2. Theory |

2.1 Contour Dynamics

2.2 Equations of Cotour Dynamics

2.3 Theory of an Elliptical Vortex

2.4 Finite-Time Lyapunov Analysis on Fluid Deformation

## 2.1 Contour Dynamics

Motion in two-dimensional nondivergent perfect fluid can be described using Lagrangian conservation law of vorticity:

(2.1)

whereu(x,t ) is the velocity, and Q(x,t ) is the vertical component of the vorticity. The vorticity distribution is assumed to be

(2.2)

for regions D_{j}( j=1,2, ... ,J ) ( D_{j1}ÇD_{j2}= j ) on an infinite plane. Each region is called vortex patch.In this situation,

Such dynamics concerning time evolution of the vortex patch is called contour dynamics.

- The vorticity in each region keeps constant in time due to Lagrangian conservation law.
- Thus time evolution of the system can be obtained by chasing the position of the contours (boundary lines) of the regions.
- Time evolution of the contour positions can be calculated from the contour positions themselves, because the velocity at an arbitrary point can be calculated by them.
- As a result, time evolution of the system is described as the contour deformation.
Advantages:

Disadvantages:

- Reducing two-dimensional problem to one-dimensional line integration decreases calculating time remarkably.
- It is easy to treat unsteady behavior.
- Reliability has been confirmed through conparative experiments with a spectral method, and it is possible to calculate in higher resolution.

- Due to the difficulty in removing the perfect fluid assumption, it is not appropriate for problems in which dissipation process is important.
- It does not have enough reality for application since it is considerably a conceptual model.

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