## Abstract

The non-normality associated with the induction equation may lead to subcritical growth of magnetic field structures even if all linear eigenmodes decay. We compute the magnitude of these transient effects for a collection of predominantly axisymmetric stationary spherical flows and find without exception the dominance of axisymmetric field growth above all other symmetries. The transient growth is robust under small flow perturbations and can be understood by simple physical mechanisms: either field line shearing or stretching. Magnetic energy amplification of is possible at magnetic Reynolds numbers of , and such effects could therefore lead the system from a principally non-magnetic state into one where the magnetic field plays a significant role in the dynamics.

## 1. Introduction

The geomagnetic field is sustained by a dynamo mechanism, which provides a means of regenerating the field against the effects of ohmic dissipation (e.g. Jacobs 1987). Despite the successful computation of many fully dynamical simulations of the geodynamo in recent years (Dormy *et al*. 2000; Jones 2000; Kono & Roberts 2002), our understanding of the physical mechanisms responsible remains incomplete, not least because the parameter-space accessible to computations is many orders of magnitude away from the real Earth. In the kinematic dynamo problem, in place of solving the Navier–Stokes equations for the fluid flow in Earth's core, the flow is instead prescribed and we merely investigate its ability to grow magnetic fields. Although, such an analysis is a gross simplification, it is ideally suited to studying magnetic field growth in isolation for a range of different core flows and other mean-field effects (Bullard & Gellman 1954; Gubbins 1973; Moffatt 1978; Dudley & James 1989). Geophysically, the flow is assumed to be generated by the dominant (non-magnetic) forces: pressure, buoyancy and Coriolis forces in Earth's core. The initial magnetic field is assumed to be weak and its associated Lorentz force negligible.

Kinematic dynamo theory is concerned with solutions of the magnetic induction equation (equation (2.1)), which is linear in the unknown magnetic field . The only parameter in the system is , the magnetic Reynolds number, which measures the strength of the flow relative to the effects of diffusion. Historically, solutions of the form have been sought, the location of the eigenvalues in the complex-plane defining the stability: if any lie in the positive-real half-plane then the corresponding eigenmodes will exponentially grow and indefinite magnetic field amplification will occur. Such a situation is a simple proxy for real dynamo action, although we expect that in reality the generated fields will saturate at some finite size due to the nonlinear Lorentz force. However, it is impossible to explain purely axisymmetric field growth using this type of solution since all eigenmodes of this symmetry necessarily decay (Cowling 1933). Therefore, no simple kinematic model, relying on eigenmode analysis, can hope to successfully model the large-scale geodynamo.

So-called subcritical transient growth has attracted much attention in the fluid dynamics community, by providing an explanation of the transition of (non-magnetic) basic laminar flows to a turbulent state (e.g. Schmid & Henningson 1994, 2001). Linearizing the system about its base state, it is found in almost all cases that eigenmode perturbations decay (i.e. the system is subcritical or linearly stable). However, non-eigenmode perturbations can still grow, sometimes being amplified by many orders of magnitude. This can occur because the underlying operator is non-normal, resulting in non-orthogonal eigenmodes. Even though they may individually decay, initially nearly cancelling modal superpositions separate in time and may cause large transient phenomena (Trefethen *et al*. 1993; Grossmann 2000). In a previous paper (Livermore & Jackson 2004) which we hereafter denote as Paper I, we studied the manifestation of the non-normality associated with the induction equation in the onset of magnetic energy growth. Our results differed markedly from eigenvalue analysis, not only in the field symmetry selected but in the critical value of required. In fact, in all flows studied, magnetic energy growth was possible at very moderate values of , despite nearly all cases being linearly stable for all . Additionally, the results were robust under perturbations in the flow, in contrast with the well-known sensitivity of the eigenspectra to the precise flow definition (Holme 1997; Gubbins *et al*. 2000*a*,*b*). In this paper, we continue the theme of the analysis and report the extent by which magnetic fields can transiently grow in a similar collection of core flows. The main focus of the work is on axisymmetric flows, principally for simplicity and computational expediency; strongly asymmetric flows are not treated. For more details on any of the methods or results the reader is referred to Livermore (2003).

Transient growth of magnetic fields is not a new idea, in fact one of the earliest papers reporting the existence of spherical fluid dynamos (Backus 1958) relied on this very mechanism. Backus considered a cyclic time sequence of various stationary flows, each linearly stable to magnetic fields. A period of stasis was imposed at each flow change-over, killing off high degree field harmonics. An initial field grew transiently in energy and before it entered its decaying phase, the flow changed and transient growth began again. Not all parts of the cycle led to magnetic field growth; indeed, in some the field was merely transformed from one structure to another, for example, by a quasi-solid body rotation. Nonetheless, despite the fact that at no time was the flow able to support growing eigenmodes, this functioning kinematic dynamo mechanism shows the importance of transient effects.

Non-normal effects also provide us with a simple mechanism for growing and sustaining axisymmetric fields. The impossibility of indefinitely sustaining such fields was proved by Cowling (1933) (and thus all axisymmetric eigenmodes necessarily decay), although in the framework of the Backus dynamo, we cannot rule out brief spells of transient growth being responsible for the generation of the dominant geomagnetic axisymmetric field component. In Paper I, we showed that in core flows dominated by axisymmetric poloidal flows (representing convective overturn) the growth of axisymmetric fields was favoured at onset. This preference for axisymmetries was also found by Farrell & Ioannou (1999*a*) in a cylindrical geometry, where they found possibilities for large (subcritical) transient axisymmetric behaviour of many orders of magnitude.

## 2. Formulation

In this section, we set out the details of the numerical method used. While there is some overlap with Paper I (which we include for completeness) particularly in §2*a–c*, the remaining analysis follows an entirely different methodology.

### (a) The geometry

In an identical set-up to Paper I, we consider a sphere *V* of constant conductivity (modelling Earth's outer core) embedded in a static infinite electrical insulator . The fluid in the sphere moves with a prescribed stationary flow . Non-dimensionalizing the radius of the sphere as the length-scale *L*, the root mean squared (r.m.s.) value of the velocity over the core as the velocity scale *U*, the magnetic diffusivity of the core as (where is the permitivity of free space) and the time-scale as , the magnetic induction equation reads(2.1)where . Typical values of are 100–1000 in Earth's core. The time-scale is that of diffusion, being around 200 000 years in Earth or dipole decay times (the slowest e-folding time of a magnetic field undergoing free ohmic decay). The relative diffusivity is unity in *V* and will be taken to be infinitely large (in a well-defined sense) in , representing a static electrical insulator. The flow is incompressible and obeys non-slip conditions on the boundary of *V*.

The magnetic energy is defined by(2.2)

### (b) Spatial discretization

We spatially discretize the induction equation by a Galerkin method, expanding in poloidal–toroidal decomposition and each of the defining scalars in spherical harmonics in and a suitable radial basis in *r*. Thus,(2.3)with(2.4)where *n* indexes the radial basis comprising recombined Chebyshev polynomials. These are chosen to not only satisfy the boundary conditions implicitly but to exploit symmetry considerations as well (see appendix A of Livermore & Jackson (2004) for full details). The notation *α* indexes , spherical harmonics of degree , order where and azimuthal dependence . The harmonics are Schmidt quasi-normalized, such that the integral over all solid angle of the square of each harmonic is given by . We will refer to vector spherical harmonics using the notation or , omitting the superscript if *m*=0, which represent, respectively, toroidal or poloidal vectors involving a single spherical harmonic.

Writing(2.5)where is the *j*th global basis function (*j* indexes both *α* and *n*), we can form an equation for by taking the inner product of equation (2.1) with each basis function in turn and integrating over all space. Note that since is determined everywhere in by its behaviour on (a consequence of being an electrical insulator) we could choose any domain over which to integrate. We choose for consistency with the definition of magnetic energy.

The resulting equation, after pre-multiplying by , is(2.6)where(2.7)For steady flows, equation (2.6) has the solution(2.8)where we may make sense of the matrix exponential by the usual Taylor series expansion. Using the vector identity , the matrix can be evaluated by noting that(2.9)In a similar manner to Paper I, this can be greatly simplified. Applying the pre-Maxwell equations on where , the (non-dimensional) electric field is and we may use continuity of both and to see that the integral over vanishes. Additionally, applying the behaviour of in a source-free electrical insulator: and ensures that the integral over is also zero. Regarding the second term, if we allow (the non-dimensional diffusivity) in so that , then the contribution from from equation (2.9) vanishes. Applying these arguments to the physically permissible field structures and , it follows that(2.10)The condition that on is not necessary but simplifies matters considerably. Without it, the continuity conditions at the boundary imply that contains extra terms involving the interaction of , and on . The matrices and are block diagonal due to the orthogonality of spherical harmonics and in fact we only have to concern ourselves with the integration in the radial direction in these cases. The poloidal and toroidal scalars defining the magnetic field are all polynomials in *r* and so Gauss–Legendre quadrature may be used with some minimal number of abscissae. To compute , we use fast Fourier transforms in *ϕ* and Gauss–Legendre quadrature in and *r*. Again, if the defining radial scalar functions of are determined in terms of polynomials, the radial integration is numerically straightforward; however, in other cases we compute the integrals using a sufficiently large number of abscissae such that the matrix entries converge. These computations only have to be performed once and stored on disk.

### (c) Harmonic selection and truncation

The induction equation admits certain symmetry separations in if is chosen appropriately. In particular, an equatorially symmetric flow gives rise to separation between and (equatorially antisymmetric) magnetic fields. Additionally, complete separation in the azimuthal wavenumber *m* occurs if the flow is axisymmetric. This separation is manifested computationally in the set of selection rules (Bullard & Gellman 1954), who detail the non-zero vector harmonic interactions that can occur if is also expressed in vector harmonics. The spherical harmonics are truncated in a triangular scheme, whereby for a given . The radial basis functions were truncated as where was typically chosen to be identical to .

### (d) Envelopes

The degree of relative transient amplification of a magnetic field from its initial state clearly depends on its initial configuration (although not on its amplitude since the induction equation is linear). We quantify the growth by considering the maximal relative magnetic energy growth at each time *t* over all possible initial fields:(2.11)The function defines an envelope that supplies an upper bound to the evolution of the relative magnetic energy growth for each possible initial field. Figure 1 shows the envelope of magnetic energy growth under the action of the (MDJ) flow (see §3) with (dashed line); the evolution of two particular initial conditions are shown as solid lines, chosen to optimize the energy growth at and 0.1. They touch the envelope at these points but in general it bounds them strictly from above. The initial condition chosen to optimize energy growth for apparently approximates the envelope for most times fairly well. In contrast, the optimal curve for does not, and decays for .

The envelope height gives a good measure of the possible transient amplification, which we find using a simple bisection algorithm. The optimizing time (the time at which the envelope maximum is achieved) also comes out of the calculations and in all cases presented it corresponds to a geophysically reasonable time-scale of years. We note that although the envelope curve is plotted as a function of time, it itself does not ‘grow’ in the same way as a particular initial magnetic field will grow—it merely bounds each such evolution from above. The envelope supplies information regarding the optimal initial field, and thus defines the maximal transient amplification and the time-scale over which it is achieved.

In the limit , the initial gradient of recovers the onset problem:(2.12)where is the maximal instantaneous growth rate of magnetic energy (of Paper I), the factor of being part of the definition.

### (e) Computing the envelope curve

We may compute the function , from equations (2.11), (2.5), (2.7) and (2.8), in the following manner (e.g. Schmid & Henningson 1994):(2.13)recalling that the matrix is the basis energy inner product matrix. Since this is symmetric and positive definite, we may write it as using the Cholesky decomposition (see Press *et al*. 1992). The above relation then becomes(2.14)where | | is the usual Euclidean 2-norm of a vector. Writing we obtain(2.15)where is the Euclidean 2-norm of a matrix, which may be effectively calculated using the singular value decomposition. The matrix exponential was computed using a Padé approximation of order 6 with scaling and squaring (see Moler & Van Loan 1978). This is identical to the algorithm used in the package Matlab.

## 3. Flows studied in this paper

Table 1 details the definitions of the flows studied in this paper. They are all defined up to a multiplicative constant which is chosen so that the r.m.s. value within *V* is unity (when ). Toroidal flows model differential rotation, thought to be an extremely important aspect of the geodynamo. Poloidal flows represent convective overturn motion, driven by buoyant material released at the inner core boundary (e.g. Fearn 1998). The (IC) flow contains an inner core (IC) of (geophysical) non-dimensional radius 0.35 which is modelled by quiescent flow (of identical conductivity) in this region, and is depicted in figure 2. The other or flows (or combinations thereof) are either modified versions of flows proposed by Dudley & James (1989) (hence MDJ) or taken directly from the axisymmetric part of the (non-axisymmetric) flow of Kumar & Roberts (1975) (hence KR). Both the (MDJ) and (KR) flows have a stagnation point at the centre of the core; in the latter case the strain rates also vanish there.

In addition to the full non-axisymmetric flow of KR, we refer to the (non-axisymmetric) flow proposed by Sarson (2003) as STW (Sarson thermal wind). Both of these flows are small perturbations to , this component making up 99% and 96%, respectively, by r.m.s.

When citing flows comprising both and components, i.e. flows, we define the ratio of the r.m.s. values between the individual parts as(3.1)

## 4. Transient magnetic energy amplification in an axisymmetric toroidal flow

Flows comprising only toroidal components can never indefinitely sustain magnetic fields and consequently only admit decaying magnetic eigenmodes (Bullard & Gellman 1954). However, they are an important aspect of many geophysical and astrophysical dynamos, generating toroidal field from poloidal by differential rotation (e.g. Moffatt 1978). We detail here the transient field amplification capability of an axisymmetric degree one toroidal flow (MDJ).

### (a) Symmetry separation

Figure 3 shows the magnetic energy envelopes for the four symmetries of largest scale. Grey shows the *m*=0 symmetry, black *m*=1, solid and dashed . It is apparent that the (axisymmetric) grey curves dominate over the time-scale shown; the highest peak being attained by the symmetry class which contains the axisymmetric dipole. A non-dimensional time of 0.2 corresponds to about two dipole diffusion times (about 40 000 years in dimensional terms).

At small times, a different field symmetry becomes important: at onset, the favoured field symmetry is (Paper I). It is therefore of note that the critical symmetry depends on the time-scale of interest.

### (b) Convergence

For this flow, truncation levels of are sufficient to resolve the envelope height and corresponding dynamics for values of , indicating that the maximizing initial field and subsequent evolution are of large scale. The time at which the envelope maxima occur are also independent of (for ), at around (7800 years).

Figure 4*a*,*b* shows the height of the envelope for the dominant symmetry and its associated optimizing time as functions of . On the log–log scale, the graph of figure 4*a* is almost perfectly linear for . This indicates a power-law scaling which we compute to be . Growth of is possible at .

### (c) The nature of the growth

The transient growth, being none other than the well-known -effect of mean-field magnetohydrodynamics (MHD), is simple to understand in this case: toroidal field is generated from poloidal by the field lines being sheared around the symmetry-axis by the flow. There is no generative mechanism for the poloidal field; it therefore decays. This is shown in figure 5*a* where the toroidal (dashed line) and poloidal (solid line) energies, respectively, are plotted for the time evolution of the initial magnetic field chosen to maximize the energy at for .

The associated axisymmetric poloidal-only initial field (whose structure is in fact unchanged with higher values of ) is shown in figure 5*b* by a selection of its field lines. It is dominated by the dipolar *l*=1 component.

### (d) A simple model

We can understand the physical mechanism and scaling laws by considering the simple model(4.1)(4.2)where the scalars and represent the poloidal and toroidal field components respectively; is the slowest poloidal decay rate and is the slowest toroidal decay rate (Backus 1958). We model the shearing of poloidal field into toroidal field by the simple source term .

The equations are trivially solved to be(4.3)(4.4)if the initial field is chosen such that and , which equate to the optimal initial configuration. Suppose now we want to find the maximal energy growth and the time at which it is attained. It is clear that the poloidal contribution merely decays and that the field will be dominated (as expected) by the toroidal component *T*, particularly for large . If we consider only the toroidal contribution to the magnetic energy, then the maximum energy scales as and the optimizing time is independent of , determined by solving as:(4.5)Despite the crudity of this model, we recover the independence of from ; indeed, its value is surprisingly close to the numerical value of 0.0407.

The simple scaling of can come about, because toroidal field is generated at a rate given approximately by , which decays only slightly over the time-scale (independent of here). Therefore, the field intensity scales as and hence the energy as . In other cases, such arguments may not apply, since the rate of field creation will not in general be constant over the relevant time-scale. Additionally, we expect that in general, will be a decreasing function of , since one might expect that the faster the flow the shorter the time-scale of magnetic field growth.

## 5. Transient magnetic energy amplification of flows

### (a) Symmetry separation

Figure 6*a–c* shows envelope calculations for flows (MDJ), (KR) and (IC), respectively, for the four independent symmetries as before with .

In each case, the field symmetry (solid grey) is the most dominant, identical to that favoured in the onset of magnetic energy growth (Paper I). There are several ‘kinks’ in the above curves (e.g. in the solid grey curve of figure 6*a* and the dashed black curve of figure 6*c*). These are not convergence errors, but can be understood by the envelope being the maximum of two different curves, one describing optimal growth for small times and one for large times. The (MDJ) flow can give a magnetic energy amplification of 52, the (KR) flow of 12 and the (IC) flow of 9. It would appear that the behaviour at the centre of *V* is of great importance in determining this maximal amplification: the more vigorous the flow at or near the centre, the greater the potential for large growth. Recall that the (KR) flow has not only a stagnation point but vanishing strain rates at the origin, and that (IC) is static for . Despite the differences in the flows, however, since the favoured symmetry is the same, we expect similar physical growth mechanisms to apply.

### (b) Convergence

We now determine the maximum envelope height for the (MDJ) flow as a function of and truncation level (for the dominant field symmetry), as shown in table 2. Convergence of the envelope maximum (as a function of truncation level) seems to ensure that the whole envelope is converged, since this point is the least well behaved. The calculations at converge surprisingly poorly, and we can only be confident that the values are correct to four significant figures which is sufficient for graphical purposes.

Immediately, we see a dramatic difference in behaviour from the envelopes of the (MDJ) flow. Apparently, the initial fields that maximize the magnetic energy growth decrease in length-scale as increases, manifested by the higher resolution required. We see also an increase of envelope height with and a decrease in the associated optimizing time.

We lastly note that this convergence behaviour is also seen in the envelope heights for the other (KR) and (IC) flows.

### (c) Scaling

We now investigate how the envelope height scales with for the three flows, for the dominant field symmetry. Table 2 shows that solutions converged to more than five significant figures are not guaranteed above . However, we shall consider up to and hope that convergence errors are not large enough to impair the general behaviour.

Figure 7*a* shows plots of the envelope height as a function of for the three flows; figure 7*b* shows the relevant optimizing times. We use the highest truncation that is computationally tractable: . Large transient growth of is possible at . We may fit power laws through the points located in the range as approximations to the asymptotic behaviour (detailed in table 3). The curves in figure 7*a* are not perfectly straight in log–log space, indicating that either numerical inaccuracies at large are important, or that the asymptotic regime (should it exist) has not yet been reached. The curves in figure 7*b* apparently reach linear asymptotes in the range shown. Indeed, the fact that an asymptote has apparently been reached suggests that similar behaviour should occur in figure 7*a*, but is affected significantly by numerical errors or truncation inaccuracies.

The (KR) and (IC) flows are themselves very similar in their asymptotic scalings (table 3); indeed, this behaviour is also identified in figure 7*a*. The envelope height for the (MDJ) flow seems to obeys a much more favourable scaling, leading to much greater potential for transient behaviour.

Despite the differences in envelope height scaling, the scalings of the optimizing time with are almost indistinguishable bearing in mind the likely errors associated with such approximate asymptotes. Therefore, generically, in the range , the envelope height scales as , where , depending on the flow chosen, and the optimizing time as . We note that these scalings lead to the increase of transient growth with , occurring on ever shorter time periods. This is intuitive: on driving the flow faster we expect that it becomes easier to generate magnetic fields and that any associated mechanism will operate more rapidly.

### (d) The nature of the transient growth

We now investigate the nature of the transient growth and the physical process responsible for the case of the (IC) flow; the other flows behave similarly. For , the envelope height is about 9 with an optimizing time , corresponding to 1800 years in dimensional terms. We forward propagate the associated initial field through 1500 years, and plot contours of (figure 8). The initial field, which is poloidal only and dominated by the axisymmetric dipole component, is advected towards the symmetry-axis immediately above and below the solid core by the flow. There, it is subject to large local vertical stretching, the orientation of the field being approximately aligned with the direction of maximal strain rate. This is a very similar mechanism to the onset of magnetic energy growth (Paper I), except in that case the field is only instantaneously stretched and does not undergo any finite time advection.

A similar sequence of images comes about from the (MDJ) and (KR) flows, although stronger stretching occurs near the centre of *V* (giving rise to the higher amplification possible). Thus, the physical mechanism responsible is the same despite the slight difference in flow geometry.

## 6. Transient magnetic energy amplification of flows

### (a) Symmetry separation

Figure 9*a–d* shows the magnetic energy envelopes for the various independent symmetries at for the flows: (MDJ), ; (MDJ), *τ*=1; (KR), and (KR), *τ*=1, respectively.

Again, we see that a robust feature of these flows is the dominance of the geophysically relevant field symmetry (shown as solid grey); this is perhaps not unexpected, since such a symmetry dominates flows consisting of the and components taken individually.

Both the (MDJ) flows out-perform the (KR) flows by a factor of four (note the vertical scale). This stems from the increased efficiency of the (MDJ) component relative to the (KR).

Two of these plots have an eigenmode that dominates at large times: the (MDJ), flow has an eigenmode that becomes unstable at and the (KR), flow has a slowly decaying eigenmode . In each case, this symmetry is highly suboptimal for (which in dimensional time is 7700 years). The existence of these growing eigenmodes is not relevant to the study due to the well-known sensitivity of the eigenspectrum on the precise definition of the flow. We refer the reader to §10 regarding this point.

### (b) Envelope height scalings

Figure 10*a*,*b* shows the variation of the envelope height for the four different flows with the associated optimizing time. We study only the dominant symmetry with a truncation of . Solid black shows (MDJ), , dashed black (MDJ), *τ*=1, solid grey (KR), and dashed grey (KR), *τ*=1.

Again, we approximate the asymptotic scalings by power laws, as shown in table 3. All of the flows have similar properties, but the two sets of (MDJ) and (KR) curves individually look and behave almost alike. Indeed, the analysis does not critically depend on the exact choice of *τ*, but shows generic behaviour.

Of the (MDJ) flows, the one with *τ*=1 (the equipartition flow) supports greater growth. This is in contrast with linear analysis, where for the symmetry, a growing eigenmode exists for but not for *τ*=1. Thus, even when a subset of the *τ* parameter-space generates linearly unstable magnetic fields, it is not coincident with that giving the maximum transient energy growth.

## 7. Transient amplification in non-axisymmetric flows

It is instructive to study the robustness of the above results by adding in a small non-axisymmetric flow component. We follow the same analysis using the KR and STW flows which are 99% and 96%, respectively, comprised by r.m.s. Both of these flows are although they introduce *m*=2 harmonics. The field now separates into only four different symmetry classes: and which are themselves divided into sets containing either odd or even values of *m*.

### (a) Symmetry separation

Figure 11*a*,*b* shows the magnetic energy envelopes for the KR and the STW flows, respectively, both at . We use a truncation level of , which is just enough to attain sufficient convergence. Since the flows are not axisymmetric, many more harmonics are needed in the symmetry classes, and in this case the matrices are of dimension . This is approximately the maximum resolution tractable, since not only are matrix exponentials computationally expensive, but numerical accuracy issues and spurious singular values may arise from large matrix calculations. In figure 11, the symmetries shown are: grey, containing even values of *m*, black, containing odd values of *m*; solid shows and dashed symmetry.

It is interesting to compare these curves with the equivalent for the (MDJ) flow (figure 3): they look almost identical. Thus, small differences in the flow leave the transient growth process unchanged, in stark contrast to the behaviour of the eigenvalues: both the KR and STW flows become eigenvalue unstable (i.e. they allow a growing magnetic field eigenfunction) as is increased and yet the (MDJ) is eigenvalue stable for all .

We could not compute any asymptotic scalings since calculations with were unconverged.

## 8. Comparison

We summarize our results in table 3 for comparison. The leftmost five columns show the maximum envelope height with optimizing time for along with their approximate asymptotic scalings. We only show the dominant field symmetry over the transient regime, which for each flow is , identical to that which dominates the geomagnetic spectrum. The flows vary considerably in amplification capability, the (MDJ) being the most effective. However, their optimizing time is similar for (as are their asymptotic scalings), and indeed the same physical mechanism is responsible for each, that of advection towards the symmetry-axis followed by stretching. In the case of the (MDJ) (and the (MDJ)) flow, transient magnetic energy growth is possible of at .

The flows again vary in envelope height, with those containing the (MDJ) component performing the best. Although those containing the (KR) component in fact show a lesser energy amplification than that of (KR) itself, the asymptotic scaling suggests that for larger this efficiency difference will reverse. It is interesting to compare these scalings with those of Farrell & Ioannou (1999*a*) for axisymmetric fields in a cylindrical geometry: they found that the magnetic energy amplification scaled as and the optimizing time as . Although the optimizing time appears to scale similarly, the spherical flows that we consider offer much more favourable amplification. We must note though that our scalings may not reflect the asymptotic regime, which may be out of reach of our computational domain. Farrell & Ioannou were able to go to a much higher truncation due to additional axial wave separation, something that is particular to their geometry.

Overall there are two main mechanisms for generating transient magnetic fields: that associated with of -effect-like field line shearing and that associated with of field line stretching. All the flows considered are either combinations of these or close to one (in r.m.s.). Small flow perturbations may alter the effectiveness of the energy amplification, but the mechanism remains the same. In addition, as increases, the ability of all the flows to grow magnetic fields concomitantly and intuitively rises.

Column six of table 3 gives the relative growth, from *t*=0 until , of the Lorentz force in r.m.s. value within *V*. This gives an idea of the likely importance of the back-reaction of the magnetic field onto the flow sustaining it in the transient regime. Again, those flows with a large (MDJ) component have an associated large increase in the Lorentz force, caused by the large amplification on small length scales. However, the relative growth in the Lorentz force does not tell us much about its actual magnitude in Earth's core, should these transient growths take effect there.

We model the importance of the generated Lorentz force by comparing its magnitude to another of the forces thought to be in quasi-balance in Earth's core: the Coriolis force. We make the assumption that the flows studied in this paper can be produced from such a geophysical force balance, although we stress that the flows are prescribed analytically and derived from qualitative and not quantitative geophysical arguments. The Coriolis force is defined by , where is Earth's angular rotation rate, (Dziewonski & Anderson 1981) is the density of the core and is the unit vector along the rotation axis. Its magnitude is computed by r.m.s. within *V*. A geophysical Lorentz force is computed by rescaling the initial magnetic field to have the same external dipole component as Earth (Livermore 2003). The dimensional velocity scale was determined by , where *L* is the radius of the outer core (3485 km), and is the dimensional magnetic diffusivity based on a core conductivity of (Gubbins & Roberts 1987). The ratio , which measures the relative importance, in r.m.s., of Lorentz to Coriolis forces in the core (and effectively is the square of a modified Elsasser number) is shown in column seven of table 3. The time at which it is computed is with the usual optimizing initial field configuration. Importantly, five of the eight flows shown have this ratio of , indicating that the Lorentz force is strong enough to enter the force balance at the transient peak (and in fact the Lorentz force itself peaks at a similar time). In the toroidal-only case, the Lorentz force generated is relatively small compared to the Coriolis force. We note that finding the optimizing time for (or effectively ) is not possible with the method as presented in this paper, because a measure of the size of the Lorentz force would be quartic in rather than quadratic (like the energy) and so cannot be written in a symmetric quadratic form.

Column eight of table 3 shows the geophysically scaled ohmic heating (in GW) , evaluated at . In each case, the ohmic heating associated with the maximal transient growth is at least an order of magnitude smaller than the value of 1–2 TW of Roberts *et al*. (2003). However, for larger (yet still geophysically realistic) values of , we find that for the (MDJ) flow the ohmic heating is 27.18 TW, much bigger than the estimated value of 1–2 TW.

## 9. Application to run-time of geodynamo simulations

Nonlinear geodynamo computations model the fully dynamical set of equations, of which the kinematic state is a simplification. The computational runs must be long enough (in model time) to ensure that any transient processes associated with the initial conditions (which are often fairly arbitrary) have decayed away. Using the kinematic framework, we can provide a lower bound on this minimum run-time by computing for the various flows the longest duration of the transient phase. This is defined as the time at which the envelope curves intersect the initial magnetic energy, i.e. . We use only the dominant symmetry in the calculations with and find the intersection of interest using a bisection method. In general, fully nonlinear transient dynamics may occur which last longer than those amenable to computation here.

Table 4 shows the non-dimensional time , the number of corresponding dipole diffusion times (given simply by the expression ) and the associated dimensional time in years. No converged solutions were found for the (MDJ) flow with nor for the KR and STW flows with .

For this value of , the minimum computational time required for a run to discount all transient effects is 3.52 dipole diffusion times (69 000 years). Such bounds depend on the particular flow studied: this figure is associated with the (MDJ) flow. It is of note that this is comparable to the analytic bound of Backus (1957) regarding the extension of the dipole diffusion time by a factor of 4 by non-normal effects. Generically we require at least three dipole diffusion times for a computational run, in line with most current simulations (see Kono & Roberts 2002).

## 10. Discussion

In Paper I, we showed that the question of magnetic energy stability in spherical dynamos was distinct from that of eigenmode stability and that transient growth was possible for , raising the issue of how large an effect this could be for larger values of . In this paper, we have demonstrated that substantial subcritical growth can occur of , attainable at geophysically plausible values of , although this depends on the flow. The physical mechanisms, involving the two principle flow types studied, are easily understood to be either field line shearing (the classical -effect) or advection followed by stretching (in the vicinity of their upwelling) by poloidal flows. All flows studied in this paper, without exception, favour the field symmetry containing the axisymmetric dipole, in stark contrast to the eigenmode problem where all fields of symmetry must necessarily decay. This is therefore suggestive that a simple kinematic description of the geodynamo is still possible, at least for the large-scale axisymmetric field. However, we stress the simplicity of our model and that it almost certainly is not a generic description of planetary dynamos—transient growth cannot explain the equatorially aligned fields seen in some fully nonlinear studies (e.g. Aubert & Wicht 2004), despite the flows being approximately axially aligned.

These results show some similarity to the onset of magnetic energy growth, which is recovered in the transient case by allowing the time-scale of interest to become vanishingly small. For poloidal flows, the same symmetries are preferred and in fact the field gets stretched in a similar fashion to the transient case. The toroidal flows favour different field symmetries, which are more amenable to instantaneous stretching than shearing.

A strong conclusion of this work is the robustness of the transient growth phase: the results are almost unchanged for small flow perturbations, either by changing the flow geometry (for example between any of the flows presented) or the addition of vector harmonic components, including those with no axisymmetry. In particular, the growth mechanism remains unaltered which means we can attach a meaningful physical interpretation to it. This is completely different from the eigenmode problem, where it is well known that small flow perturbations can catastrophically alter the stability—growing modes become decaying modes and vice versa. This is clearly an important issue, since the structure of our flows is a gross simplification of the outer core. We only hope to capture large-scale features and so only robust results can carry any weight at all. Additionally, the turnover time-scale of Earth's outer core is years (based on the core-surface flow speed of ) which is long enough for typical transient growth to take effect, but not long enough for the eigenmodes (should some be unstable) to dominate (in §6 we found this requires *O*(7000) years). In view of their sensitivity, they are not likely to survive more than a small fraction of a turnover time and despite their existence in some stationary flows, they are of no relevance to this study.

Non-normal effects may be important in understanding the long-term geodynamo mechanism. If we model the outer core flow as a cycle of stationary flows (Backus 1958), at least one of which having a large axisymmetric poloidal component, then transient growth of the axisymmetric field could occur. Provided that in other parts of the cycle other field symmetries could grow, the process could feed back on itself and such a mechanism could then plausibly indefinitely sustain Earth's magnetic field. Alternatively, statistically steady flows undergoing stochastic forcing have been shown to exhibit growth of the ‘transient’ modes rather than the eigenmodes. In a cylindrical geometry, Farrell & Ioannou (1999*b*) showed that axisymmetric fields can be sustained by such a forcing. If Earth's core is strongly turbulent yet quasi-stationary, then this assumption may hold reasonably well and extrapolating the results of this work, the geodynamo may be understood in a similar fashion.

Transient growth might also be helpful in explaining how the geomagnetic field recovers after its many global reversals (Merrill *et al*. 1998). There is evidence of a swift growth of the axisymmetric dipole component immediately after polarity transitions which are associated with a significant drop in surface field intensity to 10–20% of its usual value (Merrill & McFadden 1990; Bogue & Paul 1993). If the Lorentz force in the core concomitantly drops, then we might expect that the core adopts a force balance which is almost independent of the (weak) magnetic field. In such cases, if the flow is steady we can use the kinematic approximation with flows containing a strong convective overturn component to explain the swift regeneration of the axisymmetric field. In our models, this typically occurs over a time-scale of years which is similar to that found in the paleomagnetic data. Associated with the field recovery is the increase in Lorentz force which (according to our calculations) becomes comparable to the Coriolis force at geophysical realistic values of . Thus, the limit of the transient phase is possibly marked by the end of the kinematic regime and the subsequent alteration of the flow pattern.

## Acknowledgements

The authors would like to thank Rich Kerswell for useful discussions and the anonymous referees for their constructive criticism.

## Footnotes

↵† Present address: Institut für Geophysik, ETH Zurich, Zurich, Switzerland.

- Received November 16, 2004.
- Accepted December 14, 2005.

- © 2006 The Royal Society