= Solution
The <geodynamo> operates in the electrically conducting liquid of <Earth's outer core>. <buoyancy> from thermal and <compositional convection> supplies motion, while the <Coriolis force> organizes rotating flow. Rotation together with radial stratification permits <kinetic helicity>, typically with opposite signs in the two hemispheres. In mean-field language this allows an <alpha effect> that converts poloidal and toroidal components into one another. A large-scale <alpha-squared dynamo> is therefore a plausible regeneration mechanism even without dominant <differential rotation>.
The growth condition compares this regeneration with the sum of molecular and turbulent <magnetic diffusion> on the core scale. A spherical model must impose the appropriate magnetic matching at the core boundaries and permit spatially varying, generally anisotropic alpha and <diffusivity> tensors. Rotation selects an axis, and stratification and boundaries select positions, so homogeneous <isotropic turbulence> is only a local idealization. Such a model can favor a predominantly axial dipole, but the preferred parity and threshold must be found from its global eigenproblem rather than deduced from the plane-wave formula alone. If mean shear is significant, alpha–Omega regeneration can also contribute.
The first-order-smoothing helicity formula is a useful guide to the sign and magnitude of alpha, not an automatically controlled numerical closure for the core. The important convective motions need not have small eddy <magnetic Reynolds number> or short enough <correlation time>; scale separation can also be limited. Furthermore, the <Lorentz force> modifies the flow and its correlations once the field grows. Saturation therefore requires nonlinear feedback, including <dynamo quenching> and magnetic-helicity constraints where relevant.
The observed field varies and occasionally reverses. A constant-coefficient growing kinematic eigenmode does not itself explain these reversals or select a saturated amplitude. Spatial and temporal variations, competing modes, and nonlinear flow–field interactions must be included. \b[<mean-field dynamo theory> supplies a physical regeneration mechanism and a testable threshold, while the geodynamo requires a rotating, bounded, nonlinear realization of that mechanism.]
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