the inflows. The aspect ratio of the diffusion region is then of order unity and the maximum reconnection rate becomes This expression allows for fast reconnection and is almost independent of the Lundquist number. Theory and numerical simulations show that most of the actions of the shocks that were proposed by Petshek can be caried out by Alfvén waves and in particular rotational discontinuities (RDs). In cases of asymmetric plasma densities on the two sides of the current sheet (as at Earth's dayside magnetopause) the Alfvén wave that propagates into the inflow on higher-density side (in the case of the magnetopause the denser magnetosheath) has a lower propagation speed and so the field rotation increasingly becomes at that RD as the field line propagates away from the reconnection site: hence the magnetopause current sheet becomes increasingly concentrated in the outer, slower, RD. Simulations of resistive MHD reconnection with uniform resistivity showed the development of elongated current sheets in agreement with the Sweet–Parker model rather than the Petschek model. When a localized anomalously large resistivity is used, however, Petschek reconnection can be realized in resistive MHD simulations. Because the use of an anomalous resistivity is only appropriate when the particle mean free path is large compared to the reconnection layer, it is likely that other collisionless effects become important before Petschek reconnection can be realized. Anomalous resistivity and Bohm diffusion In the Sweet–Parker model, the common assumption is that the magnetic diffusivity is constant. This can be estimated using the equation of motion for an electron with mass and electric charge : where is the collision frequency. Since in the steady state, , then the above equation along with the definition of electric current, , where is the electron number density, yields Nevertheless, if the drift velocity of electrons exceeds the thermal velocity of plasma, a steady state cannot be achieved and magnetic diffusivity should be much larger than what is given in the above. This is called anomalous resistivity, , which can enhance the reconnection rate in the Sweet–Parker model by a factor of . Another proposed mechanism is known as the Bohm diffusion across the magnetic field. This replaces the Ohmic resistivity with , however, its effect, similar to the anomalous resistivity, is still too small compared with the observations. Stochastic reconnection In stochastic reconnection, magnetic field has a small scale random component arising because of turbulence. For the