Why current-carrying magnetic flux tubes gobble up plasma and become thin as a result

Suppose an electric current I flows along a magnetic flux tube that has poloidal flux $ψ$ and radius $a=a(z),$ where z is the axial position along the flux tube. This current creates a toroidal magnetic field $Bφ.$ It is shown that, in such a case, nonlinear, nonconservative $J×B$ forces accelerate plasma axially from regions of small a to regions of large a and that this acceleration is proportional to $∂I2/∂z.$ Thus, if a current-carrying flux tube is bulged at, say, $z=0$ and constricted at, say, $z=±h,$ then plasma will be accelerated from $z=±h$ towards $z=0$ resulting in a situation similar to two water jets pointed at each other. The ingested plasma convects embedded, frozen-in toroidal magnetic flux from $z=±h$ to $z=0.$ The counterdirected flows collide and stagnate at $z=0$ and in so doing (i) convert their translational kinetic energy into heat, (ii) increase the plasma density at $z≈0,$ and (iii) increase the embedded toroidal flux density at $z≈0.$ The increase in toroidal flux density at $z≈0$ increases $Bφ$ and hence increases the magnetic pinch force at $z≈0$ and so causes a reduction of $a(0).$ Thus, the flux tube develops an axially uniform cross section, a decreased volume, an increased density, and an increased temperature. This model is proposed as a likely hypothesis for the long-standing mystery of why solar coronal loops are observed to be axially uniform, hot, and bright. It is furthermore argued that a small number of tail particles bouncing between the approaching counterstreaming plasma jets should be Fermi accelerated to extreme energies. Finally, analytic solution of the Grad–Shafranov equation predicts that a flux tube becomes axially uniform when the ingested plasma becomes hot and dense enough to have $2μ0nκT/Bpol2=(μ0Ia(0)/ψ)2/2;$ observed coronal loop parameters are in reasonable agreement with this relationship which is analogous to having $βpol=1$ in a tokamak.