by Dan Hughes
We frequently see the simple statement, “The Laws of Physics”, invoked as the canonical summary of the status of the theoretical basis of GCMs.
We also see statements like, “The GCM models are based on the fundamental laws of conservation of mass, momentum, and energy.” Or, “GCM computer models are based on physics.” I recently ran across this summary:
How many hours have been spent verifying the Planck Law? The spectra of atmospheric gases? The laws of thermodynamics? Fluid mechanics? They make up climate models just as the equations of aerodynamics make up the airplane models.
And here’s another version:
Climate models are only flawed only if the basic principles of physics are, but they can be improved. Many components of the climate system could be better quantified and therefore allow for greater parameterisation in the models to make the models more accurate. Additionally increasing the resolution of models to allow them to model processes at a finer scale, again increasing the accuracy of the results. However, advances in computing technologies would be needed to perform all the necessary calculations. However, although the accuracy of predictions could be improved, the underlying processes of the models are accurate.
These statements present no actual information. The only possible information content is implicit, and that implicit information is at best a massive mis-characterization of GCMs, and at worst disingenuous (dishonest, insincere, deceitful, misleading, devious).
There are so many self-contradictions in the last quoted paragraph, both within a given sentence and between sentences, that it’s hard to know where to begin. The first sentence is especially self-contradictory (assuming there are degree of self-contradictions). There are a very large number of procedures and processes applied to the model equations between the continuous equations and the coded solution methods in GCMs. It is critical that the actual coding be shown to be exactly what was intended as guided by theoretical analyses of the discrete approximations and numerical solution methods.
The articles from the public press that contain such statements sometimes allude to other aspects of the complete picture such as the parameterizations that are necessarily a part of the models. But generally such public statements always present an overly simplistic picture relative to the actual characterizations and status of climate-change modeling.
It appears to me that the climate-change community is in a unique position relative to presenting such informal kinds of information. In my experience, the sole false/incomplete focus on The Laws of Physics is not encountered in engineering. Instead, the actual equations that constitute the model are presented.
The fundamental unaltered Laws of Physics that are needed for calculating the responses of Earth’s climate systems are never solved by GCMs, and certainly will never be solved by GCMs. That is a totally intractable calculation problem, both analytically and numerically. Additionally, and very importantly, the continuous equations are never solved directly for the numbers presented as calculated results. Numerical solution methods applied to discrete approximations to the continuous equations are the actual source of the presented numbers. It is important to note that it is known that the numerical solution methods used in GCM computer codes have not yet been shown to converge to the solutions of the continuous equations.
In order to gain insight from of the numbers calculated by GCMs, deep understanding of the actual source of the numbers is of paramount importance. The actual source is far removed from the implied source that is conveyed by the statements. The ultimate source of the calculated results is the numerical solutions from computer software. The numerical solutions arise from the discrete equations that are used to approximate the continuous equations that are the basis of the discrete equations. Thus a bottom-up approach for gaining an understanding of GCM reported results requires that the nitty-gritty details of what is actually in the computer codes be available for inspection and study.
The Actual Source of the Numbers
The ultimate source of the numbers calculated by GCMs is the result of the following processes and procedures:
(1) Application of assumptions and judgments to the basic fundamental “Laws of Physics” in order to formulate a calculation problem that is both (a) tractable, and (b) that captures the essence of the physical phenomena and processes important for the intended applications.
(2) Development of discrete approximations to the tractable equation system. The discrete approximations must maintain the requirements of the Laws of Physics (conservation principles, for example).
(3) Development of stable and consistent numerical solution methods for the discrete approximations. Stability plus consistency imply convergence. Yes, that’s for well-posed problems, but some of the ODEs and PDEs used in GCMs represent well-posed problems (heat conduction, for example).
(4) Coding of the numerical solution methods.
(5) Ensuring that the solution methods, and all other aspects of the software, are correctly coded.
(6) Validation of the model equations by comparisons of calculated results with data from the physical domain.
(7) Development of application procedures and user training for each of the intended applications.
Validation, demonstrating the fidelity of the resulting whole ball of wax to the physical domain, is a continuing process over the lifetime of the models, methods, and software.
The long difficult iterative path through the processes and procedures outlined above, from the basic fundamental Laws of Physics continuous equations to the calculated numbers, is critically affected by a number of factors that are seldom mentioned whenever GCMs are the subject. Among the more significant, and never mentioned, factors is the well-known user effect; item (7) above. Complex software built around complex physical domain problems requires very careful attention to the qualifications of the users for each application.
In these notes the real-world nature and characteristics of such complex software and physical domain problems are examined in the light of the extremely simplified public face of GCMs. That public face will be shown to be a largely false characterization of these models and codes.
The critically important issues are those associated with (1) the modifications and limitations of the continuous formulation of the model equation systems used in GCMs (generally, the fluid-flow model equations are not the complete fundamental form of the Navier-Stokes equations, the fundamental formulation of radiative energy transport in the presence of an interacting media, for examples), and this applies to equations used in GCMs, (2) the exact transformation of all the continuous equation formulations into discrete approximations, (3) the critically important properties and characteristics of the numerical solution methods used to solve the discrete approximations, (4) the limitations introduced at run time for each type of application and the effects of these on the response functions of interest for the application, and (5) the expertise and experience of users of the GCMs for each application area.
These matters are discussed in the following paragraphs.
Such statements as those mentioned above provide, at the very best, only a starting point relative to where the presented numbers actually come from. It is generally not possible to present an accurate and complete description of what constitutes the complete model in communications intended primarily to be informal presentations of a model and a few results. However, the overly simplistic summary that is usually presented should be tempered to more nearly reflect the reality of GCM models and methods and software.
Here are four examples of where GCM model equations depart from the fundamental Laws of Physics.
(1) In almost no practical applications of the Navier-Stokes equations are they solved to the degree of resolution necessary for accurate representation of fluid flows near and adjacent to stationary, or moving, surfaces. Two such surfaces of interest in modeling Earth’s climate systems are; (1) the air-water interface presented by the boundary between the atmosphere and oceans and (2) the interface between the atmosphere and the land. When considering the entirety of the interactions between sub-systems, including, for examples, biological, chemical, hydrodynamic, and thermodynamic interactions, the number of such interfaces is quite large.
The gradients, which appear in the fundamental formulations at these interfaces, are all replaced by algebraic approximations. The replacement occurs at the continuous equation level, even prior to making discrete approximations. These algebraic models and correlations are used to represent mass, momentum, and energy exchanges between the materials that make up the interfaces.
(2) The assumption of hydrostatic equilibrium normal to Earth’s surface is exactly that; an assumption. The fundamental Law of Physics, the complete momentum balance equation for the vertical direction, is not used.
(3) Likewise, whenever the popular description of the effects of CO2 in Earth’s atmosphere is stated, that hypothesis, too, is based on an assumption of nearly steady state balance between in-coming and out-going radiative energy exchange. And this is sometimes attributed to the Laws of Physics and conservation of energy. However, conservation of energy holds for all time and everywhere. The balance between in-coming and out-going radiative energy exchange for a system that is open to energy exchange is solely an assumption and is not related to conservation of energy.
(4) There is a singular, of upmost importance, critical difference between the, proven, fundamental Laws of Physics and the basic model equations used in GCMs. The fundamental Laws of Physics are based solely on descriptions of materials. The parameterizations that are used in GCMs are instead approximate descriptions of previous states that the materials have attained. The proven fundamental laws will not ever, as in never, incorporate descriptions of states that the materials have previously attained. Whenever descriptions of states that materials have experienced appear in equations, the results are models of the basic fundamental laws, and are not the laws as originally formulated.
A more nearly complete description of exactly what constitutes computer software developed for analyses of inherently complex physical phenomena and processes is given in the following discussions.
Characterization of the Software
Models and associated computer software intended for analyses of real-world complex phenomena and processes is generally comprised of the following models, methods, software, and user components:
1. Basic Equations Models The basic equations are generally from continuum mechanics such as the Navier-Stokes-Fourier model for mass, momentum and energy conservation in fluids, heat conduction for solids, radiative energy transport, chemical-reaction laws, the Boltzmann equation, and many others. The fundamental equations include also the constitutive equations for the behavior and properties of the associated materials: equation of state, thermo-physical and transport properties and basic material properties. Generally the basic equations refer to the behavior and properties of the materials of interest.
Even though fundamental basic equations of mass, momentum, and energy conservation are taken as the starting point for the modeling the physical phenomena and processes of importance, several assumptions and approximations are generally needed in order to make the problem tractable, even with the tremendous computing power available today. The exact radiative transfer equations for an interacting media, for example, are not solved, but instead approximations are introduced to make the problem tractable.
With almost no exceptions, the basic, fundamental laws in the form of continuous algebraic equations, ODEs and PDEs from which the models are built are not the equations that are ultimately programmed into the computer codes. Assumptions and approximations, appropriate for the intended application areas, are applied to the fundamental original form of the equations to obtain the continuous equations that will be used in the model. The approximations that are made are to more and lesser degrees important relative to the nature of the physical phenomena and processes of interest. A few examples are given in the following paragraphs.
The fluid motions of the mixtures in both the atmosphere and oceans are turbulent and there is no attempt at all to use the fundamental laws of turbulent fluid motions in GCM models/codes. For the case of two- or multi-phase turbulent flows, liquid droplets in a gaseous mixture for example, the fundamental laws are not yet known.
The exchanges of mass, momentum, and energy at the interfaces between the (atmosphere, oceans, land, biological, chemical,etc.) systems that make up the climate are, at the fundamental-law level, expressed as a coefficient multiplying the gradient of a driving potential. These are never used in the GCM models/codes because spatial resolution used in the numerical solution methods do not allow the gradients to be resolved. The gradients of the driving potentials are not calculated in the codes. Instead algebraic correlations of empirical data, based on a bulk state-to-bulk-state average potential, are used. These are almost always algebraic equations.
The modeling of radiative energy transport in an interacting media does not use the fundamental laws of radiative transport. Assumptions are applied to the fundamental law so that a reasonable and tractable approximation to the physical phenomena for the intended application is obtained.
While the fundamental equations are usually written in conservation form, not all numerical solution methods exactly conserve the physical quantities. Actually, a test of numerical methods might be that conserved quantities in the continuous partial differential equations are in fact conserved in actual calculations.
This comment should not be interpreted to mean that the basic model equations are incorrect. They are, however, incomplete representations of the fundamental Laws of Physics. Additionally, as next discussed, the algebraic equations of empirical data are often far from based on fundamental laws.
2. Engineering Models and Correlations of Empirical Data These equations generally arise from experimental data and are needed to close the basic model equations; turbulent fluid flow, heat transfer and friction factor correlations, mass exchange coefficients, for examples. Generally the engineering models and empirical correlations refer to specific states of the materials of interest, not the materials themselves, and are thus usually of much less than a fundamental nature. Many times these are basically interpolation methods for experimental data.
Models and correlations that represent states of materials and processes do not represent properties of the materials and are thus of much less of a fundamental nature than the basic conservation laws.
3. Special Purpose Models Special purpose models for phenomena and processes that are too complex or insufficiently understood to model from basic principles, or would require excessive computing resources if modeled from basic principles.
The apparently all-encompassing parameterizations used in almost all GCM models and codes fall under items 2 and 3. There are many physical phenomena and processes important to climate-change modeling that treated by use of parameterization. Some of the parameterizations are of heuristic and ad hoc nature.
Special purpose models can also include calculations of quantities that assist users, post-processing of calculated data, calculation of quality control quantities, for examples. Calculation of solution functionals, and other aspects that do not feed back to the main calculations, are examples.
4. Important Sources from Engineered Equipment Models for phenomena and processes occurring in complex engineering equipment, if a physical system of interest includes hardware. In the case of the large general GCMs, the equipment and processes involved in conversion of materials in one form and composition into other forms and compositions will use engineered equipment.
Summary of the Continuous Equations Domain The final continuous equations that are used to model the physical phenomena and processes usually arise from these first four items. The continuous equations always form a large system of coupled, non-linear partial and/or ordinary differential equations (PDEs and ODEs) plus a very large number of algebraic equations.
For the class of models of interest here, and for models of inherently-complex, real-world problems in general, the projective/predictive/extrapolative capabilities are maintained in the modeling under Items 1, 2, 3, and 4 listed above.
5. The Discrete Approximation Domain Moving to the discrete-approximation domain introduces a host of additional issues, and the ‘art’ aspects of ‘scientific and engineering’ computations complicate these. Within the linearized domain the theoretical ramifications can be computationally realized by use of idealized flows that correspond to the theoretical aspects of the analyses.
The adverse theoretical ramifications do not always prevent successful applications of the model equations. In part because the critical frequencies are not resolved in the applications, and in part due to the properties of the discrete approximations which usually have inherent implicit representations of dissipative-like terms. Such dissipative terms are also sometimes explicitly added into the discrete approximations. And sometimes these added-in terms do not have counterparts in the original fundamental equations.
Reconciliation of the theoretical results with computed results is also complicated by the basic properties of the selected solution method for the discrete approximations. The methods themselves can introduce aphysical perturbations into the calculated flows. And these are further complicated whenever the discrete approximations contain discontinuous algebraic correlations (for mass, momentum, and energy exchanges, for examples) and switches that are intended to prevent aphysical calculated results. In the physical domain any discontinuity (in pressure, velocity, temperature, EoS, thermophysical, and transport properties, for examples) has a potential to lead to growth of perturbations. In the physical domain, however, physical phenomena and processes act to limit growth of physical perturbations.
6. Numerical Solution Methods Numerical solution methods for all the equations that comprise the models are necessary. These processes are the actual source of the numbers that are usually presented as results.
Almost all complex physical phenomena are non-linear with a multitude of temporal and spatial scales, interactions, and feedbacks. Universally, numerical solution methods via finite-difference, finite-element, spectral, and other discrete-approximation approaches, are about the only alternative for solving the system of equations. When applied to the continuous PDEs and ODEs and the algebraic equations of the model these approximations give systems of coupled, nonlinear algebraic equations that are enormous in size.
Almost all of the important physical processes occur at spatial scales that are less than the discrete spatial resolution employed in all calculations. Additionally, the range of temporal scales of the phenomena and processes encountered in applications range from those associated with chemical reactions to time spans on the order of a century. In the GCM solution methods almost none of these temporal scales are resolved.
It is a fact that numerical solution methods are the dominant aspect of almost all modeling and calculation of inherently complex physical phenomena and processes in inherently complex geometries. The spatial and temporal scales of the application area of GCMs are enormous, maybe unsurpassed in all of modeling and calculations. The tremendous spatial scale of the atmosphere and oceans has so far proven to be a very limiting aspect relative to computing requirements, especially when coupled with the large temporal scale of interest; centuries of time, for example.
In GCM codes and applications, the algebraic approximations to the original continuous equations are only approximately solved. Grid independence has never been demonstrated, for example. The lack on demonstrated grid independence is proof that the algebraic equations have been only approximately solved. Evidence of independent Verification of (1) the coding and (2) the actual achieved accuracy of the numerical solution methods also have never been demonstrated.
Because numerical solutions are the source of the numbers, one of the primary focuses of discussions of GCM models and codes must be the properties and characteristics of the numerical solution methods. Some of the issues that have not been sufficiently addressed are briefly summarized here.
Summary of the Discrete Approximation Domain The discrete approximation domain ultimately determines the correspondence of the properties of the fundamental Laws of Physics and the actual numbers from GCMs. The overriding principles of conservation of mass and energy, for example, can be destroyed in this domain. One cannot simply state that the Laws of Physics insure that fundamental conservation principles are obtained.
7. Auxiliary Functional Methods Auxiliary functional methods include instructions for installation on the users’ computer system, pre- and post-processing, code input and output formats, analyses of calculated results, and other user-aids such as training for users.
Accurate understanding and presentation of calculations of inherently complex models and equally complex computer codes demands that the qualifications of the users be determined and enhanced by training. The model/code developers are generally most qualified to provide the required training.
8. Non-functional Methods Non-functional aspects of the software include its ease of, and fitness for, understandability, maintainability, extensibility and portability. Large complex codes have generally evolved, usually over decades, in contrast to being built from scratch and thus include a variety of potential sources of problems in these areas.
9. User Qualifications For real-world models of inherently complex physical phenomena and processes the software itself will generally be complex and somewhat difficult to accurately apply and the calculated results somewhat difficult to understand. Users of such software must usually receive training in applications of the software.
I think all of the above characterizations, properties, procedures, and processes, presented from a bottom-up focus, constitute a more nearly complete and correct characterization of GCM computer codes. The models and methods summarized above are incorporated into computer software for applications to the analyses for which the models and methods were designed.
Documentation of all the above characteristics, in sufficient detail to allow independent replication of the software and its applications, is generally a very important aspect of development and use of production-grade software.
Unlike a “pure” science problems, for which the unchanged fundamental Laws of Physics are solved, the simplifications and assumptions made at the fundamental-equation level, the correlations and parameterizations, and, especially, the finite-difference aspects of GCMs are the overriding concerns.
Spatial discontinuities in all fluid-state properties (density, velocity, temperature, pressure, etc.) introduce the potential for instabilities, as do discontinuities in the discrete representation of the geometry of the solution domain. Physical instabilities, captured by the equations in GCMs, and the behavior of the numerical solution methods when these are resolved becomes vitally important. The solutions are required to be demonstrated to be correct and not artifacts of numerical approximations and solution methods.
GCMs are Process Models Here’s a zeroth-order cut at differentiating a computational physics problem for The Laws of Physics from working with a process model of the same physical phenomena and processes.
A computational physics problem will have no numerical values for coefficients appearing in the continuous equations other than those that describe the material of interest.
Process models can be identified by the fact that given the same material and physical phenomena and processes, there is more than one specification for the continuous equations and more than one model.
Some processes models are based on more nearly complete usage of fundamental equations, and fewer parameterizations, than others.
The necessary degree of completeness for the continuous equations, and the level of fidelity for the parameterizations, in process models is determined by the dominant controlling physical phenomena and processes.
The sole issue for computational physics is Verification of the solution.
Process models will involve many calculations in which variations of the parameters in the model are the focus. None of these parameters will be associated with properties of the material. Instead they will all be associated with configurations that the material has experienced, or nearly so, at some time in the past.
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