Politano and Pouquet's law, a generalization of Kolmogorov's four-fifths law to incompressible MHD, makes it possible to measure the energy cascade rate in incompressible MHD turbulence by means of third-order moments. In hydrodynamics, accurate measurement of third-order moments requires large amounts of data because the probability distributions of velocity-differences are nearly symmetric and the third-order moments are relatively small. Measurements of the energy cascade rate in solar wind turbulence have recently been performed for the first time, but without careful consideration of the accuracy or statistical uncertainty of the required third-order moments. This paper investigates the statistical convergence of third-order moments as a function of the sample size N. It is shown that the accuracy of the third-moment depends on the number of correlation lengths spanned by the data set and a method of estimating the statistical uncertainty of the third-moment is developed. The technique is illustrated using both wind tunnel data and solar wind data.
Deep Dive into Accurate estimation of third-order moments from turbulence measurements.
Politano and Pouquet’s law, a generalization of Kolmogorov’s four-fifths law to incompressible MHD, makes it possible to measure the energy cascade rate in incompressible MHD turbulence by means of third-order moments. In hydrodynamics, accurate measurement of third-order moments requires large amounts of data because the probability distributions of velocity-differences are nearly symmetric and the third-order moments are relatively small. Measurements of the energy cascade rate in solar wind turbulence have recently been performed for the first time, but without careful consideration of the accuracy or statistical uncertainty of the required third-order moments. This paper investigates the statistical convergence of third-order moments as a function of the sample size N. It is shown that the accuracy of the third-moment depends on the number of correlation lengths spanned by the data set and a method of estimating the statistical uncertainty of the third-moment is developed. The tech
In the solar wind, coupling between large-and small-scale turbulence occurs at kinetic scales defined by the ion gyroradius and the ion gyro-period. At these scales, the turbulent energy cascade undergoes a transition from large magnetohydrodynamic (MHD) scales to small plasma kinetic scales where the energy is ultimately dissipated by collisionless processes. Detailed understanding of the energy cascade process at MHD-scales is a prerequisite for studies of this coupling. Here we focus on one particular aspect of MHD-scale turbulence which is of some practical importance, namely, Correspondence to: John Podesta (jpodesta@solar.stanford.edu) the determination of the energy cascade rate from measured data.
MHD-scale turbulence in the solar wind is often modeled using the theory of incompressible MHD because of its relative simplicity, even though the solar wind is known to be compressible. In the solar wind, the energy density of MHD turbulence is comparable to the plasma thermal energy at 1 AU (Belcher and Davis, 1971) and the turbulent energy cascade is believed to significantly heat the solar wind plasma as it flows from ∼ 1 AU to several tens of AU. Theoretical work has shown that plasma heating caused by dissipation of the turbulence can likely explain the observed radial temperature profile of the solar wind which decreases more slowly than would be the case if the expansion were adiabatic (Matthaeus et al., 1996;Zank et al., 1999;Matthaeus et al., 1999;Smith et al., 2001;Isenberg et al., 2003). To refine these theories, accurate measurements of the energy cascade rate are needed. Recently, the energy cascade rate ε has been directly measured for the first time in the solar wind using a generalization of Kolmogorov’s four-fifths law (MacBride et al., 2005;Sorriso-Valvo et al., 2007;MacBride et al., 2008;Marino et al., 2008). Before discussing this, it may be helpful to provide some background information on Kolmogorov’s four-fifths law.
For turbulent flows in ordinary incompressible fluids such as air or water the energy cascade rate ǫ is often measured indirectly by means of the energy dissipation rate
where ν is the kinematic viscosity and the coefficient 15 arises from the assumption that the turbulence is isotropic (Pope, 2000, p. 134). The energy cascade rate can also be measured directly by means of Kolmogorov’s four-fifths law
valid for isotropic turbulence, where
is the component of the velocity fluctuation in the direction of the displacement r and the lengthscale r lies in the inertial range (Kolmogorov, 1991;Frisch, 1995). Note that Kolmogorov’s four-fifths law (2) is independent of the kinematic viscosity ν and can be applied even when the kinematic viscosity is unknown, but the accurate evaluation of the thirdorder moment (2) requires much more data than the secondorder moment (1).
Kolmogorov’s four-fifths law was originally derived for homogeneous isotropic turbulence and a similar law was later derived by Monin for homogeneous anisotropic turbulence; see Podesta et al. (2007) for references. Politano and Pouquet (1998a,b) generalized these fundamental results of Kolmogorov and Monin from the theory of incompressible hydrodynamic turbulence to incompressible MHD turbulence. It is important to emphasize that Politano and Pouquet’s law holds for both isotropic and anisotropic turbulence, although this fact was not explicitly mentioned by Politano and Pouquet (1998a). This is especially important in MHD where statistical isotropy may not hold in the presence of an ambient magnetic field. A derivation of Politano and Pouquet’s law which is similar to Frisch’s derivation of Kolmogorov’s four-fifths law is given by Podesta (2008).
Politano and Pouquet’s law has recently been applied to obtain direct measurements of the energy cascade rate in the solar wind under the simplifying assumption that the turbulence is isotropic (MacBride et al., 2005;Sorriso-Valvo et al., 2007;MacBride et al., 2008;Marino et al., 2008). MacBride et al. (2008) have also investigated a non-isotropic 1D/2D hybrid model that is believed to be descriptive of the solar wind. The method used in all these studies consists of the evaluation of certain third-order moments which are similar to those in equation ( 2), except that for incompressible MHD turbulence the relevant thirdorder moments contain combinations of velocity and magnetic field fluctuations (or, equivalently, fluctuations in the Elsasser variables). From the linear scaling of these thirdorder moments, the energy cascade rate is obtained without any knowledge of the dissipation processes or the viscous and resistive dissipation coefficients in the the solar wind.
The solar wind studies mentioned above have not given careful consideration to the convergence properties of thirdorder moments which raises the question: how much data is required to accurately estimate the third-order moments?
The study by Sorriso-Valvo et al. (2007) used approximately 2000 da
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