ASTR/ATOC-5410: Fluid Instabilities, Waves, and Turbulence December 5, 2016, Axel Brandenburg

Handout 20: Turbulent magnetic diffusivity &

In Homework 4, we showed that

E_{i} = α_{ip}\overline{B}_{p} + η_{ipl}\overline{B}_{p,l}.  (1)

We then looked at the isotropic reduction, α = \frac{1}{3}δ_{ip}α_{ip}, and found that

α = −\frac{1}{3}τ\overline{ω · u}.  (2)

where ω = ∇ × u is the vorticity. Let us now compute the η_{ipl} tensor.

1 Analytic calculation of the η_{ipl} tensor

In Homework 4, we used

E_{i} = ϵ_{ijk}ϵ_{klm}ϵ_{mnp} τ\overline{u_{j}∂_{l}u_{n}\overline{B}_{p}}  (3)

to derive Equation (1), which defines rank 2 and rank 3 tensors. We found that α_{ip} = ϵ_{jnp} τ\overline{u_{j}u_{n,i}} − ϵ_{inp} τ\overline{u_{j}u_{n,j}}. Let us now also consider the η_{ipl} tensor, which is given by

η_{ipl} = ϵ_{ijk}ϵ_{klm}ϵ_{mnp} τ\overline{u_{j}u_{n}}.  (4)

Using ϵ_{ijk}ϵ_{klm} = δ_{il}δ_{jm} − δ_{im}δ_{jl}, we have

η_{ipl} = (δ_{il}δ_{jm} − δ_{im}δ_{jl})ϵ_{mnp} τ\overline{u_{j}u_{n}},  (5)

so

η_{ipl} = δ_{il}ϵ_{jnp} τ\overline{u_{j}u_{n}} − ϵ_{inp} τ\overline{u_{l}u_{n}}.  (6)

The first term drops out, because ϵ_{jnp} τ\overline{u_{j}u_{n}} = 0. We are then left with

η_{ipl} = −ϵ_{inp} τ\overline{u_{l}u_{n}}.  (7)

To make some sense of this, let us consider the isotropic part. As discused last time, we assume η_{ipl} = η_{t}ϵ_{ipl} and determine η_{t} = \frac{1}{6}ϵ_{ipl}η_{ipl}, i.e.,

η_{t} = −\frac{1}{6}ϵ_{ipl}ϵ_{inp} τ\overline{u_{l}u_{n}}.  (8)

Let’s rearrange the indices on ϵ_{ipl} in circular order, so

η_{t} = −\frac{1}{6}ϵ_{pli}ϵ_{inp} τ\overline{u_{l}u_{n}},  (9)

and thus

η_{t} = −\frac{1}{6} \underbrace{(δ_{pn}δ_{lp} − δ_{pp}δ_{ln})}_{−2δ_{ln}} τ\overline{u_{l}u_{n}} = \frac{1}{3}τ\overline{u^{2}}  (10)

Table 1: Typical numbers for the solar convection zone.

u_{rms} [m/s] H_{P} [m] η_{t} [ m^{2} s^{−1}] η_{t} [ cm^{2} s^{−1}]
1000  300 × 10^{3}  1 × 10^{8}  1 × 10^{12}
20  50 × 10^{6}  3 × 10^{8}  3 × 10^{12}

Typical values of η_{t} greatly exceed η if the magnetic Reynolds number Re_{M} is large. In fact, if we also estimate τ = (u_{rms}k_{f})^{−1}, then Re_{M}/3 = η_{t}/η. In the Sun, we can identify 1/k_{f} with the mixing length, which increases approximately linearly with depth from 300 km near the surface to 50 Mm near the bottom of the convection zone.

2 Magnetic quenching

If the magnetic field becomes strong (magnetic energy density comparable to the kinetic energy density), the value of η_{t} becomes quenched. How strong this effect is has been a matter of debate for decades (Piddington, 1972; Knobloch, 1978; Piddington, 1981; Cattaneo & Vainshtein, 1991). Most of this work Figure 1: Re_{M}-dependence of α̃ and η̃_{t}/k̃_{f}. is analytic, and only the work of Cattaneo & Vainshtein (1991) is numerical, but it is restricted to 2-D, where the conservation of ⟨A^{2}⟩ strongly affects the result, as was understood later (Gruzinov & Diamond, 1994). A relatively new method for computing turbulent transport coefficients such as α_{ij} and η_{ijk} is the testfield method. Figure 1 shows the dependence of η_{t} on Re_{M} (Brandenburg et al., 2008), where α̃ = α/α_{0} and k̃_{f} = k_{f}/k_{1} with α_{0} = −\frac{1}{3}u_{rms}(\overline{B}) have been used for normalization. In this simulations the field is sustained against turbulent decay by a self-consistent α effect such that λ̃ ≡ λ/(η_{t0}k_{1}^{2}) = α̃k̃_{f} − (η̃_{t} + η̃) is approximately zero.

The structure of the turbulence is determined by the vectors \overline{B} and \overline{J}, but for a Beltrami field they are aligned, so we have

α_{ij}(\overline{B}) = α_{1}(\overline{B})δ_{ij} + α_{2}(\overline{B}) ˆ\overline{B}_{i} ˆ\overline{B}_{j},  (11)
η_{ij}(\overline{B}) = η_{1}(\overline{B})δ_{ij} + η_{2}(\overline{B}) ˆ\overline{B}_{i} ˆ\overline{B}_{j},  (12)

where B̂ means the unit vector in the direction of \overline{B}.

The lifetime of sunspots is between a day (for small spots) to 3 months. The decay of sunspots can be modelled as a turbulent decay (Krause & Rüdiger, 1975; Petrovay & van Driel-Gesztelyi, 1997; Rüdiger & Kitchatinov, 2000). Estimating the sunspot decay as T_{decay} = (η_{t}k_{1}^{2})^{−1} and using η_{t} = 10^{8} m^{2} s^{−1} and 1/k_{1} = 30 Mm, we find 100 days. A 3 times smaller spot would decay 10 times faster, so these estimates are rather sensitive to size.

References

Brandenburg, A., Rädler, K.-H., Rheinhardt, M., & Subramanian, K., “Magnetic quenching of alpha and diffusivity tensors in helical turbulence,” Astrophys. J. Lett. 687, L49-L52 (2008).

Cattaneo, F., & Vainshtein, S. I., “Suppression of turbulent transport by a weak magnetic field,” Astrophys. J. Lett. 376, L21-L24 (1991).

Gruzinov, A. V., & Diamond, P. H., “Self-consistent theory of mean-field electrodynamics,” Phys. Rev. Lett. 72, 1651-1653 (1994).

Knobloch, E., “Turbulent diffusion of magnetic fields,” Astrophys. J. 225, 1050-1057 (1978).

Krause, F., & Rüdiger, G., “On the turbulent decay of strong magnetic fields and the development of sunspot areas,” Solar Phys. 42, 107-119 (1975).

Petrovay, K., & van Driel-Gesztelyi, L., “Making sense of sunspot decay. I. Parabolic decay law and Gnevyshev-Waldmeier relation,” Solar Phys. 176, 249-266 (1997).

Piddington, J. H., “Solar dynamo theory and the models of Babcock and Leighton,” Solar Phys. 22, 3-19 (1972).

Piddington, J. H., “Turbulent diffusion of magnetic fields in astrophysical plasmas,” Astrophys. J. 247, 293-299 (1981).

Rüdiger, G., & Kitchatinov, L. L., “Sunspot decay as a test of the eta-quenching concept,” Astron. Nachr. 321, 75-80 (2000).