QSAS -- a.k.a.high pressure blockings -- are \key{critical features of mid-latitude weather} and, importantly, associated with extreme events. Forecasting skill is, however, still unsatisfying. Physically, QSAS are characterized by \key{high stability}.
QSAS -- a.k.a.~high pressure blockings -- are \key{critical features of mid-latitude weather} and, importantly, associated with extreme events. Forecasting skill is, however, still unsatisfying. Physically, QSAS are characterized by \key{high stability}.
{\footnotesize Left: Blocking example with PV field (shaded), wind $>40\mathrm{\frac{m}{s}}$ (blue) 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 $>30\mathrm{\frac{m}{s}}$ (blue) and blocking region (purple). Right: Point vortex model idealizations of $\Omega$- and High-over-Low-Blockings (from [2]).}
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]
Assume at time $t$ the points $x_t^i$ are sampled from some manifold $\mathbb{M}_t$. Then, the sum of the diffusion similarities can be interpreted as a \key{Monte Carlo integral approximation} [4]
\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}
\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}
\end{align*}
\end{align*}
%$S_{\epsilon, t}$ 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 $\log(S_{\epsilon, t}) \approx \frac{d}{2}\log(\epsilon) + \log(\frac{(2\pi)^{d/2} m^2}{\mathrm{vol}(\mathbb{M}_t)})$.
%$S_{\epsilon, t}$ 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 $\log(S_{\epsilon, t}) \approx \frac{d}{2}\log(\epsilon) + \log(\frac{(2\pi)^{d/2} m^2}{\mathrm{vol}(\mathbb{M}_t)})$.
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@@ -119,9 +119,9 @@ The boundary is detected with $\alpha$\texttt{-shapes} [3] using $\alpha \approx
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@@ -119,9 +119,9 @@ The boundary is detected with $\alpha$\texttt{-shapes} [3] using $\alpha \approx
To study the \key{airstreams entering} the QSAS, sample $x_0^i$ from the QSAS. Construct $\mathbf{Q}_{\epsilon}$ and compute its dominant eigenvalues.\\
To study the \key{airstreams entering} the QSAS, sample $x_0^i$ from the QSAS. Construct $\mathbf{Q}_{\epsilon}$ and compute its dominant eigenvalues.\\
$\rightarrow$ Find spectral gap.\\
$\rightarrow$ Find spectral gap.\\
Using$k$-means, group the points into $k$ clusters, using the $k-1$ dominant eigenvectors. \vspace{14.2cm}\\
With$k$-means, group the points into $k$ clusters, using the $k-1$ dominant eigenvectors. \vspace{14.2cm}\\
\parbox{.42\boxwidth}{
\parbox{.42\boxwidth}{
The clustered trajectories differ not only wrt their \key{geometric}, but also in their \key{dynamic properties}. Specifically, two coherent sets are identified (red and pink), 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].}
The clustered trajectories differ not only wrt their \key{geometric}, but also in their \key{dynamic properties}. Specifically, two coherent sets are identified (red and pink), which feature strong vertical, cross-isentropic 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].}
