Test-field method for mean-field coefficients with MHD background

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$\begin{threeparttable}% latex2html id marker 4238 \caption{Kinematic results for... ....9428$\space &$0.4714$$^{1}$\, \\ [1mm] \hline \end{tabular}\end{threeparttable}$

Notes. ⁽¹⁾ With SOCA. Test-field wavenumber k_z=1, except in the third column where k_z=0. These results agree with those of the imposed-field method. $\tilde\alpha^{\rm {QK}}$ and $\tilde\eta_{\rm t}^{\rm {QK}}$ refer to the quasi-kinematic method.

The second line of Table 2 repeats the result for $\tilde{N}_{\rm M}=1$ , again amended by those for k_z=1 and the results of the quasi-kinematic method, which is obviously unable to produce correct answers. This is because the mean electromotive force is now given by $\overline{\mbox{$\vec{u}$ } {}_{\hspace*{-1.1pt}\,\hspace{.3mm}\overline{\!\hspace{-.3mm}B}}\times\mbox{$\vec{b}$ } {}_0}$ , which is only taken into account in the generalized method. Note further that $\eta_{\rm t}$ is positive both for hydrodynamic and magnetic forcings.

4.1.3 Hydromagnetic forcing

As already pointed out in Sect. 3.5, in the absence of a mean field, for simultaneous kinetic and magnetic Roberts forcing with $\sigma =1$ there is a solution of Eqs. (19) and (20) consisting just of the solutions $\mbox{$\vec{u}$ } {}_0$ and $\mbox{$\vec{b}$ } {}_0$ of the system when forced purely hydrodynamically and magnetically, respectively. Again, a bifurcation leading to another type of solution cannot be ruled out, but was not observed.

In contrast to what one might suppose, however, the decoupling of $\mbox{$\vec{u}$ } {}_0$ and $\mbox{$\vec{b}$ } {}_0$ lets the value of $\alpha$ for hydromagnetic forcing in general not be simply additive in the values for purely hydrodynamic and purely magnetic forcings. We denote these by $\alpha_{\rm k}~{=}~ \alpha(\mbox{$\vec{b}$ } {}_0~{=}~{\bf0})$ and $\alpha_{\rm m}~{=}~\alpha(\mbox{$\vec{u}$ } {}_0~{=}~{\bf0})$ , respectively. Beyond SOCA, the terms $(\mbox{$\vec{u}$ } {}_{\hspace*{-1.1pt}\,\hspace{.3mm}\overline{\!\hspace{-.3mm}B}}\times\mbox{$\vec{b}$ } {}_0)'$ and $(\mbox{$\vec{j}$ } {}_0\times\mbox{$\vec{b}$ } {}_{\hspace*{-1.1pt}\,\hspace{.3... ....1pt}\,\hspace{.3mm}\overline{\!\hspace{-.3mm}B}}\times\mbox{$\vec{b}$ } {}_0)'$ in Eqs. (21) and (22) provide couplings between $\mbox{$\vec{u}$ } {}_{\hspace*{-1.1pt}\,\hspace{.3mm}\overline{\!\hspace{-.3mm}B}}$ and $\mbox{$\vec{b}$ } {}_{\hspace*{-1.1pt}\,\hspace{.3mm}\overline{\!\hspace{-.3mm}B}}$ and give rise to an additional ``magnetokinetic'' part of $\alpha$ , defined as $\alpha_{\rm mk}=\alpha - \alpha_{\rm k}-\alpha_{\rm m}$ . Note that we use here lower case subscripts k, m, mk to distinguish this split of the $\alpha$ values from that introduced at the end of Sect. 3.3, which applies only to the nonlinear case. In contrast, the occurrence of $\alpha _{\rm mk}$ is a purely kinematic effect. While $\alpha _{\rm k}$ and $\alpha _{\rm m}$ are, to leading order (and hence in SOCA), quadratic in the respective background fluctuations, the magnetokinetic term is of leading fourth order and is representable in schematic form as $\alpha_{\rm mk}\propto\overline{\mbox{$\vec{u}$ } {}_0^2~\mbox{$\vec{b}$ } {}_0^2}$ .

Lines 5 and 8 of Table 2 show cases with hydromagnetic forcing and amplitudes adjusted such that we would have $\tilde\alpha_{\rm k}~{=}~\tilde\alpha_{\rm m}=1$ if SOCA were valid. In either case the preceding two lines present the corresponding purely forced cases. Lines 6 to 8 refer to the SOCA versions of the generalized methods. It can be clearly seen that the results are additive only in the latter case. The value of $\vec{\mathsf \alpha}_{\rm mk}$ as inferred from lines 3 to 5, is -0.1 resulting in a considerable reduction of $\alpha$ in comparison with the additive value. This is owing to the strong forcing amplitudes, leaving the applicability range of SOCA far behind.

$\begin{figure} \par\includegraphics[width=9cm]{14700fig3} \end{figure}$

Figure 3: $\alpha$ versus $\mbox{\rm Re}_{\rm M}=\mbox{\rm Lu}$ for hydromagnetic Roberts forcing with k_z=0 (2D case). Along with the total value the constituents $\alpha _{\rm k}$ , $\alpha _{\rm m}$ and $\alpha _{\rm mk}$ as well as $\alpha _{\rm k}+\alpha _{\rm m}$ are shown. Note the sign change in $\alpha$ at $\mbox{\rm Re}_{\rm M}\approx5.4$ . Inset: $\alpha _{\rm mk}$ in comparison to the result of a fourth order analytical calculation (solid line).

$\begin{figure} \par\includegraphics[width=18cm]{14700fig4} \end{figure}$

Figure 4: $\alpha (k_z)$ , $\eta _{\rm t}(k_z)$ , and $\phi (k_z)$ for hydromagnetic Roberts forcing with $\sigma =1$ ( left three panels), likewise $\psi (k_z)$ , but for $\sigma =0.5$ ( rightmost panel). Solid lines: SOCA results, cf. Appendix C. Curve labels refer to $\mbox{\rm Re}_{\rm M}=\mbox{\rm Lu}$ or $(\mbox{\rm Re}_{\rm M},\mbox{\rm Lu})$ .

Figure 3 shows $\alpha _{\rm mk}$ for equally strong velocity and magnetic fluctuations as a function of $\mbox{\rm Re}_{\rm M}=\mbox{\rm Lu}$ together with $\alpha _{\rm k}$ , $\alpha _{\rm m}$ , $\alpha _{\rm k}+\alpha _{\rm m}$ and the resulting total value $\alpha$ . Note the significant difference between the naive extrapolation of SOCA, $\alpha_{\rm m}+\alpha_{\rm k}$ , and the true $\alpha$ . In its inset the figure shows the numerical values of $\alpha _{\rm mk}$ in comparison to the result of a fourth order calculation $\alpha_{\rm mk}~{=} -(\sqrt{2}/64)u_{\rm rms}^2b_{\rm rms}^2$ (for the derivation see Appendix F). Clearly, the validity range of this expression extends beyond $\mbox{\rm Re}_{\rm M}=\mbox{\rm Lu}=1$ and hence further than the one of SOCA. It remains to be studied whether the magnetokinetic contribution has a significant effect also in the more general case when $\mbox{$\vec{u}$ } {}_0\not\parallel\mbox{$\vec{b}$ } {}_0$ . If so, considering $\alpha$ to be the sum of a kinetic and a magnetic part, as often done in quenching considerations, may turn out to be too simplistic.

Likewise one may wonder whether closure approaches to the determination of transport coefficients supposed to be superior to SOCA can be successful at all as long as they do not take fourth order correlations into account properly.

For the tensors $\boldsymbol{\phi}$ and $\boldsymbol{\psi}$ ,

which turn out to show up with simultaneous hydromagnetic and magnetic forcing only (in addition, $\boldsymbol{\phi}$ requires z-dependent mean fields) we have of course again isotropy, $\phi_{11}=\phi_{22}\equiv\phi$ , $\psi_{11}=\psi_{22}\equiv\psi$ .

As a peculiarity of the Roberts flow, $\psi$ vanishes in the range of validity of SOCA if the helicity is maximum ( $\sigma =1$ in (42)). For this case the first three panels of Fig. 4 show the numerically determined dependences $\alpha (k_z)$ , $\eta _{\rm t}(k_z)$ and $\phi (k_z)$ with different values of u_0rms=b_0rms(data points, dotted lines). The last panel shows $\psi (k_z)$ for $\sigma =0.5$ and the same forcing amplitudes as before. As explained above, $\mbox{$\vec{u}$ } {}_0$ and $\mbox{$\vec{b}$ } {}_0$ can now no longer be forced independently from each other. Hence, both fields cannot show exactly the geometry defined by (42) and u_0rms and b_0rmsdiverge increasingly with increasing forcing.

As demonstrated in Appendix C, $\phi(k_z)\propto k_z^2/( k_z^2+k_{\rm f}^2)$ , $\alpha(k_z),\eta_{\rm t}(k_z),\psi(k_z)\propto 1/( k_z^2+k_{\rm f}^2)$ in the SOCA limit. For comparison these functions are depicted by solid lines. Note the clear deviations from SOCA for $\mbox{\rm Re}_{\rm M}=\mbox{\rm Lu}=5$ , particularly in $\alpha$ . Note also that the expression for $\psi$ was derived under the assumption that the background has the geometry (42). It is therefore not applicable in a strict sense. The clear disagreement with the values of $\psi$ from the test-field method for high values of $\mbox{\rm Re}_{\rm M}$ and $\mbox{\rm Lu}$ are hence not only due to violating the SOCA validity constraint.

4.2 Dependence on the mean magnetic field

We now admit dynamically effective mean fields and hence have to deal with anisotropic fluctuating fields $\mbox{$\vec{u}$ } {}$ and $\mbox{$\vec{b}$ } {}$ which result in anisotropic tensors $\vec{\mathsf \alpha}$ , $\vec{\mathsf \eta}$ , $\boldsymbol{\phi}$ and $\boldsymbol{\psi}$ . For the chosen forcing, $\overline{\mbox{\boldmath$B$ }}{}$ is the only reason for anisotropy in the xy plane, so $\vec{\mathsf \alpha}$ has to have the form

$\begin{displaymath}\alpha_{ij} = \alpha_1 \delta_{ij} + \alpha_2 \,\hspace{.3mm}... ...m}B}_i \,\hspace{.3mm}\hat{\!\hspace{-.3mm}B}_j,\quad i,j=1,2, \end{displaymath}$

with $\,\hspace{.3mm}\hat{\!\hspace{-.3mm}\bm{B}}$ being the unit vector in the direction of $\overline{\mbox{\boldmath$B$ }}{}$ (here the x direction). We obtain then $\alpha_{11}=\alpha_1+\alpha_2$ and $\alpha_{22}=\alpha_1$ . Of course, the tensors $\vec{\mathsf \eta}$ , $\boldsymbol{\phi}$ and $\boldsymbol{\psi}$ are built analogously. Clearly, irrespective of whether the forcing is pure or mixed, the effects of $B_{{\rm imp}}$ prevent $\mbox{$\vec{u}$ } {}$ and $\mbox{$\vec{b}$ } {}$ from having Roberts geometry.

In general, we leave in this section safe mathematical grounds and enter empirical work. Only for vanishing magnetic background, $\mbox{$\vec{b}$ } {}_0={\bf0}$ , one version of the generalized method does coincide with the quasi-kinematic one (see Sect. 3.4, Remark (iii)) and will therefore guarantee correct results.

4.2.1 Purely hydrodynamic forcing

In this case we have $\overline{\mbox{\boldmath${\cal E}$ }}{}_{\hspace*{-1.1pt}\,\hspace{.3mm}\overl... ... E}$ }}{}_{\hspace*{-1.1pt}\,\hspace{.3mm}\overline{\!\hspace{-.3mm}B}}^{\rm K}$ and all flavors of the generalized method have to yield results which coincide with those of the quasi-kinematic method. This is valid to very high accuracy for the ju and bu versions and somewhat less perfectly so for the bb and jb versions. We emphasize that the presence of $B_{{\rm imp}}$ , being solely responsible for the occurrence of magnetic fluctuations, does not result in a failure of the quasi-kinematic method as one might conclude from the model used by Courvoisier et al. (2010).

Figure 5 presents the constituents of $\vec{\mathsf \alpha}$ as functions of the imposed field in the 2D case. We may conclude from the data that $\alpha_2$ is negative and approximately equal to $\alpha _{11}^{\rm M}$ . For values of $B_{{\rm imp}}/\eta k_1> 5$ , its modulus approaches $\alpha_{22}=\alpha_1$ and thus gives rise to the strong quenching of the effective $\alpha=\alpha_{11}$ . Indeed, $\alpha(B_{{\rm imp}})$ can be described by a power law with an exponent -4 for large $B_{{\rm imp}}$ . This is at odds with analytic results predicting either $\alpha\propto B^{-2}$ (Field et al. 1999; Rogachevskii & Kleeorin 2000) or $\propto$ B^-3(Moffatt 1972; Rüdiger 1974). By comparing with computations in which the non-SOCA term was neglected, we have checked that our discrepancy with these predictions is not a consequence of SOCA applied therein. Sur et al. (2007) suggested that the B^-2 and B^-3 dependence is likely to be valid for time-dependent and steady flows, respectively. It should be noted, however, that their numerical values for the steady ABC flow do actually exhibit the B^-4 power law; cf. their Fig. 2. They also found that an $\alpha^{\rm M}$ , defined similarly to our $\alpha _{11}$ , increases quadratically with $\overline{\mbox{\boldmath$B$ }}{}$ for weak fields and declines quadratically for strong fields. This is in agreement with our present results.

4.2.2 Purely magnetic forcing

Here, the mean electromotive force is simply $\overline{\mbox{\boldmath${\cal E}$ }}{}= \overline{\mbox{\boldmath${\cal E}$ }... ...{-1.1pt}\,\hspace{.3mm}\overline{\!\hspace{-.3mm}B}}\times\mbox{$\vec{b}$ } {}}$ . This is true as long as significant velocities in the main run occur only due to the presence of the mean field, that is, as long as $\mbox{$\vec{u}$ } {}_0={\bf0}$ (see above). While $\overline{\mbox{\boldmath$B$ }}{}$ is weak, $\overline{\mbox{\boldmath${\cal E}$ }}{}$ is approximately $\overline{\mbox{$\vec{u}$ } {}_{\hspace*{-1.1pt}\,\hspace{.3mm}\overline{\!\hspace{-.3mm}B}}\times\mbox{$\vec{b}$ } {}_0}$ . However, one could speculate that, if the imposed field reaches appreciable levels, i.e., if $\mbox{$\vec{u}$ } {}$ is sufficiently strong, $\overline{\mbox{\boldmath${\cal E}$ }}{}$ can with good accuracy be approximated by $\overline{\mbox{\boldmath${\cal E}$ }}{}_{\hspace*{-1.1pt}\,\hspace{.3mm}\overl... ...x{$\vec{b}$ } {}_{\hspace*{-1.1pt}\,\hspace{.3mm}\overline{\!\hspace{-.3mm}B}}}$ . Since the quasi-kinematic method takes just this term into account, it could then produce useful results.

In Fig. 6 we show the rms values of the resulting velocity and magnetic fields as functions of the imposed field strength for $\tilde{N}_{\rm M}=1$ , corresponding to $\mbox{\rm Lu}=1/2$ if $\mbox{$\vec{B}$ } {}_{{\rm imp}}={\bf0}$ . The data points can be fitted by expressions of the form

$\begin{displaymath}{b_{\rm rms}\over b_{0{\mathrm{rms}}}}={1\over1+B_{{\rm imp}}... ...athrm{rms}}}}={B_{{\rm imp}}/B_*\over1+B_{{\rm imp}}^2/B_*^2}, \end{displaymath}$

(46)

where $B_*/\eta k_1\approx1.8~\,\hspace{.3mm}\widetilde{\!\hspace{-.3mm}N}_{\rm M}$ . Note, that indeed the velocity fluctuations become dominant over the magnetic ones for $B_{{\rm imp}}/\eta k_1>2$ .

$\begin{figure} \par\includegraphics[width=9cm,clip]{14700fig5}\end{figure}$

Figure 5: $\alpha _{11}$ (solid line, filled circles) and $\alpha _{22}$ (dashed line, open triangles) as functions of the imposed field strength $B_{{\rm imp}}$ , compared with $-\alpha _{11}^{\rm M}$ (dotted line, small dots) and $\alpha _{22}-\alpha _{11}=-\alpha _{2}$ (dotted line, open circles) for purely kinetic Roberts forcing with $\tilde{N}_{\rm K}=1$ . $\alpha _{11}^{\rm M}\approx \alpha _2$ throughout. Note that $\alpha _{0{\rm K}}<0$ and that the $\alpha$ symbols in the legend refer to quantities that are normalized by $\alpha_{0{\rm K}}\mbox{\rm Re}_{{\rm M}0}$ and hence sign-inverted.

The resulting finding, as shown in Fig. 7, is completely analogous to the one of Sect. 4.2.1, but now we see $-\alpha_{11}^{\rm K}\approx -\alpha_2$ approaching $\alpha_{22}=\alpha_1$ with increasing $B_{{\rm imp}}$ . Hence, the idea that the quasi-kinematic method could give reasonable results for strong mean fields has not proven true as $\alpha_{11}^{\rm K}$ is not approaching $\alpha _{11}$ , despite the domination of $u_{\rm rms}$ over $b_{\rm rms}$ . Instead, the values from the quasi-kinematic method have the wrong sign and deviate in their moduli by several orders of magnitude.

$\begin{figure} \par\includegraphics[width=9cm,clip]{14700fig6} \end{figure}$

Figure 6: Root-mean-square values $u_{\rm rms}$ (open circles) and $b_{\rm rms}$ (filled circles) as functions of the imposed field strength $B_{{\rm imp}}$ for purely magnetic Roberts forcing, $\tilde{N}_{\rm M}=1$ . Solid and dashed lines represent the fits given by Eq. (46).

$\begin{figure} \par\includegraphics[width=9cm,clip]{14700fig7} \end{figure}$

Figure 7: $\alpha _{11}$ (solid line, filled circles) and $\alpha _{22}$ (dashed line, open triangles) as functions of the imposed field strength $B_{{\rm imp}}$ , compared with $-\alpha _{11}^{\rm K}$ (dotted line, small dots) and $\alpha _{22}-\alpha _{11}=-\alpha _{2}$ (dotted line, open circles) for purely magnetic Roberts forcing with $\tilde{N}_{\rm M}=1$ . Note that $\alpha _{11}^{\rm K}\approx \alpha _2$ throughout.

Table 2:

Dependence of $\tilde\alpha_{11}$ and $\tilde\alpha_{22}$ from the generalized method on $\,\hspace{.3mm}\widetilde{\!\hspace{-.3mm}B}_{\rm imp}$ for $\,\hspace{.3mm}\widetilde{\!\hspace{-.3mm}N}_{\rm K}=0$ and $\,\hspace{.3mm}\widetilde{\!\hspace{-.3mm}N}_{\rm M}=1$ together with the kinetic contribution $\tilde\alpha_{11}^{\rm K}$ and the results from the quasi-kinematic method ( $\tilde\alpha_{11}^{{\rm{QK}}}$ and $\tilde\alpha_{22}^{{\rm{QK}}}$ ).
${\tilde B}_{\rm imp}$	10^-2	1	10¹	10²
$\tilde\alpha_{11}$	$2.499\times 10^{-1}$	$1.376\times 10^{-1}$	$2.000\times 10^{-4}$	$2.131\times 10^{-8}$
$\tilde\alpha_{22}$	$2.499\times 10^{-1}$	$1.747\times 10^{-1}$	$6.161\times 10^{-3}$	$6.390\times 10^{-5}$
$\tilde\alpha_{11}^{\rm K}$	$-8.391\times 10^{-6}$	$-4.540\times 10^{-2}$	$-6.666\times 10^{-3}$	$-7.067\times 10^{-5}$
$\tilde\alpha_{11}^{{\rm{QK}}}$	$-7.858\times 10^{-6}$	$-4.350\times 10^{-2}$	$-6.657\times 10^{-3}$	$-7.067\times 10^{-5}$
$\tilde\alpha_{22}^{{\rm{QK}}}$	$-2.247\times 10^{-7}$	$-1.152\times 10^{-3}$	$-4.740\times 10^{-7}$	$-5.326\times 10^{-13}$

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