1. Introduction

Super-speeding jets in the geometry of magnetohydrodynamic (MHD) spherical Couette flow have been first noticed in the numerical simulations of Dormy et al. [1]. They have analyzed a flow of an electrically conducting fluid in a spherical gap between concentric spherical shells, rapidly and differentially rotating about a common axis in a centered axial dipolar magnetic field. The solid inner sphere, which had the same conductivity as the fluid, was spinning slightly faster than the insulating outer shell. The stationary flow obtained via DNS exhibited

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a super-rotating structure near the region, where the critical magnetic field line, henceforth denoted by C, was tangent to the outer shell. The angular velocity of the flow in that region was about 50% greater than that of the inner sphere, which was driving the flow.

Hollerbach [2] for the first time investigated numerically the effect of nonzero conductivity of the outer shell in the same spherical geometry but the outer boundary was held motionless, thus eliminating the Coriolis force from the problem. He reported that the super rotation in the1 singular region was greatly enhanced and scaled with the value of the Hartmann number <sup>M</sup>. Hollerbach [3] studied the MHD spherical Couette flow for several different topologies of the external field lines and also observed that singular points of isolated contact of the magnetic lines with boundaries result in the formation of jets.

In a following sequence of three theoretical papers, the mechanism of super-velocity formation and the effect of nonzero conductivity of the boundaries have been explained. Dormy et al. [4] performed a joint analytical and numerical study of the system analyzed previously by Hollerbach [2], where they have described the super rotating shear layer along the critical magnetic line C which grazes the outer boundary and found analytic expressions for super rotation within the scope of asymptotic theory for M ≫ 1, confirmed by the results of numerical simulations. Not only they have explained the physics of the mechanism behind the formation of super-speeding jets in the studied configuration, which relies on the enhancement of the Lorentz force accelerating the flow, due to strong currents entering the singular Hartmann boundary layer at the outer shell near the point of contact of the critical field line C with the boundary, but also their analysis set grounds for the following theoretical findings. The study of Mizerski and Bajer [5] greatly relied on that of Dormy et al. [4], although it involved geometries—planar and spherical, and the resting boundary was weakly conducting (as opposed to the previous study, where it was insulating). The two geometries studied by Mizerski and Bajer [5] are depicted on Figure 1. In the plane geometry, the bottom boundary is moving at a constant speed and has the same conductivity as the fluid, whereas the conductivity of the upper boundary relative to the fluid's conductivity e is assumed small at the order <sup>e</sup> ˜ <sup>M</sup>°<sup>1</sup> . The rest of the space was an insulator. The crucial features of the external field in this configuration are that it is potential and that there exists a critical magnetic line, which grazes the upper boundary thus creating a singularity of the Hartmann layer. The two configurations depicted on Figure 1 are planar and spherical counterparts, equivalent from the point of view of physics of the super-speeding jets which form near the singularities. The plane configuration, however, captures all the necessary physical ingredients of the problem but at the same time makes the problem more transparent, avoiding the complications resulting from the curvature of the boundaries. Mizerski and Bajer [5] utilized this simplification and demonstrated the super-velocity excess, that is, the difference between the super velocity in the cases ˜ ° <sup>ε</sup>M<sup>3</sup>=<sup>4</sup> of <sup>a</sup> conducting and insulating outer boundary scales like <sup>O</sup> . They have also studied <sup>a</sup> somewhat similar case of a strongly conducting, but thin outer shell, with conductivity being the same as that of the fluid, but the relative thickness of the conducting shell to the thickness

<sup>1</sup> He fitted an exponent of M<sup>0</sup>:<sup>6</sup> to his numerical results, which however, was later shown not to be the true asymptotic scaling law.

Figure 1. A sketch of the two situations considered: the plane case on the left (the bottom boundary is moving) and the spherical case on the right (the inner sphere is rotating). After Mizerski and Bajer [5].

<sup>1</sup> of the fluid layer <sup>δ</sup> was assumed small, at the order <sup>δ</sup> � <sup>M</sup>� . They demonstrated that both cases, <sup>e</sup> � <sup>M</sup><sup>1</sup> , <sup>δ</sup> � <sup>1</sup> and the other one <sup>e</sup> � <sup>1</sup> and <sup>δ</sup> � <sup>M</sup>�<sup>1</sup> , are exactly equivalent in terms of the flow structure and the super-velocity magnitude. In the latter case, the super-velocity excess ˜ ° <sup>δ</sup>M<sup>3</sup>=<sup>4</sup> was shown to scale like <sup>O</sup> . The same scalings were shown to pertain to the spherical geometry. The most notable contribution to the problem of super-speeding jets was the remarkable comprehensive analysis of Soward and Dormy [6] in the spherical geometry. They have emphasized the role of the parameter eM<sup>3</sup>=<sup>4</sup> (or δM<sup>3</sup>=<sup>4</sup> for the case of thin outer shell), identified in Mizerski and Bajer [5], for the general case of continuously varying relative conductivity of the outer shell e from zero to infinity. They have reported the following scaling laws for the super velocity (angular velocity) in the singular region, for the case of perfectly conducting inner boundary.

$$\begin{aligned} \Omega\_{\text{max}} &= \mathcal{O}\left(M^{1/2}\right) \quad \text{for} \quad 1 \ll \epsilon \ll \epsilon M^{3/4} \\ \Omega\_{\text{max}} &= \mathcal{O}\left(\epsilon^{2/3} M^{1/2}\right) \quad \text{for} \quad \epsilon \ll 1 \ll \epsilon M^{3/4} \\ \Omega\_{\text{max}} &= \mathcal{O}(1) \quad \text{for} \quad \epsilon \ll \epsilon M^{3/4} \ll 1 \end{aligned} \tag{1}$$

The magnitude of the super rotation Ωmax was shown to be proportional to the magnitude of current on the critical C-line, denoted by J , i.e., Ωmax � J . <sup>c</sup> <sup>c</sup>

The phenomenon of super rotation was also observed in the experimental setup called "Derviche Tourneur Sodium" (DTS) located in Grenoble at the Université Joseph-Fourier. Nataf et al. [7] conducted experiments on the spherical Couette flow of liquid sodium in an external, centered axial dipolar field, with both boundaries differentially rotating. The outer shell was only 5 mm thick, about 27 times thinner than the fluid gap and about 8 times less electrically conductive than liquid sodium. The Hartmann number in the experiment was at the order of a thousand. They observed the super-rotating jets and obtained a very good agreement with the numerical models. However, they also observed that the super-speeding jets can be destabilized and reported oscillatory motion near the singular region. More recently, Brito et al. [8] further exploited the same DTS experimental setup and explored the effects of strong inertia. They also reported strong super rotation; however, they clearly demonstrated that the Coriolis force tends to suppress the super-speeding jets.

Wei and Hollerbach [9] investigated numerically the effect of strong inertia, that is, large Reynolds number, on the spherical Couette flow configuration with the outer shell stationary. Three configurations of the external magnetic field were chosen, which resulted from a combination of dipolar and axial fields. The super-speeding jets have been destabilized by increasing the Reynolds number, whereas strengthening the filed had the opposite effect. Most recently, Hollerbach and Hulot [10] performed numerical analysis of a similar problem in cylindrical geometry, putting an emphasis on the role of conductivity of the boundaries. The field configurations were also chosen so as to create singularities in the flow. When the boundaries were electrically conducting, super-speeding jets were reported on the contrary to the case with insulating boundaries, when simply shear layers were observed in the singular regions. A curious observation is made by the introduction of a nonzero azimuthal component of the external field in which case the conductivity of the boundaries has the opposite effect to the previous case, greatly suppressing the magnitude of super rotation.

The motivation for some of the aforementioned studies was justified on geophysical grounds. The investigations of the Earth's interior reveal differential rotation of the inner core (cf. [11, 12]) and that the electrical conductivity of the lower mantle is nonnegligible [13]. Moreover, some evidence can be found for the existence of a very thin layer of anomalously high conductivity at the base of the mantle [14, 15]. It must be said, however, that the model of MHD spherical Couette flow is so idealized with respect to the true dynamics of the core, neglecting thermal and compositional driving, turbulence, the solidification processes at the inner core, etc., that no direct comparisons with the flow at the core-mantle boundary can be made. Nevertheless, it might be possible that the effect of super rotation manifests itself on the field zero isolines locally at the core mantle boundary.
