Keiji Kimura 1) and Philip J. Morrison 2)

Transcription

1 Keiji Kimura 1) and Philip J. Morrison 2) 1) Kyoto University, 2) The University of Texas at Austin Geophysical Fluid Dynamics Program 2011 (20 th June th August 2011) Woods Hole Oceanographic Institution

2 京都大学数理解析研究所博士課程 2 年木村恵二 CPS セミナー,2011 年 10 月 19 日惑星科学研究センター (CPS)

3 Research Project in GFD Program 2011 Lectures in GFD Program 2011 Life at GFD 3

4 Keiji Kimura 1) and Philip J. Morrison 2) 1) Kyoto University, 2) The University of Texas at Austin Geophysical Fluid Dynamics Program 2011 (20 th June th August 2011) Woods Hole Oceanographic Institution

5 Introduction Linear Wave Modes Energy Conservation Equilibrium States Conclusion Future Works 5

6 Magnetohydrodynamic (MHD) approximation is well-used. (HP of Prof. Z. Yoshida) (Wikipedia Sun) What are the limitations? (Glatzmeier and Roberts, 1995) 6

7 Kinetic theory Two-fluid model (taking moments) Quasi-neutrality 7

8 Kinetic theory Two-fluid model (taking moments) One-fluid model (quasi-neutrality) (Lüst, 1959) : Continuity eqn. : Momentum eqn. Hall term Collision (resistivity) Electron inertia : Generalized Ohm s law 8

9 Vasyliunas (1975) Fitzpatrick (2001) Watson Bhattacharjee et al. (1999) Shay et al. (2001) There is no term in the momentum equation! 9

10 (Ideal) MHD Hall-MHD Inertial-MHD (IMHD) 10

11 Governing eqns. : The characteristic timescale << the gyroperiod of electron (Magnetic reconnection region?) Pre-Maxwell equations 11

12 Comparing the linear wave modes between MHD and IMHD. Classifying some IMHD models in terms of the energy conservation. Considering the effect of electron inertia using IMHD, especially focusing on some equilibrium states. 12

13 Introduction Linear Wave Modes Energy Conservation Equilibrium States Conclusion Future Works 13

14 Basic state (Shear) Alfven wave : uniform : no flow Linearized compressible MHD Fast/Slow magnetosonic wave (Fitzpatrick) 14

15 MHD : Alfven speed Inertial MHD Higher wavenumber waves propagate more slowly! 15

16 Introduction Linear Wave Modes Energy Conservation Equilibrium States Conclusion Future Works 16

17 Governing equations Energy conservation 17

18 Energy conservation This is not a correct flux form but total energy H is conserved! 18

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20 The epsilon term in the momentum equation is important!! 20

21 Vasyliunas (1975) Fitzpatrick (2001) Watson Bhattacharjee et al. (1999) Shay et al. (2001) There is no term in the momentum equation! 21

22 Introduction Linear Wave Modes Energy Conservation Equilibrium States Conclusion Future Works 22

23 With no flow 1. Grad-Shafranov equation With flow (incompressible) Beltrami- Jeltrami flow 23

24 Governing equation (no flow) (Wikipedia) Axisymmetric solution : Grad-Shafranov equation What occurs with the electron inertia? 24

25 Straight torus = cylinder; torus with no curvature z Axisymmetric solution = z independent solution When, then we find that and, and obtain : G-S in straight torus What occurs when epsilon and delta are finite? 25

26 If the plasma is barotropic, i.e.,, Modified Grad-Shafranov equation in straight torus 26

27 Incompressible plasma?? I did not get any useful constraints 27

28 Straight torus Torus Incompressible Compressible 3 constraints obtained Modified G-S obtained If barotropic, 3 (incomplete) constraints obtained. If barotropic, modified GS obtained. 3 constraints obtained?????? Epsilon term is considered Epsilon and delta term are considered 28

29 With no flow 1. Grad-Shafranov equation With flow (incompressible) Beltrami- Jeltrami flow 29

30 Governing equations (incompressible IMHD) Assuming quite similar! If 30

31 With no flow 1. Grad-Shafranov equation With flow (incompressible) Beltrami- Jeltrami flow 31

32 : Beltrami flow : Jeltrami current If similar to Bernoulli s equation 32

33 Modified Alfven wave in Inertial MHD is dispersive. The epsilon term in the momentum equation is important in terms of energy conservation. Modified Grad-Shafranov equation is obtained in straight torus. In the real torus, we obtained only some constraints. Governing equations of equilibrium states with flow can be simplified in and in Beltrami- Jeltrami flow. 33

34 Physical interpretation of equilibrium state with delta term in modified Grad-Shafranov equation with curvature Studying the stability with the effect of epsilon and delta term Shocks Magnetic reconnection 34

35 Research Project in GFD Program 2011 Lectures in GFD Program 2011 Life at GFD 35

36 Shear Turbulence: Onset and Structure Fabian Waleffe (University of Wisconsin, Madison) (2011) (1990) (1988) (2011) Richard Kerswell (Bristol University) (1995) 36

37 Canonical Laminar flow Turbulent State Transition occurs suddenly, noise-dependently and dramatically! 37

38 Canonical Flow Plane Couette Critical Reynolds number Turbulent State Plane Poiseuille 5772 Hagen- Poiseuille (Pipe)? Transition can occur at lower Reynolds number! Why? What occurs? 38

39 1. General Introduction and Overview. 2. Viscous derivation of classic inviscid stability results for shear flows. Viscous instability. 3. Diffusion and damping in shear flows: a truly singular limit. Critical layers. 4. Origin and survival of 3D-ality. 5. Instability of streaky flows. Asymptotics of self-sustaining process. 6. Spatio-temporal complexity. Spots, puffs and slugs, snakes and spirals. 39

40 Transition threshold i.e. the power law of the amplitude against Re Structure (horseshoe vortex etc.) Linear and energy stabilities Squire s theorem (linear) vs. streamwise roll (energy) How to sustain the turbulent state the feedback mechanism to the roll pattern Self-Sustaining Process (SSP) and SSP method finding the exact coherent structures The boundary of the laminar and turbulent states in phase space 40

41 Waleffe (1997) 41

42 Rolls V redistribute streamwise momentum M U Streak U unstable to x-mode W nonlinear interaction of W feeds back on rolls V Mean M shears W 42

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44 1. Transition scenarios: normality vs non-normality. 2. Edge tracking walking the tightrope. 3. Triggering transition efficiently. 4. Turbulence: transient or sustained? 44

45 Supercritical or subcritical scenarios Normal or Non-normal linear operator Finding nonlinear solutions (e.g. Nagata s solution) Edge tracking Finding Minimal Seeds Is the turbulent state is a transient state or sustained one? puff, localized pattern 45

46 Minimal seed : the I.C. of smallest energy which can trigger transition to turbulence Can we identify the minimal seed by looking for an I.C. which experiences largest growth? Energy growth rate: Linear transient growth Matrix-based SVD Matrix-free Variational principle (Euler-Lagrange eq.) this method can be used to study the nonlinear transient growth! 46

47 Linearized Navier-Stokes eq. Growth rate Euler-Lagrange eqns. Dual Linearized Navier-Stokes eqn. 47

48 This method is easily extendable to the nonlinear problem! 48

49 Pringle and Kerswell (2010) Linear Nonlinear Schmid and Henningson (1994) 49

50 Lectures in GFD Program 2011 Research Project in GFD Program 2011 Life at GFD 50

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54 Matthew Chantry (Univ. of Bristol) British (23) Andrew Crosby (Univ. of Cambridge) British (26) John Platt (Harvard Univ.) British (24) Chao Ma 馬超 (Univ. of Colorado, Boulder) Chinese (28) Giulio Mariotti (Boston Univ.) Italian (27) Martin Hoecker-Martinez (Oregon State Univ.) American (32) Zhan Wang 王展 (Univ. of Wisconsin, Madison) Chinese (28) Adele Morrison (Australian National Univ.) Australian (29) Lindsey Ritchie (Corson) (Univ. of Strathclyde) British (27?) Samuel Potter (Princeton Univ.) American (29?) Keiji Kimura (Kyoto Univ.) Japanese (25) 54

55 British: 4 American: 2 Chinese: 2 Australian: 1 Italian: 1 Japanese: 1 Univ. in USA: 6 in British: 3 in Australia: 1 in Japan: 1 55

56 Genta Kawahara, Structures of low-reynolds number turbulence in a rectangular duct Friedrich Busse, Generation of magnetic fields by convection in rotating spherical fluid shells Jesse Ausubel, Self-sinking capsules to investigate Earth s interior and dispose of radioactive waste Tomoaki Itano, Coherent vortices in plane Couette flow bifurcation, symmetry and visualization Predrag Cvitanovic, What Phil Morrison would not teach us: how to reduce the symmetry of pipe flows 56

57 Martin Hoecker-Martinez: Constraints on low order models: the cost of simplicity Keiji Kimura: A One-fluid MHD Model with Electron Inertial Matthew Chantry: Traversing the edge: how turbulence decays Giulio Mariotti: A low dimensional model for shear turbulence in Plane Poiseuille Flow: an example to understand the edge Adele Morrison: Upstream basin circulation of rotating, hydraulically controlled flows Samuel Potter: Islands in locally-forced basin circulations Zhan Wang: Tow-layer viscous fluid in an inclined closed tube: Kelvin-Helmholtz instability Andrew Crosby: Chaotic interaction of vortex patches with boundaries Chao Ma: On Brownian motion in a Fluid with a Plane Boundary John Platt: Localized Solutions for Plane Couette flow: a continuation method study Lindsey Ritchie: Ascending the ridge: Maximizing the heat flux in steady porous medium convection 57

58 5 勝 3 敗!! 58

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60 得たもの 生活力, 料理の技術 アメリカ人の人生の楽しみ方スポーツの重要性度胸 ( 英語力 ) Shear Turbulence の知識 MHD の知識 失ったもの体重 (-5kg) 60

61 林先生をはじめ, 北大 神戸大 GCOE,CPS 関係者の皆様に深く感謝申し上げます (1984) 61