1. HRTW04- ２００８ Date : 11, Dec., 2008 Place : 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-Communications 1-5-1, Chofugaoka, Chofu, Tokyo, 182-8585, Japan

2. Outline Background The 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 equilibrium Influence of energy on equilibrium Mono-disperse Poly (Bi)-disperse Ex.) Point Vortex System

3. In a rotating stratified fluid, the interactions of isolated coherent vortices dominate the turbulence dynamics. Background (1) Vertical motion is suppressed due to the 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)

4. 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

5. Quasi-geostrophic Approximation Fluid motion ( : stream function ) Time-evolution under the quasi-geostrophic approximation Potential vorticity Point vortex systems( : strength, R i : location ) N ： Brunt-Vaisala frequency f 0 ： Coriolis parameter (f 0 =  sin  by at latitude  ) N ≒ 1.16×10 -2 s -1, f 0 ~ 1/(24[h]) Assuming  -function like concentration at N points, each vortex is advected by the flow field induced by other vortices. ← negligibly small

6. 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 : P 2 + Q 2, 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=Ri= Rj=Rj= Computer simulation & Statistical theory

7. 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 : 2 Performance : 330 Gflops (peak) Host Interface : PCI-X 64bit/100MHz Power Consumption : 40 W Specifications

8. Inclined Spheroidal Vortex CASL : Resolution 128 3 （ Grid Points ： 1180 ） Point Vortices: N = 2065 CASL:Resolution 256 3 （ Grid Points ： 9440 ） Pint Vortices: N = 8005 Unstable (  = 0.125,  = 0.96 [rad] )

9. Interaction between Two Spherical Vortices （ 1 ） Marginally Stable Horizontal distance ： a=3.0 Vertical distance ： h=0 Point Vortices (N = 2093) Dense Distribution CASL (Resolution : 128 3 ) Grid Points ： 1180

10. □ Full Merger Horizontal distance ： a=2.6 Vertical distance ： h=0 Point Vortices (N = 2093) Dense Distribution CASL (Resolution : 128 3 ) （ Grid Points ： 1180 ） Interaction between Two Spherical Vortices （ 2 ）

11. 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 Negative temperature region Positive temperature region Zero Inverse Temperature E=3.054×10 -2 : Positive Temperature

12. 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

13. 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 ↓ Positive temperature region Negative temperature region Case F ↓ Case|Z max |Energy Case of Zero-Inverse Temperature 1.229 3.130×10 -2 Case A1.2223.054×10 -2 Case B2.4492.323×10 -2 Case C4.8951.628×10 -2 Case D1.2222.954×10 -2 Case E1.2223.154×10 -2 Case F1.2223.291×10 -2  Case A ～ F Zero inverse temperature

14. 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),  : Probability density function Maximize under the constraints of fixed P(z), I and H. Shannon entropy

15. Zero inverse temperature:  =0 F(r, z)=P(z)/  exp(-  r 2 ) with  Positive temperature:  >0 Numerical iteration

16. 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 Entropy Direct Simulation

17. Maximum Entropy Theory for the “Patch model” Patch Model : Approximate distribution by a “Top-hat” in the Maximum Entropy process z = z FcFc r O a(z)a(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 zizi a(zi)a(zi) Potential vorticity and angular momentum of each vertical layer are reproduced.

18. Maximum Entropy Theory for the “Patch model” assume: Shannon Entropy Angular Momentum Energy The problem is reduced to find that a(z) which maximizes the entropy under the constraints of the fixed angular momentum and the fixed energy. Taking the first variation with respect to, we obtain the following integral equation,

19. [ 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 Normalized vertical height : z* Probability density near the axis (average 0 ≦ r ≦ 0.5, z*=z/z max ) averaging the numerical data during t = 50 - 300. Influence of the box height: Cases B & C (1) Case B |z max | ・・・ Twice Case C |z max | ・・・ 4 times

20. 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)

21. Energy dependence: Cases D, E and F (1) Energy : Case D < Case A < Case ZIT < Case F Case D ( E=2.954×10 -2 ） Case F ( E=3.291×10 -2 ) “Sharper End-effect” “Inverse End-effect”

22. “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

23. Physical Interpretation Energy-decrease (major) Entropy-increse A-momentum-increase spreads wider (constrained by angular momentum) F r Concentrated more closely around the axis of symmetry Energy-increase (minor) Entropy-decrease A-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

24. Influence of the vertical vorticity distribution: Parabolic P(z) (1) Parabolic vertical distribution ・ Center region ・・・ Top hat like ・ Lid region ・・・ Gaussian like Vertical distribution of angular momentum : J(z*) “Patch Model” Numerical results Equilibrium State: E=E c

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

26. 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.

27. 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