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 GFDCPS2011

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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

CPS20112011/10/195

 Magnetohydrodynamic (MHD) approximation

is well-used.

(HP of Prof. Z. Yoshida)(Wikipedia Sun) What are the limitations?

(Glatzmeier and Roberts, 1995)

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6

Kinetic theory
 Two-fluid model (taking moments)

 Quasi-neutrality

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7

Kinetic theory
 Two-fluid model (taking moments)
 One-fluid model (quasi-neutrality) (Lüst, 1959): Continuity eqn.
: Momentum eqn.

Hall term
Collision (resistivity): GeneralizedOhm’s law

Electron inertia
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Vasyliunas (1975)

Fitzpatrick (2001)
WatsonBhattacharjee et al. (1999)
Shay et al. (2001)

There is notermin the momentum equation!
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9 (Ideal) MHD

 Hall-MHD

 Inertial-MHD (IMHD)CPS20112011/10/19

10

: The characteristic timescale<< the gyroperiod of electron Governing eqns.

(Magnetic reconnection region?)

Pre-Maxwell equations
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11 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 someequilibrium states.

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12 Introduction
 Linear Wave Modes Energy Conservation
 Equilibrium States
 Conclusion
 Future Works

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

(Fitzpatrick)
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14 MHD: Alfven speed

 Inertial MHD

Higher wavenumber waves
propagate more slowly!
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15 Introduction
 Linear Wave Modes Energy Conservation
 Equilibrium States
 Conclusion
 Future Works

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 Governing equations

 Energy conservation

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17

 Energy conservation

This is not a correct flux form…
but total energy H is conserved!

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18

CPS2011

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

Vasyliunas (1975)

Fitzpatrick (2001)
WatsonBhattacharjee et al. (1999)
Shay et al. (2001)

There is notermin the momentum equation!
CPS20112011/10/1921

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

CPS20112011/10/1922 With no flow

1. Grad-Shafranov equation
 With flow (incompressible)

1.
2. Beltrami-“Jeltrami” flow

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 Governing equation (no flow)(Wikipedia)

 Axisymmetric solution: Grad-Shafranov equation

What occurs with the electron inertia?
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“Straight torus” = cylinder; torus with no curvaturez

 “Axisymmetric” solution = z independent solution

 Whenand obtain, then we find thatand,: G-S in “straight torus”What occurs when epsilon and delta are finite?
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25

 If the plasma is barotropic, i.e.,,

Modified Grad-Shafranov equation in “straight torus”

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26

 Incompressible plasma??

I did not get
any useful constraints…
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27

Incompressible

Compressible

Straight torus

Torus

3 constraints
obtained

3 constraints
obtained

Epsilon term
is considered

Modified G-Sobtained??

If barotropic, 3(incomplete)constraints
obtained.

Epsilon anddelta term
are considered?

If barotropic,modified GSobtained.

???

CPS20112011/10/1928

 With no flow

1. Grad-Shafranov equation
 With flow (incompressible)

1.
2. Beltrami-“Jeltrami” flow

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 Governing equations (incompressible IMHD) Assuming
quite similar!

 IfCPS2011

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 With no flow

1. Grad-Shafranov equation
 With flow (incompressible)

1.
2. Beltrami-“Jeltrami” flow

CPS20112011/10/1931

: Beltrami flow
: “Jeltrami” current

similar to
Bernoulli’sequation

If
CPS20112011/10/1932






Modified Alfven wave in Inertial MHD is dispersive.
The epsilon term in the momentum equation isimportant 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.

CPS20112011/10/1933

Physical interpretation of equilibrium state withdelta term in modified Grad-Shafranov equation
with curvature
Studying the stability with the effect of epsilon anddelta term

Shocks

Magnetic reconnection

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

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35

“Shear Turbulence: Onset and Structure”
Fabian Waleffe(University ofWisconsin,Madison)(2011)(1988)(1990)(2011)

Richard Kerswell(Bristol University)
CPS20112011/10/19(1995)

36

CanonicalLaminar flow

“Turbulent”
State

Transition occurs suddenly,
noise-dependently anddramatically!
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Canonical Flow

Critical Reynolds
numberPlane Couette∞

Plane Poiseuille

5772

HagenPoiseuille (Pipe)∞?

“Turbulent”
State
Transition can occur atlower Reynolds number!

Why? What occurs?
CPS20112011/10/1938

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 andspirals.

CPS20112011/10/1939

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

CPS20112011/10/1940

Waleffe (1997)
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41

Rolls V redistributestreamwise momentum
MU

Streak U
unstable tox-mode W

nonlinearinteraction of W
feeds back onrolls VCPS2011

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Mean M
shears W42

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43

1.

Transition scenarios: normality vs non-normality.

2.

Edge tracking – walking the tightrope.

3.

Triggering transition efficiently.

4.

Turbulence: transient or sustained?

CPS20112011/10/1944

 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 orsustained one?  puff, localized pattern
CPS20112011/10/1945

“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!
CPS20112011/10/1946

 Linearized Navier-Stokes eq.

 Growth rate Euler-Lagrange eqns.

Dual Linearized
Navier-Stokeseqn.
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This method is easily extendable tothe nonlinear problem!
CPS20112011/10/1948

Pringle and Kerswell (2010)

 Linear Nonlinear

Schmid and Henningson (1994)

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

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CPS2011

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51

CPS2011

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CPS2011

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Zhan Wang 王展(Univ. of Wisconsin,Madison)
Chinese (28)Andrew Crosby(Univ. of Cambridge)
British (26)John Platt(Harvard Univ.)
British (24)

Samuel Potter(Princeton Univ.)
American (29?)

Adele Morrison(Australian National Univ.)
Australian (29)

Lindsey Ritchie (Corson)(Univ. of Strathclyde)
British (27?)

Keiji Kimura(Kyoto Univ.)
Japanese (25)Matthew Chantry(Univ. of Bristol)
British (23)Chao Ma 馬超(Univ. of Colorado,Boulder)
Chinese (28)Giulio Mariotti(Boston Univ.)
Italian (27)Martin Hoecker-Martinez(Oregon State Univ.)
American (32)
CPS20112011/10/1954

British:American:Chinese:Australian:
Italian:Japanese:

4
2
2
1
1
1

Univ.
in USA:
6
in British: 3
in Australia: 1
in Japan:
1

CPS20112011/10/1955

 Genta Kawahara, “Structures of low-Reynolds

number turbulence in a rectangular duct”
 Friedrich Busse, “Generation of magnetic
fields by convection in rotating spherical fluidshells”
 Jesse Ausubel, “Self-sinking capsules toinvestigate Earth’s interior and dispose ofradioactive waste”
 Tomoaki Itano, “Coherent vortices in plane
Couette flow – bifurcation, symmetry andvisualization”
 Predrag Cvitanovic, “What Phil Morrison would
not teach us: how to reduce the symmetry ofpipe flows”

CPS20112011/10/1956

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 turbulencein 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
CPS20112011/10/1957

5勝3敗!!
CPS20112011/10/1958

CPS2011

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

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)
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60

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)CPS20112011/10/19

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