Kinematic magnetic dynamo ($D_t\mathbf{B}=(\mathbf{B}\cdot \nabla )\mathbf{u}$)

A kinematic magnetic dynamo studies magnetic field amplification by a prescribed fluid flow, neglecting the field's feedback on the velocity through the Lorentz force. With incompressibility and zero magnetic diffusivity, a Lagrangian field vector evolves by its local velocity-gradient matrix. A growth law derived in this approximation cannot hold indefinitely once magnetic stresses significantly change the flow.

For a trace-free isotropic Gaussian white noise velocity gradient with covariance $⟨{\sigma }_{im}(t){\sigma }_{jn}(t^{\prime })⟩={\kappa }_2\delta (t-t^{\prime })[{\delta }_{ij}{\delta }_{mn}-({\delta }_{im}{\delta }_{jn}+{\delta }_{in}{\delta }_{jm})/4]$, interpret the multiplicative equation as the Stratonovich integral limit of smooth velocity fluctuations. The causal Furutsu–Novikov formula has endpoint weight one half. It gives $P_t=({\kappa }_2/2){\partial }_i[(B^2{\delta }_{ij}-B_i{B}_j/2){\partial }_{j}P]$ for the vector probability density function.

Set $x=\ln (B/B_0)$ and $f=BF$ in the radial magnetic-field Fokker-Planck equation. Then $f_t=D{f}_{xx}-3D{f}_x$, so $f$ is a Gaussian distribution drifting at speed $3D$. For initial magnitude $B_0>0$, $F=[B\sqrt{4\pi Dt}{]}^{-1}\exp [-(\ln (B/B_0)-3Dt{)}^2/(4Dt)]$. This lognormal distribution has moments $B_0^q{e}^{Dq(q+3)t}$; the width of its logarithm grows as $\sqrt{2Dt}$.

Twice integrating the radial magnetic-field Fokker-Planck equation by parts gives the displayed law when moments and endpoint terms are controlled. In particular $⟨B^2⟩$ grows at rate $10D=2\gamma$, with $\gamma =5{\kappa }_2/4$. The corresponding magnetic energy density is $⟨B^2⟩/(8\pi )$ in Gaussian units.

For an isotropic initial ensemble in Gaussian white-noise magnetic stretching, $F(B)=4\pi B^{2}P(B)$ is the magnitude probability density function. The radial vector equation $P_t=D(B^2{P}_{BB}+4B{P}_B)$ gives the displayed conservative Fokker-Planck equation. Its integral is conserved under zero endpoint probability flux. For an anisotropic initial vector distribution, use the angularly averaged magnitude density instead; isotropic noise alone does not make arbitrary initial data isotropic.