frac{\gamma f_0}{\beta}}.
$$
This scaling assumes $\delta\ll L$ and that along-boundary variations occur on the basin scale.
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