[Sect. \[s:sgmod\]]{} ($256^3$ resolution) and [Sect. \[s:sghigh\]]{} ($512^3$ resolution). We summarise the paper and discuss the implications of our results in [Sect. \[s:discussion\]]{}.

Dynamical equations {#s:equations}
===================

We perform simulations solving the standard shearing box MHD/drag force/self-gravity equations for gas defined on a fixed grid and solid particles evolved as numerical superparticles. We use the Pencil Code, a sixth order spatial and third order temporal symmetric finite difference code[^1].

We model the dynamics of a protoplanetary disc in the shearing box approximation. The coordinate frame rotates at the Keplerian frequency $\varOmega$ at an arbitrary distance $r_0$ from the central star. The axes are oriented such that the $x$ points radially away from the central gravity source, $y$ points along the Keplerian flow, while $z$ points vertically out of the plane.

Gas velocity
------------

The equation of motion for the gas velocity ${{\boldsymbol u}}$ relative to the Keplerian flow is $$\begin{aligned}
\label{eq:eqmot}
  \frac{{\partial}{{\boldsymbol u}}}{{\partial}t} +
    ({{\boldsymbol u}} \cdot {{{\boldsymbol \nabla}}}) {{\boldsymbol u}} + u_y^{(0)} \frac{{\partial}{{\boldsymbol u}}}{{\partial}y} &=& 2
    \varOmega u_y {{\boldsymbol e}}_x - \frac{1}{2} \varOmega u_x {{\boldsymbol e}}_y + 2 \varOmega
    \Delta v {{\boldsymbol e}}_x \nonumber \\
    & & \hspace{-2.5cm} + \frac{1}{\rho} {{\boldsymbol J}} \times
    ({{\boldsymbol B}}+B_0\hat{{{\boldsymbol z}}}) - \frac{1}{\rho}
    {{{\boldsymbol 