Suppose we have a set of $m$\textcolor{red}{trajectories} evaluated at $T$ time instances $X :=\{x_t^i:=\Phi_tx^i_{t_0}:i=1,..., m;t\in\{t_0, ..., t_{T-1}\}\}$, with $\Phi_t$ the \textcolor{red}{flow map} of a \textcolor{red}{dynamical system} from $t_0$ to $t$. We want to find \textcolor{red}{coherent sets}, which are regions in the phase space ($\mathbb{R}^3$ here) that keep their geometric integrity to a large extent during temporal evolution. This is equivalent to finding \textcolor{red}{tight bundles of trajectories} if the points forming the trajectories are conceived as samples from such a set.
Suppose we have $m$ sample trajectories of a flow $\Phi^t$ in $\mathbb{R}^n$, evaluated at $T$ time instances
We want to find \key{coherent sets}, which are regions in the phase space that keep their geometric integrity to a large extent
during temporal evolution.\\
$\rightarrow$ Find tight bundles of trajectories.
%This is equivalent to finding \textcolor{red}{tight bundles of trajectories} if the points forming the trajectories are conceived as samples from such a set.
If $\Phi_{\epsilon, t}$ is the deterministic flow map plus some small random pertubation with variance $\epsilon$, \textcolor{red}{coherence implies robustness} in the sense of "$\Phi_{\epsilon, t}^{-1}(\Phi_{\epsilon, t}\mathbb{X})\approx\mathbb{X}$", where $\mathbb{X}\subset\mathbb{R}^3$ is a coherent set at $t_0$. We make use of this by constructing a \textcolor{red}{diffusion operator on the data points}, whose eigenvectors give a low-dimensional representation which is used to extract coherent sets (\textcolor{red}{spectral clustering}) [1].
The evolution of coherent sets should be \key{insensitive to small perturbations}. Let $D_\epsilon$ denote the diffusion operator with variance $\epsilon$. We call $\mathbb{X}$ coherent if
This heuristic is implemented using tranfer operators on the set of data-points.
%If $\Phi_{\epsilon, t}$ is the deterministic flow map plus some small random pertubation with variance $\epsilon$, \textcolor{red}{coherence implies robustness} in the sense of "$\Phi_{\epsilon, t}^{-1}(\Phi_{\epsilon, t} \mathbb{X}) \approx \mathbb{X}$", where $\mathbb{X} \subset \mathbb{R}^3$ is a coherent set at $t_0$. We make use of this by constructing a \textcolor{red}{diffusion operator on the data points}, whose eigenvectors give a low-dimensional representation which is used to extract coherent sets (\textcolor{red}{spectral clustering}) [1].
QSAS -- a.k.a. high pressure blockings -- are \textcolor{red}{critical features of mid-latitude weather} and, importantly, associated with extreme events. \textcolor{red}{Forecasting skill} is, however, still \textcolor{red}{unsatisfying}. Physically, QSAS are characterized by \textcolor{red}{high stability}.
QSAS -- a.k.a. high pressure blockings -- are \key{critical features of mid-latitude weather} and, importantly, associated with extreme events. \key{Forecasting skill} is, however, still \key{unsatisfying}. Physically, QSAS are characterized by \key{high stability}.
{\footnotesize Left: Blocking example with PV field (shaded), wind $>40\mathrm{\frac{m}{s}}$ (blue), surface pressure contours (yellow) and blocking region (purple). Right: Point vortex model idealizations of $\Omega$- and High-over-Low-Blockings (from [2]).}
{\footnotesize Left: Blocking example with PV field (shaded), wind $>40\mathrm{\frac{m}{s}}$ (blue), surface pressure contours (yellow) and blocking region (purple). Right: Point vortex model idealizations of $\Omega$- and High-over-Low-Blockings (from [2]).}
Assume (at any $t$) the $m$ points are sampled from some manifold $\mathbb{M}_t$. Then, the sum of \textcolor{red}{diffusion distances} can be interpreted as a \textcolor{red}{Monte Carlo approximation} [4] s.t. $S_{\epsilon, t} :=\sum_{i,j}\mathbf{K}_{\epsilon, t}(i,j)\approx\frac{m^2}{\mathrm{vol}(\mathbb{M}_t)^2}\int_{\mathbb{M}_t}\int_{\mathbb{M}_t}\exp(-D(x, y)/\epsilon)\dd x \dd y \approx\frac{m^2}{\mathrm{vol}(\mathbb{M}_t)}(2\pi\epsilon)^{d/2}$, where $\mathbb{R}^d$ is the tangential space of $\mathbb{M}_t$. $S_{\epsilon, t}$ is a measure of the \textcolor{red}{density} of points, with $\lim_{{\epsilon\to0}} S_{\epsilon, t}= m$ and $\lim_{{\epsilon\to\infty}}S_{\epsilon, t}= m^2$. Also $\log(S_{\epsilon, t})\approx\frac{d}{2}\log(\epsilon)+\log(\frac{(2\pi)^{d/2} m^2}{\mathrm{vol}(\mathbb{M}_t)})$.
Assume at time $t$ the points $x_t^i$ points are sampled from some manifold $\mathbb{M}_t$. Then, the sum of the diffusion similarities can be interpreted as an \key{Monte Carlo integral approximation} [4]
\vspace{6.5in}\\
\begin{align*}S_{\epsilon, t} := \sum_{i,j}\mathbf{K}_{\epsilon, t}(i,j) \approx\frac{m^2}{\mathrm{vol}(\mathbb{M}_t)^2}\int_{\mathbb{M}_t}\int_{\mathbb{M}_t}\exp (-\epsilon^{-1}\|x-y\|^2) \dd x \dd y \approx\frac{m^2}{\mathrm{vol}(\mathbb{M}_t)} (\pi\epsilon)^{d/2}
We obtain $d =\dim(\mathbb{M}_t)$ by maximizing $\frac{\dd\log(S_{\epsilon, t})}{\dd\log(\epsilon)}$ and calculate the point density measure $\rho(t) :=\frac{1}{d}\log(\frac{m}{\mathrm{vol}(\mathbb{M}_t)})$.
\end{align*}
\vspace{5.4in}\\
%$S_{\epsilon, t}$ is a measure of the \textcolor{red}{density} of points, with $\lim_{{\epsilon \to 0}} S_{\epsilon, t}= m$ and $\lim_{{\epsilon \to \infty}}S_{\epsilon, t} = m^2$. Also $\log(S_{\epsilon, t}) \approx \frac{d}{2}\log(\epsilon) + \log(\frac{(2\pi)^{d/2} m^2}{\mathrm{vol}(\mathbb{M}_t)})$.
The trajectories tend to be \textcolor{red}{denser after passage} through the QSAS, which is in line with the established notion of blockings as regions characterized by high stability.
For suited $\epsilon$, this yields a linear relationship between $\log(\epsilon)$ and $\log(S_{\epsilon, t})$.
The dimension $d$ and density $\rho(t)$ can be computed from this graph. We use $\ell(t):=\frac{1}{d}\log(\rho(t))$ as a measure of density. The trajectories tend to be \key{denser after passage} through the QSAS, which is in line with the established notion of blockings as regions characterized by high stability.
%We obtain $d = \dim (\mathbb{M}_t)$ by maximizing $\frac{\dd \log (S_{\epsilon, t})}{\dd \log (\epsilon)}$ and calculate the point density measure $\rho (t) := \frac{1}{d}\log (\frac{m}{\mathrm{vol}(\mathbb{M}_t)})$.
}
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@@ -90,39 +107,42 @@ The trajectories tend to be \textcolor{red}{denser after passage} through the QS
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@@ -90,39 +107,42 @@ The trajectories tend to be \textcolor{red}{denser after passage} through the QS
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Flow across the boundary of $\mathbb{M}_t$ is possible. Therefore, \textcolor{red}{Dirichlet boundary conditions}$\mathbf{P}_{\epsilon, t}(i, j)=0\forall x^i_t \text{ or } x^j_t \in\partial\mathbb{M}_t$ have to be applied. Boundary points are detected with $\alpha$\texttt{-shapes} [3] using $\alpha=0.001\mathrm{km}^{-1}\approx\mathcal{O}(\epsilon^{-1/2})$.
Flow across the boundary $\partial\mathbb{M}_t$ is possible. We apply \key{Dirichlet boundary conditions}
$$\mathbf{P}_{\epsilon, t}(i, j)=0\quad\forall x^i_t \text{ or } x^j_t \in\partial\mathbb{M}_t$$
The boundary is detected with $\alpha$\texttt{-shapes} [3] using $\alpha\approx\mathcal{O}(\epsilon^{-1/2})$.
To study the \textcolor{red}{airstreams entering} the QSAS, we construct $\mathbf{Q}_{\epsilon}$for $t_0, ..., t_{72}$. Eigenvaluedecay for various $\epsilon$ reveals a \textcolor{red}{spectral gap} after the 6\textsuperscript{th} eigenvalue, so we expect to find 6 coherent sets.
To study the \key{airstreams entering} the QSAS, sample $x_0^i$ from the QSAS. Construct $\mathbf{Q}_{\epsilon}$and compute its dominant eigenvalues.\\
\vspace{14.6cm}
$\rightarrow$ Find spectral gap.\\
The clustered trajectories differ not only wrt their \textcolor{red}{geometric}, but also in their \textcolor{red}{dynamic properties}. Specifically, two coherent sets are
Using $k$-means, group the points into $k$ clusters, using the$k-1$ dominant eigenvectors. \vspace{14.2cm}\\
\parbox{.42\boxwidth}{
\parbox{.42\boxwidth}{
identified, which feature strong vertical, cross-isentropical motion (latent heating) and \textcolor{red}{stabilize the QSAS} via injection of low-PV air masses. Objectively identifying this process presents an advancement of existing research [5].}
The clustered trajectories differ not only wrt their \key{geometric}, but also in their \key{dynamic properties}. Specifically, two coherent sets are identified, which feature strong vertical, cross-isentropical motion (latent heating) and \key{stabilize the QSAS} via injection of low-PV air masses. Objectively identifying this process presents an advancement of existing research [5].}
}
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\hspace*{0.5cm} [5] \textsc{Pfahl, Stephan, C. Schwierz, M. Croci-Maspoli, C. M. Grams, and H. Wernli}, \textit{Importance of Latent Heat Release in Ascending Air Streams for Atmospheric Blocking}, Nature Geoscience 8, no. 8, 2015
\hspace*{0.5cm} [5] \textsc{Pfahl, Stephan et al.}, \textit{Importance of Latent Heat Release in Ascending Air Streams for Atmospheric Blocking}, Nature Geoscience 8, no. 8, 2015