4. Results

4.1 Time Evolution of an Unstable Elliptical Vortex
4.2 Analysis on the Streamfunction
4.3 Visualization with Passive Contours
4.4 Finite-Time Lyapunov Analysis on Fluid Deformation
4.5 Check by Passive Contour Advection

4.3 Visualization with Passive Contours

To examine the motion of the fluid where the vorticity jump does not exist, passive contours are advected to get intuitive images on the advection and mixing.

Since the velocity at an arbitrary point u( x,y,t ) can be obtained by (2.11), contours put on anywhere can be advected by same numerical method without affecting on the velocity field.

Initial conditions are given as follows:

Vortex patch: same as in section 4.1 ( bold line ).
Passive contours (inside): Ellipses similar to the vortex patch with their major axis shrinking at an interval of 0.1 ( blue to light purple ).
Passive contours (outside): Ellipses similar to the vortex patch with their major axis extending at an interval of 0.1 ( blue to green, yellow, and red ).

Animation 2: Advection of passive contours (GIF animation; 1990kB).
The contour positions are displayed from t = 0.0 to t = 10.0 at an interval of 0.1 . Fluid in y < 0 at initial time is colored light blue.

- Passive contours clearly show the flow in other regions than the patch contour where little interest was put on in the previous analyses.

- The fluid are deformed even while the elliptical vortex keeps its shape. Especially at the edge of the ellipse, the fluid outside the vortex is dragged and stretched out.

- Fluid particles are stretched near the sreamlines containing the stagnation points in the co-rotating frame analyzed in section 4.2 .

- A complex flow pattern is produced after the streamer formation. There are many indications of stretching and folding.

- This stretching is also recognized as the time evolution of the length of each passive contour. ( Note that the area between two contours is preserved due to the nondivergent fluid. )

Figure 3: Time evolution of the passive contour length (click for the expanded figure).

- The contours inside the vortex are hardly stretched out.

- The contours a little outside are stretched at an almost constant rate in the early phase.

- After t = 6.0 , at which the streamer turns up to make the flow complex, the contours are stretched at a higher rate than before.

- This stretching is seen only inside the circle with its radius a little longer than the major axis of the elliptical vortex, and the contour lengths in the outside the circle changes much more slowly.

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