Influence of helicity on the Kolmogorov regime in fully developed turbulence

The influence of helicity on the stability of the Kolmogorov scaling regime in fully developed turbulence in space dimension d=3 based on the stochastic Navier-Stokes equation with the self-similar Gaussian random stirring force delta -correlated in time and with the correlator proportional to k;{4-d-2epsilon} is investigated by the field-theoretic renormalization-group technique within two-loop approximation. The two-loop renormalization constant, the beta function, and the coordinate of the fixed point are found as functions of the helicity parameter. It is shown that the presence of helicity in the system does not destroy the stability of the Kolmogorov scaling regime.

Influence of helicity on anomalous scaling of a passive scalar advected by the turbulent velocity field with finite correlation time: two-loop approximation

The influence of helicity on the stability of scaling regimes, on the effective diffusivity, and on the anomalous scaling of structure functions of a passive scalar advected by a Gaussian solenoidal velocity field with finite correlation time is investigated by the field theoretic renormalization group and operator-product expansion within the two-loop approximation. The influence of helicity on the scaling regimes is discussed and shown in the plane of exponents epsilon-eta, where epsilon characterizes the energy spectrum of the velocity field in the inertial range E proportional to k(1-2epsilon), and eta is related to the correlation time at the wave number k, which is scaled as k(-2+eta). The restrictions given by nonzero helicity on the regions with stable fixed points that correspond to the scaling regimes are analyzed in detail. The dependence of the effective diffusivity on the helicity parameter is discussed. The anomalous exponents of the structure functions of the passive scalar field which define their anomalous scaling are calculated and it is shown that, although the separate composite operators which define them strongly depend on the helicity parameter, the resulting two-loop contributions to the critical dimensions of the structure functions are independent of helicity. Details of calculations are shown.

Cascade time scales for energy and helicity in homogeneous isotropic turbulence

We extend the Kolmogorov phenomenology for the scaling of energy spectra in high-Reynolds-number turbulence, to explicitly include the effect of helicity. There exists a time scale tau(H) for helicity transfer in homogeneous, isotropic turbulence with helicity. We arrive at this time scale using the phenomenological arguments used by Kraichnan to derive the time scale tau(E) for energy transfer [J. Fluid Mech. 47, 525 (1971)]]. We show that in general tau(H) may not be neglected compared to tau(E), even for rather low relative helicity. We then deduce an inertial range joint cascade of energy and helicity in which the dynamics are dominated by tau(E) in the low wave numbers with both energy and helicity spectra scaling as k(-5/3); and by tau(H) at larger wave numbers with spectra scaling as k(-4/3). We demonstrate how, within this phenomenology, the commonly observed "bottleneck" in the energy spectrum might be explained. We derive a wave number k(h) which is less than the Kolmogorov dissipation wave number, at which both energy and helicity cascades terminate due to dissipation effects. Data from direct numerical simulations are used to check our predictions.