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Reconnection Rate Lab

Learning goal

See, in one number, why Sweet–Parker is simultaneously the most important and the most wrong model in reconnection theory. It is dimensionally inevitable, experimentally confirmed in the laboratory — and off by five orders of magnitude for a solar flare.

Things to try

  1. Solar flare (the default). \(S = 10^{13}\), so Sweet–Parker predicts \(\tau = \tau_A\sqrt{S} \approx\) one year. Flares finish in about five minutes. This factor of \(10^5\) is the reconnection problem, and it is why the field spent five decades looking for something faster.

  2. Slide S down to the laboratory value (~\(10^{3.5}\), the MRX preset). The verdict flips: Sweet–Parker now lands within a factor of a few of observation. This is not a coincidence — collisional laboratory current sheets really do reconnect at the Sweet–Parker rate. The theory is correct; it is the extrapolation to astrophysical \(S\) that fails.

  3. Watch the aspect ratio. \(\delta/L = S^{-1/2}\). At \(S = 10^{13}\) the sheet is \(3\times10^{-7}\) as thick as it is long — for a solar flare, a layer a few metres thick and 10 000 km long. The physical picture of why it is slow follows immediately: everything that comes in must squeeze out through a slot of that shape.

  4. Compare the two curves' slopes. Sweet–Parker falls as a power law, Petschek only logarithmically, because Petschek's diffusion region is short and the outflow opens into a wedge behind standing slow shocks. Petschek lands inside the observed band at every astrophysical \(S\) — its problem is not the rate but whether the required configuration survives in a uniform resistivity, which resistive-MHD simulations say it does not.

  5. Think about what is missing. Both curves assume a steady, laminar sheet. Above \(S \approx 10^4\) that sheet is itself unstable and breaks into plasmoids, and the resulting rate becomes nearly independent of \(S\) at \(M_A \approx 10^{-2}\) — inside the green band, without needing Petschek's geometry. Simulating that is a resistive-MHD problem, not a pencil-and-paper one.

The scaling in one line

\[\frac{\delta}{L} = \frac{v_{\rm in}}{v_A} = \frac{1}{\sqrt{S}}, \qquad S = \frac{\mu_0 L v_A}{\eta}, \qquad \tau_{\rm SP} = \tau_A\sqrt{S} = \sqrt{\tau_A \tau_\eta}\]

The last form is the memorable one: the Sweet–Parker time is the geometric mean of the Alfvén crossing time and the resistive diffusion time. Reconnection is a compromise between the fast process and the slow one, and a geometric mean sits inconveniently far from the fast end.

Sweet–Parker model · Magnetic reconnection · Magnetic islands · Frozen-in theorem · Lundquist number (eq.) · Sweet–Parker rate (eq.)