concerns possible extensions to the model we use here. It has to do
with the fact that while the temperature equation we use here does not
include a term that contains the pressure. It should, however:
-pressure, like gas, heats up as you compress it. Consequently,
+rock, like gas, heats up as you compress it. Consequently,
material that rises up cools adiabatically, and cold material that
sinks down heats adiabatically. We discuss this further below.
small values before picking up again in the chaotic stirring
regime. There, it remains in the range of 10-40 centimeters per year,
quite within the physically expected region.
+
+
+
+<a name="extensions"></a>
+<h3>Possibilities for extensions</h3>
+
+Apart from the various possibilities for extensions already outlined
+in the @ref step_31 "step-31", here are a few more ideas:
+
+<ul>
+ <li> The temperature field we get in our simulations after a while
+ is mostly constant with boundary layers at the inner and outer
+ boundary, and streamers of cold and hot material mixing
+ everything. Yet, this doesn't match our expectation that things
+ closer to the earth core should be hotter than closer to the
+ surface. The reason is that the energy equation we have used does
+ not include a term that describes adiabatic cooling and heating:
+ rock, like gas, heats up as you compress it. Consequently, material
+ that rises up cools adiabatically, and cold material that sinks down
+ heats adiabatically. The correct temperature equation would
+ therefore look somewhat like this:
+ @f{eqnarray*}
+ \frac{D T}{Dt}
+ -
+ \nabla \cdot \kappa \nabla T &=& \gamma + \tau\frac{Dp}{Dt},
+ @f}
+ or, expanding the advected derivative $\frac{D}{Dt} =
+ \frac{\partial}{\partial t} + \mathbf u \cdot \nabla$:
+ @f{eqnarray*}
+ \frac{\partial T}{\partial t}
+ +
+ {\mathbf u} \cdot \nabla T
+ -
+ \nabla \cdot \kappa \nabla T &=& \gamma + \tau\frac{\partial
+ p}{\partial t} + \mathbf u \cdot \nabla p.
+ @f}
+ In other words, as pressure increases in a rock volume
+ ($\frac{Dp}{Dt}>0$) we get an additional heat source, and vice
+ versa.
+
+</ul>
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