## 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.

I think you could support better your point by explaining how the models deal with:

– cloud formation

– aerosols

– sun variability

– transport properties of the fluids like viscosity, conductivities and diffusivities

– turbulence

– absorption and release of CO2 by the biosphere

– changes in albedo

– sources and sinks

Comment by JM | September 2, 2007 |

Thanks JM for your suggestion. Descriptions of all the phenomena and processes you listed would make for a very long post. With the possible exception of the transport properties and Sun variability, I think none of the phenomena and processes you list are handled even remotely by the, â€? â€¦ established physical laws, such as conservation of mass, energy and momentumâ€¦ â€?. And the degree of exactness of the transport properties will depend of assumptions about the composition of the materials of interest. Plus I am here referring to the molecular transport properties, not modeled “effective” properties. I suppose Sun variability due to the motions of Earth can be handled from basic principles.

That is not to say that I should have omitted citations to peer-reviewed literature sources for such information. However I have tried more than one time to find simple, declarative, and direct citations to the concise forms of the continuous model equations for many of the items in your list. I have not yet been successful in finding that information.

As an example, where could one find explicit citation to the peer-reviewed literature in which the exact form of the vertical-direction momentum equation used in NASA/GISS ModelE is developed. And does the continuous-equation model use the thin-shell approximation or is the complete spherical-geometry form of the equation used. Explicit identification of the continuous form of the exact model equation used in the code is the goal.

For those who proclaim that explicit citations to the peer-reviewed source for each and every continuous and discrete equation used in the ModelE model/code can be easily obtained I offer the challenge described in the previous paragraph.

The accepted level of documentation for production-level models/codes has been outlined in this document and this post.

The key point that I wanted to make was that the model equations actually “solved” by the NASA/GISS ModelE code cannot correspond to the actual physical phenomena and processes that occur in the physical system of interest. While that statement is true for almost all

modelsof inherently complex physical systems, I see it as an over-riding issue relative to Global Climate Change because the estimated physical imbalance in energy is such a small number. I have posted a comment about the magnitude of the viscous dissipation which seems to be of the same order as the imbalance due to 2xCO2.Thanks again for your comment.

Comment by Dan Hughes | September 4, 2007 |

For me it doesnt matter if we are using models or not to analyze and predict the future of our energy. What matters most is our discipline and dedication to help in our very own little ways to conserve energy for the future.

We should invest more money on the research in the actual preservation and conservation of energy.

Comment by Renewable Energy | March 16, 2008 |

Personalyl I agree with renewable energy. Too much is being placed on conserving and not finding a solution.

Comment by Dmitri | April 16, 2008 |

[…] the situation is completely consistent with this post. That is, the model for ‘conservation of energy’ used in AOLGCM codes does not account for and […]

Pingback by Another NASA/GISS ModelE Code Fragment « Models Methods Software | December 1, 2009 |

[…] yet even more interesting coding. The thread at Lucia’s is related to the posts on this Blog here, here, and here. The dissipation questions come up now and then in several […]

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