  We obtained steady solutions to the two-dimensional Boussinesq approximation
equations without mean temperature gradient. This system is referred to as free
convection in this paper. Under an external flow described by the stream
function $\mPsi = - Ayf(x)$, two types of steady solutions are found depending
on the boundary conditions. One is kept steady by the balance between the
strain of $\mPsi$ and the diffusion. The solution is similar to the Burgers
vortex layer solution. The other is done by the balance between vorticity
induced by the buoyancy and vorticity flux caused by the external flow.
Detailed argument on these two balances is presented for $f(x) = x$. Then two
examples other than $f(x) = x$ are shown to have either of the two balancing
mechanism. We discuss the relation between these solutions and long-lived fine
scale coherent structures observed in direct numerical simulations of
two-dimensional free convection turbulence.
