LCS Tutorial: Flux across an LCS

6.2 Flux across an LCS

From the previous page, recall that

L(x, xq) = ± ∥x(t) - xq(x(t), t)∥ ,

where here we have indicated the explicit functional dependencies of each variable. The derivative of L with time gives
$dL ∂ L dx ∂ L dxq ----- = ----- ⋅ -----+ -------⋅ ------ . dt ∂ x dt ∂ x dt q$ (16)

For the spatial gradient of L we have,
$〈 〉 ∂ L ±1 ∂ x ----- = ∇L = --------------- I - -----q , x - x . q ∂ x ∥x - xq ∥ ∂ x$ (17)

However, since x_q is the point on the LCS closest to point x, then the vector x - x_qmust be normal to the LCS, which implies that
$〈 〉 ∂-xq-- , x - xq = 0 ∂ x$ (18)

because the closest point on the LCS, x_q, does not change when the point xis varied in the direction normal to the curve. As an technical point, Eq. (18) is really only guaranteed to hold over an open neighborhood containing the LCS, within which each point xhas a unique and well-defined x_q. This open neighborhood, which we will denote $U ⊆ D$ , always exists because the LCS is a smooth curve, and hence its curvature is finite, implying that there always exists an open set around the LCS such that each xin this open set has a unique x_q. That is, since there are no kinks or discontinuities on the LCS, there always exists an open set $U$ containing the LCS such that for any point $x ∈ U$ , there is a unique point x_qon the LCS which is closer to xthan any other point on the LCS (for a more rigorous proof of the existence of $U$ see Shadden, et al (2005)).

Plugging Eq. (18) into Eq. (17) gives,
$---x---xq---- ∇L = ± x - x ∥ q∥ x---xq- = L = ˆn(xq, t) ,$ (19)

where $ˆn(xq, t)$ denotes a unit vector orthogonal to the LCS at time t. Similarly to above, we also have

-∂-L--- xq-------x- = = - ∇L . ∂ xq L

So we can now rewrite Eq. (16) as
$( ) dL dx dxq ----- = ∇L ⋅ ----- - ------ . dt dt dt$ (20)

On the LCS, the two points xand x_qare equal (and hence L=0); however, we can think of xas a Lagrangian, or material, point which is advected with the fluid, while x_qis a point which moves with the LCS. Therefore when evaluated on the LCS, i.e along L=0, the right-hand side of Eq. (20) represents the difference in the velocity of the two points, projected in the direction normal to the LCS. This component of the difference in velocities is precisely what contributes to particles crossing the LCS. Therefore, the total flux across the LCS is given by
$∫ ∣ dL ∣ Φ(t) = -----∣ ds , dt ∣ LCS L=0$ (21)

where s is some arc length parametrization variable.

While Eq. (21) presents a concise expression for the flux, by itself it is no more useful than just writing the flux in terms of the right hand side of Eq. (20). Typically when studying a problem we are given the velocity field v, from which we obtain the FTLE field. Therefore, to compute the flux, we should determine

∣ dL ∣ -----∣ . dt ∣ L=0

in terms of these two fields (or quantities directly measured from these fields). The following theorem provides such an expression:

Theorem 6.1 Along the LCS,
$∣ 〈 〉 〈 〉 ( ) dL ∣ ˆt, ∇ σ ∂ ˆn 1 -----∣ = ------------- ˆt, ----- - J ˆn + 𝒪 ----- . dt ∣ 〈nˆ, Σ ˆn 〉 ∂ t ∣T ∣ L=0 ◟ ---◝ ◜ ---◞ ◟ ---------◝ ◜---------◞ ◟ ----◝ ◜----◞ term 1 term 2 term 3$ (22)

where $ˆt$ is a unit vector tangent to the LCS and J is the Jacobian derivative of the velocity field v and all terms on the right-hand side are evaluated along L = 0.

The proof of this theorem is somewhat lengthy and will be covered on the next page, however, let us first interpret this result.

TERM 1

The factor

〈 〉 ∣ ˆ ∣ --t,-∇---σ---∣ ∣ , 〈 ˆn, Σ ˆn 〉 ∣ L=0

(23)

measures how well defined the ridge is, and goes to zero for sharper, well-defined ridges. From Def. 5.1 $∇ σ$ evaluated on the LCS must be tangent to the LCS, therefore the numerator of Eq. (23) can be re-written as

〈ˆ 〉 t,∇ σ = ∥∇ σ ∥

For time-independent flows, we will show on the next page that $σ$ is constant along trajectories (asymptotically). Hence for such systems, along any ridge in the FTLE field, $∇ σ = 0$ , and therefore the flux is zero. This is expected since for time-independent flows, streamlines and trajectories coincide (e.g. refer back to the double-gyre example presented earlier). Experience dictates that even for highly time-dependent flows the value of $σ$ does not vary much along ridges in the FTLE field and hence we can expect this term to typically be quite small. More precisely though, taking the derivative of the numerator $∥∇ σ ∥$ in the direction orthogonal to the LCS gives

〈 〉 ˆ 〈 ˆn, ∇ ∥ ∇ σ ∥ 〉 = ˆn, Σ t .

(24)

On the next page, we will show that the right hand side of Eq. (24) is zero, a necessary condition for a minimum. Further computation reveals that the numerator in Eq. (23) is indeed a minimum on the LCS.

Referring to Def. 5.1 we notice that the denominator of Eq. (23) is less than zero and is locally minimized (which implies its absolute value is maximized). Therefore, for a well defined ridge, we expect the magnitude of this term to be large, with a larger value the sharper the ridge. Since the numerator of Eq. (23) is locally minimized and the magnitude of the denominator is locally maximized, this implies that the magnitude of the factor given in Eq. (23) is locally minimized in the direction normal to the LCS, hence this multiplying factor is expected to be small for well defined ridges.

TERM 2

Now consider the term

〈 〉 ∣ ∂ ˆn ∣ ˆt, ----- - J ˆn ∣ ∂ t ∣ L=0

(25)

from Eq.(22). The quantity

〈 〉 ˆ ∂ˆn-- t, ∂t

represents how fast the LCS is locally rotating, which we think of as a Lagrangian rotation. This is easily seen since for an appropriate $θ$ , we can write the unit normal vector to the LCS as

ˆn = (cos θ, sinθ)

and the unit tangent vector as

ˆt = (- sin θ, cos θ)

[ ] 〈 〉 [ ] ˙ ∂-ˆn- - θ sin θ ˆt, ∂t = - sin θ cos θ θ˙ cos θ = θ˙ ,

which is the local rotation rate of the LCS. Now notice $J ˆn$ is the linearized velocity field applied to a unit vector normal to the LCS; and taking the inner product of this with the tangent to the LCS, $ˆt$ , gives the component in the direction of the LCS. That is, the term $〈 〉$ measures how much the local Eulerian field rotates vectors normal to the LCS. We therefore view this term as a local Eulerian rotation rate and hence Eq. (25) is a measure of the difference in the rotation rate of the LCS from the rotation rate induced by the (instantaneous) velocity field.

TERM 3

The last term in Eq. (22) is inversely proportional to the integration time. This tells us that the LCS is more Lagrangian, or behaves more as an invariant manifold, as the integration time T becomes longer.

Most LCS of interest are well-defined, and thus the term (23) is sufficiently small such that the product of terms (23) and (25) is negligible. Additionally, we will see in the Examples that even for relatively short integration times T, ridges in the FTLE field are practically invariant manifolds, with flux measurements on the order of numerical error.

Comments, Questions or Concerns can be sent to shawn@cds.caltech.edu