Chapter 7. The fundamental principle of spectral collocation methods is, given discrete data on a grid, to interpolate the data globally, then evaluate the derivative of the interpolant on the grid. [Tre00, p. 7]
This single sentence captures the essence of the transform method executed in a standard spectral NWP model. The discrete data are the model fields on the two-dimensional Gaussian grid (level by level in the vertical) on the sphere. They are interpolated globally using the spherical harmonics basis. The - analytically known - derivatives of the latter then serve to obtain the horizontal derivatives of the fields on the grid. Subsequent formation of products followed by finding the products' spectral representation completes the transform method.
Chapter 8. At this point we depart from many of the currently used models in which the wind field is defined at n levels and the temperature field at n-1 levels. [Lor60a, p. 370]
This departure Professor Lorenz took from the path at that time commonly traveled in numerical weather prediction has led to a strategy for the vertical arrangement of variables that is nowadays standard and that allows for ensuring the energetic consistency of the model. In fact, the paper [Lor60a] that is available at the Lorenz publications site has become a classic reference in NWP, for its pioneering character, at the same time documenting an impressive comprehesiveness and clarity in terms of its writing style.
Chapter 9. The scheme is complicated because the atmosphere is complicated. [Ric07, p. xi]
The model that Richardson used for numerical prediction contained complexities absent in standard NWP models as used today (see, [Lyn06, Ric07]). Nevertheless, even without these complexities the details and complications encountered are numerous, as is becoming particularly clear in this chapter that describes the semi-implicit time integration. It did therefore seem appropriate to me to refer in the chapter quote to the remark that Richardson made in his preface almost a century ago.
Chapter 10. I suppose that even the world itself could not contain the books that should be written. [John 21:25]
Today's science knowledge is vast and very diverse, and, in fact, locating correct information rapidly and confidently has become a major enterprise. Adding to this fact is the condition that the increase in published literature is itself accelerating, as, for example, may be measured by looking at the increase in published pages during the 1980s and the 2000s decades, for example.
Nevertheless, significant gaps may still exist in the literature in terms of proving concise, authoritative, and consolidated presentations of research results that have stood the test of time. As an example, I'd like to mention, to the best of my knowledge, the absence, within the atmospheric sciences, of an updated overview about what is referred to as normal-mode initialisation (NMI). While the concept of normal modes is included in the book in its most rudimentary form only, deep understanding of atmospheric dynamics has been gained from that NMI concept. A rather characteristic statement related to this context may be found in [Lyn06] paraphrased as "The success of today's NWP and in particular its use of the semi-implicit time integration technique hinges critically on the observation (see, e.g., Lynch 2006, p. 3) that it is apparently possible to distort the former (i.e., the fast gravity waves), without significantly affecting the latter (i.e., the slow planetary waves) that are of meteorological interest".
The above quote from the New Testament is in fact only the second part of that entire verse that fully reads "And there are also many other things which Jesus did, the which, if they should be written every one, I suppose that even the world itself could not contain the books that should be written. Amen."
Interestingly, it has been suggested to produce all books conceivably to see the light of day by an army of monkeys continuously working on typewriters thus randomly producing pages over pages over pages, thereby working on creating the "Total Library". By the Borel-Cantelli Lemma, sometimes, popularly, referred to as the Infinite Monkey Theorem, with (almost) certainty, then, within a infinite time, all known books and works, such as Shakespeare's Hamlet, will be produced, in addition to all books so far unknown. More abstractly the result may be understood by the fact that again, almost surely, the dual representation of any real number contains an infinite number of ones (and zeroes).
Of course, in the process of their writing, the monkeys produce much meaningless write-up that needs to be separated from the truly significant work. Assessing quantitatively the chances for some useful outcome has led to the statement that it might be preferable to leave the monkeys to attend to monkeys' business rather than employing them in the process of book writing.
As expressed eloquently by Feller (1968; p. 202): "Consider a message such as PROBABILITY IS FUN written in the Morse code as a finite sequence of dots and dashes. When we write H for dot and T for dash this message will appear as a finite succession of heads and tails. It follows from the preceeding example that a prolonged tossing of a coin is certain sooner or later to produce the given message and to repeat it infinitely often. By the same token the record of a prolonged coin-tossing game is bound to contain every conceivable book in the Morse code, from Hamlet to eight-place logarithmic tables. It has been suggested that an army of monkeys might be trained to pound typewriters at random in the hope that ultimately great works of literature would be produced. Using a coin for the same purpose may save feeding and training expenses and free the monkeys for other monkey business."
Chapter 11. The principal role of computers in fluid dynamics should be to give physical insight into dynamics, not to generate "realistic" flows or to extol minor virtues of one numerical scheme over another ... All reasonable standards of accuracy should be maintained and the range of applicability of any model should be clearly and explicitly understood before embarking upon a series of lengthy simulation experiments with increasing physical complexity. [OI74, pp. 281-282]
What is the role of numerical models and numerical computation ? The above quote taken from a review paper on the numerical simulation of viscous incompressible flows provides the key to the answer. The primary role of models used for numerical simulation is to obtain physical insight into the dynamics that are simulated. In this context, models claim their role as the continuous dynamics may under the usual circumstances display a complexity in terms of their behavior that is close to impossible to be unraveled by analytic attack. Thus enters a numerical model that may be used to explore such complexities that of course remain as numerous as they are in the continuous equations. The numerical model, however, offers the advantage to "see" the behavior of the equations in terms of numerical solution. Of course such numerical solution hinges on all the "reasonable standards" being employed in the development of the model. Numerical weather prediction is a prime example by itself for providing physical insight into the dynamics as the forecasts produced have to stand against nature and its evolving weather each and every day.
Chapter 12. The shallow water equations on a rotating sphere serve as a primary test problem for numerical methods used in modeling global atmospheric flows. [WDH+92, Sec. 1]
In a manner similar to the quote preceding chapter 4 on the shallow-water model, the above sentence emphasizes the importance of testing model designs on simplified sets of equations before proceeding to using them in more complex modeling situations. The present quote is taken from the paper that describes the nowadays classic suite of seven test cases developed to probe a numerical model design against "the major difficulties associated with the horizontal dynamical aspects of atmospheric modeling on the spherical earth" as stated in the abstract of [WDH+92]. Performing well in dealing with these difficulties clearly is a prerequisite for any model to be competitive with existing models when extended to simulating baroclinic flow. Within chapter 12 it is demonstated that PEAK may easily be simplified by setting a few switches to run on test situations described in [WDH+92].
Appendix A. My son, if thou wilt receive my words, and hide my commandments with thee; So that thou incline thine ear unto wisdom, and apply thine heart to understanding. [Book of Proverbs, 2:1-2]
Appendix A contains numerous results of a rather technical nature that are referred to in the main text of the book. It seemed therefore appropriate to place the above sentence from the Book of Proverbs at the beginning of this appendix, to ask you, the reader, to kindly receive these words, or rather the equations [ ... ☺ ... ], as they are stated here, by considering the equations' format to be resembling a little bit the nature of commandments.
Appendix B. Handbooks on mathematical functions are stuffed with such identities. [Boy01, Sec. 18.11]
A vast number of expressions for functions arising in various contexts in mathematical physics are available, and, as stated in the quotation opening this appendix, such expressions are filling volumes and volumes of books. Needless to say, only some of such expressions are useful for numerical computation of function values that is stable and accurate. It is of course these latter expressions, that sometimes appear in the form of recursions, that have found their way into the code of NWP models. This appendix, preceded by the above quotation, describes the adaption of such a specific recursion in order to be useful for the numerical computation of high-degree and high-order associated Legendre functions. Computation of such recursions may reach an efficiency to the extent that whenever a function value is required it is newly computed rather than retrieved from a storage location where it might have been saved as the result of a previous computation.
Appendix C. Give not over thy mind to heaviness, and afflict not thyself in thine own counsel. The gladness of the heart is the life of man, and the joyfulness of a man prolongeth his days. Love thine own soul, and comfort thy heart, remove sorrow far from thee: for sorrow hath killed many, and there is no profit therein. Envy and wrath shorten the life, and carefulness bringeth age before the time. A cheerful and good heart will have a care of his meat and diet. [Ecclesiasticus 30:21-25]
Coding of algorithms and equations in terms of a computer language may sometimes turn out to be a lengthy process. Many times coding is found to be slowed down by disturbing behavior of the code that is considered to be free of errors but apparently is not as it does not produce numbers as it should. This quote placed of the beginning of the reprinted PEAK code should serve as encouragement for anyone not to be taken over by heaviness of the mind in case such unexpected behavior of code is encountered. With confidence and patience errors can be found and removed even when present in large pieces of code, and I have offered some remarks on error searching in section 10.2.3. of the book. A strategy I have thus found most useful in the process of coding is given by continued testing of the code as it is developed, ascertaining correctness of individual portions of code first by themselves, and subsequently as these are joined and executed together.
May it be mentioned that the above quote is taken from the book Ecclesiasticus in the Old Testament and not from Ecclesiates as (unfortunately, due to a printing error) stated on p. 441 of the book (see also Errata). Ecclesiasticus, also known as Wisdom of Sirach is part of the Christian biblical canon, but was not accepted by most Protestants, and as well has not been accepted into the Hebrew Bible. Ecclesiaticus closely resembles the book of Proverbs, unlike Ecclesiates, sometimes known as and attributed to the preacher Qoheleth. In this latter book of the Old Testament, the main speaker Qoheleth expresses reflections on the meaning of life repeatedly proclaiming the senselessness and futility of man's actions. As a consequence of such total lack of meaning, the preacher suggests accepting the gifts from God, and relying on God's acceptance of one's efforts in turn, a quote I chose to stand for opening the Acknowledgments of the book.
Afterword. Then I said, I have laboured in vain, I have spent my strength for nought, and in vain. [Isaiah 49:4]
Does the result achieved through one's labors justify the efforts that were taken in reaching it? Being uncertain whether the answer is in the affirmative, may lead to expressing the pessimistic opinion of Isaiah that the effort has been spent in vain, and consequently rather should not have been undertaken. Nevertheless, without having taken the effort in the first place the result - the quality of which always remains a debatable issue - would not have been achieved either, but the strength would have been spent, disappeared, and wasted, anyway.
Sunday, 03 March 2024