[cig-commits] [commit] master: Write a section in the manual on compositional fields. (6f1b1a0)

[cig_noreply at geodynamics.org]

Repository : https://github.com/geodynamics/aspect

On branch  : master
Link       : https://github.com/geodynamics/aspect/compare/c41f9897e232c1a2ff987b46a3d31767708bf7af...7845f0356c011c6571ca6905bcdd9455ab32d3f1

>---------------------------------------------------------------

commit 6f1b1a020f0804d71ec6265b7d84c8bcba4ecab4
Author: Wolfgang Bangerth <bangerth at math.tamu.edu>
Date:   Thu Aug 7 08:53:54 2014 -0500

    Write a section in the manual on compositional fields.


>---------------------------------------------------------------

6f1b1a020f0804d71ec6265b7d84c8bcba4ecab4
 doc/manual/manual.tex | 78 ++++++++++++++++++++++++++++++++++++++++++++++++---
 1 file changed, 74 insertions(+), 4 deletions(-)

diff --git a/doc/manual/manual.tex b/doc/manual/manual.tex
index 08f4e51..f848414 100644
--- a/doc/manual/manual.tex
+++ b/doc/manual/manual.tex
@@ -1022,6 +1022,77 @@ also selected in the input parameter file.
 \subsection{Compositional fields}
 \label{sec:compositional}
 
+The last of the basic equations, \eqref{eq:compositional}, describes the
+evolution of a set of variables $c_i(\mathbf x, t), i=1\ldots C$ that we
+typically call \textit{compositional fields} and that we often aggregate into
+a vector $\mathfrak c$.
+
+Compositional fields were originally intended to track what their name
+suggest, namely the chemical composition of the convecting medium. In this
+interpretation, they composition is a quantity that is simply advected along
+passively, i.e., it would satisfy the equation
+\begin{align*}
+  \frac{\partial \mathfrak c}{\partial t} + \mathbf u \cdot \nabla \mathfrak c
+  = 0.
+\end{align*}
+However, these compositional fields participate in determining the values of
+the various coefficients as discussed in Section~\ref{sec:coefficients}.
+
+That said, over time compositional fields have shown to be a much more useful
+tool than originally intended. For example, they can be used to track where
+material comes from and goes to (see Section~\ref{sec:cookbooks-composition})
+and, if one allows for a reaction term on the right hand side,
+\begin{align*}
+  \frac{\partial \mathfrak c}{\partial t} + \mathbf u \cdot \nabla \mathfrak c
+  = \mathfrak q,
+\end{align*}
+then one can also model reactions between species -- for example to simulate
+phase changes where one compositional field indicating a particular phase
+transforms into another phase depending on pressure and temperature, or where
+several phases combine to other phases.
+
+Modeling reactions between different compositional fields often involves
+finding an equilibrium state between state between different fields because
+chemical reactions happen on a much faster time scale than transport. In other
+words, one then often assumes that there is a $\mathfrak c^\ast(p,T)$ so that
+\begin{align*}
+  \mathfrak q(p,T,\varepsilon(\mathbf u),\mathfrak c^\ast(p,T)) = 0.
+\end{align*}
+Consequently, the material model methods that deal with source terms for the
+compositional fields need to compute an \textit{increment} $\Delta\mathfrak c$
+to the previous value of the compositional fields so that the sum of the
+previous values and the increment equals $\mathfrak c^\ast$. This is opposed
+to the usual approach of evaluating the right hand side term $\mathfrak q$,
+which corresponds to a \textit{rate}, instead of an increment.
+
+On the other hand, there are other uses of compositional fields that do not
+actually have anything to do with quantities that can be considered related to
+compositions. For example, one may define a field that tracks the grain size
+of rocks. If the strain rate is high, then the grain size decreases as the
+rocks break. If the temperature is high enough, then grains heal and their size
+increases again. Such ``damage'' models would then call for an equation of the
+form (assuming one uses only a single compositional field)
+\begin{align*}
+  \frac{\partial c}{\partial t} + \mathbf u \cdot \nabla c
+  = q(T,c),
+\end{align*}
+where in the simplest case one could postulate
+\begin{align*}
+  q(T,c) = -\alpha c + \beta \max\{T-T_\text{healing},0\} c.
+\end{align*}
+One would then use this compositional field in the definition of the viscosity
+of the material: more damage means lower viscosity because the rocks are weaker.
+
+In cases like this, there is only a single compositional field and it is not
+in permanent equilibrium. Consequently, the increment implementations of
+material models in \aspect{} need to compute is typically the rate $q(T,c)$
+times the time step.
+
+Compositional fields have proven to be surprisingly versatile tools to model
+all sorts of components of models that go beyond the simple Stokes plus
+temperature set of equations. Play with them!
+
+
 \subsection{Numerical methods}
 
 There is no shortage in the literature for methods to solve the equations
@@ -1325,13 +1396,12 @@ This finally allows us to write
 \frac{1}{\rho} \nabla \rho \cdot \textbf{u} \approx \frac{1}{\rho} \frac{\partial \rho}{\partial p} \nabla p \cdot \textbf{u} \approx \frac{1}{\rho} \frac{\partial \rho}{\partial p} \nabla p_s \cdot \textbf{u} \approx \frac{1}{\rho} \frac{\partial \rho}{\partial p} \rho \textbf{g} \cdot \textbf{u} 
 \end{equation*}
 so we get
-\marginpar{There's a sign wrong here. Check with the code}
 \begin{equation}
 \label{eq:stokes-2-compressible}
-\nabla \cdot \textbf{u} = \frac{1}{\rho} \frac{\partial \rho}{\partial p} \rho \textbf{g} \cdot \textbf{u}
+\nabla \cdot \textbf{u} = -\frac{1}{\rho} \frac{\partial \rho}{\partial p} \rho \textbf{g} \cdot \textbf{u}
 \end{equation}
-where $\frac{1}{\rho} \frac{\partial \rho}{\partial p}$ is the
-compressibility.
+where $\frac{1}{\rho} \frac{\partial \rho}{\partial p}$ is often referred to
+as the compressibility.
 
 In the implementation used in \aspect{}, this equation replaces
 \eqref{eq:stokes-2}. It has the advantage that it retains the symmetry of the