Anomalous Dimensions of Leading Composite Operators in the Kinematic MHD Turbulence: Two-Loop Renormalization Group Anal
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nomalous Dimensions of Leading Composite Operators in the Kinematic MHD Turbulence: Two-Loop Renormalization Group Analysis E. Jurčišinováa, b, *, M. Jurčišina, b, **, and R. Remeckýa, b, *** aInstitute
of Experimental Physics, Slovak Academy of Sciences, Košice, 04001 Slovakia Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia *e-mail: [email protected] **e-mail: [email protected] ***e-mail: [email protected]
b
Received December 20, 2019; revised January 16, 2020; accepted January 29, 2020
Abstract—Using the field theoretic renormalization group technique and the operator product expansion, the kinematic MHD turbulence is investigated in the second order (two-loop) approximation of the corresponding perturbative expansion. The anomalous dimensions of the leading composite operators, which drive the anomalous scaling of the single-time two-point correlation functions of the passive magnetic field, are calculated. It is shown that the two-loop corrections to these anomalous dimensions are significant and lead to the more anomalous (more negative) values of the total two-loop anomalous dimensions. It also means that the anomalous scaling at the two-loop level of approximation is much more pronounced in the present model of passive vector advection than in the analogous model of passive scalar quantity advected by the turbulent velocity field driven by the stochastic Navier–Stokes equation. DOI: 10.1134/S1063779620040371
1. INTRODUCTION The problem of the anomalous scaling in developed turbulence, i.e., the existence of deviations from the predictions of the classical phenomenological Kolmogorov–Obukhov (KO) theory [1–3], remains one of the most fundamental open problems of the theory. In fact, both experimental and theoretical studies show the existence of such deviations from the predictions of the KO theory. This deviations are known as “anomalous scaling” and theoretically are explained through the existence of strong developed fluctuations of the dissipative rate (intermittency) [1–4]. Although the theoretical problem of the anomalous scaling of the turbulent velocity field is still open, the great progress has been achieved in the understanding of the properties of the anomalous scaling in the process of systematic investigation of the inertial-range scaling behavior of correlation functions of various passive quantities (scalar or vector) advected by Gaussian (e.g., in the framework of the Kraichnan model [5] and various of its descendants) as well as by non-Gaussian (e.g., driven by the stochastic Navier– Stokes equation) turbulent environments. In this respect, it is also interesting that the deviations from the classical phenomenological theory are even more strongly noticeable for passively advected fields then for the velocity field itself [4, 6].
It is also well-known that the renormalization group (RG) technique [7–9] represents an effective and powerful method for investigation of self-similar scaling behavior. The field theoretic
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