Shock waves can strongly influence magnetic reconnection as seen by the slow shocks attached to the diffusion region in Petschek reconnection. We derive necessary conditions for such shocks in a nonuniform resistive magnetohydrodynamic plasma and discuss them with respect to the slow shocks in Petschek reconnection. Expressions for the spatial variation of the velocity and the magnetic field are derived by rearranging terms of the resistive magnetohydrodynamic equations without solving them. These expressions contain removable singularities if the flow velocity of the plasma equals a certain characteristic velocity depending on the other flow quantities. Such a singularity can be related to the strong spatial variations across a shock. In contrast to the analysis of Rankine–Hugoniot relations, the investigation of these singularities allows us to take the finite resistivity into account. Starting from considering perpendicular shocks in a simplified one-dimensional geometry to introduce the approach, shock conditions for a more general two-dimensional situation are derived. Then the latter relations are limited to an incompressible plasma to consider the subcritical slow shocks of Petschek reconnection. A gradient of the resistivity significantly modifies the characteristic velocity of wave propagation. The corresponding relations show that a gradient of the resistivity can lower the characteristic Alfvén velocity to an effective Alfvén velocity. This can strongly impact the conditions for shocks in a Petschek reconnection geometry.

In space plasmas, one of the most efficient conversion mechanisms of magnetic
energy into kinetic or thermal energy is magnetic reconnection. Reconnecting magnetic field lines change the magnetic field topology due to a
finite electrical resistivity and the plasma can be highly accelerated as
described by the Sweet-Parker model

A nonuniform resistivity can significantly impact the magnetohydrodynamic
(MHD) flow by the occurrence of shocks. For an ideal MHD plasma, the
Rankine–Hugoniot relations demonstrate that a flow velocity which exceeds the
magnetosonic velocity can lead to shocks

To introduce the procedure of how the velocity conditions for shocks can be
derived from the resistive and nonuniform MHD equations, a quasi
one-dimensional situation is considered. The magnetic field is restricted to
the

Across a shock, the MHD quantities, e.g., the flow velocity, change their
values. We restrict our analysis to approximately normal shocks, i.e., the
upstream flow velocity is approximately orthogonal to the shock. Thus,

The current sheet of the diffusion region is represented by the
circles with a cross. The plasma is reconnected at the

We extend the previous analysis to investigate the velocity conditions for
shocks in a two-dimensional diffusion region in the

Thus, all components and variations with respect to

Using the substitutions

The

The coefficients in Eq. (

Note that the solution for the chosen variables

To obtain a first insight into the effects of the inflow velocity conditions
for shocks, we compare the limit of a uniform resistivity to a steep
decreasing gradient of the resistivity along the

An approach to derive necessary conditions for shocks from the stationary
resistive MHD equations was presented and applied to the slow shocks of
Petschek reconnection. The approach treats terms with partial derivatives in
the MHD equations as variables. The equations were solved with respect to
terms with spatial derivatives which are normal to a possible shock. The
resulting expressions are displayed by fractions, whereby the numerator
contains the terms with spatial derivatives tangential to the possible shock.
Across a shock, the variations of the MHD quantities are usually small
compared to variations along the shock plane. Note that this assumes the
curvature of a shock to be small, i.e., an approximately plane shock front.
Therefore, the high values of the spatial derivatives of quantities normal to
a shock, e.g., for the normal and tangential velocity components, require a
root of the denominator in our expressions. This determines a necessary
condition for a shock. The approach was introduced by a quasi one-dimensional
situation with the magnetic field perpendicular to the flow velocity. This
example is therefore limited to perpendicular shocks. Then a more general
two-dimensional situation was considered. It was assumed that the resistivity
profile is uniformly normal to a shock. The resulting necessary condition for a
shock (Eq.

This work was financially supported by the German Ministerium für Wirtschaft und Technologie and the Deutsches Zentrum für Luft- und Raumfahrt under contracts 50OC1403, 50OC1402, and 50QW1101. The topical editor, C. Owen, thanks H. Kucharek for help in evaluating this paper.