Current sheet · Alfvén speed · reconnection rate scaling with Lundquist number
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Alfvén speed Vₐ
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Lundquist # S
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Sheet half-width δ
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Inflow speed Vᵢₙ
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Outflow speed Vₒᵤₜ
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Reconnection rate M
Sweet-Parker current sheet — When oppositely directed magnetic field lines are
pushed together, they reconnect in a thin diffusion layer. The Sweet-Parker model (1958-63)
gives an analytical solution in resistive MHD.
Alfvén speed: Vₐ = B₀/√(μ₀ρ) — maximum signal speed in magnetized plasma Lundquist number: S = VₐL/η — ratio of resistive to Alfvén timescale Sheet width: δ = L/√S — set by balance of inflow and diffusion Inflow speed: Vᵢₙ = Vₐ/√S — much slower than Alfvén (bottleneck) Outflow speed: Vₒᵤₜ ≈ Vₐ — Alfvénic jets emerge from endpoints Reconnection rate: M = Vᵢₙ/Vₐ = S^{−1/2} — astrophysically too slow!
This slowness (S ~ 10¹² in solar corona → M ~ 10⁻⁶) motivates Petschek reconnection
(fast, M ~ 1/log S) and kinetic/collisionless models. Solar flares require M ~ 0.01−0.1.