Lagrangian coherent structure explained
Lagrangian coherent structures (LCSs) are distinguished surfaces of trajectories in a dynamical system that exert a major influence on nearby trajectories over a time interval of interest.[1] [2] [3] The type of this influence may vary, but it invariably creates a coherent trajectory pattern for which the underlying LCS serves as a theoretical centerpiece. In observations of tracer patterns in nature, one readily identifies coherent features, but it is often the underlying structure creating these features that is of interest.
As illustrated on the right, individual tracer trajectories forming coherent patterns are generally sensitive with respect to changes in their initial conditions and the system parameters. In contrast, the LCSs creating these trajectory patterns turn out to be robust and provide a simplified skeleton of the overall dynamics of the system.[4] [5] The robustness of this skeleton makes LCSs ideal tools for model validation, model comparison and benchmarking. LCSs can also be used for now-casting and even short-term forecasting of pattern evolution in complex dynamical systems.
Physical phenomena governed by LCSs include floating debris, oil spills,[6] surface drifters[7] [8] and chlorophyll patterns[9] in the ocean; clouds of volcanic ash[10] and spores in the atmosphere;[11] and coherent crowd patterns formed by humans[12] and animals.
While LCSs generally exist in any dynamical system, their role in creating coherent patterns is perhaps most readily observable in fluid flows.
General definitions
Material surfaces
and over a time interval
, consider a non-autonomous dynamical system defined through the flow map
\colonx0\mapstox(t,t0,x0)
, mapping initial conditions
into their position
for any time
. If the flow map
is a
diffeomorphism for any choice of
, then for any smooth set
of initial conditions in
, the set
. Borrowing terminology from
fluid dynamics, we refer to the evolving time slice
of the manifold
as a
material surface (see Fig. 1). Since any choice of the initial condition set
yields an invariant manifold
, invariant manifolds and their associated material surfaces are abundant and generally undistinguished in the extended phase space. Only few of them will act as cores of coherent trajectory patterns.
LCSs as exceptional material surfaces
In order to create a coherent pattern, a material surface
should exert a sustained and consistent action on nearby trajectories throughout the time interval
. Examples of such action are attraction, repulsion, or shear. In principle, any well-defined mathematical property qualifies that creates coherent patterns out of randomly selected nearby initial conditions.
attracting over the interval
if all small enough initial perturbations to
are carried by the flow into even smaller final perturbations to
. In classical
dynamical systems theory,
invariant manifolds satisfying such an attraction property over infinite times are called
attractors. They are not only special, but even locally unique in the phase space: no continuous family of attractors may exist.
In contrast, in dynamical systems defined over a finite time interval
, strict inequalities do not define
exceptional (i.e., locally unique) material surfaces. This follows from the
continuity of the flow map
over
. For instance, if a material surface
attracts all nearby trajectories over the time interval
, then so will any sufficiently close other material surface.
Thus, attracting, repelling and shearing material surfaces are necessarily stacked on each other, i.e., occur in continuous families. This leads to the idea of seeking LCSs in finite-time dynamical systems as exceptional material surfaces that exhibit a coherence-inducing property more strongly than any of the neighboring material surfaces. Such LCSs, defined as extrema (or more generally, stationary surfaces) for a finite-time coherence property, will indeed serve as observed centerpieces of trajectory patterns. Examples of attracting, repelling and shearing LCSs are in a direct numerical simulation of 2D turbulence are shown in Fig.2a.
LCSs vs. classical invariant manifolds
of an
autonomous dynamical system. In contrast,LCSs are only required to be invariant in the extended phase space. This means that even if the underlying dynamical system is
autonomous, the LCSs of the system over the interval
will generally be time-dependent, acting as the evolving skeletons of observed coherent trajectory patterns. Figure 2b shows the difference between an attracting LCS and a classic unstable manifold of a saddle point, for evolving times, in an
autonomous dynamical system.
[3] Objectivity of LCSs
Assume that the phase space of the underlying dynamical system is the material configuration space of a continuum, such as a fluid or a deformable body. For instance, for a dynamical system generated by an unsteady velocity field
the open set
of possible particle positions is a material configuration space. In this space, LCSs are material surfaces, formed by trajectories. Whether or not a material trajectory is contained in an LCS is a property that is independent of the choice of coordinates, and hence cannot depend of the observer. As a consequence, LCSs are subject to the basic objectivity (material frame-indifference) requirement of continuum mechanics.
[3] The objectivity of LCSs requires them to be invariant with respect to all possible observer changes, i.e., linear coordinate changes of the form
where
is the vector of the transformed coordinates;
is an arbitrary
proper orthogonal matrix representing time-dependent rotations; and
is an arbitrary
-dimensional vector representing time-dependent translations. As a consequence, any self-consistent LCS definition or criterion should be expressible in terms of quantities that are frame-invariant. For instance, the
strain rate
and the spin tensor
defined as
transform under Euclidean changes of frame into the quantities
A Euclidean frame change is, therefore, equivalent to a similarity transform for
, and hence an LCS approach depending only on the eigenvalues and eigenvectors of
[13] [14] is automatically frame-invariant. In contrast, an LCS approach depending on the eigenvalues of
is generally not frame-invariant.
A number of frame-dependent quantities, such as
,
,
, as well as the averages or eigenvalues of these quantities, are routinely used in heuristic LCS detection. While such quantities may effectively mark features of the instantaneous velocity field
, the ability of these quantities to capture material mixing, transport, and coherence is limited and a priori unknown in any given frame. As an example, consider the linear unsteady fluid particle motion
=v(x,t)=\begin \sin &2+\cos\\-2+\cos& -\sin\endx,
which is an exact solution of the two-dimensional Navier–Stokes equations. The (frame-dependent) Okubo-Weiss criterion classifies the whole domain in this flow as elliptic (vortical) because
({\vertS\vert}2-{\vertW\vert}2)<0
holds, with
referring to the Euclidean matrix norm. As seen in Fig. 3, however, trajectories grow exponentially along a rotating line and shrink exponentially along another rotating line. In material terms, therefore, the flow is hyperbolic (saddle-type) in any frame.
Since Newton’s equation for particle motion and the Navier–Stokes equations for fluid motion are well known to be frame-dependent, it might first seem counterintuitive to require frame-invariance for LCSs, which are composed of solutions of these frame-dependent equations. Recall, however, that the Newton and Navier–Stokes equations represent objective physical principles for material particle trajectories. As long as correctly transformed from one frame to the other, these equations generate physically the same material trajectories in the new frame. In fact, we decide how to transform the equations of motion from an
-frame to a
-frame through a coordinate change
precisely by upholding that trajectories are mapped into trajectories, i.e., by requiring
to hold for all times. Temporal differentiation of this identity and substitution into the original equation in the
-frame then yields the transformed equation in the
-frame. While this process adds new terms (inertial forces) to the equations of motion, these inertial terms arise precisely to ensure the invariance of material trajectories. Fully composed of material trajectories, LCSs remain invariant in the transformed equation of motion defined in the
-frame of reference. Consequently, any self-consistent LCS definition or detection method must also be frame-invariant.
Hyperbolic LCSs
(see. Fig. 4) . Similarly, a
repelling LCS can be defined as a locally strongest repelling material surface. Attracting and repelling LCSs together are usually referred to as
hyperbolic LCSs, as they provide a finite-time generalization of the classic concept of
normally hyperbolic invariant manifolds in
dynamical systems.
Diagnostic approach: Finite-time Lyapunov exponent (FTLE) ridges
Heuristically, one may seek initial positions
of repelling LCSs as set of initial conditions at which infinitesimal perturbations to trajectories starting from
grow locally at the highest rate relative to trajectories starting off of
.
[1] [15] The heuristic element here is that instead of constructing a highly repelling material surface, one simply seeks points of large particle separation. Such a separation may well be due to strong shear along the set of points so identified; this set is not at all guaranteed to exert any normal repulsion on nearby trajectories.
The growth of an infinitesimal perturbation
along a trajectory
is governed by the flow map gradient
. Let
be a small perturbation to the initial condition
, with
, and with
denoting an arbitrary unit vector in
. This perturbation generally grows along the trajectory
into the perturbation vector
{\xi}\epsilon(t1;x0)=\nabla
(x0)\epsilon{\xi}(t0)
. Then the maximum relative stretching of infinitesimal perturbations at the point
can be computed as
where
=\left[\nabla
\right]T\nabla
denotes the right Cauchy–Green strain tensor. One then concludes
[1] that the maximum relative stretching experienced along a trajectory starting from
is just
. As this relative stretching tends to grow rapidly, it is more convenient to work with its growth exponent
})/(t_1-t_0), which is then precisely the finite-time
Lyapunov exponent (FTLE)
Therefore, one expects hyperbolic LCSs to appear as codimension-one local maximizing surfaces (or ridges) of the FTLE field.[1] [16] This expectation turns out to be justified in the majority of cases: time
positions of repelling LCSs are marked by ridges of
. By applying the same argument in backward time,we obtain that time
positions of attracting LCSs are marked by ridges of the backward FTLE field
.
The classic way of computing Lyapunov exponents is solving a linear differential equation for the linearized flow map
. A more expedient approach is to compute the FTLE field from a simple finite-difference approximation to the deformation gradient.
[1] For example, in a three-dimenisonal flow, we launch a trajectory
from any element
of a grid of initial conditions. Using the coordinate representation
for the evolving trajectory
, we approximate the gradient of the flow map as
with a small vector
pointing in the
coordinate direction. For two-dimensional flows, only the first
minor matrix of the above matrix is relevant.
Issues with inferring hyperbolic LCSs from FTLE ridges
FTLE ridges have proven to be a simple and efficient tool for the visualize hyperbolic LCSs in a number of physical problems, yielding intriguing images of initial positions of hyperbolic LCSs in different applications (see, e.g., Figs. 5a-b). However, FTLE ridges obtained over sliding time windows
do not form material surfaces. Thus, ridges of
under varying
cannot be used to
define Lagrangian objects, such as hyperbolic LCSs. Indeed, a locally strongest repelling material surface over
will generally not play the same role over
and hence its evolving position at time
will not be a ridge for
. Nonetheless, evolving second-derivative FTLE ridges
[17] computed over sliding intervals of the form
have been identified by some authors broadly with LCSs. In support of this identification, it is also often argued that the material flux over such sliding-window FTLE ridges should necessarily be small.
[18] [19] [20] The "FTLE ridge=LCS" identification,[18] however, suffers form the following conceptual and mathematical problems:
- Second-derivative FTLE ridges are necessarily straight lines and hence do not exist in physical problems.[21] [22]
- FTLE ridges computed over sliding time windows
with a varying
are generally
not Lagrangian and the flux through them is generally not small.
[23] - In particular, a broadly referenced material flux formula[18] [19] for FTLE ridges is incorrect,[23] even for straight FTLE ridges
- FTLE ridges mark hyperbolic LCS positions, but also highlight surfaces of high shear. A convoluted mixture of both types of surfaces often arises in applications (see Fig. 6 for an example).
- There are several other types LCSs (elliptic and parabolic) beyond the hyperbolic LCSs highlighted by FTLE ridges
Local variational approach: Shrink and stretch surfaces
The local variational theory of hyperbolic LCSs builds on their original definition as strongest repelling or repelling material surfaces in the flow over the time interval
.
[1] At an initial point
, let
denote a unit normal to an initial material surface
(cf. Fig. 6). By the invariance of material lines, the
tangent space
is mapped into the
tangent space of
by the linearized flow map
. At the same time, the image of the normal
normal under
generally does not remain normal to
.Therefore, in addition to a normal component of length
, the advected normal also develops a tangential component of length
(cf. Fig. 7).
If
, then the evolving material surface
strictly repels nearby trajectories by the end of the time interval
. Similarly,
signals that
strictly attracts nearby trajectories along its normal directions. A
repelling (attracting) LCS over the interval
can be defined as a material surface
whose net repulsion
is pointwise maximal (minimal) with respect to perturbationsof the initial normal vector field
. As earlier, we refer to repelling and attracting LCSs collectively as
hyperbolic LCSs.
[1] Solving these local extremum principles for hyperbolic LCSs in two and three dimensions yields unit normal vector fields to which hyperbolic LCSs should everywhere be tangent.[24] [25] [26] The existence of such normal surfaces also requires a Frobenius-type integrability condition in the three-dimensional case. All these results can be summarized as follows:
+ Hyperbolic LCS conditions from local variational theory in dimensions n=2 and n=3 | LCS | Normal vector field of
for
| ODE for
for n=2 | Frobenius-type PDE for
for n=3 |
---|
Attracting |
|
(stretch lines) | \left\langle\nabla x \xi1(x0),\xi1(x0)\right\rangle=0
(stretch surfaces) |
---|
Repelling |
|
(shrink lines) | \left\langle\nabla x \xi3(x0),\xi3(x0)\right\rangle=0
(shrink surfaces) | |
---|
Repelling LCSs are obtained as most repelling shrink lines, starting from local maxima of
. Attracting LCSs are obtained as most attracting stretch lines, starting from local minima of
. These starting points serve are initial positions of exceptional saddle-type trajectories in the flow. An example of the local variational computation of a repelling LCS is shown in FIg. 8. The computational algorithm is available in LCS Tool.
In 3D flows, instead of solving the Frobenius PDE (see table above) for hyperbolic LCSs, an easier approach is to construct intersections of hyperbolic LCSs with select 2D planes, and fit a surface numerically to a large number of such intersection curves. Let us denote the unit normal of a 2D plane
by
. The intersection curve of a 2D repelling LCS surface with the plane
is normal to both
and to the unit normal
of the LCS. As a consequence, an intersection curve
satisfies the ODE
whose trajectories we refer to as reduced shrink lines.[26] (Strictly speaking, this equation is not an ordinary differential equation, given that its right-hand side is not a vector field, but a direction field, which is generally not globally orientable). Intersections of hyperbolic LCSs with
are fastest contracting reduced shrink lines. Determining such shrink lines in a smooth family of nearby
planes, then fitting a surface to the curve family so obtained yields a numerical approximation of a 2D repelling LCS.
[26] Global variational approach: Shrink- and stretchlines as null-geodesics
A general material surface experiences shear and strain in its deformation, both of which depend continuously on initial conditions by the continuity of the map
.The averaged strain and shear within a strip of
-close material lines, therefore, typically show
variation within such a strip.The two-dimensional
geodesic theory of LCSs seeks exceptionally coherent locations where this general trend fails, resulting in an order of magnitude smaller variability in shear or strain than what is normally expected across an
strip. Specifically, the geodesic theory searches for LCSs as special material lines around which
material strips show no
variability either in the material-lineaveraged shear (
Shearless LCSs) or in the material-line averaged strain (
Strainless or
Elliptic LCSs). Such LCSs turn out to be null-geodesics of appropriate
metric tensors defined by the deformation field—hence the name of this theory.
Shearless LCSs are found to be null-geodesics of a Lorentzian metric tensor
defined as
[27]
Such null-geodesics can be proven to be tensorlines of the Cauchy–Green strain tensor, i.e., are tangent to the direction field formed by the strain eigenvector fields
. Specifically,
repelling LCSs are trajectories of
starting from local maxima of the
eigenvalue field. Similarly,
attracting LCSs are trajectories of
starting from local minims of the
eigenvalue field. This agrees with the conclusion of the local variational theory of LCSs. The geodesic approach, however, also sheds more light on the robustness of hyperbolic LCSs: hyperbolic LCSs only prevail as stationary curves of the averaged shear functional under variations that leave their endpoints fixed. This is to be contrasted with parabolic LCSs (see below), which are also shearless LCSs but prevail as stationary curves to the shear functional even under arbitrary variations. As a consequence, individual trajectories are objective, and statements about the coherent structures they form should also be objective.
A sample application is shown in Fig. 9, where the sudden appearance of a hyperbolic core (strongest attracting part of a stretchline) within the oil spill caused the notable Tiger-Tail instability in the shape of the oil spill.
Elliptic LCSs
Elliptc LCSs are closed and nested material surfaces that act as building blocks of the Lagrangian equivalents of vortices, i.e., rotation-dominated regions of trajectories that generally traverse the phase space without substantial stretching or folding. They mimic the behavior of Kolmogorov–Arnold–Moser (KAM) tori that form elliptic regions in Hamiltonian systems. There coherence can be approached either through their homogeneous material rotation or through their homogeneous stretching properties.
Rotational coherence from the polar rotation angle (PRA)
As a simplest approach to rotational coherence, one may define an elliptic LCS as a tubular material surface along which small material volumes complete the same net rotation over the time intervall
of interest.
[28] A challenge in that in each material volume element, all individual material fibers (tangent vectors to trajectories) perform different rotations.
To obtain a well-defined bulk rotation for each material element, one may employ the unique left and right polar decompositions of the flow gradient in the form
where the proper orthogonal tensor
is called the rotation tensor and the symmetric, positive definite tensors
are called the left stretch tensor and right stretch tensor, respectively.
Since the Cauchy–Green strain tensor can be written asthe local material straining described by the eigenvalues and eigenvectors of
are fully captured by the singular values and singular vectors of the stretch tensors. The remaining factor in the deformation gradient is represented by
, interpreted as the bulk solid-body rotation component of volume elements. In planar motions, this rotation is defined relative to the normal of the plane. In three dimensions, the rotation is defined relative to the axis defined by the eigenvector of
corresponding to its unit eigenvalue. In higher-dimensional flows, the rotation tensor cannot be viewed as a rotation about a single axis.In two and three dimensions, therefore, there exists a
polar rotation angle (PRA)
that characterises the material rotation generated by
for a volume element centered at the initial condition
. This PRA is well-defined up to multiples of
. For two-dimensional flows, the PRA can be computed from the invariants of
using the formulas
which yield a four-quadrant version of the PRA via the formula
For three-dimensional flows, the PRA can again be computed from the invariants of
from the formulas
where
is the
Levi-Civita symbol,
is the eigenvector corresponding to the unit eigenvector of the matrix
\right]jk=\left\langle\xij,\nabla
\xik\right\rangle/{\sqrt{λk
}}.
The time
positions of elliptic LCSs are visualized as tubular level sets of the PRA distribution
. In two-dimensions, therefore, (polar) elliptic LCSs are simply closed level curves of the PRA, which turn out to be objective. In three dimensions, (polar) elliptic LCSs are toroidal or cylindrical level surfaces of the PRA, which are, however, not objective and hence will generally change in rotating frames. Coherent Lagrangian vortex boundaries can be visualized as outermost members of nested families of elliptic LCSs. Two- and three-dimensional examples of elliptic LCS revealed by tubular level surfaces of the PRA are shown in Fig. 10a-b.
Rotational coherence from the Lagrangian-averaged vorticity deviation (LAVD)
The level sets of the PRA are objective in two dimensions but not in three dimensions. An additional shortcoming of the polar rotation tensor is its dynamical inconsistency: polar rotations computed over adjacent sub-intervals of a total deformation do not sum up to the rotation computed for the full-time interval of the same deformation.[29] Therefore, while
is the closest rotation tensor to
in the
norm over a fixed time interval
, these piecewise best fits do not form a family of rigid-body rotations as
and
are varied. For this reason, rotations predicted by the polar rotation tensor over varying time intervals divert from the experimentally observed mean material rotation of fluid elements.
[30] An alternative to the classic polar decomposition provides a resolution to both the non-objectivity and the dynamic inconsistency issue. Specifically, the Dynamic Polar Decomposition (DPD) of the deformation gradient is also of the form
where the proper orthogonal tensor
is the dynamic rotation tensor and the non-singular tensors
are the left dynamic stretch tensor and right dynamic stretch tensor, respectively. Just as the classic polar decomposition, the DPD is valid in any finite dimension. Unlike the classic polar decomposition, however, the dynamic rotation and stretch tensors are obtained from solving linear differential equations, rather than from matrix manipulations. In particular,
is the deformation gradient of the purely rotational flow
and
is the deformation gradient of the purely straining flow
.
The dynamic rotation tensor
can further be factorized into two deformation gradients: one for a spatially uniform (rigid-body) rotation, and one that deviates from this uniform rotation:
As a spatially independent rigid-body rotation, the proper orthogonal relative rotation tensor
is dynamically consistent, serving as the deformation gradient of the relative rotation flow
In contrast, the proper orthogonal mean rotation tensor
is the deformation gradient of the mean-rotation flow
The dynamic consistency of
implies that the total angle swept by
around its own axis of rotation is dynamically consistent. This
intrinsic rotation angle
is also objective, and turns out to equal to one half of the
Lagrangian-averaged vorticity deviation (
LAVD). The LAVD is defined as the trajectory-averaged magnitude of the deviation of the vorticity from its spatial mean. With the vorticity
\omega(x,t)=\nabla x v(x,t)
and its spatial mean
the LAVD over a time interval
therefore takes the form
with
denoting the (possibly time-varying) domain of definition of the velocity field
. This result applies both in two- and three dimensions, and enables the computation of a well-defined, objective and dynamically consistent material rotation angle along any trajectory.
Outermost complex tubular level curves of the LAVD define initial positions of rotationally coherent material vortex boundaries in two-dimensional unsteady flows (see Fig. 11a). By construction, these boundaries may exhibit transverse filamentation, but any developing filament keeps rotating with the boundary, without global transverse departure form the material vortex. (Exceptions are inviscid flows where such a global departure of LAVD level surfaces from a vortex is possible as fluid elements preserve their material rotation rate for all times). Remarkably, centers of rotationally coherent vortices (defined by local maxima of the LAVD field) can be proven to be the observed centers of attraction or repulsion for finite-size (inertial) particle motion in geophysical flows (see Fig. 11b). In three-dimensional flows, tubular level surfaces of the LAVD define initial positions of two-dimensional eddy boundary surfaces (see Fig. 11c) that remain rotationally coherent over a time intcenter|erval
(see Fig. 11d).
Stretching-based coherence from a local variational approach: Shear surfaces
The local variational theory of elliptic LCSs targets material surfaces that locally maximize material shear over the finite time interval
of interest. This means that at initial point each point
of an elliptic LCS
, the tangent space
is the plane along which the local Lagrangian shear
is maximal (cf. Fig 7).
Introducing the two-dimensional shear vector fieldand the three-dimensional shear normal vector field
the criteria for two- and three-dimensional elliptic LCSs can be summarized as follows:[26] [31]
+Ellipitic LCS conditions from local variational theory in dimensions n=2 and n=3 | LCS | Normal vector field of
for n=3 | ODE for
for n=2 | Frobenius-type PDE for
for n=3 |
---|
Elliptic |
|
(shear lines) | \langle\nabla x n\pm(x0),n\pm(x0)\rangle=0
(shear surfaces) | |
---|
For 3D flows, as in the case of hyperbolic LCSs, solving the Frobenius PDE can be avoided. Instead, one can construct intersections of a tubular elliptic LCS with select 2D planes, and fit a surface numerically to a large number of these intersection curves. As for hyperbolic LCSs above, let us denote the unit normal of a 2D plane
by
. Again, the intersection curves of elliptic LCSs with the plane
are normal to both
and to the unit normal
of the LCS. As a consequence, an intersection curve
satisfies the reduced shear ODE
whose trajectories we refer to as
reduced shear lines.
[26] (Strictly speaking, the reduced shear ODE is not an ordinary differential equation, given that its right-hand side is not a vector field, but a direction field, which is generally not globally orientable). Intersections of tubular elliptic LCSs with
are limit cycles of the reduced shear ODE. Determining such limit cycles in a smooth family of nearby
planes, then fitting a surface to the limit cycle family yields a numerical approximation for 2D shear surface. A three-dimensional example of this local variational computation of an elliptic LCS is shown in Fig. 11.
[26] Stretching-based coherence from a global variational approach: lambda-lines
As noted above under hyperbolic LCSs, a global variational approach has been developed in two dimensions to capture elliptic LCSs as closed stationary curves of the material-line-averaged Lagrangian strain functional.[3] [32] Such curves turn out to be closed null-geodesics of the generalized Green–Lagrange strain tensor family
, where
is a positive parameter (Lagrange multiplier). The closed null-geodesics can be shown to coincide with limit cycles of the family of direction fields
Note that for
, the direction field
coincides with the direction field
for shearlines obtained above from the local variational theory of LCSs.
Trajectories of
are referred to as
-lines. Remarkably, they are initial positions of material lines that are
infinitesimally uniformly stretching under the flow map
. Specifically, any subset of a
-line is stretched by a factor of
between the times
and
. As an example, Fig. 13 shows elliptic LCSs identified as closed
-lines within the
Great Red Spot of Jupiter.
[33] Parabolic LCSs
Parabolic LCSs are shearless material surfaces that delineate cores of jet-type sets of trajectories. Such LCSs are characterized by both low stretching (because they are inside a non-stretching structure), but also by low shearing (because material shearing is minimal in jet cores).
Diagnostic approach: Finite-time Lyapunov exponents (FTLE) trenches
Since both shearing and stretching are as low as possible along a parabolic LCS, one may seek initial positions of such material surfaces as trenches of the FTLE field
.
[34] [35] A geophysical example of a parabolic LCS (generalized jet core) revealed as a trench of the FTLE field is shown in Fig. 14a.
Global variational approach: Heteroclinic chains of null-geodesics
. In contrast to hyperbolic LCSs, however, parabolic LCSs satisfy more robust boundary conditions: they remain stationary curves of the material-line-averaged shear functional even under variations to their endpoints. This explains the high degree of robustness and observability that jet cores exhibit in mixing. This is to be contrasted with the highly sensitive and fading footprint of hyperbolic LCSs away from strongly hyperbolic regions in diffusive tracer patterns.
Under variable endpoint boundary conditions, initial positions of parabolic LCSs turn out to be alternating chains of shrink lines and stretch lines that connect singularities of these line fields.[3] [27] These singularities occur at points where
, and hence no infinitesimal deformation takes place between the two time instances
and
. Fig. 14b shows an example of parabolic LCSs in Jupiter's atmosphere, located using this variational theory.
[33] The chevron-type shapes forming out of circular material blobs positioned along the jet core is characteristic of tracer deformation near parabolic LCSs.
Software packages for LCS computations
Particle advection and Finite-Time Lyapunov Exponent calculation:
See also
Further reading
- Salman . H. . Hesthaven . J. S. . Warburton . T. . Haller . G. . Predicting transport by Lagrangian coherent structures with a high-order method . 10.1007/s00162-006-0031-0 . Theoretical and Computational Fluid Dynamics . 21 . 1 . 39–58 . 2006 . 2007ThCFD..21...39S . 11159109 .
- Green . M. A. . Rowley . C. W. . Haller . G. . Detection of Lagrangian coherent structures in three-dimensional turbulence . 10.1017/S0022112006003648 . Journal of Fluid Mechanics . 572 . 111–120 . 2007 . 2007JFM...572..111G . 10.1.1.506.7756 . 1074531 .
Notes and References
- 10.1016/S0167-2789(00)00142-1. Lagrangian coherent structures and mixing in two-dimensional turbulence. Physica D: Nonlinear Phenomena. 147. 3–4. 352. 2000. Haller . G.. Yuan . G.. 2000PhyD..147..352H.
- 10.1063/PT.3.1886. Lagrangian coherent structures: The hidden skeleton of fluid flows. Physics Today. 66. 2. 41. 2013. Peacock . T. . Haller . G. . 2013PhT....66b..41P.
- 10.1146/annurev-fluid-010313-141322. Lagrangian Coherent Structures. Annual Review of Fluid Mechanics. 47. 1. 137–162. 2015. Haller . G. . 2015AnRFM..47..137H.
- 10.1016/j.physd.2013.05.003. Real-time prediction of atmospheric Lagrangian coherent structures based on forecast data: An application and error analysis. Physica D: Nonlinear Phenomena. 258. 47–60. 2013. Bozorgmagham . A. E. . Ross . S. D. . Schmale . D. G. . 2013PhyD..258...47B.
- 10.1016/j.cnsns.2014.07.011. Atmospheric Lagrangian coherent structures considering unresolved turbulence and forecast uncertainty. Communications in Nonlinear Science and Numerical Simulation. 22. 1–3. 964–979. 2015. Bozorgmagham . A. E. . Ross . S. D. . 2015CNSNS..22..964B.
- 10.1073/pnas.1118574109. 22411824. Forecasting sudden changes in environmental pollution patterns. Proceedings of the National Academy of Sciences. 109. 13. 4738–4743. 2012. Olascoaga . M. J.. Haller . G.. 2012PNAS..109.4738O. 3323984. free.
- 10.1029/2011GL048815. Surface coastal circulation patterns by in-situ detection of Lagrangian coherent structures. Geophysical Research Letters. 38. 17. 2011. Nencioli . F.. d'Ovidio . F.. Doglioli . A. M.. Petrenko . A. A.. n/a. 2011GeoRL..3817604N. free.
- 10.1002/2013GL058624. Drifter motion in the Gulf of Mexico constrained by altimetric Lagrangian coherent structures. Geophysical Research Letters. 40. 23. 6171. 2013. Olascoaga . M. J.. Beron-Vera . F. J.. Haller . G.. Triñanes . J.. Iskandarani . M.. Coelho . E. F.. Haus . B. K.. Huntley . H. S.. Jacobs . G.. Kirwan . A. D.. Lipphardt . B. L.. Özgökmen . T. M.. h. m. Reniers . A. J.. Valle-Levinson . A.. 2013GeoRL..40.6171O. free.
- 10.1029/2012GL051246. The impact of advective transport by the South Indian Ocean Countercurrent on the Madagascar plankton bloom. Geophysical Research Letters. 39. 6. 2012. Huhn . F.. von Kameke . A.. Pérez-Muñuzuri . V.. Olascoaga . M. J.. Beron-Vera . F. J.. n/a. 2012GeoRL..39.6602H. free.
- 10.1016/j.atmosenv.2011.05.053. Attracting structures in volcanic ash transport. Atmospheric Environment. 48. 230–239. 2012. Peng . J. . Peterson . R. . 2012AtmEn..48..230P.
- 10.1063/1.3624930. 21974657. Lagrangian coherent structures are associated with fluctuations in airborne microbial populations. Chaos: An Interdisciplinary Journal of Nonlinear Science. 21. 3. 033122. 2011. Tallapragada . P.. Ross . S. D.. Schmale . D. G.. 2011Chaos..21c3122T. 10919/24411. free.
- Book: 10.1109/CVPR.2007.382977. A Lagrangian Particle Dynamics Approach for Crowd Flow Segmentation and Stability Analysis. 2007 IEEE Conference on Computer Vision and Pattern Recognition. 1. 2007. Ali . S. . Shah . M. . 978-1-4244-1179-5. 10.1.1.63.4342. 8190391.
- 10.1063/1.1403336. Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence. Physics of Fluids. 13. 11. 3365–3385. 2001. Haller . G.. 2001PhFl...13.3365H.
- 10.1017/S0022112004002526. An objective definition of a vortex. Journal of Fluid Mechanics. 525. 1–26. 2005. Haller . G.. 2005JFM...525....1H. 12867087.
- 10.1016/S0167-2789(00)00199-8. Distinguished material surfaces and coherent structures in three-dimensional fluid flows. Physica D: Nonlinear Phenomena. 149. 4. 248–277. 2001. Haller . G.. 2001PhyD..149..248H. 10.1.1.331.6383.
- 10.1063/1.1477449. Lagrangian coherent structures from approximate velocity data. Physics of Fluids. 14. 6. 1851–1861. 2002. Haller . G.. 2002PhFl...14.1851H.
- Shadden . S. C. . Lekien . F. . Marsden . J. E. . Jerrold_E._Marsden. 10.1016/j.physd.2005.10.007 . Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows . Physica D: Nonlinear Phenomena . 212 . 3–4 . 271–304. 2005 . 2005PhyD..212..271S .
- 10.1063/1.2740025. Lagrangian coherent structures in n-dimensional systems. Journal of Mathematical Physics. 48. 6. 065404. 2007. Lekien . F. . Shadden . S. C. . Marsden . J. E. . 2007JMP....48f5404L.
- Web site: LCS Tutorial . Shadden, S.C. . 2005 . dead . https://web.archive.org/web/20120723092825/http://mmae.iit.edu/shadden/LCS-tutorial/overview.html . 2012-07-23 .
- 10.1063/1.3270049. A ridge tracking algorithm and error estimate for efficient computation of Lagrangian coherent structures. Chaos: An Interdisciplinary Journal of Nonlinear Science. 20. 1. 017504. 2010. Lipinski . D. . Mohseni . K. . 20370294. 2010Chaos..20a7504L.
- 10.1016/j.physd.2012.05.006. Second derivative ridges are straight lines and the implications for computing Lagrangian Coherent Structures. Physica D: Nonlinear Phenomena. 241. 18. 1475. 2012. Norgard . G. . Bremer . P. T. . 2012PhyD..241.1475N.
- Book: 10.1007/978-3-642-23175-9_15. Ridge Concepts for the Visualization of Lagrangian Coherent Structures. Topological Methods in Data Analysis and Visualization II. 221. Mathematics and Visualization. 2012. Schindler . B. . Peikert . R. . Fuchs . R. . Theisel . H. . 978-3-642-23174-2.
- 10.1016/j.physd.2010.11.010. A variational theory of hyperbolic Lagrangian Coherent Structures. Physica D: Nonlinear Phenomena. 240. 7. 574–598. 2011. Haller . G. . 2011PhyD..240..574H.
- 10.1016/j.physd.2011.09.013. Erratum and addendum to "A variational theory of hyperbolic Lagrangian coherent structures" [Physica D 240 (2011) 574–598]. Physica D: Nonlinear Phenomena. 241. 4. 439. 2012. Farazmand . M. . Haller . G. . 2012PhyD..241..439F. free.
- 10.1063/1.3690153. Computing Lagrangian coherent structures from their variational theory. Chaos: An Interdisciplinary Journal of Nonlinear Science. 22. 1. 013128. 2012. Farazmand . M. . Haller . G. . 22463004. 2012Chaos..22a3128F.
- 10.1016/j.physd.2014.01.007. Hyperbolic and elliptic transport barriers in three-dimensional unsteady flows. Physica D: Nonlinear Phenomena. 273-274. 46–62. 2014. Blazevski . D. . Haller . G. . 2014PhyD..273...46B. 1306.6497 . 44079483.
- 10.1016/j.physd.2014.03.008. Shearless transport barriers in unsteady two-dimensional flows and maps. Physica D: Nonlinear Phenomena. 278-279. 44–57. 2014. Farazmand . M. . Blazevski . D. . Haller . G. . 2014PhyD..278...44F. 1308.6136 . 44141020.
- Farazmand. Mohammad. Haller. George. Polar rotation angle identifies elliptic islands in unsteady dynamical systems. Physica D: Nonlinear Phenomena. 2016. 315. 1–12. 10.1016/j.physd.2015.09.007. 1503.05970 . 2016PhyD..315....1F . 44190280.
- Dynamic rotation and stretch tensors from a dynamic polar decomposition. Journal of the Mechanics and Physics of Solids. 86. 70–93. 10.1016/j.jmps.2015.10.002. Haller. George. 1510.05367 . 2016JMPSo..86...70H . 2016. 44073994.
- Haller. George. Hadjighasem. Alireza. Farazmand. Mohammad. Huhn. Florian. Defining Coherent Vortices Objectively from the Vorticity. Journal of Fluid Mechanics. 795. 136–173. 10.1017/jfm.2016.151. 1506.04061 . 2016JFM...795..136H . 2016. 44191318.
- 10.1016/j.physd.2012.06.012. Geodesic theory of transport barriers in two-dimensional flows. Physica D: Nonlinear Phenomena. 241. 20. 1680. 2012. Haller . G. . Beron-Vera . F. J. . 2012PhyD..241.1680H.
- 10.1017/jfm.2013.391. Coherent Lagrangian vortices: The black holes of turbulence. Journal of Fluid Mechanics. 731. R4. 2013. Haller . G.. Beron-Vera . F. J.. 2013JFM...731R...4H. 1308.2352 . 119570289.
- Hadjighasem. A.. Haller. G.. Geodesic Transport Barriers in Jupiter's Atmosphere: A Video-Based Analysis. SIAM Review. 2016. 58. 1. 69–89. 10.1137/140983665. 1408.5594. 31876317.
- 10.1063/1.3271342. 20370304. Invariant-tori-like Lagrangian coherent structures in geophysical flows. Chaos: An Interdisciplinary Journal of Nonlinear Science. 20. 1. 017514. 2010. Beron-Vera . F. J. . Olascoaga . M. A. J. . Brown . M. G. . KoçAk . H. . Rypina . I. I. . 2010Chaos..20a7514B.
- 10.1175/JAS-D-11-084.1. Zonal Jets as Meridional Transport Barriers in the Subtropical and Polar Lower Stratosphere. Journal of the Atmospheric Sciences. 69. 2. 753. 2012. Beron-Vera . F. J. . Olascoaga . M. A. J. . Brown . M. G. . Koçak . H. . 2012JAtS...69..753B. free.
- Web site: ManGen 1.4.4 . https://web.archive.org/web/20090107040806/http://www.lekien.com/~francois/software/mangen . 2009-01-07 . Lekien . Francois . Coulliette . Chad .
- Web site: LCS MATLAB Kit . Dabiri . John O. .
- Web site: FlowVC . Shadden . Shawn C. .
- Web site: cuda_ftle . https://web.archive.org/web/20110517090602/http://www.its.caltech.edu/~raymondj/LCS . 2011-05-17 . Jimenez . Raymond . Vankerschaver . Joris .
- Web site: CTRAJ . Mills . Peter .
- Web site: Newman . https://web.archive.org/web/20100613064013/http://www.cds.caltech.edu/~pdutoit/Philip_du_Toit/Software.html . 2010-06-13 . Du Toit . Philip C. .
- Ameli . Siavash . Desai . Yogin . Shadden . Shawn C.. Peer-Timo Bremer. Ingrid Hotz. Valerio Pascucci. Ronald Peikert . 2014 . Development of an Efficient and Flexible Pipeline for Lagrangian Coherent Structure Computation . Topological Methods in Data Analysis and Visualization III . https://web.archive.org/web/20141006154923/http://shaddenlab.berkeley.edu/uploads/amelidesaishadden14.pdf. 2014-10-06 . Mathematics and Visualization . 201–215 . . 10.1007/978-3-319-04099-8_13 . 978-3-319-04099-8 . 1612-3786 .