Calculating surface topography in geodynamic models is a common
numerical problem. Besides other approaches, the so-called sticky air
approach has gained interest as a free-surface proxy at the top
boundary. The often used free slip condition is thereby vertically
extended by introducing a low density, low viscosity fluid layer. This
allows the air/crust interface to behave in a similar manner to a true
free surface. We present here a theoretical analysis that provides the
physical conditions under which the sticky air approach is a valid
approximation of a true free surface. Two cases are evaluated that
characterize the evolution of topography on different timescales: (1)
isostatic relaxation of a cosine perturbation and (2) topography changes
above a rising plume. We quantitatively compare topographies calculated
by six different numerical codes (using finite difference and finite
element techniques) using three different topography calculation
methods: (i) direct calculation of topography from normal stress, (ii)
body-fitting methods allowing for meshing the topography and (iii)
Lagrangian tracking of the topography on an Eulerian grid. It is found
that the sticky air approach works well as long as the term
(?st/?ch)/(hst/L)3 is sufficiently small, where ?st and hst are the
viscosity and thickness of the sticky air layer, and ?ch and L are the
characteristic viscosity and length scale of the model, respectively.
Spurious lateral fluctuations of topography, as observed in some
marker-based sticky air approaches, may effectively be damped by an
anisotropic distribution of markers with a higher number of markers per
element in the vertical than in the horizontal direction.