  For an asymmetric sinh-Poisson problem arising as a mean field equation of
equilibrium turbulence vortices with variable intensities of interest in
hydrodynamic turbulence, we address the existence of bubbling solutions on
compact Riemann surfaces. By using a Lyapunov-Schmidt reduction, we find
sufficient conditions under which there exist bubbling solutions blowing up at
$m$ different points of $S$: positively at $m_1$ points and negatively at
$m-m_1$ points with $m\ge 1$ and $m_1\in\{0,1,...,m\}$. Several examples in
different situations illustrate our results in the sphere $\mathbb S^2$ and
flat two-torus $\mathbb T$ including non negative potentials with zero set non
empty.
