Appendix C: Velocity gradient correlation tensor

In order to check that the Vishniac-Cho correlation is the dominant correlation among the different components of the velocity gradient matrix, we define the correlation coefficient

$$C_{ijkl}={\langle u_{i,j}u_{k,l}\rangle\over\sqrt{\langle u_{i,j}^2\rangle\langle u_{k,l}^2\rangle}}\cdot$$ (C.1)

We denote by $C^{(\pm)}_{ijkl}$ the values of C_ijkl evaluated in those sub-domains where the sign of shear is locally positive or negative, respectively. We separate the coefficients that are symmetric and antisymmetric with respect to changing the sign of shear. Hence, we calculate

$$C^{(S)}_{ijkl}={\textstyle{1\over2}}[C^{(+)}_{ijkl}+C^{(-)}_{... ...}_{ijkl}={\textstyle{1\over2}}[C^{(+)}_{ijkl}-C^{(-)}_{ijkl}],$$ (C.2)

whose values are shown in Tables C.1 and C.2, respectively.

 

 

Table C.1: The coefficients C^(S)_ijkl, arranged in blocks where k increases downward and l increases to the right. Within each block i increases downward and j increases to the right. In bold are given the values that are largest by magnitude, but different from unity.

	l=1			l=2			l=3
1.00	0.04	-0.09	0.04	1.00	-0.14	-0.09	-0.14	1.00
0.02	-0.01	-0.01	-0.26	-0.03	0.03	-0.03	0.01	0.03
0.01	0.05	-0.23	0.10	0.10	-0.02	-0.54	-0.01	0.04
0.02	-0.26	-0.03	-0.01	-0.03	0.01	-0.01	0.03	0.03
1.00	-0.00	-0.03	-0.00	1.00	-0.01	-0.03	-0.01	1.00
-0.03	0.02	-0.04	0.00	0.07	-0.07	-0.03	-0.60	0.06
0.01	0.10	-0.54	0.05	0.10	-0.01	-0.23	-0.02	0.04
-0.03	0.00	-0.03	0.02	0.07	-0.60	-0.04	-0.07	0.06
1.00	-0.00	-0.03	-0.00	1.00	-0.13	-0.03	-0.13	1.00

 

 

Table C.2: Like Table C.1, but for the coefficients C^(A)_ijkl which have an antisymmetric dependence on shear. The components that enter the Vishniac-Cho correlation are shown in bold; these are also the coefficients with the largest magnitude in this table.

	l=1			l=2			l=3
0.00	-0.16	-0.01	-0.16	0.00	-0.06	-0.01	-0.06	0.00
-0.01	0.00	0.00	0.01	-0.17	0.09	0.00	0.02	-0.13
-0.15	-0.02	0.00	-0.03	-0.04	0.21	0.01	0.28	-0.07
-0.01	0.01	0.00	0.00	-0.17	0.02	0.00	0.09	-0.13
0.00	-0.04	0.11	-0.04	0.00	-0.01	0.11	-0.01	0.00
0.01	0.03	0.10	0.00	0.03	-0.00	0.05	-0.02	0.01
-0.15	-0.03	0.01	-0.02	-0.04	0.28	0.00	0.21	-0.07
0.01	0.00	0.05	0.03	0.03	-0.02	0.10	-0.00	0.01
0.00	-0.51	0.10	-0.51	0.00	0.00	0.10	0.00	0.00

In Table C.1 the largest contributions come from C^(S)_xyyx=-0.26, C^(S)_yzzy=-0.60, and C^(S)_zxxz=-0.54. In Table C.2 the largest contributions come from C^(A)_xzzy=+0.28 and C^(A)_zxzy=-0.51. These are also the coefficients that are important in the Vishniac-Cho correlation.