keiji 1)kimura and philip j. morrison...
Post on 27-Jan-2020
8 Views
Preview:
TRANSCRIPT
Keiji Kimura1) and Philip J. Morrison2)
1)Kyoto University, 2)The University of Texas at Austin
Geophysical Fluid Dynamics Program 2011
(20th June 2011—26th August 2011)
Woods Hole Oceanographic Institution
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)
Research Project in GFD Program 2011
Lectures in GFD Program 2011
Life at GFD
2011/10/19 CPS2011 3
Keiji Kimura1) and Philip J. Morrison2)
1)Kyoto University, 2)The University of Texas at Austin
Geophysical Fluid Dynamics Program 2011
(20th June 2011—26th August 2011)
Woods Hole Oceanographic Institution
Introduction
Linear Wave Modes
Energy Conservation
Equilibrium States
Conclusion
Future Works
2011/10/19 CPS2011 5
Magnetohydrodynamic (MHD) approximation is well-used.
What are the limitations?
2011/10/19 CPS2011 6
(Wikipedia Sun)
(HP of Prof. Z. Yoshida)
(Glatzmeier and Roberts, 1995)
Kinetic theory Two-fluid model (taking moments)
2011/10/19 CPS2011 7
Quasi-neutrality
Kinetic theory Two-fluid model (taking moments)
One-fluid model (quasi-neutrality) (Lüst, 1959)
2011/10/19 CPS2011 8
: Continuity eqn.
: Momentum eqn.
: Generalized Ohm’s law
Collision (resistivity)
Hall term
Electron inertia
2011/10/19 CPS2011 9
Vasyliunas (1975)
Shay et al. (2001)
Fitzpatrick (2001)
Watson
Bhattacharjee et al. (1999)
There is no term in the momentum equation!
2011/10/19 CPS2011 10
(Ideal) MHD
Hall-MHD
Inertial-MHD (IMHD)
2011/10/19 CPS2011 11
Pre-Maxwell equations
: The characteristic timescale << the gyroperiod of electron
(Magnetic reconnection region?) Governing eqns.
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.
2011/10/19 CPS2011 12
Introduction
Linear Wave Modes
Energy Conservation
Equilibrium States
Conclusion
Future Works
2011/10/19 CPS2011 13
2011/10/19 CPS2011 14
Basic state (Shear) Alfven wave
Fast/Slow magnetosonic wave
Linearized compressible MHD
: uniform
: no flow
(Fitzpatrick)
2011/10/19 CPS2011 15
MHD
Inertial MHD
: Alfven speed
Higher wavenumber waves propagate more slowly!
Introduction
Linear Wave Modes
Energy Conservation
Equilibrium States
Conclusion
Future Works
2011/10/19 CPS2011 16
2011/10/19 CPS2011 17
Governing equations
Energy conservation
2011/10/19 CPS2011 18
Energy conservation
This is not a correct flux form… but total energy H is conserved!
2011/10/19 CPS2011 19
2011/10/19 CPS2011 20
The epsilon term in the momentum equation is important!!
2011/10/19 CPS2011 21
Vasyliunas (1975)
Shay et al. (2001)
Fitzpatrick (2001)
Watson
Bhattacharjee et al. (1999)
There is no term in the momentum equation!
Introduction
Linear Wave Modes
Energy Conservation
Equilibrium States
Conclusion
Future Works
2011/10/19 CPS2011 22
With no flow 1. Grad-Shafranov equation
With flow (incompressible)
1. 2. Beltrami-“Jeltrami” flow
2011/10/19 CPS2011 23
2011/10/19 CPS2011 24
Governing equation (no flow)
Axisymmetric solution
: Grad-Shafranov equation
What occurs with the electron inertia?
(Wikipedia)
2011/10/19 CPS2011 25
“Straight torus” = cylinder; torus with no curvature
“Axisymmetric” solution = z independent solution
: G-S in “straight torus”
When , then we find that and , and obtain
z
What occurs when epsilon and delta are finite?
2011/10/19 CPS2011 26
If the plasma is barotropic, i.e., ,
Modified Grad-Shafranov equation in “straight torus”
2011/10/19 CPS2011 27
Incompressible plasma
?? I did not get
any useful constraints…
2011/10/19 CPS2011 28
Straight torus Torus
Incompressible
3 constraints obtained
3 constraints obtained
Modified G-S obtained ??
Compressible
If barotropic, 3 (incomplete) constraints obtained.
?
If barotropic, modified GS
obtained. ???
Epsilon term is considered
Epsilon and delta term
are considered
With no flow 1. Grad-Shafranov equation
With flow (incompressible)
1. 2. Beltrami-“Jeltrami” flow
2011/10/19 CPS2011 29
2011/10/19 CPS2011 30
Governing equations (incompressible IMHD)
Assuming
If
quite similar!
With no flow 1. Grad-Shafranov equation
With flow (incompressible)
1. 2. Beltrami-“Jeltrami” flow
2011/10/19 CPS2011 31
2011/10/19 CPS2011 32
: Beltrami flow
: “Jeltrami” current
If similar to Bernoulli’s equation
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.
2011/10/19 CPS2011 33
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
2011/10/19 CPS2011 34
Research Project in GFD Program 2011
Lectures in GFD Program 2011
Life at GFD
2011/10/19 CPS2011 35
2011/10/19 CPS2011 36
“Shear Turbulence: Onset and Structure”
Fabian Waleffe (University of
Wisconsin, Madison)
Richard Kerswell (Bristol University)
(2011) (1995)
(1990) (2011)
(1988)
2011/10/19 CPS2011 37
Canonical Laminar flow
“Turbulent” State
Transition occurs suddenly, noise-dependently and
dramatically!
2011/10/19 CPS2011 38
Canonical Flow Critical Reynolds number
Plane Couette ∞
Plane Poiseuille 5772 Hagen-
Poiseuille (Pipe) ∞?
“Turbulent” State
Transition can occur at lower Reynolds number!
Why? What occurs?
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.
2011/10/19 CPS2011 39
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
2011/10/19 CPS2011 40
2011/10/19 CPS2011 41
Waleffe (1997)
2011/10/19 CPS2011 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
2011/10/19 CPS2011 43
1. Transition scenarios: normality vs non-normality.
2. Edge tracking – walking the tightrope.
3. Triggering transition efficiently.
4. Turbulence: transient or sustained?
2011/10/19 CPS2011 44
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
2011/10/19 CPS2011 45
2011/10/19 CPS2011 46
“Minimal seed”: the I.C. of smallest energy which can trigger transition to “turbulence”
Energy growth rate:
Can we identify the minimal seed by looking for an I.C. which experiences largest growth?
Linear transient growth Matrix-based SVD Matrix-free Variational principle (Euler-Lagrange eq.) this method can be used to study the nonlinear transient growth!
2011/10/19 CPS2011 47
Linearized Navier-Stokes eq.
Growth rate
Euler-Lagrange eqns.
Dual Linearized Navier-Stokes eqn.
2011/10/19 CPS2011 48
This method is easily extendable to the nonlinear problem!
2011/10/19 CPS2011 49
Linear
Nonlinear
Schmid and Henningson (1994)
Pringle and Kerswell (2010)
Lectures in GFD Program 2011
Research Project in GFD Program 2011
Life at GFD
2011/10/19 CPS2011 50
2011/10/19 CPS2011 51
2011/10/19 CPS2011 52
2011/10/19 CPS2011 53
2011/10/19 CPS2011 54
Giulio Mariotti (Boston Univ.)
Italian (27)
Martin Hoecker-Martinez (Oregon State Univ.)
American (32)
Chao Ma 馬超 (Univ. of Colorado,
Boulder) Chinese (28)
Matthew Chantry (Univ. of Bristol)
British (23)
Andrew Crosby (Univ. of Cambridge)
British (26)
Zhan Wang 王展 (Univ. of Wisconsin,
Madison) Chinese (28)
Lindsey Ritchie (Corson) (Univ. of Strathclyde)
British (27?)
Adele Morrison (Australian National Univ.)
Australian (29)
Samuel Potter (Princeton Univ.) American (29?)
Keiji Kimura (Kyoto Univ.) Japanese (25)
John Platt (Harvard Univ.)
British (24)
2011/10/19 CPS2011 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
2011/10/19 CPS2011 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”
2011/10/19 CPS2011 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
2011/10/19 CPS2011 58
5勝3敗!!
2011/10/19 CPS2011 59
2011/10/19 CPS2011 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)
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
2011/10/19 CPS2011 61
(1984)
top related