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45 | 46 | <ul class="current"> |
46 | 47 | <li class="toctree-l1"><a class="reference internal" href="1_requirements.html">Requirements and Quickstart</a></li> |
47 | 48 | <li class="toctree-l1 current"><a class="current reference internal" href="#">Theory of FTLE and LCS</a><ul> |
| 49 | +<li class="toctree-l2"><a class="reference internal" href="#mathematical-framework">Mathematical Framework</a></li> |
48 | 50 | <li class="toctree-l2"><a class="reference internal" href="#steady-lcs">Steady LCS</a></li> |
49 | 51 | <li class="toctree-l2"><a class="reference internal" href="#unsteady-lcs">Unsteady LCS</a></li> |
50 | 52 | </ul> |
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84 | 86 |
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85 | 87 | <section id="theory-of-ftle-and-lcs"> |
86 | 88 | <span id="theory"></span><h1>Theory of FTLE and LCS<a class="headerlink" href="#theory-of-ftle-and-lcs" title="Link to this heading"></a></h1> |
87 | | -<p>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.</p> |
| 89 | +<p>Finite-Time Lyapunov Exponents (FTLE) and Lagrangian Coherent Structures (LCS) provide a |
| 90 | +finite-time description of transport, stirring, and material organization in fluid flows. |
| 91 | +They are especially useful when the velocity field is known only on a bounded time interval, as is |
| 92 | +typically the case for numerical simulations, laboratory measurements, and geophysical products. |
| 93 | +In this setting, one is not primarily interested in instantaneous streamlines, but in the geometry |
| 94 | +of trajectories over a prescribed interval of time. Classical invariant objects, such as stable and |
| 95 | +unstable manifolds, still provide the conceptual background, but the practical questions are |
| 96 | +finite-time and data-driven <a class="reference internal" href="9_references.html#halleryuan2000" id="id1"><span>[HallerYuan2000]</span></a> <a class="reference internal" href="9_references.html#shadden2005" id="id2"><span>[Shadden2005]</span></a> <a class="reference internal" href="9_references.html#haller2015" id="id3"><span>[Haller2015]</span></a>.</p> |
| 97 | +<section id="mathematical-framework"> |
| 98 | +<h2>Mathematical Framework<a class="headerlink" href="#mathematical-framework" title="Link to this heading"></a></h2> |
| 99 | +<p>Consider an unsteady velocity field <span class="math notranslate nohighlight">\(\mathbf{u}(\mathbf{x},t)\)</span> on a domain |
| 100 | +<span class="math notranslate nohighlight">\(U \subset \mathbb{R}^{n}\)</span> and the trajectory equation</p> |
| 101 | +<div class="math notranslate nohighlight"> |
| 102 | +\[\dot{\mathbf{x}} = \mathbf{u}(\mathbf{x},t), \qquad |
| 103 | +\mathbf{x}(t_0) = \mathbf{x}_0.\]</div> |
| 104 | +<p>The associated flow map</p> |
| 105 | +<div class="math notranslate nohighlight"> |
| 106 | +\[\varphi_{t_0}^{t_0+T} : \mathbf{x}_0 \mapsto \mathbf{x}(t_0+T; t_0, \mathbf{x}_0)\]</div> |
| 107 | +<p>transports an initial condition <span class="math notranslate nohighlight">\(\mathbf{x}_0\)</span> from time <span class="math notranslate nohighlight">\(t_0\)</span> to |
| 108 | +<span class="math notranslate nohighlight">\(t_0 + T\)</span>. The deformation of an infinitesimal perturbation |
| 109 | +<span class="math notranslate nohighlight">\(\delta \mathbf{x}_0\)</span> is described to leading order by the deformation gradient |
| 110 | +<span class="math notranslate nohighlight">\(\nabla \varphi_{t_0}^{t_0+T}\)</span> and the right Cauchy-Green strain tensor</p> |
| 111 | +<div class="math notranslate nohighlight"> |
| 112 | +\[\mathbf{C}_{t_0}^{t_0+T}(\mathbf{x}_0) |
| 113 | += |
| 114 | +\left[\nabla \varphi_{t_0}^{t_0+T}(\mathbf{x}_0)\right]^{\mathsf{T}} |
| 115 | +\nabla \varphi_{t_0}^{t_0+T}(\mathbf{x}_0).\]</div> |
| 116 | +<p>This tensor is symmetric and positive definite whenever the flow map is locally invertible. |
| 117 | +Let its eigenpairs satisfy</p> |
| 118 | +<div class="math notranslate nohighlight"> |
| 119 | +\[\mathbf{C}_{t_0}^{t_0+T}\,\boldsymbol{\xi}_i |
| 120 | += |
| 121 | +\lambda_i\,\boldsymbol{\xi}_i, |
| 122 | +\qquad |
| 123 | +0 < \lambda_1 \le \cdots \le \lambda_n.\]</div> |
| 124 | +<p>Then <span class="math notranslate nohighlight">\(\lambda_n = \lambda_{\max}\)</span> gives the largest finite-time stretching factor, while |
| 125 | +<span class="math notranslate nohighlight">\(\boldsymbol{\xi}_n\)</span> indicates the direction of maximal stretching at the initial time. |
| 126 | +Accordingly, the FTLE field is defined by</p> |
| 127 | +<div class="math notranslate nohighlight"> |
| 128 | +\[\sigma_{t_0}^{T}(\mathbf{x}_0) |
| 129 | += |
| 130 | +\frac{1}{|T|} |
| 131 | +\ln \sqrt{\lambda_{\max}\!\left(\mathbf{C}_{t_0}^{t_0+T}(\mathbf{x}_0)\right)} |
| 132 | += |
| 133 | +\frac{1}{2|T|} |
| 134 | +\ln \lambda_{\max}\!\left(\mathbf{C}_{t_0}^{t_0+T}(\mathbf{x}_0)\right).\]</div> |
| 135 | +<p>Hence FTLE measures the average exponential rate at which two initially nearby particles can |
| 136 | +separate over the finite interval <span class="math notranslate nohighlight">\([t_0,t_0+T]\)</span>. For <span class="math notranslate nohighlight">\(T>0\)</span>, the field reveals strongest |
| 137 | +forward-time repulsion; for <span class="math notranslate nohighlight">\(T<0\)</span>, it reveals strongest backward-time repulsion, which is |
| 138 | +equivalently strongest forward-time attraction. Because it is derived from the spectrum of the |
| 139 | +Cauchy-Green tensor, FTLE is an objective scalar diagnostic under time-dependent Euclidean |
| 140 | +changes of frame <a class="reference internal" href="9_references.html#shadden2005" id="id4"><span>[Shadden2005]</span></a> <a class="reference internal" href="9_references.html#haller2015" id="id5"><span>[Haller2015]</span></a>.</p> |
| 141 | +<p>In modern usage, an LCS is not merely a region of visually coherent trajectories, but a |
| 142 | +codimension-one material set that organizes nearby tracer motion. In two-dimensional flows these |
| 143 | +sets are material curves; in three-dimensional flows they are material surfaces. FTLE supplies a |
| 144 | +useful stretching diagnostic, whereas LCS theory seeks the material geometry responsible for that |
| 145 | +stretching <a class="reference internal" href="9_references.html#haller2001" id="id6"><span>[Haller2001]</span></a> <a class="reference internal" href="9_references.html#haller2011" id="id7"><span>[Haller2011]</span></a> <a class="reference internal" href="9_references.html#haller2015" id="id8"><span>[Haller2015]</span></a>.</p> |
| 146 | +</section> |
88 | 147 | <section id="steady-lcs"> |
89 | 148 | <span id="steady"></span><h2>Steady LCS<a class="headerlink" href="#steady-lcs" title="Link to this heading"></a></h2> |
| 149 | +<p>For an autonomous velocity field <span class="math notranslate nohighlight">\(\dot{\mathbf{x}}=\mathbf{u}(\mathbf{x})\)</span>, the flow map forms |
| 150 | +a one-parameter dynamical system, and the relevant organizing structures are the classical |
| 151 | +invariant sets of nonlinear dynamics. If <span class="math notranslate nohighlight">\(\mathbf{x}^{\ast}\)</span> is a hyperbolic equilibrium, its |
| 152 | +stable and unstable manifolds are</p> |
| 153 | +<div class="math notranslate nohighlight"> |
| 154 | +\[W^{s}(\mathbf{x}^{\ast}) |
| 155 | += |
| 156 | +\left\{ |
| 157 | + \mathbf{x}_0 \in U : |
| 158 | + \varphi_{t_0}^{t}(\mathbf{x}_0) \to \mathbf{x}^{\ast} |
| 159 | + \text{ as } t \to +\infty |
| 160 | +\right\},\]</div> |
| 161 | +<div class="math notranslate nohighlight"> |
| 162 | +\[W^{u}(\mathbf{x}^{\ast}) |
| 163 | += |
| 164 | +\left\{ |
| 165 | + \mathbf{x}_0 \in U : |
| 166 | + \varphi_{t_0}^{t}(\mathbf{x}_0) \to \mathbf{x}^{\ast} |
| 167 | + \text{ as } t \to -\infty |
| 168 | +\right\}.\]</div> |
| 169 | +<p>These manifolds are exact material barriers. In two dimensions they act as separatrices that divide |
| 170 | +the flow into dynamically distinct regions; in three dimensions they generalize to invariant curves |
| 171 | +and surfaces depending on the local saddle structure <a class="reference internal" href="9_references.html#halleryuan2000" id="id9"><span>[HallerYuan2000]</span></a> <a class="reference internal" href="9_references.html#haller2001" id="id10"><span>[Haller2001]</span></a>.</p> |
| 172 | +<p>From the FTLE viewpoint, steady hyperbolic manifolds appear as sharp ridges when the integration |
| 173 | +time <span class="math notranslate nohighlight">\(|T|\)</span> is sufficiently long, because initial conditions on opposite sides of the manifold |
| 174 | +experience substantially different fates. In this special setting, FTLE does not introduce a new |
| 175 | +type of coherence so much as provide a finite-time visualization of an already existing invariant |
| 176 | +geometry. Closed streamlines and invariant tori, by contrast, are associated with relatively weak |
| 177 | +net stretching and therefore do not produce the same hyperbolic ridge signature.</p> |
| 178 | +<p>This steady picture remains the correct conceptual limit for simple time-periodic or quasi-periodic |
| 179 | +flows as well: when recurrent motion is genuinely present for all times, LCS theory asymptotically |
| 180 | +connects with classical stable and unstable manifolds, KAM-type barriers, and other invariant |
| 181 | +objects from dynamical systems theory <a class="reference internal" href="9_references.html#halleryuan2000" id="id11"><span>[HallerYuan2000]</span></a> <a class="reference internal" href="9_references.html#shadden2005" id="id12"><span>[Shadden2005]</span></a> <a class="reference internal" href="9_references.html#haller2015" id="id13"><span>[Haller2015]</span></a>.</p> |
90 | 182 | </section> |
91 | 183 | <section id="unsteady-lcs"> |
92 | 184 | <span id="unsteady"></span><h2>Unsteady LCS<a class="headerlink" href="#unsteady-lcs" title="Link to this heading"></a></h2> |
| 185 | +<p>Realistic transport problems are rarely autonomous or recurrent. In aperiodic flows, and in data |
| 186 | +sets available only on a finite interval, asymptotic notions such as stable manifolds, unstable |
| 187 | +manifolds, or periodic orbits are generally not available as exact objects. The central question is |
| 188 | +therefore finite-time: which material curves or surfaces organize separation, attraction, folding, |
| 189 | +entrainment, or jet-like transport over the observation window |
| 190 | +<span class="math notranslate nohighlight">\([t_0,t_0+T]\)</span>? <a class="reference internal" href="9_references.html#shadden2005" id="id14"><span>[Shadden2005]</span></a> <a class="reference internal" href="9_references.html#haller2015" id="id15"><span>[Haller2015]</span></a>.</p> |
| 191 | +<p>The classical FTLE-based answer is to identify LCS candidates as ridges of the FTLE field. |
| 192 | +In the influential formulation of Shadden, Lekien, and Marsden, these ridges act as finite-time |
| 193 | +mixing templates: forward FTLE ridges approximate repelling structures, whereas backward FTLE |
| 194 | +ridges approximate attracting structures <a class="reference internal" href="9_references.html#shadden2005" id="id16"><span>[Shadden2005]</span></a>. This viewpoint is practically powerful |
| 195 | +because it reduces complex trajectory behavior to a scalar field derived from the flow map. |
| 196 | +It also explains why FTLE is widely used in oceanography, atmospheric transport, and |
| 197 | +experimental fluid mechanics.</p> |
| 198 | +<p>At the same time, a ridge of FTLE is only a diagnostic signature, not a complete material |
| 199 | +definition of coherence. Strong ridges often mark important transport barriers, but ridge extraction |
| 200 | +depends on the time interval, the spatial resolution, and the particular ridge criterion. Moreover, |
| 201 | +high FTLE values can arise from strong shear without identifying a uniquely most repelling |
| 202 | +material surface. For this reason, later work placed LCS theory on a stricter variational basis |
| 203 | +<a class="reference internal" href="9_references.html#haller2011" id="id17"><span>[Haller2011]</span></a> <a class="reference internal" href="9_references.html#farazmand2012" id="id18"><span>[Farazmand2012]</span></a> <a class="reference internal" href="9_references.html#haller2015" id="id19"><span>[Haller2015]</span></a>.</p> |
| 204 | +<p>In the variational theory of hyperbolic LCS, a repelling LCS is defined as a material surface whose |
| 205 | +finite-time normal repulsion is locally maximal among nearby material surfaces; an attracting LCS |
| 206 | +is obtained as the backward-time counterpart <a class="reference internal" href="9_references.html#haller2011" id="id20"><span>[Haller2011]</span></a>. This formulation links admissible |
| 207 | +LCSs directly to the eigenvalues and eigenvectors of the Cauchy-Green tensor. In two-dimensional |
| 208 | +flows, repelling and attracting LCSs can be constructed from special tensor lines of that field, |
| 209 | +which explains why the eigenstructure of <span class="math notranslate nohighlight">\(\mathbf{C}_{t_0}^{t_0+T}\)</span> is more fundamental than |
| 210 | +the FTLE scalar alone <a class="reference internal" href="9_references.html#farazmand2012" id="id21"><span>[Farazmand2012]</span></a>.</p> |
| 211 | +<p>The broader finite-time theory also distinguishes different transport mechanisms. Hyperbolic LCSs |
| 212 | +govern strongest attraction and repulsion; elliptic LCSs bound vortex-like regions that resist |
| 213 | +filamentation; parabolic LCSs act as generalized jet cores with minimal cross-stream transport |
| 214 | +<a class="reference internal" href="9_references.html#haller2015" id="id22"><span>[Haller2015]</span></a>. FTLE is most naturally tied to the hyperbolic family, because it measures |
| 215 | +exponential stretching. It is therefore an excellent first diagnostic for separation and attraction, but |
| 216 | +it is not by itself a complete theory for all coherent transport barriers.</p> |
| 217 | +<p>One further point is essential in unsteady problems: the time interval is part of the definition of |
| 218 | +the object. Changing <span class="math notranslate nohighlight">\(t_0\)</span> or <span class="math notranslate nohighlight">\(T\)</span> changes the flow map, the Cauchy-Green tensor, and |
| 219 | +hence the resulting FTLE field and LCSs. A sliding-window analysis over a long record is useful, |
| 220 | +but each window defines a distinct finite-time dynamical system; structures extracted from |
| 221 | +different windows should not automatically be interpreted as a single invariant object that simply |
| 222 | +moves in time <a class="reference internal" href="9_references.html#haller2015" id="id23"><span>[Haller2015]</span></a>. This dependence on the chosen interval is not a defect, but a direct |
| 223 | +reflection of the finite-time nature of observed transport.</p> |
| 224 | +<p>In summary, FTLE provides an objective scalar measure of finite-time stretching, while LCS theory |
| 225 | +seeks the material skeleton that gives this stretching geometric meaning. The FTLE-ridge picture |
| 226 | +offers an accessible and often informative first approximation, whereas modern variational theory |
| 227 | +clarifies when those stretching features genuinely act as repelling, attracting, vortical, or |
| 228 | +jet-defining transport barriers <a class="reference internal" href="9_references.html#shadden2005" id="id24"><span>[Shadden2005]</span></a> <a class="reference internal" href="9_references.html#haller2011" id="id25"><span>[Haller2011]</span></a> <a class="reference internal" href="9_references.html#farazmand2012" id="id26"><span>[Farazmand2012]</span></a> <a class="reference internal" href="9_references.html#haller2015" id="id27"><span>[Haller2015]</span></a>.</p> |
93 | 229 | </section> |
94 | 230 | </section> |
95 | 231 |
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