Earity-suppressing correlations must be associated with (��)-BGB-3111MedChemExpress (��)-Zanubrutinib non-Gaussian statistics. Intuitively, it is clear as sharp boundaries form between cells and these are non-spacefilling and highly concentrated. This implies nonGaussianity and high kurtosis of gradients and small-scale increments. This conclusion is also readily verified quantitatively, by evaluation of a fourth-order statistic H4 = (v ?b)2 , for two special cases. For simplicity, suppose v and b have equal and isotropic variances, and zero mean values, so that the variance of any component is 2 . For the first case, we assume both vector fields are Gaussian, generated by a jointly normal distribution, and let the the components of b and v be uncorrelated, so that v v = 0 and b b = 0 when = , and v b = 0 for all and .The concept of entropy that enters the formalism for long time relaxation in two-dimensional hydrodynamics [55] is essentially an information entropy, and must be distinguished from the Gibbs entropy in the absolute equilibrium models [41,43,44]. There have been some attempts to extend the information entropy formulation to local two-dimensional hydrodynamic relaxation within individual vortices [57]. Formulations of entropy concepts in MHD have been proposed but are less well studied [39,58].Then by direct evaluation H4 = (v ?b)2 = 3 4 . As with all jointly normal statistics, the fourth and all higher orders are determined by the first- and second-order moments, including the cross correlation, which here is zero. For the second case, assume that at every point v = , and that the distribution of each may or may not be Gaussian. After a short calculation, one finds for this case that H4 = (30 + 6) 4 , where 0 is the kurtosis of any one Cartesian component (all assumed equal by isotropy). In this case, if the fields are Gaussian (0 = 3), then the assumed pointwise RWJ 64809 biological activity correlation means that H4 is five times larger than in the fully Gaussian jointly normal case. Thus the joint distribution cannot be normal. For 0 > 3 the disparity is even greater. We conclude that the presence of strong random Alfv ic correlations distributed in patches cannot be described by jointly normal (bivariate Gaussian) statistics, as has been argued previously [52]. One may devise analogous arguments to conclude that each of the local correlations required for depression of nonlinearity requires some level of departure from Gaussianity. That is, correlations such as (b ? ?b)2 or (v ? ?v)2 must take on values other than those associated with jointly normal distributions. This has been demonstrated explicitly in three-dimensional MHD turbulence simulations [54] by computing these correlations (and others) from dynamical simulation data, and comparing the values from MHD solutions with values obtained by fixing the spectrum of all quantities and randomizing (`Gaussianizing’) the fields v and b. This provides a rather convincing quantitative demonstration of the association of depression of nonlinearity with the emergence of non-Gaussianity and intermittency. In these cases, the examination of the real space structure shows patches of relaxed magnetofluid, forming cellularized patterns (e.g. `flux tubes’) that are separated by thin high-stress boundaries, often in the form of current sheets and vortex sheets, as suggested by the examples in figure 4. The picture that emerges from studies such as those reviewed here is that the turbulent cascade is far from a structureless, self-similar or random-ph.Earity-suppressing correlations must be associated with non-Gaussian statistics. Intuitively, it is clear as sharp boundaries form between cells and these are non-spacefilling and highly concentrated. This implies nonGaussianity and high kurtosis of gradients and small-scale increments. This conclusion is also readily verified quantitatively, by evaluation of a fourth-order statistic H4 = (v ?b)2 , for two special cases. For simplicity, suppose v and b have equal and isotropic variances, and zero mean values, so that the variance of any component is 2 . For the first case, we assume both vector fields are Gaussian, generated by a jointly normal distribution, and let the the components of b and v be uncorrelated, so that v v = 0 and b b = 0 when = , and v b = 0 for all and .The concept of entropy that enters the formalism for long time relaxation in two-dimensional hydrodynamics [55] is essentially an information entropy, and must be distinguished from the Gibbs entropy in the absolute equilibrium models [41,43,44]. There have been some attempts to extend the information entropy formulation to local two-dimensional hydrodynamic relaxation within individual vortices [57]. Formulations of entropy concepts in MHD have been proposed but are less well studied [39,58].Then by direct evaluation H4 = (v ?b)2 = 3 4 . As with all jointly normal statistics, the fourth and all higher orders are determined by the first- and second-order moments, including the cross correlation, which here is zero. For the second case, assume that at every point v = , and that the distribution of each may or may not be Gaussian. After a short calculation, one finds for this case that H4 = (30 + 6) 4 , where 0 is the kurtosis of any one Cartesian component (all assumed equal by isotropy). In this case, if the fields are Gaussian (0 = 3), then the assumed pointwise correlation means that H4 is five times larger than in the fully Gaussian jointly normal case. Thus the joint distribution cannot be normal. For 0 > 3 the disparity is even greater. We conclude that the presence of strong random Alfv ic correlations distributed in patches cannot be described by jointly normal (bivariate Gaussian) statistics, as has been argued previously [52]. One may devise analogous arguments to conclude that each of the local correlations required for depression of nonlinearity requires some level of departure from Gaussianity. That is, correlations such as (b ? ?b)2 or (v ? ?v)2 must take on values other than those associated with jointly normal distributions. This has been demonstrated explicitly in three-dimensional MHD turbulence simulations [54] by computing these correlations (and others) from dynamical simulation data, and comparing the values from MHD solutions with values obtained by fixing the spectrum of all quantities and randomizing (`Gaussianizing’) the fields v and b. This provides a rather convincing quantitative demonstration of the association of depression of nonlinearity with the emergence of non-Gaussianity and intermittency. In these cases, the examination of the real space structure shows patches of relaxed magnetofluid, forming cellularized patterns (e.g. `flux tubes’) that are separated by thin high-stress boundaries, often in the form of current sheets and vortex sheets, as suggested by the examples in figure 4. The picture that emerges from studies such as those reviewed here is that the turbulent cascade is far from a structureless, self-similar or random-ph.
