Classic current sheet in x,y,z coordinates
The link between magnetic energy stored in solar plasma and structures known as "current sheets" has mesmerized solar physicists for decades. Basically, a current sheet is defined as a non-propagating boundary between two plasmas with the dominant magnetic field tangential to the boundary. The tangential field components are arbitrary in magnitude and direction, subject only to the condition that the total pressure be continuous for transverse equilibrium, or in mathematical form:P1 + B 1 2 /2m
= P2 + B 2 2 /2m
Inside solar active regions the magnetic fields are so strong that for many purposes the plasma pressures (p1, p2) can be neglected. The sketch below shows a planar current sheet in 3 dimensions (x,y,z) across which the magnetic field direction rotates from B 1 to B 2 . Note that when the field B y vanishes along the current sheet we obtain a "neutral sheet."
Because of the magnetohydrodynamics involved studies of current sheets are usually confined to one dimension. One can still apply the induction equation:
¶B / ¶ t =
curl (v x B) + h Ñ 2 B
The first term on the right hand side is the advection of the field by the plasma. The second term represents the diffusion of the field lines through the plasma. The ratio of the two terms gives the magnetic Reynolds number:
Where L is a typical length scale for the region, VA is the Alfven velocity, and h is the magnetic diffusivity. For 1-dimensional motion along x, i.e. v(x,t) with magnetic field B(x,t)y^ the induction equation reduces to:
¶B / ¶ t = - ¶ / ¶ x (vB) + h ¶ 2 B/ ¶ x2
If the plasma velocity v = 0, we obtain the pure diffusion equation:
¶B / ¶ t = h ¶ 2 B/ ¶ x2
the general solution to the above may be written:
B(x,t) = ò G(x - x', t) B(x', o) dx'
Where B(x', 0) is some initial magnetic profile and the Green's function:
G(x - x', t) = 1/ (4 ph t) 3/2 exp [ (x - x')2 / 4 h t]
When the current sheet is infinitesimally thin, with piecewise constant magnetic field (B = +B o for x > 0, B = -B o for x < 0) we find:
B(x,t) =
2B o / p 1/2 ) erf (x/ ( 4 h t)1/2]
Where: erf(e ) = ò e 0 exp(-u2) du
is the error function
The preceding solution shows the magnetic field diffuses away over time at a rate:
Where the width of the sheet is of the order (h t1/2 ). When flows are present the 1D induction equation is coupled to the equation of continuity to obtain:
¶ r/ ¶ t + ¶ / ¶ x ( r v) = 0
And an equation of motion results which yields a total pressure:
Where the right side denotes the magnetic pressure at the edge of the sheet. The formation of the basic current sheet can be traced to extreme shearing of magnetic fields such that different magnetic polarities (designated + and (-)) are br0ught into close contact with each other. Such conditions obtain in highly sheared solar fields near complex sunspots, such as exemplified in the simulation image below from my Dec. 21 post:
In the above image (from a Solar Physics paper by B.C. Low) one can see the neutral line (polarity separation line) almost rebounding onto itself, which is the telltale signature for 'squeezing' a current sheet into existence. Further work has shown the simple current sheet can evolve to a configuration more likely to trigger massive solar flares, e.g.
Where the levels depicted refer to the plasma beta b:
b = r v2 m/ B2
Note also that in high b sheared current sheets
(induced by twisting) the available energy is also specified by the pressure
gradient. That is,
Ñ p = J
X B
where J is the current density and B
the induction and J x B is the Lorentz force , i.e. produced by a
current density J perpendicular to the magnetic field B or J ^ .
This is as opposed to the current density
parallel to the magnetic field, or J ||.
Clearly, where J = J ^ (or in a
region where b > 1
) the dominant form of magnetic
free energy (the type liberated in flares) will be driven by J ^. Meanwhile, in the low b domain the dominant source of magnetic free
energy must be driven by J || .
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