Supplemental Material to “Decay law of magnetic turbulence with helicity balanced by chiral fermions”

by Axel Brandenburg, Kohei Kamada, & Jennifer Schober

March 24, 2023

Following Zhou+22 [6], the function I_{H}(t, R) can be recast as a weighted integral

I_{H}(t, R) = \frac{1}{V} ∫ \frac{d^{3}k}{(2π)^{3}} w_{R}(k) h^{∗}(k)h(k). (1)

The weight function w_{R}(k) depends on the shape of V_{R}. For a cubic region with length 2R, we have

w_{R}(k) = w_{cube}^{BC}(k) ≡ 8R^{3} ∏_{i=1}^{3} j_{0}^{2}(k_{i}R);  (2)

see Fig. 1. Here j_{0}(x) = sin x/x. For a spherical region with radius R,

w_{R}(k) = w_{sph}^{BC}(k) ≡ \frac{4πR^{3}}{3} [\frac{6j_{1}(kR)}{kR}]^{2} ,  (3)

which is the version shown in the main paper.

Alternatively, by the correlation-integral (CI) method, we have

w_{R}(k) = w_{cube}^{CI}(k) ≡ 8R^{3} ∏_{i=1}^{3} j_{0}(k_{i}R),  (4)

for a cubic region V_{R}; see Fig. 2, and

w_{R}(k) = w_{sph}^{CI}(k) ≡ \frac{4πR^{3}}{3} \frac{6j_{1}(kR)}{kR}.  (5)

for a spherical region; Fig. 3.

Figure 1: Box counting method with cubic volumes showing (a) I_{H}(R, t) versus R for different times (the normalized values of tηk_{0}^{2} are indicated by different colors), and (b) I_{H}(R, t) versus t (normalized) for two choices of R, indicated by solid (k_{0}R ≈ 1.5) and dashed (k_{0}R ≈ 6) lines in both panels. The t^{−0.16} scaling is indicated as the dashed-dotted curve for comparison.

Figure 2: Similar to Fig. 1 using the correlationintegral method for cubic volumes.

Figure 3: Similar to Fig. 1 using the correlationintegral method for spherical volumes.