Brüggemann, Nils, Eden, Carsten and Olbers, Dirk (2011) A Dynamically Consistent Closure for Zonally Averaged Ocean Models. Journal of Physical Oceanography, 41 (11). pp. 2242-2258. DOI 10.1175/JPO-D-11-021.1.

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Abstract

Simple idealized layered models and primitive equation models show that the meridional gradient of the zonally averaged pressure has no direct relation with the meridional flow. This demonstrates a contradiction in an often-used parameterization in zonally averaged models. The failure of this parameterization reflects the inconsistency between the model of Stommel and Arons and the box model of Stommel, as previously pointed out by Straub.
A new closure is proposed. The ocean is divided in two dynamically different regimes: a narrow western boundary layer and an interior ocean; zonally averaged quantities over these regions are considered. In the averaged equations three unknowns appear: the interior zonal pressure difference Delta p(i), the zonal pressure difference Delta p(b) of the boundary layer, and the zonal velocity us at the interface between the two regions. Here Delta p(i) is parameterized using a frictionless vorticity balance, Delta p(b), by the difference of the mean pressure in the interior and western boundary, and u(delta) by the mean zonal velocity of the western boundary layer.
Zonally resolved models, a layer model, and a primitive equation model validate the new parameterization by comparing with the respective zonally averaged counterparts. It turns out that the zonally averaged models reproduce well the buoyancy distribution and the meridional flow in the zonally resolved model versions with respect to the mean and time changes.

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