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We present a systematic shearing-box investigation of MRI-driven turbulencein a weakly collisional plasma by including the effects of an anisotropicpressure stress, i.e. anisotropic (Braginskii) viscosity. We constrain thepressure anisotropy ($\Delta p$) to lie within the stability bounds that wouldbe otherwise imposed by kinetic microinstabilities. We explore a broad regionof parameter space by considering different Reynolds numbers and magnetic-fieldconfigurations, including net vertical flux, net toroidal-vertical flux andzero net flux. Remarkably, we find that the level of turbulence andangular-momentum transport are not greatly affected by large anisotropicviscosities: the Maxwell and Reynolds stresses do not differ much from the MHDresult. Angular-momentum transport in Braginskii MHD still depends strongly onisotropic dissipation, e.g., the isotropic magnetic Prandtl number, even whenthe anisotropic viscosity is orders of magnitude larger than the isotropicdiffusivities. Braginskii viscosity nevertheless changes the flow structure,rearranging the turbulence to largely counter the parallel rate of strain fromthe background shear. We also show that the volume-averaged pressure anisotropyand anisotropic viscous transport decrease with increasing isotropic Reynoldsnumber (${\rm Re}$); e.g., in simulations with net vertical field, the ratio ofanisotropic to Maxwell stress ($\alpha_{\rm A} / \alpha_{\rm M}$) decreasesfrom $\sim 0.5$ to $\sim 0.1$ as we move from ${\rm Re} \sim 10^3$ to ${\rm Re}\sim 10^4$. Anisotropic transport may thus become negligible at high ${\rmRe}$. Anisotropic viscosity nevertheless becomes the dominant source of heatingat large ${\rm Re}$, accounting for $\gtrsim 50 \%$ of the plasma heating. Weconclude by briefly discussing the implications of our results for RIAFs ontoblack holes.

Shearing-box simulations of MRI-driven turbulence in weakly collisional accretion discs