P.W. Livermore $^{a}$, A. Jackson $^{a}$ and R. Kerswell $^{b}$
$^{a}$ School of Earth Sciences, University of Leeds, Leeds, UK $^{b}$ Department of Mathematics, University of Bristol, Bristol, UK
Kinematic dynamo theory has been used with great success to deduce the ability of a flow field {\bf u} in regenerating a magnetic field {\bf B} through time. The theory is based on the non-dimensionalised induction equation \[ \frac{\partial {\bf B}}{\partial t} = R_m\nabla\wedge\left({\bf u}\wedge {\bf B}\right) + \nabla^2 {\bf B} \] with magnetic Reynolds number $R_m$ being a measure of the strength of of forcing by the fluid flow {\bf u}. The normal approach to the induction equation is to seek eigenmodes of time dependence $\exp\lambda t$; self-sustaining dynamos occur for $R_m$ such that $\Re(\lambda)\ge 0$; the point at which $\Re(\lambda)=0$ is called the critical magnetic Reynolds number $R_m^c$. Whilst the eigenmodes of this equation determine the long term behaviour of the field, they are not an appropriate mechanism for understanding short term behaviour. It is perhaps not widely appreciated that the induction equation is governed by a so-called non-normal operator, associated with the non self-adjointness of the induction equation. Non-normal operators have associated with them non-orthogonal eigenfunctions, which though they form a complete set, may be almost linearly dependent. This property can lead to transient growth and decay of the field in time. We have carried out calculations to determine field behaviour in the sub critical region: $R_m\< R_m^c$, and can demonstrate magnetic energy growth for $R_m\<\< R_m^c$. For small $R_m$ the usual decay curves are shallowed, the effects acting on a time scale of the order of a dipole diffusion time. For larger $R_m$, transient growth can occur even for flows {\bf u} which are incapable of dynamo action. It is well known that the ability of a magnetic field to grow on long time scales is very sensitive to the specific form of $\bf u$. On the other hand, transient effects are much less so, and really only depend on $R_m$. Consequently these may provide a more intuitive and robust mechanism for understanding field growth. We will present results which will show the magnitudes and time scales over which these effects occur for different choices of flow {\bf u}. The physical applications of our theory will be (a) with respect to propeller-driven laboratory dynamos which use flows of Dudley-James type (b) concerned with the recovery of the field after reversals and excursions and (c) the future evolution of the current state of the magnetic field.