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

## 2.4 Finite-Time Lyapunov Analysis on Fluid Deformation

Yoden and Nomura(1993) did a finite-time Lyapunov analysis in phase space. The same analysis is done here in physical space to introduce a linear measure of deformation of a fluid percel.Trajectory of a fluid particle is given as a solution of the equation

(2.16)

Time evolution of a small perturbation from the solution can be described by a tangent linear equation of (2.16):

(2.17)

whereJis Jacobian matrix ofuatx( t ).By integrating (2.17) from t

_{0}to t_{0}+t,

(2.18)

where the matrixMis the resolvent.

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