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previews/PR399/barotropic/index.html

@@ -9,4 +9,4 @@
 w_{i+1} &= u_{i-1} + 2\Delta tF(v_i) \\
 u_i &= v_i + \frac{\nu\alpha}{2}(w_{i+1} - 2v_i + u_{i-1}) \\
 v_{i+1} &= w_{i+1} - \frac{\nu(1-\alpha)}{2}(w_{i+1} - 2v_i + u_{i-1})
-\end{aligned}\]</p><p>with the Williams filter parameter <span>$\alpha \in [0.5,1]$</span>. For <span>$\alpha=1$</span> we&#39;re back with the Robert-Asselin filter (the first two lines).</p><p>The Laplacian in the parentheses is often called a <em>displacement</em>, meaning that the filtered value is displaced (or corrected) in the direction of the two surrounding time steps. The Williams filter now also applies the same displacement, but in the opposite direction to the next time step <span>$i+1$</span> as a correction step (line 3 above) for a once-filtered value <span>$v_{i+1}$</span> which will then be twice-filtered by the Robert-Asselin filter on the next iteration. For more details see the referenced publications.</p><p>The initial Euler step (see <a href="#leapfrog">Time integration</a>, Table) is not filtered. Both the the Robert-Asselin and Williams filter are then switched on for all following leapfrog time steps.</p><h2 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h2><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>Geophysical Fluid Dynamics Laboratory, <a href="https://www.gfdl.noaa.gov/idealized-models-with-spectral-dynamics/">Idealized models with spectral dynamics</a></li><li class="footnote" id="footnote-2"><a class="tag is-link" href="#citeref-2">2</a>Geophysical Fluid Dynamics Laboratory, <a href="https://www.gfdl.noaa.gov/wp-content/uploads/files/user_files/pjp/barotropic.pdf">The barotropic vorticity equation</a>.</li><li class="footnote" id="footnote-Robert66"><a class="tag is-link" href="#citeref-Robert66">Robert66</a>Robert, André. “The Integration of a Low Order Spectral Form of the Primitive Meteorological Equations.” Journal of the Meteorological Society of Japan 44 (1966): 237-245.</li><li class="footnote" id="footnote-Asselin72"><a class="tag is-link" href="#citeref-Asselin72">Asselin72</a>ASSELIN, R., 1972: Frequency Filter for Time Integrations. Mon. Wea. Rev., 100, 487–490, doi:<a href="https://doi.org/10.1175/1520-0493(1972)100&lt;0487:FFFTI&gt;2.3.CO;2.">10.1175/1520-0493(1972)100&lt;0487:FFFTI&gt;2.3.CO;2</a></li><li class="footnote" id="footnote-Williams2009"><a class="tag is-link" href="#citeref-Williams2009">Williams2009</a>Williams, P. D., 2009: A Proposed Modification to the Robert–Asselin Time Filter. Mon. Wea. Rev., 137, 2538–2546, <a href="https://doi.org/10.1175/2009MWR2724.1">10.1175/2009MWR2724.1</a>.</li><li class="footnote" id="footnote-Amezcua2011"><a class="tag is-link" href="#citeref-Amezcua2011">Amezcua2011</a>Amezcua, J., E. Kalnay, and P. D. Williams, 2011: The Effects of the RAW Filter on the Climatology and Forecast Skill of the SPEEDY Model. Mon. Wea. Rev., 139, 608–619, doi:<a href="https://doi.org/10.1175/2010MWR3530.1">10.1175/2010MWR3530.1</a>. </li><li class="footnote" id="footnote-Williams2011"><a class="tag is-link" href="#citeref-Williams2011">Williams2011</a>Williams, P. D., 2011: The RAW Filter: An Improvement to the Robert–Asselin Filter in Semi-Implicit Integrations. Mon. Wea. Rev., 139, 1996–2007, doi:<a href="https://doi.org/10.1175/2010MWR3601.1">10.1175/2010MWR3601.1</a>. </li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../grids/">« Grids</a><a class="docs-footer-nextpage" href="../shallowwater/">Shallow water model »</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="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Saturday 7 October 2023 05:51">Saturday 7 October 2023</span>. Using Julia version 1.8.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+\end{aligned}\]</p><p>with the Williams filter parameter <span>$\alpha \in [0.5,1]$</span>. For <span>$\alpha=1$</span> we&#39;re back with the Robert-Asselin filter (the first two lines).</p><p>The Laplacian in the parentheses is often called a <em>displacement</em>, meaning that the filtered value is displaced (or corrected) in the direction of the two surrounding time steps. The Williams filter now also applies the same displacement, but in the opposite direction to the next time step <span>$i+1$</span> as a correction step (line 3 above) for a once-filtered value <span>$v_{i+1}$</span> which will then be twice-filtered by the Robert-Asselin filter on the next iteration. For more details see the referenced publications.</p><p>The initial Euler step (see <a href="#leapfrog">Time integration</a>, Table) is not filtered. Both the the Robert-Asselin and Williams filter are then switched on for all following leapfrog time steps.</p><h2 id="References"><a class="docs-heading-anchor" href="#References">References</a><a id="References-1"></a><a class="docs-heading-anchor-permalink" href="#References" title="Permalink"></a></h2><section class="footnotes is-size-7"><ul><li class="footnote" id="footnote-1"><a class="tag is-link" href="#citeref-1">1</a>Geophysical Fluid Dynamics Laboratory, <a href="https://www.gfdl.noaa.gov/idealized-models-with-spectral-dynamics/">Idealized models with spectral dynamics</a></li><li class="footnote" id="footnote-2"><a class="tag is-link" href="#citeref-2">2</a>Geophysical Fluid Dynamics Laboratory, <a href="https://www.gfdl.noaa.gov/wp-content/uploads/files/user_files/pjp/barotropic.pdf">The barotropic vorticity equation</a>.</li><li class="footnote" id="footnote-Robert66"><a class="tag is-link" href="#citeref-Robert66">Robert66</a>Robert, André. “The Integration of a Low Order Spectral Form of the Primitive Meteorological Equations.” Journal of the Meteorological Society of Japan 44 (1966): 237-245.</li><li class="footnote" id="footnote-Asselin72"><a class="tag is-link" href="#citeref-Asselin72">Asselin72</a>ASSELIN, R., 1972: Frequency Filter for Time Integrations. Mon. Wea. Rev., 100, 487–490, doi:<a href="https://doi.org/10.1175/1520-0493(1972)100&lt;0487:FFFTI&gt;2.3.CO;2.">10.1175/1520-0493(1972)100&lt;0487:FFFTI&gt;2.3.CO;2</a></li><li class="footnote" id="footnote-Williams2009"><a class="tag is-link" href="#citeref-Williams2009">Williams2009</a>Williams, P. D., 2009: A Proposed Modification to the Robert–Asselin Time Filter. Mon. Wea. Rev., 137, 2538–2546, <a href="https://doi.org/10.1175/2009MWR2724.1">10.1175/2009MWR2724.1</a>.</li><li class="footnote" id="footnote-Amezcua2011"><a class="tag is-link" href="#citeref-Amezcua2011">Amezcua2011</a>Amezcua, J., E. Kalnay, and P. D. Williams, 2011: The Effects of the RAW Filter on the Climatology and Forecast Skill of the SPEEDY Model. Mon. Wea. Rev., 139, 608–619, doi:<a href="https://doi.org/10.1175/2010MWR3530.1">10.1175/2010MWR3530.1</a>. </li><li class="footnote" id="footnote-Williams2011"><a class="tag is-link" href="#citeref-Williams2011">Williams2011</a>Williams, P. D., 2011: The RAW Filter: An Improvement to the Robert–Asselin Filter in Semi-Implicit Integrations. Mon. Wea. Rev., 139, 1996–2007, doi:<a href="https://doi.org/10.1175/2010MWR3601.1">10.1175/2010MWR3601.1</a>. </li></ul></section></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../grids/">« Grids</a><a class="docs-footer-nextpage" href="../shallowwater/">Shallow water model »</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="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 0.27.25 on <span class="colophon-date" title="Tuesday 17 October 2023 23:44">Tuesday 17 October 2023</span>. Using Julia version 1.8.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>