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Iterative spectral integration algorithm - Revision history
2026-05-19T08:33:49Z
Revision history for this page on the wiki
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http://atomix.app.uib.no/index.php?title=Iterative_spectral_integration_algorithm&diff=4507&oldid=prev
KikiSchulz at 15:37, 3 February 2023
2023-02-03T15:37:48Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:37, 3 February 2023</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l151">Line 151:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Thus, integrating the spectrum, <math>\Psi</math>, to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate? </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Thus, integrating the spectrum, <math>\Psi</math>, to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate? </div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[File:Figure 381.jpg|thumb|500px|The ratio of the true dissipation rate to the rate determined by integrating the spectrum to only 10 cpm, as a function of the 10-cpm rate for a range of kinematic viscosities, if the measured spectrum follows the Nasmyth spectrum.]]</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of <math>\varepsilon</math> This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by <math>\varepsilon^{1/4}</math>. If the fraction of the variance resolved by a particular <math>k_u</math> exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of <math>\varepsilon</math> This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by <math>\varepsilon^{1/4}</math>. If the fraction of the variance resolved by a particular <math>k_u</math> exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations. </div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l158">Line 158:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Figure_381.jpg]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math></math></div></td></tr>
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KikiSchulz
http://atomix.app.uib.no/index.php?title=Iterative_spectral_integration_algorithm&diff=4484&oldid=prev
Rolf at 20:20, 9 December 2022
2022-12-09T20:20:20Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:20, 9 December 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l158">Line 158:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[<del style="font-weight: bold; text-decoration: none;">File:epsilon_10_to_epsilon_ratio</del>.<del style="font-weight: bold; text-decoration: none;">pdf</del>]]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[<ins style="font-weight: bold; text-decoration: none;">Figure_381</ins>.<ins style="font-weight: bold; text-decoration: none;">jpg</ins>]]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math></math></div></td></tr>
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Rolf
http://atomix.app.uib.no/index.php?title=Iterative_spectral_integration_algorithm&diff=4482&oldid=prev
Rolf at 20:15, 9 December 2022
2022-12-09T20:15:24Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:15, 9 December 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l158">Line 158:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[File:<del style="font-weight: bold; text-decoration: none;">epsilon\_10\_to\_epsilon\_ratio</del>.pdf]]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[File:<ins style="font-weight: bold; text-decoration: none;">epsilon_10_to_epsilon_ratio</ins>.pdf]]</div></td></tr>
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Rolf
http://atomix.app.uib.no/index.php?title=Iterative_spectral_integration_algorithm&diff=4481&oldid=prev
Rolf at 20:12, 9 December 2022
2022-12-09T20:12:16Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:12, 9 December 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l158">Line 158:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[File:<del style="font-weight: bold; text-decoration: none;">epsilon_10_to_epsilon_ratio</del>.pdf]]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[File:<ins style="font-weight: bold; text-decoration: none;">epsilon\_10\_to\_epsilon\_ratio</ins>.pdf]]</div></td></tr>
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Rolf
http://atomix.app.uib.no/index.php?title=Iterative_spectral_integration_algorithm&diff=4480&oldid=prev
Rolf at 20:10, 9 December 2022
2022-12-09T20:10:36Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:10, 9 December 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l158">Line 158:</td>
<td colspan="2" class="diff-lineno">Line 158:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In this range, the spectrum rises in proportion to <math>\varepsilon^{2/3}k^{1/3}</math> and, thus, its level provides an estimate of the rate of dissipation. The inertial subrange is confined to wavenumbers smaller than <math>k(\nu^3/\varepsilon)^{1/4}=0.02</math>, and thus will usually use fewer spectral points than the method of spectral integration. This reduces the statistical reliability of the dissipation estimate but it does avoid the bias introduced by not fully resolving the spectrum of shear. </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div> </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[File:epsilon_10_to_epsilon_ratio.pdf]]</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math></math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
</table>
Rolf
http://atomix.app.uib.no/index.php?title=Iterative_spectral_integration_algorithm&diff=4478&oldid=prev
Rolf at 20:01, 9 December 2022
2022-12-09T20:01:10Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 20:01, 9 December 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l150">Line 150:</td>
<td colspan="2" class="diff-lineno">Line 150:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Thus, integrating the spectrum to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate? </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Thus, integrating the spectrum<ins style="font-weight: bold; text-decoration: none;">, <math>\Psi</math>, </ins>to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate? </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of <math>\varepsilon</math> This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by <math>\varepsilon^{1/4}</math>. If the fraction of the variance resolved by a particular <math>k_u</math> exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of <math>\varepsilon</math> This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by <math>\varepsilon^{1/4}</math>. If the fraction of the variance resolved by a particular <math>k_u</math> exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations. </div></td></tr>
</table>
Rolf
http://atomix.app.uib.no/index.php?title=Iterative_spectral_integration_algorithm&diff=4477&oldid=prev
Rolf at 19:58, 9 December 2022
2022-12-09T19:58:41Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<col class="diff-marker" />
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<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 19:58, 9 December 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l138">Line 138:</td>
<td colspan="2" class="diff-lineno">Line 138:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\frac{\varepsilon}{\varepsilon_{10}} = \sqrt{1+a\varepsilon_{10}} + \exp\left(-b\,\varepsilon_{10}\right) - 1 </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\frac{\varepsilon}{\varepsilon_{10}} = \sqrt{1+a\varepsilon_{10}} + \exp\left(-b\,\varepsilon_{10}\right) - 1 </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{split}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{split}</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l144">Line 144:</td>
<td colspan="2" class="diff-lineno">Line 146:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>b &= 5.5\times 10^{-8}\, \nu^{-5/2} \\</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>b &= 5.5\times 10^{-8}\, \nu^{-5/2} \\</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\varepsilon_{10} &= \frac{15}{2}\, \nu\, \int_0^{10} \Psi(k)\, \mathrm{d}k\\</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\varepsilon_{10} &= \frac{15}{2}\, \nu\, \int_0^{10} \Psi(k)\, \mathrm{d}k\\</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\varepsilon &= \frac{15}{2}\, \nu\, \int_0^{\infty} \Psi(k)\, \mathrm{d}k</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\varepsilon &= \frac{15}{2}\, \nu\, \int_0^{\infty} \Psi(k)\, \mathrm{d}k <ins style="font-weight: bold; text-decoration: none;">\ \ .</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{split}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{split}</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr>
</table>
Rolf
http://atomix.app.uib.no/index.php?title=Iterative_spectral_integration_algorithm&diff=4476&oldid=prev
Rolf at 19:57, 9 December 2022
2022-12-09T19:57:18Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 19:57, 9 December 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l135">Line 135:</td>
<td colspan="2" class="diff-lineno">Line 135:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The upper limit of spectral integration is nearly always higher than 10 cpm. For example, when <math>\varepsilon=1\times 10^{-10}\ \mathrm{W\, kg^{-1}}</math>, this limit resolves a little more than 95% of the shear variance. Less for larger dissipation rates. Thus, if we integrate a shear spectrum to 10 cpm this variance must be related to the total variance if the spectrum follows a universal form. In fact, if the measured spectrum follows the universal form exactly, then the ratio of the “true” dissipation rate, <math>\varepsilon</math>, to the rate derived from integration to 10 cpm, <math>\varepsilon_{10}</math>, is given by </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The upper limit of spectral integration is nearly always higher than 10 cpm. For example, when <math>\varepsilon=1\times 10^{-10}\ \mathrm{W\, kg^{-1}}</math>, this limit resolves a little more than 95% of the shear variance. Less for larger dissipation rates. Thus, if we integrate a shear spectrum to 10 cpm this variance must be related to the total variance if the spectrum follows a universal form. In fact, if the measured spectrum follows the universal form exactly, then the ratio of the “true” dissipation rate, <math>\varepsilon</math>, to the rate derived from integration to 10 cpm, <math>\varepsilon_{10}</math>, is given by </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\frac{\varepsilon}{\varepsilon_{10}} = \sqrt{1+a\varepsilon_{10}} + \exp\left(-b\,\varepsilon_{10}\right) - 1 </ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></math></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">where</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{split}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{split}</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">\frac{\varepsilon}{\varepsilon_{10}} &= \sqrt{1+a\varepsilon_{10}} + \exp\left(-b\,\varepsilon_{10}\right) - 1 \\</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>a &= 1.25\times 10^{-9}\, \nu^{-3} \\</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>a &= 1.25\times 10^{-9}\, \nu^{-3} \\</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>b &= 5.5\times 10^{-8}\, \nu^{-5/2} \\</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>b &= 5.5\times 10^{-8}\, \nu^{-5/2} \\</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l145">Line 145:</td>
<td colspan="2" class="diff-lineno">Line 148:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">where <math> a= 1.08 \times 10^9</math> </del>Thus, integrating the spectrum to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate? </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Thus, integrating the spectrum to 10 cpm provides the first (and quite rough) estimate of the rate of dissipation and this estimate can be used to set another upper limit to spectral integration. That is, at what wavenumber is 95% of the spectrum resolved if the dissipation rate equals this initial estimate? </div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of <math>\varepsilon</math> This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by <math>\varepsilon^{1/4}</math>. If the fraction of the variance resolved by a particular <math>k_u</math> exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations. </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The spectrum is then integrated to the lowest of these upper limits to provide an improved estimate of the rate of dissipation. It is an improved estimate because it will nearly always span a range of wavenumbers that is considerably wider than 10 cpm, which improves its statistical reliability. This improved estimate is then used to determine the fraction of the shear variance that is resolved by the upper limit of integration. The estimate is divided by this fraction to form a new estimate of <math>\varepsilon</math> This process is then repeated -- find the new fraction resolved, adjust the estimate upwards using this fraction, and so on. This iteration converges quickly because the wavenumber is non-dimensionalized by <math>\varepsilon^{1/4}</math>. If the fraction of the variance resolved by a particular <math>k_u</math> exceeds 50%, the estimate of the rate of dissipation converges to within 1% of its ultimate value in two or fewer iterations. </div></td></tr>
</table>
Rolf
http://atomix.app.uib.no/index.php?title=Iterative_spectral_integration_algorithm&diff=4475&oldid=prev
Rolf at 19:55, 9 December 2022
2022-12-09T19:55:26Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 19:55, 9 December 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l140">Line 140:</td>
<td colspan="2" class="diff-lineno">Line 140:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>a &= 1.25\times 10^{-9}\, \nu^{-3} \\</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>a &= 1.25\times 10^{-9}\, \nu^{-3} \\</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>b &= 5.5\times 10^{-8}\, \nu^{-5/2} \\</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>b &= 5.5\times 10^{-8}\, \nu^{-5/2} \\</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>\varepsilon_{10} &= \frac{15}{2}\, \nu\, \int_0^{10} \Psi(k)\, \mathrm{d}k</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>\varepsilon_{10} &= \frac{15}{2}\, \nu\, \int_0^{10<ins style="font-weight: bold; text-decoration: none;">} \Psi(k)\, \mathrm{d}k\\</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\varepsilon &= \frac{15}{2}\, \nu\, \int_0^{\infty</ins>} \Psi(k)\, \mathrm{d}k</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{split}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{split}</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr>
</table>
Rolf
http://atomix.app.uib.no/index.php?title=Iterative_spectral_integration_algorithm&diff=4474&oldid=prev
Rolf at 19:54, 9 December 2022
2022-12-09T19:54:06Z
<p></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
<col class="diff-marker" />
<col class="diff-content" />
<col class="diff-marker" />
<col class="diff-content" />
<tr class="diff-title" lang="en">
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 19:54, 9 December 2022</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l138">Line 138:</td>
<td colspan="2" class="diff-lineno">Line 138:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{split}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\begin{split}</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\frac{\varepsilon}{\varepsilon_{10}} &= \sqrt{1+a\varepsilon_{10}} + \exp\left(-b\,\varepsilon_{10}\right) - 1 \\</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\frac{\varepsilon}{\varepsilon_{10}} &= \sqrt{1+a\varepsilon_{10}} + \exp\left(-b\,\varepsilon_{10}\right) - 1 \\</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>a &= 1.25\times 10^{-9} \nu^{-3} \\</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>a &= 1.25\times 10^{-9}<ins style="font-weight: bold; text-decoration: none;">\, </ins>\nu^{-3} \\</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>b &= 5.5\times 10^{-8} \nu^{-5/2}</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>b &= 5.5\times 10^{-8}<ins style="font-weight: bold; text-decoration: none;">\, </ins>\nu^{-5/2} <ins style="font-weight: bold; text-decoration: none;">\\</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">\varepsilon_{10} &= \frac{15}{2}\, \nu\, \int_0^{10} \Psi(k)\, \mathrm{d}k</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{split}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>\end{split}</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></math></div></td></tr>
</table>
Rolf