The relationship between surface topography, gravity anomalies, and temperature structure of convection

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Abstract
Normal stresses associated with convection in a fluid layer, whose boundaries can deform, produce topography on those boundaries. When the equations of motion are linear, integral relations between topography on the boundaries and the temperature structure can be found as a function of wavelength. Expressions of this kind have been derived for the case of convection in a constant viscosity fluid when inertial effects are negligible. The total gravity anomaly is the sum of contributions due to the topography on the two boundaries and to temperature variations within the fluid, and similar integral relations between gravity and the temperature structure can also be derived. At wavelengths large compared to the depth the shape of the kernels in the integrals, particularly that for gravity, is sensitive to the boundary conditions used. The transfer function between gravity and topography is also characteristic of the boundary conditions at long wavelengths. In all the cases considered, there is a transition between short‐and long‐wavelength behavior which occurs at wavelengths proportional to the layer depth. If the bottom boundary can deform as well as the upper boundary, the gravity anomaly tends to zero at long wave‐lengths as the gravity kernel tends to zero everywhere. The surface topography kernel is always zero at the bottom boundary. The lack of any significant surface expression produced by temperature variations near the bottom of the layer provides an explanation of the similar relationships between gravity and topography associated with the dissimilar temperature structures produced by different modes of heating. At least for cellular convection, surface topography and gravity anomalies largely reflect temperature variations in the vicinity of the upper thermal boundary layer. This is consistent with an explanation of geoid anomalies over mid‐ocean swells in terms of convection beneath the lithosphere, where the lower part of the thermally defined plate acts as the upper thermal boundary layer of the convection.

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