(sec:methods:2d-models)=
# Two-dimensional models

ASPECT allows solving both two- and three-dimensional models via a parameter in the input files, see also {ref}`sec:run-aspect:2d-vs-3d`.
At the same time, the world is unambiguously three-dimensional.
This raises the question what exactly we mean when we say that we want to solve two-dimensional problems.

The notion we adopt here - in agreement with that chosen by many other codes - is to think of two-dimensional models in the following way: We assume that the domain we want to solve on is a two-dimensional cross section (parameterized by $x$ and $y$ coordinates) that extends infinitely far in both negative and positive $z$ direction.
Further, we assume that the velocity is zero in $z$ direction and that all variables have no variation in $z$ direction.
As a consequence, we ought to really think of these two-dimensional models as three-dimensional ones in which the $z$ component of the velocity is zero and so are all $z$ derivatives.

If one adopts this point of view, the Stokes equations {math:numref}`eq:stokes-1`-{math:numref}`eq:stokes-2` naturally simplify in a way that allows us to reduce the $3+1$ equations to only $2+1$, but it makes clear that the correct description of the compressible strain rate is still $\varepsilon(\mathbf u) - \frac{1}{3}(\nabla \cdot \mathbf u)\mathbf 1$, rather than using a factor of $\frac{1}{2}$ for the second term.
(A derivation of why the compressible strain rate tensor has this form can be found in {cite:t}`schubert:etal:2001`, sec. 6.5.)

It is interesting to realize that this compressible strain rate indeed requires a $3\times 3$ tensor. While under the assumptions above, we have
```{math}
\begin{aligned}
  \varepsilon(\mathbf u) = \begin{pmatrix} \tfrac{\partial u_x}{\partial x} & \tfrac 12 \tfrac{\partial u_x}{\partial y} + \tfrac 12 \tfrac{\partial u_y}{\partial x} & 0 \\
    \tfrac 12 \tfrac{\partial u_x}{\partial y} + \tfrac 12 \tfrac{\partial u_y}{\partial x} & \tfrac{\partial u_y}{\partial y} & 0 \\
    0 & 0 & 0 \end{pmatrix}
\end{aligned}
```
with the expected zeros in the last row and column, the full compressible strain rate tensor reads
```{math}
\begin{aligned}
  \varepsilon(\mathbf u) - \frac{1}{3}(\nabla \cdot \mathbf u)\mathbf 1 = \begin{pmatrix} \tfrac 23 \tfrac{\partial u_x}{\partial x}  - \tfrac 13 \tfrac{\partial u_y}{\partial y} & \tfrac 12 \tfrac{\partial u_x}{\partial y} + \tfrac 12 \tfrac{\partial u_y}{\partial x} & 0 \\
    \tfrac 12 \tfrac{\partial u_x}{\partial y} +  \tfrac 12 \tfrac{\partial u_y}{\partial x}  & \tfrac 23 \tfrac{\partial u_y}{\partial y} - \tfrac 13 \tfrac{\partial u_x}{\partial x}  & 0 \\
  0 & 0 & - \tfrac 13 \tfrac{\partial u_y}{\partial y} - \tfrac 13 \tfrac{\partial u_x}{\partial x} \end{pmatrix}.
\end{aligned}
```
The entry in the $(3,3)$ position of this tensor may be surprising.
It disappears, however, when taking the (three-dimensional) divergence of the stress, as is done in {math:numref}`eq:stokes-1`, because the divergence applies the $z$ derivative to all elements of the last row - and the assumption above was that all $z$ derivatives are zero; consequently, whatever lives in the third row of the strain rate tensor does not matter.

# Comments on the final set of equations

ASPECT solves these equations in essentially the form stated.
In particular, the form given in {math:numref}`eq:stokes-1` implies that the pressure $p$ we compute is in fact the *total pressure*, i.e., the
sum of hydrostatic pressure and dynamic pressure (however, see {ref}`sec:methods:static-v-dynamic` for more information on this, as well as the extensive discussion of this issue in {cite:t}`kronbichler:etal:2012`).
Consequently, it allows the direct use of this pressure when looking up pressure dependent material parameters.