3 An idealized model with shear and stratification, but no rotation nor helicity

In order to study the effect proposed by VC2001 in isolation we need to suppress artificially the ordinary dynamo effect due to kinetic helicity of the flow. Following VC2001 the crucial factor that is necessary for causing the anticipated correlation is shear, but not rotation. To check this we have used a simulation of externally driven turbulence with zero net helicity. This is done by using the forcing function of B2001, but with helicity whose sign changes randomly. In addition, the model is supplemented by the effects of sinusoidal shear (as in BBS2001) and vertical stratification with a conducting halo above a turbulent layer (as in BD2001).

Figure 10: Vertical distribution of the estimates of the upward magnetic helicity flux $Q_{\rm mean}^{\rm (up)}$ through planes at various locations z (negative sign means downward). The values at each zare averages over time, excluding the first five snapshots of the simulations.

3.1 Description of the model

We solve the isothermal compressible MHD equations for the logarithmic density $\ln\rho$, the velocity $\mbox{\boldmath$u$ } {}$, and the magnetic vector potential $\vec{A}$,

$${{\rm D} \, {}\ln\rho\over{\rm D} \, {}t}=-\mbox{\boldmath$\nabla$ } {}\cdot\mbox{\boldmath$u$ } {},$$ (6)

$${{\rm D} \, {}\mbox{\boldmath$u$ } {}\over{\rm D} \, {}t}=-c_... ...la$ } {}\cdot\mbox{\boldmath$u$ } {})+\mbox{\boldmath$f$ } {},$$ (7)

$${\partial\vec{A}\over\partial t}=\mbox{\boldmath$u$ } {}\time... ...\mu_0\mbox{\boldmath$J$ } {}-\mbox{\boldmath$\nabla$ } {}\phi,$$ (8)

where ${\rm D}/{\rm D}t=\partial/\partial t+\mbox{\boldmath$u$ } {}\cdot\mbox{\boldmath$\nabla$ } {}$ is the advective derivative, $\mbox{\boldmath$B$ } {}=\mbox{\boldmath$\nabla$ } {}\times\vec{A}$ is the magnetic field, and

$$\mbox{\boldmath$f$ } {}=\mbox{\boldmath$f$ } {}_{\rm turb}+\mbox{\boldmath$f$ } {}_{\rm shear}+\mbox{\boldmath$g$ } {}$$ (9)

is the sum of a random forcing function, $\mbox{\boldmath$f$ } {}_{\rm turb}$ (specified in B2001), a sinusoidal shear profile, $\mbox{\boldmath$f$ } {}_{\rm shear}=S_0(\mu/\rho)\hat{\mbox{\boldmath$y$ }} {}\sin x$, and a periodic gravity potential, $\mbox{\boldmath$g$ } {}={\textstyle{1\over2}}g_0\hat{\mbox{\boldmath$z$ }} {}\sin(z/2)$. In all calculations we assume the gauge $\phi =0$ (vanishing electrostatic potential), but for some of the analysis we also adopt the gauge $\mbox{\boldmath$\nabla$ } {}\cdot\vec{A}=0$. Instead of the dynamical viscosity $\mu$ ( $=\mbox{const.}$) we will in the following refer to $\nu\equiv~\mu/\rho_0$, where $\rho_0$ is the mean density in the domain ( $\rho_0=~\mbox{const.}$ owing to mass conservation).

We use nondimensional units where $c_{\rm s}=k_1=\rho_0=\mu_0=1$. Here, $c_{\rm s}=\mbox{const}$ is the isothermal sound speed, k₁ the smallest wavenumber of the two horizontal directions (so its size is $2\pi$ in the horizontal direction). The vertical extent of the domain is $4\pi$. Periodic boundary conditions are adopted in all three directions. The wavenumber of the forcing is $k=k_{\rm f}=5$. As in BBS2001 the forcing amplitude is f₀=0.01 and the nominal shear is $S_0\equiv\vert\partial u_y/\partial x\vert _{\max}=1$. This means that the resulting shear velocity (in the absence of magnetic stresses) is then also of order unity, i.e. close to the speed of sound, and the turbulent rms velocity is about a hundred times smaller. As in BD2001, the gravitational potential varies sinusoidally in z with an amplitude g₀=0.5, so the density contrast is $\Delta\ln\rho\approx1$. The main parameter that is varied in the models considered below is the magnetic diffusivity $\eta$, which is in the range $(1...5)\times10^{-4}$.

 

 

Table 3: Summary of the main properties of various runs. The kinetic energy, $E_{\rm kin}$, is based on the poloidal flow only, whilst $E_{\rm kin}^{\rm (tot)}$refers to the total kinetic energy (including the shear motion). In Runs A-C the resolution is $60^2\times 120$ meshpoints and in Run C2 it is $120^2\times 240$ meshpoints. The magnetic energies in Run C2 are only lower limits, because the field has not reached final saturation yet.

Run	$\nu$	$\eta$	$E_{\rm kin}$	$E_{\rm kin}^{\rm (tot)}$	$M_{\rm fluct}$	$M_{\rm mean}$
A	0.01	$5\times10^{-4}$	0.0030	2.2	0.005	0.002
B	0.01	$2\times10^{-4}$	0.0030	0.7	0.018	0.002
C	0.01	10-4	0.0035	0.23	0.013	0.001
C2	0.005	10-4	0.005	0.35	>0.012	>0.0004

Once the magnetic field becomes strong the shear motion becomes reduced significantly due to magnetic forces. We define the total kinetic energy (per unit surface) as $E_{\rm kin}^{\rm (tot)}={\textstyle{1\over2}}\int_{z_1}^{z_2} {\overline{\rho\vec{u}^2}}{\rm d} {}z$, where $z_1=-\pi$ and $z_2=\pi$ are the boundaries between halo and disc plane. (This expression for $E_{\rm kin}^{\rm (tot)}$ includes the energy contained in the shear, in contrast to $E_{\rm kin}$ which does not.) When the field is still weak we have $E_{\rm kin}^{\rm (tot)}\approx2.4$, but once the field is strong this value gets significantly reduced by magnetic stresses; see Table 3. One should keep this in mind when comparing the magnetic energies for the different runs. In the following we use rms values of velocity and magnetic field for normalization purposes; these quantities are defined in terms of $E_{\rm kin}$ and M via $u_{\rm rms}=\sqrt{2E_{\rm kin}/(\langle \rho\rangle L_z)}$and $B_{\rm rms}=\sqrt{2M/L_z}$, respectively.

3.2 The Vishniac-Cho correlation

In Fig. 11 we plot the resulting correlation between $\nabla _y u_z$( $\equiv\partial u_z/\partial y$) and $\omega _y$ for a particular snapshot at the end of Run A. The anticipated effect is relatively well pronounced - much more than in the global disc simulations. As expected (see VC2001), its sign changes where the local shear, $S=\partial u_y/\partial x$, is reversed. This supports the validity of the basic result of VC2001 that such a correlation exists owing to the presence of shear.

It turns out that the Vishniac-Cho correlation is in fact the most significant correlation coefficient that changes sign when shear changes sign. In Appendix C we have calculated, for this flow, all 81 correlation coefficients of $\langle u_{i,j}u_{k,l}\rangle$. The Vishniac-Cho correlation is given by the sum of $\langle u_{x,z}u_{z,y}\rangle$ and $-\langle u_{z,x}u_{z,y}\rangle$. These two terms are indeed the most important ones. We note that the correlators $\langle u_{i,j}u_{j,i}\rangle$ (for $i\neq j$) are also relatively large, but they do not change sign when the sign of the shear changes.

Next, we need to check whether this flow is capable of dynamo action and whether large scale fields can be generated. If so, then this effect should be associated with a significant vertical transport of magnetic helicity (VC2001). Whether or not such a flux really helps the dynamo needs of course to be seen.

Figure 11: Scatter plot showing the correlation between $\nabla _y u_z$ and $\omega _y$ for a snapshot taken at a late time of the run with $\eta =5\times 10^{-4}$. In this example the correlation coefficients are -0.66 (for $x=-\pi$, where S<0) and +0.55 (for x=0, where S>0).

3.3 Dynamo action

In this model we find dynamo action provided the magnetic Reynolds number is large enough. In Table 3 we give various parameters for the models considered. In Run A where $\eta =5\times 10^{-4}$ (the value used in BBS2001) the dynamo growth rate, $\lambda\equiv{\rm d} {}\ln B_{\rm rms}/{\rm d} {}t$, is about $\lambda =0.0012$. This is much less than in the case with helical forcing where $\lambda=0.015$. As in B2001 and BBS2001 the initial field strengths are about 10^-6. Thus, in the present case the saturation time is about $\lambda^{-1}\ln10^6\approx10^4$, compared to about 10³ in BBS2001. For $\eta =10^{-4}$ (Run C) the growth rate is about five times larger (0.0053) and the resulting saturation time correspondingly shorter. The field is concentrated at small scales and shows some loop-like pattern within the turbulent layer; see Fig. 12. In Fig. 13 we plot the evolution of magnetic energies of the mean and fluctuating field components within the turbulent zone, $\vert z\vert\le\pi$. Here, mean fields are defined with respect to averaging in the toroidal direction. Large scale magnetic fields are not present in the bulk of the turbulent layer, but can be seen in the halo, especially at later times.

Figure 12: Vertical slice of By (left) and its y-average (right) for Run C2. The boundaries between halo and disc plane are indicated by white lines. The resolution of this run is $120^2\times 240$ meshpoints and t=810, corresponding to $\lambda t=4.3$. Dark and light shades indicate negative and positive values, respectively.

In Run A where $\eta =5\times 10^{-4}$ there is significant magnetic energy in the mean magnetic field (upper panel of Fig. 13), but this mean field diminishes and the fluctuating field gains in strength as the magnetic diffusivity is lowered to $\eta =10^{-4}$ (Run C).

Figure 13: Normalized magnetic energies in the mean and fluctuating components for Runs A and C. Time is given in units of the inverse growth rate ( $\lambda =0.0012$ for Run A and $\lambda =0.0053$ for Run C).

Next we look at the evolution of magnetic and current helicities;

$$H=\int_{z_1}^{z_2}\overline{\vec{A}\cdot\vec{B}}\,{\rm d} {}z... ...C=\int_{z_1}^{z_2}\overline{\vec{J}\cdot\vec{B}}\,{\rm d} {}z,$$ (10)

where z₁ and z₂ are the boundaries of the turbulent subdomain. In the present case, $z_1=-\pi$ and $z_2=+\pi$, as indicated by white lines in Fig. 12. Overbars denote horizontal averaging, so the helicities are really volume integrals, but they are normalized by the horizontal surface area. The definition of H depends on the gauge, but by extrapolating the field onto a periodic domain using potential fields one can define a gauge independent magnetic helicity (Berger & Field 1984). This is also the one used in this paper. The gauge independent magnetic helicity of the mean-field is particularly important and requires special care due to the fact that our vector and gauge potentials are periodic in the horizontal directions; see BD2001 for details.

As expected, because there is no net helicity in the forcing of the flow, there are also no net magnetic and current helicities in the fields; see Fig. 14. The fluctuations of magnetic helicity are generally weak, but the contributions from the mean and fluctuating fields are comparable. For the current helicity there is a clear dominance of the small scale fields over the large scale fields. This is explained by the fact that the current helicity has two k-factors more than the magnetic helicity and hence the ratio of small scale to large scale contributions are larger by a factor $(k_{\rm f}/k_1)^2$ for current helicity relative to magnetic helicity.

Figure 14: Magnetic and current helicities in the mean and fluctuating components for the run with $\eta =10^{-4}$. H and C are given in units of $\mu _0\rho _0 u_{\rm rms}^2 L_z/k_1$ and $\rho _0 u_{\rm rms}^2 L_z k_1^2$, respectively.

Figure 15: Magnetic helicity flux density in Coulomb gauge (upper panel) compared with that in the $\phi =0$ gauge (lower panel). The local flux density has been averaged in y and in t (between $\lambda t=32$ and 42). Run C.

3.4 Magnetic helicity flux

Vishniac & Cho (2001) pointed out that the $\phi =0$ gauge in Eq. (8) is not suited for analysing local helicity flux densities. We recall that the procedures of Berger & Field (1984) and BD2001 only apply to helicities in a given volume and the corresponding helicity fluxes through its bounding surface. Magnetic helicity densities and the corresponding flux densities are only possible to define in a given gauge.

In the following we denote by $\vec{A}_{\rm c}$ the magnetic vector potential in the Coulomb gauge, whilst $\vec{A}$ still refers to the vector potential in the $\phi =0$ gauge. The conversion between the two is given by $\vec{A}_{\rm c}=\vec{A}-\vec{A}_0$, where $\vec{A}_0=\langle \vec{A}\rangle+\mbox{\boldmath$\nabla$ } {}\psi$and $\nabla^2\psi=\mbox{\boldmath$\nabla$ } {}\cdot\vec{A}$. In the following we use the notation $\vec{A}_0=\mbox{\boldmath$\nabla$ } {}(\nabla^{-2}\mbox{\boldmath$\nabla$ } {}\cdot\vec{A})$.

In the $\phi =0$ gauge the magnetic helicity flux density is simply $\mbox{\boldmath$E$ } {}\times\vec{A}$, but when converted into the Coulomb gauge it becomes (see Appendix D)

$$\mbox{\boldmath$J$ } {}_{\rm H}^{\rm Cou}= (\mbox{\boldmath$E... ...ox{\boldmath$\nabla$ } {}\times[2\phi(\vec{A}-\vec{A}_0)]\cdot$$ (11)

The second part does not contribute to the divergence of the magnetic helicity flux. In Fig. 15 we plot the first part, denoted by $\mbox{\boldmath$J$ } {}_{\rm H,1st}^{\rm Cou}(\mbox{fluct})$. The suffix "(fluct)'' indicates that we have only included the contribution from the fluctuating components of $\vec{A}$ and $\mbox{\boldmath$E$ } {}$ and then averaged over y and t. This term was used extensively by VC2001. We also compare with the helicity flux in the $\phi =0$ gauge, $\mbox{\boldmath$J$ } {}_{\rm H}^{\phi=0}(\mbox{fluct})$. It is striking that the two are quite different; $\mbox{\boldmath$J$ } {}_{\rm H,1st}^{\rm Cou}(\mbox{fluct})$ shows a systematic circulation pattern with noise where shear is weak, i.e. near $x=\pm\pi/2$. Such a circulation patter is absent in $\mbox{\boldmath$J$ } {}_{\rm H}^{\phi=0}(\mbox{fluct})$.

There may be additional solenoidal contributions also from $\mbox{\boldmath$J$ } {}_{\rm H,1st}^{\rm Cou}$. In general we may split $\mbox{\boldmath$J$ } {}_{\rm H}^{\rm Cou}$ into an irrotational and a solenoidal part,

$$\mbox{\boldmath$J$ } {}_{\rm H}^{\rm Cou}= \mbox{\boldmath$J$... ...,irr}^{\rm Cou}+\mbox{\boldmath$J$ } {}_{\rm H,sol}^{\rm Cou}.$$ (12)

It turns out that $\mbox{\boldmath$J$ } {}_{\rm H,1st}^{\rm Cou}$ has still a noticeable solenoidal (i.e. rotational) contribution. In particular, it is interesting to note that $\mbox{\boldmath$J$ } {}_{\rm H,1st}^{\rm Cou}$has a systematic flux through the middle of the domain with an orientation that agrees with that predicted by VC2001 (upward where S>0and downward where S<0). At x=0 (i.e. the position where shear is maximum), the solenoidal part of the averaged magnetic helicity flux is about 200 times larger than its irrotational part. The profiles of the time averaged vertical magnetic helicity flux density at x=0 is plotted in Fig. 16 as functions of z for different gauges. Clearly, the relevant irrotational part of the magnetic helicity flux does not have a systematic component.

Figure 16: Time averaged vertical magnetic helicity flux density at x=0 as a function of z in different gauges. $J_{\rm H}^{\phi =0}$ refers to the $\phi =0$ gauge and $J_{\rm H}^{\rm Cou}$ refers to the Coulomb gauge. $J_{\rm H,irr}^{\rm Cou}$ refers to the irrotational part and $J_{\rm H,1st}^{\rm Cou}$ refers to the first part in Eq. (11). Like in Fig. 15, only the contributions from the fluctuating components of $\vec{A}$ and $\mbox{\boldmath$E$ } {}$are included. Run C.

In order to assess the possibility of dynamo action we plot in Fig. 17 vertical profiles of the divergence of the helicity flux density, in units of $u_{\rm rms}B^2_{\rm rms}$. This non-dimensional quantity is reminiscent of a local dynamo number, $\alpha_{\rm BH}/(k_1\eta_{\rm T})\approx(k_{\rm f}/k_1) (\alpha_{\rm BH}/u_{\rm rms})$, where

$$\alpha_{\rm BH}=-\mbox{\boldmath$\nabla$ } {}\cdot\mbox{\bold... ...\nabla$ } {}\cdot\mbox{\boldmath$J$ } {}_{\rm H}/B^2_{\rm rms}$$ (13)

has been introduced for the $\alpha$ effect of Bhattacharjee & Hameiri (1986) and $k_{\rm f}\eta_{\rm T}\approx u_{\rm rms}$ for the turbulent magnetic diffusivity, $\eta_{\rm T}$. Our results (Fig. 17) suggest that the local dynamo number is fluctuating in space and that its rms value is somewhat larger when the magnetic Reynolds number is larger (cf. lower panel). The fact that $\alpha_{\rm BH}$ fluctuates in space (and probably also in time) is not necessarily bad; such an "incoherent'' $\alpha$ may still, together with shear and turbulent diffusion, contribute to producing large scale fields (Vishniac & Brandenburg 1997).

Figure 17 shows that the non-dimensional measure of the $\alpha_{\rm BH}$ fluctuates around $\pm0.01$. This value should be compared with the critical value of $\alpha/(\eta_{\rm T}k_1)\equiv C_\alpha$ above which dynamo action is possible. We define the dynamo number as ${\cal D}=C_\alpha C_{\rm S}$, where $C_{\rm S}=S/(\eta_{\rm T}k_1^2)$. With $S\approx S_0=1$ and $\eta_{\rm T}k_1=u_{\rm rms}k_1/k_{\rm f}=0.005$we have $C\rm _S=~200$, and since the critical dynamo number is around 2 (BBS2001) we have 0.01, which agrees with the estimate above (cf. Fig. 17). However, this estimate has been too optimistic in several ways: the actual value of S is smaller than S₀and the incoherent $\alpha$ effect dynamo will be less efficient. This may explain why the Vishniac-Cho effect does not seem to operate in the present simulations, but it may become more important at higher magnetic Reynolds numbers.

Figure 17: Divergence of the mean helicity flux, normalized to make it similar to a local dynamo number. As in Figs. 15 and 16, only the contributions from the fluctuating components of $\vec{A}$ and  $\mbox{\boldmath$E$ } {}$are included in the calculation of the y and t averaged magnetic helicity flux. Run C.