self-organization and entropy production rate in thermally ... · zonal flow and streamer is due...
TRANSCRIPT
Self-organization and entropy production rate in thermally driven turbulence
Yohei Kawazura and Zensho Yoshida
Graduate School of Frontier Sciences, The University of Tokyo
OUTLINE
Introduction
Thermodynamic Model of Self-organizing Turbulence
Numerical simulation of Thermally Driven Drift Wave Turbulence
2
[1] Z. Yoshida, S. M. Mahajan, Phys. Plasmas 15, 032307(2008).[2] Y. Kawazura and Z. Yoshida, Phys. Rev. E 82, 066403 (2010).[3] Y. Kawazura and Z. Yoshida, Phys. Plasmas 19, 012305(2012).
Large scale structure ➟ Self-organization of vortex (emergence of large scale vortex)
ex.) the Great Red Spot and zonal flow in Jupiter
In plasma, vortex structure plays an important role in terms of plasma confinement in magnetically confined fusion.
Contrary to the second law of thermodynamics.
INTRODUCTION
Vertical flow to temperature gradientzonal flow ➟ improves a confinement
[1] M. Nakata,T.-H. Watanabe and H. Sugama, Phys. Plasmas 19, 022303 (2012).
Parallel flow to temperature gradientstreamer ➟ degrade a confinement
[1]
Large Scale Structure in Plasma
3
INTRODUCTION
Baroclinic Vorticity Generation
Vorticity is defined by “curl” of momentum ⇒
Evolution of the mometum
taking curl → vorticity equation
H : combined fluid enthalpyT : temperature s : specific entropy
Only is available to generate vorticity ➟ Baroclinic
Otherwise ( ) the systems is called barotropic.
4
INTRODUCTIONThermal driving
Baroclinic term is in the form of heat .
Identifying “ ” as the change along the fluid’s streamline, we combine thermodynamic laws and fluid dynamical motion ⇒ infinite number set of cycles embedded in space [1]
Plasma converts the energy in collective motion in the
form of vorticity through baroclinic term, then change the
impedance (zonal ? or streamer ?)
➟ Thermal driving
Thermodynamic analysis of the impedance
Numerical simulation of turbulence under thermal driving
[1] Z. Yoshida, Lecture Note: Vorticity Creation and Entropy Production, Proc. in International Advanced Workshop on the Frontiers of Plasma Physics (2010).[2] Z. Yoshida, S. M. Mahajan, Phys. Plasmas 15, 032307(2008). 5
Thermodynamic Model of Self-organizing Turbulence
THERMODYNAMIC VARIATIONAL PRINCIPLES
Variational principles in dissipative systems
− Rayleigh, Onsager’s minimum dissipation principle [1,2]
Linear relation between force and flow is given by variational principle
− Prigogine’s minimum EPR principle
Dissipative structure toward turbulent structure
[1] Lord Rayleigh, Proc. math. Soc. London 4, 357 (1873), L. Onsager, Phys. Rev. 37, 405 (1931); 38, 2265 (1931)[2] P. Gransdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations (Wiley-Interscience, New York, 1971).
➟ Unable dictate convective nonlinearity dominant state, i.e. turbulence.
7
MAXIMUM ENTROPY PRODUCTION
Firstly proposed by Paltridge [6] ➟ Succeeded to predict mean temperature distribution in earth
➟ Extensively studied and applied for various self-organizing systems
Ozawa found experimental observations of fluid mechanical instabilities are predicted by MEP [2]
Yoshida and Mahajan found L - H transition of plasma boundary layer is equivalent to the bifurcation to MEP state [3]
[1] G. W. Paltridge, Q. J. R. Meteorol. Soc. 101, 475 (1975)[2] H. Ozawa, S. Shimokawa, and H. Sakuma, Phys. Rev. E 64,026303 (2001).[3] Z. Yoshida, S. M. Mahajan, Phys. Plasmas 15, 032307(2008).
BoundaryBoundaryHeat flow
Low confinement mode↓
High confinement mode
[1]
[2]
8
THERMODYNAMIC MODEL OF PLASMA BOUNDARY LAYER
Nonlinear term Carnot efficiency diffusion
Quasi-stationary boundary layer surrounded by high temperature core and cold outer region
is controlled by heat flux ➟ Flux-driven condition
Define the impedance against the flow as
P is bounded by Carnot efficiency x flowdissipationdominant
convectiondominant
1. Plasma absorbs energy from heat and converts into macroscopic flow2. Returns the energy to the heat flux by viscous process
Z. Yoshida, S. M. Mahajan, Phys. Plasmas 15, 032307(2008). 9
η0：linear impedancea ：constantP：Available power to generate macroscopic flow
THERMODYNAMIC MODEL OF PLASMA BOUNDARY LAYER
Nonlinear term Carnot efficiency diffusion
η0：linear impedancea ：constantP：Available power to generate macroscopic flow
Quasi-stationary boundary layer surrounded by high temperature core and cold outer region
is controlled by heat flux ➟ Flux-driven condition
Define the impedance against the flow as
P is bounded by Carnot efficiency x flowdissipation
dominantconvectiondominant
1. Plasma absorbs energy from heat and converts into macroscopic flow2. Returns the energy to the heat flux by viscous process
Z. Yoshida, S. M. Mahajan, Phys. Plasmas 15, 032307(2008). 10
THERMODYNAMIC MODEL OF PLASMA BOUNDARY LAYER
organizedsolution
non-organizedsolution
linearnonlinear
2 4 6 8
1
2
3
4
5
6
7linearnonlinear
EPR bifurcate to “Maximum” state
Z. Yoshida, S. M. Mahajan, Phys. Plasmas 15, 032307(2008). 11
Solutions
Zonal flow!
THERMODYNAMIC MODEL OF STREAMER
Previous model dictate high temperature difference state ➟ zonal flow
We invented the model to dictate steamer (or Benard convection).
Zonal flow ⇔ Series connection Streamer ⇔ Parallel connection
Y. Kawazura and Z. Yoshida, Phys. Plasmas 19, 012305(2012). 12
THERMODYNAMIC MODEL OF STREAMERZonal flow ⇔ Series connection Streamer ⇔ Parallel connection
Temp. driven solution Temp. driven solution
13
linearnonlinear
1.0 2.0 3.0
1
2
3
4
1 2 3 4
0.51.01.52.02.53.0
THERMODYNAMIC STABILITYWhich of the bifurcated solution (linear or nonlinear) is realized?
In order to analyze thermodynamic stability, we assume a fluctuation δT and subsequent chain event
Series connection model
⇒ unstable
Flux driven
In the same way, assuming a fluctuation δF and subsequent chain event
Temperature driven
⇒ unstable
(A)
0 2 4 6 8 100.0
1.0
2.0
3.0 (B)
2 4 6 8 10
1.0
2.0
3.0
0.0
Nonlinear solutions are stable for both case
14
if
if
THERMODYNAMIC STABILITYWhich of the bifurcated solution (linear or nonlinear) is realized?
Parallel connection model
⇒ unstable
Flux driven
Temperature driven
⇒ unstable
Nonlinear solutions are stable for both case
0.2 0.4 0.6 0.8 1.0 1.2
0.5
1.0
1.5
2.0
1.2 1.4 1.6 1.8 2.0
0.5
1.0
1.5
2.0
2.5
15
if
if
ZONAL AND STREAMER FOR FLUX-TEMPERATURE-DRIVENZonal flow ⇔ Series connection Streamer ⇔ Parallel connection
16
linearnonlinear
1.0 2.0 3.0
1
2
3
4
1 2 3 4
0.51.01.52.02.53.0
Zonal flow : F-driven → T increase T-driven → F decrease
Streamer : F-driven → T decrease
T-driven → F increase
We can construct the bifurcation matrix of EPR
COMPARISON OF EPR IN PARALLEL AND SERIES SYSTEMS
Flux driven Temperature drivenZonal flow Max Min
Bénard convection Min Max
0.2 0.6 1.0 1.4
0.20.40.60.8
2 4 6 81234567
2 4 6 8 10
2468
1.5 2.5 3.5
0.51.01.52.0
17Similar results are proposed by Niven (2010) for fluid pipe flow.
THERMODYNAMIC POTENTIAL
We introduce the potential functions that give operating point
or
as its extremum.
The following potential functions give operating point.
Here, equals to EPR in stationary state.
or ⇒
，
⇒
18
THERMODYNAMIC POTENTIALLinear theory (Onsager)
,
Min EPR = Min Dissipation function
， ：Dissipation function
，
，
，
‣ EPR is not the target function to be extremized‣ EPR appears as Legendre transform of and
Noninear theory
19
HYSTERESIS IN H-MODE
[1] A. E. Hubbard et al.Plasma Phys. Control. Fusion 44 (2002) A359–A366
Hysteresis is sometimes observed in L-H transition. However, our previous series model do not have hysteresis
linearnonlinear
10 20 30 40 50
1234567
Ramp up and down in Alcator C-Mod [1]
20
HYSTERESIS IN H-MODE
⇒From implicit function theorem, when
is satisfied, T(F) has three nonlinear solutions. ➟ If a(T) has steep gradient, hysteresis occursThis statement is verified from experimental data
The intermediate solution is unstable because,
10 20 30 40 50
1234567
⇒
21
SUMMARY
We extended the thermodynamic model to describe streamer (Benard convection) type self-organization.
From the stability analysis, nonlinear solutions are stable for all case
Each of nonlinear solutions seem to be correct phenomenologically
Min/Max of EPR depends on “1. how plasma organize (zonal or stremer)” and “2. how plasma is driven (flux-driven or force-driven)”
EPR is not target functional of variation in nonlinear regime, but appears in Legendre transformation of dissipation functions.
Flux driven Temperature drivenZonal flow Max Min
Bénard convection Min Max
22
SUMMARY
These thermodynamic models are mechanism free
We made assumption : the power P available to generate flow is determined by Carnot cycle’s efficiency
➟ turbulence should be almost ideal heat engine!
We need to verify it → numerical simulation with specific mechanism
23
Self-organization in Thermally Driven Drift Wave Turbulence
UNDERLYING MECHANISM OF MEPWhy maximization of EPR and structure formation coexists?
➟ Scale hierarchy of turbulence
Preliminary example of 2D Navier-Stokes equation
Two constants of motion ➟ Energy
Enstrophy
Energy and enstrophy have different scale ➟ Selective decay and inverse cascade
[1] D. Biskamp and H. Welter, Phys. Fluids B 1, 1964 (1989)
Scale separation enables the coexistence of “order” and “disorder” [2]
[2] Z. Yoshida, S. M. Mahajan, Phys. Plasmas 15, 032307(2008). 25
BAROCLINIC VORTICITY GENERATION
However, inverse cascade theory do not have baloclinic vorticity generation and coupling with thermodynamics ( )
Many of the preceding studies have been indicated that the emergence of zonal flow and streamer is due to the self–organization of drift wave turbulence. However there remains thermodynamic discussion.
In our previous thermodynamic discussion, we assumed that coherent flow (= vortex structure) is effectively excited by heat flow
In this section, we investigate how the structure in plasma is thermally excited numerically.
26
HASEGAWA - MIMA EQUATION
Consider 2 dimensional compressible barotropic motion
Boltzmann relation and charge neutrality
Drift approximation
Identical to Chaney equation dictating Rossby wave in atmospheric flow
Vorticity eq. ,
Continuity eq.
A. Hasegawa and K. Mima, Phys. Fluids 21, 1 (1978). 27
BAROCLINIC HASEGAWA - MIMA (BHM)
Including baroclinicity
Vorticity eq.
Continuity eq.
Entropy eq.Normalization
In all the other preceding works, normalization is not by L but Ln, therefore β reduces to unity.Freedom of (α, β) is important for zonal flow bifurcation
28
ENERGY BALANCE OF BHM
Total energy of BHM is given as
Assuming closed boundary conditions, the evolution of E is
In quasi-stationary state, the energy injection from background gradient and dissipation balance.
29
HAMILTONIAN STRUCTURE OF BHM
Hamiltonian
Poisson bracket
Casimir invariant
⇒ there is no enstrophy like quadratic invariant
⇒ invariant
⇒ satisfies Jacobi identity
30
SIMULATION CONDITIONS AND RESULTS
Simulation conditions
x : Dirichlet ➟ central differencey : Periodic ➟ pseudo spectral method4th-order Adams Bashforth method for time integralGrid points : 256 x 256 ~ 512 x 512
CASE I : (α, β) = (1, 1)
CASE II : (α, β) = (0.21, 1.6)
31
SIMULATION CONDITIONS AND RESULTS
CASE I : (α, β) = (1, 1) CASE II : (α, β) = (0.21, 1.6)⇒ non zonal ⇒ zonal
32
Closed boundary results
ENERGY EVOLUTION
10-510-410-310-210-1100101102103104105
0 100 200 300 400 500 600 700 800 900
CASE II : (α, β) = (0.21, 1.6)⇒ zonal
33
POWER SPECTRUM OF ZONAL CASE
Inverse cascade
10-6
10-5
10-4
10-3
10-2
10-1
0.1 1 1010-2
10-1
100
101
102
103
104
0.1 1 10
Energy Injection
Linear state Noninear state
Inverse cascade is observed. As zonal structure becomes clear, the energy
of low wave number mode grows.
34
THERMAL DRIVE
HO
T
CO
LD
Back ground inhomogeneity
Feed back
Instability is driven by background inhomogeneity
Heat flows in and out at the boundary by turbulence
Turbulent field change original background inhomogeneity ⇒ “perturbation feedback”
35
THERMAL DRIVE
HO
T
CO
LD
Back ground inhomogeneity
Feed back
Instability is driven by background inhomogeneity
Heat flows in and out at the boundary by turbulence
Turbulent field change original background inhomogeneity ⇒ “perturbation feedback”
36
Flux Driven Boundary conditions for φ
0 mode ➟ Dirichlet. Other modes (turbulent modes) → random oscilation → Turbulent component of velocity penetrates the boundary. we can change the intensity of driving by changing the amplitude of the oscilation
ENERGY INJECTION AT BOUNDARIES
Considering boundary conditions, the evolution of energy is written as
Time evolution of incoming power through the boundary.
-60
-40
-20
0
20
40
60
80
100
120
0 100 200 300 400 500 600 700 800
Energy Injection
through inner boundary
Strongly drivenWeakly driven
We consider strongly driven and weakly driven case by boundaries37
ENERGY INJECTION AT BOUNDARIES
Strongly drivenWeakly driven
0
1
2
3
4
5
6
0 10 20 30 40 50 60 0
1
2
3
4
5
6
0 10 20 30 40 50 60
Strong thermal driving → Large scale mode is enhanced → Large temperature gradientTurbulence convert injected heat to large scale vortex → efficient heat engine
T0 at t = 0
38
T0 at final
Radial profile
SUMMARY
We performed nonlinear simulation of baroclinic drift wave turbulence
The bifurcation of zonal flow depends on the parematers (α, β)
We introduced thermal driving boundary condition
39
→ Therefore the assumption that turbulence is almost ideal heat engine seems correct
- Large scale structure is enhanced by heat injection at the boundary
- Large mean temperature gradient is induced by strong heat injection at the boundary
→ Consistent with the result from the thermodynamic model
Our thermodynamic model is appropriate for plasma zonal flow