Small-scale dynamo action in rotating compressible convection
SSmall-scale dynamo action in rotating compressibleconvection
B. Favier ∗ and P.J. BushbySchool of Mathematics and Statistics, Newcastle University,Newcastle upon Tyne NE1 7RU, UK Abstract
We study dynamo action in a convective layer of electrically-conducting, compressiblefluid, rotating about the vertical axis. At the upper and lower bounding surfaces, perfectly-conducting boundary conditions are adopted for the magnetic field. Two different levels ofthermal stratification are considered. If the magnetic diffusivity is sufficiently small, the con-vection acts as a small-scale dynamo. Using a definition for the magnetic Reynolds number R M that is based upon the horizontal integral scale and the horizontally-averaged velocity atthe mid-layer of the domain, we find that rotation tends to reduce the critical value of R M above which dynamo action is observed. Increasing the level of thermal stratification withinthe layer does not significantly alter the critical value of R M in the rotating calculations, butit does lead to a reduction in this critical value in the non-rotating cases. At the highestcomputationally-accessible values of the magnetic Reynolds number, the saturation levels ofthe dynamo are similar in all cases, with the mean magnetic energy density somewhere be-tween 4 and 9% of the mean kinetic energy density. To gain further insights into the differencesbetween rotating and non-rotating convection, we quantify the stretching properties of eachflow by measuring Lyapunov exponents. Away from the boundaries, the rate of stretching dueto the flow is much less dependent upon depth in the rotating cases than it is in the corre-sponding non-rotating calculations. It is also shown that the effects of rotation significantlyreduce the magnetic energy dissipation in the lower part of the layer. We also investigatecertain aspects of the saturation mechanism of the dynamo. In a hydromagnetic dynamo, motions in an electrically-conducting fluid lead to the ampli-fication of a seed magnetic field. Dynamo action can only occur if the inductive effects ofthe fluid motions outweigh the dissipative effects of magnetic diffusion. Provided that themagnetic energy density of the seed field is very much less than the kinetic energy density ofthe flow, the early stages of evolution of the dynamo process are effectively kinematic, whichimplies that the magnetic energy in the system grows exponentially with time (although themagnetic energy will tend to fluctuate about this exponential trajectory if the velocity fieldis time-dependent). Eventually, however, the nonlinear feedback of the magnetic field uponthe flow (via the Lorentz force) becomes dynamically significant. This halts the growth of thedynamo-generated magnetic field, leading to a saturated nonlinear dynamo. There are manyexamples of natural dynamos. For example, it is generally accepted that the 11 year solar mag-netic cycle (see, for example, Stix, 2004), is driven by an oscillatory dynamo. Similar dynamomechanisms drive cyclic magnetic activity on other stars (Baliunas et al. , 1996). Dynamoaction is also responsible for sustaining the Earth’s magnetic field (see, for example, Roberts& Glatzmaier, 2000). It is also possible to drive hydromagnetic dynamos in laboratories, asdemonstrated by recent liquid metal experiments (such as the French VKS experiment, seefor example, Monchaux et al. , 2009).Convection plays a crucial role in many natural dynamos, particularly in the context ofsolar, stellar and planetary dynamos. Although it is possible to study global dynamo mod-els numerically (Brun et al. , 2004), most of the theoretical work on this problem has been ∗ Corresponding author: [email protected] a r X i v : . [ a s t r o - ph . S R ] O c t ased upon local models of convection, where a fluid layer is heated from below and cooledfrom above. In the context of Boussinesq convection, this classical setup has been studiednumerically by Meneguzzi & Pouquet (1989) and Cattaneo (1999). In highly-conducting flu-ids, Boussinesq convection can act as an efficient dynamo, producing small-scale, intermittentmagnetic fields. In the Boussinesq approximation, the effects of compressibility are neglected.More recent studies have focused upon dynamo action in local models of convection in fullycompressible fluids (V¨ogler & Sch¨ussler, 2007; Brummell et al. , 2010; Bushby et al. , 2010). Itis now well-established that these compressible models can also drive small-scale dynamos.Previous calculations have clearly demonstrated that rotation is not a necessary ingredientfor small-scale dynamo action in convectively-driven flows. However, rotation is present in mostnatural dynamos, and this additional physical effect can profoundly influence the behaviourof a convective layer. For example, when the rotation vector is aligned with the vertical axis,rapid rotation not only inhibits convection, but it also reduces the preferred horizontal scaleof the convective instability (Chandrasekhar, 1961). Even far from onset, when compressibleconvection is fully turbulent, rotation can also play an important role: helical convectiveplumes tend to become aligned with the rotation vector. This is particularly apparent whenthe axis of rotation is tilted away from the vertical (Brummell et al. , 1996, 1998 b ). Given thesehydrodynamical considerations, we might reasonably expect the dynamo properties of rotatingconvection to differ from the equivalent non-rotating case. There have been numerous studiesof this problem in Boussinesq convection (Childress & Soward, 1972; St. Pierre, 1993; Jones& Roberts, 2000; Stellmach & Hansen, 2004; Cattaneo & Hughes, 2006), but the compressiblecase is less well understood. Existing studies have generally adopted complex models, withmultiple polytropic layers, inclined rotation vectors, or additional physical effects such as animposed velocity shear (see, for example, Brandenburg et al. , 1996; K¨apyl¨a et al. , 2008, 2009).In fact, the simpler problem of dynamo action in a single layer of turbulent compressibleconvection, rotating about the vertical axis, is still not fully understood. Therefore, the aimof this paper is to study the ways in which rotation and compressibility influence the dynamoproperties of convection in a simple polytropic layer.The paper is organised as follows. The governing equations, boundary conditions andparameters, together with the numerical methods are discussed in the next section. Consid-erations about the dimensionless numbers of interest, and the Lyapunov exponents of hydro-dynamic convection are presented in Section 3. In Section 4, we discuss results from a set ofdynamo calculations. Finally, in Section 5, we present our conclusions. We consider the evolution of a plane-parallel layer of compressible fluid, bounded above andbelow by two impenetrable, stress-free walls, a distance d apart. The upper and lower bound-aries are held at fixed temperatures, T and T + ∆ T respectively. Taking ∆ T > x and y corresponding to the horizontal coordinates. The z -axis points verticallydownwards, parallel to the constant gravitational acceleration g = g e z . The layer is rotatingabout the z -axis, with a constant angular velocity Ω = Ω ˆ z . The horizontal size of the fluiddomain is defined by the aspect ratio λ so that the fluid occupies the domain 0 < z < d and 0 < x, y < λd . The physical properties of the fluid, namely the specific heats c p and c v ,the dynamic viscosity µ , the thermal conductivity K , the magnetic permeability µ and themagnetic diffusivity η , are assumed to be constant. The model is identical to that used byMatthews et al. (1995), except for the addition of rotation.It is convenient to introduce dimensionless variables, so we adopt the scalings describedin Bushby et al. (2008). Lengths are scaled with the depth of the layer d . The temperature T and the density ρ are scaled with their values at the upper surface, T and ρ respectively.The velocity u is scaled with the isothermal sound speed √ R ∗ T at the top of the layer, where R ∗ = c p − c v is the gas constant. We adopt the same scaling for the Alfv´en speed, whichimplies that the magnetic field B is scaled with √ µ ρ R ∗ T . Finally, we scale time by anacoustic time scale d/ √ R ∗ T .We now express the governing equations in terms of these dimensionless variables. The quation for conservation of mass is given by ∂ρ∂t = −∇ · ( ρ u ) . (1)Similarly, the dimensionless momentum equation can be written in the following form, ρ (cid:18) ∂ u ∂t + κσT a / ˆ z × u (cid:19) = −∇ (cid:18) P + B (cid:19) + θ ( m + 1) ρ ˆ z + ∇ · ( BB − ρ uu + κσ τ ) , (2)where P is the pressure, given by the equation of state P = ρT , and τ is the stress tensordefined by τ ij = ∂u i ∂x j + ∂u j ∂x i − δ ij ∂u k ∂x k . (3)Several non-dimensional parameters appear in Equation (2). The parameter θ = ∆ T /T is thedimensionless temperature difference across the layer, whilst m = gd/R ∗ ∆ T − κ = K/dρ c p ( R ∗ T ) / and σ = µc p /K is the Prandtl number. Finally, T a is the standard Taylor number, T a =4 ρ Ω d /µ , evaluated at the upper boundary. The induction equation for the magnetic fieldis ∂ B ∂t = ∇ × ( u × B − ζ κ ∇ × B ) , (4)where ζ = ηc p ρ /K is the ratio of magnetic to thermal diffusivity at the top of the layer.The magnetic field is solenoidal so that ∇ · B = 0 . (5)Finally, the heat equation is ∂T∂t = − u · ∇ T − ( γ − T ∇ · u + κγρ ∇ T + κ ( γ − ρ (cid:0) στ / ζ |∇ × B | (cid:1) , (6)where γ = c p /c v .To complete the specification of the model, some boundary conditions must be imposed.In the horizontal directions, all variables are assumed to be periodic. As has already beendescribed, the upper and lower boundaries are assumed to be impermeable and stress-free,which implies that u x,z = u y,z = u z = 0 at z = 0 (the upper boundary) and z = 1 (thelower boundary). Having non-dimensionalised the system, the thermal boundary conditionsat these surfaces correspond to fixing T = 1 at z = 0 and T = 1 + θ at z = 1. For the magneticfield boundary conditions, two choices have typically been made in previous studies. One canconsider the upper and lower boundaries to be perfect electrical conductors or one can adopt avertical field boundary condition. We choose appropriate conditions for perfectly-conductingboundaries, which implies that B z = B x,z = B y,z = 0 at z = 0 and z = 1. This is partlyto facilitate comparison with the Boussinesq study of Cattaneo & Hughes (2006), but this isnot the only motivation for adopting boundary conditions that enforce B z = 0 at the uppersurface. Strong concentrations of vertical magnetic flux at the upper surface tend to becomepartially evacuated (see, for example, Bushby et al. , 2008), which dramatically increases thelocal Alfv´en speed (as well as significantly reducing the local thermal diffusion time scale).This can impose very strong constraints upon the time-step in any explicit numerical scheme.Hence, there are also numerical reasons for adopting perfectly-conducting boundary conditions.In this context, it is worth noting that it has been shown in the Boussinesq case (Thelen &Cattaneo, 2000) that the dynamo efficiency of convection is largely insensitive to the detailedchoice of boundary conditions for the magnetic field (see also the compressible calculationsof K¨apyl¨a et al. , 2010). Although not reported here, we have carried out a small number ofsimulations with vertical field boundary conditions, and the results are qualitatively similarin that case. With the given boundary conditions, it is straightforward to show that these governing equa-tions have a simple equilibrium solution, corresponding to a hydrostatic, polytropic layer: T = 1 + θz , ρ = (1 + θz ) m , u = , B = . (7) Ra T a θ κ ReR × . R . × . R × . R ×
10 0 .
02 152Table 1: The set of parameters for the four different cases. Note that the (mid-layer) Taylor numberis defined by
T a = 4(1 + θ/ m ρ Ω d /µ . See the main part of the text for the definitions of theglobal Reynolds number, Re , and the Rayleigh number, Ra . This polytropic layer (coupled with a small thermal perturbation) is an appropriate initialcondition for simulations of hydrodynamic convection. All the hydrodynamic flows that areconsidered in this paper are obtained by evolving the governing equations from this basicinitial condition. Once a statistically-steady state has been reached, the dynamo properties ofthese flows can be investigated by inserting a weak (seed) magnetic field into the domain.There are many non-dimensional parameters in this system. Since it is not viable tocomplete a systematic survey of the whole of parameter space, we only vary a subset of theavailable parameters. Throughout this paper, the polytropic index is fixed at m = 1, whilstthe ratio of specific heats is given by γ = 5 / σ = 1. In order to study the effects of varying thestratification, we adopt two different values of θ . The case of θ = 3 corresponds to a weakly-stratified layer, whilst θ = 10 represents a highly-stratified case in which the temperature anddensity both vary by an order of magnitude across the layer.The main aim of this study is to address the effects of rotation and compressibility upondynamo action in a convective layer. So, for each value of θ , we consider a rotating and anon-rotating case (which implies four different cases overall). Note that some care is neededwhen comparing simulations of rotating convection at different values of θ . The Taylor numberthat appears in the governing equations, T a , corresponds to the Taylor number at the topof the domain. Given the differing levels of stratification, it makes more sense to specify thesame mid-layer Taylor number for each of the rotating cases. Since the Taylor number isproportional to ρ , the mid-layer Taylor number (in the unperturbed polytrope) is defined by T a = T a (1 + θ/ m . Two different values of T a are considered,
T a = 0 (the non-rotatingcases) and
T a = 10 . Further discussion regarding the depth-dependence of the Taylor numberis given in the next section.Another parameter that needs to be specified is the dimensionless thermal diffusivity, κ .However, rather than prescribing values for κ , we define the mid-layer Rayleigh number Ra = ( m + 1 − mγ ) (1 + θ/ m − ( m + 1) θ κ γσ , (8)which is inversely proportional to κ and measures the destabilising effect of the superadiabatictemperature gradient relative to the stabilising effect of (viscous and thermal) diffusive pro-cesses. As described in the Introduction, rotation tends to stabilise convection, whilst largervalues of θ also have a stabilising effect (Gough et al. , 1976). Therefore it does not make senseto keep Ra constant in all cases. Instead, we vary the Rayleigh number from one calculationto the other, ranging from Ra = 3 × up to Ra = 6 × . The aim was to reach similarvalues of the Reynolds number in all cases. We shall discuss another possible definition forthe Reynolds number in the next section, but for now we define a global Reynolds number,based upon the rms velocity ( U rms ), the kinematic viscosity at the mid-layer ( κσ/ρ mid , where ρ mid is the mean density at the mid-layer of the domain) and the depth of the layer (whichequals unity in these dimensionless units). Hence this global Reynolds number is given by Re = ρ mid U rms κσ . (9)The choices for Ra that we have used imply that Re is approximately 150 for each of the fourcases. A summary of our choice of parameters for each case is given in Table 1.All the parameters that have been discussed so far relate to the hydrodynamic propertiesof the convection. If all other non-dimensional parameters in the system are fixed, the earlyevolution of any weak magnetic field in the domain depends solely upon value of ζ , which is parameter that can be set as the field is introduced. This parameter is proportional to themagnetic diffusivity, η , so we require low values of ζ for dynamo action. Equivalently, we couldsay that dynamo action is only expected in the high magnetic Reynolds number regime. Asfor the Reynolds number, we shall discuss an alternative definition for the magnetic Reynoldsnumber in Section 3.1. However, for the moment, we make an analogous global definition: R M = U rms κζ . (10)We choose a range of values for ζ which give values of R M that vary between approximately30 and 480 (0 . ≤ ζ ≤ . θ = 3 and 0 . ≤ ζ ≤ . θ = 10). This range ofvalues is restricted by numerical constraints: smaller values of ζ (higher magnetic Reynoldsnumbers) would require a higher level of numerical resolution, which would greatly increasethe computational expense. Solving the equations of compressible convection in the presence of a magnetic field is moredemanding numerically than equivalent Boussinesq calculations, so it is important to use awell optimised code. The given set of equations is solved using a modified version of the mixedpseudo-spectral/finite difference code originally described by Matthews et al. (1995). Due toperiodicity in the horizontal direction, horizontal derivatives are computed in Fourier spaceusing fast Fourier transforms. In the vertical direction, a fourth-order finite differences schemeis used, using an upwind stencil for the advective terms. The time-stepping is performed by anexplicit third-order Adams-Bashforth technique, with a variable time-step. For all calculationspresented here, the aspect ratio is λ = 4. The resolution is 256 grid-points in each horizontaldirections and 120 grid-points in the vertical direction. A poloidal-toroidal decomposition isused for the magnetic field in order to ensure that the field remains solenoidal. As explainedlater, we also aim to calculate Lagrangian statistics. Trajectories of fluid particles are thereforecomputed using the following equation: ∂ x p ∂t = u ( x p , t ) , (11)where x p is the position of the particle. The velocity at the particle position is interpolatedfrom the grid values using a sixth-order Lagrangian interpolation scheme. The boundariesare treated with a decentred scheme and we carefully check that all the particles remain inthe fluid domain. The time-stepping used to solve Equation (11) is the same as for the otherevolution equations in the system. As we have already described, fully-developed hydrodynamic convection is used as a startingpoint for all dynamo calculations. In this section, we consider the properties of the hydrody-namical flows that are obtained in each of four cases that are listed in Table 1.
In a stratified layer, most quantities of interest will be a function of depth. This is true notonly for the thermodynamic quantities in the flow, but also for some of the parameters inthe system. In the previous section, we defined the Taylor number and the Reynolds numberusing mid-layer values for the density, in addition to using the layer depth for the characteristiclength scale. These are certainly valid definitions for these quantities, but further insight can begained by considering the depth-dependence of these parameters, particularly when comparingcalculations with different levels of stratification. This is a point that we consider in detail inthis subsection.For each value of z , it is possible to define local dimensionless numbers by consideringhorizontally-averaged quantities. Under such circumstances, it is more sensible to define theseparameters in terms of a horizontal length-scale rather than using the depth of the layer (whichequals unity in this dimensionless system). We choose here the horizontal integral length scale,defined by l ( z ) = 1 (cid:104) u i ( x , t ) u i ( x , t ) (cid:105) z (cid:90) λ (cid:104) u i ( x , t ) u i ( x + r e x , t ) (cid:105) z d r (12) D e p t h z Integral length scale l ( z ) T a = T a = R1R2R3R4 00.20.40.60.81 0 0.2 0.4 0.6 0.8 1 1.2Root mean square velocity U rms( z ) T a = (a) (b)Figure 1: Variation of (a) the integral length scale l ( z ) and (b) the root mean square velocity U rms ( z ) with depth for the four calculations considered here. The rotating cases correspond to theempty blue symbols whereas the non-rotating cases correspond to the red crosses. (where we have assumed the the flow is horizontally-isotropic). Here, and in the following,the brackets < . > z mean a statistical average over horizontal coordinates and time at a givendepth z , whereas the brackets < . > (with no subscript) mean a statistical average over timeand all spatial coordinates. We note that this horizontal scale (unlike the vertical dimensionof the domain) will vary not only with depth, but also from one flow to another.Figure 1 shows the variation with depth of both the integral scale, l ( z ), and the localrms velocity, U rms ( z ) = < | u ( x , t ) | > / z , for the four cases. Each of these horizontally-averaged quantities is averaged over more than 100 acoustic crossing times. In all four cases,the trends are similar. Near the surface, the integral scale l ( z ) decreases with depth as theflow becomes more turbulent, and therefore less correlated. As the flow reaches the lowerpart of the layer, the integral length scale increases again, presumably due to the presence ofboundaries. Comparing the two non-rotating cases, one observes that the effect of increasing θ is to decrease the integral scale of the flow. However, whatever the value of θ is, the lengthscales are very similar in the two rotating cases, and are always smaller than the correspondingintegral scales in the non-rotating calculations. This mirrors the result from linear theory thatrotation tends to decrease the size of the convective cells (Chandrasekhar, 1961). The rmsvelocity is comparatively independent of depth, except close to the lower boundary where (likethe integral scale) it increases. Note that U rms ( z ) is larger in both the highly-stratified cases.At fixed stratification, the rms velocity is smaller in the rotating cases.We can now use these velocity and length scales to define local dimensionless parametersfor this system. Unlike their corresponding global values (based upon the depth of the layerand the depth-averaged rms velocity), these are always functions of z . We now define the local Reynolds number to be Re ( z ) = (cid:104) ρ ( x , t ) (cid:105) z U rms ( z ) l ( z ) κσ . (13)Similarly, we can define the local Rossby number to be Ro ( z ) = U rms ( z )2Ω l ( z ) , (14)whilst T a ( z ) = 4 (cid:104) ρ ( x , t ) (cid:105) z Ω l ( z ) κ σ (15)gives a local definition for the Taylor number. Although we are focusing upon the hydrody-namic properties of convection in this section, this is also a convenient place to define the local D e p t h z Re ( z ) Ta = R M ( z ) 00.20.40.60.81 0 0.1 0.2 0.3 0.4 Ro ( z ) 00.20.40.60.81 0 5 . Ta ( z ) Figure 2: Dimensionless numbers versus depth. From left to right: Reynolds number Re ( z ),magnetic Reynolds number R M ( z ) (the global value of R M is 30 in each of the four cases shownhere), Rossby number Ro ( z ) and Taylor number T a ( z ). magnetic Reynolds number, R M ( z ) = U rms ( z ) l ( z ) κζ . (16)In the dynamo calculations in the next section, we shall often refer to the mid-layer value ofthe local magnetic Reynolds number, i.e. R M (0 . R M = 30), the local Rossby number andthe local Taylor number is shown in Figure 2. First of all, it is clear that the local Reynoldsnumber is an increasing function of depth, which is a clear indication that the flow is becomingmore turbulent as z increases. This variation with depth is partly due to the fact that thedynamic viscosity µ is held constant in all our simulations. An important observation is thatthe local Reynolds number (and the local magnetic Reynolds number) is always smaller inthe rotating cases. This is despite the fact that the global Reynolds numbers and magneticReynolds numbers (as defined in the previous subsection) are the same in all cases. Thisreduction in the local Reynolds number is due to the fact that rotation leads to motions witha smaller characteristic horizontal length scale (as well as a lower rms velocity). So, if theselocal definitions for the Reynolds numbers give a clearer indication of the level of turbulencein the flow, the rotating calculations are actually slightly less turbulent than suggested by thevalues of the global Reynolds numbers. We shall return to this point in Section 4. Returningto Figure 2, we see that the Rossby number is roughly constant across the layer, but is smallerin the high θ case. Although we could conclude that this implies that the effects of the Coriolisforce with respect to inertia are more important in the highly stratified case, the key point isthat Ro ( z ) is always much less that unity, which implies that the flows are both rotationallydominated. Finally, the local Taylor number is largely independent of depth (and θ ) in themiddle of the domain, but increases near the boundaries. Note that this behaviour is verydifferent from the global Taylor number, which actually increases with depth. This apparentdiscrepancy is due to the fact that the local Taylor number is strongly influenced by variationsin the integral scale (varying as l ( z )), which tend to outweigh any local increases in the meandensity. The Taylor number (in particular) highlights some of the difficulties that must befaced when defining appropriate dimensionless numbers in highly-stratified fluids. From the point of view of dynamo action, it is interesting to quantify the amount of stretchingin the flow. This can be achieved by measuring the corresponding Lyapunov exponents.Several previous studies have explored the relationship between dynamo action and Lyapunov D e p t h z Maximum Lyapunov exponent λ e θ = θ =
10 R1R2R3R400.20.40.60.81 0 0.5 1 1.5 2 2.5 3 D e p t h z Normalized maximum Lyapunov exponent λ e l / U rms R1R2R3R4
Temperature Enstrophy STLE (a)(b)Figure 3: (a) Maximum Lyapunov exponents, λ e , versus depth. (b) Maximum Lyapunov exponentsscaled by the mean turn-over time l /U rms . Right column: Comparison between non-rotating (twofirst rows) and rotating convection (two last rows). From left to right: Temperature fluctuations,enstrophy and short time Lyapunov exponents (STLE) at a given depth z . From top to bottom: z = 0 . z = 0 . z = 0 . z = 0 . θ = 3 in all cases. exponents (see, for example, Finn & Ott, 1988; Brandenburg et al. , 1995; Brummell et al. ,1998 a ; Tanner & Hughes, 2003), and we would expect regions with large Lyapunov exponentsto be playing a dominant role in the dynamo process. To estimate the short-time Lyapunovexponent, we release 5 × fluid particles randomly within the convective layer. For eachparticle, we also release a second particle at a distance d = 10 − apart from the first particle,in a random direction. This pair of particles is then followed as it moves through the fluid.The short-time Lyapunov exponent λ e (Eckhardt & Yao, 1993) is then calculated using thefollowing expression: λ e = 1 t log d ( t ) d , (17)where d ( t ) is the distance between the two particles at time t . The Lyapunov exponent iscomputed after a fixed time (approximately equal to one crossing time) and the distancebetween the particles is then renormalised to d . The results appear to be insensitive tothe choice of the initial separation distance d . Furthermore, they do not appear to dependupon the details of the method that is used to reinitialise the particle positions. All initialseparations will quickly align with the expanding manifold, so we would expect to see verylittle dependence upon the initial condition for sufficiently large time, t .To study the depth dependence of the stretching within the flow, we consider the horizontalaverage of the largest Lyapunov exponent at each value of z . In addition, we time-average theresulting Lyapunov exponent over approximately 20 convective turnover times. Figure 3(a)shows the mean maximum Lyapunov exponent versus depth for each of the four cases. In allcases, the stretching increases with depth (as the flow becomes more and more turbulent),with the Lyapunov exponents taking their maximum values at the bottom of the convective omain. Unsurprisingly, this is reminiscent of the depth-dependence of the local Reynoldsnumber. This depth-dependence has important implications for the dynamo problem: wewould expect the magnetic field to be mostly generated in the lower part of the layer, wherethe rate of stretching is maximal. Compared to the equivalent non-rotating cases, in rotatingconvection we see higher values of the Lyapunov exponent at the top of the layer and lowervalues at the bottom of the layer. In other words, the maximum Lyapunov exponent isless depth-dependent in the rotating cases than it is in the non-rotating calculations. Froma dynamo perspective, this suggests that we should expect the magnetic fields to be morehomogeneously distributed across the layer in the rotating cases than they are in the non-rotating simulations. Figure 3(b) shows a plot in which the Lyapunov exponents are scaledby the local convective turnover time l ( z ) /U rms ( z ). We use a similar scaling for the dynamogrowth rates in the next section, where the implications of this plot will be discussed in moredetail. Here we simply note two key features of this scaled plot. Firstly, this scaling causes thefour curves to collapse onto a single curve in the upper part of the layer. Secondly, we notethat the scaled Lyapunov exponents suggest that there is less stretching in the lower part ofthe domain in the rotating cases.On the right-hand side of Figure 3, we show a snapshot of the temperature, enstrophyand the short time Lyapunov exponent (STLE) in a horizontal plane at two different depths( z = 0 . z = 0 .
8) for both non-rotating (top) and rotating (bottom) convection in theweakly-stratified ( θ = 3) case. In the temperature plot, brighter contours correspond towarmer regions of fluid. As expected from our considerations of the horizontal integral scale,it is clear that the horizontal scale of convection is smaller in the rotating case than it is innon-rotating convection. This reduction in horizontal scale is also apparent in the enstrophyplot where bright regions, corresponding to areas of high (squared) vorticity, outline the edgesof the convective cells. The STLE map is obtained by releasing 10 particle pairs at a givendepth. The particles are followed for approximately one crossing time. The STLE is thencomputed using Equation (17) and plotted at the initial particle pair position. Comparingthe STLE map with the enstrophy distribution, it is clear that there is a correlation betweenzones of strong stretching and regions of high vorticity. Given this correlation, it is naturalto conclude that the higher (unscaled) Lyapunov exponent in the upper part of the rotatingsimulations, as observed in Figure 3(a), is mostly due to the larger number of convective cellsthat are present. The scaled plot that is shown in Figure 3(b) tends to support this conclusion:the scaling that has been used here takes into account this “filling factor” effect, which explainswhy all the scaled Lyapunov exponent curves collapse onto a single curve near the surface inthat case. In this section, we investigate the dynamo properties of the four convective flows that are beingconsidered in this paper. Each dynamo calculation is initialised by introducing a seed magneticfield into statistically-steady hydrodynamic convection. The initial spatial structure of thismagnetic field is given by B ( x, y, z ) =( A cos( k i y ), A cos( k i x ), 0), where k i = 2 π/λ and A is thepeak amplitude of the initial perturbation. This is almost the simplest possible magnetic fieldconfiguration that is consistent with the imposed boundary conditions, whilst also ensuringthat there is no net (horizontal or vertical) magnetic flux across the computational domain.Test calculations suggest that the evolution of the dynamo is largely insensitive to the precisechoice of initial conditions. Initially, we consider the kinematic dynamo regime in which the seed magnetic field is assumedto be weak. This implies that the magnetic field tends to grow (or decay) exponentially ata rate that is determined by the value of magnetic Reynolds number. Given that the actualgrowth of the magnetic energy fluctuates in time, long time-averages are often needed in orderto accurately measure growth rates. This kinematic phase of evolution can be indefinitelyprolonged by switching off the Lorentz force terms in the momentum equation and the ohmicheating terms in the temperature equation. This is the procedure that is adopted in thissubsection. For each velocity field, we carry out 6 kinematic calculations at different valuesof ζ . This parameter is chosen so that the global magnetic Reynolds number ranges fromapproximately 30 to 480 in each case. This implies that the magnetic Prandtl number (the atio of the global magnetic Reynolds number to the global Reynolds number) varies fromapproximately 0 . . θ = 3 and θ = 10) of the kinematic growth rate of themagnetic energy, λ , versus the global magnetic Reynolds number. In all cases, λ has beenscaled by the mid-layer turnover time of the turbulence, l (0 . /U rms (0 . θ = 3 case that there is a suggestion of a slightly higher critical global magneticReynolds number (although the error bars in the growth rate close to criticality are largeenough to suggest that this difference may not be significant). Adopting this definition for themagnetic Reynolds number, we see that the scaled growth rates in the highly stratified caseseem to be rather insensitive to the presence of rotation. It is only in the weakly stratifiedcase, at small values of the global magnetic Reynolds number, that significant differences areseen between the rotating and the non-rotating cases. So it is tempting to conclude from theseresults that the dynamo properties of compressible convection (particularly at higher valuesof the global magnetic Reynolds number) are largely insensitive to the presence of rotation,as well as variations in the level of stratification.However, some caution is needed when interpreting these results. As we discussed in theprevious section, a global definition for the magnetic Reynolds number takes no account of thehorizontal scale of convection (a quantity that varies significantly between the four cases thatare being considered). So we would argue that it is more sensible to consider representativevalues of the local magnetic Reynolds number for the purposes of this comparison. The lowerpart of Figure 4 also shows the growth rate curves for the four different cases, but this time λ has been plotted against the mid-layer value of the local magnetic Reynolds number, R M (0 . θ without rotation. For θ = 3, the critical value for R M (0 .
5) isabout 420, whereas for θ = 10, the critical value for R M (0 .
5) is about 290. Therefore, anincrease in the level of stratification reduces the critical value of the local magnetic Reynoldsnumber. Let us now consider the two rotating cases. A striking feature of the plots in thelower part of Figure 4 is how similar the growth rate curves of the two rotating cases are.For both values of θ , the critical value of R M (0 .
5) is about 220. Therefore, independentlyof the level of stratification, rotation tends to reduce the critical value for the local magneticReynolds number.Whatever definition is adopted for the magnetic Reynolds number, it is clear that thegrowth rate curves exhibit certain characteristic features. The dotted lines on Figure 4, whichare shown on these plots for indication, correspond to a logarithmic scaling (see, for example,Rogachevskii & Kleeorin, 1997) and a square-root scaling (Schekochihin et al. , 2004) of thegrowth rate with the magnetic Reynolds number. A logarithmic scaling seems to be valid(at least for all magnetic Reynolds numbers considered here) for the non-rotating θ = 3 case.This situation is less clear in the highly-stratified non-rotating case, although an R / M scalingmay be more appropriate here. The other possibility is that there are two different regimes inthis case, with a logarithmic scaling holding only for larger values of the magnetic Reynoldsnumber. The scalings are again not completely convincing in the rotating cases, but the dataseem to be compatible with an R / M scaling, irrespective of the value of θ . In both plots inthe lower part of Figure 4 it is clear that the rotating cases always have higher growth ratesthan the equivalent non-rotating calculations. The largest difference in growth rates betweenthe rotating and non-rotating calculations can be seen at low magnetic Reynolds number inthe θ = 3 case.The observed variations in the growth rate curves suggest that rotation is beneficial fordynamo action. However, as discussed in the previous section, it could be argued that therotating calculations are (in some sense) less turbulent that their non-rotating counterparts:although the global Reynolds numbers are similar in all cases, the local Reynolds number issignificantly smaller in the rotating calculations. So we have to consider the possibility thatit is actually the difference in the local Reynolds numbers that is giving the impression ofenhanced dynamo action in the rotating cases. In order to address this issue, we performedan additional set of kinematic simulations for a new convective flow (referred to here as case R b ). Setting T a = 0, we choose θ = 3 since this it is in this case that we observe the N o r m a li ze dg r o w t h r a t e λ l / U r m s Magnetic Reynolds number R M (0 . ∼ log R M ∼ R / M -1.2-1-0.8-0.6-0.4-0.200.20.4 10 100 1000Magnetic Reynolds number R M (0 . ∼ R / M ∼ log R M -1.2-1-0.8-0.6-0.4-0.200.20.4 10 100 1000 N o r m a li ze dg r o w t h r a t e λ l / U r m s Global magnetic Reynolds number R M = U rms d /κζ θ = ∼ log R M ∼ R / M -1.2-1-0.8-0.6-0.4-0.200.20.4 10 100 1000Global magnetic Reynolds number R M = U rms d /κζ θ = ∼ R / M ∼ log R M Figure 4: Growth rates of the magnetic energy versus the magnetic Reynolds number for θ = 3(left) and θ = 10 (right). In the upper plots, the global magnetic Reynolds number has been used.In the lower plots, the same growth rates have been plotted against the mid-layer value of the localmagnetic Reynolds number, R M (0 . l (0 . /U rms (0 . R R b and R
2. (b) Growthrates of the magnetic energy versus the mid-layer value of the local magnetic Reynolds number, R M (0 . θ = 3 simulations. The rotating results, from simulation R
2, are plotted withtriangles whereas the non-rotating results correspond to the circles (filled for R
1, and empty for R l (0 . /U rms (0 . greatest difference between the dynamo properties of rotating and non-rotating convection.All parameters are identical to case R Ra = 5 × . The global Reynolds number is now significantly smallerthan in all other calculations reported in this paper. However, the local value now has asimilar depth dependence to the equivalent rotating case, R
2, as illustrated in Figure 5(a).Figure 5(b) shows a comparison of the kinematic dynamo growth rates for all the θ = 3calculations ( R R b and R R R M (0 .
5) in the rotating cases cannot be explained simplyby the differences between the local Reynolds numbers. Therefore this change in the criticallocal magnetic Reynolds number must be due to the effects of rotation.Before concluding this section on kinematic growth rates, it is worth commenting on themagnitudes of the growth rates in the dynamo regime. Obviously the range of magneticReynolds numbers is restricted by numerical constraints. However, it is worth noting that ourpositive growth rates (when properly normalised) are comparable to the values reported byK¨apyl¨a et al. (2008), who also considered simulations of compressible convection (albeit withovershoot and shear), and those found in the forced homogeneous turbulence calculations ofHaugen et al. (2004). It is also of interest to compare the peak kinematic growth rates tothe scaled Lyapunov exponents that are shown in Figure 3(b). In all cases, it is clear thatthe growth rates are significantly smaller than the corresponding Lyapunov exponents. Thisis unsurprising given that the magnetic Reynolds numbers are comparatively modest in thesenumerical simulations, and we would expect to see higher growth rates at higher values of R M . Nevertheless our results would appear to be consistent with the findings of Brandenburg et al. (1995) who, for a related dynamo calculation, found that the Lyapunov exponents weresystematically larger than the observed kinematic growth rates. We now consider the fully nonlinear set of governing equations (Equations (1) to (6)) includingthe back-reaction of the magnetic field on the velocity field. For each of the four cases under N o r m a li ze dk i n e ti ca nd m a gn e ti ce n e r g i e s Time (crossing time unit) θ = KineticMagnetic (x5)
T a = T a = θ = Kinetic Magnetic (x5)
T a = T a = (a) (b)Figure 6: The time evolution of the normalised mean kinetic and mean magnetic energy densitiesfor (a) θ = 3 and (b) θ = 10. Both quantities are normalised by the mean kinetic energy densityduring the saturated phase ( i.e. from t = 200 to t = 500 in the θ = 3 cases, and from t = 100 to t = 250 in the θ = 10 cases). The magnetic energy has been multiplied by 5. consideration, we choose the values of ζ corresponding to the largest values of the magneticReynolds number, in order to maximise the growth rate of the dynamo, thus minimising theduration of the kinematic phase. In the following, the time t = 0 corresponds to the insertiontime of the small magnetic perturbation. The initial ratio between the total magnetic energy inthe seed field and the total kinetic energy in the flow is roughly 10 − in all cases. This impliesthat the initial field is weak enough to be kinematic, without unnecessarily extending thekinematic phase in these nonlinear calculations. Each nonlinear calculation has been evolvedover a significant fraction of the ohmic decay time (based upon the magnetic diffusivity andthe horizontal integral scale).Figure 6 shows the evolution of the mean kinetic and mean magnetic energy densities forthe four cases. All of these dynamos are highly time-dependent, but there are clear patternsof behaviour. During the early stages of evolution, the magnetic perturbation grows. How-ever, the kinematic phase of the dynamo is extremely brief in these calculations, as the seedfield rapidly becomes dynamically significant. After the short kinematic phase, the dynamoundergoes a more extended period of nonlinear growth, eventually settling down to a time-dependent saturated state. Conservation of energy implies that the kinetic energy decreasesduring the nonlinear phase of the dynamo, reaching a final level that is clearly lower than thekinetic energy of the initial hydrodynamic state. Across all simulations, the mean magneticenergy density, (cid:10) | B | (cid:11) /
2, saturates at a level that is somewhere between 4 and 9% of themean kinetic energy density, (cid:10) ρ | u | (cid:11) /
2. Although not shown here, a corresponding nonlinearcalculation for the R b case (described in the previous section) saturates at a similar level.These nonlinear results suggest that the saturation level of the dynamo is comparatively in-sensitive to the level of stratification within the domain. There is weak evidence to suggestthat the rotating cases are saturating at a slightly higher level (particularly for θ = 3), despitethe fact that the mid-layer values of the local magnetic Reynolds number are actually smallerin the rotating calculations than they are in the corresponding non-rotating cases. If we werecomparing nonlinear calculations at similar values of the (mid-layer) local magnetic Reynoldsnumber, as opposed to similar values of the global R M , we would expect the rotating cases tosaturate at a higher level. However, we did not investigate this parameter regime here giventhat this would require higher spatial resolution in the rotating cases.Looking again at Figure 6, it is worth noting that the time-dependence of the magneticenergy in the non-rotating θ = 3 case is strongly intermittent. This is clearly seen at t ≈ D e p t h z h B x i z + h B y i z B eq θ = θ = R R R R h B z i z B eq R R R R D e p t h z D B x E z + D B y E z R R R R B eq ( z ) = D ρ u E z R R R R (a) (b)(c) (d)Figure 7: (a) The horizontal magnetic energy, normalised by the equipartition energy B eq ( z ) = (cid:10) ρ | u | (cid:11) z . (b) The equivalent plot for the vertical magnetic energy density. (c) The (unnormalised)horizontal magnetic energy density. (d) A plot of B eq ( z ) as a function of depth. Note that figures(c) and (d) are presented in semi-log scale. dynamo saturates in this case. As discussed in the previous subsection, this non-rotating θ = 3case may have a slightly higher critical global magnetic Reynolds number than the other threecases. If the dynamo is indeed closer to criticality than the other cases, we would expect it tobe more sensitive to time-dependent variations in the flow. This would explain the observedbehaviour.A more detailed description of the magnetic field in the saturated phase can be obtainedby considering the horizontal and vertical components of the magnetic energy density. Whencomparing different cases, it is useful to normalise the magnetic energy densities by the localequipartition energy defined by B ( z ) = (cid:10) ρ | u | (cid:11) z . The normalised horizontal magnetic en-ergy density (cid:10) B x (cid:11) z + (cid:10) B y (cid:11) z and the normalised vertical magnetic energy density (cid:10) B z (cid:11) z areshown in Figures 7(a) and (b). The unnormalised horizontal magnetic energy density is shownin Figure 7(c), whilst the depth dependence of the equipartition energy B ( z ) is shown in Fig-ure 7(d). In the middle of the layer, the horizontal magnetic energy density is comparable tothe vertical magnetic energy density, whereas it clearly dominates close to the boundaries, asexpected from the chosen boundary conditions. It is clear that the (unnormalised) horizontalmagnetic energy density is stronger at the lower boundary than it is at the upper boundary,although these are more comparable when they are scaled in terms of the equipartition fieldstrength. Note that all of these magnetic energy densities are sub-equipartition. Althoughthere are some quantitative differences in these curves, the variation with depth of each ofthese quantities is broadly similar in most cases. The main difference to note is that the = 0 . z = 0 . T a = 0
T a = 10 Figure 8: Grey-scale plots of the horizontal component B x of the magnetic field at two differentdepths, z = 0 . z = 0 .
8, for both rotating and non-rotating cases in the saturated phase. Thethermal stratification is θ = 3 in all cases. Contours are evenly spaced between B x = − . B x = 0 . T a = 0 and between B x = − . B x = 0 . T a = 10 . Light and dark tonescorrespond to opposite polarities. 15 D e p t h z ζ κ j θ = T a = T a = ζ κ j θ = T a = T a = (a) (b)Figure 9: The horizontally-averaged ohmic dissipation, ζ κ (cid:10) j (cid:11) z , for (a) θ = 3 and (b) θ = 10. vertical component of the magnetic energy density tends to be larger in the rotating cases.Figure 8 shows some examples of contours of the horizontal component of the magneticfield, B x , during the saturated phase. In all cases, θ = 3, and two different depths areconsidered ( z = 0 . z = 0 . R R
2. It is useful to compare thisfigure with the corresponding temperature and enstrophy plots in Figure 3. Near the upperboundary ( i.e. at z = 0 . i.e. at z = 0 . ζ κ | j | ,where j = ∇ × B is the current density. This expression for the ohmic dissipation can bederived by taking the scalar product of the induction equation (4) with B , and then integratingover the domain. Note that the rate of dissipation is also proportional to the ohmic heatingterm in Equation (6). The rate of ohmic dissipation in the nonlinear phase, averaged over timeand horizontal coordinates, is plotted in Figure 9(a) for θ = 3 and Figure 9(b) for θ = 10. Thedissipation is much larger in the θ = 10 case, since both the kinetic and the magnetic energydensities are greater by roughly one order of magnitude. For each value of θ , the magnitudeof the dissipation term near the top of the layer is roughly the same, whether or not the layeris rotating. On the other hand, rotation clearly reduces the dissipation of magnetic energynear the lower boundary. Put another way, the dissipation term is more weakly dependentupon depth in the rotating cases than it is in the corresponding non-rotating calculations.A similar reduction of the magnetic dissipation by rotation has already been observed inrotating homogeneous magnetohydrodynamic turbulence by Favier et al. (2011). It is alsoworth noting that this reduction in magnetic dissipation is not a nonlinear effect: it canalso be observed in the kinematic phase (although it is more difficult to quantify such areduction since it becomes necessary to normalise the dissipation term by an exponentiallygrowing field in order to take an appropriate time-average). The regions of strongest ohmicdissipation in the non-rotating cases coincide with the regions of strongest shear. We havealready seen, in Figures 3(a) and 3(b), that the stretching is more homogeneously distributed θ = 3 (the results are similar for θ = 10). The solid lines correspondto the saturated phase whereas the dotted lines correspond to the kinematic phase. The kineticenergy spectrum E K ( k ⊥ ) is averaged over time and vertical coordinate. The magnetic energyspectrum is normalised by the time-average of the magnetic energy for the saturated dynamo. across the layer in the rotating calculations. As a result, the dynamo-generated magnetic fieldorganises itself in such a way that the rate of dissipation in these cases is lower than it is inthe equivalent non-rotating calculations. So even though the scaled Lyapunov exponents, asshown in Figure 3(b), suggest that there is generally less stretching in the rotating cases, thisreduction in dissipation explains why rotating convection can still act as an efficient dynamo(at least at moderate magnetic Reynolds numbers). In the previous subsection, we discussed nonlinear results, without really saying anythingabout the saturation mechanism for these dynamo calculations. Due to the complexity of thesedynamo models, it is difficult to say anything definitive about this. However, some insightsinto the saturation process can be gained by comparing certain aspects of the kinematic andnonlinear phases of the dynamo.Firstly, we consider some of the global properties of the dynamo. We define the magneticenergy spectrum in the following way: E M ( k ⊥ ) = 12 (cid:88) z (cid:88) k ⊥ ˆ B ( k x , k y , z ) · ˆ B ∗ ( k x , k y , z ) (18)where k ⊥ = k x + k y is the horizontal wave number, ˆ B ( k x , k y , z ) is the two-dimensional Fouriertransform of B ( x, y, z ) and the star denotes the complex conjugation. Similarly, the kineticenergy spectrum is defined as follows: E K ( k ⊥ ) = 14 (cid:88) z (cid:88) k ⊥ (cid:98) u ( k x , k y , z ) · (cid:99) ρ u ∗ ( k x , k y , z ) + (cid:98) u ∗ ( k x , k y , z ) · (cid:99) ρ u ( k x , k y , z ) . (19)In Figure 10, we show the energy spectra corresponding to the θ = 3 calculations ( R R i.e. for 1 < k ⊥ <
3) in the rotating case. Thisis connected to the rotationally-induced reduction in the horizontal scale of motion. Whetheror not rotation is present, the magnetic energy spectra always peak at small scales, which is D e p t h z Maximum Lyapunov exponent λ e T a = θ = θ = λ e T a = θ = θ = (a) (b)Figure 11: Horizontally-averaged maximum Lyapunov exponents versus depth during the nonlinearphase (solid lines). The dotted lines correspond to the previous kinematic results (see figure 3(a)). consistent with the fact that we do not observe a large-scale dynamo in these simulations.Comparing the kinematic and nonlinear phases, we see that the kinetic energy spectra areonly weakly perturbed by the magnetic fields. So there is no evidence here to suggest thatsaturation is accompanied by a significant variation in the kinetic energy spectrum. Moreinterestingly, there is a small (but perhaps significant) alteration of the magnetic energy spectraas the dynamo saturates. There is a clear reduction of magnetic energy at small scales anda corresponding increase at large scales. This is observed in both the rotating and the non-rotating cases. Although not shown here, the same trend is observed in the θ = 10 calculations.We note that this is consistent with the increase of the Taylor microscale of the magnetic fieldreported by Brandenburg et al. (1996).When considering potential saturation mechanisms, one of the most interesting things toconsider is whether or not the stretching properties of the flow are modified by the magneticfields. Local variations in the stretching would probably not lead to significant variations in thekinetic energy spectrum. However, we would expect to see changes in the Lyapunov exponentsif the stretching properties of the flow are modified in the nonlinear dynamo regime. Using theflows from our nonlinear calculations, we evaluate the Lyapunov exponents using exactly thesame procedure as for the kinematic phase (as described in Section 3.2). The particle pairs arefollowed from t = 300 to t = 500 in the θ = 3 cases, and from t = 150 to t = 250 in the θ = 10cases. The (horizontally-averaged) maximum Lyapunov exponents for the four cases are shownin Figure 11 by the solid lines. The dotted lines show the corresponding Lyapunov exponentsfrom the kinematic calculations. In the non-rotating cases, regardless of the value of θ , theLyapunov exponents are slightly lower in the nonlinear phase everywhere except near the upperand lower boundaries. In the rotating cases, the Lyapunov exponent is slightly reduced almosteverywhere, even close to the lower boundary. So there is some indication of a suppression ofstretching due the presence of magnetic fields. Furthermore, given the intermittent nature ofthe magnetic field distribution, we might expect the local reduction in stretching (in regionsof strong magnetic fields) to be greater than that suggested by these horizontally-averagedquantities. However, this does not mean that the saturation of the dynamo can be explainedsimply by a reduction in stretching. Cattaneo & Tobias (2009) have shown that a dynamo-saturated velocity field in Boussinesq convection can still act as a kinematic dynamo (providedthat the new seed field is not aligned with the original magnetic field). Although this has notbeen tested here, the depth-dependence of the Lyapunov exponents in the nonlinear regime isqualitatively similar to that of the original hydrodynamic flows, so we would speculate thatthese dynamos would exhibit similar behaviour. Certainly we do not see the drastic reductionin the Lyapunov exponents that Cattaneo et al. (1996) observed in their model of nonlineardynamo action in a much simpler one-scale flow. The saturation process in convectively-driven R u and B .We separate contributions from points where | B | > B rms (solid lines) and | B | < B rms (dottedlines). (b) Same as (a) but for the angle between B and e , where e is the eigenvector associatedto the largest eigenvalue of the rate of strain tensor. dynamos appears to be more subtle than this.We also consider the possibility that the correlation between u and B plays some role inthe saturation process. The alignment between these two vectors, and the influence that thishas upon some of the key nonlinearities in the dynamo system, has been extensively studiedin recent years (see, for example, Servidio et al. , 2008). Figure 12(a) shows the probabilitydensity function of the cosine of the angle between u and B :cos( u , B ) = u · B | u || B | . (20)The pdfs have been obtained by averaging over space (between z = 0 . z = 0 . | B | > B rms , where B rms is the rms magnetic field, the other for those mesh points that fall below this threshold fieldstrength. Despite the fact that this flow is compressible, inhomogeneous and anisotropic, wesee remarkable alignment between u and B . Perhaps unsurprisingly, the alignment is alwaysslightly more pronounced for strong magnetic fields than for the weaker fields. However, thestrong alignment between the two fields is observed in both the kinematic and the nonlinearphases, and is therefore a property of the induction equation. Hence the saturation of theseconvectively-driven dynamos cannot be explained by a modified alignment between u and B .Following Brandenburg et al. (1996), we also consider the alignment between B and theeigenvectors of the rate of strain tensor S . The symmetric matrix S ij = ( u i,j + u j,i ) is com-puted at each mesh point and the three corresponding eigenvalues are ordered from the smallestto the largest. The smallest eigenvalue is always negative and its eigenvector corresponds tothe direction of compression. The magnetic field is found to be mostly perpendicular to thisdirection (as already reported by Brandenburg et al. , 1996), in both the kinematic and thenonlinear phases. The largest eigenvalue is always positive and its eigenvector, denoted hereby e , corresponds to the direction of maximum stretching. The probability density functionof cos( B , e ), is shown in Figure 12(b). Note that we again separate the contributions fromthe weak field and strong field regions. During the kinematic phase, the strongest magneticfields are mostly aligned with the direction of maximum stretching. Note that there is alsosome indication of preferential alignment for the weaker field regions, but this is much less ronounced. In the saturated phase, the pdf corresponding to the weak field regions showsa very modest reduction in alignment. However, there is a dramatic reduction in the align-ment between B and e in the strong field regions. Note that the magnetic fields for which | B | > B rms only represent 2% of the total number of points, so that the loss of alignmentbetween the direction of maximum stretching and the magnetic field is only observed very lo-cally. This modification of the alignment between B and e is observed in all our simulations,with or without rotation, and for both thermal stratifications. This indicates that this processis rather robust. Given that this modified alignment will reduce the efficiency of the dynamoin regions of strong magnetic fields, we conclude that this effect plays a role in the saturationof convectively-driven dynamos. In this paper, we have investigated small-scale dynamo action in rotating compressible convec-tion. Regardless of the level of stratification within the domain, both rotating and non-rotatingconvective flows can sustain a small-scale dynamo if the magnetic diffusivity is small enough.Using a mid-layer value for the magnetic Reynolds number (based upon the integral scale ofthe turbulence rather than the layer depth), rotation seems to reduce the value of the magneticReynolds number above which dynamo action is observed. Increasing the thermal stratifica-tion also reduces the critical value of the local magnetic Reynolds number in the non-rotatingcase. At high values of the magnetic Reynolds number, the growth rate of the magnetic energyof the dynamo appears to have a logarithmic dependence upon R M in the weakly-stratified,non-rotating simulation. It is more difficult to fit a scaling law in the other cases, but an R / M scaling may be more appropriate here. At a given value of the mid-layer (local) magneticReynolds number, the kinematic growth rates are always larger in rotating convection thanthey are in the corresponding non-rotating cases. This dependence upon rotation cannot beattributed solely to the fact that the local Reynolds number is smaller in the rotating cases.So we conclude that the Coriolis force plays a key role in determining the kinematic dynamoproperties of rotating convection. At the highest computationally accessible values of the mag-netic Reynolds number, we find similar levels of saturation in all of our nonlinear calculations(with the magnetic energy saturating at about 4 −
9% of the final kinetic energy). At firstsight, this result is slightly surprising given that the Lyapunov exponents suggest that thereis less stretching in the rotating cases (particularly near the lower boundary). However, thisis compensated by the fact that magnetic dissipation seems to be much less efficient in therotating calculations. It is difficult to say anything definitive about the saturation mecha-nism for these convectively-driven dynamos. However, a comparison between the kinematicdynamo regimes and the nonlinear saturated states show a slight reduction in the Lyapunovexponents in the nonlinear regime (due to a local reduction of the stretching properties of theflow). Furthermore, there is some evidence to suggest that a reduction in alignment betweenthe strongest magnetic fields and the direction of maximum stretching also plays a role in thesaturation process.The next natural step in this study is to explore the parametric dependence of this systemin more detail. Of course, given the complexity of this numerical model, it is not feasibleto conduct a complete survey of parameter space, although any further calculations that canbe carried out will obviously enhance our understanding of this problem. We also intend toinvestigate the ability of rotating compressible convection to produce a magnetic field on muchlarger scales than the scales that are associated with the convective motions. The question ofhow hydromagnetic dynamos are able to generate large-scale magnetic fields is undoubtedlyone of the most challenging issues in modern dynamo theory. In the standard formulation ofmean-field dynamo theory (Moffatt, 1978), this regenerative process relies upon the presenceof helical motions, which are invariably produced when a flow is influenced by rotation. Hence,rotating convection should be able to drive a large-scale dynamo. However, despite predictionsfrom mean-field dynamo theory, Cattaneo & Hughes (2006) failed to find evidence for a large-scale dynamo in their Boussinesq model. Interestingly, K¨apyl¨a et al. (2009) did find large-scalemagnetic fields in their compressible model. They argue that the absence of large-scale fieldsin the Boussinesq model of Cattaneo and Hughes can be attributed to a rotation rate that istoo slow. This is a possibility, however the effects of compressibility may also be playing arole. The flow is (locally) strongly helical in the Boussinesq calculations of Cattaneo & Hughes(2006), but the mid-layer symmetry of their setup implies that the mean helicity distributionis antisymmetric about the mid-plane. It may be important to break this symmetry in order o generate large-scale fields. In work that is currently in progress, we are investigating thisissue by carrying out simulations of rapidly-rotating convection in a wide compressible layer.The size of the layer is important, because it is necessary to have a clear separation in scalesbetween the small-scale fields and any large scale magnetic fields that may be generated. Thecomputational domain that was considered in the present study was too small to allow for thisseparation in scales, which may explain why no large scale fields were observed here (despitethe fact that the helicity of the flow is asymmetric about the mid-plane in these stratifiedrotating calculations). Of course, other physical ingredients could also be included into thissystem once the basic ingredients of rotation and stratification are properly understood. Forexample, the role of the penetrative layer in the model of K¨apyl¨a et al. (2009) is also unclear.This may be promoting the formation of large-scale fields in some way. It may also be ofinterest to consider the effect of including a shear flow (see, for example, Hughes & Proctor,2009) in this compressible model. Acknowledgements
The authors wish to thank the anonymous referees for their helpfulcomments and suggestions, which improved the quality of the manuscript. B.F. wishes to thankA.W. Baggaley for helpful discussions. This work has been supported by the Engineering andPhysical Sciences Research Council through a research grant (EP/H006842/1). All numericalcalculations have been carried out using the HECToR supercomputing facility.
References
Baliunas, S.L., Nesme-Ribes, E., Sokoloff, D. & Soon, W.H.
Astrophys. J. , 848–854.
Brandenburg, A., Jennings, R.L., Nordlund, ˚A., Rieutord, M., Stein, R.F. & Tuomi-nen, I.
J. Fluid Mech. , 325–352.
Brandenburg, A., Klapper, I. & Kurths, J.
Phys. Rev. E , R4602–R4605. Brummell, N.H., Cattaneo, F. & Tobias, S.M. a Linear and nonlinear dynamoaction.
Phys. L. A , 437–442.
Brummell, N.H., Hurlburt, N.E. & Toomre, J.
Astrophys. J. , 494–513.
Brummell, N.H., Hurlburt, N.E. & Toomre, J. b Turbulent compressible convectionwith rotation. II. Mean flows and differential rotation.
Astrophys. J. , 955–969.
Brummell, N.H., Tobias, S.M. & Cattaneo, F.
Geophys. Astrophys. Fluid Dyn. ,565–576.
Brun, A.S., Miesch, M.S. & Toomre, J.
Astrophys. J. , 1073–1098.
Bushby, P.J., Houghton, S.M., Proctor, M.R.E. & N.O.Weiss
Mon. Not. R. Astron. Soc. , 698–706.
Bushby, P.J., Proctor, M.R.E. & Weiss, N.O.
Numerical Modeling of Space Plasma Flows, Astronum-2009 (ed.N. V. Pogorelov, E. Audit, & G. P. Zank),
Astronomical Society of the Pacific ConferenceSeries , vol. 429, pp. 181–186.
Cattaneo, F.
Astrophys. J. , L39–L42.
Cattaneo, F. & Hughes, D.W.
J. FluidMech. , 401–418.
Cattaneo, F., Hughes, D.W. & Kim, E.
Phys. Rev. Lett. , 2057–2060. attaneo, F. & Tobias, S.M. J. Fluid Mech. , 205–214.
Chandrasekhar, S.
Hydrodynamic and hydromagnetic stability . Oxford UniversityPress.
Childress, S. & Soward, A.M.
Phys. Rev.Lett. , 837–839. Eckhardt, B. & Yao, D.
Physica D ,100–108. Favier, B.F.N., Godeferd, F.S. & Cambon, C.
Geo. Astro. Fluid Dyn.
Finn, J.M. & Ott, E.
Phys. Fluids ,2992–3011. Gough, D.O., Moore, D.R., Spiegel, E.A. & Weiss, N.O.
Astrophys. J. , 536–542.
Haugen, N.E.L., Brandenburg, A. & Dobler, W.
Phys. Rev. E , 016308. Hughes, D.W. & Proctor, M.R.E.
Phys. Rev. Lett. , 044501.
Jones, C.A. & Roberts, P.H.
J.Fluid Mech. , 311–343.
K¨apyl¨a, P.J., Korpi, M.J. & Brandenburg, A.
A.&A. , 353–362.
K¨apyl¨a, P.J., Korpi, M.J. & Brandenburg, A.
Astrophys. J. , 1153–1163.
K¨apyl¨a, P.J., Korpi, M.J. & Brandenburg, A.
A.& A. , A22.
Matthews, P.C., Proctor, M.R.E. & Weiss, N.O.
J. Fluid Mech. , 281–305.
Meneguzzi, M. & Pouquet, A.
J. FluidMech. , 297–398.
Moffatt, H.K.
Magnetic field generation in electrically conducting fluids . CambridgeUniversity Press.
Monchaux, R., Berhanu, M., Aumaˆıtre, S., Chiffaudel, A., Daviaud, F., Dubrulle,B., Ravelet, F., Fauve, S., Mordant, N., P´etr´elis, F., Bourgoin, M., Odier, P.,Pinton, J.-F., Plihon, N. & Volk, R.
Phys. Fluids (3), 035108. Roberts, P.H. & Glatzmaier, G.A.
Reviews ofModern Physics , 1081–1123. Rogachevskii, I. & Kleeorin, N.
Phys. Rev. E , 417–426. Schekochihin, A.A., Cowley, S.C., Taylor, S.F., Maron, J.L. & McWilliams, J.C.
Astrophys. J. , 276–307.
Servidio, S., Matthaeus, W.H. & Dmitruk, P.
Phys. Rev. Lett. , 095005. t. Pierre, M.G. Solarand Planetary Dynamos (ed. M. R. E. Proctor, P. C. Matthews, & A. M. Rucklidge), pp.295–302.
Stellmach, S. & Hansen, U.
Phys. Rev. E , 056312. Stix, M.
The Sun: an introduction . Springer, Berlin.
Tanner, S.E.M. & Hughes, D.W.
Astrophys. J. , 685–691.
Thelen, J.-C. & Cattaneo, F.
Mon. Not. R. Astron. Soc. , L13–L17.
V¨ogler, A. & Sch¨ussler, M.
A.&A. , L43–L46., L43–L46.