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Magnetic reconnection in coronal plasmas
I.J.D Craig Depar tment of Mathematics University of Waikato Hamilton New Zealand

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Why reconnection?
Reconnection is the only mechanism that can alter the magnetic field topology. It is required in two fundamental problems: Dynamo theory; How to develop the strongs fields from a weak "seed field". Coronal heating; How to heat the corona and account for explosive flare release. Here we concentrate on developing exact anlytic models for 3D coronal reconnection.

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The flare problem
In solar flares around 1030 ergs are released rapidly, in 100s or so. Magnetic reconnection--a resistive process involving the cutting and rejoining of field lines--is the accepted release mechanism. But weak coronal resistivity generally leads to energy loss rates that are too slow to account for flare observations. How can the rate of reconnection be speeded up?

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Coronal parameters
In reconnection theory we are dealing with 3D magnetic and velocity fields. Scale the problem using the typical values Bc = 102 G lc = 109.5 cm nc = 109 cm
-3

and employ AlfvИnic units . Times are measured in units of A = lc /vA where vA 109 cm s-1 is the AlfvИn speed. Energy losses have the units B2 3 lc /A = 4 в 1030 8 Modest flares require around 1027 erg/s. ergs/s

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The coronal resistivity
In these units the resistivity is 10-
14

Contrast this with the viscosity coefficient 10-3 . However multiplies the highest derivatives in the MHD equations and so cannot be neglected. This difficulty meant no exact reconnection solutions were discovered until the mid 1990's. Now 3D solutions separator (no-null) Even so, it is diffic rate 1/2 of energy are available that cover spine, fan and reconnection. ult to go beyond the slow Sweet-Parker (1958) release.

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Governing equations
These are the (MHD) momentum and induction equations t V + (V · )V = J в B - P + · S , t B = в (V в B) - в J, plus constraints · B = · V = 0. Here P is the plasma pressure, · S is the viscous force and J=вB the current density. Resistive effects require huge J gradients.
(3) (1)

(2)

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Key questions
Can analytic solutions valid for arbitrary be constructed? Are the models physically realistic? Can resistive scaling laws be deduced for the model? Can the spine and fan geometry of the null be exploited, as kinematic studies would suggest?

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Fig.1: Field skeleton

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The field skeleton
The skeleton defines the eigenstructure of B close to a null. So a current free X -point, say P = (x, y , -2z ), i = (1, 1, -2).

Positive eigenvalues corresponding to outflow (say) must be balanced by inflow along the spine (the z -axis). Now superpose a disturbance field Q onto skeleton P. Identically we must have в [ в Q) в Q] = 0. for consistency with momentum equation (1). If Q bends the spine fan currents appear; Q distorting the fan implies tubular currents along the spine. (4)

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Fig.2a: Fan current reconnection

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Fig. 2b: Spine current reconnection

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Constructing 3D reconnection solutions
For > 0, = 0 a typical construction is V = P(x) + v(x, t), B = P(x) + b(x, t),

with > 0. Note that defines "shear" (Craig & Henton 1995). The prototype fan and spine forms (Craig & Fabling 1996) ^ bS = Z (x, y , t)z, ^ ^ bF = X (x, t)x + Y (x, t)y, have reduced dimensionality due to condition (4). Cylindrical models also follow this scheme (Watson & Craig 2002, Tassi et al 2002, Pontin & Craig 2006).

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Steady fan solution
The simplest model is the two dimensional fan solution: Q = [0, Y (x), 0] P = [-x, y , 0]. The formal solution is E ^ B = P + Daw(µx)y µ E ^ Daw(µx)y V = P + µ where E (the flux transfer rate) is constant and Ї 2 - 2 = > 0. µ= 2 2 (2)

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Fan solution 2D

Figure 3: fan solution

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Resistive scalings
The Dawson function identifies xs = / as the current sheet Ї thickness. Less formally, since the disturbance field satisfies E - xY = Yx Ї we can equate outer and inner approximations, namely Yo
ut

E x Ї

and

E Yin = x

to get the same result for xs . The field in the sheet Ys E ( ) Ї .

therefore increases with for fixed E !

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Ohmic losses
This leads to Ohmic losses that diverge with W = Ys2 J 2 dV xs
-1/2

.

The problem--common to all exact analytic solutions--is that the flow can only maintain sheets with Ys . Therefore Ys has to be limited to the flow amplitude Ys . When this is done W
1/2

Ys

5/2

.

(5)

The rate differs from Sweet-Parker by the flux pile-up factor Ys 5/2 which could exceed 102 (600 Gauss fields). This result is found to hold for all fan-reconnection soluti ons.

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Summary
Exact reconnection models can be constructed in 2D and 3D. To enhance the Ohmic dissipation rate (5) can invoke a curren t limiting resistivity, ef f 106 10-8 . Then W 10-2 which equates to 4 в 1028 erg/s for a flux pile-up factor of one hundred. Other possible enhancements include: Multiple null solutions; Inclusion of Hall and viscous effects; Using 3D turbulence models. Flare-like release rates can be approached using these modifications but no model is yet accepted!

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