[cig-commits] r15872 - in doc/geodynamics.org/benchmarks/trunk: . mc mc/2d-cartesian mc/2d-cartesian/images mc/3d-spherical
luis at geodynamics.org
luis at geodynamics.org
Fri Oct 23 17:18:41 PDT 2009
Author: luis
Date: 2009-10-23 17:18:37 -0700 (Fri, 23 Oct 2009)
New Revision: 15872
Added:
doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/images/
doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/images/Vx.png
doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/images/get-images.sh
doc/geodynamics.org/benchmarks/trunk/mc/notes.header
doc/geodynamics.org/benchmarks/trunk/mc/notes.html
Modified:
doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/index.html
doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/index.rst
doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite1.html
doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite1.rst
doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite2.html
doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite2.rst
doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite3.html
doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite3.rst
doc/geodynamics.org/benchmarks/trunk/mc/3d-spherical/index.html
doc/geodynamics.org/benchmarks/trunk/mc/3d-spherical/index.rst
doc/geodynamics.org/benchmarks/trunk/mc/notes.rst
doc/geodynamics.org/benchmarks/trunk/sources.txt
Log:
Formatted mantle convection pages.
Added: doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/images/Vx.png
===================================================================
(Binary files differ)
Property changes on: doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/images/Vx.png
___________________________________________________________________
Name: svn:mime-type
+ image/png
Added: doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/images/get-images.sh
===================================================================
--- doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/images/get-images.sh (rev 0)
+++ doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/images/get-images.sh 2009-10-24 00:18:37 UTC (rev 15872)
@@ -0,0 +1,4 @@
+#!/bin/bash
+
+rm -f Vx.png
+wget http://www.geodynamics.org/cig/Members/tan2/benchmarks/Vx.png
Property changes on: doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/images/get-images.sh
___________________________________________________________________
Name: svn:executable
+ *
Modified: doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/index.html
===================================================================
--- doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/index.html 2009-10-24 00:18:25 UTC (rev 15871)
+++ doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/index.html 2009-10-24 00:18:37 UTC (rev 15872)
@@ -12,53 +12,63 @@
<div class="document" id="general-description-of-the-benchmark-problem">
<h1 class="title">General Description of the Benchmark Problem</h1>
-<p>The benchmark will be similar to the benchmark of Blankenbach <em>et al.</em>
-(1989) in methodology.</p>
-<p>The benchmark problem is 2-D thermal convection of a non-rotating
-anelastic liquid of infinite Prandtl number in a Cartesian, closed, unit
-cell. The governing equations is based on Truncated An-Elastic
-Approximation (TALA).</p>
+<p>The benchmark will be similar to the benchmark of
+Blankenbach <em>et al.</em> (1989) in methodology.</p>
+<p>The benchmark problem is 2-D thermal convection of a
+non-rotating anelastic liquid of infinite Prandtl number
+in a Cartesian, closed, unit cell. The governing equations
+are based on the Truncated An-Elastic Approximation (TALA).</p>
<p>(attach equations as image here...)</p>
-<p>We will have several cases, steady or unsteady, constant or variable
-viscosity, bottom or internal heated, heat or mechanically driven.</p>
+<p>We will have several cases, steady or unsteady,
+constant or variable viscosity, bottom or internal heated,
+heat or mechanically driven.</p>
<div class="section" id="grids">
<h1>Grids</h1>
-<p>Each case will be run at <strong>3 different resolutions</strong> (grid resolution
-32x32, 64x64, 128x128, or higher if needed) to quantify the convergence
-asymptotically. By comparing the asymptotically converged result, we
-probably can negate the need of mesh refinement near the boundary and
-reduce the uncertainty associated with various interpolation/extrapolation
-schemes in calculating derived information (e.g. geoid).</p>
+<p>Each case will be run at <strong>3 different resolutions</strong>
+(grid resolution 32×32, 64×64, 128×128, or higher if needed)
+to quantify the convergence asymptotically. By comparing the
+asymptotically converged result, we can probably negate the need
+for mesh refinement near the boundary and reduce the uncertainty
+associated with various interpolation/extrapolation schemes in
+calculating derived information (e.g. geoid).</p>
</div>
<div class="section" id="velocity-bc-s">
<h1>Velocity BC's</h1>
-<p>All boundaries (top, bottom, left, right) are <strong>impermeable</strong> (i.e., zero
-normal velocity) and <strong>free-slip</strong> (i.e., zero tangential stress), except
-for the mechanically driven case, where the top boundary is impermeble
-and zero-slip (i.e. fixed horizontal velocity).</p>
+<p>All boundaries (top, bottom, left, right) are
+<strong>impermeable</strong> (i.e., zero normal velocity) and
+<strong>free-slip</strong> (i.e., zero tangential stress), except
+for the mechanically driven case, where the top boundary
+is impermeable and zero-slip (i.e. fixed horizontal velocity).</p>
</div>
<div class="section" id="temperature-bc-s">
<h1>Temperature BC's</h1>
-<p>All non-dimensional numbers are defined at the top surface. There are
-five non-dimensional numbers:</p>
+<p>All non-dimensional numbers are defined at the top surface.
+There are five non-dimensional numbers:</p>
<ul class="simple">
-<li><strong>Ra</strong>: Rayleigh number</li>
-<li><strong>H</strong>: volumentric heat production number, <strong>H = 0</strong>, except for
+<li>[;Ra;]: Rayleigh number</li>
+<li>[;H;]: volumetric heat production number. [;H = 0;], except for
internal heated cases.</li>
-<li><strong>Di</strong>: Dissipation number</li>
-<li><strong>Gamma</strong>: Gruneisen parameter</li>
-<li><strong>T_0</strong>: Surface temperature, <strong>T_0 = 0.1</strong> for all cases.</li>
+<li>[;Di;]: Dissipation number</li>
+<li>[;\Gamma;]: Gruneisen parameter</li>
+<li>[;T_0;]: Surface temperature, [;T_0 = 0.1;] for all cases.</li>
</ul>
</div>
<div class="section" id="reference-state">
<h1>Reference State</h1>
-<p>The reference density profile is $rho_ref(z) = exp((1-z)*Di/Gamma)$</p>
-<p>The reference temperature profile is $T_ref(z) = T_0 * exp((1-z) * Di) T_0$</p>
-<p>These physical properties are constant: thermal diffusivity, coefficient
-of thermal expansion, gravitational acceleration.</p>
+<p>The reference density profile is</p>
+<blockquote>
+[;\rho_{ref}(z) = \rho_0 \exp\left((1-z)*Di/\Gamma\right);]</blockquote>
+<p>The reference temperature profile is</p>
+<blockquote>
+[;T_{ref}(z) = T_0 * \exp\left((1-z) * Di\right);]</blockquote>
+<p>These physical properties are constant:
+thermal diffusivity,
+coefficient of thermal expansion,
+gravitational acceleration.</p>
</div>
-<div class="section" id="required-information-all-quantities-are-non-dimensional-unless-specified">
-<h1>Required Information (all quantities are non-dimensional unless specified)</h1>
+<div class="section" id="required-information">
+<h1>Required Information</h1>
+<p>All quantities are non-dimensional unless specified.</p>
<ul>
<li><p class="first">Nusselt number</p>
</li>
@@ -66,9 +76,9 @@
</li>
<li><p class="first">Total Kinetic Energy</p>
</li>
-<li><p class="first">RMS(V_x at top surface)</p>
+<li><p class="first">RMS([;{\smashmargin2{V_x}};] at top surface)</p>
</li>
-<li><p class="first">Max(V_x at top surface)</p>
+<li><p class="first">Max([;{\smashmargin2{V_x}};] at top surface)</p>
</li>
<li><p class="first">Total Dissipation Heating</p>
</li>
@@ -80,13 +90,13 @@
<p>The following dimensional constants (in SI units) are used for the
calculation of geoid:</p>
<ul class="simple">
-<li>Gravitational constant <strong>G</strong> = 6.673x10^-11</li>
-<li>depth of the box <strong>R</strong> = 3x10^6</li>
-<li>density at surface <strong>rho_0</strong> = 4000</li>
-<li>coefficient of thermal expansion <strong>alpha_0</strong> = 3x10^-5</li>
-<li>temperature contrast <strong>Delta_T</strong> = 3000</li>
-<li>viscosity <strong>visc_0</strong> = 10^21</li>
-<li>thermal diffusivity <strong>kappa_0</strong> = 10^-6</li>
+<li>Gravitational constant: [;G=6.673\times10^{-11};]</li>
+<li>Depth of the box: [;R=3\times10^{6};]</li>
+<li>Density at surface: [;\rho_0 = 4000;]</li>
+<li>Coefficient of thermal expansion: [;\alpha_0 = 3\times10^{-5};]</li>
+<li>Temperature contrast: [;\Delta{T} = 3000;]</li>
+<li>Viscosity: [;\eta_0 = 10^{21};]</li>
+<li>Thermal diffusivity: [;\kappa_0 = 10^{-6};]</li>
</ul>
</li>
</ul>
Modified: doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/index.rst
===================================================================
--- doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/index.rst 2009-10-24 00:18:25 UTC (rev 15871)
+++ doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/index.rst 2009-10-24 00:18:37 UTC (rev 15872)
@@ -1,81 +1,97 @@
+.. |times| unicode:: U+00D7
+.. |32x32| replace:: 32\ |times|\ 32
+.. |64x64| replace:: 64\ |times|\ 64
+.. |128x128| replace:: 128\ |times|\ 128
+
General Description of the Benchmark Problem
============================================
-The benchmark will be similar to the benchmark of Blankenbach *et al.*
-(1989) in methodology.
+The benchmark will be similar to the benchmark of
+Blankenbach *et al.* (1989) in methodology.
-The benchmark problem is 2-D thermal convection of a non-rotating
-anelastic liquid of infinite Prandtl number in a Cartesian, closed, unit
-cell. The governing equations is based on Truncated An-Elastic
-Approximation (TALA).
+The benchmark problem is 2-D thermal convection of a
+non-rotating anelastic liquid of infinite Prandtl number
+in a Cartesian, closed, unit cell. The governing equations
+are based on the Truncated An-Elastic Approximation (TALA).
(attach equations as image here...)
-We will have several cases, steady or unsteady, constant or variable
-viscosity, bottom or internal heated, heat or mechanically driven.
+We will have several cases, steady or unsteady,
+constant or variable viscosity, bottom or internal heated,
+heat or mechanically driven.
Grids
-----
-Each case will be run at **3 different resolutions** (grid resolution
-32x32, 64x64, 128x128, or higher if needed) to quantify the convergence
-asymptotically. By comparing the asymptotically converged result, we
-probably can negate the need of mesh refinement near the boundary and
-reduce the uncertainty associated with various interpolation/extrapolation
-schemes in calculating derived information (e.g. geoid).
+Each case will be run at **3 different resolutions**
+(grid resolution |32x32|, |64x64|, |128x128|, or higher if needed)
+to quantify the convergence asymptotically. By comparing the
+asymptotically converged result, we can probably negate the need
+for mesh refinement near the boundary and reduce the uncertainty
+associated with various interpolation/extrapolation schemes in
+calculating derived information (e.g. geoid).
Velocity BC's
-------------
-All boundaries (top, bottom, left, right) are **impermeable** (i.e., zero
-normal velocity) and **free-slip** (i.e., zero tangential stress), except
-for the mechanically driven case, where the top boundary is impermeble
-and zero-slip (i.e. fixed horizontal velocity).
+All boundaries (top, bottom, left, right) are
+**impermeable** (i.e., zero normal velocity) and
+**free-slip** (i.e., zero tangential stress), except
+for the mechanically driven case, where the top boundary
+is impermeable and zero-slip (i.e. fixed horizontal velocity).
Temperature BC's
----------------
-All non-dimensional numbers are defined at the top surface. There are
-five non-dimensional numbers:
+All non-dimensional numbers are defined at the top surface.
+There are five non-dimensional numbers:
-* **Ra**: Rayleigh number
+* [;Ra;]: Rayleigh number
-* **H**: volumentric heat production number, **H = 0**, except for
+* [;H;]: volumetric heat production number. [;H = 0;], except for
internal heated cases.
-* **Di**: Dissipation number
+* [;Di;]: Dissipation number
-* **Gamma**: Gruneisen parameter
+* [;\\Gamma;]: Gruneisen parameter
-* **T_0**: Surface temperature, **T_0 = 0.1** for all cases.
+* [;T_0;]: Surface temperature, [;T_0 = 0.1;] for all cases.
Reference State
---------------
-The reference density profile is $rho_ref(z) = exp((1-z)*Di/Gamma)$
+The reference density profile is
-The reference temperature profile is $T_ref(z) = T_0 * exp((1-z) * Di) T_0$
+ [;\\rho_{ref}(z) = \\rho_0 \\exp\\left((1-z)*Di/\\Gamma\\right);]
-These physical properties are constant: thermal diffusivity, coefficient
-of thermal expansion, gravitational acceleration.
+The reference temperature profile is
+ [;T_{ref}(z) = T_0 * \\exp\\left((1-z) * Di\\right);]
-Required Information (all quantities are non-dimensional unless specified)
---------------------------------------------------------------------------
+These physical properties are constant:
+thermal diffusivity,
+coefficient of thermal expansion,
+gravitational acceleration.
+
+Required Information
+--------------------
+
+All quantities are non-dimensional unless specified.
+
* Nusselt number
* Mean Temperature
* Total Kinetic Energy
-* RMS(V_x at top surface)
+* RMS([;{\\smashmargin2{V_x}};] at top surface)
-* Max(V_x at top surface)
+* Max([;{\\smashmargin2{V_x}};] at top surface)
* Total Dissipation Heating
@@ -88,19 +104,24 @@
The following dimensional constants (in SI units) are used for the
calculation of geoid:
- * Gravitational constant **G** = 6.673x10^-11
+ .. |G| replace:: 6.673\ |times|\ 10\ :sup:`-11`
+ .. |R| replace:: 3\ |times|\ 10\ :sup:`6`
+ .. |alpha| replace:: 3\ |times|\ 10\ :sup:`-5`
+ .. |kappa_0| replace:: 10\ :sup:`-6`
- * depth of the box **R** = 3x10^6
+ * Gravitational constant: [;G=6.673\\times10^{-11};]
- * density at surface **rho_0** = 4000
+ * Depth of the box: [;R=3\\times10^{6};]
- * coefficient of thermal expansion **alpha_0** = 3x10^-5
+ * Density at surface: [;\\rho_0 = 4000;]
- * temperature contrast **Delta_T** = 3000
+ * Coefficient of thermal expansion: [;\\alpha_0 = 3\\times10^{-5};]
- * viscosity **visc_0** = 10^21
+ * Temperature contrast: [;\\Delta{T} = 3000;]
- * thermal diffusivity **kappa_0** = 10^-6
+ * Viscosity: [;\\eta_0 = 10^{21};]
+ * Thermal diffusivity: [;\\kappa_0 = 10^{-6};]
+
The density is 0 above the top of the box and is 2 below the bottom of
the box.
Modified: doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite1.html
===================================================================
--- doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite1.html 2009-10-24 00:18:25 UTC (rev 15871)
+++ doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite1.html 2009-10-24 00:18:37 UTC (rev 15872)
@@ -12,42 +12,52 @@
<div class="document">
-<p>Testing the implementation of driving forces, isoviscous, comparing with
-analytical solutions.</p>
-<p>The non-dimensional numbers are moderately low in this case (Ra=10^5,
-Di=0.25, gamma=1.5). The viscosity is constant. The purpose of this suite
-is to ensure the driving forces are implemented correctly. This set of
-non-dimensional numbers will give low compressibility and slow
-convection. Therefore, most of the codes should behave well in this
-parameter range.</p>
+<p>Testing the implementation of driving forces, isoviscous,
+comparing with analytical solutions.</p>
+<p>The non-dimensional numbers are moderately low in this case
+([;Ra=10^5;], [;Di=0.25;], [;\Gamma=1.5;]). The viscosity
+is constant. The purpose of this suite is to ensure the driving
+forces are implemented correctly. This set of non-dimensional
+numbers will give low compressibility and slow convection.
+Therefore, most of the codes should behave well in this parameter
+range.</p>
<div class="section" id="case-1a-bottom-heated">
<h1>Case 1a: Bottom Heated</h1>
<p>The temperature at the bottom is fixed at 1. The initial temperature
condition is</p>
<blockquote>
-T = 0.5 everwhere, except at z = 0.5,
-where T = 0.5 + cos(pi * x) * 0.001 * elz</blockquote>
-<p>where elz is the number of elements in the z-direction.</p>
-<p>The initial temperature perturbation mimics a delta function. However,
-the amplitude of the perturbation, with a factor of 1/1000, doesn't match
-with a delta function. A comparison with analytical solution is possible
-for the 0th-step velocity.</p>
+[;
+T = \begin{cases}
+0.5 & \text{ everywhere, } \\
+0.5 + \cos(\pi x) \cdot 0.001 \cdot \mathtt{\bf elz}
+& \text{ except at $z=0.5$ }
+\end{cases}
+;]</blockquote>
+<p>where <tt class="docutils literal"><span class="pre">elz</span></tt> is the number of elements in the z-direction.</p>
+<p>The initial temperature perturbation mimics a delta function.
+However, the amplitude of the perturbation, with a factor of
+1/1000, doesn't match with a delta function. A comparison with
+analytical solution is possible for the 0th-step velocity.</p>
</div>
<div class="section" id="case-1b-internal-heated">
<h1>Case 1b: Internal Heated</h1>
-<p>The temperature BC at the bottom is no-heatflux. The initial temperature
-is the same as Case 1a. H = 1.</p>
+<p>The temperature BC at the bottom is no-heatflux.
+The initial temperature is the same as Case 1a.
+In this case, [;H = 1;].</p>
</div>
<div class="section" id="case-1c-mechanically-driven">
<h1>Case 1c: Mechanically Driven</h1>
-<p>The temperature at the bottom is fixed at 0. The initial temperature is
-zero everywhere. The horizontal velocity BC at the top boundary is</p>
+<p>The temperature at the bottom is fixed at 0.
+The initial temperature is zero everywhere.
+The horizontal velocity BC at the top boundary is</p>
<blockquote>
-V_x = 1000 * x^2 * (x-1)^2</blockquote>
-<p>so that V_x = 0 at x = 0 and 1, and dV_x/dx = 0 at x = 0 and 1.
+[;{\smashmargin2{V_x}} = 1000 x^2 (1-x)^2;]</blockquote>
+<p>so that both [;{\smashmargin2{V_x}}=0;] and [;\frac{d{\smashmargin2{V_x}}}{dx}=0;]
+whenever [;x=0;] and [;x=1;].
(This case is optional.)</p>
-<div class="figure">
-<img alt="Vx.pngHorizontalvelocityboundarycondition" src="Vx.pngHorizontalvelocityboundarycondition" />
+<div align="center" class="figure">
+<img alt="images/Vx.png" src="images/Vx.png" />
+<p class="caption">Horizontal velocity boundary condition</p>
</div>
</div>
</div>
Modified: doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite1.rst
===================================================================
--- doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite1.rst 2009-10-24 00:18:25 UTC (rev 15871)
+++ doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite1.rst 2009-10-24 00:18:37 UTC (rev 15872)
@@ -1,12 +1,13 @@
-Testing the implementation of driving forces, isoviscous, comparing with
-analytical solutions.
+Testing the implementation of driving forces, isoviscous,
+comparing with analytical solutions.
-The non-dimensional numbers are moderately low in this case (Ra=10^5,
-Di=0.25, gamma=1.5). The viscosity is constant. The purpose of this suite
-is to ensure the driving forces are implemented correctly. This set of
-non-dimensional numbers will give low compressibility and slow
-convection. Therefore, most of the codes should behave well in this
-parameter range.
+The non-dimensional numbers are moderately low in this case
+([;Ra=10^5;], [;Di=0.25;], [;\\Gamma=1.5;]). The viscosity
+is constant. The purpose of this suite is to ensure the driving
+forces are implemented correctly. This set of non-dimensional
+numbers will give low compressibility and slow convection.
+Therefore, most of the codes should behave well in this parameter
+range.
Case 1a: Bottom Heated
@@ -15,35 +16,45 @@
The temperature at the bottom is fixed at 1. The initial temperature
condition is
- T = 0.5 everwhere, except at z = 0.5,
- where T = 0.5 + cos(pi * x) * 0.001 * elz
+ [;
+ T = \\begin{cases}
+ 0.5 & \\text{ everywhere, } \\\\
+ 0.5 + \\cos(\\pi x) \\cdot 0.001 \\cdot \\mathtt{\\bf elz}
+ & \\text{ except at $z=0.5$ }
+ \\end{cases}
+ ;]
-where elz is the number of elements in the z-direction.
+where ``elz`` is the number of elements in the z-direction.
-The initial temperature perturbation mimics a delta function. However,
-the amplitude of the perturbation, with a factor of 1/1000, doesn't match
-with a delta function. A comparison with analytical solution is possible
-for the 0th-step velocity.
+The initial temperature perturbation mimics a delta function.
+However, the amplitude of the perturbation, with a factor of
+1/1000, doesn't match with a delta function. A comparison with
+analytical solution is possible for the 0th-step velocity.
Case 1b: Internal Heated
------------------------
-The temperature BC at the bottom is no-heatflux. The initial temperature
-is the same as Case 1a. H = 1.
+The temperature BC at the bottom is no-heatflux.
+The initial temperature is the same as Case 1a.
+In this case, [;H = 1;].
Case 1c: Mechanically Driven
----------------------------
-The temperature at the bottom is fixed at 0. The initial temperature is
-zero everywhere. The horizontal velocity BC at the top boundary is
+The temperature at the bottom is fixed at 0.
+The initial temperature is zero everywhere.
+The horizontal velocity BC at the top boundary is
- V_x = 1000 * x^2 * (x-1)^2
+ [;{\\smashmargin2{V_x}} = 1000 x^2 (1-x)^2;]
-so that V_x = 0 at x = 0 and 1, and dV_x/dx = 0 at x = 0 and 1.
+so that both [;{\\smashmargin2{V_x}}=0;] and [;\\frac{d{\\smashmargin2{V_x}}}{dx}=0;]
+whenever [;x=0;] and [;x=1;].
(This case is optional.)
-.. figure:: Vx.png
+.. figure:: images/Vx.png
+ :align: center
+
Horizontal velocity boundary condition
Modified: doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite2.html
===================================================================
--- doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite2.html 2009-10-24 00:18:25 UTC (rev 15871)
+++ doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite2.html 2009-10-24 00:18:37 UTC (rev 15872)
@@ -12,25 +12,21 @@
<div class="document" id="steady-state-basal-heated">
<h1 class="title">Steady state, basal heated</h1>
-<p>Gamma = 1.1 in this suite.</p>
-<p>Temperature dependent viscosity: eta = exp(-5 * T)</p>
+<p>[;\Gamma = 1.1;] in this suite.</p>
+<p>Temperature dependent viscosity:</p>
+<blockquote>
+[;;\eta = \exp(-5 T);]</blockquote>
<div class="section" id="case-2a">
<h1>Case 2a</h1>
-<ul class="simple">
-<li>Ra = 3x10^5, Di = 0.5</li>
-</ul>
+<p>[;Ra = 3\times{}10^5;], [;Di = 0.5;]</p>
</div>
<div class="section" id="case-2b">
<h1>Case 2b</h1>
-<ul class="simple">
-<li>Ra = 10^6, Di = 0.75 (not sure whether this case can reach steady state)</li>
-</ul>
+<p>[;Ra = 10^6;], [;Di = 0.75;] (not sure whether this case can reach steady state)</p>
</div>
<div class="section" id="case-2c">
<h1>Case 2c</h1>
-<ul class="simple">
-<li>Ra = 3x10^6, Di = 1.0 (not sure whether this case can reach steady state)</li>
-</ul>
+<p>[;Ra = 3\times{}10^6;], [;Di = 1.0;] (not sure whether this case can reach steady state)</p>
</div>
</div>
</body>
Modified: doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite2.rst
===================================================================
--- doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite2.rst 2009-10-24 00:18:25 UTC (rev 15871)
+++ doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite2.rst 2009-10-24 00:18:37 UTC (rev 15872)
@@ -1,26 +1,28 @@
Steady state, basal heated
==========================
-Gamma = 1.1 in this suite.
+[;\\Gamma = 1.1;] in this suite.
-Temperature dependent viscosity: eta = exp(-5 * T)
+Temperature dependent viscosity:
+ [;;\\eta = \\exp(-5 T);]
+
Case 2a
-------
-* Ra = 3x10^5, Di = 0.5
+[;Ra = 3\\times{}10^5;], [;Di = 0.5;]
Case 2b
-------
-* Ra = 10^6, Di = 0.75 (not sure whether this case can reach steady state)
+[;Ra = 10^6;], [;Di = 0.75;] (not sure whether this case can reach steady state)
Case 2c
-------
-* Ra = 3x10^6, Di = 1.0 (not sure whether this case can reach steady state)
+[;Ra = 3\\times{}10^6;], [;Di = 1.0;] (not sure whether this case can reach steady state)
Modified: doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite3.html
===================================================================
--- doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite3.html 2009-10-24 00:18:25 UTC (rev 15871)
+++ doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite3.html 2009-10-24 00:18:37 UTC (rev 15872)
@@ -13,7 +13,7 @@
<h1 class="title">Steady state, internal heated</h1>
<p>This suite will have several cases taken from Jarvis and McKenzie (1980)</p>
-<p>H = 1, no-heatflux for the bottom boundary.</p>
+<p>[;H = 1;], no-heatflux for the bottom boundary.</p>
</div>
</body>
</html>
Modified: doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite3.rst
===================================================================
--- doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite3.rst 2009-10-24 00:18:25 UTC (rev 15871)
+++ doc/geodynamics.org/benchmarks/trunk/mc/2d-cartesian/suite3.rst 2009-10-24 00:18:37 UTC (rev 15872)
@@ -3,5 +3,5 @@
This suite will have several cases taken from Jarvis and McKenzie (1980)
-H = 1, no-heatflux for the bottom boundary.
+[;H = 1;], no-heatflux for the bottom boundary.
Modified: doc/geodynamics.org/benchmarks/trunk/mc/3d-spherical/index.html
===================================================================
--- doc/geodynamics.org/benchmarks/trunk/mc/3d-spherical/index.html 2009-10-24 00:18:25 UTC (rev 15871)
+++ doc/geodynamics.org/benchmarks/trunk/mc/3d-spherical/index.html 2009-10-24 00:18:37 UTC (rev 15872)
@@ -12,13 +12,13 @@
<div class="document">
-<p>Presents initial benchmark results for CitcomS and comparisons with Stemmer et
-al {2006} and Ratcliff et al {1996} and Yoshida and Kageyama {2004} when the
-solutions are available from these codes. The community's comments and
-suggestions are solicited (use "add comment" feature at the bottom of the
-page).</p>
+<p>Presents initial benchmark results for CitcomS and
+comparisons with Stemmer et al {2006} and Ratcliff et al {1996}
+and Yoshida and Kageyama {2004} when the solutions are available
+from these codes. The community's comments and suggestions are
+solicited (use "add comment" feature at the bottom of the page).</p>
<div class="section" id="introduction">
-<h1>(1) Introduction</h1>
+<h1>1. Introduction</h1>
<p>There have been quite a lot of new developments in 3D spherical mantle
convection models in the last few years, particularly in Japan and Europe.
It seems to be a good time now to start doing some benchmarks for these
@@ -32,184 +32,226 @@
{2004, GRL} and the previous discussions form a basis for benchmark
calculations presented here and for discussions of our future benchmark
studies.</p>
-<p>Here we present some initial benchmark results for CitcomS (20 cases so
-far) and comparisons with Stemmer et al {2006} and Ratcliff et al {1996}
-and Yoshida and Kageyama {2004} when the solutions are available (9 of
-them) from these codes. We define benchmark cases and present averaged
-global quantities as well as local properties for benchmark cases. At the
-moment, all the cases are for entirely basal heating and steady state or
-weakly time-dependent results, and viscosity variations from
-temperature-dependence vary from 1 (mobile-lid regime) to 1e6 (stagnant lid
-regime). And Ra<sub>0.5</sub> defined by viscosity at T=0.5 varies from 7e3
-to 1e5 (i.e., to achieve steady state or weakly time-dependent state).</p>
+<p>Here we present some initial benchmark results for CitcomS
+(20 cases so far) and comparisons with Stemmer et al {2006}
+and Ratcliff et al {1996} and Yoshida and Kageyama {2004} when
+the solutions are available (9 of them) from these codes. We define
+benchmark cases and present averaged global quantities as well as
+local properties for benchmark cases. At the moment, all the cases
+are for entirely basal heating and steady state or weakly time-dependent
+results, and viscosity variations from temperature-dependence vary from
+1 (mobile-lid regime) to 10<sup>6</sup> (stagnant lid regime). And [;Ra_{0.5};]
+defined by viscosity at [;T=0.5;] varies from 7×10<sup>3</sup> to 10<sup>5</sup> (i.e., to
+achieve steady state or weakly time-dependent state).</p>
<p>We hope that the community will join our effort and discuss what other
benchmark cases should be included in our future benchmarks (e.g., internal
heating, time-dependent cases at relatively high Ra, ...).</p>
</div>
<div class="section" id="definition-of-benchmark-cases">
-<h1>(2) Definition of Benchmark Cases</h1>
+<h1>2. Definition of Benchmark Cases</h1>
<div class="section" id="general-features">
-<h2>(2.1) General features</h2>
+<h2>2.1 General features</h2>
<p>All the cases assume the Boussinesq approximation and constant material
properties except for viscosity. There are no phase transitions, no
compositional anomalies, and no internal heating (i.e., entirely basal
heating) for all the cases. The top and bottom boundaries have radii
-r<sub>t</sub>=1 and r<sub>b</sub>=0.55, respectively.</p>
+[;r_t=1;] and [;r_b=0.55;], respectively.</p>
</div>
<div class="section" id="viscosity-rayleigh-number-and-scalings">
-<h2>(2.2) Viscosity, Rayleigh number and scalings.</h2>
+<h2>2.2 Viscosity, Rayleigh number and scalings.</h2>
<p>The viscosity can be temperature-dependent and the non-dimensional
viscosity, eta, is given by</p>
<blockquote>
-image:: visc.gif (1)</blockquote>
-<p>where E is the activation energy and T is non-dimensional temperature.
-Rayleigh number is defined as</p>
+<!-- from visc.gif (1) -->
+[;\eta = \exp\left[E(0.5 - T)\right],\qquad(1);]</blockquote>
+<p>where [;E;] is the activation energy and [;T;] is non-dimensional
+temperature.</p>
+<p>The Rayleigh number is defined as</p>
<blockquote>
-image:: Ra.gif (2)</blockquote>
-<p>where d is the thickness of the mantle (i.e., the length scale) and <font
-face="Symbol">h</font><sub>0.5</sub> is the viscosity at T=0.5. That is, Ra
-is what we sometimes call Ra<sub>0.5</sub>. This also implies that time and
-velocity scalings are defined using the thickness of the mantle, which is
-somewhat different from those in CitcomS which uses planetary radius as the
-length scale. All the CitcomS results in this benchmark study are re-scaled
-to use the mantle thickness as length scaling, to be consistent with those
-used in other studies.</p>
+<!-- from Ra.gif (2) -->
+[;Ra =
+\frac{\rho g \alpha \Delta{T} d^3}{\eta_{0.5}\kappa}
+\qquad(2);]</blockquote>
+<p>where [;d;] is the thickness of the mantle (i.e., the length scale)
+and [;\eta_{0.5};] is the viscosity at [;T=0.5;]. That is, [;Ra;]
+is what we sometimes call [;Ra_{0.5};]. This also implies that time
+and velocity scalings are defined using the thickness of the mantle,
+which is somewhat different from those in CitcomS which uses planetary
+radius as the length scale. All the CitcomS results in this benchmark
+study are re-scaled to use the mantle thickness as length scaling,
+to be consistent with those used in other studies.</p>
</div>
<div class="section" id="boundary-conditions">
-<h2>(2.3) Boundary conditions</h2>
+<h2>2.3 Boundary conditions</h2>
<p>Boundary conditions are free-slip at the top and bottom boundaries and
isothermal with non-dimensional temperatures of 0 and 1 at the top and
bottom boundaries, respectively.</p>
</div>
<div class="section" id="initial-conditions">
-<h2>(2.4) Initial conditions.</h2>
-<p>Initial conditions come in two types. While most cases use initial
-temperature that is given as a function of coordinates (i.e., perturbation
-at some given spherical harmonics superimposed on a conductive temperature
-profile), some other cases use temperature field from other calculations as
-initial temperature. For the first type of initial condition, the initial
-temperature is given by</p>
+<h2>2.4 Initial conditions</h2>
+<p>Initial conditions come in two types. While most cases use an
+initial temperature that is given as a function of coordinates
+(i.e., perturbation at some given spherical harmonics superimposed
+on a conductive temperature profile), some other cases use temperature
+field from other calculations as initial temperature. For the first
+type of initial condition, the initial temperature is given by</p>
<blockquote>
-image:: IC.gif (3)</blockquote>
-<p>where <font face="Symbol">e</font><sub>c</sub> and <font
-face="Symbol">e</font><sub>s</sub> are the magnitudes of perturbation for
-cosine and sinine terms respectively, l and m are spherical harmonic degree
-and order, r, <font face="Symbol">q</font>, and <font
-face="Symbol">f</font> are radial, co-latitude and longitude coordinates,
-and p<sub>lm</sub> is a normalized associated Legendre polynomial that is
-related to the associated Legendre polynomial P<sub>lm</sub> as:</p>
+<!-- from IC.gif (3) -->
+[;T(r, \theta, \phi) =
+\frac{r_b}{(r_b - r_t)}\left(1 - r_t/r\right) +
+\left[
+\varepsilon_{c} \cos(m\phi) +
+\varepsilon_{s} \sin(m\phi)
+\right]
+p_{lm}(\theta, \phi)
+\sin\left[\pi\left(\frac{r-r_b}{r_t-r_b}\right)\right]
+\qquad(3)
+;]</blockquote>
+<p>where [;\varepsilon_{c};] and [;\varepsilon_{s};] are the magnitudes
+of perturbation for cosine and sine terms respectively, [;l;] and [;m;]
+are spherical harmonic degree and order, [;r;], [;\theta;], and [;\phi;]
+are radial, co-latitude and longitude coordinates, and [;p_{lm};] is a
+normalized associated Legendre polynomial that is related to the
+associated Legendre polynomial [;P_{lm};] as:</p>
<blockquote>
-image:: plm.gif (4)</blockquote>
-<p>The magnitude of the perturbations (i.e., <font
-face="Symbol">e</font><sub>c</sub> and <font
-face="Symbol">e</font><sub>s</sub>) are 0.01, but for some cases, we only
-use the cosine term with <font face="Symbol">e</font><sub>s</sub>=0.0. For
-some cases, perturbations can be given at two different sets of harmonics
-(e.g., l=4, m=0 and l=4, m=4 for all the cubic symmetry cases). Detailed
-initial conditions for each case will be discussed later.</p>
+<!-- from plm.gif (4) -->
+[;p_{lm}(\theta, \phi) =
+\sqrt{\frac{(2l+1)(l-m)!}{(1+\delta_{m0})2\pi(l+m)!}}
+P_{lm}(\theta, \phi)
+\qquad(4)
+;]</blockquote>
+<p>The magnitude of the perturbations (i.e., [;\varepsilon_{c};] and
+[;\varepsilon_{s};]) are 0.01, but for some cases, we only use
+the cosine term with [;\varepsilon_{s}=0.0;]. For some cases,
+perturbations can be given at two different sets of harmonics
+(e.g., [;l=4;], [;m=0;] and [;l=4;], [;m=4;] for all the cubic
+symmetry cases). Detailed initial conditions for each case
+will be discussed later.</p>
<p>We do not use random perturbations for benchmark studies here.</p>
</div>
</div>
<div class="section" id="quantifying-outputs-of-benchmarks">
-<h1>(3) Quantifying Outputs of Benchmarks</h1>
+<h1>3. Quantifying Outputs of Benchmarks</h1>
<p>All the cases are computed to a steady state for global properties. Only
steady state results are quantified. For some cases, we also give an output
file that lists time-dependence of the results for better comparisons with
other codes.</p>
<div class="section" id="global-properties">
-<h2>(3.1) Global properties</h2>
+<h2>3.1 Global properties</h2>
<p>For every 20 timesteps, we compute Nusselt numbers for both the top and
-bottom boundaries, averaged temperature for the whole mantle or shell <font
-face="Symbol">&#225;</font>T<font face="Symbol">&#241;</font>, and averaged
-RMS velocity <font face="Symbol">&#225;</font>v<sub>rms</sub><font
-face="Symbol">&#241;</font>.</p>
+bottom boundaries, averaged temperature for the whole mantle or shell
+[;<T>;], and averaged RMS velocity [;<{\smashmargin2{v_{rms}}>;].</p>
<blockquote>
-<p>image:: T.gif (5)</p>
-<p>image:: vrms.gif (6)</p>
+<!-- from T.gif (5) -->
+<p>[;<T> =
+\frac{\int_{\Omega}\ T\ d\Omega}{\int_{\Omega} d\Omega},
+\qquad(5)
+;]</p>
+<!-- from vrms.gif (6) -->
+<p>[;<{\smashmargin2{v_{rms}}}> =
+\left[
+\frac{\int_{\Omega}\ v^2\ d\Omega}{\int_{\Omega} d\Omega}
+\right]^{1/2}
+\qquad(6)
+;]</p>
</blockquote>
<p>When a case reaches a steady state, we then compute the time-averaged
values and standard deviation of Nusselt numbers, averaged temperature, and
averaged RMS velocity over a certain period of time.</p>
</div>
<div class="section" id="local-properties">
-<h2>(3.2) Local properties</h2>
+<h2>3.2 Local properties</h2>
<p>For every 20 timesteps, we also compute the maximum and minimum radial
velocity and temperature at the middle depth of the mantle (i.e., at
-r=0.775). Again, we compute their time-averaged values and standard
+[;r=0.775;]). Again, we compute their time-averaged values and standard
deviations over a certain period of time after a steady state is reached.</p>
<p>These global and local properties are also used in Stemmer et al {2006}
in their benchmark study.</p>
</div>
</div>
<div class="section" id="results">
-<h1>(4) Results</h1>
+<h1>4. Results</h1>
<p>Two sequences of cases are presented here, one associated with tetrahedral
-symmetry and the other with cubic symmetry. Ra<sub>0.5</sub> ranges from
+symmetry and the other with cubic symmetry. [;Ra_{0.5};] ranges from
7e3 to 1e5 and viscosity variations due to temperature-dependence ranges
-from 1 (i.e., isoviscous) to 1e6 in stagnant lid regime. These cases are
+from 1 (i.e., isoviscous) to 10<sup>6</sup> in stagnant lid regime. These cases are
similar to what Ratcliff et al {1996} and Stemmer et al. {2006} had
presented. However, we added more cases with larger viscosity contrasts.
-For cases with Ra<sub>0.5</sub>=7e3, we use 12x32x32x32 elements, and for
-cases with Ra<sub>0.5</sub>=1e5, 12x48x48x48 elements are used. Redial
-resolution is refined near the top and bottom boundary layers.</p>
+For cases with [;Ra_{0.5} = 7\times{}10^{3};], we use 12×32×32×32 elements,
+and for cases with [;Ra_{0.5} = 10^{5};], 12×48×48×48 elements are used.
+Radial resolution is refined near the top and bottom boundary layers.</p>
+<blockquote>
+</blockquote>
<p>In addition to those outputs in Tables, we also provide files that list the
full time-dependence of our standard outputs. For some cases, thermal
structure images are also provided.</p>
<div class="section" id="the-cases-with-tetrahedral-symmetry-and-its-variations">
-<h2>(4.1) The cases with tetrahedral symmetry and its variations</h2>
-<p>For all the cases in this category, we use Ra<sub>0.5</sub>=7e3, but these
-cases have different temperature dependent viscosity, <font
-face="Symbol">Dh</font> of 1, 10, 20, 100, 1000, 1e4, 1e5, and 1e6 (Cases
-BM1A-BM1H in <a href=table1>Table 1</a> for definitions of all cases. The
-initial condition is the same for these cases with <font
-face="Symbol">e</font><sub>c</sub>=<font
-face="Symbol">e</font><sub>s</sub>=0.01 in equation 3. The first three
-cases, BM1A, 1B, and 1C were also presented in Stemmer et al {2006},
-Ratcliff et al. {1996}, and Yoshida and Kageyama {2004}; and Zhong et al.
-{2000} and Bercovici et al. {1989} computed case BM1A. Only the first five
-cases display tetrahedral symmetry. The last two cases, BM1G and 1H are in
-stagnant-lid regime, showing a large number of small plumes. Case BM1F with
-<font face="Symbol">Dh</font> of 1e4 is a transitional case.</p>
+<h2>4.1 The cases with tetrahedral symmetry and its variations</h2>
+<p>For all the cases in this category, we use
+[;Ra_{0.5}=7\times{}10^3;],
+but these cases have different temperature dependent viscosity,
+[;\Delta\eta;] of 1, 10, 20, 100, 1000, 10<sup>4</sup>, 10<sup>5</sup>, and 10<sup>6</sup>
+(Cases BM1A-BM1H in <a href=table1>Table 1</a> for definitions
+of all cases.) The initial condition is the same for these cases
+with [;\varepsilon_{c}=\varepsilon_{s}=0.01;] in equation 3.
+The first three cases, BM1A, 1B, and 1C were also presented in
+Stemmer et al {2006}, Ratcliff et al. {1996},
+and Yoshida and Kageyama {2004}; and Zhong et al. {2000}
+and Bercovici et al. {1989} computed case BM1A.
+Only the first five cases display tetrahedral symmetry.
+The last two cases, BM1G and 1H are in stagnant-lid regime,
+showing a large number of small plumes. Case BM1F with
+[;\Delta\eta;] of 10<sup>4</sup> is a transitional case.</p>
<p><a href=table2>Table 2</a> shows the output results and comparisons with
previous studies when available. Note that in addition to average values,
Table 2 also gives the time duration over which the averages are computed
and standard deviations. It is clear that for Cases BM1A, 1B and 1C,
CitcomS' results compare well with Stemmer et al. {2006}, Ratcliff et al.
{1996} and Yoshida and Kageyama {2004}.</p>
-<p>Files for time-dependence of the outputs from t=0 to steady state and
+<p>Files for time-dependence of the outputs from [;t=0;] to steady state and
representative snapshots of residual temperature field for selective cases
-can be downloaded at <a
-href="<a class="reference external" href="http://anquetil.colorado.edu/szhong/CitcomS.html">http://anquetil.colorado.edu/szhong/CitcomS.html</a>">Thermal Convection
+can be downloaded at
+<a href="<a class="reference external" href="http://anquetil.colorado.edu/szhong/CitcomS.html">http://anquetil.colorado.edu/szhong/CitcomS.html</a>">Thermal Convection
Benchmarks for CitcomS</a>. The format of these output files is also
explained there.</p>
</div>
<div class="section" id="the-cases-with-cubic-symmetry-and-its-variations">
-<h2>(4.2) The cases with cubic symmetry and its variations</h2>
-<p>For cubic symmetry cases with Ra<sub>0.5</sub>=7e3, we use the initial
-condition similar to Ratcliff et al. {1996}:</p>
+<h2>4.2 The cases with cubic symmetry and its variations</h2>
+<p>For cubic symmetry cases with [;Ra_{0.5} = 7\times{}10^{3};],
+we use the initial condition similar to Ratcliff et al. {1996}:</p>
<blockquote>
-image:: IC40.gif (7)</blockquote>
-<p>where sin(m <font face="Symbol">f</font>) is excluded. We have computed
-cases with temperature dependent viscosity, <font face="Symbol">Dh</font>
-of 1, 20, 30, 100, 1000, 1e4, 1e5 and 1e6 (Cases BM2A-BM2H in <a
-href=table1>Table 1</a> for definitions of all cases). The first three
-cases, BM2A, 2B, and 2C, can again be compared with previous studies. The
-first six cases (BM2A-BM2F) display cubic symmetry with 6 upwelling plumes.
-Different from tetrahedral cases, now BM2F with <font
-face="Symbol">Dh</font> of 1e4 also shows flow cubic symmetry, the same as
+<!-- IC40.gif (7) -->
+[;T(r, \theta, \phi) =
+\frac{r_b}{(r_b - r_t)}\left(1 - r_t/r\right) +
+\varepsilon_{c}
+\left[
+p_{40}(\theta,\phi) +
+\frac{5}{7} \cos(4\phi) p_{44}(\theta,\phi)
+\right]
+p_{lm}(\theta,\phi)
+\sin\left[\pi\left(\frac{r-r_b}{r_t-r_b}\right)\right]
+;]</blockquote>
+<p>where [;\sin(m\phi);] is excluded. We have computed
+cases with temperature dependent viscosity, [;\Delta\eta;]
+of 1, 20, 30, 100, 1000, 10<sup>4</sup>, 10<sup>5</sup> and 10<sup>6</sup> (Cases BM2A-BM2H in
+<a href=table1>Table 1</a> for definitions of all cases).
+The first three cases, BM2A, 2B, and 2C, can again be compared with
+previous studies. The first six cases (BM2A-BM2F) display cubic symmetry
+with 6 upwelling plumes. Different from tetrahedral cases, now BM2F
+with [;\Delta\eta;] of 10<sup>4</sup> also shows flow cubic symmetry, the same as
cases with smaller viscosity contrast. The last two cases BM2G and BM2H are
again in stagnant lid regime, with flow pattern breaking away from the
cubic symmetry. Notice that the slight difference in Nu between BM1 and BM2
-cases for the same Ra.</p>
+cases for the same [;Ra;].</p>
<p><a href=table3>Table 3</a> shows the output results and comparisons with
previous studies when available. Again, CitcomS compares well with all the
previous studies for BM2A, BM2B and BM2C.</p>
-<p>For cases with Ra<sub>0.5</sub>=1e5, we have computed cases with
-temperature dependent viscosity, <font face="Symbol">Dh</font> of 1, 10,
-30, and 100, so far (cases BM3A-3D, in <a href=table1>Table 1</a>). We use
-the same initial condition as in equation 7 only for the case with
-uniform viscosity that leads to cubic symmetry flow. For other
-Ra<sub>0.5</sub>=1e5 cases with variable viscosity, we use the final
+<p>For cases with [;Ra_{0.5} = 10^{5};], we have computed cases with
+temperature dependent viscosity, [;\Delta\eta;] of 1, 10,
+30, and 100, so far (cases BM3A-3D, in <a href=table1>Table 1</a>).
+We use the same initial condition as in equation 7 only for the case
+with uniform viscosity that leads to cubic symmetry flow. For other
+[;Ra_{0.5} = 10^{5};] cases with variable viscosity, we use the final
temperature field from either the uniform viscosity case (i.e., BM3A)
or other cases as initial condition to help preserve the cubic symmetry
flow. <a href=table1>Table 1</a> lists the detailed initial conditions.
@@ -227,30 +269,30 @@
{1996} used the same resolution (i.e., 32, 64, and 128 cells in radial,
co-latitude and longitude directions, respectively, for cubic symmetry
cases) for different Ra cases. We think that cases with
-Ra<sub>0.5</sub>=1e5 should require higher resolution.</p>
+[;Ra_{0.5} = 10^{5};] should require higher resolution.</p>
<p>Again, files for time-dependence of the outputs from t=0 to steady state
and representative snapshots of residual temperature field for selective
-cases can be downloaded at <a
-href="<a class="reference external" href="http://anquetil.colorado.edu/szhong/CitcomS.html">http://anquetil.colorado.edu/szhong/CitcomS.html</a>">Thermal Convection
+cases can be downloaded at
+<a href="<a class="reference external" href="http://anquetil.colorado.edu/szhong/CitcomS.html">http://anquetil.colorado.edu/szhong/CitcomS.html</a>">Thermal Convection
Benchmarks for CitcomS</a>.</p>
</div>
</div>
<div class="section" id="conclusions">
-<h1>(5) Conclusions</h1>
+<h1>5. Conclusions</h1>
<p>Here we present results of Nussult numbers, RMS velocity, averaged
temperature, and maximum and minimum flow velocity and temperature at the
mid-mantle depth for 20 cases from CitcomS. For these cases,
-Ra<sub>0.5</sub> is either 7e3 or 1e5, and viscosity contrast varies from 1
-to 1e6. The style of convection varies from mobile lid to stagnant lid
-regimes. For nine of the 20 cases, we could compare with previously
-published results {Stemmer et al., 2006; Ratcliff et al., 1996; Yoshida and
-Kageyama, 2004}. Comparisons show that CitcomS's results are generally
-consistent with these previous studies, although at high Ra, the relatively
-low resolution from previous studies may degrade the agreement. Time
-dependent results from CitcomS are compiled into a file for each case, and
-all of them can be downloaded for more detailed comparisons. Compared with
-benchmarks in Zhong et al. {2000}, here we added significantly more thermal
-convection cases.</p>
+[;Ra_{0.5};] is either 7×10<sup>3</sup> or 10<sup>5</sup>, and viscosity contrast varies
+from 1 to 10<sup>6</sup>. The style of convection varies from mobile lid to
+stagnant lid regimes. For nine of the 20 cases, we could compare with
+previously published results {Stemmer et al., 2006; Ratcliff et al., 1996;
+Yoshida and Kageyama, 2004}. Comparisons show that CitcomS's results are
+generally consistent with these previous studies, although at high [;Ra;],
+the relatively low resolution from previous studies may degrade the agreement.
+Time dependent results from CitcomS are compiled into a file for each case,
+and all of them can be downloaded for more detailed comparisons. Compared
+with benchmarks in Zhong et al. {2000}, here we added significantly more
+thermal convection cases.</p>
<p>We encourage our colleagues to send in their results for these cases. We
think that more benchmark cases are needed to test time-dependent
convection at even higher Ra, and internal heating convection. We welcome
Modified: doc/geodynamics.org/benchmarks/trunk/mc/3d-spherical/index.rst
===================================================================
--- doc/geodynamics.org/benchmarks/trunk/mc/3d-spherical/index.rst 2009-10-24 00:18:25 UTC (rev 15871)
+++ doc/geodynamics.org/benchmarks/trunk/mc/3d-spherical/index.rst 2009-10-24 00:18:37 UTC (rev 15872)
@@ -1,12 +1,21 @@
-Presents initial benchmark results for CitcomS and comparisons with Stemmer et
-al {2006} and Ratcliff et al {1996} and Yoshida and Kageyama {2004} when the
-solutions are available from these codes. The community's comments and
-suggestions are solicited (use "add comment" feature at the bottom of the
-page).
-(1) Introduction
-=================
+.. |times| unicode:: U+00D7
+.. |1e4| replace:: 10\ :sup:`4`
+.. |1e5| replace:: 10\ :sup:`5`
+.. |1e6| replace:: 10\ :sup:`6`
+.. |7e3| replace:: 7\ |times|\ 10\ :sup:`3`
+
+Presents initial benchmark results for CitcomS and
+comparisons with Stemmer et al {2006} and Ratcliff et al {1996}
+and Yoshida and Kageyama {2004} when the solutions are available
+from these codes. The community's comments and suggestions are
+solicited (use "add comment" feature at the bottom of the page).
+
+
+1. Introduction
+===============
+
There have been quite a lot of new developments in 3D spherical mantle
convection models in the last few years, particularly in Japan and Europe.
It seems to be a good time now to start doing some benchmarks for these
@@ -21,164 +30,212 @@
calculations presented here and for discussions of our future benchmark
studies.
-Here we present some initial benchmark results for CitcomS (20 cases so
-far) and comparisons with Stemmer et al {2006} and Ratcliff et al {1996}
-and Yoshida and Kageyama {2004} when the solutions are available (9 of
-them) from these codes. We define benchmark cases and present averaged
-global quantities as well as local properties for benchmark cases. At the
-moment, all the cases are for entirely basal heating and steady state or
-weakly time-dependent results, and viscosity variations from
-temperature-dependence vary from 1 (mobile-lid regime) to 1e6 (stagnant lid
-regime). And Ra<sub>0.5</sub> defined by viscosity at T=0.5 varies from 7e3
-to 1e5 (i.e., to achieve steady state or weakly time-dependent state).
+Here we present some initial benchmark results for CitcomS
+(20 cases so far) and comparisons with Stemmer et al {2006}
+and Ratcliff et al {1996} and Yoshida and Kageyama {2004} when
+the solutions are available (9 of them) from these codes. We define
+benchmark cases and present averaged global quantities as well as
+local properties for benchmark cases. At the moment, all the cases
+are for entirely basal heating and steady state or weakly time-dependent
+results, and viscosity variations from temperature-dependence vary from
+1 (mobile-lid regime) to |1e6| (stagnant lid regime). And [;Ra_{0.5};]
+defined by viscosity at [;T=0.5;] varies from |7e3| to |1e5| (i.e., to
+achieve steady state or weakly time-dependent state).
We hope that the community will join our effort and discuss what other
benchmark cases should be included in our future benchmarks (e.g., internal
heating, time-dependent cases at relatively high Ra, ...).
-(2) Definition of Benchmark Cases
-=================================
+2. Definition of Benchmark Cases
+================================
-(2.1) General features
-----------------------
+2.1 General features
+--------------------
All the cases assume the Boussinesq approximation and constant material
properties except for viscosity. There are no phase transitions, no
compositional anomalies, and no internal heating (i.e., entirely basal
heating) for all the cases. The top and bottom boundaries have radii
-r<sub>t</sub>=1 and r<sub>b</sub>=0.55, respectively.
+[;r_t=1;] and [;r_b=0.55;], respectively.
-(2.2) Viscosity, Rayleigh number and scalings.
-----------------------------------------------
+2.2 Viscosity, Rayleigh number and scalings.
+--------------------------------------------
The viscosity can be temperature-dependent and the non-dimensional
viscosity, eta, is given by
- image:: visc.gif (1)
+ .. from visc.gif (1)
-where E is the activation energy and T is non-dimensional temperature.
-Rayleigh number is defined as
+ [;\\eta = \\exp\\left[E(0.5 - T)\\right],\\qquad(1);]
- image:: Ra.gif (2)
+where [;E;] is the activation energy and [;T;] is non-dimensional
+temperature.
-where d is the thickness of the mantle (i.e., the length scale) and <font
-face="Symbol">h</font><sub>0.5</sub> is the viscosity at T=0.5. That is, Ra
-is what we sometimes call Ra<sub>0.5</sub>. This also implies that time and
-velocity scalings are defined using the thickness of the mantle, which is
-somewhat different from those in CitcomS which uses planetary radius as the
-length scale. All the CitcomS results in this benchmark study are re-scaled
-to use the mantle thickness as length scaling, to be consistent with those
-used in other studies.
+The Rayleigh number is defined as
-(2.3) Boundary conditions
--------------------------
+ .. from Ra.gif (2)
+ [;Ra =
+ \\frac{\\rho g \\alpha \\Delta{T} d^3}{\\eta_{0.5}\\kappa}
+ \\qquad(2);]
+
+where [;d;] is the thickness of the mantle (i.e., the length scale)
+and [;\\eta_{0.5};] is the viscosity at [;T=0.5;]. That is, [;Ra;]
+is what we sometimes call [;Ra_{0.5};]. This also implies that time
+and velocity scalings are defined using the thickness of the mantle,
+which is somewhat different from those in CitcomS which uses planetary
+radius as the length scale. All the CitcomS results in this benchmark
+study are re-scaled to use the mantle thickness as length scaling,
+to be consistent with those used in other studies.
+
+
+2.3 Boundary conditions
+-----------------------
+
Boundary conditions are free-slip at the top and bottom boundaries and
isothermal with non-dimensional temperatures of 0 and 1 at the top and
bottom boundaries, respectively.
-(2.4) Initial conditions.
--------------------------
-Initial conditions come in two types. While most cases use initial
-temperature that is given as a function of coordinates (i.e., perturbation
-at some given spherical harmonics superimposed on a conductive temperature
-profile), some other cases use temperature field from other calculations as
-initial temperature. For the first type of initial condition, the initial
-temperature is given by
+2.4 Initial conditions
+----------------------
- image:: IC.gif (3)
+Initial conditions come in two types. While most cases use an
+initial temperature that is given as a function of coordinates
+(i.e., perturbation at some given spherical harmonics superimposed
+on a conductive temperature profile), some other cases use temperature
+field from other calculations as initial temperature. For the first
+type of initial condition, the initial temperature is given by
-where <font face="Symbol">e</font><sub>c</sub> and <font
-face="Symbol">e</font><sub>s</sub> are the magnitudes of perturbation for
-cosine and sinine terms respectively, l and m are spherical harmonic degree
-and order, r, <font face="Symbol">q</font>, and <font
-face="Symbol">f</font> are radial, co-latitude and longitude coordinates,
-and p<sub>lm</sub> is a normalized associated Legendre polynomial that is
-related to the associated Legendre polynomial P<sub>lm</sub> as:
+ .. from IC.gif (3)
- image:: plm.gif (4)
+ [;T(r, \\theta, \\phi) =
+ \\frac{r_b}{(r_b - r_t)}\\left(1 - r_t/r\\right) +
+ \\left[
+ \\varepsilon_{c} \\cos(m\\phi) +
+ \\varepsilon_{s} \\sin(m\\phi)
+ \\right]
+ p_{lm}(\\theta, \\phi)
+ \\sin\\left[\\pi\\left(\\frac{r-r_b}{r_t-r_b}\\right)\\right]
+ \\qquad(3)
+ ;]
-The magnitude of the perturbations (i.e., <font
-face="Symbol">e</font><sub>c</sub> and <font
-face="Symbol">e</font><sub>s</sub>) are 0.01, but for some cases, we only
-use the cosine term with <font face="Symbol">e</font><sub>s</sub>=0.0. For
-some cases, perturbations can be given at two different sets of harmonics
-(e.g., l=4, m=0 and l=4, m=4 for all the cubic symmetry cases). Detailed
-initial conditions for each case will be discussed later.
+where [;\\varepsilon_{c};] and [;\\varepsilon_{s};] are the magnitudes
+of perturbation for cosine and sine terms respectively, [;l;] and [;m;]
+are spherical harmonic degree and order, [;r;], [;\\theta;], and [;\\phi;]
+are radial, co-latitude and longitude coordinates, and [;p_{lm};] is a
+normalized associated Legendre polynomial that is related to the
+associated Legendre polynomial [;P_{lm};] as:
+ .. from plm.gif (4)
+
+ [;p_{lm}(\\theta, \\phi) =
+ \\sqrt{\\frac{(2l+1)(l-m)!}{(1+\\delta_{m0})2\\pi(l+m)!}}
+ P_{lm}(\\theta, \\phi)
+ \\qquad(4)
+ ;]
+
+The magnitude of the perturbations (i.e., [;\\varepsilon_{c};] and
+[;\\varepsilon_{s};]) are 0.01, but for some cases, we only use
+the cosine term with [;\\varepsilon_{s}=0.0;]. For some cases,
+perturbations can be given at two different sets of harmonics
+(e.g., [;l=4;], [;m=0;] and [;l=4;], [;m=4;] for all the cubic
+symmetry cases). Detailed initial conditions for each case
+will be discussed later.
+
We do not use random perturbations for benchmark studies here.
-(3) Quantifying Outputs of Benchmarks
-=====================================
+3. Quantifying Outputs of Benchmarks
+====================================
+
All the cases are computed to a steady state for global properties. Only
steady state results are quantified. For some cases, we also give an output
file that lists time-dependence of the results for better comparisons with
other codes.
-(3.1) Global properties
------------------------
+3.1 Global properties
+---------------------
+
For every 20 timesteps, we compute Nusselt numbers for both the top and
-bottom boundaries, averaged temperature for the whole mantle or shell <font
-face="Symbol">á</font>T<font face="Symbol">ñ</font>, and averaged
-RMS velocity <font face="Symbol">á</font>v<sub>rms</sub><font
-face="Symbol">ñ</font>.
+bottom boundaries, averaged temperature for the whole mantle or shell
+[;<T>;], and averaged RMS velocity [;<{\\smashmargin2{v_{rms}}>;].
- image:: T.gif (5)
+ .. from T.gif (5)
- image:: vrms.gif (6)
+ [;<T> =
+ \\frac{\\int_{\\Omega}\\ T\\ d\\Omega}{\\int_{\\Omega} d\\Omega},
+ \\qquad(5)
+ ;]
+
+ .. from vrms.gif (6)
+
+ [;<{\\smashmargin2{v_{rms}}}> =
+ \\left[
+ \\frac{\\int_{\\Omega}\\ v^2\\ d\\Omega}{\\int_{\\Omega} d\\Omega}
+ \\right]^{1/2}
+ \\qquad(6)
+ ;]
+
When a case reaches a steady state, we then compute the time-averaged
values and standard deviation of Nusselt numbers, averaged temperature, and
averaged RMS velocity over a certain period of time.
-(3.2) Local properties
-----------------------
+3.2 Local properties
+--------------------
+
For every 20 timesteps, we also compute the maximum and minimum radial
velocity and temperature at the middle depth of the mantle (i.e., at
-r=0.775). Again, we compute their time-averaged values and standard
+[;r=0.775;]). Again, we compute their time-averaged values and standard
deviations over a certain period of time after a steady state is reached.
These global and local properties are also used in Stemmer et al {2006}
in their benchmark study.
-(4) Results
-===========
+4. Results
+==========
+
Two sequences of cases are presented here, one associated with tetrahedral
-symmetry and the other with cubic symmetry. Ra<sub>0.5</sub> ranges from
+symmetry and the other with cubic symmetry. [;Ra_{0.5};] ranges from
7e3 to 1e5 and viscosity variations due to temperature-dependence ranges
-from 1 (i.e., isoviscous) to 1e6 in stagnant lid regime. These cases are
+from 1 (i.e., isoviscous) to |1e6| in stagnant lid regime. These cases are
similar to what Ratcliff et al {1996} and Stemmer et al. {2006} had
presented. However, we added more cases with larger viscosity contrasts.
-For cases with Ra<sub>0.5</sub>=7e3, we use 12x32x32x32 elements, and for
-cases with Ra<sub>0.5</sub>=1e5, 12x48x48x48 elements are used. Redial
-resolution is refined near the top and bottom boundary layers.
+For cases with [;Ra_{0.5} = 7\\times{}10^{3};], we use |12x32x32x32| elements,
+and for cases with [;Ra_{0.5} = 10^{5};], |12x48x48x48| elements are used.
+Radial resolution is refined near the top and bottom boundary layers.
+ .. |12x32x32x32| replace:: 12\ |times|\ 32\ |times|\ 32\ |times|\ 32
+ .. |12x48x48x48| replace:: 12\ |times|\ 48\ |times|\ 48\ |times|\ 48
+
In addition to those outputs in Tables, we also provide files that list the
full time-dependence of our standard outputs. For some cases, thermal
structure images are also provided.
-(4.1) The cases with tetrahedral symmetry and its variations
-------------------------------------------------------------
-For all the cases in this category, we use Ra<sub>0.5</sub>=7e3, but these
-cases have different temperature dependent viscosity, <font
-face="Symbol">Dh</font> of 1, 10, 20, 100, 1000, 1e4, 1e5, and 1e6 (Cases
-BM1A-BM1H in <a href=table1>Table 1</a> for definitions of all cases. The
-initial condition is the same for these cases with <font
-face="Symbol">e</font><sub>c</sub>=<font
-face="Symbol">e</font><sub>s</sub>=0.01 in equation 3. The first three
-cases, BM1A, 1B, and 1C were also presented in Stemmer et al {2006},
-Ratcliff et al. {1996}, and Yoshida and Kageyama {2004}; and Zhong et al.
-{2000} and Bercovici et al. {1989} computed case BM1A. Only the first five
-cases display tetrahedral symmetry. The last two cases, BM1G and 1H are in
-stagnant-lid regime, showing a large number of small plumes. Case BM1F with
-<font face="Symbol">Dh</font> of 1e4 is a transitional case.
+4.1 The cases with tetrahedral symmetry and its variations
+----------------------------------------------------------
+For all the cases in this category, we use
+[;Ra_{0.5}=7\\times{}10^3;],
+but these cases have different temperature dependent viscosity,
+[;\\Delta\\eta;] of 1, 10, 20, 100, 1000, |1e4|, |1e5|, and |1e6|
+(Cases BM1A-BM1H in <a href=table1>Table 1</a> for definitions
+of all cases.) The initial condition is the same for these cases
+with [;\\varepsilon_{c}=\\varepsilon_{s}=0.01;] in equation 3.
+The first three cases, BM1A, 1B, and 1C were also presented in
+Stemmer et al {2006}, Ratcliff et al. {1996},
+and Yoshida and Kageyama {2004}; and Zhong et al. {2000}
+and Bercovici et al. {1989} computed case BM1A.
+Only the first five cases display tetrahedral symmetry.
+The last two cases, BM1G and 1H are in stagnant-lid regime,
+showing a large number of small plumes. Case BM1F with
+[;\\Delta\\eta;] of |1e4| is a transitional case.
+
<a href=table2>Table 2</a> shows the output results and comparisons with
previous studies when available. Note that in addition to average values,
Table 2 also gives the time duration over which the averages are computed
@@ -186,44 +243,56 @@
CitcomS' results compare well with Stemmer et al. {2006}, Ratcliff et al.
{1996} and Yoshida and Kageyama {2004}.
-Files for time-dependence of the outputs from t=0 to steady state and
+Files for time-dependence of the outputs from [;t=0;] to steady state and
representative snapshots of residual temperature field for selective cases
-can be downloaded at <a
-href="http://anquetil.colorado.edu/szhong/CitcomS.html">Thermal Convection
+can be downloaded at
+<a href="http://anquetil.colorado.edu/szhong/CitcomS.html">Thermal Convection
Benchmarks for CitcomS</a>. The format of these output files is also
explained there.
-(4.2) The cases with cubic symmetry and its variations
-------------------------------------------------------
-For cubic symmetry cases with Ra<sub>0.5</sub>=7e3, we use the initial
-condition similar to Ratcliff et al. {1996}:
+4.2 The cases with cubic symmetry and its variations
+----------------------------------------------------
- image:: IC40.gif (7)
+For cubic symmetry cases with [;Ra_{0.5} = 7\\times{}10^{3};],
+we use the initial condition similar to Ratcliff et al. {1996}:
-where sin(m <font face="Symbol">f</font>) is excluded. We have computed
-cases with temperature dependent viscosity, <font face="Symbol">Dh</font>
-of 1, 20, 30, 100, 1000, 1e4, 1e5 and 1e6 (Cases BM2A-BM2H in <a
-href=table1>Table 1</a> for definitions of all cases). The first three
-cases, BM2A, 2B, and 2C, can again be compared with previous studies. The
-first six cases (BM2A-BM2F) display cubic symmetry with 6 upwelling plumes.
-Different from tetrahedral cases, now BM2F with <font
-face="Symbol">Dh</font> of 1e4 also shows flow cubic symmetry, the same as
+ .. IC40.gif (7)
+
+ [;T(r, \\theta, \\phi) =
+ \\frac{r_b}{(r_b - r_t)}\\left(1 - r_t/r\\right) +
+ \\varepsilon_{c}
+ \\left[
+ p_{40}(\\theta,\\phi) +
+ \\frac{5}{7} \\cos(4\\phi) p_{44}(\\theta,\\phi)
+ \\right]
+ p_{lm}(\\theta,\\phi)
+ \\sin\\left[\\pi\\left(\\frac{r-r_b}{r_t-r_b}\\right)\\right]
+ ;]
+
+where [;\\sin(m\\phi);] is excluded. We have computed
+cases with temperature dependent viscosity, [;\\Delta\\eta;]
+of 1, 20, 30, 100, 1000, |1e4|, |1e5| and |1e6| (Cases BM2A-BM2H in
+<a href=table1>Table 1</a> for definitions of all cases).
+The first three cases, BM2A, 2B, and 2C, can again be compared with
+previous studies. The first six cases (BM2A-BM2F) display cubic symmetry
+with 6 upwelling plumes. Different from tetrahedral cases, now BM2F
+with [;\\Delta\\eta;] of |1e4| also shows flow cubic symmetry, the same as
cases with smaller viscosity contrast. The last two cases BM2G and BM2H are
again in stagnant lid regime, with flow pattern breaking away from the
cubic symmetry. Notice that the slight difference in Nu between BM1 and BM2
-cases for the same Ra.
+cases for the same [;Ra;].
<a href=table3>Table 3</a> shows the output results and comparisons with
previous studies when available. Again, CitcomS compares well with all the
previous studies for BM2A, BM2B and BM2C.
-For cases with Ra<sub>0.5</sub>=1e5, we have computed cases with
-temperature dependent viscosity, <font face="Symbol">Dh</font> of 1, 10,
-30, and 100, so far (cases BM3A-3D, in <a href=table1>Table 1</a>). We use
-the same initial condition as in equation 7 only for the case with
-uniform viscosity that leads to cubic symmetry flow. For other
-Ra<sub>0.5</sub>=1e5 cases with variable viscosity, we use the final
+For cases with [;Ra_{0.5} = 10^{5};], we have computed cases with
+temperature dependent viscosity, [;\\Delta\\eta;] of 1, 10,
+30, and 100, so far (cases BM3A-3D, in <a href=table1>Table 1</a>).
+We use the same initial condition as in equation 7 only for the case
+with uniform viscosity that leads to cubic symmetry flow. For other
+[;Ra_{0.5} = 10^{5};] cases with variable viscosity, we use the final
temperature field from either the uniform viscosity case (i.e., BM3A)
or other cases as initial condition to help preserve the cubic symmetry
flow. <a href=table1>Table 1</a> lists the detailed initial conditions.
@@ -242,31 +311,32 @@
{1996} used the same resolution (i.e., 32, 64, and 128 cells in radial,
co-latitude and longitude directions, respectively, for cubic symmetry
cases) for different Ra cases. We think that cases with
-Ra<sub>0.5</sub>=1e5 should require higher resolution.
+[;Ra_{0.5} = 10^{5};] should require higher resolution.
Again, files for time-dependence of the outputs from t=0 to steady state
and representative snapshots of residual temperature field for selective
-cases can be downloaded at <a
-href="http://anquetil.colorado.edu/szhong/CitcomS.html">Thermal Convection
+cases can be downloaded at
+<a href="http://anquetil.colorado.edu/szhong/CitcomS.html">Thermal Convection
Benchmarks for CitcomS</a>.
-(5) Conclusions
-===============
+5. Conclusions
+==============
+
Here we present results of Nussult numbers, RMS velocity, averaged
temperature, and maximum and minimum flow velocity and temperature at the
mid-mantle depth for 20 cases from CitcomS. For these cases,
-Ra<sub>0.5</sub> is either 7e3 or 1e5, and viscosity contrast varies from 1
-to 1e6. The style of convection varies from mobile lid to stagnant lid
-regimes. For nine of the 20 cases, we could compare with previously
-published results {Stemmer et al., 2006; Ratcliff et al., 1996; Yoshida and
-Kageyama, 2004}. Comparisons show that CitcomS's results are generally
-consistent with these previous studies, although at high Ra, the relatively
-low resolution from previous studies may degrade the agreement. Time
-dependent results from CitcomS are compiled into a file for each case, and
-all of them can be downloaded for more detailed comparisons. Compared with
-benchmarks in Zhong et al. {2000}, here we added significantly more thermal
-convection cases.
+[;Ra_{0.5};] is either |7e3| or |1e5|, and viscosity contrast varies
+from 1 to |1e6|. The style of convection varies from mobile lid to
+stagnant lid regimes. For nine of the 20 cases, we could compare with
+previously published results {Stemmer et al., 2006; Ratcliff et al., 1996;
+Yoshida and Kageyama, 2004}. Comparisons show that CitcomS's results are
+generally consistent with these previous studies, although at high [;Ra;],
+the relatively low resolution from previous studies may degrade the agreement.
+Time dependent results from CitcomS are compiled into a file for each case,
+and all of them can be downloaded for more detailed comparisons. Compared
+with benchmarks in Zhong et al. {2000}, here we added significantly more
+thermal convection cases.
We encourage our colleagues to send in their results for these cases. We
think that more benchmark cases are needed to test time-dependent
Added: doc/geodynamics.org/benchmarks/trunk/mc/notes.header
===================================================================
--- doc/geodynamics.org/benchmarks/trunk/mc/notes.header (rev 0)
+++ doc/geodynamics.org/benchmarks/trunk/mc/notes.header 2009-10-24 00:18:37 UTC (rev 15872)
@@ -0,0 +1,17 @@
+id: notes
+title: Notes on Mantle Convection Benchmarks
+subject:
+description: description here
+contributors:
+creators: luis
+effectiveDate: None
+expirationDate: None
+language:
+rights:
+creation_date: 2009/10/18 03:26:50.907 GMT-7
+modification_date: 2009/10/23 14:47:54.749 GMT-7
+excludeFromNav: False
+relatedItems:
+allowDiscussion: None
+Content-Type: text/x-rst
+
Added: doc/geodynamics.org/benchmarks/trunk/mc/notes.html
===================================================================
--- doc/geodynamics.org/benchmarks/trunk/mc/notes.html (rev 0)
+++ doc/geodynamics.org/benchmarks/trunk/mc/notes.html 2009-10-24 00:18:37 UTC (rev 15872)
@@ -0,0 +1,84 @@
+<?xml version="1.0" encoding="utf-8" ?>
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en">
+<head>
+<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+<meta name="generator" content="Docutils 0.5: http://docutils.sourceforge.net/" />
+<title>Notes On Mantle Convection Benchmarks</title>
+<script type="text/javascript" src="http://geodynamics.org/cig/portal_javascripts/Plone%20Default/textheworld6.user.js"></script>
+<link rel="stylesheet" href="../css/voidspace.css" type="text/css" />
+</head>
+<body>
+<div class="document" id="notes-on-mantle-convection-benchmarks">
+<h1 class="title">Notes On Mantle Convection Benchmarks</h1>
+
+<ol class="arabic simple">
+<li>Ra = 3×10<sup>6</sup> (I think this is high enough to give some
+real time dependence without pushing available resolution
+very much).</li>
+<li>Constaint properties (thermal expansivity, thermal diffusivity,
+density, gravity, viscosity, internal heat generation)
+to keep things very simple.</li>
+<li>Free slip upper and lower boundaries.</li>
+<li>Radius ratio = 0.546 (cmb/surface radius).</li>
+<li>Purely internally heated.</li>
+<li>Insulating at cmb, constant temperature at surface</li>
+<li>Model resolution: 65 nodes (64 layers) radially, with some
+packing of nodes near the top and bottom boundaries. (We'll
+send you the actual radii we use, assuming you can vary them
+at will.)</li>
+<li>Initial diagnostics: (basically, these are just to get started
+and see if we're in the same universe)<ol class="loweralpha">
+<li>Nu vs. time (this should square with the internal heating
+in a time-average sense).</li>
+<li>Radial temperature profile vs. time - this is effectively a
+measure of the efficiency of heat transfer, or equivalent
+of Nu for bottom heated cases.</li>
+<li>Spherical harmonic expansion of temperature field at all
+radial levels at beginning and ending time (see below).</li>
+<li>Peak velocity and peak temperature in each radial layer vs. time</li>
+<li>For now, let's ignore dynamic topography, since it's derived from
+primitive results</li>
+</ol>
+</li>
+<li>Initial conditions and run time: This is a bit thorny, so here's a
+proposal. We can run TERRA to equilibrium under the specified model
+conditions. Equilibrium is where Nu has settled down to fluctuations
+about a steady mean value. At some point, call it time = 0.0, we'll
+stop the code and output the full temperature field in the form of a
+spherical harmonic expansion up to degree 128, which corresponds
+to the highest model resolution. We can then restart both TERRA
+and CitcomS using this spherical harmonic expansion (NOT the full
+temperature field at each node, since this would prejudice things
+with regard to the particular horizontal discretization.) Then both
+codes can run for a defined amount of model time, keeping track of
+Nu, peak T, and peak V as a function of time as indicated above.
+At the end of this time, or at several times along the way, we can
+output spherical harmonic representations of T at each layer for
+comparison.</li>
+</ol>
+<p>I added the following comments:</p>
+<ol class="arabic simple">
+<li>We use some analytic expressions for initial conditions
+(e.g., some radial profile superimposed with a small perturbation
+of a given harmonic function). In this way, others, if they want
+to benchmark their codes, do not need to get the Terra output.
+Also in case some summary report comes out of this effort, we can
+simply write down the initial conditions.</li>
+<li>We aim to reproduce four benchmark cases in steady of just one.
+The four cases at the moment in my mind can be: three constant
+property cases with purely basal heating at Ra=1e5 (case 1),
+and Ra=1e6 (case 2), and purely internal heating at Ra=1e6 (case 3),
+and one temperature dependent viscosity and purely basal heating
+at Ra=1e6 (case 4).</li>
+</ol>
+<p>Case 1 will likely reach to a steady state, which is always a good thing
+for a benchmark. Cases 2 and 3 are almost identical to what you have
+suggested recently, and they are most likely time-dependent. The 1e6 Ra
+is smaller than what you suggested today but is consistent with your
+earlier suggestion. With Ra=1e6, we may not need grid refinement, which
+is also good for benchmark purposes (again, others can do it later).
+Case 4 is obviously of interest too.</p>
+</div>
+</body>
+</html>
Modified: doc/geodynamics.org/benchmarks/trunk/mc/notes.rst
===================================================================
--- doc/geodynamics.org/benchmarks/trunk/mc/notes.rst 2009-10-24 00:18:25 UTC (rev 15871)
+++ doc/geodynamics.org/benchmarks/trunk/mc/notes.rst 2009-10-24 00:18:37 UTC (rev 15872)
@@ -1,3 +1,5 @@
+.. |times| unicode:: U+00D7
+.. |3e6| replace:: 3\ |times|\ 10\ :sup:`6`
Notes On Mantle Convection Benchmarks
=====================================
@@ -2,67 +4,74 @@
-(1) Ra = 3x10**6 (I think this is high enough to give some real time
-dependence without pushing available resolution very much).
+(1) Ra = |3e6| (I think this is high enough to give some
+ real time dependence without pushing available resolution
+ very much).
+
(2) Constaint properties (thermal expansivity, thermal diffusivity,
-density, gravity, viscosity, internal heat generation) to keep things
-very simple.
+ density, gravity, viscosity, internal heat generation)
+ to keep things very simple.
(3) Free slip upper and lower boundaries.
-(4) Radius ratio = 0.546 (cmb/surface radius)
+(4) Radius ratio = 0.546 (cmb/surface radius).
-(5) purely internally heated
+(5) Purely internally heated.
-(6) insulating at cmb, constant temperature at surface
+(6) Insulating at cmb, constant temperature at surface
-(7) model resolution: 65 nodes (64 layers) radially, with some packing of
-nodes near the top and bottom boundaries. (We'll send you the actual
-radii we use, assuming you can vary them at will.)
+(7) Model resolution: 65 nodes (64 layers) radially, with some
+ packing of nodes near the top and bottom boundaries. (We'll
+ send you the actual radii we use, assuming you can vary them
+ at will.)
-(8) initial diagnostics: (basically, these are just to get started and
-see if we're in the same universe)
+(8) Initial diagnostics: (basically, these are just to get started
+ and see if we're in the same universe)
-* (a) Nu vs. time (this should square with the internal heating in a
- time-average sense)
+ (a) Nu vs. time (this should square with the internal heating
+ in a time-average sense).
-* (b) Radial temperature profile vs. time - this is effectively a
- measure of the efficiency of heat transfer, or equivalent of Nu for
- bottom heated cases.
+ (b) Radial temperature profile vs. time - this is effectively a
+ measure of the efficiency of heat transfer, or equivalent
+ of Nu for bottom heated cases.
-* (c) Spherical harmonic expansion of temperature field at all radial
- levels at beginning and ending time (see below).
+ (c) Spherical harmonic expansion of temperature field at all
+ radial levels at beginning and ending time (see below).
-* (d) peak velocity and peak temperature in each radial layer vs. time
+ (d) Peak velocity and peak temperature in each radial layer vs. time
-* (e) for now, let's ignore dynamic topography, since it's derived from
- primitive results
+ (e) For now, let's ignore dynamic topography, since it's derived from
+ primitive results
(9) Initial conditions and run time: This is a bit thorny, so here's a
-proposal. We can run TERRA to equilibrium under the specified model
-conditions. Equilibrium is where Nu has settled down to fluctuations
-about a steady mean value. At some point, call it time = 0.0, we'll stop
-the code and output the full temperature field in the form of a spherical
-harmonic expansion up to degree 128, which corresponds to the highest
-model resolution. We can then restart both TERRA and CitcomS using this
-spherical harmonic expansion (NOT the full temperature field at each
-node, since this would prejudice things with regard to the particular
-horizontal discretization.) Then both codes can run for a defined amount
-of model time, keeping track of Nu, peak T, and peak V as a function of
-time as indicated above. At the end of this time, or at several times
-along the way, we can output spherical harmonic representations of T at
-each layer for comparison.
+ proposal. We can run TERRA to equilibrium under the specified model
+ conditions. Equilibrium is where Nu has settled down to fluctuations
+ about a steady mean value. At some point, call it time = 0.0, we'll
+ stop the code and output the full temperature field in the form of a
+ spherical harmonic expansion up to degree 128, which corresponds
+ to the highest model resolution. We can then restart both TERRA
+ and CitcomS using this spherical harmonic expansion (NOT the full
+ temperature field at each node, since this would prejudice things
+ with regard to the particular horizontal discretization.) Then both
+ codes can run for a defined amount of model time, keeping track of
+ Nu, peak T, and peak V as a function of time as indicated above.
+ At the end of this time, or at several times along the way, we can
+ output spherical harmonic representations of T at each layer for
+ comparison.
+
I added the following comments:
-(1) We use some analytic expressions for initial conditions (e.g., some
-radial profile superimposed with a small perturbation of a given harmonic
-function). In this way, others, if they want to benchmark their codes, do
-not need to get the Terra output. Also in case some summary report comes
-out of this effort, we can simply write down the initial conditions.
+(1) We use some analytic expressions for initial conditions
+ (e.g., some radial profile superimposed with a small perturbation
+ of a given harmonic function). In this way, others, if they want
+ to benchmark their codes, do not need to get the Terra output.
+ Also in case some summary report comes out of this effort, we can
+ simply write down the initial conditions.
-(2) We aim to reproduce four benchmark cases in steady of just one. The
-four cases at the moment in my mind can be: three constant property cases
-with purely basal heating at Ra=1e5 (case 1), and Ra=1e6 (case 2), and
-purely internal heating at Ra=1e6 (case 3), and one temperature dependent
-viscosity and purely basal heating at Ra=1e6 (case 4).
+(2) We aim to reproduce four benchmark cases in steady of just one.
+ The four cases at the moment in my mind can be: three constant
+ property cases with purely basal heating at Ra=1e5 (case 1),
+ and Ra=1e6 (case 2), and purely internal heating at Ra=1e6 (case 3),
+ and one temperature dependent viscosity and purely basal heating
+ at Ra=1e6 (case 4).
Modified: doc/geodynamics.org/benchmarks/trunk/sources.txt
===================================================================
--- doc/geodynamics.org/benchmarks/trunk/sources.txt 2009-10-24 00:18:25 UTC (rev 15871)
+++ doc/geodynamics.org/benchmarks/trunk/sources.txt 2009-10-24 00:18:37 UTC (rev 15872)
@@ -17,6 +17,7 @@
./magma/milestone4.rst
./magma/milestone5.rst
./mc/index.rst
+./mc/notes.rst
./mc/2d-cartesian/index.rst
./mc/2d-cartesian/suite1.rst
./mc/2d-cartesian/suite2.rst
More information about the CIG-COMMITS
mailing list