Preface. Here, on the level sand, between the sea and land, what shall I build or write, against the fall of night? [Har40, Sec. 7]
This quote taken from G.H. Hardy's famous A Mathematician's Apology underlines the seriousness and difficulty of leaving behind something of a permanent and lasting value. As Hardy points out, it is to some extent ambition that drives the desire to do so. But, also, a driving motivation is to leave behind results of years of work and research in an accessible form for others who may thus find such remnants being of some value.
However, what is it that can stand against sand and the fall of night? Examples one may think of is, in mathematics, the eternal and objective existence of an infinity of prime numbers, or, in physics, the existence of a quantity called energy that no one so far has been able to create or destroy. Similarly, the Euler equation governing the momentum balance of fluid flow forms a cornerstone in describing the physics of the Earth's atmosphere. The firm rooting of atmospheric physics and NWP within classical physics and mathematics, together with the ambition to present accessibly the outcome of many years of NWP research may thus provide the rationale to write something that can stand on level sand against the fall of night.
Acknowledgments. Go thy way, eat thy bread with joy, and drink thy wine with a merry heart; for God now accepteth thy works. [Ecclesiastes 9:7]
Confidence and assurance of being able to reach goals set are of course prerequisite when setting out to accomplish something. However, the setting of goals - as frequently inspired by ideas - requires additional liberty of thought. Good ideas will apparently not be coming to one's mind on command, but in my experience come at most unexpected moments. It is then when goals leading to work 'accepteth by God' can be formulated given such ideas. Not accidentially does the Greek word idea denote in Plato's view a perfect, eternal, and immutable concept. It might be added that the suggestion that the preacher Qoheleth formulates in the sentence that is quoted above arises from the perceived senselessness of all of man's actions. Thus, as Qoheleth suggests, enjoying simple pleasures considered to be received as gifts from God is a viable strategy, at the same time hoping for and believing in His acceptance of one's works.
Chapter 1. At the heart of spectral methods is the fact that any nice enough function $u(x)$ can be expanded in the form $u= \sum_{|n|<\infty} \hat u_n \phi_n(x)$ where $\phi$ are the global basis functions. [HGG07, p. 66]
The succinctness of this statment is difficult to surpass. When applying spectral methods to the solution of differential equations the unknown function u(x) appears as a finite collection of expansion coefficients un that are evolving over time as governed by the model dynamics. While a duality exists between such spectral representation un and the grid-point representation of u(x) the spectral approach possesses properties that are very distinct and, in some respects, superior to properties found for a grid point-approach. Within NWP, one such advantage relates to the spherical geometry of modeling global flow on the sphere.
Chapter 2. The sun also ariseth, and the sun goeth down, and hasteth to his place where he arose. [Ecclesiastes 1:5]
The deep philosophical contents of this sentence that expresses a seemingly eternal, unchanging, and fully predictable nature of the world surrounding us has led me to include this quote from the Old Testament, when setting out, in chapter two of the book, with describing the physical laws of the atmosphere. The Sun, standing here for the most reliable and most predictable natural phenomenon, is, with its radiative power of 3.84 x 1026 W that amounts to 63 MW/m2 the driving force for all activities on Earth. While the sun's regularity is unsurpassed, the irregularity of the activities, events, and fluid motions in the atmosphere and oceans it causes through its energy supply is difficult to surpass, as well. The preacher thus expresses a deep mystery relating to our understanding of the physics of nature as it surrounds us, reflected by its regularity and simplicity and its irregularity and complexity, both of them present at the same time.
Chapter 3. The angular-momentum and energy principles are fundamental in any treatment of the circulation. If in some approximate formulation of the equations they are not retained, the results are likely to be unrealistic. A spurious energy source may, for example, cause the wind to increase without limit. [Lor67, p. 15]
This quote taken from Professor Lorenz may by itself be considered as a fundamental principle. Be they continuous or discrete, formulations of equations that are modeling the atmosphere need to respect the principles that have been raised to axioms of physics as never has anyone seen them to be violated by processes in nature. The monograph on "The nature and theory of the general circulation of the atmosphere" [Lor67] is available at the Lorenz publications site a place highly recommended to be visited.
Chapter 4. The single-layer model is one of the simplest useful models in geophysical fluid dynamics, because it allows for consideration of the effects of rotation in a simple framework without the complicating effects of stratification. [Val06, Ch. 3]
Important features of the rotational and divergent dynamics of three-dimensional baroclinic flow are found as well in two-dimensional barotropic flow governed by the single-layer model, also referred to as the shallow-water model. For this reason the SWM provides for a framework for realistic testing of components of NWP models in a simplified environment without the need to consider the vertical extent of the fluid. Not surprisingly, the classic set of tests for NWP models suggested by Williamson et al. (1992) refers to the SWM, as does the test suggested by Galewsky et al. (2006). Preliminary validation of NWP model components on the basis of these test suites builds confidence in the correctness of the code and allows for tossing out useless code before bringing it into more complex simulation models. Beyond the usefulness of the SWM for numerical modeling, of course, the continuous properties of the SWM resembling much of the continuous HPEs make the former a much more tractable environment for gaining physical insight into the dynamics from analytical investigations.
Chapter 5. Rossby never, in any of his writings used an equation of motion written in spherical coordinates. He always began with the basic physical ideas expressed in the simplest possible geometry. [Phi98, p. 1103]
The essence of the physics described by an equation shouldn't be clouded by terms of secondary importance, such as metric terms. This principle is vividly illustrated by the above quote. It is of course important to consider all terms within an equation when carrying out detailed numerical or analytical work. The physical understanding, however, might, usually be easier to achieve by dropping terms of minor importance. Of course, recognizing which terms to discard and which ones to keep is an art itself representing the key to a successful interpretation of the physical contents. A prime example of such successful simplification is not only given by the barotropic vorticity equation discussed in chapter 5, but by the quasigeostrophic set of equations considered in some detail in chapter 6.
Chapter 6. The B-29 Crew stuck to their fantastic story although the intelligence officer scoffed. "We couldn't get in to the target," they insisted. "We came up to the coast of Japan. The closer we got, the less progress we made. We finally saw the coast outlined through instruments. Then we began flying backwards. Finally we lost sight of the coast althogether and we never did see it again." Ribbed by other crews, the men looked sheepish, but they stuck to the story. Another crew came back with an equally fantastic story. "We were slugging our way along when we came to the coast. With all four engines roaring full blast, we couldn't gain an inch. We weren't flying backwards, but we weren't going forward either. We just hovered motionless over the coast. Finally we gave up and came back." The G-2 boys shook their heads, as did the aeronautical engineers. Only a couple of weeks after the first successful raid on Tokyo - on Thanksgiving Day - they had run head-on onto the biggest single obstacle to high altitude precision bombing of Japan: weather. [DS45, pp. 3-4]
The nearly geostrophic nature of large-scale flow in the free atmosphere, being fundamentally related to the rotation of the Earth, implies that winds are observed to be more or less perpendicular to and proportional to the pressure gradient. Given decreasing pressure towards the poles, predominantly westerly flow is implied that increases up to a certain height in conjunction with the pressure gradient tightening as height increases. While close-to-geostrophic conditions near the surface had been observed for much over a century, little was known about the implications of geostrophy in the free atmosphere up to the times of the second world war.
The above quote vividly illustrates the force of the jet stream as encountered by military air planes that may reach speeds well above a hundred meters per second nowadays being exploited by commercial airplanes to reduce fuel consumption when flying in an eastward direction. Interestingly, even climatological conditions are such that jet stream maxima are located above the eastern seaboards of the North America and Asia.
The quote is taken from the opening paragraphs of the cover article of the BRIEF issue published on 10 April 1945. BRIEF was an official publication of the Army Air Forces in the Pacific Ocean Areas. This 10 April 1945 issue apparently was available then for 15 cents and is available online nowadays.
Sunday, 03 March 2024