The field that cannot get out

A magnetic field in a good conductor is difficult to get rid of. Not because the field is strong, but because changing it induces currents that oppose the change, and in a conductor with no resistance those currents are whatever they need to be. The limiting case is exact: in a perfectly conducting fluid, the magnetic flux through any closed loop that moves with the material never changes at all.

Where it comes from #

Two ingredients, both already in place. Faraday’s law says that a changing flux through a loop drives an electromotive force around it. Ohm’s law says that an electromotive force in a conductor drives a current, $J = \sigma E$.

Now take $\sigma \to \infty$. A finite current requires $E \to 0$ in the material’s own frame, so the electric field in the frame moving with the fluid vanishes, so the electromotive force around any comoving loop vanishes, so the flux through it cannot change.

The mechanism in its familiar form is a loop moving through a field region: the flux through it changes, a current is induced, and a force opposes the motion. Flux freezing is that statement taken to the limit of perfect conductivity, where the induced current is whatever it needs to be — so the opposition becomes total and the flux through any loop of fluid simply cannot change at all.

The result is due to Alfvén in 1942, and it is the founding statement of the subject that treats a conducting fluid and a magnetic field as a single object rather than two.

Flux is a count, and the count does not change #

The clean way to picture it is to think of flux as a number of lines threading a loop, and to notice that the number is what is conserved rather than any property of a line.

Flux through a surface is the right object because it is a count rather than a field strength. The field can be compressed, stretched, twisted and moved, and the number of lines threading a given loop of fluid stays what it was — so the conserved quantity is topological in character, which is why the statement survives being carried around by a turbulent flow that no one can compute.

There is one trap in the picture and it is worth naming, because it is this collection’s fourth verdict again. Field lines have no identity from one moment to the next. No experiment follows a particular line, and asking which line a given line “became” is a question about the drawing. The dragging image is an excellent calculational habit and it is not a statement about anything real; what is real is the number attached to each material loop.

It is worth being careful about which loops the statement applies to, because the qualifier does all the work. The flux through a loop fixed in the laboratory changes freely — that is ordinary induction, and it is how every transformer works. What cannot change is the flux through a loop drawn on the material and carried along by it, deforming as the material deforms. Two loops occupying the same place at the same moment can therefore disagree about whether their flux is changing, and both are right, because they are not the same loop a moment later.

What it does under a collapse #

The scaling follows immediately. Flux is field times area; area goes as $R^2$; so if the flux is fixed, $B \propto 1/R^2$.

That is a violent exponent. A collapse by a factor of ten multiplies the field by a hundred; by a thousand, a million. The hero figure’s compression of $7\times10^4$ gives $4.9\times10^9$.

Explosive flux compression is the honest laboratory version, and it is instructive precisely because it is lossy. The field achieved is smaller than the ideal scaling, by a factor that depends on how fast the compression is compared with the leakage. Which is the general rule: the frozen-flux estimate is an upper bound on what compression can deliver, and how close a real system gets is a question about a competition between two rates.

The field pushes back, and the work has to come from somewhere #

Compression is not free, and the reason is that a magnetic field carries an energy density $B^2/2\mu_0$ that behaves mechanically like a pressure.

Follow the bookkeeping through a collapse. The energy density goes as $B^2$, which goes as $1/R^4$; the volume goes as $R^3$; so the total magnetic energy goes as $1/R$. Halving the radius doubles the stored magnetic energy, and that energy is supplied by whatever is doing the squeezing, against a pressure that is rising as the fourth power of the compression.

Put numbers on the laboratory case. One tesla is a magnetic pressure of $4\times10^5$ Pa, four atmospheres — noticeable and not dramatic. At 100 tesla it is $4\times10^9$ Pa, which exceeds the strength of any material, and this is precisely why fields above about a hundred tesla cannot be produced in a coil that survives: the coil is torn apart by its own field. The only way past that limit is to accept the destruction and measure fast, which is what a flux-compression device does.

At the hero figure’s $4.9\times10^7$ tesla the magnetic pressure is around $10^{21}$ Pa. There is no material sense in which that is a pressure a container could hold; what holds it is self-gravity, which is the only agent available at those numbers.

This also gives the honest limit on the whole scaling argument. The compression can only continue while whatever drives it is stronger than the magnetic pressure resisting it, and since the resistance rises as $1/R^4$ while gravitational compression rises far more slowly, there is always a radius at which the field wins and the collapse stops or becomes non-spherical. The $1/R^2$ line in the figures is what happens until that point and not beyond it.

The rate that decides whether the picture applies #

Real conductivity is finite, so the field does leak. The relevant equation contains two terms — one carrying the field along with the fluid, one letting it diffuse through the material — and the ratio of the two is a dimensionless number:

$$R_m = \frac{UL}{\eta},$$

with $U$ a flow speed, $L$ a size, and $\eta$ the magnetic diffusivity. When $R_m \gg 1$ the field is frozen in; when $R_m \ll 1$ it diffuses as though the fluid were not moving.

The magnetic diffusivity of copper is about $10^{-2}$ m²/s, so a field diffuses out of a metre of copper in a few hundred seconds. That makes a laboratory experiment resistive unless it is very fast: the freezing picture applies for milliseconds and then stops.

Scale the same numbers up and the conclusion reverses violently, because the diffusion time goes as $L^2/\eta$. A conducting region a million times larger has a diffusion time $10^{12}$ times longer. That is why frozen flux is an excellent approximation for large bodies of ionised gas and a poor one for anything on a bench — not because the material is better, but because the geometry is bigger. The same size dependence appears in every diffusion problem, from momentum spreading across a fluid to heat crossing a slab, and it is the reason a dimensionless ratio is a more useful thing to know than either quantity on its own.

The field of a current loop is what any of these configurations reduces to at a distance, and the flux threading that loop is the conserved number. What decides whether the freezing holds is a comparison of two rates — how fast the fluid moves against how fast the field diffuses through it — and that ratio is the magnetic Reynolds number, which is enormous in almost every astrophysical setting and small in almost every laboratory one.

Where the field cannot simply be squeezed out #

There is a second consequence, at least as important as the compression one, and it is about topology rather than magnitude.

If flux through every material loop is conserved, then two regions of fluid threaded by different field lines cannot be exchanged, and a field configuration cannot be simplified without moving material. In the frozen limit, the topology of the field is conserved — knots stay knotted, and two oppositely directed fields pressed together cannot merge.

The static analogue is a conductor whose charges rearrange until the interior field is zero. Both cases are enforced by mobile carriers responding to whatever field appears, and both give a statement that looks like a law and is really a limit — perfect conduction in one case, infinite conductivity in the other. Where the carriers are not mobile enough, both fail in the same way and at a rate set by the same resistivity.

This is where the model’s failure becomes the interesting part. Fields pressed together in real systems do merge, in a process called reconnection, and they do it by developing a very thin layer in which the length scale is small enough that even a tiny resistivity matters: $R_m$ is a ratio containing a length, and shrink the length enough and the ratio drops below one however good the conductor is. The consequence is a system that behaves as though flux were exactly frozen almost everywhere, and violates it catastrophically in thin sheets — which is the mechanism behind the sudden release of stored magnetic energy in every plasma that does it.

Where the idea was tested on a bench #

Alfvén’s 1942 paper is short and was not immediately believed. What made the subject a laboratory one rather than a speculative one was the discovery that a conducting fluid with a field in it supports a wave nobody had predicted from the fluid equations alone: the field supplies a tension, the fluid supplies the inertia, and a transverse disturbance travels along the field lines at $B/\sqrt{\mu_0\rho}$.

That wave is a direct consequence of the frozen condition — without it there is nothing to tie the fluid to the field, and no restoring force. It was measured in liquid mercury in the late 1940s and in ionised gases shortly after, and the measured speed matched the prediction. Alfvén received a Nobel Prize for the work in 1970.

The same physics underlies every device that confines a hot ionised gas with a magnetic field. If the flux is frozen to the material, then a particle cannot cross field lines freely, and a suitably shaped field is a container with no walls. Whether it is a good container is a question about instabilities, and the answer took several decades of unpleasant surprises — but the reason the idea was worth pursuing at all is the statement on this page.

There is also a nice inversion available. Because the flux is conserved and the diffusion time goes as $L^2$, a measurement of how long a field has persisted in a conducting body puts a lower bound on the body’s size or its conductivity. The argument runs in both directions, and it is one of the few ways to learn about the inside of something from a field measured outside.

The spiral the Sun draws #

The most directly observed consequence of frozen flux is a shape, and it fills the solar system.

The Sun emits a wind: a continuous supersonic outflow of ionised gas, moving radially outward at around four hundred kilometres a second. The wind is a good conductor, so the Sun’s field is frozen into it and is carried away.

The Sun also rotates, once in about twenty-five days at its equator. So a parcel of wind leaving one point on the surface remains magnetically connected to that point while travelling radially outward, and the point moves round. The field line joining the parcel to its footpoint is therefore not radial: it is the locus of every parcel that has left that footpoint since, and successive parcels left from positions progressively further round.

The result is an Archimedean spiral, and the everyday version of the same construction is a rotating garden sprinkler, whose water travels radially while the stream traces a curve.

The angle the spiral makes with the radial direction follows from two speeds. At a distance $r$ the footpoint has moved sideways at $\Omega r$ while the parcel has moved outward at $v$, so the tangent of the angle is $\Omega r/v$. At the Earth’s orbit that ratio is very nearly one, and the field arrives at about forty-five degrees to the Sun–Earth line.

Every spacecraft that has carried a magnetometer has measured it, beginning with Mariner 2 in 1962, and found the predicted angle. Parker worked it out in 1958 in the same paper that argued the solar wind must exist at all, against considerable resistance.

The consequence that matters practically is about where particles go. Energetic particles released by a solar flare are charged, so they travel along the field rather than across it — which means they follow the spiral. A flare on the Sun’s western limb sits at the footpoint of the spiral that reaches the Earth, and its particles arrive within tens of minutes; a flare at the same distance on the eastern limb is connected to a quite different place, and its particles arrive late or not at all. Forecasting a radiation storm is therefore partly a question of geometry drawn by a frozen field.

Further out the spiral tightens, because $\Omega r/v$ grows with distance. Beyond Jupiter the field is very nearly azimuthal, and the heliosphere’s outer regions are wound like a coil.

The flux a star has to lose #

The hero figure computes what a collapse does to a field if the flux is exactly conserved, and it was described as a lower bound with an assumption attached. It is worth doing the arithmetic for a real collapse, because the answer shows just how badly the assumption fails.

A star forms from a core inside a molecular cloud: a region perhaps ten thousand astronomical units across, threaded by a measured field of a few tens of microgauss. Collapse that to the size of the Sun and the linear compression is about two million, so the field should rise by four times $10^{12}$ — from $3\times10^{-9}$ tesla to something above $10^4$ tesla.

The Sun’s surface field averages about $10^{-4}$ tesla. The prediction is wrong by eight orders of magnitude, and the missing factor is not a detail of the geometry.

Almost all of the flux must be shed during the collapse, and how is a standing problem in star formation. Two mechanisms are known to contribute and neither is obviously sufficient. The first is that a molecular cloud is only very slightly ionised — perhaps one part in $10^7$ — so the field is frozen to the ions and not to the gas, and the neutral majority drifts slowly through the ionised minority carrying no flux with it. That is ambipolar diffusion, and it is slow, which is part of why star formation is slow. The second is reconnection, which disposes of flux by changing the field’s connectivity in thin layers.

The same frozen field is, at the same time, doing something indispensable. A cloud core’s angular momentum, if conserved through the collapse, would give a star spinning far above the rate at which it would fly apart, by a similarly enormous factor. What removes it is a torque transmitted along the field lines to the surrounding cloud — magnetic braking — and that torque exists precisely because the field is frozen to both the core and its surroundings.

So the same approximation is catastrophically too effective in one respect and essential in another, in the same object, at the same time. That is a fair summary of what an idealisation of this kind is for: it says what a process cannot avoid doing, and the interesting physics is in the mechanisms that get round it.

What it costs, and where the model stops #

Perfect conductivity is never exact, and the failures concentrate. The magnetic Reynolds number can be $10^{10}$ over the bulk of a system and below one in a current sheet a metre thick inside it. Any argument that treats the flux as frozen everywhere will therefore be right about the storage of energy and wrong about its release.

Compression is assumed to be uniform and spherical. A real collapse is neither, and a field that is compressed along one axis and not another does not gain by the square of anything. The $1/R^2$ scaling is the answer for the idealised case and an order-of-magnitude statement otherwise.

Nothing here generates a field. Flux freezing conserves what is already present and can amplify it geometrically. Sustained amplification against dissipation is a dynamo, which requires flow of a particular kind and is a genuinely harder problem: it is possible to prove that certain simple flows cannot maintain a field at all.

A superconductor is a different case that is often run together with this one. A perfect conductor holds the flux it had; a superconductor expels it, ending with zero field inside whatever it started with, which is the Meissner effect and is a genuinely quantum-mechanical statement rather than a limit of Ohm’s law. The two behaviours are distinguishable by an experiment — cool first and then apply the field, or apply the field and then cool — and they disagree in the second case. Type-II superconductors do something else again, admitting flux in quantised tubes and pinning them in place, which is the mechanism behind magnetic levitation demonstrations.

The fluid is assumed to be a fluid. In a sufficiently dilute plasma the ions and electrons stop moving together, and the field is frozen to the electron fluid rather than to the material as a whole. The frozen-flux statement then acquires additional terms and stops being one line long.

What the condition is worth as an approximation #

A closing word on how to use a statement like this, since “perfect conductor” is an idealisation and every real system violates it.

The useful comparison is between the two timescales the problem contains: the time over which the configuration is changing, and the diffusion time $L^2/\eta$ over which the field would leak out if nothing moved. Frozen flux is a good approximation when the first is short compared with the second, and this is a statement about a particular process, not about a material. The same copper block is an excellent flux conserver for a microsecond pulse and a poor one for a slow squeeze.

That framing also explains why the idealisation is more useful than its exactness would suggest. Because the diffusion time carries $L^2$, moving to a system a thousand times larger buys a factor of a million in the ratio, and there is no corresponding gain available from improving the material. Conductivity varies over a few orders of magnitude across everything available; size varies over twenty. So whether flux is frozen in practice is nearly always decided by geometry, and an argument that begins by asking how big the system is will get the answer right more often than one that begins by asking what it is made of.

The ladder from here #

Later rungs on this anchor: the induction equation derived properly, with the two terms and the number that compares them; magnetic pressure and tension, which turn a field into something with mechanical properties; Alfvén waves, in which the tension supplies a restoring force and the fluid supplies the inertia; reconnection, where the frozen condition fails on purpose; and the dynamo problem, which asks what flows can sustain a field rather than merely carry one.

The neighbouring ladders are induction, whose Lenz’s law is the mechanism here taken to its limit, and conductors in electrostatics, where mobile charges enforce a different condition by the same means.