Magnetic reconnection - 2D X-point and current sheet

Sweet-Parker layer

The Alfvén speed is v_A = B / √(μ₀ n m_i). Plasma flows into a sheet of length L and thickness δ, and mass conservation (v_in L = v_out δ) plus energy balance (½ρ v_out² = B²/2μ₀) give v_out = v_A, δ/L = S^-1/2 and v_in/v_A = S^-1/2, where the Lundquist number S = L v_A / η_m and η_m = η/μ₀ with Spitzer η ≈ 7.8×10^-4 T_eV^-3/2 Ωm. Half the incoming Poynting flux becomes jet kinetic energy, half becomes heat.

Why flares are faster

For astrophysical S > 10⁴ the thin sheet breaks into plasmoids and the rate saturates near 0.01 v_A, independent of S. For the solar-flare preset this turns a Sweet-Parker time of years into a few hundred seconds.

Honest simplifications: the field-line picture is a schematic hyperbolic X-point, not a solved MHD flow; the tracer flow is a divergence-free model flow normalised to the Sweet-Parker numbers; Spitzer resistivity uses a fixed Coulomb logarithm of 15; the plasmoid rate 0.01 is an empirical value and collisionless (Hall) reconnection is not modelled. Numbers are order-of-magnitude.

• Press Laboratory experiment (S in the hundreds): a slow, thick Sweet-Parker sheet.
• Press Solar flare: S is enormous, the sheet is far thinner than a pixel, and only the plasmoid rate explains a flare.
• Lower the temperature to raise the resistivity and watch the sheet thicken.