Holographic parity violating charged fluid dual to Chern-Simons modified gravity

We discuss the $(2+1)$-dimensional parity violating charged fluid on a finite cutoff surface $\Sigma_c$, dual to the nondynamical and dynamical Chern-Simons (CS) modified gravities. Using nonrelativistic long-wavelength expansion method, the field equations are solved up to $\mathcal{O}(\epsilon^2)$ in the nondynamical model. It is shown that there exists nonvortical dual fluid with shear viscosity $\eta$ and Hall viscosity $\eta_A$ on the cutoff surface $\Sigma_c$. The ratio of shear viscosity over entropy density $\eta/s$ of the fluid takes the universal value $1/{4\pi}$, while the ratio of Hall viscosity over entropy density $\eta_A/s$ depends on the $\Sigma_c$ and black brane charge $q$. Moreover the nonvortical dual fluid obeys the magnetohydrodynamic (MHD) equation. However, these kinematic viscosities $\nu$ and $\nu_A$ related to $\eta$ and $\eta_A$ do not appear in this MHD equation, due to the constraint condition $\tilde{\partial}^2\beta_j=0$ for the $(2+1)$-dimensional dual fluid. Then, we extend our discussion to the dynamical CS modified gravity and show that the dual vortical fluid possesses another so-called Curl viscosity $\zeta_A$, whose ratio to entropy density $\zeta_A/s$ also depends on the $\Sigma_c$ and $q$. Moreover, the value of $\eta/s$ still equals to $1/4\pi$ and the result of $\eta_A/s$ agrees to the previous result under the probe limit of the pseudo scalar field at the infinite boundary in the charged black brane background for the dynamical CS modified gravity. This vortical dual fluid corresponds to the magnetohydrodynamic (MHD) turbulence equation in plasma physics.