W04- W04- 2008 2008 : 11, Dec., 2008 e : 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
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HRTW04- 2008 Date : 11, Dec., 2008 Place : Isaac Newton Institute Equilibrium States of Quasi-geostrophic Point Vortices Takeshi Miyazaki, Hidefumi Kimura.
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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)
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
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).