## Internal Variability=Weather and Numerical Artifacts

This post is based on some notes related to verification and numerical artifacts that I made back in early July.

The theoretical order of the discrete approximations to the continuous PDEs and ODEs that comprise parts of GCM models and methods can almost always be determined. For a single continuous ODE or PDE the math is relatively straightforward, although care should be exercised at the boundaries; and actually that caveat applies for the continuous equations, too. As the number of equations increases due to inclusion of models for additional physical phenomena and processes, the problem can become more difficult. Sometimes, a computer software tool might have to be developed in order to determine the theoretical order of the approximations for very complex model-equation systems.

Whenever a mathematical model includes algebraic-model elements, the effects of these on the theoretical order of convergence are frequently overlooked. GCMs are known to contain numerous algebraic-model elements, as are almost all models and codes developed for inherently complex physical phenomena and processes. The following are some examples of how these can completely over-ride the theoretical order of convergence for the ODEs and PDEs.

Consider an algebraic switch that checks, say, the temperature of a chunk of material against a reference value for the temperature that represents a change in the response of the material to its surroundings. Changes of phase are examples. Another example might be a change from representation of laminar friction to turbulent friction factors. There are a very large number of these in models and codes such as GCMs.

Take the first case in the above paragraph and assume that an initial calculation has been completed and that a time-step-size sensitivity study is underway. A second calculation using a different time-step size will cause the temperature of the material to be different from that of the initial calculation. When the temperature is compared to the reference value the response of the material will be different. This behavior represents zeroth-, or at most first,-order characteristics. Not the higher-order properties expected based on theoretical analyses of the numerical solution method.

The effects of algebraic switches, such as might appear in important parameterizations used in GCMs, can be minimized by careful consideration of the equations in discrete-approximation and numerical-solution spaces. Continuity in the first and second derivatives of the switches is the required objective.

The same behavior will be seen as spatial resolution is increased and the same time-step size is used. In this case, it’s the amount of material that is different between the calculations and the temperature response will also be different. [ Note that there are formal methodologies that should be followed for these kinds of studies, but I will not discuss them here. ]

The same behavior will obtain for what initially would seem to be a very straightforward processes. Let’s say the gird is refined near the Earth’s surface in a region that has significant variations in topology. As the gird is refined some surface features that were completely lost in a calculation using a rough grid will begin to appear. Other features that were roughly approximated will begin to appear in greater detail. And, if the grid is not refined in a manner that is consistent with the discrete approximations, the surface features will appear to move around locally in discrete-approximation space. So long as details of these features continue to change as the grid changes, zeroth-, or at most first,-order characteristics for the discrete approximations will be obtained.

It is possible that in local regions of the solution domain that some aspects of the calculated solutions will be dominated by this property of the calculation. Of course it is also possible that the important calculated results are not significantly impacted by these numerical artifacts.

A critically important aspect of these behaviors and characteristics is that algebraic switches have a potential to introduce artificial changes in mass, momentum and energy into the solution domain that are in no way related to the physical response of the material. These are usually exhibited in the calculations by an oscillatory response of the materials following the activation of the switch. If an noisy oscillatory response is the expected / accepted response of the material it is easy to assume that these pure numerical artifacts are in fact correct representations of the response of the materials.

Well, that was kind of long. Let me know if my representation is not correct.

Here are some questions.

(1) Are there papers in which any of the above have been investigated for any GCMs? Note that merely making a few additional calculations at different temporal and spatial resolutions is an incomplete approach for investigating these issues. Sufficient calculations, accompanied by analyses of the results, to demonstrate that the actual-application-calculation order of convergence is in agreement with theoretical values are necessary.

If approach toward the theoretical values is not obtained, the root-cause of the differences must be explicitly identified. Rough, hand-waving declarations of the causes, without explicit demonstration, do not count for anything.

(2) Are there any good and proven reasons to know that the calculated GCM responses are in fact actual solutions to the discrete equations and not results due to algebraic switches and / or other purely numerical artifacts? Rough statistical comparisons of an ensemble mean of a bunch of calculations with meta-global-solution-functionals, such as some kind of global-average temperature, do not count for anything. Mapping the results of evaluation of models for maybe hundreds of physical phenomena and processes for maybe thousands of time steps for thousands of grid locations to a single annual-averaged number is not how any other codes and methods are verified.

(3) If all aspects of the numerical solution methods used in GCMs have not yet been rigorously verified, on what basis can it be thought that the model equations have been solved?

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