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3 | 3 | Theory of FTLE and LCS |
4 | 4 | ====================== |
5 | 5 |
|
6 | | -This page introduces the theory of Finite-Time Lyapunov Exponents (FTLE) and Lagrangian Coherent Structures (LCS). It covers the mathematical foundations and applications of these concepts in fluid dynamics and other fields. |
| 6 | +Finite-Time Lyapunov Exponents (FTLE) and Lagrangian Coherent Structures (LCS) provide a |
| 7 | +finite-time description of transport, stirring, and material organization in fluid flows. |
| 8 | +They are especially useful when the velocity field is known only on a bounded time interval, as is |
| 9 | +typically the case for numerical simulations, laboratory measurements, and geophysical products. |
| 10 | +In this setting, one is not primarily interested in instantaneous streamlines, but in the geometry |
| 11 | +of trajectories over a prescribed interval of time. Classical invariant objects, such as stable and |
| 12 | +unstable manifolds, still provide the conceptual background, but the practical questions are |
| 13 | +finite-time and data-driven [HallerYuan2000]_ [Shadden2005]_ [Haller2015]_. |
| 14 | + |
| 15 | +Mathematical Framework |
| 16 | +---------------------- |
| 17 | + |
| 18 | +Consider an unsteady velocity field :math:`\mathbf{u}(\mathbf{x},t)` on a domain |
| 19 | +:math:`U \subset \mathbb{R}^{n}` and the trajectory equation |
| 20 | + |
| 21 | +.. math:: |
| 22 | +
|
| 23 | + \dot{\mathbf{x}} = \mathbf{u}(\mathbf{x},t), \qquad |
| 24 | + \mathbf{x}(t_0) = \mathbf{x}_0. |
| 25 | +
|
| 26 | +The associated flow map |
| 27 | + |
| 28 | +.. math:: |
| 29 | +
|
| 30 | + \varphi_{t_0}^{t_0+T} : \mathbf{x}_0 \mapsto \mathbf{x}(t_0+T; t_0, \mathbf{x}_0) |
| 31 | +
|
| 32 | +transports an initial condition :math:`\mathbf{x}_0` from time :math:`t_0` to |
| 33 | +:math:`t_0 + T`. The deformation of an infinitesimal perturbation |
| 34 | +:math:`\delta \mathbf{x}_0` is described to leading order by the deformation gradient |
| 35 | +:math:`\nabla \varphi_{t_0}^{t_0+T}` and the right Cauchy-Green strain tensor |
| 36 | + |
| 37 | +.. math:: |
| 38 | +
|
| 39 | + \mathbf{C}_{t_0}^{t_0+T}(\mathbf{x}_0) |
| 40 | + = |
| 41 | + \left[\nabla \varphi_{t_0}^{t_0+T}(\mathbf{x}_0)\right]^{\mathsf{T}} |
| 42 | + \nabla \varphi_{t_0}^{t_0+T}(\mathbf{x}_0). |
| 43 | +
|
| 44 | +This tensor is symmetric and positive definite whenever the flow map is locally invertible. |
| 45 | +Let its eigenpairs satisfy |
| 46 | + |
| 47 | +.. math:: |
| 48 | +
|
| 49 | + \mathbf{C}_{t_0}^{t_0+T}\,\boldsymbol{\xi}_i |
| 50 | + = |
| 51 | + \lambda_i\,\boldsymbol{\xi}_i, |
| 52 | + \qquad |
| 53 | + 0 < \lambda_1 \le \cdots \le \lambda_n. |
| 54 | +
|
| 55 | +Then :math:`\lambda_n = \lambda_{\max}` gives the largest finite-time stretching factor, while |
| 56 | +:math:`\boldsymbol{\xi}_n` indicates the direction of maximal stretching at the initial time. |
| 57 | +Accordingly, the FTLE field is defined by |
| 58 | + |
| 59 | +.. math:: |
| 60 | +
|
| 61 | + \sigma_{t_0}^{T}(\mathbf{x}_0) |
| 62 | + = |
| 63 | + \frac{1}{|T|} |
| 64 | + \ln \sqrt{\lambda_{\max}\!\left(\mathbf{C}_{t_0}^{t_0+T}(\mathbf{x}_0)\right)} |
| 65 | + = |
| 66 | + \frac{1}{2|T|} |
| 67 | + \ln \lambda_{\max}\!\left(\mathbf{C}_{t_0}^{t_0+T}(\mathbf{x}_0)\right). |
| 68 | +
|
| 69 | +Hence FTLE measures the average exponential rate at which two initially nearby particles can |
| 70 | +separate over the finite interval :math:`[t_0,t_0+T]`. For :math:`T>0`, the field reveals strongest |
| 71 | +forward-time repulsion; for :math:`T<0`, it reveals strongest backward-time repulsion, which is |
| 72 | +equivalently strongest forward-time attraction. Because it is derived from the spectrum of the |
| 73 | +Cauchy-Green tensor, FTLE is an objective scalar diagnostic under time-dependent Euclidean |
| 74 | +changes of frame [Shadden2005]_ [Haller2015]_. |
| 75 | + |
| 76 | +In modern usage, an LCS is not merely a region of visually coherent trajectories, but a |
| 77 | +codimension-one material set that organizes nearby tracer motion. In two-dimensional flows these |
| 78 | +sets are material curves; in three-dimensional flows they are material surfaces. FTLE supplies a |
| 79 | +useful stretching diagnostic, whereas LCS theory seeks the material geometry responsible for that |
| 80 | +stretching [Haller2001]_ [Haller2011]_ [Haller2015]_. |
7 | 81 |
|
8 | 82 | .. _steady: |
9 | 83 |
|
10 | 84 | Steady LCS |
11 | | --------------- |
| 85 | +---------- |
| 86 | + |
| 87 | +For an autonomous velocity field :math:`\dot{\mathbf{x}}=\mathbf{u}(\mathbf{x})`, the flow map forms |
| 88 | +a one-parameter dynamical system, and the relevant organizing structures are the classical |
| 89 | +invariant sets of nonlinear dynamics. If :math:`\mathbf{x}^{\ast}` is a hyperbolic equilibrium, its |
| 90 | +stable and unstable manifolds are |
| 91 | + |
| 92 | +.. math:: |
| 93 | +
|
| 94 | + W^{s}(\mathbf{x}^{\ast}) |
| 95 | + = |
| 96 | + \left\{ |
| 97 | + \mathbf{x}_0 \in U : |
| 98 | + \varphi_{t_0}^{t}(\mathbf{x}_0) \to \mathbf{x}^{\ast} |
| 99 | + \text{ as } t \to +\infty |
| 100 | + \right\}, |
| 101 | +
|
| 102 | +.. math:: |
| 103 | +
|
| 104 | + W^{u}(\mathbf{x}^{\ast}) |
| 105 | + = |
| 106 | + \left\{ |
| 107 | + \mathbf{x}_0 \in U : |
| 108 | + \varphi_{t_0}^{t}(\mathbf{x}_0) \to \mathbf{x}^{\ast} |
| 109 | + \text{ as } t \to -\infty |
| 110 | + \right\}. |
| 111 | +
|
| 112 | +These manifolds are exact material barriers. In two dimensions they act as separatrices that divide |
| 113 | +the flow into dynamically distinct regions; in three dimensions they generalize to invariant curves |
| 114 | +and surfaces depending on the local saddle structure [HallerYuan2000]_ [Haller2001]_. |
| 115 | + |
| 116 | +From the FTLE viewpoint, steady hyperbolic manifolds appear as sharp ridges when the integration |
| 117 | +time :math:`|T|` is sufficiently long, because initial conditions on opposite sides of the manifold |
| 118 | +experience substantially different fates. In this special setting, FTLE does not introduce a new |
| 119 | +type of coherence so much as provide a finite-time visualization of an already existing invariant |
| 120 | +geometry. Closed streamlines and invariant tori, by contrast, are associated with relatively weak |
| 121 | +net stretching and therefore do not produce the same hyperbolic ridge signature. |
| 122 | + |
| 123 | +This steady picture remains the correct conceptual limit for simple time-periodic or quasi-periodic |
| 124 | +flows as well: when recurrent motion is genuinely present for all times, LCS theory asymptotically |
| 125 | +connects with classical stable and unstable manifolds, KAM-type barriers, and other invariant |
| 126 | +objects from dynamical systems theory [HallerYuan2000]_ [Shadden2005]_ [Haller2015]_. |
12 | 127 |
|
13 | 128 | .. _unsteady: |
14 | 129 |
|
15 | 130 | Unsteady LCS |
16 | | --------------- |
| 131 | +------------ |
| 132 | + |
| 133 | +Realistic transport problems are rarely autonomous or recurrent. In aperiodic flows, and in data |
| 134 | +sets available only on a finite interval, asymptotic notions such as stable manifolds, unstable |
| 135 | +manifolds, or periodic orbits are generally not available as exact objects. The central question is |
| 136 | +therefore finite-time: which material curves or surfaces organize separation, attraction, folding, |
| 137 | +entrainment, or jet-like transport over the observation window |
| 138 | +:math:`[t_0,t_0+T]`? [Shadden2005]_ [Haller2015]_. |
| 139 | + |
| 140 | +The classical FTLE-based answer is to identify LCS candidates as ridges of the FTLE field. |
| 141 | +In the influential formulation of Shadden, Lekien, and Marsden, these ridges act as finite-time |
| 142 | +mixing templates: forward FTLE ridges approximate repelling structures, whereas backward FTLE |
| 143 | +ridges approximate attracting structures [Shadden2005]_. This viewpoint is practically powerful |
| 144 | +because it reduces complex trajectory behavior to a scalar field derived from the flow map. |
| 145 | +It also explains why FTLE is widely used in oceanography, atmospheric transport, and |
| 146 | +experimental fluid mechanics. |
| 147 | + |
| 148 | +At the same time, a ridge of FTLE is only a diagnostic signature, not a complete material |
| 149 | +definition of coherence. Strong ridges often mark important transport barriers, but ridge extraction |
| 150 | +depends on the time interval, the spatial resolution, and the particular ridge criterion. Moreover, |
| 151 | +high FTLE values can arise from strong shear without identifying a uniquely most repelling |
| 152 | +material surface. For this reason, later work placed LCS theory on a stricter variational basis |
| 153 | +[Haller2011]_ [Farazmand2012]_ [Haller2015]_. |
| 154 | + |
| 155 | +In the variational theory of hyperbolic LCS, a repelling LCS is defined as a material surface whose |
| 156 | +finite-time normal repulsion is locally maximal among nearby material surfaces; an attracting LCS |
| 157 | +is obtained as the backward-time counterpart [Haller2011]_. This formulation links admissible |
| 158 | +LCSs directly to the eigenvalues and eigenvectors of the Cauchy-Green tensor. In two-dimensional |
| 159 | +flows, repelling and attracting LCSs can be constructed from special tensor lines of that field, |
| 160 | +which explains why the eigenstructure of :math:`\mathbf{C}_{t_0}^{t_0+T}` is more fundamental than |
| 161 | +the FTLE scalar alone [Farazmand2012]_. |
| 162 | + |
| 163 | +The broader finite-time theory also distinguishes different transport mechanisms. Hyperbolic LCSs |
| 164 | +govern strongest attraction and repulsion; elliptic LCSs bound vortex-like regions that resist |
| 165 | +filamentation; parabolic LCSs act as generalized jet cores with minimal cross-stream transport |
| 166 | +[Haller2015]_. FTLE is most naturally tied to the hyperbolic family, because it measures |
| 167 | +exponential stretching. It is therefore an excellent first diagnostic for separation and attraction, but |
| 168 | +it is not by itself a complete theory for all coherent transport barriers. |
| 169 | + |
| 170 | +One further point is essential in unsteady problems: the time interval is part of the definition of |
| 171 | +the object. Changing :math:`t_0` or :math:`T` changes the flow map, the Cauchy-Green tensor, and |
| 172 | +hence the resulting FTLE field and LCSs. A sliding-window analysis over a long record is useful, |
| 173 | +but each window defines a distinct finite-time dynamical system; structures extracted from |
| 174 | +different windows should not automatically be interpreted as a single invariant object that simply |
| 175 | +moves in time [Haller2015]_. This dependence on the chosen interval is not a defect, but a direct |
| 176 | +reflection of the finite-time nature of observed transport. |
| 177 | + |
| 178 | +In summary, FTLE provides an objective scalar measure of finite-time stretching, while LCS theory |
| 179 | +seeks the material skeleton that gives this stretching geometric meaning. The FTLE-ridge picture |
| 180 | +offers an accessible and often informative first approximation, whereas modern variational theory |
| 181 | +clarifies when those stretching features genuinely act as repelling, attracting, vortical, or |
| 182 | +jet-defining transport barriers [Shadden2005]_ [Haller2011]_ [Farazmand2012]_ [Haller2015]_. |
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