HRTW04-HRTW04- ２００８２００８ Date : 11, Dec., 2008Place : Isaac Newton Institute

Equilibrium States of Quasi-geostrophic Point Vortices

Takeshi Miyazaki, Hidefumi Kimura and Shintaro Hoshi

Dept. Mech. Eng. Int'l Systems.,University of Electro-Communications1-5-1, Chofugaoka, Chofu, Tokyo, 182-8585, Japan

Outline

BackgroundThe relevance of vortices in Geophysical flows

Quasi-geostrophic Approximation Hierarchy of approximation and introducing the QG approximation

Statistical Mechanics of QG Point Vortices “Mono-disperse” (All vortices of identical strength) and “Poly-disperse” (Bi-disperse: = {0.5, 1},{1, -1} etc.)

Simulation results “Equilibrium”The comparison with the “Maximum entropy theory”

　　　 　 further simplified theory “Patch Model”Influence of vertical vorticity distribution on equilibriumInfluence of energy on equilibrium

Mono-disperse Poly (Bi)-disperse

Ex.) Point Vortex System

In a rotating stratified fluid, the interactions of isolated coherent vortices dominate the turbulence dynamics.

Background (1)

Vertical motion is suppressed due tothe Coriolis force and stable stratification.

Geophysical flows …

Gulf Stream (numerical simulation, NASA) Instant value of the surface temperature in the north-east Pacific (The Earth Simulator by JAMSTEC)

Largest-ever Ozone Hole over Antarctica (NASA)

Background (2)

Two-dimensional point vortex systems

lowest order approximation of geophysical flows

Many studies have been made on purely 2D flows.

Statistical mechanics

L. Onsager (1949), negative temperature

D. Montgomery and G. Joyce (1974), canonical ensemble

Y. B. Pointin and T. S. Lundgren (1976), micro canonical ensemble

Yatsuyanagi et al. (2005), very large numerical simulation (N = 6724)

The actual geophysical flows are 3D. The fluid motions are almost confined within a horizontal plane Different motions are allowed on different horizontal planes.

‘Quasi-geostrophic approximation’ (next order approximation)⇒The QG-approximation confines the motion in different horizontal planes with “interacting” two dimensional behavior.

N degrees of freedom Point vortex

Spheroidal vortex

Ellipsoidal vortex

2N degrees of freedom

Miyazaki et al. (2005)

Li et al. (2006)3N degrees of freedom

Vortex models

J. C. McWilliams et al. (1994) Coherent vortex structures in QG turbulence

Quasi-geostrophic Approximation

Fluid motion ( : stream function)

Time-evolution under the quasi-geostrophic approximation

Potential vorticity

Point vortex systems( : strength, Ri : location )

N ： Brunt-Vaisala frequency f0 ： Coriolis parameter (f0 = sinby at latitude )

N ≒ 1.16×10-2 s-1, f0 ~ 1/(24[h])

Assuming-function like concentration at N points, each vortex is advected by the flow field induced by other vortices.

← negligibly small

Hamiltonian of QG N point vortex system (invariant)

Center of the Vorticity (invariant) : shift the coordinate origin to the vorticity center

Angular momentum (invariant)

Equation of Motion for Quasi-geostrophic Point Vortices

Poisson commutable invariants : P2+Q2, I, H Chaotic behavior (According to the Liouville-Arnol’d theorem)

Canonical equations of motion for the i−th vortex

Computation on MDGRAPE-3 Special Purpose Computer &Time Integration with LSODE (6 significant digits)

t : dimensionless time (in units of the inverse potential vorticity)

Canonical Variables : X, Y (Z=constant)

Length scale is normalized using I .

interaction energy Ri=

Rj=

Computer simulation & Statistical theory

MDGRAPE-3

© http://mdgrape.gsc.riken.jp

The architecture of MDGRAPE-3 is quite similar to its predecessors, the GRAPE (GRAvity PipE) systems. The GRAPE systems are special-purpose computers for gravitational N-body simulations and molecular dynamics simulations developed in University of Tokyo. Its predecessor MDM (Molecular Dynamics Machine), developed by RIKEN, achieved 78 Tflops peak performance in 2000. The GRAPE systems won seven Gordon Bell prizes in total. It consists of 20 force calculation pipelines, a j-particle memory unit, a cell-index controller, a master controller, and a force summation unit.

Number of MFGRAPE-3 Chip : 2Performance : 330 Gflops (peak)Host Interface : PCI-X 64bit/100MHzPower Consumption : 40 W

Specifications

Inclined Spheroidal Vortex

CASL: Resolution 1283

（ Grid Points ：　 1180 ）Point Vortices: N = 2065

CASL:Resolution 2563

（ Grid Points ：　　 9440 ）

Pint Vortices: N = 8005

Unstable ( = 0.125, = 0.96 [rad] )

Interaction between Two Spherical Vortices （ 1 ）

Marginally StableHorizontal distance ： a=3.0Vertical distance ： h=0

Point Vortices (N = 2093)

Dense Distribution

CASL (Resolution : 1283 )

Grid Points ：　 1180

□ Full MergerHorizontal distance ： a=2.6Vertical distance ： h=0

Point Vortices (N = 2093)

Dense Distribution

CASL (Resolution : 1283 )

（ Grid Points 　：　 1180 ）

Interaction between Two Spherical Vortices （ 2 ）

Initial Distribution: Uniform Top Hat (Vertical) 　 and 　　　 Uniform or Normal (Horizontal)

N=2000-8000

Initial Distribution

Energy : E =3.054×10-2, 3.130×10-2

(Uniform , Normal) 　

Highest Multiplicity

Uniform 3.054×10-2

Normal 3.130×10-2

Equilibrium• Time for a single turn

of the vortex cloud O(t = 1)• Equilibrium state after t =10 – 20

Negativetemperature

region

Positivetemperature

region

Zero Inverse Temperature

E=3.054×10-2: Positive Temperature

Zero Inverse Temperature and Positive Temperature

Center region

Lid region

r : the distance from the axis of symmetry

| z | : vertical height

averaging the numerical data during t = 50 - 300.Hoshi, S. and Miyazaki, T., FDR (2008)

Equilibrium Distribution

“End-effect”Uniform: E=3.054×10-2

Positive Temperature

Normal: E=3.130×10-2

Zero inverse Temperature

Vertically Uniform and Radially Gaussian 　

Other Computations

(P,Q)=(0,0)

axisymmetric Case A

(P,Q)=(0,0)

axisymmetric

(P,Q)=(0,0)

axisymmetric

Case B Case C

Case A　　　　↓

　 Case D

　　　　↓

Case E

　↓

Positivetemperature

region

Negativetemperature

region

Case F↓

Case |Zmax| Energy

Case of Zero-Inverse Temperature

1.229 3.130×10-2

Case A 1.222 3.054×10-2

Case B 2.449 2.323×10-2

Case C 4.895 1.628×10-2

Case D 1.222 2.954×10-2

Case E 1.222 3.154×10-2

Case F 1.222 3.291×10-2

　 Case A ～ F 　

Zero inverse temperature

Vertical distribution of vortices

Angular momentum

Energy

Maximum entropy theory : Mean field equation

Two-dimensional point vortices by Kida ( J. P. S. J., 39(5) (1975), pp.1395-1404 )

Lagrange multipliers of the maximum entropy optimization : (z),

)(rF : Probability density function

Maximize

under the constraints of fixed P(z), I and H.

Shannon entropy

Zero inverse temperature: =0

F(r, z)=P(z)/ exp(-r2)

with

Positive temperature: >0

Numerical iteration

assume ：

Maximum entropy theory (converged result )

Good agreement within statistical error bar Statistics becomes worse close to the lids due to the lower particle density.

Drawback of the Maximum Entropy Method Iteration is influenced by the initial values Computationally too intensive

Quest for a “simpler” theory“Patch Model”

for parameter scanning

Maximum EntropyDirect Simulation

Maximum Entropy Theory for the “Patch model”

Patch Model : Approximate distribution by a “Top-hat” in the Maximum Entropy process

z = zFc

rOa(z)

Can the patch model capture the ‘end-effect’ ? Yes⇒

z = 1.20z = 0

Maximum entropy in “cubic case”Patch model

z = 0.636

r

z

O

zi

a(zi)

Potential vorticity and angular momentum of each vertical layer are reproduced.

[ Probability Density Function (PDF) ]

Center region

Lid region

r : the distance from the axis of symmetry

⇒　 Elongated (rectangular) systems are more affected due to the larger distance between “center” and “lids”.

| z | : vertical height

*)(zFav

Normalized vertical height : z*

• Probability density

near the axis(average 0 ≦ r 0.5 , ≦ z*=z/zmax)

averaging the numerical data during t = 50 - 300.

Influence of the box height: Cases B & C (1)

Case B |zmax| ・・・　 TwiceCase C |zmax| ・・・　 4 times

Vertical distribution of angular momentum : J(z*)

As the vertical height increases,

Lid region 　　　　　 ・・・　 More evident ‘end-effect’

Center region 　 ・・・ Two-dimensional

Influence of the box height: Cases B & C (2)

Energy dependence: Cases D, E and F (1)

Energy : Case D < Case A < CaseZIT < Case F

Case D ( E=2.954×10-2 ） Case F ( E=3.291×10-2 )

“Sharper End-effect” “Inverse End-effect”

“End-effect” is more evident for lower energy.“ Inverse End-effect” occurs for higher energy.

Energy dependence: Cases D, E and F (2)

Vertical distribution of the angular momentum

Energy lowered

Physical Interpretation

Energy-decrease (major)Entropy-increseA-momentum-increase

spreads wider (constrained by angular momentum)

F

r

Concentrated more closely around the axis of symmetry

Energy-increase (minor)Entropy-decreaseA-momentum-decrease

The distribution in the center region expands radially for lower energy 　 and shrinks for higher energy.

In order to keep the angular momentum unchanged, the distribution near the lids should shrink for lower energy and should expand for higher energy.

Patch model and physical interpretation

Patch model

Qualitative agreement with the numerical results

Energy lowered

Influence of the vertical vorticity distribution: Parabolic P(z) (1)

Parabolic vertical distribution

・ Center region ・・・ Top hat like・ Lid region ・・・ Gaussian likeVertical distribution of angular momentum : J(z*)

“Patch Model” Numerical results

Equilibrium State: E=Ec

Angular momentum distribution: J(z*) Patch radius: a(z*)

A spheroidal patch of uniform potential vorticity seems to be the maximum entropy state in the limit of lowest energy.

Energy loweredEnergy lowered

Influence of the vertical vorticity distribution: Parablic P(z) (2)

Summary

Vertically uniform (top hat) distribution Axisymmetric equilibrium state At zero inverse temperature “Gaussian in the radial direction” Positive temperature “End-effect near the upper and lower lids”

• Elongated (Rectangular system) is more affected due to 　 the larger distance between “center” and “lids”.

Negative temperature “Inverted End-effect ” Maximum entropy theory: “Good agreement for the direct simulation” Patch model reproduces the vertical distribution of angular momentum.

•The energy increases, if the vortices in the center region concentrate tightly near the axis, whereas the vortices in the lids spread in order to keep the total angular momentum unchanged. Physical interpretation of “End-effect”

Parabolic vertical distribution “End-effect” at the top and bottom is more evident. In the limit of lowest energy, a spheroidal vortex patch of uniform potential vorticity seems to be the maximum entropy state.

Thank you for your attention.

Dr. Hiroki Matsubara (RIKEN, Japan)We thank for computational resources of the RIKEN Super Combined Cluster (RSCC).

All members of Miyazaki Laboratory

Acknowledgements