University of Groningen
Dynamics of the Lorenz-96 model
van Kekem, Dirk Leendert
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van Kekem, D. L. (2018). Dynamics of the Lorenz-96 model: Bifurcations, symmetries and waves. Rijksuniversiteit Groningen.
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D Y N A M I C S
of the
L O R E N Z - 9 6 M O D E L
Bifurcations, symmetries and waves
u
The research for this doctoral dissertation has been carried out at the Faculty of Science and Engineering, University of Groningen, The Netherlands, within the Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence.
Dynamics of the Lorenz-96 model Bifurcations, symmetries and waves
© Copyright Dirk L. van Kekem, 2018 PhD Thesis Rijksuniversiteit Groningen isbn 978-94-034-0979-5 (printed version) isbn 978-94-034-0978-8 (electronic version)
Dynamics of the Lorenz-96 model
Bifurcations, symmetries and waves
Proefschrift
ter verkrijging van de graad van doctor aan de Rijksuniversiteit Groningen
op gezag van de
rector magnificus prof. dr. E. Sterken en volgens besluit van het College voor Promoties.
De openbare verdediging zal plaatsvinden op vrijdag 12 oktober 2018 om 14:30 uur
door
Dirk Leendert van Kekem
geboren op 8 november 1989 te Ede
Promotor Prof. dr. H.W. Broer Copromotor Dr. A.E. Sterk Beoordelingscommissie Prof. dr. H. Waalkens Prof. dr. B.W. Rink Prof. dr. S.M. Wieczorek
The fear of the LORD is the beginning of wisdom: and the knowledge of the holy is understanding. — proverbs 9:10
P R E FA C E A N D A C K N O W L E D G E M E N T S
p r e f a c e On the basis of this thesis lies the monoscale Lorenz-96model, constructed by Edward Lorenz to study the predictabil-ity of the atmosphere. It turns out that the Lorenz-96 model con-tains an extraordinarily rich structure of bifurcations, worth to examine in greater mathematical detail. The aim of this thesis is to provide a systematic study into the dynamics of the model, focusing on the bifurcation structures before chaos.
The first chapter of this thesis is devoted to the scientific work of Lorenz and introduces the Lorenz-96 model as the main subject of this thesis. The chapters 2–5 describe the results of the mathe-matical analysis of the Lorenz-96 model. In chapter 6, we present the main conclusions of our research, accompanied by a scientific summary.
a c k n o w l e d g e m e n t s First and above all, it suits me to thank God who gave me the strength and light to do my research and to write this thesis. The logo of this university expresses this with the words verbum domini lucerna pedibus nostris — i.e. The word of the Lord is a lamp unto our feet, a reference to Psalm 119, verse 105 — by which the founders of our university confessed that they want to be guided by the Word of God.
Besides that, this thesis would not have been written without the help, trust and support of many people, either directly or in-directly. I would like to thank all of them. In particular, I want to express my gratitude to the following persons:
a l e f s t e r k Thank you very much for giving me the opportu-nity to do my PhD. It was a real pleasure to work with you and I am fortunate that I had you as a daily supervisor. You have vii
viii p r e f a c e a n d a c k n o w l e d g e m e n t s
always been very kind and supportive to me. During these four years, I learned a lot from you on doing research, writing scientific papers and teaching. Working together with a work-aholic means that I had a hard job in matching you in terms of working hours per year... I really wish that your relentless zeal will pay out in the end.
h e n k b r o e r I am grateful to you, as my supervisor and pro-motor. This gave me the opportunity to benefit from your knowledge and experience to become a better researcher. You learned me (amongst others) to distinguish main issues from side issues and how to get the message across in science. a s s e s s m e n t c o m m i t t e e I would like to thank the members
of the assessment committee, Professors Holger Waalkens, Bob Rink and Sebastian Wieczorek for reading the manuscript of my thesis and for their valuable remarks that helped me im-proving the thesis.
I want to express my gratitude to all people that showed in-terest in my research. I want to acknowledge Andre Vanderbauw-hede for his clarifications regarding equivariant bifurcation theory. I am in debt to my former and current officemates: Vladimir, Mohammad (Khan) and Gabriël. Thank you for creating the pleas-ant, friendly and stimulating atmosphere. Also, I wish to thank all others who stayed in our office for a shorter period.
Furthermore, I would like to thank all helpful and friendly col-leagues of the (Johann) Bernoulli Institute. Besides food and sci-ence, there was enough lunch- and coffeetime left for conversa-tions about other things transcending science. Thanks for making this time enjoyable! Mahdi, thank you for being my paranimph.
Vanaf deze plaats wil ik ook mijn ouders, schoonouders, ver-dere familie en vrienden bedanken voor hun interesse in en steun tijdens mijn onderzoek. Voor een luisterend oor is het niet ver-moeiend om nòg een keer in alledaags Nederlands uit te leggen waar mijn onderzoek over gaat. Als mijn pogingen tot (enig) be-grip hebben geleid, stemt dat tot verheuging.
p r e f a c e a n d a c k n o w l e d g e m e n t s ix
Ten slotte: mijn Maatje! Je was achtereenvolgens mijn vriendin, verloofde en vrouw gedurende deze periode. Ik ben immens dankbaar om jou aan mijn zijde te hebben. In het bijzonder dank ook voor het grondig proeflezen van het manuscript. Zonder jou zou ik niet zijn wie ik nu ben. Jij mag met recht een πα%ανιµϕoς,
paranimf1, genoemd worden. 1
Dat wil zeggen, iemand die naast de bruidegom staat
DvK Groningen, August 2018.
C O N T E N T S
p r e f a c e a n d a c k n o w l e d g e m e n t s vii
1 i n t r o d u c t i o n 1
1.1 Predictability: Lorenz’s voyage . . . 2
1.1.1 Chaos (un)recognised . . . 2
1.1.2 Unpredictable atmosphere and weather fore-cast . . . 3
1.1.3 Lorenz’s contribution to mathematics . . . . 8
1.2 Lorenz-96 model . . . 11
1.2.1 The monoscale Lorenz-96 model . . . 11
1.2.2 Applications . . . 14
1.2.3 Setting of the problem . . . 14
1.3 Overview of this thesis . . . 17
1.3.1 Main results . . . 18 1.3.2 Outline . . . 23 A N A LY S I S O F T H E L O R E N Z - 9 6 M O D E L 27 2 t h e l o r e n z-96 model 29 2.1 General properties . . . 30 2.2 Two-parameter model . . . 35 2.3 Symmetries . . . 38 2.3.1 Zn-Equivariance . . . 39 2.3.2 Invariant manifolds . . . 40 3 b i f u r c at i o n a na ly s i s 47 3.1 Bifurcations for positive F . . . 48
3.2 Unfolding: two-parameter model . . . 56
3.2.1 General dimensions . . . 56
3.2.2 Unfolding for n=12 . . . 60
xii c o n t e n t s
3.3 Bifurcations for negative F . . . 63
3.3.1 Preliminaries . . . 63
3.3.2 First pitchfork bifurcation . . . 65
3.3.3 Second pitchfork bifurcation . . . 68
3.3.4 Hopf bifurcations . . . 70
3.3.5 Multiple pitchfork bifurcations . . . 77
4 wav e s t r u c t u r e s 85 4.1 Approximation of periodic orbit . . . 86
4.2 Travelling waves for positive F . . . 87
4.2.1 Spatiotemporal properties . . . 88
4.2.2 Symmetric waves . . . 91
4.3 Travelling and stationary waves for negative F . . . 92
4.3.1 Spatiotemporal properties . . . 92
4.3.2 Symmetric waves . . . 98
4.4 Coexistence of multiple waves . . . 102
5 r o u t e s t o c h a o s 107 5.1 Individual routes to chaos . . . 108
5.1.1 Dimension n=4 . . . 108 5.1.2 Dimension n=5 . . . 112 5.1.3 Dimension n=6 . . . 113 5.1.4 Dimension n=7 . . . 116 5.1.5 Dimension n=8 . . . 117 5.1.6 Dimension n=12 . . . 118 5.1.7 Dimension n=36 . . . 121 5.1.8 Dimension n=40 . . . 125 5.2 Patterns . . . 125 5.2.1 General dimensions . . . 125 5.2.2 Dimensions n=5k . . . 127 6 s u m m a r y a n d c o n c l u s i o n s 131 6.1 Summary of results . . . 132 6.2 Major conclusions . . . 136 6.3 Further research . . . 137
c o n t e n t s xiii
A P P E N D I C E S 139
a au x i l i a r y m at e r i a l 141
a.1 Multiscale version . . . 141
a.2 Elementary equivariant dynamical systems theory . 142
a.2.1 Equivariant dynamical systems . . . 143
a.2.2 Equivariant branching lemma . . . 146
b l o n g p r o o f s 149
b.1 Proof of Lemma3.1 . . . 149
b.2 Proof of Theorem3.5 . . . 153
b.2.1 Simplifying the expression . . . 154
b.2.2 Sign of the first Lyapunov coefficient . . . . 160
b.3 Proofs by center manifold reduction . . . 163
b.3.1 Proof of Theorem3.14 . . . 163 b.3.2 Proof of Lemma3.18 . . . 170 b i b l i o g r a p h y 171 s a m e n vat t i n g 185 s.1 Ons onderzoek . . . 185 s.2 Onze resultaten . . . 187 c u r r i c u l u m v i tæ 193
