The free surface of a viscous fluid is a source of convective
flow (Marangoni convection) if its surface tension is distributed
non-uniformly. Such non-uniformity arises from the dependence of the
surface tension on a scalar quantity, either surfactant concentration
or temperature. The surface-tension-induced velocity redistributes
the scalar forming a closed-loop interaction. It is shown that under
the assumptions of (i) small Reynolds number and (ii) vanishing
diffusivity this nonlinear process is described by a single
self-consistent two-dimensional evolution equation for the scalar
field at the free surface that can be derived from the
three-dimensional basic equations without approximation. The
formulation of this equation for a particular system requires only
the knowledge of the closure law, which expresses the surface
velocity as a linear functional of the active scalar at the free
surface. We explicitly derive these closure laws for various systems
with a planar non-deflecting surface and infinite horizontal extent,
including an infinitely deep fluid, a fluid with finite depth, a
rotating fluid, and an electrically conducting fluid under the
influence of a magnetic field. For the canonical problem of an
infinitely deep layer we demonstrate that the dynamics of singular
(point-like) surfactant or temperature distributions can be further
reduced to a system of ordinary differential equations, equivalent to
point-vortex dynamics in two-dimensional perfect fluids. We further
show, using numerical simulations, that the dynamical evolution of
initially smooth scalar fields leads in general to a finite-time
singularity. The present theory provides a rational framework for a
simplified modelling of strongly nonlinear Marangoni convection in
high-Prandtl-number fluids or systems with high Schmidt
number.