Skip to content

Sweet–Parker Model

Source lecture(s): pc368_lec17_sweetparker

Intuition

In the Sweet–Parker picture, reconnection takes place in a long, thin current sheet. Plasma from far upstream is squeezed into the sheet, diffuses across resistively, and shoots out the other side. The sheet length \(L\) is macroscopic, but its thickness \(\delta\) is microscopic (\(\delta \ll L\)). The result is a very slow reconnection rate unless the Lundquist number is modest.

Formal Definition

The Sweet–Parker model gives the steady-state reconnection rate in a long, thin resistive current sheet:

\[V_{\text{rec}} = V_{\text{in}} \sim \eta \frac{L}{\delta} \sim \frac{v_A}{\sqrt{S}}\]

where \(S = \mu_0 v_A L / \eta\) is the Lundquist number.

Mathematical Formulation

From the four sheets (inflow, transition, outflow, current):

  1. Mass conservation: \(U_{\text{in}} L = U_{\text{out}} \delta\)
  2. Transition balance: \(U_{\text{in}} B_{\text{in}} \sim \eta B_{\text{in}} / (\mu_0 \delta)\)
  3. Induction in sheet: \(\partial \psi / \partial t + U\cdot\nabla\psi = \eta/\mu_0 \nabla^2\psi\)

Combining (1) and (2):

\[\frac{U_{\text{out}}}{U_{\text{in}}} = \frac{L}{\delta}, \qquad U_{\text{in}} \sim \frac{\eta}{\mu_0 \delta}\]

With \(v_A = B_{\text{in}}/\sqrt{\mu_0 \rho}\), the reconnection rate is:

\[M_A = \frac{U_{\text{in}}}{v_A} \sim \frac{1}{\sqrt{S}}, \qquad S = \frac{\mu_0 v_A L}{\eta}\]

Derivation

  1. Inflow region: ideal Ohm’s law \(\mathbf{E} + \mathbf{U}_{\text{in}}\times\mathbf{B} = 0\). At the separatrices, \(\mathbf{E}_{\text{rec}} = -U_{\text{in}} B_{\text{in}}\).
  2. Inside sheet: resistive Ohm’s law \(\eta \mathbf{J} = \mathbf{E} + \mathbf{U}_{\text{out}}\times\mathbf{B}\). With \(\mathbf{U}_{\text{out}} \approx 0\) in the sheet, \(E_{\parallel} = \eta J_{\parallel}\).
  3. Transition layer: match \(|\mathbf{U}\times\mathbf{B}| \sim |\eta\mathbf{J}|\).
  4. Using Ampère’s law \(J \sim B_{\text{in}}/(\mu_0\delta)\) gives sheet thickness.
  5. Mass conservation ties \(U_{\text{in}}\) and \(U_{\text{out}}\).
  6. Eliminate \(\delta\) to express rate purely in terms of \(S\).

Worked Example

Solar corona reconnection: For \(L = 10^7\) m, \(v_A = 10^6\) m/s, \(\eta = 1\) m\(^2\)/s, compute \(S\) and \(U_{\text{in}}\). \(S = \mu_0 v_A L / \eta \sim 10^{13}\). Then \(M_A = S^{-1/2} \sim 10^{-6.5}\). \(U_{\text{in}} \sim 10\) cm/s. This is far slower than observed reconnection rates, motivating Petschek fast-mode or collisionless diffusion models.

Common Mistakes

  • Sweet–Parker is always slow. It is only slow when \(S \gg 1\); for \(S \sim 10^2\) the rate is already substantial.
  • Applying it to collisionless plasmas. The original model assumes resistive

    MHD; kinetic effects are needed for magnetospheric reconnection.

  • Assuming steady state. Unsteady plasmoid instability tears the sheet into

    many small islands, enhancing the rate.

Quiz Questions

  1. Conceptual: Why does a longer current sheet reconnect more slowly in the Sweet–Parker regime?
  2. Computational: Show that plasmoid instability grows when \(S > S_c \sim 10^4\).
  3. MCQ: The Lundquist number compares:
  4. A) Inertia to pressure
  5. B) Advection to diffusion
  6. C) Hall term to resistivity
  7. D) Gravity to Lorentz force

Further Reading

  • P. A. Sweet, * Nuggets of Plasma Astrophysics*.