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HADLEY CELL EXPANSION IN TODAYÕS CLIMATE AND PALEOCLIMATES Bill Langford University Professor Emeritus Department of Mathematics and Statistics University of Guelph, Canada Presented to the BioM&S Symposium on Climate Change and Ecology University of Guelph April 28, 2011

Joint work with: Greg Lewis University of Ontario Institute of Technology 2

OUTLINE Present-day climate changes Greenhouse and icehouse climate modes Questions from paleoclimatology Mathematical model of Hadley convection Numerical analysis by Greg Lewis Implications of the model The Pliocene Paradox and future work 3

Mean Circulation of TodayÕs Atmosphere 4

Meridional Streamfunction 1979-2001: Annually and Longitudinally Averaged [European Centre for Medium Range Weather Forecasts] Note: Hadley, Ferrel and Polar Cells. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5 0 -50 -50 -50 -1 0 -10 -10 -10 -1 0 -1 0 -10 -10 -10 -10 -1 0 5 5 5 5 5 5 5 5 5 5 5 30 30 30 30 3 0 3 0 1000 900 800 700 600 500 400 300 200 100 80 O N 6 0 O N 4 0 O N 2 0 O N 0 O 20 O S 4 0 O S 6 0 O S 8 0 O S Annual mean -300 -250 -200 -150 -100 -75 -50 -30 -20 -10 -5 Pressure (hPa) Mean meridional streamfunction 10 9 kg/sec 5 10 20 30 50 75 100 150 200 250 300 5

Zonal (Longitudinal) Wind 1979-2001: Annually and Longitudinally Averaged [European Centre for Medium Range Weather Forecasts] Note: Trade Winds and Jet Streams. 0 0 0 0 0 0 0 0 0 -2 -2 -2 -2 -2 -2 2 2 2 2 2 2 2 2 2 2 2 15 15 15 15 15 15 15 1000 900 800 700 600 500 400 300 200 100 Pressure (hPa) 80 O N 6 0 O N 4 0 O N 2 0 O N 0 O 20 O S 4 0 O S 6 0 O S 8 0 O S Zonal mean wind Annual mean -100 -80 -60 -50 -40 -35 -30 -25 -20 -15 -10 -5 -2 2 5 10 15 20 25 30 35 40 50 60 80 100 m/sec 6

PRESENT-DAY CLIMATE CHANGES Intergovernmental Panel on Climate Change [IPCC Report 2007 and beyond] The mean global temperature is rising. The poles are warming faster than the tropics. More rain in equatorial regions, less in subtropics. The Hadley cells are expanding poleward. The Hadley circulation is slowing. The jet streams are moving poleward. 7

Understanding the Causes of TodayÕs Climate Change Mean global warming is believed to be driven by greenhouse gas buildup (anthropogenic). Enhanced polar warming is caused by positive feedbacks, such as a decrease in albedo due to the melting of ice caps. There is no consensus on the causes of the changes in Hadley cells and jet streams. 8

PALEOCLIMATES The Earth has experienced dramatically different climate ÒmodesÓ in its geological history. The two dominant modes of the past half- billion years are often called Ò Greenhouse Climate Ó and Ò Icehouse Climate Ó. Better knowledge of paleoclimate changes will help us to understand modern day climate changes, and vice-versa.

Greenhouse Climate Mode Mean annual temperature (MAT) was a few degrees warmer than today. (But means can mislead.) The global climate was more EQUABLE than today. Equable climate means: 1. Warmer winters without much warmer summers; i.e. low seasonality . 2. Low temperature gradient, pole-to-equator. Most of the higher MAT is due to the warmer winters and warmer polar regions.

Greenhouse Climate dominated the Mesozoic Ò Age of DinosaursÓ 240-65 Mya

Icehouse Climate Mode Permanent polar icecaps (all year). Large pole-to-equator temperature gradient. Cold winters and warm/hot summers for mid-latitudes of Earth; i.e. high seasonality . Equatorial region has climate similar to that in the greenhouse climate mode. This Òicehouse climate modeÓ has been dominant only in the past 30 million years. 12

Outstanding Questions of Paleoclimatology How can such different but stable climate modes both exist on the same Earth? Why has the Earth ÒpreferredÓ greenhouse to icehouse climate for most of 250 My? What has caused abrupt changes between greenhouse and icehouse climate modes? 13

Our Mathematical Model

Question 15 Can the mean behaviour of the Hadley cells and atmospheric ßow be replicated in a simple model based on convection, rotation and spherical geometry ?

Basic Components of the Mathematical Model Navier-Stokes PDE in rotating spherical shell. Boussinesq ßuid [density varies linearly with temperature]. Incompressible ßuid for convenience. Convection is driven by the latitudinal temperature gradient on the inner boundary. 16

Rotating spherical shell of ßuid differentially heated on the interior boundary ! " r # T $ Assume rotational symmetry and north-south reßectional symmetry. 17

Averaged solar heating of the rotating tilted earth 18 ! !"# $ $"# $"# % %"# & &"# ' '"# ( ()*)( ! ()*)( ! ) ) )()+,-.%) / 0,1223456,7 Fix this as the boundary temperature on the surface of the Earth.

Navier-Stokes Boussinesq Equations Lewis a nd La ngfo rd 7 the ßuid is assumed to b e ! = ! 0 (1 ! " ( T ! T r )) , (1 ) where ! is the densit y o f the ßuid, T is the temp erat ure, " is the (co ns t an t) co e ! c ien t of therma l expa nsion, a nd ! 0 is t he densit y a t a ref er e nce temp e r ature T r . The d ime nsio nle ss qua n tit y " ( T ! T r ) is assumed t o b e s ma ll. In the Boussines q a ppro x imatio n the ßuid ca n b e considered to b e incom pres sible, w hic h is a sig niÞcan t simpliÞcatio n. The ßuid is con tained within a spherical shell w it h inn e r sphere of ra dius r a and o uter sphere o f radi us r b . W e a ss ume gra vit y a c t s ev erywhere in the radia l directio n. The spherical shell r ota tes a t rate " ab out t he p ola r axis, and the inner and ou ter s pheres rot ate at the same rat e. The equa tio ns a re writ ten in spherical p ola r co or dinates in a frame of referenc e co-rot ating at rat e " with the shell. The ra dial, p olar , an d azim utha l co o rdinates a re denoted r , # , and $ , res p ectiv ely , with unit v e ct ors e r , e ! and e " ; s ee Figure 1. The Na vier-Stok e s B o uss inesq e q uatio ns desc r ibing the ev o lution of the v e cto r ßuid v e - lo cit y , u = u ( r , # , $ , t ) = w e r + v e ! + u e " and t he t e mp era ture of the ßuid, T = T ( r , # , $ , t ) ar e : % u % t = & " 2 u ! 2 ! # u + [ g e r + ! # ( ! # r )] " ( T ! T r ) ! 1 ! 0 " p ! ( u á " ) u , (2 ) % T % t = ' " 2 T ! ( u á " ) T , (3 ) " á u = 0 , (4 ) where ! = " (cos # e r ! sin # e ! ) is the rota tion v ec t or, " = | ! | is the rate of ro tat ion ab o ut the p o lar a xis, p is the pres sure devia tion fro m p 0 = ! 0 g ( R ! r ) + ! 0 " 2 r 2 sin 2 # / 2, r = r e r + # e ! + $ e " , & is the k ine ma tic viscosit y , ' is the co e ! cien t of therma l di # usivit y , g is the gra vita tiona l acceleratio n, " is the usual gr adien t o p era to r in s pher ic a l co ordina tes, u is the azim utha l ßuid v elo cit y , often r ef err e d to as t he zonal v elo cit y , v is the p o lar ßuid v elo cit y , a nd w is the radia l ßuid v elo cit y . The spatia l doma in is de Þned b y r a < r < r b , 0 $ $ < 2 ( , a nd 0 < # < ( . Th us, # = 0 , ( corr e sp ond t o the nor th and sout h p o les o f the shell res p ectiv e ly , while # = ( / 2 co rresp o nds to the equa tor . 1 The e q uatio ns c a n b e rewritten in planeta ry co o rdinat e s b y p erforming the c ha nge of v ar iable # % ( / 2 ! # . T he v alues of & and ' ar e c hosen to b e tho s e of the ßuid at the reference temp eratur e T r , and it is assume d that the di # erence b et w een the t emp era ture o f the ßuid and T r is ev ery w here small eno ugh so that & and ' can b e considered a s c o nstan t s . W e ha v e inc luded t he e # ects of cen trifug al b o uy a ncy in th e equa tions via the term ! # ( ! # r ). All dimens io nal qua n tities ar e measured in CGS units. 1 Pl anetary co ord ina te s d i ! e r f rom s p herical p ol ar c o ordi nates onl y in th e ran ge of th e p olar c o ord inate. where u is the velocity vector, T is temperature, ! is the rotation vector, p is pressure deviation, " is kinematic viscosity, # is thermal diffusivity, g is gravitational acceleration and ! is the gradient operator. Here is the buoyancy force. Lewis a nd La ngfo rd 7 the ßuid is assumed to b e ! = ! 0 (1 ! " ( T ! T r )) , (1 ) where ! is the densit y o f the ßuid, T is the temp erat ure, " is the (co ns t an t) co e ! c ien t of therma l expa nsion, a nd ! 0 is t he densit y a t a ref er e nce temp e r ature T r . The d ime nsio nle ss qua n tit y " ( T ! T r ) is assumed t o b e s ma ll. In the Boussines q a ppro x imatio n the ßuid ca n b e considered to b e incom pres sible, w hic h is a sig niÞcan t simpliÞcatio n. The ßuid is con tained within a spherical shell w it h inn e r sphere of ra dius r a and o uter sphere o f radi us r b . W e a ss ume gra vit y a c t s ev erywhere in the radia l directio n. The spherical shell r ota tes a t rate " ab out t he p ola r axis, and the inner and ou ter s pheres rot ate at the same rat e. The equa tio ns a re writ ten in spherical p ola r co or dinates in a frame of referenc e co-rot ating at rat e " with the shell. The ra dial, p olar , an d azim utha l co o rdinates a re denoted r , # , and $ , res p ectiv ely , with unit v e ct ors e r , e ! and e " ; s ee Figure 1. The Na vier-Stok e s B o uss inesq e q uatio ns desc r ibing the ev o lution of the v e cto r ßuid v e - lo cit y , u = u ( r , # , $ , t ) = w e r + v e ! + u e " and t he t e mp era ture of the ßuid, T = T ( r , # , $ , t ) ar e : % u % t = & " 2 u ! 2 ! # u + [ g e r + ! # ( ! # r )] " ( T ! T r ) ! 1 ! 0 " p ! ( u á " ) u , (2 ) % T % t = ' " 2 T ! ( u á " ) T , (3 ) " á u = 0 , (4 ) where ! = " (cos # e r ! sin # e ! ) is the rota tion v ec t or, " = | ! | is the rate of ro tat ion ab o ut the p o lar a xis, p is the pres sure devia tion fro m p 0 = ! 0 g ( R ! r ) + ! 0 " 2 r 2 sin 2 # / 2, r = r e r + # e ! + $ e " , & is the k ine ma tic viscosit y , ' is the co e ! cien t of therma l di # usivit y , g is the gra vita tiona l acceleratio n, " is the usual gr adien t o p era to r in s pher ic a l co ordina tes, u is the azim utha l ßuid v elo cit y , often r ef err e d to as t he zonal v elo cit y , v is the p o lar ßuid v elo cit y , a nd w is the radia l ßuid v elo cit y . The spatia l doma in is de Þned b y r a < r < r b , 0 $ $ < 2 ( , a nd 0 < # < ( . Th us, # = 0 , ( corr e sp ond t o the nor th and sout h p o les o f the shell res p ectiv e ly , while # = ( / 2 co rresp o nds to the equa tor . 1 The e q uatio ns c a n b e rewritten in planeta ry co o rdinat e s b y p erforming the c ha nge of v ar iable # % ( / 2 ! # . T he v alues of & and ' ar e c hosen to b e tho s e of the ßuid at the reference temp eratur e T r , and it is assume d that the di # erence b et w een the t emp era ture o f the ßuid and T r is ev ery w here small eno ugh so that & and ' can b e considered a s c o nstan t s . W e ha v e inc luded t he e # ects of cen trifug al b o uy a ncy in th e equa tions via the term ! # ( ! # r ). All dimens io nal qua n tities ar e measured in CGS units. 1 Pl anetary co ord ina te s d i ! e r f rom s p herical p ol ar c o ordi nates onl y in th e ran ge of th e p olar c o ord inate. 19

Boundary Conditions Inner Boundary T varies with Non-slip BC for velocity u Outer Boundary T is insulated Stress-free BC for velocity u Lewis a nd La ngfo rd 8 As describ ed in the I n tro ductio n, the b ounda ry c o nditio ns ar e u = 0 , T = T r ! ! T cos(2 ! ) o n r = r a , " u " r = 0 , " v " r = 0 , w = 0 , " T " r = 0 on r = r b , (5 ) with 2 # -p erio dicit y in t he azim uthal v a riable $ . In this pa p e r w e will in v es t igat e ßo ws tha t preserv e the symmetries of the mo del, tha t is, ar e in v aria n t und e r ro tatio n a b o ut t he p olar ax is (i.e. they a re indep e nden t o f the azim uthal v aria ble $ ), a nd in v ar ian t under r e ßectio n a c r oss the equa to r (i.e. across t he line deÞned b y ! = # / 2) . Therefor e , w e study solut ions of ( 2) Ð (5) in the f o rm u = u ( r , ! , t ) = u ( r , # ! ! , t ) , v = v ( r , ! , t ) = v ( r , # ! ! , t ) , w = w ( r , ! , t ) = w ( r , # ! ! , t ) , T = T ( r , ! , t ) = T ( r , # ! ! , t ) , (6 ) In th is con text , solutio ns tha t a re indep enden t of $ ar e often c a lled axisymmetric . T he assumed symmetries signi Þc a n tly s imp lif y the a nalysis. W e ma y u s e t he ana lysis of the symmetric s y s t e m as a sta rting p oin t fo r a n a nalysis o f the full system. Altho ugh it is not written explicitly , t he solut ions a lso de p end on the pa rameters. If w e s ca le the radia l co o rdina te a s r " R r ! , (7 ) where R = r b ! r a is t he ga p widt h, write T " T ! + T r ! ! T cos(2 # ) , (8 ) substitute in to ( 2) Ð (4 ), and dr op the pr imes , w e obt ain t he f o llo wing equat ions describing the ev o lution o f t he ßuid v elo cit y u = w ( r , ! , t ) e r + v ( r , ! , t ) e ! + u ( r , ! , t ) e " , pressure de v iatio n p = p ( r , ! , t ) and temp eratur e devia tion T = T ( r , ! , t ): " u " t = % s # 2 0 u ! % s 1 r 2 sin 2 ! u ! 2 " (sin ! w + cos ! v ) ! 1 R ! ( u á # 0 ) u + cos ! r sin ! uv + uw r " (9 ) " v " t = % s # 2 0 v ! % s # 1 r 2 sin 2 ! v ! 2 r 2 " w " ! $ + 2 " cos ! u ! 1 & 0 R r " p " ! ! % ' " 2 R r sin ! cos ! & ( T ! ! T cos 2 ! ) ! 1 R ! ( u á # 0 ) v ! cos ! r sin ! u 2 + v w r " (1 0) " w " t = % s # 2 0 w ! % s # 2 r 2 cos ! sin ! v + 2 r 2 " v " ! + 2 r 2 w $ + 2 " sin ! u ! 1 & 0 R r " p " r ! ' % " 2 R r sin 2 ! + g & ( T ! ! T cos 2 ! ) ! 1 R ! ( u á # 0 ) w ! 1 r % u 2 + v 2 & " (1 1) " T " t = ( s # 2 0 T + 4 ! T ( s r 2 % cos 2 ! + cos 2 ! & + 2 ! T R r sin 2 ! v ! 1 R ( u á # 0 ) T (1 2) # 0 á u = " w " r + 2 r w + 1 r " v " ! + cos ! r sin ! v = 0 (1 3) 20

Numerical Analysis Discretize on NxN grid. Get sparse nonlinear equations. Solve nonlinear system by Newton iteration. Use Keller continuation from the trivial solution, which is known exactly at . Lewis a nd La ngfo rd 11 If w e w r ite u = u ! + u 0 , ! = ! ! + ! 0 , T = T ! + T 0 , (1 9) where u 0 , ! 0 , T 0 is a s t e a dy so lution, and s ubstit ute in to the t hree e q uatio ns for u , ! , and T , w e o btain the p ertur batio n equa tions in u ! , ! ! , and T ! . T he trivia l so lution s a tisÞes the p erturba tion e q uatio ns , and it corr e sp onds t o u 0 , ! 0 , T 0 . I f t he p ert urbati on equa tions a re linearized and w e a s sume that the unkno w n functions ma y b e w r itten a s u ! ( r , " , t ) = e ! t # u ( r , " ) , ! ! ( r , " , t ) = e ! t # " ( r , " ) , T ! ( r , " , t ) = e ! t # T ( r , " ) , (2 0) then a linear eigen v alue problem is o btained. Consequen tly , the eigen v alues $ can b e found from the generalized eigen v alue pr oblem of t he form $ A 0 ! = L 0 ! , (2 1) where ! = ! " # u # " # T , # $ is t he eigenfunction, and A 0 and L 0 ar e 3 ! 3 mat rice s of linear di " e r e n tial op era tors. 4 N umer i cal metho ds 4 .1 Di scr et izat io n Be ca us e it is not p ossible to Þnd ana lytic solu tions for either the stea dy solut ion o r the eigen v alue pro ble m, the solutio ns a re appro ximat e d n um e r ic a lly . Second order cen tered Þnite di " erenc ing is used to discretize the s pa tia l deriv at iv es. W e appro xima te t he v alue of t he unkno w n f unctio ns at the lo cat ions of N ! N uniforml y spaced gr id p oin ts in the in terio r o f the d omain. Su # c ien t accura cy is o btained f r om a ppro ximat ing t he solutio ns on a uniform g rid b ecause the b o undary la y ers in the stea dy solut ion a re not sev ere (see b elo w). The v alues o f u and T on the o uter b o undary and on the equat or, a s w ell a s those o f T at the p ol e , ar e not deter mine d b y the b o undary conditio ns, a nd m ust a ls o b e considered as unkno wns. This leads to dis cretized solutio n v ectors o f size 3 N 2 + 5 N . Discretizat ion of the steady equa tions f o r u , ! , and T leads to a system o f no nline a r algebra ic equa tions that ca n b e s o lv ed b y Ne wto n iterat ion a nd K eller con tin ua tion a s expla ined in Sec t ion 4.2 , to Þnd an app ro x imatio n o f the steady solutio n. F or the n umerical a ppro ximatio n of the eigen v alues, the linearized p erturba tion equa tions ar e dis cr e t iz ed, a nd th us the v a lue s o f the steady solutio n ar e only needed a t sp e ciÞc lo c a tions (the gr id p oin ts) a nd the compu ted a ppro x imatio ns a re used. That is, the linea rizatio n is made a b out t he appro xima te s o lutio n. Th us, up on discretizat ion the pa rtia l di " e r e n tial eigen v alue problem b ec o mes a genera liz ed mat rix e ig en v alue pr oblem. Lewis a nd La ngfo rd 12 4. 2 So lut io n tec hni ques W e are in terested in computing the s t e a dy s o lution for a w ide rang e of para meter v alues. T o do t his , w e implemen t pseudo-arcleng th con tin uatio n with the Ke ller co rrection co ndition [7], a nd use a New t on metho d to solv e the resulting equat ions. If a s o lution is kno wn for a par ticular set o f para meter v a lues , then this metho d ca n b e used e ! ec t iv ely to follo w s o lutions as a para meter is v aried, i.e . t o Þnd a solut ion curv e (with r e sp ect to the pa ramet e r ). Here, w e kno w tha t fo r " T = 0, the t rivial solutio n sa tisÞe s the equa tions for u , ! and T . Th us, for " T small, the trivia l s o lution is a reasona ble pr e dictio n of the solu tion, and NewtonÕs me t ho d is us ed for the cor rec t ion. I n psue d o-arclength con t iuatio n, the para me- ter is considered as a n unkno w n, and init ial g ue sses of the solutio n are fo und b y follo wing the tang en t, or a sec a n t line a ppro ximat ion, to the solutio n curv e. I ncremen ts are made appr o xim ately a long the solutio n curv e , a nd not b y incremen ting the pa rameter. The K eller conditio n e nsures that the corr e ctio ns to t he initia l guess es o cc ur a ppro ximately p erp en- dicularly t o the tang en t. This me t ho d is pa rticular ly us eful b e ca us e it is a ble to compute solutio ns alo ng t he solutio n curv e ev e n whe n there is a limit p oin t o n the curv e , i.e. when the solutio n cur v e t urns ba c k on itse lf. In pr actice, the ev alua tion of the Ja cobian is e x p ensiv e, and therefore, in order to reduc e the n um b er o f Jacobia n ev aluat ions, w e use a qua s i-Newton metho d instead of Ne wto nÕs metho d. The genera liz ed matrix eigen v alue pro blem tha t results fro m the discretizat ion o f (21 ) is solv ed in Mat lab using the implicitly r e sta rted Arno ldi metho d [17 ], whic h is a memory - e # cie n t iterat iv e met ho d f o r Þnding a sp e ciÞed n um b er of the larg e st eigen v alu e s. A g ene r - alized Ca yley transfor matio n [7] is made s o tha t the Arno ldi itera tion Þnds the eig e n v a lue s of in terest. The pa rameter s of t he tra nsf o rmat ion c a n b e c ho se n to impro v e co n v erg enc e pro p ert ies . In p articula r, t he g e nera lize d Ca y ley t ransforma tion C ( L , A ) = ( L ! " 1 A ) ! 1 ( L ! " 2 A ) (2 2) maps eigen v alues # of the genera liz ed ma trix eigen v alue pro ble m # A v = L v to eig en v a lues $ of t he transfor me d matrix C ( L , A ), suc h t hat t he e ig en v a lues # with R e a l( # ) > ( " 1 + " 2 ) / 2 ar e ma pp e d to the e ig en v alues $ with | $ | > 1, whe r e " 1 and " 2 ar e th e rea l para me t e r s o f t he Ca y le y tr ansforma tion. The ma trix C ( L , A ) do es no t ha v e to b e fo rmed explicitly , b ec a use the Ar noldi it e r atio n only requires ma trix-v e cto r pr o ducts in v olving C ( L , A ) [17]. Th us, the full spa rsene ss prop erties o f L and A can b e explo ited, and comput e r memor y requiremen ts can b e reduced. 21

Bifurcation Analysis Use the temperature difference as both bifurcation parameter and Keller continuation parameter. Monitor eigenvalues of the linearized system to determine bifurcation and stability as varies. Identify both symmetry-preserving and symmetry-breaking bifurcations. Lewis a nd La ngfo rd 12 4. 2 So lut io n tec hni ques W e are in terested in computing the s t e a dy s o lution for a w ide rang e of para meter v alues. T o do t his , w e implemen t pseudo-arcleng th con tin uatio n with the Ke ller co rrection co ndition [7], a nd use a New t on metho d to solv e the resulting equat ions. If a s o lution is kno wn for a par ticular set o f para meter v a lues , then this metho d ca n b e used e ! ec t iv ely to follo w s o lutions as a para meter is v aried, i.e . t o Þnd a solut ion curv e (with r e sp ect to the pa ramet e r ). Here, w e kno w tha t fo r " T = 0, the t rivial solutio n sa tisÞe s the equa tions for u , ! and T . Th us, for " T small, the trivia l s o lution is a reasona ble pr e dictio n of the solu tion, and NewtonÕs me t ho d is us ed for the cor rec t ion. I n psue d o-arclength con t iuatio n, the para me- ter is considered as a n unkno w n, and init ial g ue sses of the solutio n are fo und b y follo wing the tang en t, or a sec a n t line a ppro ximat ion, to the solutio n curv e. I ncremen ts are made appr o xim ately a long the solutio n curv e , a nd not b y incremen ting the pa rameter. The K eller conditio n e nsures that the corr e ctio ns to t he initia l guess es o cc ur a ppro ximately p erp en- dicularly t o the tang en t. This me t ho d is pa rticular ly us eful b e ca us e it is a ble to compute solutio ns alo ng t he solutio n curv e ev e n whe n there is a limit p oin t o n the curv e , i.e. when the solutio n cur v e t urns ba c k on itse lf. In pr actice, the ev alua tion of the Ja cobian is e x p ensiv e, and therefore, in order to reduc e the n um b er o f Jacobia n ev aluat ions, w e use a qua s i-Newton metho d instead of Ne wto nÕs metho d. The genera liz ed matrix eigen v alue pro blem tha t results fro m the discretizat ion o f (21 ) is solv ed in Mat lab using the implicitly r e sta rted Arno ldi metho d [17 ], whic h is a memory - e # cie n t iterat iv e met ho d f o r Þnding a sp e ciÞed n um b er of the larg e st eigen v alu e s. A g ene r - alized Ca yley transfor matio n [7] is made s o tha t the Arno ldi itera tion Þnds the eig e n v a lue s of in terest. The pa rameter s of t he tra nsf o rmat ion c a n b e c ho se n to impro v e co n v erg enc e pro p ert ies . In p articula r, t he g e nera lize d Ca y ley t ransforma tion C ( L , A ) = ( L ! " 1 A ) ! 1 ( L ! " 2 A ) (2 2) maps eigen v alues # of the genera liz ed ma trix eigen v alue pro ble m # A v = L v to eig en v a lues $ of t he transfor me d matrix C ( L , A ), suc h t hat t he e ig en v a lues # with R e a l( # ) > ( " 1 + " 2 ) / 2 ar e ma pp e d to the e ig en v alues $ with | $ | > 1, whe r e " 1 and " 2 ar e th e rea l para me t e r s o f t he Ca y le y tr ansforma tion. The ma trix C ( L , A ) do es no t ha v e to b e fo rmed explicitly , b ec a use the Ar noldi it e r atio n only requires ma trix-v e cto r pr o ducts in v olving C ( L , A ) [17]. Th us, the full spa rsene ss prop erties o f L and A can b e explo ited, and comput e r memor y requiremen ts can b e reduced. Lewis a nd La ngfo rd 12 4. 2 So lut io n tec hni ques W e are in terested in computing the s t e a dy s o lution for a w ide rang e of para meter v alues. T o do t his , w e implemen t pseudo-arcleng th con tin uatio n with the Ke ller co rrection co ndition [7], a nd use a New t on metho d to solv e the resulting equat ions. If a s o lution is kno wn for a par ticular set o f para meter v a lues , then this metho d ca n b e used e ! ec t iv ely to follo w s o lutions as a para meter is v aried, i.e . t o Þnd a solut ion curv e (with r e sp ect to the pa ramet e r ). Here, w e kno w tha t fo r " T = 0, the t rivial solutio n sa tisÞe s the equa tions for u , ! and T . Th us, for " T small, the trivia l s o lution is a reasona ble pr e dictio n of the solu tion, and NewtonÕs me t ho d is us ed for the cor rec t ion. I n psue d o-arclength con t iuatio n, the para me- ter is considered as a n unkno w n, and init ial g ue sses of the solutio n are fo und b y follo wing the tang en t, or a sec a n t line a ppro ximat ion, to the solutio n curv e. I ncremen ts are made appr o xim ately a long the solutio n curv e , a nd not b y incremen ting the pa rameter. The K eller conditio n e nsures that the corr e ctio ns to t he initia l guess es o cc ur a ppro ximately p erp en- dicularly t o the tang en t. This me t ho d is pa rticular ly us eful b e ca us e it is a ble to compute solutio ns alo ng t he solutio n curv e ev e n whe n there is a limit p oin t o n the curv e , i.e. when the solutio n cur v e t urns ba c k on itse lf. In pr actice, the ev alua tion of the Ja cobian is e x p ensiv e, and therefore, in order to reduc e the n um b er o f Jacobia n ev aluat ions, w e use a qua s i-Newton metho d instead of Ne wto nÕs metho d. The genera liz ed matrix eigen v alue pro blem tha t results fro m the discretizat ion o f (21 ) is solv ed in Mat lab using the implicitly r e sta rted Arno ldi metho d [17 ], whic h is a memory - e # cie n t iterat iv e met ho d f o r Þnding a sp e ciÞed n um b er of the larg e st eigen v alu e s. A g ene r - alized Ca yley transfor matio n [7] is made s o tha t the Arno ldi itera tion Þnds the eig e n v a lue s of in terest. The pa rameter s of t he tra nsf o rmat ion c a n b e c ho se n to impro v e co n v erg enc e pro p ert ies . In p articula r, t he g e nera lize d Ca y ley t ransforma tion C ( L , A ) = ( L ! " 1 A ) ! 1 ( L ! " 2 A ) (2 2) maps eigen v alues # of the genera liz ed ma trix eigen v alue pro ble m # A v = L v to eig en v a lues $ of t he transfor me d matrix C ( L , A ), suc h t hat t he e ig en v a lues # with R e a l( # ) > ( " 1 + " 2 ) / 2 ar e ma pp e d to the e ig en v alues $ with | $ | > 1, whe r e " 1 and " 2 ar e th e rea l para me t e r s o f t he Ca y le y tr ansforma tion. The ma trix C ( L , A ) do es no t ha v e to b e fo rmed explicitly , b ec a use the Ar noldi it e r atio n only requires ma trix-v e cto r pr o ducts in v olving C ( L , A ) [17]. Th us, the full spa rsene ss prop erties o f L and A can b e explo ited, and comput e r memor y requiremen ts can b e reduced. 22

Stability Analysis Calculate the leading eigenvalues of the linearized system. Use implicitly restarted Arnoldi method. Find critical (zero) eigenvalues. Over 3000 lines of code. 23

OUR MODEL RESULTS Ref. Gregory M. Lewis and William F. Langford (2008). Hysteresis in a rotating differentially heated spherical shell of Boussinesq ßuid. SIAM J. Applied Dynamical Systems, V. 7, pp. 1421-1444. 24

HADLEY CELL CHANGES As increases, the Hadley cell shrinks toward the equator and the Ferrel and polar cells appear. x y ! T =0.0006 K 0 5 10 15 5 10 15 x y ! T =0.0015 K 0 5 10 15 5 10 15 x y ! T =0.0018 K 0 5 10 15 5 10 15 x y ! T =0.0022 K 0 5 10 15 5 10 15 x y ! T =0.0028 K 0 5 10 15 5 10 15 x y ! T =0.0036 K 0 5 10 15 5 10 15 Lewis a nd La ngfo rd 12 4. 2 So lut io n tec hni ques W e are in terested in computing the s t e a dy s o lution for a w ide rang e of para meter v alues. T o do t his , w e implemen t pseudo-arcleng th con tin uatio n with the Ke ller co rrection co ndition [7], a nd use a New t on metho d to solv e the resulting equat ions. If a s o lution is kno wn for a par ticular set o f para meter v a lues , then this metho d ca n b e used e ! ec t iv ely to follo w s o lutions as a para meter is v aried, i.e . t o Þnd a solut ion curv e (with r e sp ect to the pa ramet e r ). Here, w e kno w tha t fo r " T = 0, the t rivial solutio n sa tisÞe s the equa tions for u , ! and T . Th us, for " T small, the trivia l s o lution is a reasona ble pr e dictio n of the solu tion, and NewtonÕs me t ho d is us ed for the cor rec t ion. I n psue d o-arclength con t iuatio n, the para me- ter is considered as a n unkno w n, and init ial g ue sses of the solutio n are fo und b y follo wing the tang en t, or a sec a n t line a ppro ximat ion, to the solutio n curv e. I ncremen ts are made appr o xim ately a long the solutio n curv e , a nd not b y incremen ting the pa rameter. The K eller conditio n e nsures that the corr e ctio ns to t he initia l guess es o cc ur a ppro ximately p erp en- dicularly t o the tang en t. This me t ho d is pa rticular ly us eful b e ca us e it is a ble to compute solutio ns alo ng t he solutio n curv e ev e n whe n there is a limit p oin t o n the curv e , i.e. when the solutio n cur v e t urns ba c k on itse lf. In pr actice, the ev alua tion of the Ja cobian is e x p ensiv e, and therefore, in order to reduc e the n um b er o f Jacobia n ev aluat ions, w e use a qua s i-Newton metho d instead of Ne wto nÕs metho d. The genera liz ed matrix eigen v alue pro blem tha t results fro m the discretizat ion o f (21 ) is solv ed in Mat lab using the implicitly r e sta rted Arno ldi metho d [17 ], whic h is a memory - e # cie n t iterat iv e met ho d f o r Þnding a sp e ciÞed n um b er of the larg e st eigen v alu e s. A g ene r - alized Ca yley transfor matio n [7] is made s o tha t the Arno ldi itera tion Þnds the eig e n v a lue s of in terest. The pa rameter s of t he tra nsf o rmat ion c a n b e c ho se n to impro v e co n v erg enc e pro p ert ies . In p articula r, t he g e nera lize d Ca y ley t ransforma tion C ( L , A ) = ( L ! " 1 A ) ! 1 ( L ! " 2 A ) (2 2) maps eigen v alues # of the genera liz ed ma trix eigen v alue pro ble m # A v = L v to eig en v a lues $ of t he transfor me d matrix C ( L , A ), suc h t hat t he e ig en v a lues # with R e a l( # ) > ( " 1 + " 2 ) / 2 ar e ma pp e d to the e ig en v alues $ with | $ | > 1, whe r e " 1 and " 2 ar e th e rea l para me t e r s o f t he Ca y le y tr ansforma tion. The ma trix C ( L , A ) do es no t ha v e to b e fo rmed explicitly , b ec a use the Ar noldi it e r atio n only requires ma trix-v e cto r pr o ducts in v olving C ( L , A ) [17]. Th us, the full spa rsene ss prop erties o f L and A can b e explo ited, and comput e r memor y requiremen ts can b e reduced. 25

One-Cell Pattern for Small A single large Hadley cell extends from equator to pole. Note the jet stream at high altitude and trade winds in the tropics. 26 Lewis a nd La ngfo rd 12 4. 2 So lut io n tec hni ques W e are in terested in computing the s t e a dy s o lution for a w ide rang e of para meter v alues. T o do t his , w e implemen t pseudo-arcleng th con tin uatio n with the Ke ller co rrection co ndition [7], a nd use a New t on metho d to solv e the resulting equat ions. If a s o lution is kno wn for a par ticular set o f para meter v a lues , then this metho d ca n b e used e ! ec t iv ely to follo w s o lutions as a para meter is v aried, i.e . t o Þnd a solut ion curv e (with r e sp ect to the pa ramet e r ). Here, w e kno w tha t fo r " T = 0, the t rivial solutio n sa tisÞe s the equa tions for u , ! and T . Th us, for " T small, the trivia l s o lution is a reasona ble pr e dictio n of the solu tion, and NewtonÕs me t ho d is us ed for the cor rec t ion. I n psue d o-arclength con t iuatio n, the para me- ter is considered as a n unkno w n, and init ial g ue sses of the solutio n are fo und b y follo wing the tang en t, or a sec a n t line a ppro ximat ion, to the solutio n curv e. I ncremen ts are made appr o xim ately a long the solutio n curv e , a nd not b y incremen ting the pa rameter. The K eller conditio n e nsures that the corr e ctio ns to t he initia l guess es o cc ur a ppro ximately p erp en- dicularly t o the tang en t. This me t ho d is pa rticular ly us eful b e ca us e it is a ble to compute solutio ns alo ng t he solutio n curv e ev e n whe n there is a limit p oin t o n the curv e , i.e. when the solutio n cur v e t urns ba c k on itse lf. In pr actice, the ev alua tion of the Ja cobian is e x p ensiv e, and therefore, in order to reduc e the n um b er o f Jacobia n ev aluat ions, w e use a qua s i-Newton metho d instead of Ne wto nÕs metho d. The genera liz ed matrix eigen v alue pro blem tha t results fro m the discretizat ion o f (21 ) is solv ed in Mat lab using the implicitly r e sta rted Arno ldi metho d [17 ], whic h is a memory - e # cie n t iterat iv e met ho d f o r Þnding a sp e ciÞed n um b er of the larg e st eigen v alu e s. A g ene r - alized Ca yley transfor matio n [7] is made s o tha t the Arno ldi itera tion Þnds the eig e n v a lue s of in terest. The pa rameter s of t he tra nsf o rmat ion c a n b e c ho se n to impro v e co n v erg enc e pro p ert ies . In p articula r, t he g e nera lize d Ca y ley t ransforma tion C ( L , A ) = ( L ! " 1 A ) ! 1 ( L ! " 2 A ) (2 2) maps eigen v alues # of the genera liz ed ma trix eigen v alue pro ble m # A v = L v to eig en v a lues $ of t he transfor me d matrix C ( L , A ), suc h t hat t he e ig en v a lues # with R e a l( # ) > ( " 1 + " 2 ) / 2 ar e ma pp e d to the e ig en v alues $ with | $ | > 1, whe r e " 1 and " 2 ar e th e rea l para me t e r s o f t he Ca y le y tr ansforma tion. The ma trix C ( L , A ) do es no t ha v e to b e fo rmed explicitly , b ec a use the Ar noldi it e r atio n only requires ma trix-v e cto r pr o ducts in v olving C ( L , A ) [17]. Th us, the full spa rsene ss prop erties o f L and A can b e explo ited, and comput e r memor y requiremen ts can b e reduced. x y stream function ! 051015 2 4 6 8 10 12 14 x y azimuthalvelocity u + ! 051015 2 4 6 8 10 12 14 x y temperature T ! + 051015 2 4 6 8 10 12 14

Three-Cell Pattern for Large Lewis a nd La ngfo rd 12 4. 2 So lut io n tec hni ques W e are in terested in computing the s t e a dy s o lution for a w ide rang e of para meter v alues. T o do t his , w e implemen t pseudo-arcleng th con tin uatio n with the Ke ller co rrection co ndition [7], a nd use a New t on metho d to solv e the resulting equat ions. If a s o lution is kno wn for a par ticular set o f para meter v a lues , then this metho d ca n b e used e ! ec t iv ely to follo w s o lutions as a para meter is v aried, i.e . t o Þnd a solut ion curv e (with r e sp ect to the pa ramet e r ). Here, w e kno w tha t fo r " T = 0, the t rivial solutio n sa tisÞe s the equa tions for u , ! and T . Th us, for " T small, the trivia l s o lution is a reasona ble pr e dictio n of the solu tion, and NewtonÕs me t ho d is us ed for the cor rec t ion. I n psue d o-arclength con t iuatio n, the para me- ter is considered as a n unkno w n, and init ial g ue sses of the solutio n are fo und b y follo wing the tang en t, or a sec a n t line a ppro ximat ion, to the solutio n curv e. I ncremen ts are made appr o xim ately a long the solutio n curv e , a nd not b y incremen ting the pa rameter. The K eller conditio n e nsures that the corr e ctio ns to t he initia l guess es o cc ur a ppro ximately p erp en- dicularly t o the tang en t. This me t ho d is pa rticular ly us eful b e ca us e it is a ble to compute solutio ns alo ng t he solutio n curv e ev e n whe n there is a limit p oin t o n the curv e , i.e. when the solutio n cur v e t urns ba c k on itse lf. In pr actice, the ev alua tion of the Ja cobian is e x p ensiv e, and therefore, in order to reduc e the n um b er o f Jacobia n ev aluat ions, w e use a qua s i-Newton metho d instead of Ne wto nÕs metho d. The genera liz ed matrix eigen v alue pro blem tha t results fro m the discretizat ion o f (21 ) is solv ed in Mat lab using the implicitly r e sta rted Arno ldi metho d [17 ], whic h is a memory - e # cie n t iterat iv e met ho d f o r Þnding a sp e ciÞed n um b er of the larg e st eigen v alu e s. A g ene r - alized Ca yley transfor matio n [7] is made s o tha t the Arno ldi itera tion Þnds the eig e n v a lue s of in terest. The pa rameter s of t he tra nsf o rmat ion c a n b e c ho se n to impro v e co n v erg enc e pro p ert ies . In p articula r, t he g e nera lize d Ca y ley t ransforma tion C ( L , A ) = ( L ! " 1 A ) ! 1 ( L ! " 2 A ) (2 2) maps eigen v alues # of the genera liz ed ma trix eigen v alue pro ble m # A v = L v to eig en v a lues $ of t he transfor me d matrix C ( L , A ), suc h t hat t he e ig en v a lues # with R e a l( # ) > ( " 1 + " 2 ) / 2 ar e ma pp e d to the e ig en v alues $ with | $ | > 1, whe r e " 1 and " 2 ar e th e rea l para me t e r s o f t he Ca y le y tr ansforma tion. The ma trix C ( L , A ) do es no t ha v e to b e fo rmed explicitly , b ec a use the Ar noldi it e r atio n only requires ma trix-v e cto r pr o ducts in v olving C ( L , A ) [17]. Th us, the full spa rsene ss prop erties o f L and A can b e explo ited, and comput e r memor y requiremen ts can b e reduced. 27 Hadley, Ferrel and Polar cells all exist. The jet stream has moved toward the equator. x y stream function ! 051015 2 4 6 8 10 12 14 x y azimuthalvelocity u + ! 051015 2 4 6 8 10 12 14 x y temperature T ! + 051015 2 4 6 8 10 12 14

Compare the Model with Real Data x y stream function ! 051015 2 4 6 8 10 12 14 x y azimuthalvelocity u + ! 051015 2 4 6 8 10 12 14 x y temperature T ! + 051015 2 4 6 8 10 12 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -5 0 -50 -50 -50 -1 0 -10 -10 -10 -1 0 -1 0 -10 -10 -10 -10 -1 0 5 5 5 5 5 5 5 5 5 5 5 30 30 30 30 3 0 3 0 1000 900 800 700 600 500 400 300 200 100 80 O N 6 0 O N 4 0 O N 2 0 O N 0 O 20 O S 4 0 O S 6 0 O S 8 0 O S Annual mean -300 -250 -200 -150 -100 -75 -50 -30 -20 -10 -5 Pressure (hPa) Mean meridional streamfunction 10 9 kg/sec 5 10 20 30 50 75 100 150 200 250 300 0 0 0 0 0 0 0 0 0 -2 -2 -2 -2 -2 -2 2 2 2 2 2 2 2 2 2 2 2 15 15 15 15 15 15 15 1000 900 800 700 600 500 400 300 200 100 Pressure (hPa) 80 O N 6 0 O N 4 0 O N 2 0 O N 0 O 20 O S 4 0 O S 6 0 O S 8 0 O S Zonal mean wind Annual mean -100 -80 -60 -50 -40 -35 -30 -25 -20 -15 -10 -5 -2 2 5 10 15 20 25 30 35 40 50 60 80 100 m/sec

Implications for TodayÕs Climate Change A decrease in pole-to-equator temperature gradient can cause: 1. Poleward expansion of the Hadley cells. 2. Slowing of the Hadley circulation. 3. Poleward movement of jet streams. All of these changes are occurring today. 29

Changes in other parameters (rotation rate, radii, ...) do not alter the conclusions. This behaviour of Hadley cells is robust in the model. The changes in Hadley circulation depend strongly on small changes in the temperature gradient. Furthermore: 30

Implications for Paleoclimate Change Major changes in the Hadley cells could have caused dramatic changes in paleoclimate. For very small $ T our model has a single large Hadley cell from equator to pole. We propose that a single large Hadley cell would yield an equable climate similar to that of the Mesozoic Era. 31

But there is more to our model! The mathematical model exhibits hysteresis bifurcation . Hysteresis is a nonlinear phenomenon in which there is co-existence of two different stable states (or modes), with abrupt jumps from either state to the other state. In the model, the two states could represent greenhouse and icehouse climates. 32

HYSTERESIS BIFURCATION (Cusp) R ! T A small change in can cause a jump in state. Lewis a nd La ngfo rd 12 4. 2 So lut io n tec hni ques W e are in terested in computing the s t e a dy s o lution for a w ide rang e of para meter v alues. T o do t his , w e implemen t pseudo-arcleng th con tin uatio n with the Ke ller co rrection co ndition [7], a nd use a New t on metho d to solv e the resulting equat ions. If a s o lution is kno wn for a par ticular set o f para meter v a lues , then this metho d ca n b e used e ! ec t iv ely to follo w s o lutions as a para meter is v aried, i.e . t o Þnd a solut ion curv e (with r e sp ect to the pa ramet e r ). Here, w e kno w tha t fo r " T = 0, the t rivial solutio n sa tisÞe s the equa tions for u , ! and T . Th us, for " T small, the trivia l s o lution is a reasona ble pr e dictio n of the solu tion, and NewtonÕs me t ho d is us ed for the cor rec t ion. I n psue d o-arclength con t iuatio n, the para me- ter is considered as a n unkno w n, and init ial g ue sses of the solutio n are fo und b y follo wing the tang en t, or a sec a n t line a ppro ximat ion, to the solutio n curv e. I ncremen ts are made appr o xim ately a long the solutio n curv e , a nd not b y incremen ting the pa rameter. The K eller conditio n e nsures that the corr e ctio ns to t he initia l guess es o cc ur a ppro ximately p erp en- dicularly t o the tang en t. This me t ho d is pa rticular ly us eful b e ca us e it is a ble to compute solutio ns alo ng t he solutio n curv e ev e n whe n there is a limit p oin t o n the curv e , i.e. when the solutio n cur v e t urns ba c k on itse lf. In pr actice, the ev alua tion of the Ja cobian is e x p ensiv e, and therefore, in order to reduc e the n um b er o f Jacobia n ev aluat ions, w e use a qua s i-Newton metho d instead of Ne wto nÕs metho d. The genera liz ed matrix eigen v alue pro blem tha t results fro m the discretizat ion o f (21 ) is solv ed in Mat lab using the implicitly r e sta rted Arno ldi metho d [17 ], whic h is a memory - e # cie n t iterat iv e met ho d f o r Þnding a sp e ciÞed n um b er of the larg e st eigen v alu e s. A g ene r - alized Ca yley transfor matio n [7] is made s o tha t the Arno ldi itera tion Þnds the eig e n v a lue s of in terest. The pa rameter s of t he tra nsf o rmat ion c a n b e c ho se n to impro v e co n v erg enc e pro p ert ies . In p articula r, t he g e nera lize d Ca y ley t ransforma tion C ( L , A ) = ( L ! " 1 A ) ! 1 ( L ! " 2 A ) (2 2) maps eigen v alues # of the genera liz ed ma trix eigen v alue pro ble m # A v = L v to eig en v a lues $ of t he transfor me d matrix C ( L , A ), suc h t hat t he e ig en v a lues # with R e a l( # ) > ( " 1 + " 2 ) / 2 ar e ma pp e d to the e ig en v alues $ with | $ | > 1, whe r e " 1 and " 2 ar e th e rea l para me t e r s o f t he Ca y le y tr ansforma tion. The ma trix C ( L , A ) do es no t ha v e to b e fo rmed explicitly , b ec a use the Ar noldi it e r atio n only requires ma trix-v e cto r pr o ducts in v olving C ( L , A ) [17]. Th us, the full spa rsene ss prop erties o f L and A can b e explo ited, and comput e r memor y requiremen ts can b e reduced. Lewis a nd La ngfo rd 20 ! " #$%&'()*+,-$./, 0 1 2 034 1 134 2 ! " '5.(+$6'7)*7+.8)9&7/-.$" : ! 0 1 2 034 1 134 2 ! " $&(;&%'$+%&)8&9.'$./, ! : 0 1 2 034 1 134 2 Figur e 11 : An e x am p le of a t w o- cell cir c u lation pa tte r n obse rv ed for h e atin g p arame t e r ! T =, and gap widt h R = 35. (a) The stream f un c ti on ! ; ßo w tend s to f ollo w con tour s , (b ) th e azim uth al (or zona l v e l o c i t y) u , an d (c) the te mp e r atur e deviat ion T fr om th e temp eratu re p res cri b ed on th e lo w e r b ou ndar y . Th e in ner and outer b o und aries h a v e b e en m a pp e d to r = 1 an d r = 2 , r e sp e ctiv ely . L 0 is g iv en b y L 0 V = LV + N ( V , U 0 ) + N ( U 0 , V ) . (2 5) That is, w e ha v e L 0 ! = 0 , (2 6) where ! is the eig e nfunctio n co rresp o nding to the ze r o e ig en v alue. Under c ert ain conditio ns on L 0 , the dep enden t v a riable U can b e writt e n in the fo rm U = w ! + " , (2 7) where w ! " and th us w ! ! span { ! } , and " ! E s . He r e E s is called the sta ble subs pa ce , and is t he space spanned b y all eigenfunc t ions co rresp o nding to eigen v alues w it h nega tiv e real par t. If w e write U as in (2 7) then under certa in t e c h nic a l c o nditio ns , a cen tr e ma nifold and nor mal fo rm reduction c a n b e p erformed o n (2 3) t o o btain t he e q uatio n on the ce n tre ma n- ifold in n orma l for m ú w = ! 1 + ! 2 w + aw 2 + cw 3 + O ( w 4 ) , (2 8) where a and c ar e co e # cie n ts o f the no rmal form and ! 1 and ! 2 ar e unfolding pa rameters tha t a re i n g e nera l functio ns o f the para me t e r s $ T and R . It ca n b e sho wn t hat if c # = 0, then ne g lecting the t e r ms o f O ( w 4 ) do not c ha nge t he qua litat iv e feat ures of the s o lutions. The cen tre manifold and norma l fo rm t he o ries stat e tha t fo r ( $ T , R ) near ( $ T c , R c ) and when t he solut ions are in so me sense small, then t he dyna mics o f (2 3) can b e de duced from (2 8). In pa rticula r, so lutions of ( 28 ) a re in one-to-o ne corr e sp ondence w it h tho se of (2 3).

CUSP BIFURCATION IN THE MODEL !"!#!$ !"!#!% !"!#!& ! #! ! ' ! ()#! ! % *+,-./012-)3+45)106,-74)6-01)8064 ! )9 :0()6-01; " < !"!#!$ !"!#!% !"!#!& # #"' ()#! ! = >?.4+.204+?.)?@)74-0AB)7?124+?. # ! C-11 D ! C-11 ! )9 EE) # )EE D !"!# !"!#$% !"!#% !"!#&% !"!$ '"& '"( )*#! % +,-./0123.*4,56*217-.85*7.12*9175 *: ;1)*7.12< = !"!# !"!#$% !"!#% !"!#&% !"!$ > ? % ( )*#! > @A/5,/315,A/*AB*85.1CD*8A235,A/ # E.22 $ E.22 *: FF* *FF $ R =0 . 5 R =3 . 5 34

Hysteresis Bifurcation Theorem A hysteresis bifurcation point exists if: 1. There is an equilibrium point. 2. The linearization at the equilibrium point has a simple zero eigenvalue. 3. The coefÞcient of the second-order term of the normal form on the center manifold vanishes. 4. Certain other dominant terms in the normal form are nonzero. 35

Result of Greg Lewis A hysteresis bifurcation point EXISTS in this model. It yields discontinuities in ÒclimateÓ, as the pole-to-equator temperature difference varies. Proof: Lewis showed that the conditions of the Hysteresis Bifurcation Theorem are satisÞed by the model equations. 36

The model suggests: Polar cooling may cause a global climate bifurcation , in which climate jumps abruptly from one climate mode to another. 37

Where are we today? In the model, we are on the upper branch, moving to the left (the poles are warming). Next question: How long before we fall over the edge? R ! T 38

R ! T Sudden transition from icehouse to greenhouse? !

The Pliocene Paradox (3-5 million years ago) Greenland was ice-free, with palm trees on its southern coast [D. Greenwood]. There were temperate rain forests on CanadaÕs far northern arctic coastline [J. Basinger]. Yet, all the Òdriving forcesÓ were essentially the same as todayÕs: CO 2 levels, solar radiation, EarthÕs axis tilt and orbit, continent positions, ocean currents, etc. 40

Pliocene Paradox 2.0 The south pole switched from greenhouse climate to icehouse climate about 28 Mya (Oligocene). The north pole remained in a greenhouse climate until about 3 Mya (early Pliocene). Thus the north-south symmetry of the EarthÕs climate was broken for 25M years. Symmetry was restored when the north pole switched to icehouse climate about 3 Mya.

Work in Progress Drop the assumption of north-south symmetry in the mathematical model. Investigate the existence and stability of asymmetric modes. Study the relationship between asymmetric Hadley cells and asymmetric climates.

FUTURE WORK Add compressibility; assume an ideal gas law; break the rotational symmetry. Include in the model the Òatmospheric conveyor beltÓ positive feedback. (Hadley cells carry heat from the equator to higher latitudes.) Add greenhouse gases and albedo to the model and compute Hopf bifurcations (ice age cycles). 43

Thank you! References: !!Gregory M. Lewis and William F. Langford (2008). Hysteresis in a rotating differentially heated spherical shell of Boussinesq ßuid. SIAM J. Applied Dynamical Systems, V. 7, pp. 1421-1444. !!William F. Langford and Gregory M. Lewis (2009). Poleward expansion of Hadley cells. Canadian Applied Mathematics Quarterly, V. 17, No.1, pp. 105-119. 44