Flowvectors, Parcel Trajectories, Air Pollution, Meteorology, Wavefunctions, Matrices, Patterns, Vector Spaces, Complex Numbers,
Mechanics – Continuum, Quantum and Singular, Computer Algebra Systems, EigenAnalysis
Interested? Want to learn how these topics interrelate? Want to learn how one can “connect the dots” in a way that provides the foundation for a new
approach to the computation of particle motion in a turbulent media—in effect a new mechanics?
Called “singular mechanics”, the subject features a methodology that unites aspects of the deterministic, classical macroworld and the probabilistic, quantum microworld.
A new book, Patterns in the Wind-Volume I, Methods, examines the various “keyword” topics and develops the singular approach in intriguing detail, in 306 pages, with four appendices.
Patterns in the Wind is not a popularized text. It is a serious attempt to develop applications for the practicing meteorologist. It could be used as a textbook.
In particular the book aims to support the air pollution meteorologist who must sort out source-receptor relationships and calculate downwind concentrations of noxious pollutants from unplanned releases. The content unavoidably includes a certain amount of mathematics. Symbols and equations are dealt with, and some knowledge of linear algebra and the calculus is desirable
Volume I is now available, (in soft cover only). Interested parties may place orders for Patterns in the Wind-I as follows;
1) Order through Amazon—click on www.amazon.com, locate the book with a search on either the author’s name or the book’s title, add the item to the shopping cart, and then check out. If more than one copy is desired, Amazon makes it easy to specify the number required.
2) Telephone the publisher in San Francisco at 415-986-2425. Determine California Publishing Company’s payment procedures and shipping and handling charges. Place an order for one or more copies.
Volume I may be purchased for $9.95 (recently reduced price), plus shipping and handling. (California residents only must add 8.5% for state sales tax.). Orders placed through Amazon books can use any one of their several payment options, including a credit card.
Click on Front Cover Art to display the front cover. If necessary, click on the result to enlarge the picture.
Here are the first dozen or so pages, including the Dedication, Preface, and Table of Contents.
Volume I—Methods, with Appendices
Lewis H. Robinson
Copyright © 2001–2005 by Lewis H. Robinson
All rights reserved, Volume I and Volume II may not be reproduced, distributed or stored electronically without the prior written permission of the author. Original Manuscript Prepared July 21, 2001. Manuscripts Revised January 22, 2002, April 4, 2003, November 20, 2003, April 8, 2004, and June 30, 2005.
Printed on acid-free paper
Library of Congress Catalog Number: 2005930594
ISBN 0-9771264-0-4
FOR INFORMATION ONLY
Website: http://www.flowvector.com
Email: Lewis11@cwnet.com
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
Printed and Manufactured by
California Publishing Company
989 Howard Street
San Francisco, CA 94103
Dedicated
to the memory of all who have died from poison gas attacks in wartime and from
poison gas disasters in peacetime. When death rides the wind what shall we do?
Scatter. Some will live.
Preface
Volumes I and II
The object of Volume I is to present a unified examination of a new mechanics in order to provide a practical scheme for the calculation of advection and diffusion of atmospheric pollutants. Called “singular mechanics,” the approach collects and logically arranges for the reader scattered results from meteorology, linear algebra, and calculus that bear directly on the problem of understanding atmospheric mixing processes. The mathematical setting is finite-dimensional vector spaces. Two kinds of vectors are involved—complex matrices and flow vectors. The latter are just the horizontal winds converted into complex scalars, which in turn may be thought of as complex 2-tuples, or also as 2-by-2 matrices that commute.
The approach is to exploit the inner and outer product forms of matrix multiplication and matrix decompositions to attack the problem of Lagrangian trajectory calculation from synoptic networks of Eulerian wind surface observations. In the process of solving the problem, the singular mechanics viewpoint provides a new way to think about atmospheric mixing processes in general. Although presently limited to horizontal surface wind observations, (two-dimensional), the concept has the potential to become an all-inclusive (three-dimensional) one that enfolds all flow types, from laminar to turbulent, into a single method of computation.
The object of Volume II is to apply the results of Volume I to actual observations of surface winds from mesoscale networks. To obtain the necessary numerical computing muscle, Volume II relies heavily on the computer algebra system known as Mathematica®. As of this writing—late June 2005, Volume II is still a work in progress. Mathematica is a registered trademark of Wolfram Research, Inc.
The essence of singular mechanics is to apply the complex singular value decomposition algorithm to mesoscale wind fields to represent them as a finite sum of complex tensor products in a complex, finite-dimensional vector space. The complex tensor products, also called singular modes, are composed from time-varying left singular vectors and spatially varying right singular vectors.
For n greater than 2, all n-tuples of complex numbers are treated as n-by-1 or 1-by-n matrices. Every singular vector and singular mode thus becomes a matrix. This unconventional, apparently minor, technical issue turns out to be not so minor because of the substantial amounts of matrix algebra that are necessary to develop singular mechanics. Reasons are provided that justify this “every big tuple is a matrix” approach.
The representation also provides a matrix analog to the wave function of quantum mechanics. This object, called the singular wave function, is simply the diagonal matrix containing the decomposition’s real, positive singular values. Suitably and easily normalized, these quantities become akin to the probability amplitudes of quantum mechanics. Changes with time in the singular wave function may be tracked.
The decomposition itself has a summation form that is simply a linear combination (linear superposition) of all possible modes. A physical interpretation comes from examining a line in the decomposition’s left- and right-side tableaux. Such a line equates a set of wind observations on the left to a linear combination of invariant wind fields on the right. These wind fields are the spatial factors in the space-time tensor products that comprise singular modes. The coefficient multipliers in the linear combination are complex numbers from the temporal factors in the tensor products. Being temporal quantities, they change from line to line. The wind fields do not so change. They are the same for each line, but with a different set of complex coefficient multipliers in the linear combination.
A particle released into the mesoscale flow passing through the region of observation moves along an actual, real-world trajectory. This single path traverses the observed winds in the left-hand side of the tableaux. The superimposed singular modes on the right-hand side of the tableaux are presumed to act individually, one at a time, on the particle. Virtual copies of the particle are presumed to exist simultaneously on the right-hand side of the tableaux, one virtual particle in each of the right-hand invariant wind fields. Each copy or virtual particle traces a virtual path in its own virtual wind field. The particle’s real path requires that a selection be made from one of its virtual paths. This selection is presumed to happen according to the probabilities implicit in the singular wave function.
The state of a quantum system is a notion from quantum mechanics. It presumes that a quantum particle has an indefinite, potential existence everywhere and “everywhen” in its wave-packet. Somehow the state is an enfolded system of probabilities of finding the particle somewhere at some time. This mysterious aspect of quantum theory becomes even more mysterious with the claim that the act of observation, an act combining thoughts with laboratory hardware, causes a real number to materialize out of the cloud of probability amplitudes that comprises the wave-packets of the quantum wave function.
The preceding descriptions and presumptions are equivalent to saying that, for the fixed period between observations, a random choice from all possible modes controls a particle’s actual trajectory and, further, that the singular wave function controls this mode selection according to a quantum-mechanical-like interpretation of the singular wave function probability amplitudes as probabilities. Thus, the active mode must be selected randomly from the finite, quantized set of singular modes, and its selection must be according to the squared magnitude of the singular wave function probability amplitude associated with the mode. Singular wave function normalization insures that this magnitude is a proper probability.
Accordingly, from a fixed source at a fixed time, repeated Lagrangian trajectory calculations will almost inevitably display differing tracks. This must be true because, selected independently and probabilistically, no two sequences of controlling modes are likely to be identical. The end result is a natural calculation of Lagrangian transport with its high-probability modes intermixed with Lagrangian dispersion with its low-probability modes.
Pre-multiplied by its associated real, positive amplitude, a mode is a projection of the original mesoscale wind field onto a lower-dimensional vector space. Because such a mode retains all the physical properties of a mesoscale wind field, efficient numerical calculation of part of a particle trajectory is possible. The final trajectory is pieced together across time and space from the various path fragments obtained from the probabilistically generated sequence of controlling modes.
This project began life as a survey article of a few dozen pages, grew uncontrollably into a monograph of over fifty pages, and continued its uncontrollable growth into the present two-volume effort. The author expects that, in the final count, Volumes I and II of Patterns in the Wind will exceed five hundred pages.
Mathematica procedures that embody the
methods of singular mechanics have successfully analyzed large matrices of
mesoscale wind data. Necessary for the content of Volume II, this work has required
the author to write substantial amounts of Mathematica
code. More coding and writing remains to be done and the effort is
time-consuming. Volume II exists now only as a partial draft. It will likely
not be completed until mid or late 2006.
*****************End of Sample of Opening Pages**************
Compressed Table of Contents
CHAPTER
1
Introduction...1
Weather Forecasting at Regional and Local Scales: A
Short History of Numerical and Statistical Approaches...1
The Fashionable Statistical Approach: the Karhunen-
Loève Expansion—a Quick Overview Thereof...2
Volume I, Methods...5
Volume II, Applications...7
Volumes I and II, General Comments...7
CHAPTER
2
Continuum
Mechanics...8
A Review of an Idealization...8
Practical Considerations in Meteorology...9
CHAPTER
3
Matrices
and De-Eulerized Forms...11
De-Eulerizing W Matrices...11
CHAPTER
4
The
De-Eulerized W Matrix in Summation
Form...17
A Convention, a Rule, and a Notation...17
A Convention...17
A Rule...19
A Notation...19
The CSVD as a Finite Sum...21
The Normal Mode—Singular Mode Analogy...24
Waves—Real, Ideal, and Singular—And Superposition...25
A Flawed Metaphor...26
An Amended Metaphor...27
The Quantum Mechanical Principle of Superposition...30
Superposition of Singular Waves Versus Classical
Nonlinear Waves—A Paradox...31
The Ideal Waves of Classical Physics...32
The Addition of Amplitudes—Ideal Waves Versus
Real Waves...33
Singular Modes and Ideal Waves—A Tabular
Comparison...35
CHAPTER
5
The
De-Eulerized Matrix W in Product Form...37
CHAPTER
6
Singular
Mechanics...40
What Is Singular Mechanics?...40
Potential for Confusion: Vectors, Matrices, and Wave
Functions—Singular and Otherwise...40
How Temporal Subscripts Denote Windows in Time...42
Several Good Reasons for Treating Singular Vectors as
Matrices...43
The Sixty–Fold Way of Singular Mechanics...46
Mathematical Setting...46
Physics...46
Computational Physics...47
Assumptions, Definitions, and Conclusions...47
CHAPTER
7
Five
Topics Combining Meteorological Assumptions With
Definitions and Conclusions From Linear Algebra Into a
Meaningful Singular Mechanics...49
CHAPTER 8
Twenty–Seven
Topics Providing the Essential Tools From
Linear Algebra to Manage the Complex Matrices in Singular
Mechanics...51
CHAPTER 9
Eleven
Topics Discussing How Singular Mechanics Is Similar
to Quantum Mechanics and How It Is Different From
Quantum Mechanics, With an Emphasis on Notational
Differences...85
CHAPTER 10
Fourteen
Topics Using the Essential Tools From Linear
Algebra to Arrive at the Essence of Singular Mechanics for
Complex Matrices...93
CHAPTER 11
Three
Miscellaneous Topics ..118
CHAPTER 12
Discussion—Dimensional
Analysis and Basic Plots...119
CHAPTER
13
Physical
Idealizations...122
A Multiplicity of Additive Flow Patterns...122
Expectations About How the Flow Patterns in Singular
Modes Vary As Their Index Numbers Increase...127
Surface Winds and Singular Mechanics—What about the
Missing Vertical Motions?...129
CHAPTER
14
Trajectory
Calculations: Overview of Methodology...134
CHAPTER 15
Trajectory
Calculations: Methodology Details...142
CHAPTER 16
Trajectory
Calculations: Interpolation Issues...154
CHAPTER 17
Trajectory
Calculations: The Final Word...157
CHAPTER 18
Probability,
Expectation, Prediction, Interpolation...159
Probabilities from Probability Amplitudes Via Born
Conversion...159
An Upscale Cascade of Probabilities...166
Probabilities from Probability Amplitudes Via the
Hadamard Product...168
Expected Values...170
Prediction, a Viewpoint, and a Question...172
Analog Prediction...172
Time Series Prediction...175
Interpolation and the Unitary Property...188
CHAPTER 19
Uncertainty
in Singular Mechanics...190
CHAPTER 20
Linearity,
Non-Linearity, and the Energy Cascade...194
APPENDIX
A: NOTATION
Terminology—Base
Symbols, Enumerators, Decorators,
Glyphs—Syntax, and Semantics...199
The Treatment of Vectors: A Convention...199
Design of a Notation: Handling a Sampling Challenge, the
“Collapse” Conundrum, and Basis Universality...201
A Rule Regarding Italic Usage...201
A Notation for Patterns in the Wind...203
Part I: Base Symbols—Overview...204
Four Main Categories of Base Symbols...205
The Quintessential: Base Symbols That Denote the
Pure and Abstract Essence of an Idea, Such as Hilbert
Space...205
The Unobserved: Base Symbols That Denote the
Intangible, Such as Future Observations or Simple
Abstract Expressions...205
The Observed: Base Symbols That Denote the
Tangible, Such as Actual Observations or Computable
Expressions...206
The Mixed: Unobserved or Observed Base Symbols
with Mixed Scripting...206
Part II: Enumerator Attachments...206
Treatment of Orphaned Enumerators...207
Part III: Decorator Attachments...208
Treatment of Orphaned Decorators...209
Decorators That Are Containers...210
Treatment of Containers and Certain Decorators: Never
in Italics...211
Decorators Considered in a Broader Context...211
Decorator Listings...212
Four Decorators Used to Signify Four Kinds of
Products...215
Part IV: Three Examples with Listings...215
First Example: Comprehensive...216
Second Example: Application...219
Third Example: Glyphs Only, with Short Descriptions...223
Part V: Listings by Category...224
List of Quintessentially Abstract Symbols in
Appendix D...224
List of the Unobserved (Symbols for the Intangible)...224
List of the Observed (Symbols for the Tangible)...230
A Few Mixed Symbols and Expressions...232
The Change of Basis Matrix...232
The Properties of
P...232
The Linear Form for a Component of a Matrix after
Changing Its Basis...236
APPENDIX B: COMMENTARY
Poetry
and Meteorology...240
Poetry and Singular Mechanics: “Because It’s There”...240
A Matrix Curiosity...241
Metaphysics, Mathematics, and Quantum Mechanics...243
History of the Singular Value Decomposition, Real and
Complex...248
APPENDIX C: STREAM LINES AND TRAJECTORIES
APPENDIX
D, PART I: REVIEW, VECTORS
Overview,
Fashions in Mathematical Writing...254
Overview: The Abstract Setting...255
Boldface Notation and Issues Concerning Its Usage...256
The Axioms...257
Vector Varieties...258
Linear Independence; Linear Combination; Spanning Set;
Basis and Dimension of a Linear Vector Space; the Coordinate
Representation of an Abstract Vector; the Change of Basis
Matrix..259
The Abstract Concept of a Mapping Between Sets: An
Isomorphism Is a Special Mapping...264
Linear Mappings in Linear Vector Spaces Are Called Linear
Transformations; Linear Functionals Are Special Cases of
Linear Transformations; An Inner Product Is a Special Case of
a Linear Functiona...264
Examples of Linear Mappings...266
The Product of Two Complex Matrices; Definition of a
Complex Matrix Inner Product; Determination of an
Orthonormal Basis of Complex Matrices...270
The Linear Vector Space of Observed Complex
Matrices over the Complex Number Field
; Vector
Space
Bases and Dimensions; The Rank of the Observed Complex
Matrix...274
APPENDIX D, PART II: REVIEW, WAVES
Overview...286
A Pure Spatial Sinusoid—Motionless...286
A Pure Temporal Sinusoid...287
Temporal Modulation of a Spatial Sinusoid...288
Setting a Spatial Sinusoid Into Motion...288
Euler’s Relation...290
The Expression for the Sinusoid Generalized...290
Expressions for the Moving Cosine Wave—All
Equivalent...291
The Plane Wave...292
REFERENCES—VOLUME I...297
TOPICS INDEX...300
ABOUT THE AUTHOR...306
Volume I has four Appendices, A through D. Appendix A provides notational details. It contains lists of symbol tables with descriptive text. Appendix B is a digression into elegance; the poetry to be found in the science of meteorology, the axiom systems of mathematics, and the equations of physics. Appendix C reviews the definitions and equations for vector field, trajectory and streamline. It includes the differential equations for the latter and briefly describes how Mathematica devises their numerical solution from an observed vector field. Appendix D is a limited review of vector space concepts, definitions, terminology, and results.
(Volume II, applications, is still in preparation. It is about 30% complete.)
Expect updated information to be posted to this web
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