Question 4.3.7: Derive the singular ideal MHD equilibrium, which is formed u......

The respective solution containing the current sheet (as shown in Figure 4.13(d)) can be constructed in the following way. Any potential planar magnetic field [B_{x}(x, y),B_{y}(x, y)] can be represented as B_{x} − iB_{y} = f(z) = u(x, y) + iv(x, y), where f(z) is an analytical function of the complex variable z = x + iy. Indeed, the divergence-free and curl-free requirements for \vec{B} , namely \frac{∂B_{x}}{∂x} = − \frac{∂B_{y}}{∂y} and \frac{∂B_{x}}{∂y} = \frac{∂B_{y}}{∂x} , are equivalent to the Cauchy-Riemann equations for the real and imaginary parts of f(z): \frac{∂u}{∂x} = \frac{∂v}{∂y} , \frac{∂v}{∂x} = − \frac{∂u}{∂y} . For example, the initial field, specified by the flux function (4.77), corresponds to f(z) = iB_{0}z/R. Then, the deformed magnetic field with the current sheet shown in Figure 4.13(d) can be described as

ψ_0(r,θ)=\frac{B_0r^2}{2R}\cos 2θ                      (4.77)

B_{x} − iB_{y} = \frac{iB_{0}}{R} (z − ia)^{1/2}(z − ib)^{1/2},        (4.83)

where the current sheet corresponds to the branch cut in the complex plane between z = ia and z = ib, where two neutral Y -points are formed. In order to find the values of a and b, consider the magnetic field of Equation (4.83) far enough from the current sheet, i.e., for |z| ≫ a, b. Thus,

B_{x} − iB_{y} = \frac{iB_{0}}{R} z \left(1 − \frac{ia}{z} \right)^{1/2} \left( 1 − \frac{ib }{z}\right)^{1/2} ≈ \frac{iB_{0}}{R} z + \frac{B_{0}}{2R} (a + b),

which means that this current sheet generates there an additional uniform magnetic field B_{x} = B_{0}(a + b)/2R. Therefore, in order to comply with the dipole contribution to the boundary flux perturbation, this extra field should be equal to B_{1x} = B_{0}δ_{0}/2R, i.e., (a + b) = δ_{0}. In other words, the current sheet is centered at y = y_{1} = δ_{0}/2, the location of the X-point of the regular solution of Figure 4.13(c). Then, if ε is a half-length of the current sheet, i.e., a = δ_{0}/2 − \epsilon, b = δ_{0}/2 + \epsilon, one can re-write (4.83) as

B_{x} − iB_{y} = \frac{iB_{0}}{R} \left[ z − i \left(\frac{δ_{0}}{2} − \epsilon \right) \right]^{1/2} \left[ z − i \left(\frac{δ_{0}}{2} + \epsilon \right) \right]^{1/2}        (4.84)

Another requirement applied to this singular solution, which allows to derive \epsilon, is that all separatrix lines and, therefore, the current sheet from which they originate should correspond to ψ = ψ_{s} = 0. This ensures that the separatrix lines end up at the boundary r = R at the points (S^{′}_{1}, S^{′}_{2}, S^{′}_{3}, S^{′}_{4}), preserving in this way the field lines’ connectivity of the initial magnetic field (see Figure 4.13(d)). Thus, consider magnetic field (4.84) on the horizontal line y = δ_{0}/2, which stretches from the center of the current sheet to the point A at the boundary circle as shown in Figure 4.13(d). It follows from expression (4.84) that along this line, where z = x + iδ_{0}/2, B_{y}(x) = − \frac{B_{0}}{R} (x^{2} + \epsilon^{2})^{1/2}, hence

ψ_{A} − ψ_{s} = − \int_{0}^{x_{A}} B_{y}dx = \frac{B_{0}}{R} \int_{0}^{x_{A}} (x^{2} + \epsilon^{2})^{1/2}dx =

ψ^{(R)}(θ_{A}) = ψ^{(R)}_{0} (θ_{A}) + ψ^{(R)}_{1} (θ_{A}) ≈ \frac{B_{0}R}{2},        (4.85)

where x_{A} = (R^{2}−δ^{2}_{0}/4)^{1/2}. A straightforward derivation of the integral in (4.85) yields

\int_{0}^{x_{A}} (x^{2} + \epsilon^{2})^{1/2} dx ≈ \frac{1}{2} [R^{2} − \frac{δ^{2}_{0}}{4} + \epsilon^{2} sinh^{−1}(R/\epsilon)],

which being inserted into equation (4.85) results in

\epsilon ≈ \frac{δ_{0}}{2\mathcal{L}^{1/2}},        (4.86)

where \mathcal{L} = \ln(R/δ_{0}) ≫ 1.

Thus, two different magnetostatic equilibria are associated with the dipoletype deformation of the initial X-point magnetic configuration (4.77). There is the regular equilibrium shown in Figure 4.13(c), and the singular current sheet equilibrium of Figure 4.13(d). The latter corresponds to a perfectly conducting fluid, where connectivity of the magnetic field-line footpoints is preserved. However, presence of a very small but finite resistivity leads to magnetic reconnection, which eliminates the magnetic field singularity (the current sheet), and enables transition to the regular equilibrium with a lower magnetic energy.

Therefore, this process is similar to forced magnetic reconnection discussed in Problems 4.3.1 and 4.3.2.