3. The shear layer along the C-line

We will now briefly introduce the reader into the mathematical approach to the analysis of the shear layer structure, which is based on the singular perturbation theory. To take into account of the curvature of the C-line, it is more suitable to use different variables. As mentioned, the symmetries of the system imply that it is enough to limit the analysis to the interval 0 < x < π=2λ. Thus, the magnetic field lines can be represented parametrically in the following way

$$\begin{aligned} x(\tau) &= \lambda A(\tau - 1) + \frac{\pi}{2\lambda} \\ \exp[\lambda z(\tau)] &= \frac{1}{\lambda A} \sin \left[ \lambda^2 A(\tau - 1) + \frac{\pi}{2} \right] \end{aligned} \tag{16}$$

and the point <sup>x</sup> <sup>¼</sup> <sup>π</sup> , <sup>z</sup> <sup>¼</sup> <sup>1</sup> where the Hartman layer singularity occurs (referred to as the point <sup>2</sup><sup>λ</sup> S (see Figure 5)) is defined by A ¼ A<sup>C</sup> and τ ¼ 1. Introducing a measure of distance along the critical C-line

$$\gamma(\tau) = -\int\_{\mathcal{S}} \mathbf{B} \cdot \mathbf{dr} = -\int\_{\mathcal{S}} d\mathbf{\varPhi} = \frac{e^{-\lambda z(\mathbf{r}')}}{\lambda} \cos[\lambda x(\mathbf{r}')] \Big|\_{1}^{\mathbb{T}} = -A \text{tan} \left[\lambda^2 A(\tau - 1)\right],\tag{17}$$

Figure 5. The shear layer along the critical magnetic field line <sup>C</sup>. The point <sup>S</sup> is the point <sup>x</sup> <sup>¼</sup> <sup>π</sup> , <sup>z</sup> <sup>¼</sup> <sup>1</sup> at which the line <sup>2</sup><sup>λ</sup> grazes the upper boundary and the point of intersection of the critical line with the lower boundary is denoted by I. After Mizerski and Bajer [5].

and letting Γ ¼ γ τð Þ <sup>I</sup> be the distance between point S and the point of intersection of the field line C with the lower boundary (referred to as the point I (see Figure 5)) we introduce a new set of coordinates

$$d = 1 - \frac{\gamma(\pi)}{\Gamma} = 1 - \frac{e^{-\lambda z}}{\lambda \Gamma} \cos(\lambda x) \tag{18}$$

$$m = M^{\ddagger} \sqrt{\Gamma} (A\_{\mathcal{C}} - A) = M^{\ddagger} \sqrt{\Gamma} \left( A\_{\mathcal{C}} - \frac{e^{-\lambda z}}{\lambda} \sin(\lambda x) \right) \tag{19}$$

where l is the coordinate along the basic magnetic field lines which has the property that l ¼ 1 at S and l ¼ 0 at I, whereas n is a measure of distance between other field lines and the C-line within the shear layer of thickness M�1=<sup>2</sup> (cf. [19, 20]). The so-defined coordinate n has the properties, that it is 0 on the C-line, positive in region II and negative in region I; moreover pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup> <sup>Γ</sup><sup>2</sup> <sup>A</sup><sup>C</sup> <sup>¼</sup> expð�λÞ=<sup>λ</sup> <sup>¼</sup> <sup>1</sup> � <sup>λ</sup> <sup>=</sup>λ.

For the simplest case of poorly conducting or insulating upper/outer boundary, with the use of the shear layer coordinates ðl; nÞ, the Eq. (8) can now be written at the leading order in the form

$$\frac{\partial V\_{\pm}}{\partial l} \pm \frac{\partial^2 V\_{\pm}}{\partial n^2} = 0 \tag{20}$$

where

$$V\_{\pm} = \mu \pm Mb.\tag{21}$$

In the spherical geometry, the analogous formulation leads to the coupling of the two equations for V� through nonzero curvature terms on the right-hand sides

Super-Speeding Jets in MHD Couette Flow 37 http://dx.doi.org/10.5772/intechopen.79005

$$\frac{\partial V\_{\pm}}{\partial l} \pm \frac{\partial^2 V\_{\pm}}{\partial n^2} = \frac{1}{s\_c} \frac{\mathbf{d}s\_c}{\mathbf{d}l} V\_{\mp}. \tag{22}$$

The coupling term, however, may be neglected if the narrow gap limit is assumed. More importantly, however, the two equations, in both—planar and spherical configurations, are coupled through the boundary conditions at l ¼ 0, 1, thus at the points I and S. Eqs. (20) and (22) are diffusion equations (with a source in the spherical case) valid for all ∞ < n < ∞, with the variable l corresponding to time variable from standard diffusion processes in the case of V� and 1 � l corresponding to time in the case of Vþ. The following solving procedure of Eqs. (20) or (22) can be applied. One can utilize the Green's formula for the diffusion equation and first solve for V<sup>þ</sup> by the use of the "initial condition" at l ¼ 1 (1 � l ¼ 0). Then introduce the obtained expression for V<sup>þ</sup> into the "initial condition" for V� at l ¼ 0 and utilize the Green's formula again. Finally, matching the two solutions through the condition at l ¼ 1 again yields an integral equation for the super velocities. Such procedure leads to an integral equation of Fredholm type, which has been solved numerically in Dormy et al. [4] and Mizerski and Bajer [5] for the two cases <sup>e</sup> <sup>¼</sup> <sup>0</sup> and <sup>e</sup> <sup>¼</sup> <sup>M</sup>�<sup>1</sup> .

However, when the boundaries are perfectly conducting, the problem becomes more complicated. The same equations as (20) and (22) are obtained for

$$V\_{\pm} = M^{-1/2} u \pm M^{1/2} b. \tag{23}$$

(cf. Eq. (3.20) in [6]), but the problem becomes analytically intractable due to the complications arising from vanishing of the current component parallel to the boundary at l ¼ 1. Nevertheless, Soward and Dormy [6] have managed to show that for perfectly conducting boundaries, the strong current leakage from the shear layer into the outer boundary in the vicinity of the ˜ ° <sup>M</sup><sup>1</sup>=<sup>2</sup> critical point causes strong super rotation <sup>Ω</sup> <sup>¼</sup> <sup>O</sup> .
