Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 75 / 3

Interpret the multiplicative Gaussian white noise as the limit of a smooth short-correlation-time velocity field, hence in the Stratonovich integral convention. Put ${\partial }_i=\partial /\partial B_i$. For a realization, differentiating the field-space Dirac delta gives its field-space continuity equationThe velocity-gradient covariance has zero trace variance: contracting $T_{mn}^{ij}$ with $m=i,n=j$ gives $3-(9+3)/4=0$. Thus the gradient is trace free, consistently with the incompressible ideal induction equation.

A noise impulse in component ${\sigma }_{jn}$ changes ${\tilde{B}}_j$ by ${\tilde{B}}_n$ times that impulse. Its instantaneous retarded action on the Dirac delta is thereforeOnly earlier times affect the present field. In the white-noise limit the integral over this causal interval has ${\int }_0^t\delta (t-t^{\prime })d{t}^{\prime }=1/2$, corresponding to the symmetric short-time regularization. Applying the Furutsu–Novikov formula and then averaging givesSubstitute into the averaged field-space continuity equation:The differentiated $B_n$ term vanishes because ${\sum }_j{T}_{mj}^{ij}=0$, and the remaining contraction isConsequently the closed vector Fokker-Planck equation for Gaussian white-noise magnetic stretching isThis is a positive diffusion operator: the displayed matrix has eigenvalues $B^2$ on the two transverse directions and $B^2/2$ along $\mathbf{B}$. Replacing the endpoint half-weight by one would double the diffusion and fail to reproduce the later rate.

For isotropic initial statistics, isotropic noise preserves $P=P(B,t)$. Isotropy of the noise alone does not turn an arbitrary anisotropic initial vector distribution into an isotropic one; alternatively take the angularly averaged magnitude distribution. The radial gradient satisfies ${\partial }_{j}P=P_B{B}_j/B$, so the diffusion-vector contraction in the preceding equation is $B{B}_i{P}_B/2$. Its divergence givesNow $F=4\pi B^{2}P$ is the magnitude probability density function, normalized by ${\int }_0^{\infty }FdB=1$. Sincewe obtain the radial magnetic-field Fokker-Planck equationThe outward Fokker-Planck probability current is $J=-D(B^2{F}_B-2BF)$. Zero probability flux at zero and infinity preserves normalization. For positive initial magnitude the solution below remains on $B>0$; an atom initially at $B=0$ instead remains an absorbing zero-field component.

In Gaussian units, the ensemble-averaged magnetic energy density isLet $M_q={\int }_0^{\infty }{B}^{q}FdB$, and assume the relevant moment exists and endpoint terms vanish. Twice applying integration by parts to the conservative magnitude equation givesIn particular, the magnetic stretching moment growth for $q=2$ isThis is a kinematic magnetic dynamo result with prescribed velocity statistics and no Ohmic loss. Once the Lorentz force appreciably changes the flow, this indefinitely exponential kinematic approximation is no longer self-consistent.

For $B_0>0$, change to logarithmic magnitude and its correctly transformed probability density function:Using ${\partial }_B=B^{-1}{\partial }_x$, the radial equation becomesThis is diffusion with constant drift. Set $y=x-3Dt$ and $g(y,t)=f(y+3Dt,t)$; then $g_t=D{g}_{yy}$. The initial Dirac delta transforms to $f(x,0)=\delta (x)$, so the heat-kernel solution givesDividing by $B$ yields the lognormal magnetic-field amplification lawExpanding the square and using $B=B_0{e}^x$ gives the equivalent requested formThe logarithm is a Gaussian random variable with mean $3Dt$ and variance $2Dt$, which proves normalization and spreading directly. The exponential moments of that Gaussian distribution give $⟨B^q⟩=B_0^q{e}^{Dq(q+3)t}$, independently checking the energy growth. The limit at $t\downarrow 0$ is the initial Dirac delta distribution, not a pointwise finite function.