Geometry, Not Conductivity: How a Solar Flare Beats the Million-Year Diffusion Limit — Epoche C1
A solar flare converts stored magnetic energy into heat, bulk motion and fast particles in a few hundred seconds. The store is real and easy to size: the energy density of a magnetic field of strength $B$ is $B^{2}/8\pi$ in Gaussian units, so a coronal field of $100\ \mathrm{G}$ holds about $400\ \mathrm{erg\,cm^{-3}}$, and a flaring loop $10^{9}\ \mathrm{cm}$ on a side holds some $4\times10^{29}\ \mathrm{erg}$; the largest flares, with stronger fields over larger volumes, reach $10^{32}\ \mathrm{erg}$. The difficulty is not where the energy comes from but how it gets out. The solar corona — the Sun's outer atmosphere, a hydrogen plasma at roughly a million kelvin — is a superb electrical conductor, and in such a conductor the magnetic field is frozen in: Alfvén showed in 1942 that when resistance is negligible the field lines move with the gas and cannot break or exchange partners. Surface motions load the field with energy that the frozen-in condition should never let go quickly. The resolution developed below is geometric rather than material. The corona never becomes a worse conductor; it builds a structure thin enough that a tiny resistivity finally matters. Where flux freezing comes from, and where it can fail The governing equation follows from three standard ones, and assembling it shows precisely which term carries the freezing and which can break it. Faraday's law in Gaussian units is $\partial\mathbf{B}/\partial t = -c\,\nabla\times\mathbf{E}$. Ohm's law for a conductor moving at velocity $\mathbf{v}$ relates the current density to the field the moving matter actually feels: $\mathbf{J} = \sigma(\mathbf{E} + \mathbf{v}\times\mathbf{B}/c)$, with $\sigma$ the electrical conductivity. Ampère's law without the displacement current — negligible for motions far slower than light — gives $\mathbf{J} = c\,\nabla\times\mathbf{B}/4\pi$. Eliminating $\mathbf{E}$ and $\mathbf{J}$ between the three, and using $\nabla\cdot\mathbf{B} = 0$ to turn $\nabla\times(\nabla\times\mathbf{B})$ into $-\nabla^{2}\mathbf{B}$, produces the induction equation: $$\frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v} \times \mathbf{B}) + \eta\,\nabla^{2}\mathbf{B}, \qquad \eta \equiv \frac{c^{2}}{4\pi\sigma},$$ where $\eta$ is the magnetic diffusivity, the rate at which field slips through the plasma. The first term carries the field with the flow — set $\eta = 0$ and the magnetic flux through any surface moving with the fluid is conserved, which is Alfvén's theorem and the reason field lines can be spoken of as material objects. The second term is the only one that lets field lines break and rejoin. Balancing the time derivative against it over a region of size $L$ gives the diffusion time $\tau_{\eta} = L^{2}/\eta$, and the factor of $L^{2}$ is where the whole argument will eventually turn. The number that makes it a paradox The conductivity of a fully ionised plasma is set by Coulomb collisions between electrons and ions, and Spitzer computed it in a form still used. Its temperature dependence has a short physical explanation: the Coulomb scattering cross-section falls as the fourth power of relative speed, so the collision frequency goes as $n v^{-3}$, and since $v \propto T^{1/2}$ the resistivity — and with it the diffusivity — falls as $T^{-3/2}$. Hot plasmas are excellent conductors. At $T = 10^{6}\ \mathrm{K}$, with the Coulomb logarithm near $20$, this gives $\eta \approx 10^{4}\ \mathrm{cm^{2}\,s^{-1}}$. Now insert the scale of a flaring loop, $L \approx 10^{9}\ \mathrm{cm}$ — ten thousand kilometres. Then $$\tau_{\eta} = \frac{L^{2}}{\eta} = \frac{(10^{9})^{2}}{10^{4}} = 10^{14}\ \mathrm{s},$$ about three million years, since a year is $3.2\times10^{7}\ \mathrm{s}$. A flare takes minutes to tens of minutes; call it $10^{3}\ \mathrm{s}$. Resistive diffusion is too slow by eleven orders of magnitude. Hence the common belief that a plasma this highly conducting should be unable to release its magnetic energy quickly — and hence the long resistance to the idea that flares are magnetic at all. That they are was settled observationally rather than theoretically: Masuda and colleagues reported in 1994, from the Yohkoh satellite's hard X-ray telescope, a compact source of hard X-rays sitting above the top of the bright flare loop, where the energy release must therefore be occurring, in the geometry a reconnecting field would produce. The geometric escape The escape hides in the $L^{2}$. Conductivity is a fixed material property of a plasma at a given temperature, but the length scale over which the field reverses is not fixed by anything: it is whatever the dynamics makes it. Where oppositely directed field lines are pressed together, the entire reversal concentrates into a thin, intensely current-carrying layer — a current sheet . Across a sheet of thickness $\delta$ the relevant diffusion time is $\delta^{2}/\eta$, which can be short even while $L^{2}/\eta$ is astronomical. Inside the sheet, and only there, field lines break and rejoin — they reconnect — while remaining frozen in everywhere else. Such layers are not exotic; the corona is driven to make them. Coronal field lines are anchored at both ends in the photosphere, where convection shuffles the footpoints on timescales of hours — four orders of magnitude longer than the five-second Alfvén crossing time computed below. The field therefore has ample time to relax towards force balance after each nudge, and passes through a sequence of near-equilibria while its topology is progressively braided. Since the frozen-in condition forbids the braid from untying itself, regions of oppositely directed field are pushed together with nothing to stop them, and the reversal between them is squeezed into ever shorter distances. Parker's argument in 1957 was that this squeezing does not stop at some comfortable scale: it proceeds until diffusion becomes competitive, which is a statement about $\delta$, not about $\eta$. Sweet and Parker turned thi