2.1 Governing equations and boundary conditions

We consider a three-dimensional Cartesian model of rotating convection for an electrically conducting Boussinesq fluid. Motions are driven by an initially uniform temperature gradient, imposed by fixing the temperature difference $\unicode[STIX]{x0394}T$ between the top and bottom boundaries. The box depth is $d$ . The horizontal dimensions of the computational domain are equal in the $x$ and $y$ directions, with the ratio of horizontal to vertical dimensions denoted by $\unicode[STIX]{x1D706}$ . The acceleration due to gravity is constant, $\boldsymbol{g}=-g\boldsymbol{e}_{z}$ . The rotation vector is $\unicode[STIX]{x1D6FA}\boldsymbol{e}_{z}$ . The fluid has kinematic viscosity $\unicode[STIX]{x1D708}$ , thermal diffusivity $\unicode[STIX]{x1D705}$ , magnetic diffusivity $\unicode[STIX]{x1D702}$ , density $\unicode[STIX]{x1D70C}$ , thermal expansion coefficient $\unicode[STIX]{x1D6FC}$ and magnetic permeability $\unicode[STIX]{x1D707}_{0}$ , all of which are constant. We solve the momentum, temperature and magnetic induction equations in dimensionless form, obtained by scaling lengths with $d$ , times with $1/(2\unicode[STIX]{x1D6FA})$ , temperature with $\unicode[STIX]{x0394}T$ , and magnetic field with $2\unicode[STIX]{x1D6FA}d(\unicode[STIX]{x1D70C}\unicode[STIX]{x1D707}_{0})^{1/2}$ . The resulting system of dimensionless governing equations is

where $\boldsymbol{u}=(u_{x},u_{y},u_{z})$ is the velocity, $p$ the pressure, $\unicode[STIX]{x1D703}$ the temperature perturbation relative to a linear background profile, and $\boldsymbol{B}=(B_{x},B_{y},B_{z})$ the magnetic field. The dimensionless parameters are the Rayleigh number,

the Ekman number,

and the thermal and magnetic Prandtl numbers,

We assume that all variables are periodic in the horizontal directions. In the vertical direction, the upper and lower boundaries are taken to be perfect thermal and electrical conductors, impermeable and stress free, i.e.

It is worth noting that the implementation of stress-free boundary conditions provides the best opportunity for the development of horizontal flows of large amplitude. For the Ekman numbers considered here, no-slip boundary conditions inhibit the formation of LSVs (Stellmach et al. Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014).

On occasion, we shall refer to results obtained for the kinematic dynamo problem. This is governed by equations (2.1)–(2.5), but with the Lorentz force (the final term on the right-hand side of (2.1)) omitted; there is then no feedback from the magnetic field onto the flow and the problem becomes linear in the magnetic field $\boldsymbol{B}$ .

2.2 Numerical method

Equations (2.1)–(2.5) are solved numerically using a parallel pseudospectral code developed by Cattaneo, Emonet & Weiss (Reference Cattaneo, Emonet and Weiss2003). The temperature perturbation and all the components of the velocity and magnetic field are transformed from configuration space (containing $N_{x}\times N_{y}\times N_{z}$ collocation points) to phase space by a discrete Fourier transform of the form

where $f$ and $\hat{f}$ are the functions in configuration and phase spaces respectively and $\text{c.c.}$ denotes complex conjugate. The function $\unicode[STIX]{x1D719}(s)$ is governed by the boundary conditions: $\unicode[STIX]{x1D719}(s)=\cos (s)$ for $u_{x}$ , $u_{y}$ , $B_{x}$ and $B_{y}$ , whereas $\unicode[STIX]{x1D719}(s)=\sin (s)$ for $u_{z}$ , $B_{z}$ and $\unicode[STIX]{x1D703}$ . Further details concerning the numerical methods can be found in Cattaneo et al. (Reference Cattaneo, Emonet and Weiss2003). Table 1 gives the parameter values and numerical resolution of the simulations discussed in this paper.

Table 1.

Summary of the parameter values and numerical resolution for the simulations performed at $Ek=5\times 10^{-6}$ and $\unicode[STIX]{x1D706}=1$ .

2.3 Output parameters for the flow and field

The convective dynamo problem is formulated unambiguously by the input parameters described in § 2.1, together with the initial conditions. It is helpful also to introduce additional output parameters that characterise the resulting velocity and magnetic field.

We define the Reynolds number in terms of the vertical velocity, which is a representative velocity of the convective flow, and the box depth, i.e.

where $\langle \cdot \rangle _{a}$ denotes an average over the direction $a$ . The magnetic Reynolds number is defined by $Rm=RePm$ .

We define the energy spectrum of the horizontal velocity, $\boldsymbol{u}_{h}=(u_{x},u_{y},0)$ , by

where $k_{h}=\sqrt{k_{x}^{2}+k_{y}^{2}}$ is the horizontal wavenumber and $C=(1+\unicode[STIX]{x1D6FF}_{k_{x}0}\unicode[STIX]{x1D6FF}_{k_{y}0})(1+\unicode[STIX]{x1D6FF}_{k_{z}0})$ . Similarly, the energy spectrum of the vertical velocity, $(0,0,u_{z})$ , is defined by

The energy spectra are obtained by binning the energy into rings of radius $k_{h}$ with $\unicode[STIX]{x0394}k_{h}=1/\unicode[STIX]{x1D706}$ . In the same way, we denote the energy spectra of the horizontal magnetic field, $\boldsymbol{B}_{h}=(B_{x},B_{y},0)$ , and the vertical magnetic field, $(0,0,B_{z})$ , by $E_{b}^{h}$ and $E_{b}^{v}$ respectively.

Finally, the integral horizontal wavenumber of the convective flow is defined as

where $k_{u}^{v}$ is also averaged in time. We use a similar formula to calculate the integral wavenumbers of $\boldsymbol{B}_{h}$ (denoted by $k_{b}^{h}$ ) and $B_{z}$ ( $k_{b}^{v}$ ).

3.1 Large-scale dynamo near the onset of convection

Figure 4.

Mean magnetic field produced by the dynamo near the onset of convection ( $Ra=1.2\times 10^{8}$ and $Pm=0.5$ ). (a) Snapshot of the vertical structure of the mean magnetic field $(\langle B_{x}\rangle _{xy},\langle B_{y}\rangle _{xy},0)$ ; (b) space–time diagram of $\langle B_{x}\rangle _{xy}$ and $\langle B_{y}\rangle _{xy}$ .

For Rayleigh numbers just above $Ra_{c}$ , the convection takes the form of elongated columns, which have a small horizontal cross-section and are aligned with the rotation axis. In the non-magnetic case, the integral convective wavenumber (defined by (2.16)) is $k_{u}^{v}=12.6$ for $Ra=1.2\times 10^{8}$ . For this Rayleigh number, the convective flows maintain a dynamo for $Pm\gtrsim 0.5$ , which corresponds to $Rm\gtrsim 24$ . The magnetic field is dominated by a mean (i.e. horizontally averaged) component; this is clearly seen in the horizontal cross-section of $B_{x}$ in figure 3(d). Note that, owing to the magnetic boundary conditions (2.10), the mean field can have only a horizontal component. This type of large-scale dynamo, which operates near the onset of convection, was first described analytically by Childress & Soward (Reference Childress and Soward1972) and Soward (Reference Soward1974), and was later studied through detailed numerical simulations of rotating convection by Stellmach & Hansen (Reference Stellmach and Hansen2004) and by consideration of specific flow planforms by Favier & Proctor (Reference Favier and Proctor2013) and Calkins et al. (Reference Calkins, Julien, Tobias, Aurnou and Marti2016). Details of this dynamo are recalled here for comparison with the dynamos found at larger values of $Ra$ . The dynamo process works as a two-scale mechanism, whereby the mean field is generated by the electromotive force (e.m.f.) produced by the collective action of small-scale convective columns. Figure 4(a) shows the vertical structure of the mean magnetic field $(\langle B_{x}\rangle _{xy},\langle B_{y}\rangle _{xy},0)$ , revealing a spiral staircase structure that is antisymmetric with respect to the mid-plane. Figure 4(b) shows the space–time diagram of $\langle B_{x}\rangle _{xy}$ and $\langle B_{y}\rangle _{xy}$ . The components of the mean field have a well-defined periodicity corresponding to a clockwise rotation of the entire staircase structure, with the maxima moving in time from the boundaries towards the mid-plane (Stellmach & Hansen Reference Stellmach and Hansen2004; Favier & Proctor Reference Favier and Proctor2013).

In figure 2, the green dashed line represents the transition between the regime of this mean-field, large-scale dynamo found close to the onset of convection and a regime in which there is no dynamo action; this boundary was deduced from consideration of the kinematic dynamo growth rates of neighbouring points in parameter space. The most significant, and somewhat counter-intuitive, feature of this dynamo is that increasing the Rayleigh number, and hence the strength of the convection and the magnetic Reynolds number, eventually kills off the dynamo process. Mean-field dynamo action relies crucially on a high degree of spatial and temporal coherence of the small-scale velocity in order to provide an effective e.m.f. (see, for example, Courvoisier, Hughes & Tobias Reference Courvoisier, Hughes and Tobias2009). As the convection becomes more vigorous, although there is more kinetic energy available to drive a dynamo, the coherence of the convective columns is lost, leading to the failure of the large-scale dynamo (Cattaneo & Hughes Reference Cattaneo and Hughes2006; Hughes & Cattaneo Reference Hughes and Cattaneo2008; Tilgner Reference Tilgner2012). Critically, this occurs even though the convective vortices remain rotationally constrained. Hence this large-scale dynamo is confined very close to the onset of convection, i.e. $Ra\lesssim 1.5Ra_{c}$ for $Ek=5\times 10^{-6}$ . Provided the magnetic Reynolds number is sufficiently high, the convective flows that are unable to generate a mean (large-scale) magnetic field are then able to generate only small-scale fields (i.e. fields with a size comparable with or smaller than the convective scale) (Cattaneo & Hughes Reference Cattaneo and Hughes2006).

3.2 Dynamos from LSVs

For sufficiently large thermal driving, an LSV forms in non-magnetic, rapidly rotating convection; an example is shown in figure 3(b). The LSV consists of a concentrated cyclone and a more dilute anticyclone; details of the formation and stability of the LSV are discussed in Favier et al. (Reference Favier, Silvers and Proctor2014), Guervilly et al. (Reference Guervilly, Hughes and Jones2014), Rubio et al. (Reference Rubio, Julien, Knobloch and Weiss2014) and Stellmach et al. (Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014). Starting from such convective flows, two very different types of dynamo can then be generated, depending on the values of $Rm$ , or, equivalently, on $Pm$ for fixed $Re$ ; in one the LSV is preserved by the magnetic field, whereas in the other it is destroyed.

Figure 5.

Time-averaged horizontal power spectra of (a) the horizontal magnetic energy $E_{b}^{h}$ and (b) the vertical magnetic energy $E_{b}^{v}$ , for different $Pm$ and $Ra=5\times 10^{8}$ .

Figure 5 shows the time-averaged horizontal power spectra of the magnetic energies for $Ra=5\times 10^{8}$ and for different values of $Pm$ . For $Pm=0.2$ , the energy of the horizontal field is dominated by small wavenumbers, $k_{h}\lesssim 10$ ; the vertical magnetic field is significantly weaker than the horizontal field for $k_{h}<12$ . As $Pm$ is increased beyond unity, the magnetic energy moves towards larger wavenumbers, peaking around the integral convective wavenumber $k_{u}^{v}\approx 12.1$ . For $Pm=0.2$ , the mean magnetic field (corresponding to the wavenumber $k_{h}=0$ ) contains on average 5 % of the total magnetic energy, while for $Pm=2.5$ , it represents only 0.3 %.

Figure 6.

Integral horizontal wavenumbers as a function of $Pm$ with $Ra=5\times 10^{8}$ .

Figure 6 shows the integral wavenumbers of the horizontal magnetic field, $k_{b}^{h}$ , and the vertical magnetic field, $k_{b}^{v}$ , as a function of $Pm$ with $Ra=5\times 10^{8}$ . For comparison, the integral convective wavenumber, $k_{u}^{v}$ , is also plotted. As expected from inspection of the magnetic energy spectrum, both $k_{b}^{h}$ and $k_{b}^{v}$ increase with $Pm$ . For $Pm\lesssim 1$ , the integral magnetic wavenumbers are smaller than the convective wavenumber, so most of the magnetic energy is contained in the large scales. Also in these cases, $k_{b}^{h}$ is systematically smaller than $k_{b}^{v}$ , so the horizontal magnetic field is dominated by scales larger than those of the vertical field. For $Pm\gtrsim 1$ , the integral magnetic wavenumbers are larger than the convective wavenumber, so most of the magnetic energy is contained in scales smaller than the convective scale; we refer to these dynamos as small-scale dynamos. In these cases, the integral wavenumbers of the horizontal and vertical magnetic fields are comparable. Hence there is a transition from a dynamo that generates a large-scale magnetic field (i.e. an LSV dynamo) to a small-scale dynamo; for this particular Rayleigh number ( $Ra=5\times 10^{8}$ ), the transition occurs for $Pm$ in the range $0.5<Pm<1$ . Figure 2 shows the magnetic Prandtl number for this transition (i.e. when $k_{b}^{h}=k_{u}^{v}$ ) for different values of $Ra$ . Note that the transition between the two dynamo regions is continuous.

The difference in the spatial structure of the magnetic field generated by the LSV dynamo at small $Pm$ and that of the small-scale dynamo at large $Pm$ is clearly demonstrated in the horizontal slices of $B_{x}$ shown in figure 3(e,f). The magnetic field for $Pm=0.2$ is organised into large-scale structures, concentrated in horizontal bands localised in the shear layers surrounding the LSV. In each band of concentrated field, the horizontal magnetic field has one main direction. In the core of the cyclone, the field intensity is weak. By contrast, for $Pm=2.5$ , the magnetic field is dominated by small scales; no large-scale organisation of the field is apparent.

Figure 7.

Mean magnetic field produced by the LSV dynamo ( $Ra=5\times 10^{8}$ and $Pm=0.2$ ). (a) Snapshot of the vertical structure of $(\langle B_{x}\rangle _{xy},\langle B_{y}\rangle _{xy},0)$ ; (b) space–time diagram of $\langle B_{x}\rangle _{xy}$ and $\langle B_{y}\rangle _{xy}$ .

Figure 7(a) shows the vertical structure of the mean magnetic field $(\langle B_{x}\rangle _{xy},\langle B_{y}\rangle _{xy},0)$ for the LSV dynamo at $Pm=0.2$ . It has a large-scale variation along $z$ with a structure similar to a spiralling staircase, but which is more complex than that of the mean field for the large-scale dynamo near the onset of convection (cf. figure 4 a). Figure 7(b) shows the space–time diagram of $\langle B_{x}\rangle _{xy}$ and $\langle B_{y}\rangle _{xy}$ . The direction and amplitude of the mean magnetic field evolves in time in an approximately periodic manner, with a tendency to drift from the boundaries towards the mid-plane. The averaged fields $\langle B_{x}\rangle _{xy}$ and $\langle B_{y}\rangle _{xy}$ are preferentially antisymmetric with respect to the mid-plane. The entire staircase structure tends to rotate clockwise, in a similar manner to that of the large-scale dynamo operating close to the onset of convection (figure 4 b), but here with a slower rotation rate.

Figure 8.

Time-averaged horizontal power spectra of (a) the horizontal kinetic energy $E_{u}^{h}$ , and (b) the vertical kinetic energy $E_{u}^{v}$ , for the same cases as in figure 5, together with the hydrodynamical case ( $Pm=0$ ).

Figure 8 shows the time-averaged horizontal power spectra of the kinetic energies for $Ra=5\times 10^{8}$ and for different values of $Pm$ , including the comparison with the spectrum in the hydrodynamical case (denoted by $Pm=0$ ). The spectrum for the horizontal flow at $Pm=0.2$ shows that the velocity is dominated by the LSV (i.e. $k_{h}=1$ ), in a similar fashion to the hydrodynamical case. As $Pm$ is increased, the energy at large scales is progressively diminished, for both the horizontal and vertical flows. For example, the energy of the LSV is decreased by a factor $50$ for $Pm=2.5$ compared with its value for $Pm=0.2$ . Figure 3(c) shows the axial vorticity for $Pm=2.5$ ; the flow is dominated by a multitude of convective vortices, with no LSV visible. For sufficiently large $Pm$ ( $Pm\gtrsim 1$ for $Ra=5\times 10^{8}$ ), the LSV is thus destroyed by the magnetic field. Guervilly et al. (Reference Guervilly, Hughes and Jones2015) argued that it is the generation of the small-scale magnetic field that yields the suppression of the LSVs. We study this question in detail in § 5. For small values of $Pm$ , a weaker small-scale magnetic field is generated, and consequently the LSV is at least partially preserved. Indeed, the presence of the LSV is vital to the generation of the large-scale magnetic field, as we shall see in § 4.

Figure 9.

(a) Horizontal and vertical cross-sections of axial vorticity for purely hydrodynamic convection with a small aspect ratio ( $\unicode[STIX]{x1D706}=0.25$ ) and $Ra=5\times 10^{8}$ . (b) Time evolution of the magnetic energy for $\unicode[STIX]{x1D706}=0.25$ , $Ra=5\times 10^{8}$ and three values of  $Pm$ .

To conclude this section, we determine the boundaries in parameter space of small-scale and LSV dynamo action. First we calculate the small-scale dynamo threshold. To do this, we perform simulations in a box of aspect ratio $\unicode[STIX]{x1D706}=0.25$ , chosen to be sufficiently small that no LSV forms, as can be seen from the horizontal cross-sections of the axial vorticity shown in figure 9(a). Figure 9(b) shows the time evolution of the magnetic energy for $Ra=5\times 10^{8}$ and three different values of $Pm$ with this small aspect ratio. For $Pm\lesssim 0.5$ , the magnetic energy decays, whereas for $Pm=1$ , the magnetic energy increases and eventually saturates. The sustained magnetic field is dominated by small scales. The small-scale dynamo threshold is thus located between $Pm=0.5$ and $Pm=1$ . To complete the diagram of figure 2, we computed dynamo simulations with $\unicode[STIX]{x1D706}=0.25$ for a range of $Ra$ . From these we find that the small-scale dynamo threshold is always in agreement with the dynamo transition identified earlier for $\unicode[STIX]{x1D706}=1$ , which was calculated by comparing the integral convective and magnetic wavenumbers (marked as diamonds in figure 2). The boundary between the two types of dynamo corresponds to the line where $Rm\approx 550$ ; this is plotted in figure 2. It should though be noted that this line is indicative; the actual transition between the two types of dynamo is continuous. We can also define a local magnetic Reynolds number, $Rm_{l}=Rm/k_{u}^{v}$ . The curve $Rm_{l}=45$ lies close to that of $Rm=550$ for all $Ra$ , since the integral convective wavenumber remains close to $k_{u}^{v}=12$ for $Ra$ increasing up to $Ra=10^{9}$ . The small-scale dynamo threshold can therefore be defined equivalently by the local magnetic Reynolds number taking the value $Rm_{l}\approx 45$ . This threshold is independent of the formation of the LSV and, indeed, exists for $Ra<3\times 10^{8}$ , where no LSVs are present.

The lower threshold of the LSV dynamo is determined from consideration of kinematic dynamo action as $Pm$ is decreased; this is shown as the red dashed line in figure 2. For $Ra>5\times 10^{8}$ , the threshold remains fixed at $Pm\approx 0.15$ , at least up to the highest Rayleigh number calculated, $Ra=10^{9}$ . For larger $Ra$ , there are two reasons why the LSV dynamo might fail: the LSVs themselves might cease to exist at large Rossby number (cf. figure 1), or the small-scale dynamo threshold intersects that of the LSV dynamo. From hydrodynamical simulations, the strength of the LSV declines when the local Rossby number, $Ro_{l}$ , exceeds a value of approximately $0.1$ . For the Ekman number considered here, this occurs for $Ra>1.8\times 10^{9}$ (i.e. $Ra/Ra_{c}>18$ ). Since it was not possible to pursue dynamo simulations for $Ra>10^{9}$ , we were unable to locate directly the intersection of the small-scale and LSV dynamo thresholds. However, we could perform hydrodynamical simulations for larger $Ra$ , up to $Ra=2.2\times 10^{9}$ ( $Ra/Ra_{c}\approx 22$ ), thereby allowing us to make a prediction about the demise of the LSV dynamo under the assumption that the small-scale dynamo threshold continues to obey $Rm_{l}\approx 45$ for $Pm<0.5$ . We find that this boundary intersects the line $Pm=0.15$ for some value of $Ra$ in the range $2\times 10^{9}<Ra<2.2\times 10^{9}$ . For this value of $Ek$ , the two critical Rayleigh numbers – one marking the demise of the LSV, the other the preference for small-scale dynamo action – are, coincidentally, of the same order of magnitude.

4.1 LSV filtering

We first examine the role played by the LSV in the dynamo process: in particular, does the LSV velocity field contribute directly to the magnetic induction, or does it play a more subtle role through modifying the smaller-scale flows in such a way that they become favourable to dynamo action? To address this question, we consider the velocity field obtained by removing only the modes $(k_{x},k_{y})\leqslant (1,1)$ from the velocity (LSV filtering). A snapshot of this filtered flow is shown in figure 10(d), illustrating that the effect of the LSV on the rest of the flow is kept intact; for comparison, the unfiltered flow is shown in figure 10(a). The result of the kinematic dynamo simulation is shown in figure 11(b) (labelled LSV-filter), leading to the interesting result that the filtered flow fails to act as a dynamo. Consequently, despite its inability to sustain a dynamo by itself, the LSV must play a critical role in the magnetic induction; in particular, its importance for dynamo action is not confined to its hydrodynamical influence on the other parts of the flow.

Figure 11.

Time series of the magnetic energy of the filtered simulations ( $Ra=5\times 10^{8}$ , $Pm=0.3$ ): (a) short-wavelength cutoff; (b) long-wavelength cutoff and LSV filtration.

Figure 12.

Snapshots of $\langle B_{x}^{2}+B_{y}^{2}\rangle _{z}$ for (a) the unfiltered flow (kinematic dynamo), and flows with (b) short-wavelength cutoff ( $k_{f}=10$ ), and (c) long-wavelength cutoff ( $k_{f}=4$ ).

Figure 13.

Time average of $(B_{x}^{2}+B_{y}^{2})/\text{max}(B_{x}^{2}+B_{y}^{2})$ (top row) and $B_{z}^{2}/\text{max}(B_{z}^{2})$ (bottom) in a horizontal slice ( $z=0.25$ ) for short-wavelength filtered dynamo simulations: (a)  $k_{f}=10$ ; (b)  $k_{f}=30$ , (c) $k_{f}=40$ ; (d) unfiltered flow (kinematic dynamo).

4.2 Short-wavelength filtering

We now consider the nature of the dynamo action that arises from short-wavelength filtration of the velocity field. The cutoff horizontal wavenumber is denoted by $k_{f}$ : all the spectral modes for which either $k_{x}>k_{f}$ or $k_{y}>k_{f}$ are set to zero. Examples of short-wavelength filtered flows are shown in figure 10(b, c). Figure 11(a) shows time series of the magnetic energy from simulations in which the cutoff wavenumber is varied from $k_{f}=30$ to $k_{f}=2$ ; the magnetic energy of the kinematic dynamo of the unfiltered flow is also plotted for comparison. As $k_{f}$ is decreased from the unfiltered flow to $k_{f}=10$ , the dynamo is successfully maintained by the filtered flows; indeed, the growth rate of the magnetic energy increases, indicating that the small scales are simply providing enhanced dissipation. For $k_{f}=8$ , the flow still acts as a dynamo, but with a smaller growth rate than that of $k_{f}=10$ , thereby indicating that the filtration has started to impede the dynamo mechanism. For $k_{f}=6$ , the magnetic energy is essentially neither decreasing nor increasing, i.e. this case is close to the dynamo threshold. For $k_{f}\leqslant 5$ , the dynamo fails. The intermediate modes of scale between that of the box size and that given by $k_{h}=O(8{-}10)$ are therefore absolutely vital for dynamo action at $Pm=0.3$ . Visual comparison of a flow that is capable of dynamo action (figure 10 b for $k_{f}=10$ ) with one that is not (figure 10 c for $k_{f}=6$ ), suggests that the elongated flow structures that are produced by the horizontal shear from the LSV might be crucial for the dynamo process.

Figure 12(b) shows the structure of the horizontal magnetic field for the filtered simulation with $k_{f}=10$ ; for comparison, the field produced by the unfiltered flow in the kinematic case is shown in figure 12(a). The plotted quantity is a snapshot of $\langle B_{x}^{2}+B_{y}^{2}\rangle _{z}$ , twice the depth-averaged horizontal magnetic energy. As in the unfiltered case, the horizontal magnetic field appears on a large scale and is organised into bands around the cyclone. However, unlike the unfiltered case, the horizontal field is also strong in the core of the cyclone. This observation indicates that magnetic induction does occur in the core of the cyclone, but that the small-scale components of the unfiltered flow (i.e. $k_{h}>k_{f}$ ) must subsequently expel the magnetic field from the core into the shear layer.

The destruction of the field inside the core of the cyclone is portrayed in figure 13, which shows the time-averaged values of $B_{x}^{2}+B_{y}^{2}$ and $B_{z}^{2}$ in a given horizontal slice, for different filtered simulations as well as for the unfiltered case. Since here we are considering kinematic dynamos, the values of $B_{x}^{2}+B_{y}^{2}$ and $B_{z}^{2}$ have been normalised by their maximum value at each time step before averaging. The time averages are taken over $400$ snapshots, during which the LSV drifts slightly in the horizontal plane; this explains why the averaged values are not as sharp as in the snapshot of figure 12(b). For $k_{f}\leqslant 30$ , both the horizontal and vertical magnetic fields are concentrated mainly in the cyclone. As the filtration is relaxed ( $k_{f}\geqslant 40$ ), the horizontal and vertical magnetic fields are expelled from the core of the cyclone. It is therefore the flow corresponding to large wavenumbers ( $k_{h}>30$ ) that leads to the magnetic flux expulsion. The dilute large-scale anticyclone always corresponds to a region of weak magnetic field; here, the horizontal shear is weaker than in the concentrated large-scale cyclone and seems unable to promote field amplification.

4.3 Long-wavelength filtering

In performing long-wavelength filtration, the amplitudes of the modes for which either $k_{x}<k_{f}$ or $k_{y}<k_{f}$ are set to zero, except, crucially, for the modes corresponding to the large-scale flow, i.e. $(k_{x},k_{y})\leqslant (1,1)$ . We have seen from § 4.1 that the LSV plays a critical role in the magnetic induction. The aim here therefore is to determine whether a large-scale dynamo of ‘ $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D714}$ ’ type is at work, i.e. a two-scale dynamo driven by a combination of the shearing at large scale (the $\unicode[STIX]{x1D714}$ -effect) and the production of a coherent e.m.f. by interactions of the small scales (the $\unicode[STIX]{x1D6FC}$ -effect). Examples of flows obtained from a long wavelength filtration are shown in figure 10(e, f). Figure 11(b) shows the temporal evolution of the magnetic energy from a series of filtered simulations in which the cutoff wavenumber is varied from $k_{f}=10$ to $k_{f}=4$ . For $k_{f}\gtrsim 6$ the dynamo fails, but with $k_{f}=4$ the flow is able to sustain a magnetic field. These experiments thus demonstrate that the dynamo requires the presence of intermediate scales up to $k_{h}=4$ , i.e. a scale somewhat larger than a characteristic convective scale (see figure 6), but that larger scales (with the exception of $k_{h}=1$ ) are not necessary for dynamo action at $Pm=0.3$ . There is therefore no rigorous scale separation between that of the shear exerted by the LSV and that of the modes that are significant for dynamo action ( $k_{f}\approx 4{-}10$ ); as such, this dynamo cannot strictly be categorised as an $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D714}$ -dynamo. Figure 12(c) shows the structure of the horizontal magnetic field for the case $k_{f}=4$ . The field is similar to that produced in the unfiltered case, with bands of strong magnetic intensity concentrated in the shear layers surrounding the LSV, and with no magnetic field in the core of the cyclone.

In summary, the filtering exercise helps us to identify that the LSV dynamo mechanism (at least in the kinematic regime) relies on the presence of the LSV together with velocity modes of scale intermediate between the box size and the dominant convective scale. The LSV itself plays a crucial role in the magnetic induction, and not simply via its action on the smaller-scale flows. The filtered simulations also show that the magnetic field is generated in the core of the LSV and in the surrounding shear layers. The magnetic field is subsequently expelled from the core by small-scale vortices.

Figure 14.

Time series of the kinetic and magnetic energies of the modes $k_{h}=0$ , $k_{h}=1$ and $k_{h}>1$ for the dynamos at $Ra=5\times 10^{8}$ , (a)  $Pm=2.5$ and (b)  $Pm=0.2$ . The dotted vertical lines indicate the approximate times at which the kinetic energy of the LSV starts to decrease.

5.1 Suppression of the LSV

In this subsection we consider the dynamo resulting from the convective flow with $Ra=5\times 10^{8}$ and $Pm=2.5$ ; as can be seen from figure 2, it lies well within the small-scale dynamo regime. Figure 14(a) shows the time series of the kinetic and magnetic energies of the horizontal wavenumbers $k_{h}=0$ (for the magnetic field only), $k_{h}=1$ and $k_{h}>1$ . The initial condition is one of purely hydrodynamic convection, to which a seed magnetic field is added at $t=0$ . When the magnetic energy grows to a sufficiently large value, the kinetic energy of the LSV ( $k_{h}=1$ ) decays rapidly, before reaching a new saturated level, more than two orders of magnitude smaller than in the hydrodynamical case. In the meantime, the kinetic energy of the modes with $k_{h}>1$ decreases by about $25\,\%$ , which is mainly attributable to the decrease in the energy of modes $k_{h}=2{-}3$ . In the saturated phase, the kinetic energy of the mode $k_{h}=1$ is smaller than the total energy of the other modes by a factor of $30$ (see also the kinetic energy spectra of figure 8). Although the magnetic field does not completely eliminate the kinetic energy at the largest scale, no coherent LSV is apparent. The magnetic field therefore has an unfavourable effect on the LSV, while leaving the amplitude of the convective flows relatively unchanged. Since no large-scale magnetic field is generated by this dynamo, the unavoidable conclusion is that the small-scale field is responsible for the suppression of the LSV. It is of interest to note that this suppression does not require a strong magnetic field; at the point when the amplitude of the LSV starts to decrease, the magnetic energy is only approximately one tenth of the kinetic energy of the convective flow.

5.2 Oscillations of the LSV dynamo

Here we consider the same convective flow as in § 5.1 (i.e. $Ra=5\times 10^{8}$ ), but at the much lower magnetic Prandtl number of $Pm=0.2$ . Time series of the kinetic and magnetic energies are shown in figure 14(b); in contrast to figure 14(a), only the saturated (dynamic) phase is shown, because the kinematic dynamo growth rate is slow. Large oscillations in the kinetic energy of the LSV are observed. Such oscillations are absent in the purely hydrodynamic case, and so are indisputably of magnetic origin. The kinetic energy of the modes with $k_{h}>1$ remains mostly unchanged except for small variations that follow the oscillations of the LSV. Oscillations of the magnetic energy are associated with the oscillations of the LSV. These oscillations are nearly anticorrelated, but a systematic time lag exists between maxima of the LSV amplitude and minima of the magnetic energy. The oscillations are not quite periodic as their duration and amplitude varies, those with the largest variation having the longest duration. All scales of the magnetic field are affected by the oscillations, with a seemingly similar evolution. Once the magnetic field is amplified to a critical strength by the dynamo, the LSV decays. For each oscillation, a dotted vertical line indicates the approximate time at which the LSV starts to decay. The vertical lines highlight that this critical value of the magnetic energy is similar for each oscillation ( $O(10^{-6})$ for the modes $k_{h}>1$ ) and is also of the same order as the critical value observed for $Pm=2.5$ . The decay of the LSV is shortly followed by the decrease of the magnetic energy at all scales. Subsequently, once the field becomes sufficiently weak, the LSV regenerates, eventually leading to a new phase of growth of the magnetic field. While the oscillations of the magnetic energy have sharp minima and maxima, the oscillations of the LSV have rounded maxima and sharp minima. The rounded maxima are a consequence of the slow growth of the magnetic field, which allows enough time during the recovery phase of the LSV for it to saturate hydrodynamically before the magnetic energy reaches the critical value of approximately $10^{-6}$ . The periodicity of the oscillations is controlled by the growth and decay rates of the magnetic field and also by the growth rate of the LSV because the dynamo needs the LSV to have sufficient velocity to start to operate. In the LSV dynamo – for which $Rm$ is below the threshold for small-scale dynamo action – the small-scale magnetic field is produced by the interactions of the large-scale field with the convective flows. The evolution of the small-scale field therefore closely follows that of the large-scale field. For $Pm=2.5$ , the suppression of the LSV can be attributed solely to the small-scale field; it is plausible that this is also the case for the dynamo at $Pm=0.2$ , although the direct influence of the large-scale field cannot be entirely ruled out.

Figure 15.

Range of the oscillations of (a) kinetic and (b) magnetic energy for several values of $Ra$ . The dashed red lines correspond to the threshold of the LSV dynamo and the dashed blue lines to $Rm=550$ , the threshold of the small-scale dynamo (cf. figure 2). For comparison, the relative amplitude of the oscillations of the kinetic energy in the hydrodynamical case is represented at $Pm=0$ .

Figure 15 shows the amplitude of the oscillations of the energies for both the LSV dynamos and small-scale dynamos. The transition between the two types of dynamo is identified by the dashed blue line, which corresponds to $Rm=550$ (cf. figure 2). For comparison, the oscillations of the kinetic energy in the absence of the magnetic field are also represented at $Pm=0$ . In the absence of the magnetic field, LSVs are present for $Ra\gtrsim 3\times 10^{8}$ and the fluctuations in the kinetic energy are small in this case. By contrast, the variations of the kinetic energy are much larger in the dynamo cases. In particular, the oscillations of both energies are largest for small $Pm$ . Indeed, for small $Pm$ , the maximum value of the kinetic energy can even attain the level of the purely hydrodynamic case, implying that the LSV is temporarily fully restored. As $Pm$ increases, however, the maximum of the kinetic energy remains well below the hydrodynamic level. For the LSV dynamos, the maximum of the magnetic energy is comparable at all values of $Pm$ for a given value of $Ra$ . This is consistent with the existence of a critical value of the magnetic field at which the LSV is systematically disrupted. As $Pm$ increases, this critical value is reached more quickly; as such, the suppression of the LSV occurs before it can reach its full amplitude. Consequently the oscillations tend to have shorter duration as $Pm$ increases.

5.3 Disruption of the Reynolds stresses

In the absence of magnetic field, LSVs are maintained by the persistent action of the Reynolds stresses, and in particular, the interaction of $z$ -dependent convective vortices (Rubio et al. Reference Rubio, Julien, Knobloch and Weiss2014). In this subsection, we examine the means by which the magnetic field can lead to the suppression of the LSV by the magnetic field. There are two possible mechanisms: (i) the magnetic field destroys the Reynolds stresses that are the source terms of the LSV by decreasing either their amplitude or their spatio-temporal coherence, or (ii) the Maxwell stresses cancel out the Reynolds stresses. To investigate how the Reynolds stresses are modified in the presence of the magnetic field, we focus on two velocity components of the LSV, $\langle u_{x}\rangle _{xz}$ and $\langle u_{y}\rangle _{yz}$ . The equations for the temporal evolution of these components are

where the components of the Reynolds and Maxwell stresses are respectively

and where we have assumed that $\langle u_{x}\rangle _{yz}\approx 0$ and $\langle u_{y}\rangle _{xz}\approx 0$ . The velocity and magnetic fields are decomposed into $z$ -average and $z$ -dependent parts, where $v^{\prime }=v-\langle v\rangle _{z}$ . The stress components $R$ and $M$ can then be decomposed into

Figure 16.

Time series of the r.m.s. values of the components of the Reynolds and Maxwell stresses as defined in (5.4) and (5.5) for the same time series as in figure 14(a).

The term of particular interest is $R_{f}$ , which represents the interaction of $z$ -dependent vortices, and thus, the driving term of the LSV. Figure 16 shows the time evolution of the r.m.s. values of $R_{m}$ , $R_{f}$ and $M_{f}$ for the small-scale dynamo at $Ra=5\times 10^{8}$ and $Pm=2.5$ . This time series can be compared with the evolution of the energies in figure 14(a). Since the magnetic field does not have a large-scale component, the r.m.s. value of $M_{m}$ is small, and hence is not plotted in figure 16. As the small-scale field grows, the most notable change to the Reynolds stresses is the decrease of the r.m.s. value of $R_{m}$ by approximately two orders of magnitude. This is an expected consequence of the decrease of the velocity of the LSV by an order of magnitude since the major contribution to $R_{m}$ comes from the self-interaction of the LSV. Of greater interest are the more subtle changes to the r.m.s. value of $R_{f}$ : the amplitude of $R_{f}$ remains at a similar level in the presence of the magnetic field but its temporal variations change, marked in particular by the disappearance of the high frequencies. The suppression of the LSV cannot therefore be ascribed to a decrease of the amplitude of its driving term $R_{f}$ . As the magnetic field increases in strength, $M_{f}$ increases, eventually reaching a level approximately half the averaged r.m.s. value of $R_{f}$ . The amplitude of $M_{f}$ is therefore too small to cancel completely $R_{f}$ , but partial cancellation might occur. To determine this, the r.m.s. value of the residual of the components of the Reynolds and Maxwell stresses, $R-M$ , is also plotted in figure 16. $R-M$ has the same amplitude as $R_{f}$ in the saturated phase of the dynamo, thereby indicating that the suppression of the LSV is not due to a straightforward cancellation of the Reynolds stresses by the Maxwell stresses. Instead, it follows that the LSV is suppressed by a modification of the driving term $R_{f}$ by the magnetic field, although its amplitude remains relatively unchanged.

Figure 17.

Space–time diagram of $\langle R_{f}\rangle _{x}$ and $\langle R_{f}\rangle _{y}$ for the same series as figure 16: (a) the kinematic phase, and (b) the saturated phase of the dynamo. The plot on the right of each diagram represents the time average.

Figure 17 shows the spatio-temporal evolution of $\langle R_{f}\rangle _{x}$ and $\langle R_{f}\rangle _{y}$ during the kinematic and saturated phases of the dynamo. To remove the variations of high temporal frequency in the data, we have performed a moving average over $30$  time units. During the kinematic phase, $\langle R_{f}\rangle _{x}$ and $\langle R_{f}\rangle _{y}$ maintain good temporal coherence of the modes $k_{y}=1$ and $k_{x}=1$ respectively. The time series is short enough that the LSV does not drift appreciably in the horizontal plane. The time averages therefore have a well-defined signal at $k_{y}=1$ or $k_{x}=1$ . Thus, in the kinematic regime, the LSV is forced consistently by the convective vortices. By contrast, during the saturated phase, this temporal coherence is lost, so the LSV cannot be forced consistently. Thus, significantly, the demise of the LSV is due not to a reduction in the amplitude of the Reynolds stresses that feed the LSV, nor by their cancellation by the Maxwell stresses, but instead to the more subtle influence of the small-scale field in hindering the coherence of the Reynolds stresses. We note that the study of Rubio et al. (Reference Rubio, Julien, Knobloch and Weiss2014) in non-magnetic convection shows that the feedback of the LSV itself plays a positive role on the energy transfer from small to large scales. The loss of the positive feedback of the LSV might therefore be partly responsible for the loss of the coherence in the Reynolds stresses, although its primary cause is the disruption by the magnetic field.