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Third moment in MHD SHINE 2006 Miriam Forman 1 Third moment of MHD fluctuations: a new interpretation for anisotropic turbulence in the solar wind… and.

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Presentation on theme: "Third moment in MHD SHINE 2006 Miriam Forman 1 Third moment of MHD fluctuations: a new interpretation for anisotropic turbulence in the solar wind… and."— Presentation transcript:

1 Third moment in MHD SHINE 2006 Miriam Forman 1 Third moment of MHD fluctuations: a new interpretation for anisotropic turbulence in the solar wind… and a definition of intermittency Miriam Forman Charles W. Smith Benjamin.T. MacBride (see poster MacBride, et al.)

2 Third moment in MHD SHINE 2006 Miriam Forman 2 What’s a moment, and what’s scaling and what’s intermittency Turbulence is much more than the power spectrum… Important points of this talk

3 Third moment in MHD SHINE 2006 Miriam Forman 3 What’s a moment, and what’s scaling and what’s intermittency Turbulence is much more than the power spectrum… Politano and Pouquet theorem about third moment in Elsasser variables Important points of this talk

4 Third moment in MHD SHINE 2006 Miriam Forman 4 What’s a moment, and what’s scaling and what’s intermittency Turbulence is much more than the power spectrum… Politano and Pouquet theorem about third moment in Elsasser variables Valid in spite of anisotropy! Important points of this talk

5 Third moment in MHD SHINE 2006 Miriam Forman 5 What’s a moment, and what’s scaling and what’s intermittency Turbulence is much more than the power spectrum… Politano and Pouquet theorem about third moment in Elsasser variables Valid in spite of anisotropy! Application to solar wind Important points of this talk

6 Third moment in MHD SHINE 2006 Miriam Forman 6 What’s a moment, and what’s scaling and what’s intermittency Turbulence is much more than the power spectrum… Politano and Pouquet theorem about third moment in Elsasser variables Valid in spite of anisotropy! Application to solar wind Fundamental measure of turbulent energy cascade rates Important points of this talk

7 Third moment in MHD SHINE 2006 Miriam Forman 7 What’s a moment, and what’s scaling and what’s intermittency Turbulence is much more than the power spectrum… Politano and Pouquet theorem about third moment in Elsasser variables Valid in spite of anisotropy! Application to solar wind Fundamental measure of turbulent energy cascade rates Implications for intermittency, anisotropy and power spectrum Important points of this talk

8 Third moment in MHD SHINE 2006 Miriam Forman 8 Finding some order in turbulent fluctuations, through “scaling” measures fluctuations look a complete mess at first does obey physics does have an equation: in ordinary fluid, the Navier-Stokes Navier-Stokes is non-linear >>Can deduce some properties of the fluctuations exactly >>Continue to look for more order, make predictions and interpretive models (I’ll get to MHD soon)

9 Third moment in MHD SHINE 2006 Miriam Forman 9 Derivation: Play around with the N-S at different places in the fluid Subtract and multiply by One exact relation, from Navier-Stokes: Von Karman-Howarth-Monin relation (following Frisch, 1995, section 6.2)

10 Third moment in MHD SHINE 2006 Miriam Forman 10 Von Karman-Howarth-Monin relation for hydrodynamic turbulence (following Frisch, 1995, section 6.2) is the scale vector Ensemble-average gives functions of L Use homogeneity and incompressibility, and Energy input  J/(kg-s) on scale >>L Energy not input or extracted on scale L do NOT use isotropy Exact vector equation, even in anisotropic turbulence, direct from Navier-Stokes Wow! This odd moment is not only not zero, it has a sign, and It is probably the fundamental theorem of real turbulence!!

11 Third moment in MHD SHINE 2006 Miriam Forman 11 Kolmogorov’s 4/5 law for Isotropic Hydrodynamic Turbulence (can be derived separately, or from KHM) Exact for isotropic turbulence, direct from Navier-Stokes Wow! This odd moment is not only not zero, it has a sign, and It is probably the fundamental theorem of real turbulence!!

12 Third moment in MHD SHINE 2006 Miriam Forman 12 MAFIGPP Astrophysics Conference, Palm Springs, CA February 10, 2003MAFUmd IPST, Feb 24, 2003 June 2000 ACE radial velocities

13 Third moment in MHD SHINE 2006 Miriam Forman 13 Third velocity moment linear scaling has profound implications 5/3 spectrum! These inferences are not as exact as K-H-M, but a good guess. Big failings: They ignore intermittency and anisotropy. No intermittency! One might think that

14 Third moment in MHD SHINE 2006 Miriam Forman 14 Definitions: Moments, Scaling, and Intermittency in Isotropic Hydro-turbulence n th moment at scale L moment scaling Kolmogorov 1941 scaling (not intermittent) BUT, moments of vectors are actually tensors… scaling is this simple only in isotropy

15 Third moment in MHD SHINE 2006 Miriam Forman 15 Moments, Scaling, and Intermittency in Isotropic Hydro-turbulence Intermittent scaling: ζ is not linear, but has negative curvature in n:  Kurtosis of pdf of fluctuations increases toward smaller scales It’s the law, even in intermittency! Frisch, 1995 is a good reference Quite a few “models” around to predict ζ(n), but none are exact. All predict zero third moment. All increase slope of spectrum a little. Don’t believe any model, at n > 6. Below 6, they are all the same anyway! Negative curvature of ζ(n)

16 Third moment in MHD SHINE 2006 Miriam Forman 16

17 Third moment in MHD SHINE 2006 Miriam Forman 17 MAFUmd IPST, Feb 24, 2003

18 Third moment in MHD SHINE 2006 Miriam Forman 18 Structure function and its exponent; the usual way of looking at intermittency

19 Third moment in MHD SHINE 2006 Miriam Forman 19 Another exact hydrodynamic result: turbulent convection of a scalar, Yaglom’s law Turbulent convection + “molecular” diffusion Same algebra as K-H-M derivation yields Yaglom’s Law Unfortunately, usually written in spherical isotropy as And called “Yaglom’s 4/3 law” Transporting vector field Transported passive scalar

20 Third moment in MHD SHINE 2006 Miriam Forman 20 What about third moment in MHD? Politano and Pouquet, 1997 derived the K-H-M law for MHD MHD equations have the same non-linear structure… P&P exact result:

21 Third moment in MHD SHINE 2006 Miriam Forman 21 P&P exact result: What does this vector equation really mean? Is it TRUE? How do we handle it in anisotropic MHD? How should we use it in the solar wind? Use to measure turbulent energy injection /dissipation rate? What does it tell us to do to measure intermittency in MHD? In the solar wind? Unfortunately, P&P went right to isotropy, which they don’t need, and doesn’t occur anyway, sadly obscuring their wonderful result.

22 Third moment in MHD SHINE 2006 Miriam Forman 22 Review of  V and vector scale L Ensemble average, over a homogeneous volume, various L Ensemble average, over a homogeneous volume, various L lengths and directions

23 Third moment in MHD SHINE 2006 Miriam Forman 23 Review of  V and vector scale L Ensemble average, over a homogeneous volume, various L lengths and directions But, in solar wind Ensemble average, along the solar wind direction, L = Vt, only 1 choice of direction. Be careful about using too many intervals or lose homogeneity

24 Third moment in MHD SHINE 2006 Miriam Forman 24 Looks a lot like Look at how the parallel and perpendicular fluxes change with parallel and perpendicular scale.

25 Third moment in MHD SHINE 2006 Miriam Forman 25 In anisotropy defined by fluctuations parallel and perpendicular to mean magnetic field, divergence is We don’t know how the terms go separately, so guess that:

26 Third moment in MHD SHINE 2006 Miriam Forman 26 Because we take the differences as z(t+  )-z(t), which is backwards in the sense of the solar wind direction, our sign for data is the opposite of the theorem. Parallel “1D” Perpendicular “2D”

27 Third moment in MHD SHINE 2006 Miriam Forman 27 Some of the ACE data analysis by Ben MacBride and Chuck Smith: See poster!!

28 Third moment in MHD SHINE 2006 Miriam Forman 28 ACE, MHD turbulent energy dissipation rates, 10^4 Joules per second per kg Evaluated with P&P vector equation Very preliminary numbers, maybe to 30%!! It appears that the outward waves’ energy flux cascade is to smaller perpendicular scales, and larger parallel scales, while the inward waves’ (smaller) energy flux is to larger perpendicular scales and smaller parallel scales

29 Third moment in MHD SHINE 2006 Miriam Forman 29 Parallel scale -> Perpendicular scale -> z+ energy flux z- energy flux

30 Third moment in MHD SHINE 2006 Miriam Forman 30 What’s a moment, and what’s scaling and what’s intermittency Turbulence is much more than the power spectrum… Politano and Pouquet theorem about third moment in Elsasser variables Valid in spite of anisotropy! Application to solar wind Fundamental measure of turbulent energy cascade rates Implications for intermittency, anisotropy and power spectrum Important points of this talk

31 Third moment in MHD SHINE 2006 Miriam Forman 31 Summary P&P exact result: Handles anisotropy very well. Worth a lot more attention and application. Basis for a meaningful intermittency measure for MHD? Long-term turbulent energy dissipation rate at 1 AU is about 10^4 J/kg-s.

32 Third moment in MHD SHINE 2006 Miriam Forman 32

33 Third moment in MHD SHINE 2006 Miriam Forman 33


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