# Heat pinch

The concept of heat pinch is related to the convective term proportional to V in the (electron) heat transport equation:

$q_e = -n_e\chi \nabla T_e + n_e V T_e$

Using modulation experiments, the contributions to (electron) transport due to the heat conductivity χ and the convection V can be distinguished.  In such experiments, it was found that V is often large and directed inward (hence the term 'pinch'), which raises the question which mechanism is responsible for this effect that drives heat up the gradient.

## Coupling between transport equations

To answer this question, first it should be noted that the heat transport equation only captures part of the transport problem in fusion plasmas. A more complete description also considers ion heat transport and electron and ion density (particle) transport. The mentioned set of transport equations is coupled (cf. Neoclassical transport), which may give rise to an apparent pinch.  This coupling between transport equations has successfully been invoked to explain some salient features of propagating heat pulses.  Even so, the coupling between transport equations by itself does not generally seem capable of explaining the observed heat pinch.

## Fick versus Fokker Planck

Part of the problem may be due to the use of a Fickian transport equation, the use of which is only recommended in homogeneous systems. In inhomogenous systems (such as fusion plasmas), the Fokker-Planck formulation seems more appropriate.  Within the Fokker-Planck formulation, the radial gradient of the heat conductivity produces a 'natural' heat pinch. By way of simplified example, one may write the Fokker-Planck heat transport equation

$q_e = - \nabla(n_e \chi T_e) + n_eUT_e$

Setting U = 0 and assuming ∇ ne = 0, comparison with the above 'Fickian' heat transport equation shows that

$V = -\nabla \chi$

I.e., the gradient of the heat conductivity produces a 'natural' pinch. It should be noted that at the descriptive level, the Fick and Fokker-Planck equations are fully equivalent and capable of describing the same phenomena (with one exception). It is only at the interpretative level that this difference appears: a 'pinch' may seem mysterious when using Fick's equation, while it appears 'natural' when using the Fokker-Planck equation.

## Mesoscopic and microscopic mechanisms

The preceding paragraphs only discuss the 'macroscopic' fluid level description. But the fluid scale description should of course arise from underlying 'mesoscopic' or 'microscopic' level descriptions through an appropriate averaging procedure. At the microscopic level, several mechanisms may be operative, such as Turbulent Equipartition.   At the mesoscopic level, critical gradients can provide strong inward transport