Sign-changing bubble tower solutions for sinh-Poisson type equations on pierced domains

For asymmetric sinh-Poisson type problems with Dirichlet boundary condition arising as a mean field equation of equilibrium turbulence vortices with variable intensities of interest in hydrodynamic turbulence, we address the existence of sign-changing bubble tower solutions on a pierced domain ${\mathrm{\Omega }}_{\epsilon }:=\mathrm{\Omega }\setminus {\displaystyle \overline{B(\xi ,\epsilon )}}$, where $\mathrm{\Omega }$ is a smooth bounded domain in ${\mathbb{R}}^2$ and $B(\xi ,\epsilon )$ is a ball centered at $\xi \in \mathrm{\Omega }$ with radius $\epsilon >0$. Precisely, given a small parameter $\rho >0$ and any integer $m\ge 2$, there exist a radius $\epsilon =\epsilon (\rho )>0$ small enough such that each sinh-Poisson type equation, either in Liouville form or mean field form, has a solution $u_{\rho }$ with an asymptotic profile as a sign-changing tower of $m$ singular Liouville bubbles centered at the same $\xi$ and with $\epsilon (\rho )\to 0^{+}$ as $\rho$ approaches to zero.