Stably Stratified Spherical Couette Flow

• Stably stratified spherical Couette flow refers to the rotation of a conducting fluid layer between concentric spheres, relevant for Earth's core dynamics and stellar phenomena.
• Key dynamics include the interaction of rotation, magnetic fields, and stable stratification, leading to phenomena like super-rotating shear layers and magnetohydrodynamic instabilities.
• Applications range from exploring geomagnetic jerks on Earth to understanding magnetic fields in stars, aided by numerical simulations and asymptotic analyses.

Stably stratified spherical Couette flow refers to the differential rotation of a stably stratified, electrically conducting fluid confined between two concentric spheres, with rotation imposed on at least one sphere. The system is of key relevance to geophysical and astrophysical contexts where such fluid layers, embedded in global magnetic fields, are observed—specifically, the Earth’s outer core and stellar radiative regions. Rotation, stratification, magnetic field, and boundary forcing together generate a complex spectrum of hydrodynamic and magnetohydrodynamic regimes, including the emergence of thin, equatorial super-rotating shear layers and associated instabilities (Philidet et al., 2019).

1. Governing Equations and Formulation

The system is formulated under the Boussinesq approximation in spherical coordinates $(r, \theta, \phi)$, accounting for rotation with angular velocity $\Omega_o$, magnetic induction from a dipolar field, and a controlled temperature contrast generating stable stratification. The nondimensionalized governing equations comprise:

• Momentum Equation: Includes Coriolis, Lorentz, and buoyancy forces:

$$\frac{\partial\bm v}{\partial t} + (\bm v\cdot\nabla)\bm v = -\nabla P - 2\,\bm e_z\times\bm v + E\,\Delta\bm v + \frac{E}{Pm}\,(\nabla\times\bm B)\times\bm B + E\,\widetilde{Ra}\,\Theta\,\bm e_r$$

• Induction Equation:

$$\frac{\partial\bm B}{\partial t} = \nabla\times(\bm v\times\bm B) + \frac{E}{Pm}\,\Delta\bm B$$

• Heat Equation (for temperature perturbation $\Theta$):

$$\frac{\partial \Theta}{\partial t} + (\bm v\cdot\nabla)\,\Theta = \frac{E}{Pr}\,\Delta\,\Theta$$

• Solenoidal Constraint (Incompressibility):

$\nabla\cdot\bm v=0$

Variables are normalized by characteristic quantities: $r_o$ for length, $\Omega_o^{-1}$ for time, $B_0$ for magnetic field, and the imposed temperature difference $\Omega_o$0 for $\Omega_o$1.

2. Dimensionless Parameters

The dynamics are governed by several key nondimensional numbers:

Parameter	Definition	Physical Meaning
$\Omega_o$2	$\Omega_o$3	Ekman number (rotation/viscosity)
$\Omega_o$4	$\Omega_o$5	Reynolds number (inertia/viscosity)
$\Omega_o$6	$\Omega_o$7	Rossby number (differential rotation)
$\Omega_o$8	$\Omega_o$9	Froude number (inertia/stratification)
$$\frac{\partial\bm v}{\partial t} + (\bm v\cdot\nabla)\bm v = -\nabla P - 2\,\bm e_z\times\bm v + E\,\Delta\bm v + \frac{E}{Pm}\,(\nabla\times\bm B)\times\bm B + E\,\widetilde{Ra}\,\Theta\,\bm e_r$$0	$$\frac{\partial\bm v}{\partial t} + (\bm v\cdot\nabla)\bm v = -\nabla P - 2\,\bm e_z\times\bm v + E\,\Delta\bm v + \frac{E}{Pm}\,(\nabla\times\bm B)\times\bm B + E\,\widetilde{Ra}\,\Theta\,\bm e_r$$1	Magnetic Prandtl number
$$\frac{\partial\bm v}{\partial t} + (\bm v\cdot\nabla)\bm v = -\nabla P - 2\,\bm e_z\times\bm v + E\,\Delta\bm v + \frac{E}{Pm}\,(\nabla\times\bm B)\times\bm B + E\,\widetilde{Ra}\,\Theta\,\bm e_r$$2	$$\frac{\partial\bm v}{\partial t} + (\bm v\cdot\nabla)\bm v = -\nabla P - 2\,\bm e_z\times\bm v + E\,\Delta\bm v + \frac{E}{Pm}\,(\nabla\times\bm B)\times\bm B + E\,\widetilde{Ra}\,\Theta\,\bm e_r$$3	Magnetic Ekman number
$$\frac{\partial\bm v}{\partial t} + (\bm v\cdot\nabla)\bm v = -\nabla P - 2\,\bm e_z\times\bm v + E\,\Delta\bm v + \frac{E}{Pm}\,(\nabla\times\bm B)\times\bm B + E\,\widetilde{Ra}\,\Theta\,\bm e_r$$4	$$\frac{\partial\bm v}{\partial t} + (\bm v\cdot\nabla)\bm v = -\nabla P - 2\,\bm e_z\times\bm v + E\,\Delta\bm v + \frac{E}{Pm}\,(\nabla\times\bm B)\times\bm B + E\,\widetilde{Ra}\,\Theta\,\bm e_r$$5	Elsasser number (magnetism/rotation)
$$\frac{\partial\bm v}{\partial t} + (\bm v\cdot\nabla)\bm v = -\nabla P - 2\,\bm e_z\times\bm v + E\,\Delta\bm v + \frac{E}{Pm}\,(\nabla\times\bm B)\times\bm B + E\,\widetilde{Ra}\,\Theta\,\bm e_r$$6	$$\frac{\partial\bm v}{\partial t} + (\bm v\cdot\nabla)\bm v = -\nabla P - 2\,\bm e_z\times\bm v + E\,\Delta\bm v + \frac{E}{Pm}\,(\nabla\times\bm B)\times\bm B + E\,\widetilde{Ra}\,\Theta\,\bm e_r$$7	Stratification-rotation interplay

Where: $$\frac{\partial\bm v}{\partial t} + (\bm v\cdot\nabla)\bm v = -\nabla P - 2\,\bm e_z\times\bm v + E\,\Delta\bm v + \frac{E}{Pm}\,(\nabla\times\bm B)\times\bm B + E\,\widetilde{Ra}\,\Theta\,\bm e_r$$8 is kinematic viscosity; $$\frac{\partial\bm v}{\partial t} + (\bm v\cdot\nabla)\bm v = -\nabla P - 2\,\bm e_z\times\bm v + E\,\Delta\bm v + \frac{E}{Pm}\,(\nabla\times\bm B)\times\bm B + E\,\widetilde{Ra}\,\Theta\,\bm e_r$$9 is magnetic diffusivity; $$\frac{\partial\bm B}{\partial t} = \nabla\times(\bm v\times\bm B) + \frac{E}{Pm}\,\Delta\bm B$$0 is magnetic Prandtl number; $$\frac{\partial\bm B}{\partial t} = \nabla\times(\bm v\times\bm B) + \frac{E}{Pm}\,\Delta\bm B$$1 is thermal expansion coefficient; $$\frac{\partial\bm B}{\partial t} = \nabla\times(\bm v\times\bm B) + \frac{E}{Pm}\,\Delta\bm B$$2 is Brunt–Väisälä frequency $$\frac{\partial\bm B}{\partial t} = \nabla\times(\bm v\times\bm B) + \frac{E}{Pm}\,\Delta\bm B$$3; $$\frac{\partial\bm B}{\partial t} = \nabla\times(\bm v\times\bm B) + \frac{E}{Pm}\,\Delta\bm B$$4 quantifies stratification via $$\frac{\partial\bm B}{\partial t} = \nabla\times(\bm v\times\bm B) + \frac{E}{Pm}\,\Delta\bm B$$5.

3. Geometric Configuration and Boundary Conditions

The system comprises a spherical fluid shell between inner radius $$\frac{\partial\bm B}{\partial t} = \nabla\times(\bm v\times\bm B) + \frac{E}{Pm}\,\Delta\bm B$$6 and outer radius $$\frac{\partial\bm B}{\partial t} = \nabla\times(\bm v\times\bm B) + \frac{E}{Pm}\,\Delta\bm B$$7 with two primary shell aspect ratios of interest: the "thin-shell" ($$\frac{\partial\bm B}{\partial t} = \nabla\times(\bm v\times\bm B) + \frac{E}{Pm}\,\Delta\bm B$$8) relevant for planetary interiors and the "thick-shell" ($$\frac{\partial\bm B}{\partial t} = \nabla\times(\bm v\times\bm B) + \frac{E}{Pm}\,\Delta\bm B$$9) for stellar contexts.

Boundary conditions:

• Mechanical: No-slip at both boundaries:

$\Theta$0

• Thermal: Prescribed temperature at the boundaries: $\Theta$1, $\Theta$2.
• Magnetic: The inner sphere ($\Theta$3) is perfectly conducting and carries an imposed dipole; the exterior ($\Theta$4) is insulating such that the field matches a potential solution.

4. Flow Regimes and Super-Rotating Layers

In the non-magnetic setting, the dynamics transition between three regimes, controlled by the parameter $\Theta$5:

• For $\Theta$6: The flow is strongly rotation-dominated with cylindrical (Taylor–Proudman) jets.
• For $\Theta$7: Buoyancy dominates, driving near-radial, spherically symmetric circulation.
• For $\Theta$8: The flow exhibits mixed geometry.

The inclusion of a sufficiently strong dipole ($\Theta$9) generates a thin, equatorial "super-rotating" shear layer in the stably stratified region. This layer is produced by azimuthal Lorentz torque due to the misalignment of imposed dipole field lines and currents induced in the insulating outer boundary. The maximum angular velocity in this region reaches approximately $$\frac{\partial \Theta}{\partial t} + (\bm v\cdot\nabla)\,\Theta = \frac{E}{Pr}\,\Delta\,\Theta$$0 for $$\frac{\partial \Theta}{\partial t} + (\bm v\cdot\nabla)\,\Theta = \frac{E}{Pr}\,\Delta\,\Theta$$1 and $$\frac{\partial \Theta}{\partial t} + (\bm v\cdot\nabla)\,\Theta = \frac{E}{Pr}\,\Delta\,\Theta$$2, with the shear layer thickness $$\frac{\partial \Theta}{\partial t} + (\bm v\cdot\nabla)\,\Theta = \frac{E}{Pr}\,\Delta\,\Theta$$3 decreasing sharply with lower Ekman number, $$\frac{\partial \Theta}{\partial t} + (\bm v\cdot\nabla)\,\Theta = \frac{E}{Pr}\,\Delta\,\Theta$$4 to $$\frac{\partial \Theta}{\partial t} + (\bm v\cdot\nabla)\,\Theta = \frac{E}{Pr}\,\Delta\,\Theta$$5 in the accessible numerical regime.

5. Linear Stability and Local Dispersion Analysis

Linear WKB analysis near the equator for axisymmetric disturbances yields a fourth-order dispersion relation,

$$\frac{\partial \Theta}{\partial t} + (\bm v\cdot\nabla)\,\Theta = \frac{E}{Pr}\,\Delta\,\Theta$$6

with explicit parameter-dependent coefficients, incorporating the effects of rotation, stratification, magnetic field, and velocity shear.

In the magnetostrophic regime ($$\frac{\partial \Theta}{\partial t} + (\bm v\cdot\nabla)\,\Theta = \frac{E}{Pr}\,\Delta\,\Theta$$7, $$\frac{\partial \Theta}{\partial t} + (\bm v\cdot\nabla)\,\Theta = \frac{E}{Pr}\,\Delta\,\Theta$$8), the dispersion simplifies for vertical wavenumbers to a quadratic:

$$\frac{\partial \Theta}{\partial t} + (\bm v\cdot\nabla)\,\Theta = \frac{E}{Pr}\,\Delta\,\Theta$$9

The fastest-growing vertical MRI-like instability exhibits normalized growth rate:

$\nabla\cdot\bm v=0$0

where $\nabla\cdot\bm v=0$1 is the local shear rate. For Earth-like parameters, the corresponding growth time is approximately 3 years, matching observed timescales for geomagnetic jerks.

6. Physical Mechanisms and Astrophysical Applications

Stable stratification suppresses radial motions and disrupts Taylor–Proudman columns, producing a shift from cylindrical to spherical flow as $\nabla\cdot\bm v=0$2 increases. Magnetohydrodynamic coupling confines the Lorentz-induced shear to a thinner equatorial layer, re-establishing “super-rotation” even for strong rotation. In the Earth’s core, this super-rotation atop the stratified region may launch magneto-Archimedes–Coriolis (MAC) waves, contribute to length-of-day variations, and trigger rapidly growing MHD instabilities consistent with secular variations and jerks in the geomagnetic field.

For stellar radiative interiors, the shear–field–stratification interplay selectively enables axisymmetric MRI-like instabilities for fields within a particular amplitude window. A plausible implication is an explanation for the observed magnetic dichotomy, or "magnetic desert," seen among intermediate-mass stars, connecting fossil field strengths to coupled hydro-magnetic instabilities.

7. Theoretical and Computational Advances

The combination of direct numerical simulations and asymptotic theory by Philidet et al. establishes a direct linkage between classical spherical Couette dynamics and global planetary or stellar observational phenomena. The parameterizations, flow regime maps, and stability analyses provide predictive tools for interpreting fluid and field behavior in stably stratified, rapidly rotating bodies (Philidet et al., 2019). This underlines the continuing role of global and local analyses in uncovering the interplay between hydrodynamic shear, stable stratification, and MHD instabilities across geophysical and astrophysical scales.