Sweet–Parker Reconnection Rate Calculator

Formula: Equations used

The Alfvén speed is computed from the upstream field and density:

Define the Lundquist number (based on L):

S = (L V_A) / η

Sweet–Parker scaling gives the normalized thickness and reconnection rate:

δ/L = S^{-1/2} and M_A = v_in / V_A = S^{-1/2}

So the current sheet thickness and inflow speed are:

δ = L / sqrt(S) = sqrt(η L / V_A)

v_in = V_A (δ/L) = V_A / sqrt(S) = sqrt(η V_A / L)

If you want an order-of-magnitude reconnection electric field (SI):

E ≈ v_in B (V/m)

How to interpret the results

• Alfvén speed V_A: the characteristic outflow speed in the Sweet–Parker picture (often v_out ≈ V_A).
• Current sheet thickness δ: the predicted resistive layer half-thickness (order-of-magnitude). Sweet–Parker requires δ ≪ L to be self-consistent.
• Inflow speed v_in: how fast magnetic flux is convected into the sheet. In Sweet–Parker, this is much smaller than V_A when S is large.
• Reconnection rate M_A: the dimensionless inflow Mach number relative to Alfvén speed. In many applications, “fast reconnection” corresponds to M_A ~ 0.01–0.1; Sweet–Parker typically predicts M_A = S^{-1/2}, which becomes extremely small for large S.
• Electric field E (if computed): a convenient measure of the potential for particle acceleration and flux transfer, since E is tied to the rate of change of magnetic flux.

Worked example

Suppose you choose (SI units):

• B = 0.01 T
• ρ = 1e-12 kg/m³
• η = 1 m²/s
• L = 1e6 m

1) Compute Alfvén speed:

V_A = B / sqrt(μ0 ρ). Using μ0 ≈ 4π×10^{-7} H/m, this gives V_A on the order of ~ 9×10^6 m/s.

2) Lundquist number:

S = L V_A / η ≈ (1e6)(9e6)/1 ≈ 9e12.

3) Reconnection rate:

M_A = S^{-1/2} ≈ 1/sqrt(9e12) ≈ 3.3e-7.

4) Inflow speed:

v_in = M_A V_A ≈ (3.3e-7)(9e6) ≈ 3 m/s.

5) Sheet thickness:

δ = L M_A ≈ (1e6)(3.3e-7) ≈ 0.33 m.

This illustrates the central Sweet–Parker message: when S is huge, the predicted inflow is extremely slow and the sheet becomes extremely thin.

Sweet–Parker vs other reconnection regimes (high-level comparison)

Assumptions & limitations

The calculator output should be treated as an order-of-magnitude Sweet–Parker estimate under the following assumptions:

• Resistive MHD applies, with a single (scalar) uniform magnetic diffusivity η.
• Steady-state, 2D geometry with a long, laminar current sheet of length L and thickness δ.
• Incompressible (or weakly compressible) flow so that simple mass continuity leads to v_in L ~ v_out δ.
• Outflow at Alfvénic speed: v_out ≈ V_A based on upstream B and ρ.
• Thin-sheet ordering: the model requires δ ≪ L. If your inputs yield δ comparable to L, Sweet–Parker is not self-consistent.
• No guide-field / 3D effects are included explicitly; turbulence, shear, line-tying, and kinetic physics can change rates dramatically.
• Parameter meaning: ensure η is magnetic diffusivity in m²/s. If you instead have electrical conductivity σ, convert via η = 1/(μ0 σ).

Inputs (with units)

• Magnetic field B (Tesla, T)
• Mass density ρ (kg/m³)
• Magnetic diffusivity η (m²/s) — note: this is not electrical resistivity (Ω·m). In SI MHD, magnetic diffusivity is related to resistivity by η = 1/(μ₀ σ).
• System size L (m) — typically the current-sheet length (or a global scale comparable to it).

Practical notes

• If your computed S is extremely large, Sweet–Parker will predict extremely small M_A; this is the classical “Sweet–Parker is too slow” result.
• For very large S, real sheets may become plasmoid-unstable, invalidating laminar Sweet–Parker scaling.
• Use this calculator as a baseline comparison when evaluating whether additional physics is required to explain an observed reconnection rate.