Integrated Computation of Finite Time Lyapunov Exponent During ...

Definition and typical computation of the FTLE
1 Release a grid of tracers and see where they go: This gives us the The flow map,
Φ(x 0 , t 0 , )
∫ t1
Φ t 1
t0
(x 0 , t 0 ) = x 0 + u(x(τ), τ)dτ (1)
t 0
2 Differentiate the flow map W.R.T. x 0 : Then, compose the right Cauchy-Green deformation
tensor, ∆
⎡
⎤∗ ⎡ ⎤
∆ t 1
t0
(x 0 , t 0 ) = ⎣ dΦt 1
t0
(x 0 , t 0 )
⎦ ⎣ dΦt 1
t0
(x 0 , t 0 )
⎦ , (2)
dx 0
dx 0
3 Determine the largest eigenvalue of this tensor. This is then used to compute the
(maximum) FTLE field.
√
σ t 1
1
t0
(x 0 , t 0 ) =
|t 1 − t 0 | log λ max (∆ t 1
t0
(x 0 , t 0 )) (3)
4 Detect FTLE ridges visually or analytically: Forward time ridges ⇒ repelling LCS
candidates. Backward time ridges ⇒ attracting LCS candidates. Recent, more exact
theory [Haller, 2011]
Note: Separate computations for forward (t 1 > t 0 ) and backward (t 1 < t 0 ) time
Note: Let T be the absolute value of the integration time, |t 1 − t 0 |
We can then write forward and backward time T flow maps as Φ t 0+T
t and Φ t 0−T
0
t 0
Finn & Apte FTLE & DNS Integration 4