Dynamic topography and gravity anomalies for fluid layers whose viscosity varies exponentially with depth

We have derived analytic integral relations between boundary topography and temperature as a function of wavenumber for a fluid layer whose viscosity varies exponentially with depth. Similar relations between gravity and temperature are also derived. It is found that when the viscosity changes little over the depth of the layer (less than an order of magnitude), the topography kernels for both the surface and bottom boundaries are similar to those for the isoviscous case, the differences being more pronounced at intermediate wavelengths (λ∼ twice the layer depth). At these wavelengths, the extrema of the gravity kernel can be altered by as much as 30 per cent for a factor of two increase or decrease in viscosity over the depth of the layer. At wavelengths less than the layer depth, there are dependences on both the rate of growth of viscosity and the surface boundary condition, the latter dependency vanishing in the isoviscous case. For greater viscosity variation (several orders of magnitude), topography and gravity kernels at all wavelengths are strongly affected. Viscosity increasing rapidly with depth can approximate a rigid lower boundary condition, causing gravity kernels calculated with a free lower boundary condition to be negative for a significant range of source depths. For very rapidly varying viscosity, the sign of the perturbation pressure near the layer boundaries can be reversed from the isoviscous case, resulting in topography kernels that are negative for some source depths.