The author thanks anonymous reviewers for helpful comments. This study was partially supported by Grant-in-Aid for Scientific Research from MEXT, Japan, and JSPS (21540447, 22106006, and 23340139). The GFD DENNOU library was used for drawing the figures.
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The longitudinal oscillation of the boundary, which is defined by q2 = Q2W, at the outcrop latitude would cause ventilation of fluid with slightly different potential thickness into the pool zone, because the second-layer depth at (λW, θ1) also oscillates. However, because the second-layer potential thickness anomalies are only advected by the mean flow in the pool zone and this is a linear theory, we do not need to consider this small anomaly confined to a very narrow band along the boundary.