LARGE-SCALE DYNAMOS AT LOW MAGNETIC PRANDTL NUMBERS

Visualizations of one component of the magnetic field show the emergence of a large-scale magnetic field for small values of $\mbox{\rm Pr}_M$. This is clearly demonstrated in Figure 1, where we see for $\mbox{\rm Pr}_M=0.01$ a large-scale pattern with a systematic variation in the y direction. For $\mbox{\rm Pr}_M=0.1$ there is no such variation, although there are some extended patches in which the field orientation is the same. For $\mbox{\rm Pr}_M=1$ even this is no longer the case and the field appears completely random with small-scale variations only.

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We emphasize that random and patch-like structures only occur in the kinematic regime. In the saturated regime a large-scale field emerges in all cases. This will be discussed in Section 3.7.

The growth rate is calculated as the average of the instantaneous growth rate, dln B_rms/dt. Examples are shown in Figure 2 for runs with $\mbox{\rm Re}=670$ and different values of $\mbox{\rm Pr}_M$ using 512³ meshpoints. In Figure 3, we show growth rates normalized by u_rmsk_f (inverse turnover times) as a function of $\mbox{\rm Re}_M$ for three values of $\mbox{\rm Re}_M$ and compare with the corresponding results for nonhelical turbulence forced at larger scales in the wavenumber interval 1 ⩽ k/k₁ ⩽ 2 (Haugen et al. 2004). In that case we use k_f = 1.5 for the average value. The small-scale dynamo is then only excited when $\mbox{\rm Re}_M\ge 35$. For $\mbox{\rm Re}_M\ge 100$ the growth rates for helical turbulence with k_f = 4 are quite similar to those of nonhelical turbulence with k_f = 1.5. We have also calculated growth rates for the nonhelical case with k_f = 4 and find the same values as in the helical case. We note that in all cases, and even for small values of $\mbox{\rm Pr}_M$, the growth rates based on the rms magnetic field are equal to those based on the rms values of the mean fields obtained by averaging over any two coordinate directions.

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In both helical and nonhelical cases, when $\mbox{\rm Re}_M$ is large enough, λ increases like $\mbox{\rm Re}_M^{1/2}$, as expected (Schekochihin et al. 2004). This is because the eddy turnover rate at the resistive scale is ∝k^2/3_η, but because $k_\eta /k_{\it f}\propto \mbox{\rm Re}_M^{3/4}$ we have $\lambda \propto \mbox{\rm Re}_M^{1/2}$. However, when there is also large-scale dynamo action, one expects there to be a lower bound for λ given by the growth rate for the large-scale dynamo, λ_LS. Using the theory for an α² dynamo (Moffatt 1978, Krause & Rädler 1980), we have

$$\begin{equation} \lambda _{\rm LS}=|\alpha k|-(\eta +\eta _{\it t})k^2, \end{equation} \tag{ 4 }$$

where α is a pseudo scalar (the α effect) and η_t is the turbulent magnetic diffusivity. For fully helical turbulence we have |α| ≈ u_rms/3 and η_t ≈ u_rms/3k_f (Sur et al. 2008), so we can write the growth rate as

$$\begin{equation} {\lambda _{\rm LS}\over u_{\rm rms}k_{\it f}}=\mbox{$\frac{1}{3}$}{k_1\over k_{\it f}} \left[1-{k_1\over k_{\it f}}\left(1+3\mbox{\rm Re}_M^{-1}\right)\right], \end{equation} \tag{ 5 }$$

where we have put |k| = k₁. Over the parameter range considered in this paper ($\mbox{\rm Re}_M\ge 6.7$ and k_f/k₁ = 4), λ_LS/u_rmsk_f increases only slightly from 0.053 to 0.062 as $\mbox{\rm Re}_M$ increases.

The result shown in Figure 3 gives values that are systematically below λ_LS. There could be two reasons for this discrepancy. On the one hand, the accuracy of the estimates |α| ≈ u_rms/3 and η_t ≈ u_rms/3k_f may not be good enough. On the other hand, Equation (4) is only an approximation in cases where λ_LS ≠ 0, because then memory effects become important. This means that, when allowing α and η_t to be integral kernels in time, they are no longer proportional to δ functions, but have finite widths in time. This effect has recently been studied by Hubbard & Brandenburg (2008) and can be quite dramatic in some cases.

In Figure 3, we have varied $\mbox{\rm Re}_M$ by changing $\mbox{\rm Pr}_M$ and keeping $\mbox{\rm Re}={\rm const} {}=670$. We can therefore also consider this graph as a representation of the magnetic Prandtl number dependence of λ. However, $\mbox{\rm Pr}_M$ can also be changed while keeping $\mbox{\rm Re}_M={\rm const} {}=6.7$. The corresponding result is shown in Figure 4 and compared with the previous case. The two graphs are in reasonable agreement for small values of $\mbox{\rm Pr}_M$, but the rise of λ for $\mbox{\rm Re}=670$ around $\mbox{\rm Pr}_M=1$ is not seen in the case with $\mbox{\rm Re}_M=6.7$. This suggests that the transition from a purely large-scale turbulent dynamo to a mixed large-scale and small-scale turbulent dynamo requires values of $\mbox{\rm Re}_M$ above some critical value (somewhere between 10 and 100), and is not just determined by the value of $\mbox{\rm Pr}_M$.

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The transition from a purely large-scale turbulent dynamo to a mixed large-scale and small-scale turbulent dynamo is accompanied by characteristic changes in the spectral properties of the magnetic field. In the following we employ shell-integrated spectra of kinetic and magnetic energy, E(k) and M(k), respectively, as well as of kinetic and magnetic helicities, F(k) and H(k), respectively. These spectra are normalized such that $\int E(k)\,{\rm d} {}k={\textstyle {1\over 2}}\langle \mbox{\boldmath $U$} {}^2\rangle \equiv E$, $\int M(k)\,{\rm d} {}k={\textstyle {1\over 2}}\langle \mbox{\boldmath $B$} {}^2\rangle \equiv M$, $\int F(k)\,{\rm d} {}k=\langle \mbox{\boldmath $W$} {}\,{\bm \cdot}\, \mbox{\boldmath $U$} {}\rangle$, and $\int H(k)\,{\rm d} {}k=\langle \mbox{\boldmath $A$} {}\,{\bm \cdot }\, \mbox{\boldmath $B$} {}\rangle$, where $\mbox{\boldmath $W$} {}=\mbox{\boldmath $\nabla $} {}\times \mbox{\boldmath $U$} {}$ is the vorticity.

In Figure 5, we plot E(k) and M(k) for three cases with $\mbox{\rm Pr}_M=1$, 0.1, and 0.01, keeping $\mbox{\rm Re}=670$ in all cases. The magnetic energy spectra are compensated by exp(−λt), where λ is the numerically determined growth rate for each run, and then averaged in time. We consider here only the kinematic regime when the magnetic energy is weak. The kinetic energy spectra are then always the same. Since the magnetic energy is weak, we have scaled the magnetic energy spectra for different $\mbox{\rm Pr}_M$ to an arbitrarily chosen reference value of 10⁻⁶ below the kinetic energy spectrum. For $\mbox{\rm Pr}_M=1$ the magnetic energy seems to follows an approximate Kazantsev (1968) spectrum with a range proportional to k^3/2 and is peaked at the resistive scale near k/k₁ = 50. For smaller values of $\mbox{\rm Pr}_M$ the peak of magnetic energy moves to smaller wavenumbers.

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In order to judge the correspondence with various power-law scalings we label, in Figure 6, various compensated spectra as follows:

$$\begin{equation} \mbox{label E}:\quad E(k)\epsilon _K^{-2/3}k^{5/3}, \end{equation} \tag{ 6 }$$

$$\begin{equation} \mbox{label F}:\quad |F(k)|\epsilon _K^{-2/3}k^{5/3}/2k_{\it f}, \end{equation} \tag{ 7 }$$

$$\begin{equation} \mbox{label M}:\quad M(k)k_{\it f}(k/k_*)^{-3/2}/M, \end{equation} \tag{ 8 }$$

$$\begin{equation} \mbox{label H}:\quad |H(k)|k_{\it f}\epsilon _K^{-2/3}k^{5/3} E/2M, \end{equation} \tag{ 9 }$$

where k_* = ∫kM(k) dk/M is the wavenumber where the magnetic energy spectrum peaks and _K is the kinetic energy dissipation per unit mass. The compensated kinetic energy spectrum shows a bottleneck that is clearly stronger than in the case without helicity (e.g., Kaneda et al. 2003, Haugen & Brandenburg 2006). The kinetic helicity spectrum shows a similar spectrum that also has a strong bottleneck, which is particularly evident when it is compensated by k^5/3. The existence of a k^5/3 subrange for the modulus of the kinetic helicity spectrum is well known from early closure calculations (André & Lesieur 1977), and has also been seen in direct numerical simulations (Borue & Orszag 1997; Brandenburg & Subramanian 2005a) and in shell model calculations (Ditlevsen & Giuliani 2001). Such a scaling implies that the relative spectral kinetic helicity,

$$\begin{equation} {\cal R}_K(k)\equiv F(k)/2k E(k), \end{equation} \tag{ 10 }$$

decreases toward small scales like k⁻¹ and has a maximum at k = k_f with ${\cal R}_K(k_{\it f})=0.96$. At that scale, the relative magnetic helicity,

$$\begin{equation} {\cal R}_M(k)\equiv kH(k)/2 M(k), \end{equation} \tag{ 11 }$$

is −0.15, −0.08, and +0.42 for $\mbox{\rm Pr}_M=1$, 0.1, and 0.01, respectively. The realizability condition implies that the moduli of ${\cal R}_K(k)$ and ${\cal R}_M(k)$ are less than unity (Moffatt 1969). The positive sign for $\mbox{\rm Pr}_M=0.01$ agrees with the idea that the helical driving of the flow imprints a helical field of the same sense at the same scale. Owing to an inverse cascade of magnetic helicity (Pouquet et al. 1976), H(k) is of opposite sign at large scales. While this is very clearly established in the nonlinear regime (B01) or for small values of $\mbox{\rm Pr}_M$, a larger range of scales attains negative values during the linear stage when $\mbox{\rm Pr}_M=0.1$ and 1.

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For the magnetic helicity we also find an approximate |H(k)| ∼ k^−5/3 spectrum, which is different from the nonlinear case when the current helicity, C(k) = k²H(k) shows a k^−5/3 spectrum (Brandenburg & Subramanian 2005a). Assuming that M(k) ∼ k^3/2, the relative spectral helicity would seem to decrease now more rapidly like k|H(k)|/2M(k) ∼ k^−13/6. Figure 7 shows that the correspondingly compensated magnetic helicity to energy ratio changes now less strongly in the range 4 ⩽ k/k₁ ⩽ 40.

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Eventually the initial exponential growth comes to a halt and is followed by a resistively long saturation phase during which a large-scale magnetic field develops at wavenumber k₁, regardless of the value of k_f. Owing to the use of periodic boundary conditions, this large-scale field tends to be force-free and fully helical, and its energy per unit volume is by a factor k_f/k₁ = 4 larger than the value at k_f, which in turn is comparable to the kinetic energy per unit volume. For details see B01. Here we only consider the end of this slow saturation phase. Compensated kinetic and magnetic energy spectra are shown in Figure 8 for magnetic Prandtl numbers ranging from 1 to down to 10⁻³.

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In the final saturated state, and especially for $\mbox{\rm Pr}_M=1$, the M(k) and E(k) spectra are nearly on top of each other with M(k) being slightly larger than E(k) by 20%, which is qualitatively similar to the nonhelical case (cf. Haugen et al. 2003). There are indications of a somewhat shallower spectrum due to a bottleneck effect both for kinetic and magnetic energies just before the two enter the viscous and resistive dissipation ranges. Also for $\mbox{\rm Pr}_M=0.1$ there is a short range where M(k) exceeds E(k), but then, not surprisingly, M(k) turns into the dissipation range before E(k) does. This is even more clearly the case for $\mbox{\rm Pr}_M=0.01$.

Low-$\mbox{\rm Pr}_M$ turbulence has the interesting property that for given numerical resolution much larger fluid Reynolds numbers can be achieved than for $\mbox{\rm Pr}_M=1$. This is simply because almost all the energy is dissipated resistively, and the energy that continues along the kinetic energy cascade is comparatively weak, so not much viscosity is needed for dissipating the remaining kinetic energy. In fact, for $\mbox{\rm Pr}_M=0.01$ we were able to go to $\mbox{\rm Re}=2300$ with a resolution of only 512³ meshpoints. For $\mbox{\rm Pr}_M=0.1$ and 1 and the same resolution we could only go to $\mbox{\rm Re}=1200$ and 450, respectively.

As has recently been stressed by Mininni (2007), an increasing fraction of energy is being dissipated via Joule dissipation, as $\mbox{\rm Pr}_M$ decreases, In Figure 9, we plot the dependence of the kinetic and magnetic energy dissipation rates per unit mass, $\epsilon _K=\langle 2\rho \nu \mbox{\boldmath ${\sf S}$} {}^2\rangle /\rho _0$ and $\epsilon _M=\langle \eta \mu _0\mbox{\boldmath $J$} {}^2\rangle /\rho _0$, relative to the total dissipation, _T = _K + _M, versus $\mbox{\rm Pr}_M$. The data are well described by a power-law fit of the form

$$\begin{equation} \epsilon _K/\epsilon _T\approx 0.37\,\mbox{\rm Pr}_M^{1/2}. \end{equation} \tag{ 12 }$$

Thus, for $\mbox{\rm Pr}_M=1$ about the 37% of the energy is dissipated into viscous heat, while 63% is dissipated via Joule dissipation. This is similar to the case of nonhelical hydromagnetic turbulence (Haugen et al. 2003), where these numbers are about 30% and 70%, respectively.

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In turbulence the energy dissipation is generally proportional to U³/L, where U is the typical velocity and L is a typical length scale. Conventionally one defines a dimensionless dissipation parameter as

$$\begin{equation} C_\epsilon ={\epsilon _T\over U^3/L}, \end{equation} \tag{ 13 }$$

where U is the one-dimensional rms velocity, which is related to u_rms via U² = u²_rms/3, and L is the integral scale and is related to k_f via ${3\over 4}\pi /k_{\it f}$. In nonhelical turbulence this value is typically around 0.5 (see also Pearson et al. 2004), but this value has never been determined for hydromagnetic turbulence with helicity. An exception is the work of Blackman & Field (2008), who considered a range of power-law scalings for kinetic and magnetic energy spectra to calculate analytically the dissipation rates.

It turns out that for our runs, C ≈ 1.5, i.e. ≈3 times larger than the usual value; see Figure 10. Let us now discuss possible reasons for this difference. In the definition of the quantity C one assumes that the energy flux scales with U³/L. However, U is based on the typical rms velocity. In the presence of a strong dynamo-generated magnetic field it may be sensible to base it on a combination of typical velocity and magnetic field strength. In our case we have $\langle \mbox{\boldmath $B$} {}^2/\mu _0\rangle /\langle \rho \mbox{\boldmath $U$} {}^2\rangle \approx 2$, so U would need to be scaled up by a factor $\sqrt{3}$, which reduces C by a factor 3^3/2 ≈ 5 to about 0.3. This value is nearly independent of the value of $\mbox{\rm Pr}_M$, which was also found by Blackman & Field (2008) under plausible assumptions.

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As mentioned before, in the nonlinear regime both kinetic and current helicities, F(k) and C(k) = k²H(k), respectively, are expected to display a forward cascade with a k^−5/3 spectrum. If this is true, we would expect that within some wavenumber interval $|{\cal R}_K(k)|$ and $|{\cal R}_M(k)|$ decrease with increasing k like k⁻¹. In Figure 11 we show the correspondingly compensated relative kinetic and magnetic helicity spectra. It turns out that they are surprisingly similar regardless of the value of $\mbox{\rm Pr}_M$. For $\mbox{\rm Pr}_M=1$ and 0.1 the compensated profiles of ${\cal R}_M(k)$ and ${\cal R}_K(k)$ are reasonably flat in the range 6 ⩽ k/k₁ ⩽ 14. However, for $\mbox{\rm Pr}_M=10^{-2}$ and 10⁻³ the compensated profiles show an increase proportional to k^1/2. The fact that the anticipated k⁻¹ scaling occurs only for magnetic Prandtl numbers down to 0.1 and only over an extremely short range may indicate that our Reynolds numbers are still too small to yield conclusive results. Especially at smaller scales, and certainly in the runs with the smallest $\mbox{\rm Pr}_M$, the compensated relative kinetic and magnetic helicity spectra are compatible with a k^1/2 slope. This would imply a k^−7/6 spectrum for the kinetic and current helicities, which is shallower than that anticipated for a forward cascade, but still steeper than that in the case of equipartition. We emphasize again that this applies to the resistively controlled regime.

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In Figure 11 we see that at k = k_f both ${\cal R}_K$ and ${\cal R}_M$ are close to unity. This indicates that velocity and magnetic fields are nearly fully helical. However, for k > k_f the velocity and magnetic fields become less helical, because the compensated relative helicities in Figure 11 increase with k not faster than to the 1/2 power. On the other hand, for k = k₁ the magnetic field is again fully helical, i.e. $(k_1/k_{\it f})|{\cal R}_M(k_1)|$ is equal to k₁/k_f = 1/4, but its helicity has the opposite sign.

In Table 1 we compare the values of ${\cal R}_M(k)$ during the linear and nonlinear stages at the wavenumbers k₁ and k_f for the three or four values of $\mbox{\rm Pr}_M$. Note that during the nonlinear stage ${\cal R}_M(k_1)$ and ${\cal R}_M(k_{\it f})$ are of opposite sign. The former is close to −1 while the latter increases from 0.52 to 0.72 as $\mbox{\rm Pr}_M$ decreases. As already indicated in Section 3.3, during the linear stage, the two are of opposite sign only for $\mbox{\rm Pr}_M=0.01$, while for larger values of $\mbox{\rm Pr}_M$ a larger range of scales appears to be affected by the inverse transfer of magnetic helicity causing ${\cal R}_M(k_{\it f})$ to be negative. It would be tempting to try and model this behavior using, for example, the four-scale helical dynamo model of Blackman (2003).

Table 1. Comparison of ${\cal R}_M(k_1)$ and ${\cal R}_M(k_{\it f})$ During the Linear and Nonlinear Stages for Different Values of $\mbox{\rm Pr}_M$

	Linear		Nonlinear
$\mbox{\rm Pr}_M\quad$	k1	kf	k1	kf
10−3			−0.993	+0.72
10−2	−0.88	+0.42	−0.994	+0.66
10−1	−0.59	−0.08	−0.993	+0.59
1	−0.41	−0.15	−0.993	+0.52

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In Figure 12 we compare visualizations of B_z and U_z for all four values of $\mbox{\rm Pr}_M$. The velocity and magnetic field patterns are surprisingly similar for all four values of $\mbox{\rm Pr}_M$. Only for $\mbox{\rm Pr}_M=10^{-2}$ and 10⁻³ the magnetic field appears noticeably smoother than in the other two cases. The velocity field shows a marked anisotropy with small-scale elongated patterns aligned with the local direction of the mean magnetic field, which is here of the form $\overline{\mbox{\boldmath $B$}}{}\sim (0,\sin k_1x,-\cos k_1x)$. The anisotropy in the velocity can still be seen for small values of $\mbox{\rm Pr}_M$, but the small-scale patterns are slightly smoother.

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