Mass and Energy Conservation
When the basis of the mathematical models used in AOLGCM models/codes are discussed it is almost always stated that the ‘fundamental laws of conservation and mass and energy’ are at the foundations of the models. This is an incomplete, and somewhat incorrect, statement on several levels.
Conservation of mass and energy is the focus of the following discussion.
Here is how the models are summarized in the AR4. First from Section 8.1.3 of Chapter 8:
8.1.3 How Are Models Constructed?
The fundamental basis on which climate models are constructed has not changed since the TAR, although there have been many specific developments (see Section 8.2). Climate models are derived from fundamental physical laws (such as Newtonâ€™s laws of motion), which are then subjected to physical approximations appropriate for the large-scale climate system, and then further approximated through mathematical discretization. Computational constraints restrict the resolution that is possible in the discretized equations, and some representation of the large-scale impacts of unresolved processes is required (the parametrization problem).
And also in the FAQ for Chapter 8.
Frequently Asked Question 8.1
How Reliable Are the Models Used to Make Projections of Future Climate Change?
Climate models are mathematical representations of the climate system, expressed as computer codes and run on powerful computers. One source of confidence in models comes from the fact that model fundamentals are based on established physical laws, such as conservation of mass, energy and momentum, along with a wealth of observations.
The citation for the above information is as follows, just as the IPCC states.
This chapter should be cited as:
Randall, D.A., R.A. Wood, S. Bony, R. Colman, T. Fichefet, J. Fyfe, V. Kattsov, A. Pitman, J. Shukla, J. Srinivasan, R.J. Stouffer, A. Sumi and K.E. Taylor, 2007: Cilmate Models and Their Evaluation. In: Climate Change 2007: The Physical Science Basis. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change [Solomon, S., D. Qin, M. Manning, Z. Chen, M. Marquis, K.B. Averyt, M.Tignor and H.L. Miller (eds.)]. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA.
Note that the summary in Section 8.1.3 is a little more nearly complete than that given in the FAQ. But only in the sense that a limited number of qualifications are simply listed without any elaborations at all. It is important to note that some of the parameterizations of phenomena and processes alluded to in Section 8.1.3 include those associated with mass and energy transport, storage, and conversion. The parameterizations at best are sometimes based on algebraic correlations of the phenomena and processes of interest. Many, on the other hand, are heuristic and ad hoc, at best.
The focus of these notes will be on the hypothesis of conservation of mass and energy as mentioned in the AR4 summaries above. The momentum equations are also of interest, but the conservation of mass and energy is relatively more important because one of the major objectives of the GCMs are calculations of the transport, storage, and conversions of mass and energy, and associated thermodynamic and hydrodynamic processes. Conversions between the various phases of water substance and various forms of energy are significant processes relative to calculation of the mean thermal state of the climate system. Conversions of mass (phase change, for example) always involve transport and storage of thermal energy.
As noted in the AR4 the fundamental and complete forms of the basic mathematical statements of conservation of mass, momentum, and energy are not the equations actually used in the GCM models. Such complete and fundamental equations are always subjected to assumptions and simplifications at both the continuous and discrete stages of models and methods and software development. As one result, the equations are models of conservation of mass and energy. The nature and necessity of the assumptions and simplifications have been discussed in detail here.
Because they are only models of conservation of mass and momentum, the model equations, even at the continuous-equation stage, do not contain a precise accounting of the real-world processes and phenomena that is necessary for mass and energy to be conserved, and thus cannot calculate mass and energy conservation that corresponds to the actual real-world systems.
The lack of conservation is further compromised at the discrete-equation and numerical solution level because it is known and readily acknowledged that the solution methods cannot be shown to provide convergence of the solutions of the discrete equations to the solutions of the continuous equations. In effect, the so-called fundamental laws are functions of the size of the discrete increments (or series expansions) used in the numerical solution methods. And, because convergence is not ever tested, the actual truncation-order error in the discrete approximations at solution time is unknown. It is very likely that the truncation error is larger than the theoretical analysis indicates. An example is given in this paper.
The, “established physical laws, such as conservation of mass, energy and momentum” can never be functions of the sizes of the discrete increments used for numerical solutions of the discrete approximations. The dependency of the discrete increments means for example that the basic laws of thermodynamics as used in applications of GCM models and codes are functions of quantities that are never mentioned in the laws.
So, in effect, mass and energy are not conserved in GCMs calculations on two levels; (1) at the continuous level because of the assumptions and approximations applied to the basic fundamental equations, and (2) at the discrete-approximation level because the discrete equations are not actually solved by the application procedures. Both of these deficiencies introduce errors, relative to the actual physical phenomena and processes, into the calculated results.
I consider these issues to be important for the following ideas. The estimated net energy imbalance resulting from the phenomena and processes related to addition of CO2 into the atmosphere is relatively a small number. Although I do not yet have any good data points, it seems that the known approximations and errors in the continuous-equation models and solution methods could very likely induce errors into the calculations that are very likely larger than the expected real-world energy imbalance.
A calculation can be shown to conserve (or not) mass and energy for a given model and methods. Such illustrations, however, are not in any way related to conservation of mass and energy in the physical systems of interest.
That GCM models and methods conserve mass and energy of the physical systems of interest is yet another untested and un-falsifiable hypothesis in Climate Science.