\nabla}}}P -\frac{\rho_{\rm p}/\rho_{\rm g}}{\tau_{\rm f}}
    ({{\boldsymbol u}}-\overline{{{\boldsymbol v}}}) + {{\boldsymbol f}}_\nu({{\boldsymbol u}}) \, .\end{aligned}$$ The left hand side includes advection both by the velocity field ${{\boldsymbol u}}$ itself and by the linearised Keplerian flow $u_y^{(0)}=-(3/2)\varOmega x$. The first two terms on the right hand side represent the Coriolis force in the $x$- and $y$-directions, modified in the $y$-component by the radial advection of the Keplerian flow, $\dot{u}_y=-u_x {\partial}u_y^{(0)}/{\partial}x$. The third term mimics a global radial pressure gradient which reduces the orbital speed of the gas by the positive amount $\Delta v$. The fourth and fifth terms in [Eq. (\[eq:eqmot\])]{} are the Lorentz and pressure gradient forces. The current density is calculated from Ampère’s law $\mu_0{{\boldsymbol J}}={{{\boldsymbol \nabla}}}\times{{\boldsymbol B}}$. The Lorentz force is modified to take into account a mean vertical field component of strength $B_0$. The sixth term is a drag force term is described in Sect.\[s:drag\].

The high-order numerical scheme of the Pencil Code has very little numerical dissipation from time-stepping the advection term [@Brandenburg2003], so we add explicit viscosity through the term ${{\boldsymbol f}}_\nu({{\boldsymbol u}})$ in [Eq. (\[eq:eqmot\])]{}. We use sixth order hyperviscosity with a constant dynamic viscosity $\mu_3=\nu_3\rho$, $${{\boldsymbol f}}_\nu = \frac{\mu_3}{\rho_{\rm g}} \nabla^6 {{\boldsymbol u}} \, 