The secondary splitting of zero-gradient points in a scalar field

The mechanisms related to the secondary splitting of zero-gradient points of scalar fields are analyzed using the two-dimensional case of a scalar extreme point lying in a region of local strain. The velocity field is assumed to resemble a stagnation-point flow, cf. Gibson (Phys Fluids 11:2305–2315, 1968), which is approximated using a Taylor expansion up to third order. The temporal evolution of the scalar field in the vicinity of the stagnation point is derived using a series expansion, and it is found that the splitting can only be explained when the third-order terms of the Taylor expansion of the flow field are included. The non-dimensional splitting time turns out to depend on three parameters, namely the local Péclet number Pe _δ based on the initial size of the extreme point δ and two parameters which are measures of the rate of change of the local strain. For the limiting casePe _δ → 0, the splitting time is found to be finite but Péclet-number independent, while for the case of Pe _δ → ∞ it increases logarithmically with the Péclet number. The physical implications of the two-dimensional mathematical solution are discussed and compared with the splitting times obtained numerically from a Taylor–Green vortex.