Benjamin MacBride presentation

Slide1: 

Application of the Kolmogorov 4/5 Law to Interplanetary Data (A Study in Progress to Explain the Observed Heating of the Solar Wind) Benjamin T. MacBride1, Miriam A. Forman2, and Charles W. Smith1 1Physics Department, University of New Hampshire 2State University of New York, Stonybrook New York Abstract: Interplanetary space is filled by a hot gas of electrically charged particles called a plasma that originates at the sun and expands outwards to beyond the planets. The actual source of this gas, called the solar wind, is not fully understood, but it is tied to an as yet undetermined heating source several solar radii above the photosphere. There is evidence from the Voyager spacecraft that the solar wind continues to undergo heating as it expands beyond the orbit of the Earth. This work is designed to assess the rate of that heating and perhaps learn something of the heating mechanism responsible for accelerating the solar wind. Hydrodynamic Background: In the traditional view of hydrodynamic turbulence there exists a self-organizing inertial range characterized by a -5/3 power law spectrum. In this range of spatial scales, energy from large scales cascades to small scales and is dissipated to heat the background fluid. The form of the spectrum in the inertial range is determined by the cascade and not by the energy source that stirs the fluid. Kolmogorov’s study of hydrodynamic fluids postulates that the heating rate due to the dissipation of turbulent flow is proportional to the third moment of fluctuations at lag L. This is Kolmogorov’s “4/5 law” which states that the averaged third moment of longitudinal velocity fluctuations (3rd order structure function for parallel fluctuations) is equal to minus 4/5 times the length scale times the energy dissipation rate per unit mass, : D3HD  [VR(t+L/V)  VR(t)]3 =  4/5  L. (1) Longitudinal fluctuations, VR, are the component of velocity parallel to the lag direction, e.g. fluctuations in radial velocity parallel to the solar wind. The symbols  …  denote averages computed over the ensemble. Solar Wind Extension: The key feature of extending the above hydrodynamic concepts to the solar wind revolves around the fact that the solar wind plasma is a magnetohydrodynamic (MHD) fluid that supports electrical currents and magnetic fields. MHD equivalents of the 4/5 law using the Elsasser variables, have been proposed by Politano and Pouquet [1998] and others and discussed in the Biskamp [2003] book on MHD turbulence. Elsasser variables are defined by: Z(x)  V(x)  B(x)/4 where  is the mass density of the fluid. From this definition of variables, the MHD formula for the dissipation rate per unit mass is as follows: D3MHD  [ZR(t+L/V) - ZR(t)] [Z-/+i(t+L/V) - Z-/+i(t)]2 = -4/d  L (2) where repeated subscripts “i” are summed and d is the dimension of the turbulence (most likely d=2). The separate equations for + and  represent the dissipation rate for the Z+ and Z components. The total energy dissipation rate per unit mass is given by: T = (+ + ) / 2. (3) Since the third moment has such a crucial role in turbulence theory, we thought we should find out how it actually behaves in the solar wind, and if it is related to the energy dissipation rate as the theory implies. Method: We use ACE solar wind data from the MAG and SWEPAM instruments to calculate signed third moments at lags from 64 seconds to days. Magnetic, density, and velocity data are combined to compute Elsasser variables for the years 1998 through late-2004 and the expressions for energy dissipation rates in the hydrodynamic limit and full MHD descriptions are evaluated. In the process we explore the possible roles of detrending and stationarity as tools in the analysis that may enable this method to become more widely applicable to spacecraft measurements throughout the heliosphere. Conclusions: The hydrodynamic and MHD expressions for the 3rd order structure functions are indeed proportional to lag and positive in sign. Dissipation rates inferred from these moments compare well with the energy dissipation rates computed with other measures of the heating of the solar wind as inferred from Helios and Voyager and from the observed power spectrum of IMF fluctuations. The data we work with is often interrupted by sudden spikes in wind speed that are caused by shocks associated with solar flares and coronal mass ejections. The data also contains heliospheric current sheet crossings, rarefaction regions, stream interfaces, and assorted other imposed heliospheric structures that constitute inhomogeneities that contribute potential errors in the application of eqn. 1 & 2. The focus of this study is to analyze the statistics of the fluctuations within a given region and not the large-scale structure of the region itself. These events produce errors in the calculations that we wish to remove. In order to analyze the true nature of the solar wind fluctuations, we perform a low-order detrending analysis that removes much of the more gradual structure (such as ramping wind speeds, slow rotations of the magnetic field, etc.). This is followed by a stationarity test on the data, designed to remove the abrupt boundary intervals such as shocks which remain after detrending. This rejects approximately ¼ of the data intervals before computing the third moment so that the large-scale structures in the data are removed from the calculations. The radial component of the solar wind velocity plays a major role in the computation of the third moment, D3MHD. This is shown by the closeness of the hydrodynamic form and the MHD form. Because of the importance of the radial velocity, it is important that we analyze it correctly. In order to do so, we want to remove the background trend, and keep the fluctuation relative to this trend because it is the fluctuation that forms the basis of our calculation. In order to remove the trend and preserve the fluctuation, we perform a detrending analysis on the data before analyzing it. Calculations for the signed third moment using equations 1 & 2 with solar wind data from the month of June of 2000 (above) give particularly nice results. From the graphs, it is evident how closely the hydrodynamic and MHD forms are related. The Blue curve is the hydrodynamic form while the Black curve is the MHD equivalent. This strongly suggests that variations in the wind speed dominate the computed expressions for the dissipation rates and emphasizes the need to treat this component carefully. We divide June 2000 into 91 intervals suitable for spectral analysis and extract estimates for the local heating rate based on the Kolmogorov [1941] spectral theory where P = CK2/3k-5/3. The probability distribution function (pdf) of these values is plotted (right) and found to be lognormally distributed. The error that results if we do not detrend is readily apparent to the right. The figure to the right shows ACE observations for a 48-hour interval contained within the month of June 2000 (those results shown to the left). Note that the wind speed, VR, changes by ~100 km/s over 2 days. If we compute the 3rd-order structure function, D3HD, from this interval the large-scale trend in VR will contribute according to (VR)3/(173k) ~ 6 which is comparable to the computed value of D3/ shown to the left. This may explain why the hydrodynamic and MHD formalisms give such remarkable agreement! The error from the uncorrected wind speed dominates both results. In order to eliminate this error source, we adopt a method of low-order detrending followed by a stationarity test to pass/reject the detrended data. The goal is to remove “gentle” trends in the data while strongly inhomogeneous intervals are left to fail and be rejected. These intervals, such as shocks, are strong sources of plasma gradients and lie outside the range of this theory. Application of the detrending and stationarity analyses to the June 2000 interval results in the figure below. To our surprise, the four estimates for  converge and the hydrodynamic and MHD expressions exhibit even less difference than in the earlier example. However, it is clear that an important source of error has been removed and it may well be the case that the spectral cascade in the solar wind is well-described by the hydrodynamic expression. At the very least, the estimate for D3/ is now flat and independent of  as the theory predicts. Comparisons of the figure below with the distribution of  and with Voyager observations suggest a dimension of d=2. A great deal more remains to be done. The best implementation of the stationarity test remains to be found. The code is presently attempting to run multiple years of data, but at present the resulting estimates fail to converge. The goal for this project is to develop a tool that can be applied throughout the heliosphere to data sets from many spacecraft to study the heating of the solar wind under diverse conditions and at many locations. Although much remains to be done, the early results are encouraging.