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@@ -1 +1 @@
-{"documenter":{"julia_version":"1.11.1","generation_timestamp":"2024-11-08T16:13:06","documenter_version":"1.7.0"}}
+{"documenter":{"julia_version":"1.11.1","generation_timestamp":"2024-11-08T16:14:08","documenter_version":"1.7.0"}}

dev/Examples/index.html

@@ -35,7 +35,7 @@
     colormap = :balance,
     aspect_ratio=1,
     xlims = (-Lx/2, Lx/2),
-    ylims = (-Ly/2, Ly/2))</code></pre><img src="80d9d539.svg" alt="Example block output"/><p>Note that we transpose <span>$\psi$</span> when plotting as <span>$x$</span> corresonds to the first dimension of <span>$\psi$</span>.</p><h2 id="Example-2:-multi-layer-QG"><a class="docs-heading-anchor" href="#Example-2:-multi-layer-QG">Example 2: multi-layer QG</a><a id="Example-2:-multi-layer-QG-1"></a><a class="docs-heading-anchor-permalink" href="#Example-2:-multi-layer-QG" title="Permalink"></a></h2><p>This example considers a 3-layer solution and introduces the concept of active and passive layers. We define an active layer to be a layer with a closed streamline at <span>$x^2 + y^2 = \ell^2$</span> whereas a passive layer has no closed streamlines. Therefore, fluid within the vortex in an active layer remains trapped in the vortex. Conversely, fluid in the passive layer is not trapped in a vortex core but can still be affected through the change in layer thickness associated with the streamfunction in neighbouring layers. Passive layers have <span>$F_i(z) = (\beta_i/U) z$</span> everywhere and hence have no eigenvalue, <span>$K_i$</span>, to solve for. Further, the coefficients within a passive layer are zero though the solution may still be non-zero due to the coefficients in neighbouring layers. Therefore, the corresponding linear system can be simplified by removing rows and columns corresponding to passive layers and solving the reduced system for the active layers only.</p><p>We&#39;ll start by defining some parameters:</p><pre><code class="language-julia hljs">using QGDipoles
+    ylims = (-Ly/2, Ly/2))</code></pre><img src="0747b68f.svg" alt="Example block output"/><p>Note that we transpose <span>$\psi$</span> when plotting as <span>$x$</span> corresonds to the first dimension of <span>$\psi$</span>.</p><h2 id="Example-2:-multi-layer-QG"><a class="docs-heading-anchor" href="#Example-2:-multi-layer-QG">Example 2: multi-layer QG</a><a id="Example-2:-multi-layer-QG-1"></a><a class="docs-heading-anchor-permalink" href="#Example-2:-multi-layer-QG" title="Permalink"></a></h2><p>This example considers a 3-layer solution and introduces the concept of active and passive layers. We define an active layer to be a layer with a closed streamline at <span>$x^2 + y^2 = \ell^2$</span> whereas a passive layer has no closed streamlines. Therefore, fluid within the vortex in an active layer remains trapped in the vortex. Conversely, fluid in the passive layer is not trapped in a vortex core but can still be affected through the change in layer thickness associated with the streamfunction in neighbouring layers. Passive layers have <span>$F_i(z) = (\beta_i/U) z$</span> everywhere and hence have no eigenvalue, <span>$K_i$</span>, to solve for. Further, the coefficients within a passive layer are zero though the solution may still be non-zero due to the coefficients in neighbouring layers. Therefore, the corresponding linear system can be simplified by removing rows and columns corresponding to passive layers and solving the reduced system for the active layers only.</p><p>We&#39;ll start by defining some parameters:</p><pre><code class="language-julia hljs">using QGDipoles
 
 # Set problem parameters
 
@@ -161,7 +161,7 @@
     colormap = :balance,
     aspect_ratio=1,
     xlims = (-Lx/2, Lx/2),
-    ylims = (-Ly/2, Ly/2))</code></pre><img src="cff2e881.svg" alt="Example block output"/><p>If we look at <span>$K$</span>, we find that <span>$K \approx 7.34205$</span> which is not the value we&#39;d expect for the usual dipole solution. Instead, if we look at our plot, we see that it&#39;s a different solution with a mode 2 structure in the radial direction.</p><p>In addition to these wrapper functions, the functions <code>CreateLCD</code> and <code>CreateLCD</code> implement the Lamb-Chaplygin dipole<sup class="footnote-reference"><a id="citeref-2" href="#footnote-2">[2]</a></sup> and Larichev-Reznik dipole<sup class="footnote-reference"><a id="citeref-3" href="#footnote-3">[3]</a></sup> directly using the analytical solution for these cases.</p><h2 id="Example-5:-A-GeophysicalFlows.jl-simulation"><a class="docs-heading-anchor" href="#Example-5:-A-GeophysicalFlows.jl-simulation">Example 5: A GeophysicalFlows.jl simulation</a><a id="Example-5:-A-GeophysicalFlows.jl-simulation-1"></a><a class="docs-heading-anchor-permalink" href="#Example-5:-A-GeophysicalFlows.jl-simulation" title="Permalink"></a></h2><p>This package is designed to be compatible with <code>GeophysicalFlows.jl</code><sup class="footnote-reference"><a id="citeref-4" href="#footnote-4">[4]</a></sup> and provide a means of generating dipolar vortex initial conditions for layered QG and surface QG simulations. Here, we&#39;ll discuss a simple example of how to setup a 1-layer simulation in <code>GeophyiscalFlows.jl</code> using the Lamb-Chaplygin dipole as the initial condition. We&#39;ll also see that, as expected, the dipole retains it&#39;s form during the evolution and hence is a steady solution in a co-moving frame. Let&#39;s begin by defining some parameters for our vortex initial condition and our numerical simulation:</p><pre><code class="language-julia hljs">using GeophysicalFlows, QGDipoles
+    ylims = (-Ly/2, Ly/2))</code></pre><img src="d1946918.svg" alt="Example block output"/><p>If we look at <span>$K$</span>, we find that <span>$K \approx 7.34205$</span> which is not the value we&#39;d expect for the usual dipole solution. Instead, if we look at our plot, we see that it&#39;s a different solution with a mode 2 structure in the radial direction.</p><p>In addition to these wrapper functions, the functions <code>CreateLCD</code> and <code>CreateLCD</code> implement the Lamb-Chaplygin dipole<sup class="footnote-reference"><a id="citeref-2" href="#footnote-2">[2]</a></sup> and Larichev-Reznik dipole<sup class="footnote-reference"><a id="citeref-3" href="#footnote-3">[3]</a></sup> directly using the analytical solution for these cases.</p><h2 id="Example-5:-A-GeophysicalFlows.jl-simulation"><a class="docs-heading-anchor" href="#Example-5:-A-GeophysicalFlows.jl-simulation">Example 5: A GeophysicalFlows.jl simulation</a><a id="Example-5:-A-GeophysicalFlows.jl-simulation-1"></a><a class="docs-heading-anchor-permalink" href="#Example-5:-A-GeophysicalFlows.jl-simulation" title="Permalink"></a></h2><p>This package is designed to be compatible with <code>GeophysicalFlows.jl</code><sup class="footnote-reference"><a id="citeref-4" href="#footnote-4">[4]</a></sup> and provide a means of generating dipolar vortex initial conditions for layered QG and surface QG simulations. Here, we&#39;ll discuss a simple example of how to setup a 1-layer simulation in <code>GeophyiscalFlows.jl</code> using the Lamb-Chaplygin dipole as the initial condition. We&#39;ll also see that, as expected, the dipole retains it&#39;s form during the evolution and hence is a steady solution in a co-moving frame. Let&#39;s begin by defining some parameters for our vortex initial condition and our numerical simulation:</p><pre><code class="language-julia hljs">using GeophysicalFlows, QGDipoles
 
 # Define vortex parameters
 
@@ -212,4 +212,4 @@
 		aspect_ratio=1,
 		xlims = (-Lx/2, Lx/2),
 		ylims = (-Ly/2, Ly/2))
-</code></pre><p>Note that we need to move our fields back to the CPU prior to plotting. The two plots are shown below and are approximately identical. Therefore, we observe that the vortex remains centred at the origin. Over long times, numerical error will result in the vortex moving at a slightly different speed to <code>U</code> and hence moving away from the origin.</p><p><img src="../Ex_5a.png" alt="image"/><img src="../Ex_5b.png" alt="image"/></p><p>See the <code>GeophyiscalFlows.jl</code> documentation <a href="https://fourierflows.github.io/GeophysicalFlowsDocumentation/stable/">here</a> for more details on how to run QG simulations.</p><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a><a href="https://doi.org/10.1017/jfm.2023.87">Johnson, E. R., and M. N. Crowe, 2023, Oceanic dipoles in a surface quasigeostrophic model, J. Fluid Mech., 958, R2</a>.</li><li class="footnote" id="footnote-2"><a class="tag is-link" href="#citeref-2">2</a><a href="https://archive.org/details/hydrodynamics00lamb">Lamb, H., 1932, Hydrodynamics. Cambridge University Press</a>.</li><li class="footnote" id="footnote-3"><a class="tag is-link" href="#citeref-3">3</a><a href="https://www.researchgate.net/publication/248173065_Two-dimensional_solitary_Rossby_waves">Larichev, V.D. &amp; Reznik, G.M., 1976, Two-dimensional solitary Rossby waves, Dokl. Akad. Nauk SSSR, 12–13</a>.</li><li class="footnote" id="footnote-4"><a class="tag is-link" href="#citeref-4">4</a><a href="https://joss.theoj.org/papers/10.21105/joss.03053">Constantinou et al., 2021, GeophysicalFlows.jl: Solvers for geophysical fluid dynamics problems in periodic domains on CPUs &amp; GPUs, JOSS, 6(60), 3053</a>.</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../Installation/">« Installation</a><a class="docs-footer-nextpage" href="../Functions/">List of Functions »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.7.0 on <span class="colophon-date" title="Friday 8 November 2024 16:13">Friday 8 November 2024</span>. Using Julia version 1.11.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+</code></pre><p>Note that we need to move our fields back to the CPU prior to plotting. The two plots are shown below and are approximately identical. Therefore, we observe that the vortex remains centred at the origin. Over long times, numerical error will result in the vortex moving at a slightly different speed to <code>U</code> and hence moving away from the origin.</p><p><img src="../Ex_5a.png" alt="image"/><img src="../Ex_5b.png" alt="image"/></p><p>See the <code>GeophyiscalFlows.jl</code> documentation <a href="https://fourierflows.github.io/GeophysicalFlowsDocumentation/stable/">here</a> for more details on how to run QG simulations.</p><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a><a href="https://doi.org/10.1017/jfm.2023.87">Johnson, E. R., and M. N. Crowe, 2023, Oceanic dipoles in a surface quasigeostrophic model, J. Fluid Mech., 958, R2</a>.</li><li class="footnote" id="footnote-2"><a class="tag is-link" href="#citeref-2">2</a><a href="https://archive.org/details/hydrodynamics00lamb">Lamb, H., 1932, Hydrodynamics. Cambridge University Press</a>.</li><li class="footnote" id="footnote-3"><a class="tag is-link" href="#citeref-3">3</a><a href="https://www.researchgate.net/publication/248173065_Two-dimensional_solitary_Rossby_waves">Larichev, V.D. &amp; Reznik, G.M., 1976, Two-dimensional solitary Rossby waves, Dokl. Akad. Nauk SSSR, 12–13</a>.</li><li class="footnote" id="footnote-4"><a class="tag is-link" href="#citeref-4">4</a><a href="https://joss.theoj.org/papers/10.21105/joss.03053">Constantinou et al., 2021, GeophysicalFlows.jl: Solvers for geophysical fluid dynamics problems in periodic domains on CPUs &amp; GPUs, JOSS, 6(60), 3053</a>.</li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../Installation/">« Installation</a><a class="docs-footer-nextpage" href="../Functions/">List of Functions »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.7.0 on <span class="colophon-date" title="Friday 8 November 2024 16:14">Friday 8 November 2024</span>. Using Julia version 1.11.1.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>