[cig-commits] commit 2315 by bangerth to /var/svn/dealii/aspect

[dealii.demon at gmail.com]

Revision 2315

Provide more discussion.

A   trunk/aspect/doc/manual/cookbooks/shell_simple_2d/avg-temperature-noshear.png
A   trunk/aspect/doc/manual/cookbooks/shell_simple_2d/avg-temperature.png
A   trunk/aspect/doc/manual/cookbooks/shell_simple_2d/heat-flux-noshear.png
A   trunk/aspect/doc/manual/cookbooks/shell_simple_2d/heat-flux.png
U   trunk/aspect/doc/manual/manual.tex
U   trunk/aspect/doc/manual.pdf


http://www.dealii.org/websvn/revision.php?repname=Aspect+Repository&path=%2F&rev=2315&peg=2315

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@@ -4391,7 +4391,10 @@
 scientifically invalid results.}
 
 The model we have chosen has a number of inadequacies that
-make it not very realistic. Let us discuss these issues in the following.
+make it not very realistic (some of those happened more as an accident while
+playing with the input file and weren't a purposeful experiment, but we left
+them in because they make for good examples to discuss below).
+Let us discuss these issues in the following.
 
 \paragraph{Dimension.} This is a cheap shot but it is nevertheless true that the
 world is three-dimensional whereas the simulation here is 2d. We will address
@@ -4480,7 +4483,108 @@
 functions that do not depend on the pressure, and only the density varies
 (slightly) with the temperature. We know that this is not the case in nature.
 
-As mentioned, we will address some of these issues in the following sections.
+\paragraph{Shear heating.}
+When we set up the input file, we started with a model that includes the
+shear heating term $2\eta arepsilon(\mathbf u):arepsilon(\mathbf u)$ in
+eq.~\eqref{eq:temperature}. In hindsight, this may have been the wrong decision,
+but it provides an opportunity to investigate whether we think that the results of our
+computations can possibly be correct. 
+
+We first realized the issue when looking at the heat flux that the
+	exttt{heat flux statistics} postprocessor computes. This is shown in the left
+panel of Fig.~
ef{fig:simple-shell-2d-heatflux}.%
+ootnote{The 	exttt{heat flux statistics} postprocessor computes heat fluxes
+through parts of the boundary in 	extit{outward} direction, i.e., from the
+mantle to the air and to the core. However, we are typically interested in the
+flux from the core into the mantle, so the figure plots the negative of the
+computed quantity.}
+There are two issues one should notice here.
+The more obvious one is that the flux from the mantle to the air is consistently
+higher than the heat flux from core to mantle. Since we have no radiogenic
+heating switched on (see the 	exttt{Radiogenic heating rate}
+\index[prmindex]{Radiogenic heating rate}
+\index[prmindexfull]{Model settings!Radiogenic heating rate}
+parameter in the 	exttt{Model settings} section of the input file; see also
+Section~
ef{parameters:Model_20settings}), in the long run the heat output
+of the mantle must equal the input, unless is cools. Our misconception was that
+after the 800 million year transition, we believed that we had reached a steady
+state where the average temperature remains constant and convection simply
+moves heat from the core-mantle boundary the surface. One could also be tempted
+to believe this from the right panel in Fig.~
ef{fig:simple-shell-2d-rms} where
+it looks like the average temperature does at least not change dramatically.
+But, it is easy to convince oneself that that is not the case: the
+	exttt{temperature statistics} postprocessor we had previously selected also
+outputs data about the mean temperature in the model, and it looks like shown in
+the left panel of Fig.~
ef{fig:simple-shell-2d-temperature}. Indeed, the
+average temperature drops over the course of the 1.2 billion years shown here.
+We could now convince ourselves that indeed the loss of thermal
+energy in the mantle due to the drop in average temperature is exactly what
+fuels the persistently imbalanced energy outflow. In essence, what this would
+show is that if we kept the temperature at the boundaries constant, we would
+have chosen a mantle that was initially too hot on average to be sustained by
+the boundary values and that will cool until it will be in energetic balance and
+on longer time scales, in- and outflow of thermal energy would balance each
+other.
+
+
+egin{figure}[tb]
+  \includegraphics[width=.48	extwidth]{cookbooks/shell_simple_2d/heat-flux}
+  \hfill
+  \includegraphics[width=0.48	extwidth]{cookbooks/shell_simple_2d/heat-flux-noshear}
+