Sweet–Parker Reconnection Rate Calculator

Formula: Sweet–Parker equations used

The Sweet–Parker relations in this calculator start from the upstream Alfvén speed set by the magnetic field and density:

From there, the Lundquist number is built on the chosen system size:

S = (L V_A) / η

In the Sweet–Parker limit, the sheet narrows and the inflow slows as S grows:

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

So the current sheet thickness and inflow speed become:

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 a back-of-the-envelope reconnection electric field in SI units, the calculator uses:

E ≈ v_in B (V/m)

How to interpret Sweet–Parker results

The Sweet–Parker outputs are best read as a consistency check on how thin and how slow a resistive current sheet becomes for your inputs.

• Alfvén speed V_A: the upstream outflow benchmark. In the Sweet–Parker picture, plasma leaves the sheet at roughly this speed, so it sets the natural scale for comparing reconnection cases.
• Current sheet thickness δ: the predicted half-thickness of the resistive layer. If δ is not much smaller than L, the geometry is no longer the long, thin Sweet–Parker sheet assumed by the model.
• Inflow speed v_in: how fast magnetic flux is driven into the layer. For large S, the Sweet–Parker inflow is far below V_A, which is why the model is often described as slow.
• Reconnection rate M_A: the dimensionless inflow Mach number. Very small values are normal for laminar Sweet–Parker reconnection; much larger rates usually point to extra physics such as plasmoids, Hall effects, or turbulence.
• Electric field E (if computed): a rough SI proxy for flux-transfer strength. It is useful for comparing scenarios, but it remains a scaling estimate rather than a full kinetic prediction.

Worked Sweet–Parker example

To see the Sweet–Parker scaling in action, try the following SI inputs and watch how quickly the sheet becomes thin when the Lundquist number is large:

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

1) Alfvén speed:

V_A = B / sqrt(μ0 ρ). Using μ0 ≈ 4π×10^{-7} H/m, the inputs give V_A of about 8.9×10^6 m/s.

2) Lundquist number:

S = L V_A / η ≈ 8.9×10^12, which is enormous for a laminar resistive sheet.

3) Reconnection rate:

M_A = S^{-1/2} ≈ 3.4×10^{-7}.

4) Inflow speed:

v_in = M_A V_A ≈ 3.0 m/s.

5) Sheet thickness:

δ = L / sqrt(S) ≈ 0.33 m.

6) Electric field:

E ≈ v_in B ≈ 0.03 V/m.

The takeaway is simple: even with a strong magnetic field, a large Sweet–Parker system and a small diffusivity can still produce an extremely slow inflow and an extremely thin current sheet.

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

This comparison keeps the calculator in context by showing where the Sweet–Parker baseline sits relative to faster reconnection ideas.

Assumptions & limitations for Sweet–Parker reconnection

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

• Resistive MHD applies, with a single, 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 the upstream B and ρ.
• Thin-sheet ordering: the model requires δ ≪ L. If your inputs yield δ comparable to L, the Sweet–Parker assumptions are 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 for Sweet–Parker reconnection (with units)

Use one consistent SI set so the Sweet–Parker formulas stay dimensionally correct and the result can be compared across scenarios.

• 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 Sweet–Parker notes

These quick checks help you decide whether the laminar Sweet–Parker picture is a useful baseline or a sign that more physics is needed.

• If your computed S is extremely large, Sweet–Parker will predict an extremely small M_A; this is the classic "Sweet–Parker is too slow" result.
• For very large S, real sheets may become plasmoid-unstable, which breaks the simple laminar scaling the calculator is using.
• Use this calculator as a reference point when deciding whether a reconnection observation looks ordinary for resistive MHD or whether additional effects are likely required.