Lecture 4 Flux Freezing and the Ideal MHD Limit

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Lecture 4
Flux Freezing and the Ideal MHD Limit

Overview

This lecture isolates the geometric content of ideal MHD.
1.
It identifies the ordering that turns the resistive induction equation into the ideal one.
2.
It derives the line-stretching form for \(\B /\rho \).
3.
It states precisely what is meant by frozen-in magnetic flux, and just as importantly, what is not meant.

The induction equation (1.13) already contains the whole story in compressed form: advection tries to move and stretch magnetic flux, while resistivity tries to diffuse it. Lecture 2 exposed the dissipative term; here we examine the opposite asymptotic limit in which that term is negligible over the scales of interest.

Historical Perspective

Alfvén emphasized early that a conducting fluid can support field-guided dynamics, and Newcomb later sharpened the geometric meaning of the ideal limit by relating magnetic evolution to the motion of material elements [Alfvén, 1950, 1942, Newcomb, 1961, 1962, Moffatt, 1978].

Caution

Flux freezing is an asymptotic statement, not a mystical one. Large magnetic Reynolds number means resistive diffusion is weak on the scale being considered. It does not mean that non-ideal physics is absent everywhere, nor that field lines are literal material strings.

4.1 Resistive Induction and the Magnetic Reynolds Number

Starting from (2.12), define the magnetic diffusivity

\eta _{\mathrm {m}} \equiv \frac {\eta }{\muo } = \frac {1}{\muo \sigma }. \tag{4.1}

The induction equation is then

\pp {\B }{t} = \curl (\uvec \times \B ) + \eta _{\mathrm {m}} \lap \B . \tag{4.2}

Comparing the advection term to the diffusion term suggests the magnetic Reynolds number

Rm = \frac {U L}{\eta _{\mathrm {m}}}. \tag{4.3}

When \(Rm\gg 1\), advection dominates diffusion in the bulk. The ideal-MHD induction equation is the formal reduction of (4.2) obtained by dropping the diffusive term.

4.2 Convective Form of the Induction Equation

Using the vector identity

\curl (\uvec \times \B ) = \uvec (\divergence \B ) - \B (\divergence \uvec ) + (\B \cdot \grad )\uvec - (\uvec \cdot \grad )\B \tag{4.4}

and the solenoidal constraint (1.12), equation (4.2) becomes

\frac {D\B }{Dt} + \B \,\divergence \uvec = (\B \cdot \grad )\uvec + \eta _{\mathrm {m}} \lap \B . \tag{4.5}

Now combine (4.5) with the continuity equation (3.1). Since

\frac {D\rho }{Dt} = -\rho \,\divergence \uvec , \tag{4.6}

we obtain

\boxed { \frac {D}{Dt} \left ( \frac {\B }{\rho } \right ) = \left ( \frac {\B }{\rho }\cdot \grad \right )\uvec + \frac {\eta _{\mathrm {m}}}{\rho }\lap \B . } \tag{4.7}

Equation (4.7) lays the geometry bare: velocity gradients stretch and rotate the field, while magnetic diffusion smooths it.

4.3 Ideal MHD

In ideal MHD the non-ideal term in Ohm’s law is neglected, so (1.9) reduces to

\boxed { \E = -\uvec \times \B . } \tag{4.8}

Substituting (4.8) into Faraday’s law (1.10) gives

\boxed { \pp {\B }{t} = \curl (\uvec \times \B ), \qquad \divergence \B =0. } \tag{4.9}

Likewise, (4.7) becomes

\boxed { \frac {D}{Dt} \left ( \frac {\B }{\rho } \right ) = \left ( \frac {\B }{\rho }\cdot \grad \right )\uvec . } \tag{4.10}

Figure 4.1: Cartoon of ideal advection: a material surface deforms with the flow while the magnetic flux through it remains unchanged.

4.4 Frozen-In Flux

Let \(S(t)\) be a material surface bounded by a material curve \(C(t)\), and define the magnetic flux

\Phi (t) = \int _{S(t)} \B \cdot d\vect {S}. \tag{4.11}

For a loop moving with the fluid, Faraday’s law takes the form

\frac {d\Phi }{dt} = - \oint _{C(t)} \left ( \E +\uvec \times \B \right )\cdot d\vect {\ell }. \tag{4.12}

Using the ideal Ohm law (4.8), the integrand vanishes, and therefore

\boxed { \frac {d}{dt} \int _{S(t)} \B \cdot d\vect {S} = 0. } \tag{4.13}

Equation (4.13) is the precise statement of frozen-in magnetic flux. It means that the flux through any material surface is conserved under ideal evolution.

4.5 What Frozen-In Does Not Mean

Three cautions matter.

1.: Field lines are not material strings. They are integral curves of \(\B \) drawn at an instant.
2.: The ideal approximation may hold in the bulk and fail in thin layers. Reconnection is exactly the classic example.
3.: Large \(Rm\) is necessary for the ideal picture, but it is not sufficient to suppress Hall physics, pressure anisotropy, or electron inertia. Those are separate orderings and reappear in Lecture 5.

Takeaways

The ideal limit of Ohm’s law is (4.8), and the corresponding induction equation is (4.9).

The line-stretching equation (4.10) is the local differential statement behind the frozen-in picture.

Flux freezing, (4.13), is exact only in ideal MHD and only as long as the assumptions leading to (4.8) are respected.

Bibliography

Hannes Alfvén. Cosmical Electrodynamics. Clarendon Press, Oxford, 1950.

Hannes Alfvén. Existence of electromagnetic-hydrodynamic waves. Nature, 150:405–406, 1942. doi:10.1038/150405d0.

W. A. Newcomb. Convective instability induced by gravity in a plasma with a frozen-in magnetic field. Physics of Fluids, 4:391–396, 1961. doi:10.1063/1.1706342.

W. A. Newcomb. Lagrangian and hamiltonian methods in magnetohydrodynamics. Nuclear Fusion Supplement, Part 2, pages 451–463, 1962.

H. K. Moffatt. Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press, Cambridge, England; New York, NY, USA, 1978. ISBN 9780521216401. Cambridge Monographs on Mechanics and Applied Mathematics.

Problems

Problem 4.1. Frozen-in Magnetic Flux

(a): Starting from the ideal-MHD induction equation (4.9), derive the evolution equation \[ \frac {D}{Dt}\left (\frac {\B }{\rho }\right ) = \left (\frac {\B }{\rho }\cdot \grad \right )\uvec . \]
(b): Consider two infinitesimally separated fluid elements connected by a magnetic field line. Show that their separation vector obeys the same line-stretching law as \(\B /\rho \).
(c): Using the moving-loop form (4.12), show that the magnetic flux through a material surface moving with the fluid is conserved in ideal MHD.
(d): Give two distinct physical mechanisms, beyond scalar resistivity, by which flux freezing can be broken in real plasmas.

Problem 4.2. When Does MHD Break Down?

For each of the following regimes, state which MHD assumptions fail and why:

(a): collisionless magnetic-reconnection layers,
(b): the solar wind near 1 AU,
(c): a tokamak edge pedestal,
(d): a liquid-metal laboratory dynamo.

Indicate whether the dominant correction is expected to be Hall physics, pressure anisotropy, finite-Larmor-radius effects, electron inertia, or essentially none.