Mathematical Structure of the Singularities at the Transitions Between Steady States in Hydrodynamic Systems

286 Pages·1983·3.063 MB·English

by  Hampton N. Shirer, Robert Wells (eds.)

Lecture Notes
in Physics
Edited by H. Araki, Kyoto, J. Ehlers, MDnchen, K. Hepp, Ziirich
R. Kippenhahn, MOnchen, H. A. Weidenmiiller, Heidelberg
and J. Zittartz, Kijln
185
Hampton N. Shirer
Robert Wells
Mathematical Structure
of the Singularities
at the Transitions
Between Steady States
in Hydrodynamk Systems
S pri nger-Verlag
Berlin Heidelberg New York Tokyo 1983
Authors
Hampton N. Shirer
Department of Meteorology
The Pennsylvania State University
University Park, PA 16802, USA
Robert Wells
Department of Mathematics
The Pennsylvania State University
University Park, PA 16802, USA
AMS Subject Classifications (1980): 58C 27, 58C 28, 76 E30
ISBN 3-540-l 2333-4 Springer-Verlag Berlin Heidelberg New York Tokyo
ISBN O-387-1 2333-4 Springer-Verlag New York Heidelberg Berlin Tokyo
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0 by Springer-Verlag Berlin Heidelberg 1983
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2153/3140-543210
Dedicated to our wives, Becky Shirer and Valerie Wells, without whose
encouragement this monograph would not have been completed.
PREFACE
Since its introduction by Rene Thom, catastrophe theory has been a potentially
valuable instrument for discovering the nature of transitional behavior in physical
systems. In the excitement generated by his new view of the world, a central
technical obstacle was considerably underrated, ~-Ith the result that now an army of
critics has replaced the original multitude of proponents.
The great promise of catastrophe theory is that with it, out of a vast number
of influences on an evolving system, we may select a small number through which the
rest will act to control the transitional behavior of that system. From this
situation, we may obtain the classical, canonical pictures of surfaces of steady
solutions and sets of bifurcation points, parameterlzed by a few numbers quantifying
the few controlling influences. However, to apply catastrophe theory as it was
originally formulated, we must have a potential or Lyapunov function for our
evolutionary system. This requirement is the central technical obstacle to the
rigorous application of catastrophe theory, the obstacle which was not overcome
adequately in the early applications. Unfortunately it is in general, extremely
difficult, if not impossible, to show that such a function exists, let alone to
produce it. Consequently, most attempts at realization of the full promise of the
theory cannot even get started.
Yet the canonical surfaces and singularity sets of catastrophe theory have
appeared, independently of that theory, in the description of the behavior of a wide
variety of physical systems. This fact is closely related to a singularity theory
originated by John Mather during his work to establish the mathematical foundations
of catastrophe theory. Besides enjoying the inestimable advantage of being
mathematically rigorous, this generalization completely by-passes the central
technical difficulty of catastrophe theory: Mather's Theory requires no Lyapunov
function, and yet it can do everything that catastrophe theory, in the presence of a
Lyapunov function, can do. In fact, now the appearance of the canonical surfaces
and singularity sets in systems not regulated by a Lyapunov function is explained
completely by Mather's Theory.
Unfortunately, Mather's Theory also includes an obstacle; it is as inaccessible
to an applied physicist as anything in mathematics can be. Accordingly, to fill the
gap between theory and utillzatlon~ in this monograph we first describe Mather's
Theory operationally using examples instead of proofs, and then we develop a
procedure for Its application to physical problems whose dynamics are governed by
systems of ordinary differential equations. We demonstrate the utility of our
procedure by applying it to three different hydrodynamic systems. We show first how
to identify the crucial parameters in the equations and then how to associate them
with the corresponding physical effects. Consequently, by finding these parameters,
we obtain systems that no longer must be unrealistically ideal because certain of
their crucial parameters need not remain identically zero. The strength of our
Vl
application of singularity theory is that we obtain a theoretical model whose
solutions are directly comparable with experimental observations.
An apparent defect of Mather's Theory is that it does not, as it stands,
describe the stability characteristics of the stationary solutions of a dynamical
system. In particular, it does not respect Hopf bifurcations. However, it is
readily extendable to a theory which does describe the stability characteristics,
and we describe this extension in the final chapter.
We are deeply grateful to Professor John A. Dutton for the encouragement and
advice freely given us during the lengthy evolution of this monograph from a jumble
of ideas to six chapters of organized material. We also thank him for his many
constructive criticisms of earlier versions of this manuscript that allowed better
presentation of its contents.
We greatly appreciate the interest and useful comments given us by our
colleagues. In particular, we thank Mr. David A. Yost for his help in unraveling
the subtleties of horizontally and vertically heated convection, Dr. Kenneth E.
Mitchell for his advice concerning quasi-geostrophic flow in a channel, and Dr.
Peter Kloeden for directing us to the appropriate low-order model of rotating
convection.
Finally, we are indebted to Mrs. Lori Weaver for her patient and meticulous
efforts in typing the nearly unending stream of revisions of this manuscript, and to
Mr. Victor King for his excellent drafting of the figures.
The research reported here was sponsored by the National Science Foundation
through grants ATM 78-02699, ATM 79-08354, and ATM 81-13223 and by the National
Aeronautic and Space Administration through grants NSG-5347 and NAS8-33794.
May 1983 Hampton N. Shirer
Robert Wells
TABLE OF CONTENTS
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
I.I Transitions in Hydrodynamics . . . . . . . . . . . . . . . . . . 1
1.2 Modeling Observed Transitions . . . . . . . . . . . . . . . . . . 3
2. INTRODUCTION TO CONTACT CATASTROPHE THEORY . . . . . . . . . . . . . . 7
2.1 The Stationary Phase Portrait . . . . . . . . . . . . . . . . . . 7
Example i. The cusp and hysteresis . . . . . . . . . . . . . . 8
2.2 The Definitions of Mather's Theory . . . . . . . . . . . . . . . 12
Example 2. A contact map to the cusp:
embedding and hysteresis . . . . . . . . . . . . . . 14
Example 3. A contact map to the cusp:
embedding and bifurcation . . . . . . . . . . . . . 14
Example 4. A contact map to the cusp: extension ....... 15
Example 5. A contact map to the cusp:
transformation of coordinates . . . . . . . . . . . 17
Example 6. Destruction of information:
loss of periodic solutions . . . . . . . . . . . . . 18
Example 7. Versal unfolding of f(x) = x . . . . . . . . . . . 20
Example 8. A versal unfolding of the Lorenz (1963) model:
a preview . . . . . . . . . . . . . . . . . . . . . 21
2.3 Mather's Theorems . . . . . . . . . . . . . . . . . . . . . . . . 24
Example 9. The cusp and Mather's Theorem I . . . . . . . . . . 25
Example I0. A versal unfolding of the Lorenz model:
Mather's Theorem II ...... . . . . . . . . . . 28
2.4 Altering Versal Unfoldings . . . . . . . . . . . . . . . . . . . 30
Example II. Codimension and the cusp . . . . . . . . . . . . . . 32
Example 12. Versal unfoldlngs of the Lorenz model:
elementary alterations . . . . . . . . . . . . . . . 33
Example 13. Versal unfoldings of the Lorenz model:
alterations . . . . . . . . . . . . . . . . . . . . 36
2.5 The Lyapunov-Schmidt Splitting Procedure . . . . . . . . . . . . 38
Example 14. A versal unfolding of the Lorenz model:
splitting and reducing lemmas . . . . . . . . . . . 45
2.6 Vector Spaces and Contact Computations . . . . . . . . . . . . . 47
Example 15. Codimenslon: Propositions 2.2 and 2.3 ....... 48
Example 16. The dimension of ~(n)/~2(n): quotient spaces . . , 49
Example 17. Codimension of x3: versal unfoldings ....... 51
Example 18. Unfoldings of ± x k, k > 2: minimal versal
forms in codimension 1 . . . . . . . . . . . . . . . 52
IIIV
TABLE OF CONTENTS (Con't)
Example 19. The hyperbolic umbillc:
minimal versal unfoldings . . . . . . . . . . . . . 53
Example 20. The elliptic umbillc:
minimal versal unfoldings . . . . . . . . . . . . . 56
2.7 Classification of Singularities . . . . . . . . . . . . . . . . . 57
Example 21. A versal unfolding of a nonpolynomial function:
contact equivalence to a polynomial ........ 58
Table 2.1 Corank i unfoldings . . . . . . . . . . . . . . . . 61
Table 2.2 Corank 2 unfoldings . . . . . . . . . . . . . . . . 61
2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3. RAYLEIGH-BENARD CONVECTION . . . . . . . . . . . . . . . . . . . . . . 67
3.1 Classification of the SingUlarity . . . . . . . . . . . . . . . . 69
3.2 Physical Interpretation of the Unfolding . . . . . . . . . . . . 73
4. QUASI-GEOSTROPHIC FLOW IN A CHANNEL . . . . . . . . . . . . . . . . . 82
4.1 Heating at the Middle Wavenumber Only . . . . . . . . . . . . . . 83
4.2 Singularities in the Vickroy and Dutton Model .......... 91
4.3 Butterfly Points in the Rossby Regime . . . . . . . . . . . . . . 94
5. ROTATING AXISYMMETRIC FLOW . . . . . . . . . . . . . . . . . . . . . . 114
5.1 The Butterfly Points . . . . . . . . . . . . . . . . . . . . . . 116
5.2 Unfolding about the Butterfly Point: The Hadley Problem .... 121
5.3 Unfolding about the Butterfly Point: The Rotating
Raylelgh-Benard Problem . . . . . . . . . . . . . . . . . . . . . 123
5.4 Dynamic Similarity . . . . . . . . . . . . . . . . . . . . . . . 125
5.4.1 Horizontal heating . . . . . . . . . . . . . . . . . . . . 131
5.4.2 Tilting domain . . . . . . . . . . . . . . . . . . . . . . 135
5.4.3 Other candidates . . . . . . . . . . . . . . . . . . . . . 137
5.4.4 Final comments . . . . . . . . . . . . . . . . . . . . . . 144
6. STABILITY AND UNFOLDINGS . . . . . . . . . . . . . . . . . . . . . . . 145
6.1 Invarlant Sets of Matrices . . . . . . . . . . . . . . . . . . . 146
Example i. Some invariant subsets of M 2 . . . . . . . . . . . . 148
6.2 Smooth Submanifolds of R n . . . . . . . . . . . . . ....... 153
Example 2. The sphere: a 2-submanlfold of R 3 ......... 154
Example 3. The double cone: a subset which is not a
submanifold of R 3 . . . . . . . . . . . . . . . . . 155
Example 4. The cone: a subset which is not a smooth
submanifold of R 3 157
Example 5. Invarlant submanifolds . . . . . . . . . . . . . . . 159
Example 6. The orbit of a matrix . . . . . . . . . . . . . . . 159
Example 7. Some orbits in M 2 . . . . . . . . . . . . . . . . . 164
IX
TABLE OF CONTENTS (Con't)
6.3 Transversality and Tangent Space . . . . . . . . . . . . . . . . 167
Example 8. Transversal curves and surfaces . . . . . . . . . . 167
Example 9. Transversality of two circles in the plane ..... 169
Example i0. The tangent space at the fold on a cusp
surface . . . , . . , , . , . . , . . . . . , . . . 176
Example II. The tangent space of Orb(F) . . . . . . . . . . . . 177
Example 12. The spaces associated with transversallty of
a map on the cusp surface . . . . . . . . . . . . . . 180
Example 13. Computational verification of transversallty
of a map on the cusp surface . . . . . . . . . . . . 183
Example 14. Transversality of maps associated with the
hyperbolic umbillc . . . . . . . . . . . . . . . . . 184
6.4 Versal Unfoldlngs and Contact Transformations of the
First Order . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
Example 15. An extended hyperbolic umbillc . . . . . . . . . . . 191
Example 16. First-order contact transformations of the
extended hyperbolic umbillc . . . . . . . . . . . . 196
Example 17. First-order contact transformation of the
hyperbolic umbillc . . . . . . . . . . . . . . . . . 196
6.5 Stability and First-Order Versal Unfoldings and Contact
Transformations . . . . . . . . . . . . . . . , , . . ..... . 201
Example 18. The modified Lorenz system unfolded further .... 203
Example 19. The stability phase portrait of a flrst-order
versal unfolding of the Lorenz system . . . . . . . 213
Example 20. The stability phase portrait of the original
unfolding of the modified Lorenz system ...... 217
6.6 First-Order Mather Theory . . . . . . . . . . . . . . . . . . . . 223
Example 21. The flrst-order Versal unfolding of x n . . . . . . . 233
Example 22. First-Order Versal unfolding of a fold . . . . . . . 235
Example 23. The stability phase portrait of a general
first-order versal unfolding of
g(x) = x2, - Xl, x32T . . . . . . . . . . . . . . 247
6.7 Conclusion . , . . . . . . . . . . . . . . . ..... . . . . . 253
APPENDIX SUMMARY OF SPECTRAL MODELS . . . . . . . . . . . . . . . . . . . 256
A.I The Lorenz Model . . . . . . . . . . . . . . . . . . . . . . . . 256
Table A.I Dimensional Variables: Lorenz Model . . . . . . . . . 257
Table A.2 Nondlmensional Variables & Parameters:
Lorenz Model . . . . . . . . . . . . . . . . . . . . . 258
TABLE OF CONTENTS (Con't)
A.2 The vickroy and Dutton Model . . . . . . . . . . . . . . . . . . 259
Table A.3 Nondlmensional Variables & Parameters:
Vickroy and Dutton Model . . . . . . . . . . . . . . . 261
A.3 The Charney and DeVore Model ................... 265
Table A.4 Dimensional Variables: Charney and
DeVore Model . . . . . . . . . . . . . . . . . . . . . 265
Table A.5 Nondimenslonal Variables & Parameters:
Charney and DeVore Model . . . . . . . . . . . . . . . 266
A.4 The Veronis Model . . . . . . . . . . . . . . . . . . . . . . . . 268
Table A.6 Dimensional Variables: Veronis Model ........ 269
Table A.7 Nondimensional Variables & Parameters:
Veronis Model . . . . . . . . . . . . . . . . . . . . 270
Table A.8 Butterfly Points in the Veronis Model ........ 273
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
"There is an almost forgotten branch of mathematics, called catastrophe theory,
which could make meteorology a really precise science."
--from a conversation between the Venerable Parakarma and Mahnayake Thero
in The Foundation of Paradise by Arthur C. Clarke, 1978.