Turbulence with pressure: Anomalous scaling of a passive vector field

Abstract

The field theoretic renormalization group (RG) and the operator product expansion are applied to the model of a transverse (divergence-free) vector quantity, passively advected by the "synthetic" turbulent flow with a finite (and not small) correlation time. The vector field is described by the stochastic advection-diffusion equation with the most general form of the inertial nonlinearity; it contains as special cases the kinematic dynamo model, linearized Navier-Stokes (NS) equation, the special model without the stretching term that possesses additional symmetries and has a close formal resemblance with the stochastic NS equation. The statistics of the advecting velocity field is Gaussian, with the energy spectrum E(k) ∝ k 1−ε and the dispersion law ω ∝ k −2+η , k being the momentum (wave number). The inertial-range behavior of the model is described by seven regimes (or universality classes) that correspond to nontrivial fixed points of the RG equations and exhibit anomalous scaling. The corresponding anomalous exponents are associated with the critical dimensions of tensor composite operators built solely of the passive vector field, which allows one to construct a regular perturbation expansion in ε and η; the actual calculation is performed to the first order (one-loop approximation), including the anisotropic sectors. Universality of the exponents, their (in)dependence on the forcing, effects of the large-scale anisotropy, compressibility and pressure are discussed. In particular, for all the scaling regimes the exponents obey a hierarchy related to the degree of anisotropy: the more anisotropic is the contribution of a composite operator to a correlation function, the faster it decays in the inertial-range. The relevance of these results for the real developed turbulence described by the stochastic NS equation is discussed.

Multifractal structure of fully developed turbulence

“…Similarly, the viscous term should be added into the right-hand side of (19). Changing to the new variable q = xe −at , we get…”

“…To illustrate this, we consider one particular solution of (19), for which the Fourier transform is easy to count analytically. This time we do not demand the 'strong' boundary condition (21); as we have found in previous subsection, it is enough for ω(t, 1) not to grow exponentially.…”

“…An alternative way is to introduce stochasticity as a multiplicative noise; it is used in papers on passive scalar and vector fields (e.g., [17][18][19]). We apply this approach to the NSE and consider large-scale velocity perturbations (not forces) as given random process.…”

“…A multiplicative noise instead of additive random forcing was used in papers concerned to dynamics of passive scalar and vector fields [17][18][19]. We use this approach to introduce randomness into the NSE or Euler equation; this allows us to consider the evolution of small-scale velocity perturbations in the 'external' field produced by random large-scale velocity (the next Section).…”

Anomalous Scaling Exponents in Nonlinear Models of Turbulence

“…(2) becomes a linear equation for w, passively advected by u. This linear problem was referred to before as a "passive vector with pressure" [5,6,7]. It is known to exhibit anomalous scaling with exponents that are universal to the forcing.…”