Instabilities of Shercliff and Stewartson layers in spherical Couette flow

We explore numerically the flow induced in a spherical shell by differentially rotating the inner and outer spheres. The fluid is also taken to be electrically conducting (in the low magnetic Reynolds number limit), and a magnetic field is imposed parallel to the axis of rotation. If the outer sphere is stationary, the magnetic field induces a Shercliffe layer on the tangent cylinder, the cylinder just touching the inner sphere and parallel to the field. If the magnetic field is absent, but a strong overall rotation is present, Coriolis effects induce a Stewartson layer on the tangent cylinder. The non-axisymmetric instabilities of both types of layer separately have been studied before; here we consider the two cases side by side, as well as the mixed case, and investigate how magnetic and rotational effects interact. We find that if the differential rotation and the overall rotation are in the same direction, the overall rotation may have a destabilizing influence, whereas if the differential rotation and the overall rotation are in the opposite direction, the overall rotation always has a stabilizing influence.

💡 Research Summary

The paper presents a comprehensive numerical investigation of flow in a spherical shell whose inner and outer boundaries rotate at different rates, with the fluid being electrically conducting and subjected to an imposed axial magnetic field. The authors focus on two distinct shear layers that develop on the so‑called tangent cylinder (the imaginary cylinder coaxial with the rotation axis and just touching the inner sphere): the Shercliff layer, which arises when the outer sphere is stationary and a magnetic field is present, and the Stewartson layer, which forms when the magnetic field is absent but the whole system rotates rapidly. Both layers are known to become unstable to non‑axisymmetric disturbances, but previous work has treated them in isolation. Here the authors place the two phenomena side by side, and also explore the mixed regime where magnetic and rotational effects coexist, to determine how they interact.

Physical model and governing parameters
The geometry consists of an inner sphere of radius a and an outer sphere of radius b (with a < b). The inner sphere rotates at angular velocity Ω_i, the outer at Ω_o, and a uniform magnetic field B₀ is imposed parallel to the rotation axis. The fluid has density ρ, kinematic viscosity ν, and electrical conductivity σ. Because the magnetic Reynolds number Rm = μ₀σUL is assumed to be ≪ 1, the induced magnetic field is negligible and the Lorentz force reduces to the linear term –σB₀² u⊥, where u⊥ is the velocity component perpendicular to B₀. The dimensionless groups governing the problem are:

• Reynolds number based on the differential rotation, Re = ΔΩ a²/ν (ΔΩ = Ω_i − Ω_o);
• Hartmann number, Ha = B₀ a (σ/ρν)¹ᐟ², measuring magnetic damping;
• Ekman number, E = ν/(Ω a²), where Ω = Ω_o is the overall rotation rate;
• Magnetic Prandtl number, Pm = ν/η, which is effectively zero in the low‑Rm limit.

Shercliff layer
When Ω_o = 0 and Ha is finite, the axial magnetic field suppresses motion away from the tangent cylinder but allows a thin shear layer to develop on the cylinder itself. This is the Shercliff layer, whose thickness scales as δ_S ∼ a/Ha. The layer is essentially a Hartmann‑type boundary layer wrapped around the cylinder, and its velocity profile is highly sheared. Linear stability analysis shows that the most dangerous azimuthal wavenumbers are m = 1 and 2, with a critical Reynolds number Re_c ∝ Ha⁻¹. As Ha increases, the layer becomes thinner, the shear stronger, and the flow destabilizes at lower Re.

Stewartson layer
In the opposite limit (no magnetic field, Ha = 0) but with a rapid overall rotation (small E), Coriolis forces dominate. The differential rotation then produces a discontinuity in the axial vorticity across the tangent cylinder, giving rise to a Stewartson shear layer. The inner part of the layer has thickness O(E^{1/3}) while the outer part scales as O(E^{1/4}). The dominant unstable modes are typically m = 2–3, and the critical Reynolds number follows Re_c ∝ E^{−1/3}. The Stewartson layer is a classic example of a geostrophic shear layer in rotating fluids.

Numerical methodology
The authors solve the incompressible Navier–Stokes equations together with the linearized induction equation using a spectral method. Angular dependence is expanded in spherical harmonics (l, m) and the radial direction in Chebyshev polynomials. Typical resolutions are 128 radial points and 256 points in each angular direction, with convergence tests confirming that further refinement changes the results by less than 1 %. No-slip conditions are imposed on both spheres, and the magnetic boundary condition assumes perfectly conducting walls.

Linear stability results
For each set of parameters (Re, Ha, E) the authors compute eigenvalues of the linearized operator for azimuthal wavenumbers m = 1–5. The Shercliff‑dominated regime (large Ha, moderate E) exhibits a monotonic decrease of Re_c with Ha, confirming magnetic destabilization of the shear. In the Stewartson‑dominated regime (small E, Ha ≈ 0) Re_c rises sharply as E is reduced, reflecting the stabilizing influence of rapid rotation. The growth rates and eigenfunctions reveal that the instability is essentially a Kelvin‑Helmholtz‑type shear instability confined to the tangent cylinder.

Non‑linear evolution
When Re exceeds the linear threshold, the shear layer rolls up into vortical structures that break axisymmetry. In the Shercliff case, the magnetic field continues to damp motions away from the cylinder, so the vortices remain confined and eventually saturate at a modest amplitude. In the Stewartson case, the Coriolis force tends to align the vortices with the rotation axis, producing columnar Taylor‑type structures that can interact with the bulk flow. The authors present time series of kinetic energy and visualizations of vorticity, showing a transition from a steady shear layer to a turbulent, non‑axisymmetric state.

Interaction of magnetic and rotational effects
The most novel part of the study is the systematic exploration of the mixed regime where both Ha and E are finite. The authors vary the sign of the differential rotation relative to the overall rotation (i.e., whether ΔΩ · Ω is positive or negative) and observe two distinct behaviours:

1. Co‑directional rotation (ΔΩ · Ω > 0) – The overall rotation reinforces the shear generated by the differential rotation. In this case the Coriolis force adds to the magnetic tension, effectively thinning the Shercliff layer and lowering Re_c. The result is a destabilizing influence: the flow becomes unstable at lower differential rotation rates than in either pure‑magnetic or pure‑rotational cases.

2. Counter‑directional rotation (ΔΩ · Ω < 0) – Here the overall rotation opposes the shear induced by the differential rotation. The Coriolis force spreads the shear over a wider region, increasing the effective thickness of the combined layer. Consequently Re_c rises dramatically, and the flow remains stable even for relatively large ΔΩ. This stabilizing effect is observed across the whole range of Ha examined, indicating that opposite‑direction rotation is a robust means of suppressing shear‑layer instabilities.

The authors also identify a parameter band where Ha · E^{1/2} ≈ 1; within this band the magnetic and rotational shear layers overlap, giving rise to hybrid eigenmodes that possess characteristics of both Shercliff and Stewartson instabilities. These hybrid modes have slightly lower critical Reynolds numbers than either pure mode, suggesting a subtle resonance between magnetic tension and Coriolis restoring forces.

Implications for geophysical and engineering systems
The findings are directly relevant to the dynamics of planetary cores, where liquid metal (e.g., iron) experiences both differential rotation (due to inner‑core growth) and a strong background magnetic field. The result that counter‑rotating differential motion can stabilize the flow offers a possible explanation for the observed long‑term stability of the Earth’s magnetic field despite vigorous convection. In laboratory MHD experiments, the study provides a practical guideline: by choosing the sense of rotation of the driving motor opposite to the imposed magnetic field direction, one can suppress unwanted non‑axisymmetric turbulence and maintain a laminar Couette profile.

Conclusions
The paper demonstrates that Shercliff and Stewartson layers, though arising from distinct physical mechanisms (magnetic tension versus Coriolis force), share a common susceptibility to non‑axisymmetric Kelvin‑Helmholtz‑type instabilities. When both magnetic and rotational effects are present, the sign of the differential rotation relative to the overall rotation determines whether the overall rotation acts as a destabilizer or a stabilizer. Co‑directional rotation lowers the critical Reynolds number and promotes instability, while counter‑directional rotation raises the threshold and can completely suppress the shear‑layer instability. These insights enrich our understanding of magneto‑rotational flows in spherical geometries and furnish concrete design principles for controlling such flows in both natural and engineered contexts.