Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 36 / 2

Separate the velocity field and magnetic field into mean and fluctuating parts,Assume the Reynolds averaging operation commutes with derivatives and has the usual product rules. Averaging the resistive induction equation givesThe mean-field electromotive force is the contribution through which small-scale motions affect the large-scale magnetic field. If the mean field varies slowly compared with the fluctuation scales, its response can be expanded in the mean field and its spatial derivatives. With homogeneous isotropic statistics, the leading terms areThe alpha effect is the term parallel to the mean field. Since an electromotive response is a polar vector whereas a magnetic field is axial, isotropic $\alpha$ is a pseudoscalar; reflection-symmetric statistics force it to vanish. Nonzero kinetic helicity supplies the required handedness. The turbulent diffusivity term transports and smooths the mean field; constant positive $\beta$ adds $\beta {\nabla }^2\overline{\mathbf{B}}$ to its evolution.

One controlled calculation is first-order smoothing. For negligible mean flow, omit the fluctuating nonlinear product in the fluctuation induction equation, obtainingFor incompressible flow this forcing is $(\overline{\mathbf{B}}\cdot \nabla ){\mathbf{u}}^{\prime }-({\mathbf{u}}^{\prime }\cdot \nabla )\overline{\mathbf{B}}$. A short-memory approximation of duration $\tau$ replaces the inverse temporal response by multiplication by $\tau$. Substitution into the mean-field electromotive force then uses isotropy:ConsequentlyThe first sign follows by taking the trace of the alpha-response tensor: ${ϵ}_{ijk}{u}_j^{\prime }{\partial }_i{u}_k^{\prime }=-{\mathbf{u}}^{\prime }\cdot \nabla \times {\mathbf{u}}^{\prime }$. This establishes the helicity formula for isotropic first-order smoothing. Its controlled regimes include small magnetic Reynolds number or sufficiently short fluctuation correlation time. Outside such a closure, the coefficients involve response-weighted time correlations; an instantaneous helicity formula is not universal.

An explicit example avoids an unspecified correlation time. For each coordinate axis ${\mathbf{e}}_j$, choose transverse unit vectors ${\mathbf{a}}_j,{\mathbf{b}}_j$ with ${\mathbf{e}}_j\times {\mathbf{a}}_j={\mathbf{b}}_j$, and take the three real circular polarizationsThe propagation directions are mutually perpendicular and all three waves have the same handedness. Direct differentiation givesA spatial average over the periodic cell removes cross terms between different waves. Hence the mean kinetic helicity of their sum isThe sum has isotropic second moments, $⟨u_i{u}_j⟩=U^2{\delta }_{ij}$.

For a locally uniform test magnetic field $\overline{\mathbf{B}}$, the periodic first-order smoothing response obeysSet $a=\eta k^2>0$. Since ${\partial }_t=-\Omega {\partial }_{\varphi }$ and ${\partial }_{\varphi }^2{\mathbf{u}}_j=-{\mathbf{u}}_j$, direct substitution gives the response after transients decay:The term proportional to ${\mathbf{u}}_j$ contributes no cross product with that same wave. Cross terms between distinct waves average to zero. Thereforeand the isotropic alpha effect of three helical traveling waves isThis is the alpha-helicity relation with the exact response time ${\tau }_{\mathrm{eff}}=a/(a^2+{\Omega }^2)$ for these monochromatic waves. In the quasistatic limit $\Omega =0$, $\alpha =-\mathcal{H}/(3\eta k^2)$. A single wave would give an anisotropic response along its propagation direction; the three equal waves make the response tensor a scalar multiple of the identity. The calculation retains the validity assumptions of first-order smoothing, rather than extrapolating the formula to arbitrary fluctuation amplitude.

With no mean flow and constant isotropic coefficients, the mean-field dynamo equation isIf the symbol $\beta$ already denotes the total diffusivity in the chosen mean-field convention, replace ${\eta }_T$ by $\beta$. Let $\nabla \times {\mathbf{B}}_0=\kappa {\mathbf{B}}_0$. This Beltrami field is solenoidal for $\kappa \ne 0$, and taking its curl again gives ${\nabla }^2{\mathbf{B}}_0=-{\kappa }^2{\mathbf{B}}_0$. ThusFor example, ${\mathbf{B}}_0=(\cos kz,-\sin kz,0)$ has $\kappa =k$, whereas reversing the sine sign gives $\kappa =-k$. Choose the helicity sign to agree with $\alpha$. The homogeneous alpha-squared dynamo growth criterion is thenOn an unbounded or suitably periodic domain with freely selectable wavenumber, this gives exponentially growing modes for every $\alpha \ne 0$. Maximizing their growth rate gives ${\kappa }_{\ast }=\alpha /(2{\eta }_T)$ and $s_{\max }={\alpha }^2/(4{\eta }_T)$. In a specified bounded domain, the permitted modes and boundary conditions impose a threshold; the optimal wavelength must also remain large compared with the turbulent scale for mean-field electrodynamics to apply.

A nonzero alpha effect is necessary in this constant-coefficient model. If $\alpha =0$, the mean equation is pure diffusion and the claimed growing modes do not exist. Positivity of $\beta$ alone does not imply growth.

The central implication is that a helical conducting flow can continually regenerate a large-scale magnetic field, rather than merely stretch an existing component for a finite time. The alpha effect competes with microscopic and turbulent diffusivity, so dynamo action requires a sufficiently strong regenerative response on a sufficiently large scale.

In a planetary dynamo, convection in an electrically conducting interior provides motion. Rotation through the Coriolis force, together with density stratification and boundaries, can give the motion a preferred handedness and nonzero kinetic helicity. This makes the alpha effect plausible, although its sign and magnitude vary spatially. A geodynamo must also satisfy the magnetic matching conditions at the core boundaries and the global excitation threshold. Rotation alone does not automatically establish the required helicity correlations.

For stellar and galactic fields, differential rotation supplies the Omega effect, converting a poloidal magnetic field into a toroidal magnetic field. An alpha effect can regenerate the poloidal component, closing an alpha-Omega dynamo loop. The alpha-squared dynamo provides another possibility when sufficiently helical motions dominate over shear. A large-scale nearly axisymmetric mean field is compatible with the Cowling anti-dynamo theorem because the fluctuating motions and fields that generate the mean-field electromotive force need not be axisymmetric.

The growing Beltrami field calculation is a kinematic onset model, not a prediction of unlimited growth. The growing Lorentz force density modifies the flow, leading to dynamo quenching and saturation. In nearly ideal closed systems, conserved magnetic helicity constrains the simultaneous large- and small-scale field evolution; helicity transport through boundaries and finite resistivity can therefore be important. Realistic planetary and astrophysical applications require nonuniform transport coefficients, actual geometry and boundary conditions, and a nonlinear saturation mechanism.