.$$ This form of the viscosity conserves momentum. The $\nabla^6$ operator is defined as $\nabla^6={\partial}^6/{\partial}x^6 + {\partial}^6/{\partial}y^6 + {\partial}^6/{\partial}z^6$. It was shown by [@Johansen+etal2009a] that hyperviscosity simulations show zonal flows and pressure bumps very similar to simulations using Navier-Stokes viscosity.

Gas density
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The continuity equation for the gas density $\rho$ is $$\frac{{\partial}\rho}{{\partial}t} + ({{\boldsymbol u}} \cdot {{{\boldsymbol \nabla}}}) \rho + u_y^{(0)} \frac{{\partial}\rho }{{\partial}y} = -\rho {{{\boldsymbol \nabla}}}\cdot {{\boldsymbol u}} + f_D(\rho) \, .$$ The diffusion term is defined as $$f_{\rm D} = D_3 \nabla^6 \rho \, ,$$ where $D_3$ is the hyperdiffusion coefficient necessary to suppress Nyquist scale wiggles arising in regions where the spatial density variation is high. We adopt an isothermal equation of state with pressure $P=c_{\rm s}^2 \rho$ and (constant) sound speed $c_{\rm s}$.

Induction equation
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The induction equation for the magnetic vector potential ${{\boldsymbol A}}$ is [see @Brandenburg+etal1995 for details] $$\frac{{\partial}{{\boldsymbol A}}}{{\partial}t} + u_y^{(0)} \frac{{\partial}{{\boldsymbol A}}}{{\partial}y} =
  {{\boldsymbol u}}\times({{\boldsymbol B}}+B_0\hat{{{\boldsymbol z}}}) + \frac{3}{2} \varOmega A_y
  \hat{{{\boldsymbol x}}} + {{\boldsymbol f}}_\eta({{\boldsymbol A}}) \, .$$ The resistivity term is $${{\boldsymbol f}}_\eta = \eta_3 \nabla^6 {{\boldsymbol A}} \, ,$$ where $\eta_3$ is the hyperresistivity coefficient. The magnetic field is calculated from ${{\boldsymbol B}}={{{\boldsymbol \nabla}}}\times{{\boldsymbol A}}$.

Particles and 