## Abstract

The problem of the steady symmetric motion of a Boussinesq fluid is considered for a system with small aspect ratio. It is assumed that the motion is driven by applying a periodic heat flux to the horizontal boundaries. Solutions are first found for a non-rotating system in which nonlinear effects are small, but not zero. The solutions show that if the fluid is heated from above, the meridional circulation tends to be concentrated near the upper boundary at the point where the cooling is a maximum; when the fluid is heated from below the meridional circulation tends to be concentrated near the lower boundary at the paint where the heating is a maximum.

Then, it is shown for a non-rotating system that when nonlinear effects are dominant, vertical boundary layers must form. These vertical boundary layers form at points where the horizontal velocity is zero, and are characterized by small horizontal velocities and temperature gradients, but large vertical velocities and horizontal diffusion. By means of scaling analysis, the scales and magnitudes of the variables are determined for both the internal boundary layers and the boundary layers along the horizontal boundaries, when nonlinear effects are dominant.

Next, the effect of rotation is considered, and it shown that exactly the same sorts of vertical boundary layers will form in a rotating system. Scaling analysis is again used to show that in this case the horizontal boundary layers near the internal boundary layers are of the same kind as in the non-rotating case, but far enough away from the internal boundary layers they merge into a nonlinear Ekman layer.

Finally, some possible geophysical applications are considered. The model of the atmospheric circulations on Venus proposed by Goody and Robinson is found to agree qualitatively with the results presented here, but the quantitative results for the internal boundary layer, or mixing region, are found to differ considerably. Also, estimates are made for the internal boundary layer which would accompany a Hadley cell similar to that found in the earth's tropical region. It is found that the rising motions will occur over a region about 200 km in width. This result suggests that the nonlinear process which produces these internal boundary layers may be one of the important processes in determining the structure of the Intertropical Convergence Zone. Finally, the identification of the narrow sinking regions as another example of the kind of internal boundary layer studied here is considered, but in this case the magnitudes and scales are not plausible.