Passive scalar spectrum in high-Schmidt-number stationary and nonstationary turbulence.

Chasnov [Phys. Fluids 10, 1191 (1998)] reviewed the results for passive scalar spectra in high-Schmidt-number stationary turbulence as derived by Kraichnan [J. Fluid Mech. 64, 737 (1974)] and generalized them to simple nonstationary flows. In two-dimensional turbulence, the Kraichnan spectra are usually fitted by numerically solving the spectral equation using the derived asymptotic behavior for small and large wave numbers. In this Brief Report, we show that the Kraichnan passive scalar spectrum over the entire range of k is essentially a modified Bessel function of the second kind. We also present analytical forms of the spectra in three-dimensional nonstationary turbulence, where as shown by Chasnov, the nonstationarity can be responsible for different asymptotic behavior than the usual Kraichnan's three-dimensional stationary form. Our results considerably simplify the "fitting" of passive scalar spectra from experimental and numerical data, with the simple analytical form valid for the whole range of k , instead of just the asymptotes, which are usually valid only for a small fraction of resolved wave numbers.