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For an asymmetric sinh-Poisson problem arising as a mean field equation ofequilibrium turbulence vortices with variable intensities of interest inhydrodynamic turbulence, we address the existence of bubbling solutions oncompact Riemann surfaces. By using a Lyapunov-Schmidt reduction, we findsufficient 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 indifferent situations illustrate our results in the sphere $\mathbb S^2$ andflat two-torus $\mathbb T$ including non negative potentials with zero set nonempty.

Bubbling solutions for mean field equations with variable intensities on compact Riemann surfaces