Influence of helicity on scaling regimes in model of passive scalar advected by the turbulent velocity field with finite correlation time

Abstract

The advection of a passive scalar quantity by incompressible helical turbulent flow has been investigated in the frame of an extended Kraichnan model. Statistical fluctuations of the velocity field are assumed to have the Gaussian distribution with zero mean and defined noise with finite time-correlation. Actual calculations have been done up to twoloop approximation in the frame of the field-theoretic renormalization group approach. It turned out that the space parity violation (helicity) of a stochastic environment does not affect anomalous scaling which is the peculiar attribute of corresponding model without helicity. However, stability of asymptotic regimes, where anomalous scaling takes place, and the effective diffusivity strongly depend on the amount of helicity.

Anomalous scaling of a passive vector advected by the Navier–Stokes velocity field

“…The model was also intensively investigated by the field theoretic RG technique, where systematic perturbation expansion for the anomalous exponents was constructed, and the exponents were calculated to the second [25] and third [26] orders. Besides, various descendants of the Kraichnan rapid-change model, namely, models with inclusion of smallscale anisotropy [27], compressibility [28], the finite correlation time of velocity field [29,30] and helicity [31] were studied by the field theoretic RG approach. Moreover, advection of the passive vector field by the Gaussian self-similar velocity field (with and without large-and small-scale anisotropies, pressure, compressibility and a finite correlation time) has been also investigated, all possible asymptotic scaling regimes and crossover among them have been classified, and anomalous scaling was analyzed [24,32,33] (see also survey paper [34]).…”

“…Thus, as was discussed in [29], the perturbative expansion in the parameter which characterizes the energy spectrum of the velocity field in the inertial range is potentially dangerous even in the case with Gaussian spatial statistics of the velocity field. On the other hand, the Kraichnan rapid-change model is Galilean invariant and is free of sweeping effects but the model is so simple that it is not possible to describe some features of genuine turbulence within it (e.g., helical effects cannot be investigated within the Kraichnan model [31]).…”

Anomalous scaling of a passive scalar advected by a turbulent velocity field with finite correlation time and uniaxial small-scale anisotropy

“…( 37) and γ * 1 means that γ 1 is taken at the corresponding fixed point. From above discussion of the possible scaling regimes we have We are working only in one-loop approximation but the anomalous dimension γ * ν is already exact for all fixed points at one-loop level [30,34], i.e., it has no loop corrections of higher order, therefore the critical dimensions of frequency ω and of fields Φ ≡ {v, θ, θ ′ } are also found exactly at one-loop level approximation [30]. In our notation they read…”

“…Afterwards, various descendants of the Kraichnan model, namely, models with inclusion of small scale anisotropy [27], compressibility [28,29], finite correlation time of velocity field [30,31,32,33], and helicity [34] were studied by field theoretic approach. Moreover, advection of passive vector field by Gaussian self-similar velocity field (with and without large and small scale anisotropy, pressure, compressibility, and finite correlation time) has been also investigated and all possible asymptotic scaling regimes and cross-over among them have been classified and anomalous scaling was investigated [35].…”

Influence of finite-time velocity correlations on scaling properties of the magnetic field in the Kazantsev-Kraichnan model: Two-loop renormalization group analysis

“…Здесь необходимо подчеркнуть, что приближения с большим количеством петель особенно важны в ситуациях, когда однопетелевого приближения недостаточно для описания и понимания некоторых свойств турбулентных систем. В этом отношении типичными примерами являются турбулентные системы с нарушением пространственной четности (спиральность), поскольку в рамках теоретико-полевого подхода РГ влияние спиральности на свойства таких систем можно исследовать начиная только с двухпелевого приближения [10]- [13].…”