Small-Scale Structure of a Scalar Field Convected by Turbulence

Abstract

Batchelor's theory of the turbulent straining of small-spatial-scale amplitude variations of a convected scalar field is re-examined to see the effects of fluctuation of the rates of strain in space and time. The k−1 viscous-convective-range spectrum is unaltered, except for the constant of proportionality, but spectrum level in the viscous-diffusive range displays a sensitivity to fluctuations which increases with wavenumber. The Gaussian cutoff found by Batchelor is replaced by more gently decreasing dependences of spectrum level on wavenumber. The scalar spectrum is also treated by the Lagrangian-history direct-interaction approximation. The k−5/3 inertial-convective range of Obukhov and Corrsin, the k−1 viscous-convective range, and the k−17/3 inertial-diffusive range of Batchelor, Howells, and Townsend all are recovered. For a given rate of spectral transport of scalar variance, the predicted spectrum levels in the k−5/3 and k−1 ranges are too small, in comparison with experiment, by numerical factors in the neighborhood of one to three. The largest error is for the k−1 range at large Reynolds number.

Evolution of localized magnetic field perturbations and the nature of turbulent dynamo

“…Thus, δ B = 3D/2 is the exact damping decrement of the passive scalar. This result coincides with that obtained in 15,28 . One can check that for any δ ≤ 3D/2 the positive solution exists, so there are no solutions decreasing slower than δ B .…”

Evolution of localized magnetic field perturbations and the nature of turbulent dynamo

“…Thus, δ B = 3D/2 is the exact damping decrement of the passive scalar. This result coincides with that obtained in 15,28 . One can check that for any δ ≤ 3D/2 the positive solution exists, so there are no solutions decreasing slower than δ B .…”

Transition phenomena in unstably stratified turbulent flows

“…Kraichnan (1968) derived Eq. ( 4) for a passive scalar regime using the Lagrangian-history direct-interaction approximation for small Peclet numbers [66]. Equation ( 4) is in agreement with the corresponding equation derived by means of the path integral approach in [67] for the passive scalar advected by the delta-correlated in time Gaussian smooth velocity field.…”

Magnetic buoyancy in simulated galactic discs with a realistic circumgalactic medium

“…The magnetic field is amplified by turbulent motion on small scales and transported to large scales by an inverse energy cascade, as predicted by dynamo theory. We note, that the power spectrum on the large scales is fully consistent with a Kazantsev-spectrum (Kazantsev 1968;Kraichnan 1968), known from a small scale turbulent dynamo (Brandenburg & Subramanian 2005;Tobias et al 2011). This is also consistent with the findings of other simulations of isolated disc galaxies (Butsky et al 2017;Rieder & Teyssier 2016, 2017a, as well as those of cosmological zoom-in simulations (Pakmor et al 2017;Rieder & Teyssier 2017b).…”

Evolution of localized magnetic field perturbations and the nature of turbulent dynamo

“…Thus, δ B = 3D/2 is the exact damping decrement of the passive scalar. This result coincides with that obtained in 15,28 . One can check that for any δ ≤ 3D/2 the positive solution exists, so there are no solutions decreasing slower than δ B .…”

Transition phenomena in unstably stratified turbulent flows

“…Kraichnan (1968) derived Eq. ( 4) for a passive scalar regime using the Lagrangian-history direct-interaction approximation for small Peclet numbers [66]. Equation ( 4) is in agreement with the corresponding equation derived by means of the path integral approach in [67] for the passive scalar advected by the delta-correlated in time Gaussian smooth velocity field.…”

Magnetic buoyancy in simulated galactic discs with a realistic circumgalactic medium

“…The magnetic field is amplified by turbulent motion on small scales and transported to large scales by an inverse energy cascade, as predicted by dynamo theory. We note, that the power spectrum on the large scales is fully consistent with a Kazantsev-spectrum (Kazantsev 1968;Kraichnan 1968), known from a small scale turbulent dynamo (Brandenburg & Subramanian 2005;Tobias et al 2011). This is also consistent with the findings of other simulations of isolated disc galaxies (Butsky et al 2017;Rieder & Teyssier 2016, 2017a, as well as those of cosmological zoom-in simulations (Pakmor et al 2017;Rieder & Teyssier 2017b).…”

Evolution of localized magnetic field perturbations and the nature of turbulent dynamo

“…Thus, δ B = 3D/2 is the exact damping decrement of the passive scalar. This result coincides with that obtained in 15,28 . One can check that for any δ ≤ 3D/2 the positive solution exists, so there are no solutions decreasing slower than δ B .…”

Transition phenomena in unstably stratified turbulent flows

“…Kraichnan (1968) derived Eq. ( 4) for a passive scalar regime using the Lagrangian-history direct-interaction approximation for small Peclet numbers [66]. Equation ( 4) is in agreement with the corresponding equation derived by means of the path integral approach in [67] for the passive scalar advected by the delta-correlated in time Gaussian smooth velocity field.…”

Magnetic buoyancy in simulated galactic discs with a realistic circumgalactic medium

“…The magnetic field is amplified by turbulent motion on small scales and transported to large scales by an inverse energy cascade, as predicted by dynamo theory. We note, that the power spectrum on the large scales is fully consistent with a Kazantsev-spectrum (Kazantsev 1968;Kraichnan 1968), known from a small scale turbulent dynamo (Brandenburg & Subramanian 2005;Tobias et al 2011). This is also consistent with the findings of other simulations of isolated disc galaxies (Butsky et al 2017;Rieder & Teyssier 2016, 2017a, as well as those of cosmological zoom-in simulations (Pakmor et al 2017;Rieder & Teyssier 2017b).…”