Question 4.3.2: Derive magnetic energy of the two magnetostatic equilibria o......

The magnetic energy of the system under consideration is equal (per unit area in the (y, z) plane) to W_{M} = (8π)^{−1} \int^{l}_{−l} < B^{2} > dx, where the symbols < > mean averaging over field variation along the y-coordinate. Since, for a weak perturbation, the change of magnetic energy of each of the two equilibria, ψ^{(i)}_{1} and ψ^{(r)}_{1} , is of the order of (a/l)^{2}; their straightforward derivations would require knowledge of the second order corrections to the deformed magnetic field. However, these somewhat cumbersome calculations can be bypassed by exploring the following energy balance consideration. Indeed, the sought after magnetic energies, W^{(i,r)}_{M} , are equal to W^{(i,r)}_{M} = W^{(0)}_{M} + \Delta W^{(i,r)}, where W^{(0)}_{M} = B^{2}_{0}2l/8π is magnetic energy of the initial field (4.38), and \Delta W^{(i,r)} is the work of an external force which provides the boundary deformation (4.39).

\vec{B}^{(0)} = [0,B_{0} \sin(αx), B_{0} \cos(αx)]        (4.38)

x^{(+)}_{b} = l + a \cos(ky),  a ≪ l,        (4.39)

Thus, assume that the latter is formed by a gradual quasistatic increase of the boundary perturbation amplitude, δ(t), from δ = 0 at t = 0 to δ = a at t → ∞. Then, in the course of the boundary deformation the external force is balanced by the internal magnetic pressure; hence, the work under consideration is equal to

\Delta W = − \int_{0}^{∞} < \frac{B^{2}(x^{(+)}_{b})}{8π} \frac{dδ}{dt} \cos(ky) > dt      (4.44)

Under the above quasistatic deformation the magnetic field remains forcefree at any instance of time. Therefore, the magnetic pressure in (4.44) can be derived by using the respective perturbed equilibria ψ^{(i,r)}. Moreover, since the integrand in (4.44) already contains the factor dδ/dt, in doing so it is sufficient to use the flux perturbation ψ^{(i,r)}_{1} obtained in the linear approximation.

Thus, the procedure is as follows. The deformed magnetic field is \vec{B} = \vec{B}^{(0)} + \vec{b}, where the field perturbation \vec{b} is related to the flux function perturbation ψ_{1} as

b_{x} = \frac{∂δψ}{∂y} = −kψ_{1}(x) \sin(ky),

b_{y} = −\frac{∂δψ}{∂x} = −ψ^{′}_{1}(x) \cos(ky),

b_{z} = αδψ = αψ_{1}(x) \cos(ky)

Then, the required magnetic pressure at the deformed boundary takes the form

\frac{B2(x^{(+)}_{b}}{8π} = \frac{B^{2}_{0}}{8π} + \frac{B_{0}}{4π} [αψ_{1}(l) \cos(αl) − ψ^{′}_{1} (l) \sin(αl)] \cos(ky)        (4.45)

Obviously, only the last term on the right-hand side of (4.45) makes a non zero contribution in (4.44). By using expressions (4.42) and (4.43) for ψ^{(r,i)}_{1} (with a now replaced by δ(t)), equations (4.44) and (4.45) yield for \Delta W^{(i,r)}:

ψ_1^{(r)}(x)=\frac{aB_0\ sin(αl)}{sin(2κl)}sin[κ(x+l)]             (4.42)

ψ_1^{(i)}(x)=\frac{aB_0\ sin(αl)}{sin(κl)}sin(κx)             (4.43)

\Delta W^{(i)} = \frac{B^{2}_{0} \sin^{2}(αl)}{16πl} a^{2}[(κl) cot(κl) − (αl) cot(αl)]

\Delta W^{(r)} = \frac{B^{2}_{0} \sin^{2}(αl)}{16πl} a^{2}[(κl) cot(2κl) − (αl) cot(αl)]      (4.46)

It follows from the above expressions that the reconnected magnetic configuration always has a lower magnetic energy than its ideal MHD counterpart:

\Delta W_{M} = \Delta W^{(i)}− \Delta W^{(r)} = \frac{B^{2}_{0} \sin^{2}(αl)}{16πl} a^{2}(κl)[cot(κl)−cot(2κl)] > 0,       (4.47)

which is, therefore, released in the process of magnetic reconnection. Moreover, the released energy, \Delta W_{M}, can greatly exceed the amount of energy, \Delta W^{(i)}, that needs to be supplied to the system by an external force in order to form the ideal MHD equilibrium with the current sheet (note that, according to (4.46), the energy difference \Delta W^{(i)} is always positive). Indeed, as seen from (4.47), the released energy formally tends to infinity when the parameter (κl) approaches π/2, and the system becomes MHD unstable in respect to reconnective tearing mode (see Problem 4.3.5). This means that the released energy \Delta W_{M} is not supplied externally but is tapped from the excess magnetic energy stored in the initial magnetic configuration (4.38). Therefore, even a weak external perturbation can trigger a substantial internal reconnective magnetic relaxation in a highly conducting fluid. An overall scenario of this process, called forced magnetic reconnection, occurs as follows (Figure 4.9). A system undergoes external perturbation, during which (t ∼ \Delta t) a small plasma resistivity still plays no role. Therefore, an ideal MHD equilibrium with a current sheet and slightly increased magnetic energy W_{0}+ \Delta W^{(i)} is formed. Then, on a much longer time scale τ_{r} (see next Problem), resistive effects intervene, destroying the current sheet and causing transition to an equilibrium with different magnetic topology and lower magnetic energy.