  The statistically steady humidity distribution resulting from an interaction
of advection, modeled as an uncorrelated random walk of moist parcels on an
isentropic surface, and a vapour sink, modeled as immediate condensation
whenever the specific humidity exceeds a specified saturation humidity, is
explored with theory and simulation. A source supplies moisture at the
deep-tropical southern boundary of the domain, and the saturation humidity is
specified as a monotonically decreasing function of distance from the boundary.
The boundary source balances the interior condensation sink, so that a
stationary spatially inhomogeneous humidity distribution emerges. An exact
solution of the Fokker-Planck equation delivers a simple expression for the
resulting probability density function (PDF) of the water vapour field and also
of the relative humidity. This solution agrees completely with a numerical
simulation of the process, and the humidity PDF exhibits several features of
interest, such as bimodality close to the source and unimodality further from
the source. The PDFs of specific and relative humidity are broad and
non-Gaussian. The domain averaged relative humidity PDF is bimodal with
distinct moist and dry peaks, a feature which we show agrees with middleworld
isentropic PDFs derived from the ERA interim dataset.
