Engineering & Science
Sweet-Parker Magnetic Reconnection Rate, Lundquist Number (S) & Sheet Thickness
Calculate the dimensionless MHD Lundquist number S = (L * vA) / eta_m, Sweet-Parker reconnection inflow speed v_in = vA / SQRT(S), and resistive current-sheet half-thickness delta = L / SQRT(S).
Understand solar flares, magnetotail substorms, and the fast-reconnection puzzle!
When oppositely directed magnetic field lines are pressed together across a current sheet of half-length L in a solar flare, Earth's magnetotail, or a tokamak plasma, magnetic tension snaps and reconfigures topology—hurling plasma outward at the Alfvén speed vA! Peter Sweet (1958) and Eugene Parker (1957) combined MHD mass conservation, magnetic induction, and energy balance to relate reconnection speed to the dimensionless Lundquist number S = (L * vA) / eta_m, where eta_m = 1 / (mu0 * sigma) is magnetic diffusivity (m^2/s)!
Sweet-Parker theory predicts dimensionless reconnection Mach rate M_SP = v_in / vA = 1 / SQRT(S) and current-sheet thickness delta = L / SQRT(S)! In a solar coronal loop (L = 1.0e7 m, vA = 1.0e6 m/s, turbulent/anomalous diffusivity eta_m = 10 m^2/s -> S = 10^12), reconnection inflow proceeds at v_in = 1.00 m/s across a razor-thin 10.0 m current sheet—revealing why classical Spitzer resistivity (S ~ 10^14) is slow until plasmoid / Hall reconnection kicks in!
| MHD Current Sheet (L = 10^7 m, vA = 10^6 m/s) | Magnetic Diffusivity eta_m | Lundquist Number S | Inflow v_in & Sheet Width delta |
|---|---|---|---|
| High-Conductivity Coronal Sheet | eta_m = 10 m^2/s | S = 1.0e12 | v_in = 1.00 m/s, delta = 10.0 m |
| Anomalous Micro-Turbulent Sheet | eta_m = 1,000 m^2/s | S = 1.0e10 | v_in = 10.00 m/s, delta = 100.0 m |
| Invalid Diffusivity eta_m <= 0 | Division by Zero Guard | Guarded Input Error | "Invalid MHD Sheet Inputs" |
Dimensionless MHD Lundquist Number S = (L * vA) / eta_m
Ratio of resistive magnetic diffusion time L^2 / eta_m to Alfvén transit time L / vA.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Invalid MHD Sheet Inputs", ROUND((B2 * B3) / B4, 0))Sweet-Parker Reconnection Inflow Speed v_in = vA / SQRT(S) (m/s)
Computes resistive steady-state inflow velocity vA / SQRT((L * vA) / eta_m).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Invalid MHD Sheet Inputs", ROUND(B3 / SQRT((B2 * B3) / B4), 2))Sweet-Parker Current-Sheet Half-Thickness delta = L / SQRT(S) (Meters)
Calculates the resistive dissipation layer thickness across which magnetic field reverses.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0), "Invalid MHD Sheet Inputs", ROUND(B2 / SQRT((B2 * B3) / B4), 2))Try the example
Use the original values or change them to test your understanding.