Ts br are treated on equal footing. LOXO-101MedChemExpress Larotrectinib Without delving more deeply into the background theory, one may1 The association of enhanced viscous dissipation with enhanced spatial derivatives of velocity requires no hypothesis, as the local rate of viscous dissipation for incompressible hydrodynamics is (x) = (/2)Sij Sij with Sij i uj + j ui .(a)(b)2.0 1.5 zp 1.0 0.(c)1.0 PDF(d v) 10? 10? PDF(d v) t = 0.2 h t = 0.9 hrsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 373:…………………………………………………3 zp10? t = 1.44 h 10? ? ? 0 dv 2 4 ? t = 23.04 h 0 dv 210 12 14 16 18 p4 pFigure 2. Evidence for intermittency, in the form of multi-fractal scalings of structure functions of increasing order. (a) (p) from hydrodynamic experiments. (Adapted from Anselmet et al. [14].) (b) (p) from an MHD simulation. (Adapted from Biskamp M ler [15].) When the scaling exponent (p) exhibits anomalous behaviour (i.e. is a nonlinear function of p, the order of the structure function), the scaling is described as multi-fractal [6]. An alternative and more direct approach to characterize intermittency is to compare the PDFs of increments at different spatial lags, finding that fatter non-Gaussian tails appear in the PDFs of the smaller lags. (c) An example from solar wind data, where spatial lag is proportional to time lag. (Adapted from Sorriso-Valvo et al. [16].)simply adopt a perspective based on analogy with equation (2.6) and proceed to evaluate higher order statistics and their scalings with spatial lag. From this emerges a picture quite reminiscent of hydrodynamics, as illustrated in figure 2. Despite what might appear to be an encouraging similarity in the scaling of higher order moments in hydrodynamics and in MHD, there are in fact at least two main impediments to a direct extension of the KRSH to MHD, and at least one additional major problem in extending it to kinetic buy (��)-BGB-3111 plasma, as follows. First, for both MHD and plasma, there is ambiguity regarding the choice of fluid-scale variables as there are now at least two vector fields involved. These are the velocity v and magnetic field b, or equivalently the two Elsasser fields z+ = v + b/ 4 and z- = v – b/ 4, where is the mass density. (When the magnetic quantities are expressed in Alfv speed units, the Elsasser variables take the form z?= v ?b.) Is the local dissipation related to vr ? To br ? More properly, based on the structure of the third-order law for MHD [17,18], perhaps relations + analogous to equation (2.6) should be written separately for two local dissipation functions r – and r , in terms of the increment combinations (z+ |z- |2 )1/3 and (z- |z+ |2 )1/3 [19]. There have also been suggestions that even more information could be embedded in the primitive increment functions entering the MHD KRSH, for example by allowing for a scaling of alignment angles between v and b [20]. All such suggestions are permissible from a dimensional standpoint, but we have not yet seen in the literature either a precise statement of an MHD KRSH (however, see [21,22]) or a full statistical test of any such hypothesis, as has been carried out repeatedly in the hydrodynamic case [23,24]. Second, there is increasing evidence that MHD turbulence lacks a universal character [19,25], so that it is not clear that there is a single, simple answer to the questions posed in the previous paragraph. Non-universality is related not only to the multiplicity of independent field variables in MHD bu.Ts br are treated on equal footing. Without delving more deeply into the background theory, one may1 The association of enhanced viscous dissipation with enhanced spatial derivatives of velocity requires no hypothesis, as the local rate of viscous dissipation for incompressible hydrodynamics is (x) = (/2)Sij Sij with Sij i uj + j ui .(a)(b)2.0 1.5 zp 1.0 0.(c)1.0 PDF(d v) 10? 10? PDF(d v) t = 0.2 h t = 0.9 hrsta.royalsocietypublishing.org Phil. Trans. R. Soc. A 373:…………………………………………………3 zp10? t = 1.44 h 10? ? ? 0 dv 2 4 ? t = 23.04 h 0 dv 210 12 14 16 18 p4 pFigure 2. Evidence for intermittency, in the form of multi-fractal scalings of structure functions of increasing order. (a) (p) from hydrodynamic experiments. (Adapted from Anselmet et al. [14].) (b) (p) from an MHD simulation. (Adapted from Biskamp M ler [15].) When the scaling exponent (p) exhibits anomalous behaviour (i.e. is a nonlinear function of p, the order of the structure function), the scaling is described as multi-fractal [6]. An alternative and more direct approach to characterize intermittency is to compare the PDFs of increments at different spatial lags, finding that fatter non-Gaussian tails appear in the PDFs of the smaller lags. (c) An example from solar wind data, where spatial lag is proportional to time lag. (Adapted from Sorriso-Valvo et al. [16].)simply adopt a perspective based on analogy with equation (2.6) and proceed to evaluate higher order statistics and their scalings with spatial lag. From this emerges a picture quite reminiscent of hydrodynamics, as illustrated in figure 2. Despite what might appear to be an encouraging similarity in the scaling of higher order moments in hydrodynamics and in MHD, there are in fact at least two main impediments to a direct extension of the KRSH to MHD, and at least one additional major problem in extending it to kinetic plasma, as follows. First, for both MHD and plasma, there is ambiguity regarding the choice of fluid-scale variables as there are now at least two vector fields involved. These are the velocity v and magnetic field b, or equivalently the two Elsasser fields z+ = v + b/ 4 and z- = v – b/ 4, where is the mass density. (When the magnetic quantities are expressed in Alfv speed units, the Elsasser variables take the form z?= v ?b.) Is the local dissipation related to vr ? To br ? More properly, based on the structure of the third-order law for MHD [17,18], perhaps relations + analogous to equation (2.6) should be written separately for two local dissipation functions r – and r , in terms of the increment combinations (z+ |z- |2 )1/3 and (z- |z+ |2 )1/3 [19]. There have also been suggestions that even more information could be embedded in the primitive increment functions entering the MHD KRSH, for example by allowing for a scaling of alignment angles between v and b [20]. All such suggestions are permissible from a dimensional standpoint, but we have not yet seen in the literature either a precise statement of an MHD KRSH (however, see [21,22]) or a full statistical test of any such hypothesis, as has been carried out repeatedly in the hydrodynamic case [23,24]. Second, there is increasing evidence that MHD turbulence lacks a universal character [19,25], so that it is not clear that there is a single, simple answer to the questions posed in the previous paragraph. Non-universality is related not only to the multiplicity of independent field variables in MHD bu.