The output of Earth system models is inherently uncertain. It is very difficult to find corresponding error bars because there are various different error sources.

• external errors (incomplete subsystems, wrong initial conditions, imperfect parametrisations ...)

• internal errors (choice of grid, discretisation of operators, interpolation, truncation ...)

One essential ingredient of every Earth system model is the general circulation model, i.e. the dynamic kernel. In our work we try to find a method that yields error estimates for outputs of so-called target functions on the flow field solution (e.g. global potential energy, ...).

• Model solution

• Local residual estimator

• Adjoint solution

• Integral over space / time domain

General approach

Our testbed consists of geophysical fluid test cases with known analytical solution [3,4]. The target quantity is the global potential energy at the end of the run. We define a quality parameter and test the quality of the adjoint and the local residual estimators separately.

Quality parameter

The “true” error is defined as the difference between the target function values of the analytical solution and the calculated solution on different resolutions. The quality parameter is the ratio between this true error and the corresponding error estimate on the respective level.

First Results

We have shown that our adjoint solution is perfectly usable, given a good local residual estimator (figure 2). We are currently working on various local residual estimators.

Figure 2: The ratio between estimated error and true error is shownon the y-axis. Our solution converges towards 1.

Local residual estimators

In principle, all LREs try to find an estimate of the difference of the computation at the local computational grid in space and time and the possible “truth”, be it analytical, high resolution, or different model solution. Roughly, one can divide potential LREs into two classes

• geometrical estimators that calculate differences / means / interpolations on different grid resolutions

• local multi-model approaches that use different model formulations / discretisations to calculate the residual of flow solutions

The neighbor approach (geometrical)

We use the difference between an interpolated “low resolution” field and the cell itself as an indicator of the true residual. This low resolution field can be constructed with the Nearest Neighbor values (NN) or even with a bigger stencil. The estimate ratio approaches 100% with increasing resolution (figure 3), but the behavior at high resolutions is still not satisfactory (no convergence, as can be seen in figure 4). This result shows the need for high resolution runs to test the convergence behavior of our local residual estimators.

The gradient approach

We use the the mean of the difference in height field values between neighboring cells as an indicator of the residual at a given cell. While this method works comparably well at low resolutions, the high resolution behavior is similar to the NN approach.