---
title: "How a thin current sheet sets the pace"
description: "Derive the Sweet–Parker estimate from mass flow and electric-field balance. Keep sheet length, viscosity and instability growth distinct."
canonical_url: "https://sandboxphysics.com/stories/solar-magnetic-reconnection/how-a-sheet-sets-the-pace/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/solar-magnetic-reconnection/how-a-sheet-sets-the-pace/"
last_modified: "2026-11-03"
---

[From magnetic stress to reconnection](https://sandboxphysics.com/stories/solar-magnetic-reconnection/)

L2 · Magnetic reconnection and solar-flare signals

# How a thin current sheet sets the pace

Sandbox Physics · 2026-11-03

A steady thin current sheet must admit the same mass that its ends expel. Combining that geometric requirement with resistive diffusion gives the Sweet–Parker estimate. Its assumptions tell us when a thinner sheet implies slower inflow and when a different instability question begins.

## Before you begin

Use the moving-flux and diffusivity definitions from When magnetic flux follows the fluid. You need rectangular areas, mass flow and square roots; the following argument is a scaling estimate.

## Draw the rectangle and name both half-sizes

Let $x$ run along the outflow and $y$ across the sheet. Its half-length is $\ell$ and half-thickness is $\delta$. Magnetic fields of magnitude $B_u$ approach from opposite sides, pointing in opposite directions along the sheet. Use positive speeds $v_{\rm in}$ and $v_{\rm out}$ for inflow and outflow.

Assume a stationary, nearly rectangular layer, equal constant upstream and downstream density, and no accumulating mass. Per unit depth, the two long faces admit mass; the two short ends expel it. The factors of two from the faces and full side lengths cancel.

A narrow exit then requires a faster outflow than inflow. This is a geometric statement about a steady layer. It has not yet told us how either speed depends on resistance.

$$
4\rho\ell v_{\rm in}=4\rho\delta v_{\rm out}\quad\Longrightarrow\quad v_{\rm in}\ell=v_{\rm out}\delta
$$

The same density and invariant depth occur on both sides. Compression would change this relation.

Sources: [Loureiro, Schekochihin & Cowley (2007), astro-ph/0703631v1, equations 3 and 18–20, Figure 4](https://arxiv.org/abs/astro-ph/0703631v1)

## Match electric field outside and inside the layer

In a steady two-dimensional magnetic configuration, the out-of-plane electric field has the same value across the layer. Upstream, ideal advection gives its magnitude as inflow speed times field strength. Near the center, resistive current supplies it.

The current-density magnitude is of order $B_u/(\mu_0\delta)$. Multiplying by electrical resistivity, then using $\eta_{\rm m}=\eta_{\rm e}/\mu_0$, makes the field strength cancel. The resulting relation says that inflow across the sheet must balance magnetic diffusion through it.

Use magnitudes in this argument: the sign of the actual electric field depends on the chosen magnetic and coordinate directions. No signed current was replaced by a positive heating rate.

$$
|E_z|\sim v_{\rm in}B_u\sim\eta_{\rm e}|J_z|\sim\eta_{\rm m}\frac{B_u}{\delta}
$$

This is a steady-layer scaling with uniform scalar magnetic diffusivity.

$$
v_{\rm in}\sim\frac{\eta_{\rm m}}{\delta}
$$

A thinner diffusion region can balance faster inflow if the other conditions are maintained.

Sources: [Loureiro, Schekochihin & Cowley (2007), astro-ph/0703631v1, equations 3 and 18–20, Figure 4](https://arxiv.org/abs/astro-ph/0703631v1) · [Zhou et al. (2019), Magnetic island merger as a mechanism for inverse magnetic energy transfer, Physical Review Research 1, 012004(R)](https://doi.org/10.1103/PhysRevResearch.1.012004)

## Magnetic tension supplies the outflow scale

The relaxing field pulls plasma along the layer. Comparing upstream magnetic energy per volume with the bulk kinetic-energy scale gives an outflow speed of order the Alfvén speed. Pressure, geometry and dissipative partition determine the numerical factor; this argument does not set it exactly.

Insert that speed into mass continuity, and combine with the diffusion relation. The sheet thickness and inflow rate then follow together. Define the Lundquist number using the half-length we chose, so every factor has a known convention.

$$
v_{\rm out}\sim v_A=\frac{B_u}{\sqrt{\mu_0\rho}},\qquad S_\ell=\frac{\ell v_A}{\eta_{\rm m}}
$$

The Lundquist number compares diffusion and Alfvén travel times over the same half-length.

$$
\frac{\delta}{\ell}\sim S_\ell^{-1/2},\qquad\frac{v_{\rm in}}{v_A}\sim S_\ell^{-1/2}
$$

These Sweet–Parker scalings assume a steady, thin, weakly viscous layer. Geometry and dissipation determine their prefactors.

Sources: [Loureiro, Schekochihin & Cowley (2007), astro-ph/0703631v1, equations 3 and 18–20, Figure 4](https://arxiv.org/abs/astro-ph/0703631v1)

## Work an estimate and test its assumptions

Suppose the hypothetical layer has $S_\ell=10^4$. Its half-thickness is about one hundredth of its half-length, and inflow is about one hundredth of the Alfvén speed. The inferred thinness agrees with the thin-layer assumption.

Increasing this Lundquist number by a factor of one hundred lowers both ratios by a factor of ten. Taking it down to order one would make the layer as thick as it is long. The thin-sheet derivation has then removed its own justification.

Viscosity introduces another ratio, the magnetic Prandtl number $P_{\rm m}=\nu/\eta_{\rm m}$, where $\nu$ is kinematic viscosity. Strong viscosity can reduce the outflow. The interactive viscous case changes the solved fluid motion and heat, so changing its speed is a physical result. It is not a test of the low-viscosity steady prefactor.

| Specified estimate | Thickness ratio | Inflow ratio |
| --- | --- | --- |
| $S_\ell=10^4$ | $\delta/\ell\sim0.01$ | $v_{\rm in}/v_A\sim0.01$ |
| $S_\ell=10^6$ | $\delta/\ell\sim0.001$ | $v_{\rm in}/v_A\sim0.001$ |

Sources: [Loureiro, Schekochihin & Cowley (2007), astro-ph/0703631v1, equations 3 and 18–20, Figure 4](https://arxiv.org/abs/astro-ph/0703631v1) · [Zhou et al. (2019), Magnetic island merger as a mechanism for inverse magnetic energy transfer, Physical Review Research 1, 012004(R)](https://doi.org/10.1103/PhysRevResearch.1.012004)

## A sheet’s instability is a new question

The steady estimate becomes thinner as diffusivity falls. Such a thin equilibrium may itself be unstable. Asking how fast a perturbation grows is different from asking for the steady inflow speed. These two questions can give different trends without contradicting one another.

Loureiro and colleagues use the full sheet length $L_{\rm CS}=2\ell$. Their definitions imply $S_{\rm full}=2S_\ell$ and the thickness parameter below. Keeping those conventions prevents a missing square-root-of-two factor when reading the paper.

The advanced guide solves that paper’s linear dispersion relation. A linear growth curve identifies small-perturbation behavior around the assumed layer. A nonlinear chain of merging magnetic islands requires a resolved evolving calculation. The periodic initial-stress experiment supplies a transient energy-conversion case, with its own grid checks and endpoints.

$$
\epsilon=\frac{2\delta}{L_{\rm CS}}\sim\sqrt{\frac{2}{S_{\rm full}}},\qquad\Gamma_0=\frac{2v_A}{L_{\rm CS}}
$$

These are the full-length and flow-rate conventions used by the fixed 2007v1 reading and reference script.

Sources: [Loureiro, Schekochihin & Cowley (2007), astro-ph/0703631v1, equations 3 and 18–20, Figure 4](https://arxiv.org/abs/astro-ph/0703631v1) · [Zhou et al. (2019), Magnetic island merger as a mechanism for inverse magnetic energy transfer, Physical Review Research 1, 012004(R)](https://doi.org/10.1103/PhysRevResearch.1.012004)

## Work through the question

For fixed half-length and upstream Alfvén speed, increase magnetic diffusivity by a factor of nine. What happens in the Sweet–Parker estimate?

The Lundquist number falls by nine. Both $\delta/\ell$ and $v_{\rm in}/v_A$ rise by three, provided the layer remains thin and the other steady assumptions hold.

Why can a larger Lundquist number mean slower steady inflow but faster instability growth?

The inflow estimate follows mass flow and diffusion in a steady sheet. A linear instability asks how perturbations grow around that sheet, whose thickness also changes with Lundquist number. They are different quantities and different calculations.

May a transient periodic snapshot be used to measure a universal Sweet–Parker coefficient?

A transient snapshot supplies a time-dependent comparison. Inferring a steady coefficient additionally requires a demonstrated rectangular layer, fixed upstream conditions and the intended weak-viscosity outflow.

### [Compare diffusion with viscous damping](https://sandboxphysics.com/labs/magnetic-reconnection/)

1. Predict whether increasing magnetic diffusion and increasing viscosity should modify the same equation.
2. Compare their ready cases at the middle saved state. Move the probe across the current concentration.
3. Use the energy curves to distinguish magnetic smoothing from damping of fluid motion.

## Primary sources

1. [Zhou et al. (2019), Magnetic island merger as a mechanism for inverse magnetic energy transfer, Physical Review Research 1, 012004(R)](https://doi.org/10.1103/PhysRevResearch.1.012004)
2. [Loureiro, Schekochihin & Cowley (2007), astro-ph/0703631v1, equations 3 and 18–20, Figure 4](https://arxiv.org/abs/astro-ph/0703631v1)

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