The scaling properties of two-dimensional compressible magnetohydrodynamic turbulence

Abstract

Understanding the phenomenology captured in direct numerical simulation ͑DNS͒ of magnetohydrodynamic ͑MHD͒ turbulence rests upon models and assumptions concerning the scaling of field variables and dissipation. Here compressible MHD turbulence is simulated in two spatial dimensions by solving the isothermal equations of resistive MHD on a periodic square grid. In these simulations it is found that the energy spectrum decreases more slowly with k, and the viscous cutoff length is larger, than would be expected from the 1941 phenomenology of Kolmogorov ͑K41͒. Both these effects suggest that the cascade time is modified by the presence of Alfvén waves as in the phenomenology of Iroshnikov and Kraichnan ͑IK͒. Motivated by this, these scaling exponents are compared with those of the IK-based model of Politano and Pouquet ͓Phys. Rev. E 52, 636 ͑1995͔͒, which is an extension of the model of She and Leveque ͓Phys. Rev. Lett. 72, 336 ͑1994͔͒. However, the scaling exponents from these simulations are not consistent with the model of Politano and Pouquet, so that neither IK nor K41 models would appear to describe the simulations. The spatial intermittency of turbulent activity in such simulations is central to the observed phenomenology and relates to the geometry of structures that dissipate most intensely via the scaling of the local rate of dissipation. The framework of She and Leveque implies a scaling relation that links the scaling of the local rate of dissipation to the scaling exponents of the pure Elsässer field variables ͑z ± = v ± B / ͱ o ͒. This scaling relation is conditioned by the distinct phenomenology of K41 and IK. These distinct scaling relations are directly tested using these simulations and it is found that neither holds. This deviation suggests that additional measures of the character of the dissipation may be required to fully capture the turbulent scaling, for example, pointing towards a refinement of the phenomenological models. It may also explain why previous attempts to predict the scaling exponents of the pure Elsässer fields in two-dimensional magnetohydrodynamic turbulence by extending the theory of She and Leveque have proved unsuccessful.

Intermittency, dissipation, and scaling in two-dimensional magnetohydrodynamic turbulence

“…21͒. 4,5,7 The scaling found in both driven and decaying 3D MHD turbulence simulations 8,9 shows good agreement with the intermittency correction of She and Leveque. 7 These simulations show an extended range of scaling under ESS as in Eq.…”

“…As we discuss below, GRS has been tested for the first time in the context of 2D MHD by the recent DNS of driven 2D MHD turbulence. [2][3][4]7 Furthermore, the adoption of IK MHD phenomenology ͑a =4͒ to extend SL to MHD ͑SL-IK͒, 20 is found to decrease the level of agreement between simulation and model. ͑2͒ which however does not conform to GRS.…”

Vorticity scaling and intermittency in drift-interchange plasma turbulence

“…This technique is known as extended self-similarity (ESS) 32 and is increasingly applied to plasma physics simulations. 14,15,18 We use ESS to evaluate the scaling of all fluctuating fields, by providing an estimate of the normalised scaling exponent fðpÞ=fð3Þ. Our main focus is on the degree of intermittency, characterised by deviation of fðpÞ from linearity, and not on the exact values of scaling exponents.…”

“…Studies include direct measurements in the solar wind, 22,23 and the outputs of numerical simulations of turbulence in plasmas using various levels of description. [9][10][11][12][13][14][15][16] We present the first higher order analysis of the velocity, density, and vorticity fluctuations for the extended H-W equations, which are capable of modelling and comparing drift-interchange turbulence on the high field side (HFS) and low field side (LFS) of a tokamak. 17 In the context of MCF plasma turbulence, the scaling properties of fluctuations are important for several reasons.…”

“…These changes are of particular practical interest in relation to vorticity, which is known to play a key role in the transport of energy and particles in plasmas and fluids. [10][11][12][13][14][15] The ability to extrapolate transport properties between experiments of different sizes, or between simulation and experiment, rests on understanding how the statistical properties of the physical processes that drive transport scale with length and time. Insofar as the scaling is nonlinear, practical extrapolation becomes more difficult.…”