A trigger for the internal q = 1 disruption in tokamaks

A variational method is used to study the shape of magnetic surfaces and particularly of the q = 1 separatrix during the resistive evolution of the q = 1 island. It is shown that, in the zero beta limit and in cylindrical geometry, the angle at the X-point is always zero for a monotonic safety factor profile and, therefore, it is unlikely that evolution of the q = 1 island on a resistive time-scale is possible with such a profile. Conversely, if the q profile has an inflexion point or a plateau, the angle decreases monotonically with increasing island size. If the angle would reach a value of zero, a singular current layer would appear. This is a non-physical situation which must be removed by some other mechanism, such as turbulence (resistive microinstabilities), viscosity or inertia effects. The impact of approaching the singularity on the onset of the sawtooth crash in the case of 'standard' Ohmic sawteeth is discussed and it is suggested that similar methods can be used to compute the appearance of singular current layers in astrophysical problems (for example solar flares).