page_title: Pinch (plasma physics)
section_title: The Carlqvist relation
hierarchical_section_title: Pinch (plasma physics) / Formal treatment / The Carlqvist relation
context_page_description: A pinch is the compression of an electrically conducting filament by magnetic forces. The conductor is usually a plasma, but could also be a solid or liquid metal. Pinches were the first type of device used for controlled nuclear fusion.
  The phenomenon may also be referred to as a Bennett pinch, electromagnetic pinch, magnetic pinch, pinch effect or plasma pinch.
  Pinches occur naturally in electrical discharges such as lightning bolts, the aurora, current sheets, and solar flares.
context_section_description: The Carlqvist relation, published by Per Carlqvist in 1988, is a specialization of the generalized Bennett relation (above), for the case that the kinetic pressure is much smaller at the border of the pinch than in the inner parts. It takes the form
  and is applicable to many space plasmas.
  The Carlqvist relation can be illustrated (see right), showing the total current (I) versus the number of particles per unit length (N) in a Bennett pinch. The chart illustrates four physically distinct regions. The plasma temperature is quite cold (Tᵢ = Tₑ = Tₙ = 20 K), containing mainly hydrogen with a mean particle mass 3×10⁻²⁷ kg. The thermokinetic energy Wₖ >> πa² pₖ(a). The curves, ΔWBz show different amounts of excess magnetic energy per unit length due to the axial magnetic field Bz. The plasma is assumed to be non-rotational, and the kinetic pressure at the edges is much smaller than inside.
  Chart regions: (a) In the top-left region, the pinching force dominates. (b) Towards the bottom, outward kinetic pressures balance inwards magnetic pressure, and the total pressure is constant. (c) To the right of the vertical line ΔWBz = 0, the magnetic pressures balances the gravitational pressure, and the pinching force is negligible. (d) To the left of the sloping curve ΔWBz = 0, the gravitational force is negligible. Note that the chart shows a special case of the Carlqvist relation, and if it is replaced by the more general Bennett relation, then the designated regions of the chart are not valid.
  Carlqvist further notes that by using the relations above, and a derivative, it is possible to describe the Bennett pinch, the Jeans criterion (for gravitational instability, in one and two dimensions), force-free magnetic fields, gravitationally balanced magnetic pressures, and continuous transitions between these states.
caption: The Bennett pinch showing the total current (I) versus the number of particles per unit length (N). The chart illustrates four physically distinct regions. The plasma temperature is 20 K, the mean particle mass 3×10−27 kg, and ΔWBz is the excess magnetic energy per unit length due to the axial magnetic field Bz. The plasma is assumed to be non-rotational, and the kinetic pressure at the edges is much smaller than inside.
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