Flow Structures of Jupiter s Great Red Spot Extracted by Using Optical Flow Method

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1 Flow Structures of Jupiter s Great Red Spot Extracted by Using Optical Flow Method Tianshu Liu & Bo Wang Department of Mechanical & Aeronautical Engineering Western Michigan University, Kalamazoo, MI David Choi Department of Planetary Sciences University of Arizona, Tucson, AZ 85721

2 Objective To study the flow structures of Great Red Spot (GRS) from high-resolution velocity fields extracted by using the physicsbased optical flow method from the Galileo 1996 and 2000 images of the GRS

3 Jupiter s Atmosphere Images Taken by Voyager Spacecraft Great Red Spot White Ovals

4 Current GRS Velocity Field Database Manual tracking by Dowling & Ingersoll (1988) from Voyager images

5 Current GRS Velocity Field Database Correlation-based method by Choi et al. (2007) from Galileo 2000 images (G28)

6 Physics-Based Optical Flow Method Modeling of Projection through Fluid Flows Projection onto Image: Geometrical (Perspective) Radiometric

7 Physics-Based Optical Flow Equation When the projected motion equation is expressed in the image coordinates, we obtain a physics-based optical flow equation g / t u 1 2 g f ( x,x,g ) Optical flow has a clear physical meaning: u ( u U 1,u2 ) 12 Diffusion and boundary terms: f ( x,x,g ) D g B( 1, 2 ) Path-averaged velocity: U U d X d X 3 3

8 Variational Formulation Functional for Minimization: J( u ) A g / t g u f dx1dx2 u1 u2 A dx 1 dx 2 Euler-Lagrange Equation: Smooth Constraint 2 g / t ( gu ) f 0 g u Neumann Boundary Condition: u/ n 0 on Numerical solution: Finite difference & Jacob iteration A

9 Galileo 1996 Images of the GRS (G1) t = 0 s t = 4320 s

10 Global Structures of the GRS High-speed, near-elliptical, anti-cyclonical collar Low-speed inner region Relative Vorticity Velocity Vectors (resolution reduced by 4)

11 Zonal Velocity Profile across the GRS along the Minor Axis Cyclonic, counter-rotational motion near the center It will be pointed out that this structure is intrinsic.

12 Meridional Velocity Profile across the GRS along the Major Axis

13 High-Speed Collar

14 High-Speed Collar

15 High-Speed Collar Q T S / 2 tr S S where 2 T S tr

16 Mean Transverse Velocity Profile across Collar Elliptical Coordinate System and Dividing Ellipse G1: Bickley Jet Distribution U( n ) / Umax sech ( n / L0 ) 6 U max 122 m / s L m Arclength: m

17 Cross-Cut-Averaged Vorticity Variation along Collar

18 Power Spectra of Vorticity Variation as the Vortical Structures Travel at Umax/2 Inner Ring s High Frequency: Hz k 2.5 k L m Outer Ring s High Frequency: Hz Most Amplifying Bickley Jet Even Mode: k L k m -1 k L

19 Mechanism for the low-frequency component? St Most Amplifying Kelvin-Helmholtz Instability Mode: f / U Wavelength 7 m m 3.5 Number of structures along the collar m = 3 predicted by Marcus (1993) for the instability of a circular vortex layer Strouhal Number of the Low-Frequency Component: St The reduced Strouhal number could be caused by vortex-merging process (Marcus 1993)

20 Absolute Vorticity as a Function of the Latitude along the Dividing Ellipse Absolute vorticity: f c f c Coriolis frequency: 2 sin Rotational rate of Jupiter: j j Conservation of potential vorticity: ( fc ) / H

21 (resolution reduced by 4) Inner Region

22 Inner Region

23 Inner Region

24 Inner Region Q T S / 2 tr S S where 2 T S tr

25 Isolated Singular Points in Region 1

26 ,, Isolated Singular Points in Region 2 There are 3 cyclonic source nodes: N1 N 9 N10

27 Cyclonic Motion and Source Nodes near the Center The cyclonic, counterrotational motion near the center is associated with the cyclonic spiraling source nodes with positive divergence. N9 N10

28 Cyclonic Source Node & Convention Instability In a Lagrangian sense by following a fluid parcel along z to a source node, div( u ) where f N N 2 t 0 N ( 2 g / [ z(t' T ) )]dt' T 0 / z g / (N is the buoyancy frequency) The surface vorticity at a source node: s 2 o 2 s div( u ) 0 c p Cyclonic Rotation Convection Instability The relative vorticity is intensified by the convection-induced stretching of the planetary vorticity.

29 Topological Constraint on Inner Region of the GRS The Poincare-Bendixson Index Formula: # N # S 1 ( # Z # Z ) / 2 Topological Constraint on the Inner Region of the GRS: # N # S 1 Counting gives 17 nodes and 16 saddles. The Poincare- Bendixson index formula is satisfied, indicating the topological consistency of flow measurement.

30 Consequence of Topological Constraint # N # S 1 (1) There is at least one node in the inner region. (2) The GRS would no longer exist without this node. In other words, the existence of this node is necessary for the maintenance of the GRS. (3) This node is long-lived.

31 Quasi-2D Turbulence: Kinetic Energy Spectrum Kraichnan s 2D turbulence theory (1967) predicts: Forward enstrophy cascade inertial range: E( k ) k 3 Inverse energy cascade inertial range: E( k ) k 5 / 3? Forcing The instability mode in the high-speed collar: k m k Breaking wavenumber m The flow instability in the collar provides random forcing to quasi-2d turbulence in the inner region of the GRS

32 Quasi-2D Turbulence: Enstrophy Spectrum The vortical structures from the collar drift into the inner region, and entrain kinetic energy.

33 Conclusions Global structures of the GRS are extracted, which confirms the cyclonic, counter-rotating motion near the center. The high-speed collar has the mean transverse velocity distribution described by the Bickley jet distribution. The high-frequency component in the collar is close to the most amplifying even mode of the Bickley jet instability. The Kelvin-Helmholtz instability could be responsible to the low-frequency component.

34 Conclusions The topological structures in the inner region of the GRS are revealed, and the topological constraint is given. The topological consistency of the flow structures is confirmed. According to the topological constraint, there is at least one node in the inner region. The existence of this node is necessary for the maintenance of the GRS. The anti-cyclonic high-speed collar and the cyclonic rotation motion near the center of the GRS must coexist.

35 Conclusions Statistically, the flow in the inner region behaves as quasi-2d turbulence that sustains by forcing from the shear-flow instability in the high-speed collar.