1 Comparison with analytic SOCA calculations

In Rädler & Stepanov (2006, referred to as RS06 in the following) the mean electromotive force has been calculated in the second–order correlation approximation for a generally inhomogeneous turbulence in a incompressible rotating fluid showing a position–dependent mean motion. In this context the second–order correlation approximation was understood as neglect of terms in the induction equation as well as in the momentum balance. Both the Coriolis force and the derivatives of the mean velocity were assumed to be small enough so that the mean electromotive force is linear in the angular velocity Ω and the gradient tensor of \overline{U}. Detailed results were obtained for a special correlation function for the background turbulence.

Let us apply the results to the situations considered in this paper. In the case of rotation without shear we obtain

\frac{δ}{η_{t}} = \frac{\sqrt{π}}{4\sqrt{2}}Co Re_{M}(λ_{c}k_{f})^{2}\sqrt{q} δ^{o}(q, Pr_{M})  (1)

with Co, Re_{M} and Pr_{M} as defined above, q = λ_{c}^{2}/ητ_{c}, and λ_{c} and τ_{c} being correlation length and time. When introducing the Strouhal number St = u_{rms}k_{f}τ_{c}, we have q = Re_{M}/St. The function δ^{o} approaches unity if Pr_{M} = 1 and q → 0. It can be calculated according to

δ^{o}(q, Pr_{M}) = \frac{δ^{o(Ω)}(q, Pr_{M}) + κ^{o(Ω)}(q, Pr_{M})}{2β^{o(0)}(q)} ,  (2)

from the functions δ^{o(Ω)}, κ^{o(Ω)} and β^{o(0)} defined and plotted in RS06.

Proceeding to the case of shear without rotation we note first that due the above–mentioned assumption on the linearity in the mean–velocity gradient, that is in S, both κ_{11} and κ_{22} are equal to zero. Further we have

\frac{η_{12}}{η_{t}} = −\frac{19}{20}Sh Re_{M}(λ_{c}k_{f})^{2}η_{12}^{o}(q, Pr_{M}) , \frac{η_{21}}{η_{t}} = −\frac{7}{20}Sh Re_{M}(λ_{c}k_{f})^{2}η_{21}^{o}(q, Pr_{M})

(3)

with functions η_{12}^{o} and η_{21}^{o} which approach unity if Pr_{M} = 1 and q → 0. They are given by

η_{12}^{o}(q, Pr_{M}) = \frac{13κ^{o(D)}(q, Pr_{M}) + 5δ^{o(W)}(q, Pr_{M}) + κ^{o(W)}(q, Pr_{M})}{19β^{o(0)}(q)} ,  (4)
η_{21}^{o}(q, Pr_{M}) = \frac{13κ^{o(D)}(q, Pr_{M}) − 5δ^{o(W)}(q, Pr_{M}) − κ^{o(W)}(q, Pr_{M})}{7β^{o(0)}(q)} (5)

with the functions κ^{o(D)}, δ^{o(W)}, κ^{o(W)} and β^{o(0)} of RS06.

[From (??) we conclude that ln(δ/η_{t}) = ln Co + const if all relevant parameters except Co are constant. The data with Co ≤ 1.15 given in Fig.1 correspond to ln(δ/η_{t}) ≈ 0.86 ln Co + const. We further conclude that ln(δ/η_{t}) = ln Re_{M} + const if all relevant parameters (including q) except Re_{M} are constant. The data of Fig.2 with 0.65 ≤ Re_{M} ≤ 5.0 lead to ln(δ/η_{t}) ≈ 0.76 ln Re_{M} + const. It is hard to evaluate Fig.3 in that sense. From (??) we conclude that ln(η_{12}/η_{t}) = ln Re_{M} + const and ln(η_{21}/η_{t}) = ln Re_{M} + const if all relevant parameters except Re_{M} are constant. From the data for 0.7 ≤ Re_{M} ≤ 10 in Fig.4 we conclude that ln(η_{12}/η_{t}) ≈ 0.87 ln Re_{M} + const, from the data for 0.7 ≤ Re_{M} ≤ 7 there ln(η_{21}/η_{t}) ≈ 0.61 ln Re_{M} + const. All evaluations of the figures are very crude, maybe erroneous. Moreover I am not sure to which extent the assumptions on the constancy of the other parameters are justified. These considerations say, of course, nothing about, e.g., the dependencies on λ_{c}k_{f}, q or Pr_{M}.]