ABSTRACT
In the case of negligible dissipation, the evolution equation for thermal pressure reduces to the adiabatic law, namely
d
dt s(p, ρ) = 0
In the equation of motion (4.1) the relative role of the thermal pressure p is characterized by the non-dimensional parameter β = 8πp/B2. In the case of small β (i.e., when β ≪ 1), the magnetic force dominates. In this regime the dynamic timescale is the Alfven time, τA = L/vA, where L is the spatial scale of the system, and vA = B/(4πρ)
1/2 is the Alfven velocity. In the magnetic induction equation (4.2) the relative role of magnetic
diffusion due to finite fluid resistivity η is given by the magnetic Reynolds number Rm = Lv/η. It is also common to use another non-dimensional parameter called the Lundquist number, S. This is the ratio of τA and the resistive diffusion timescale, τη = L
2/η, so that S = τη/τA. If these parameters are large, which is the case for a wide class of laboratory and astrophysical applications, the induction equation (4.2) reduces to
∂ ~B
∂t = ~∇× (~v × ~B) (4.3)
Second Edition
This is the limit of ideal magnetohydrodynamics (MHD), when magnetic field is “frozen” to the fluid flow. It implies conservation of the magnetic field lines connectivity and topology, as well as fixing the magnetic flux through any surface moving with the fluid. Formally, the “frozen-in” condition can be formulated as follows: if two infinitely close fluid particles are on the same magnetic line of force at some time t0, they will remain on the same line of force in the course of fluid motion. The separation between these fluid elements, δ~l(t), is related to the magnetic field, ~B(t), as
Bi = ρ
δli
, (4.4)
where ~B(0), δ~l(0), and ρ(0) correspond to t = t0. As can be seen from equation (4.4), in a perfectly conducting fluid the magnetic field can be amplified by fluid compression and/or stretching of magnetic field lines.
