Models Methods Software

Dan Hughes

The Purpose of Computing

“The purpose of computing is insight, not numbers.” Richard Hamming

In order to gain insight, deep understanding of the source of the numbers is of paramount importance. Until deep understanding of the source of the numbers is attained, the numbers do not provide insight. In my opinion, deep understanding of the source of the numbers is best attained by a bottom-up approach. Merely comparing line, contour, or 3-dimensional plots of some calculated numbers does not constitute either understanding or verification/validation.

Introduction and Background
Typically we see statements like this, “The models are based on the fundamental laws of conservation of mass, momentum, and energy.” Or, “Computer models based on physics.” The articles from the public press that contain such statements sometimes allude to other aspects of the complete picture, but generally always present an optimistic picture relative to the present status of climate-change modeling.

Such statements provide, at the very best, only a starting point relative to where the numbers presented actually come from. Of course 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. It appears to me that the climate-change community is in a unique position relative to presenting such informal kinds of information. Unfortunately, it seems that there is no complete documentation to which readers can be referred so as to get a fuller picture of the source of the numbers.

As one example, and there are many others, the equations usually presented do not represent either a complete or accurate statement of the problem; all fluid flows encountered in real-world problems are turbulent flows. The equations usually presented, the three-dimensional Navier-Stokes equations, are not the complete description for practical turbulent fluid flows. Additionally, 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 are the surface of the Earth and at the air-water interface presented by the boundary between the atmosphere and oceans. More will be said about the continuous equations later in these notes.

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. Finally, it is extremely important to note that it is known that the numerical solution methods used in AOLGCM computer codes have not yet been shown to converge to the solutions of the continuous equations. More will be said about the discrete equations later in these notes.

The ultimate source of the calculated results are the numerical solutions from computer software. Thus a bottom-up approach for gaining an understanding of the reported results requires that the nitty-gritty details of what is actually in the computer codes be available for inspection and study.

The numerical solutions arise from the discrete equations that are used to approximate the continuous equations that are the actual basis of the discrete equations. Detailed documentation of the discrete equations and the associated solution methods is required in order to understand the source of the reported results.

The actual continuous equations that comprise the models arise from the fundamental continuous equations and laws that describe the physical phenomena and processes of importance for the intended application areas of the models and software. The actual continuous equations are generally not the same as the fundamental equations for a variety of valid assumptions and approximations applied to the fundamental equations and laws.

Finally, some of the models, methods, and equations used to generate the numbers have no basis in either theory or empirical data.

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. These properties are also discussed in this discussion paper.

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 and methods components:

Fundamental Basic Model Equations
1. These equations are generally from continuum mechanics such as the Navier-Stokes for mass, momentum and energy conservation, heat conduction, 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 material of interest.

Even though fundamental basic equations of mass, momentum, and energy conservation are taken as the starting point for the modeling of a few of 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 scalar mass and energy equations are typically less effected than the vector momentum equations in this regard. The exact radiative transfer equations, for example, are not solved, but instead approximations are introduced to make the problem tractable. A few examples are given in the following paragraphs.

With almost no exceptions whatsoever, 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 basic 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 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 AOLGCM models/codes. For the case of two- or multi-phase flows, liquid droplets in a gaseous mixture for example, the fundamental laws are not known.

The exchanges of mass, momentum, and energy at the interfaces between the (atmosphere, oceans, land, biological, 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 AOLGCM 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. Reasonable assumptions are applied to the fundamental law so that a reasonable approximation to the physical phenomena for the intended application is obtained.

Finally, 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. Additionally, as next discussed, the algebraic equations of empirical data are often far from based on fundamental laws.

Engineering Models and Correlations of Empirical Data
2. 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.

Special Purpose Models
3. 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 AOLGCM 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.

Important Sources from Human-Made Equipment
4. 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 AOLGCMs, the equipment and processes involved in conversion of materials in one form and composition into other forms and compositions.

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.

” All predictions are extrapolations.” … unknown to me

Numerical Solution Methods
5. Analytical and numerical solution methods for all the equations that comprise the models. 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 which are enormous in size.

Almost all important physical processes occur at spatial scales which 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 AOLGCM solution methods almost none of these temporal scales are resolved.

It is a true 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 AOLGCMs 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 AOLGCM 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 AOLGCM 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.

Auxiliary Functional Methods
6. Auxiliary functional methods include instructions for installation on the users’ computer system, 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.

The results from the software must be independent of the various operating systems and compilers that are used to employ the codes.

Non-functional Requirements
7. Non-functional aspects of the software include its ease of, and fitness for, understandability, maintainability, extensibility and portability.

Large complex codes have generally evolved, sometimes over decades, in contrast to being built from scratch and thus include a variety of potential sources of problems in these areas.

Summary
I think all of the above properties and processes, presented from a bottom-up focus, constitute a more nearly complete and correct characterization of AOLGCM computer codes. The models and methods are incorporated into computer software for use and application to the analyses for which the models and methods were designed to be applied.

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. Such documentation is not available for any AOLGCM models/codes. Additional discussions of the depth of documentation required of production-level software is available, and some documentation issues specific to AOLGCM models/codes have been summarized here.

” Somebody’s got to pony up some equations.” … Unknown to me.

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.

Unlike a “pure” science problem, such as solution of the incompressible Navier-Stokes equations to resolve directly turbulent motions for which the basic equations are solved, the simplifications and assumptions made at the fundamental-equation level, the correlations and parameterizations, and especially the finite-difference aspects of AOLGCMs 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. Additionally, physical instabilities are known to be captured by the equations in AOLGCMs, and the behavior of the numerical solution methods when these are resolved becomes vitally important.

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February 27, 2007 - Posted by | Uncategorized |

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