It is well known that substantial difference of the sea surface temperature between the western and the eastern part of the Pacific Ocean causes the longitudinal-oriented mean circulation called the Walker circulation. In spite of its theoretical importance and availability of the data, a simple theoretical framework of the Walker circulation is not known, which is intended to be constructed in the present paper. A motivation for the present work is simulations of a Walker--like circulation using a cloud-resolving model (CRM) by Wojciech Grabowski (NCAR). A two--dimensional configuration with a horizontal domain size of 4000km is used in the standard run for the numerical economy. Hence, a simple theoretical framework is required to compare these idealized simulations with the observed Walker circulation.
To the leading order, the thermodynamics of the mean state tropical atmosphere is described by the balance between the vertical advection and diabatic heating (radiative and convective heating). The horizontal advection is negligible to the leading order because the horizontal temperature gradient is substantially smaller than the vertical gradient. Hence, the vertical velocity is simply determined from the total diabatic heating from this balance, and the whole Walker circulation is determined from the continuity. (We neglect the meridional circulation for simplicity in this study.) Furthermore, because the subsidence region is overall convection free (in good approximation for the CRM case, and as a crude but an acceptable approximation for the observation), the mean subsidence (and its vertical structure) is totally defined by the mean radiative cooling of the area with the given mean stratification. The mean vertical motion of the ascending region is simply estimated from the continuity. Hence, the mean convective heating is determined from the radiative heating rate once the ratio of the ascending region to the total domain is specified.
A four box model is constructed based on this physical principle to further compute the moist entropy (or moisture) distribution of this system using a bulk mass--flux formulation similar to the one used by Riehl and Malkus. Here, the whole domain is divided into the descending and the ascending regions, and vertically a two-layer formulation is adopted, which leads to a four box description of the system. A reasonable agreement with both the CRM results and the observation is attained.
However, the most tricky part of this problem is the leading order thermodynamic balance aforementioned. It is in fact degenerated in the sense that both the vertical motion as well as the mean temperature vertical gradient are to be determined from this single relationship with a given radiative heating rate. A higher order perturbation problem is considered to resolve this degeneracy. Interestingly, it is shown that the perturbation temperature field (the part that defines a weak horizontal gradient) is vertically homogeneous, if the convective heating is totally defined at the leading order by radiation field. The mean circulation is shown to be determined from the surface thermodynamic conditions and the perturbation convective heating rate.
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12th Conference on Atmospheric and Oceanic Fluid Dynamics