CARMEN
HERRERO
CARMEN HERRERO
RELAXATION MODELS
APPLIED TO
PALEOCLIMATE DYNAMICS:
SOUTHERN OCEAN
MECHANISMS CONTROLLING
GLACIAL- INTERGLACIAL CYCLES
RELAXATION MODELS APPLIED TO PALEOCLIMATE DYNAMICS: SOUTHERN OCEAN MECHANISMS CONTROLLING GLACIAL- INTERGLACIAL CYCLES
Ba celona
Oc obe 2015
UNIVERSIDAD DE LAS PALMAS DE GRAN
CANARIA
Facul ad de Ciencias del Ma
ANEXO I
Dª MARÍA ISABEL PADILLA LEÓN, SECRETARIA DE LA FACULTAD
DE CIENCIAS DEL MAR, ÓRGANO RESPONSABLE DEL
PROGRAMA DE DOCTORADO EN OCEANOGRAFÍA, DE LA
UNIVERSIDAD DE LAS PALMAS DE GRAN CANARIA.
CERTIFICA
Que el Consejo de Doc o es del P og ama de Doc o ado en
Oceanog a ía, en su sesión de echa 28 de oc ub e de 2015, omó el
acue do de da el consen imien o pa a la ami ación, de la esis doc o al
i ulada:
“Relaxa ion Models Applied To Paleoclima e Dynamics: Sou he n Ocean
Mechanisms Con olling Glacial-In e glacial Cycles ”
p esen ada po la doc o anda: Dª Ca men He e o Na a o
di igida po el Doc o D. An onio Ga cía-Oli a es Rod íguez
Y pa a que así cons e, a e ec os de lo p e is o en el A º 6 del
Reglamen o pa a la elabo ación, ibunal de ensa y e aluación de esis
doc o ales de la Uni e sidad de Las Palmas de G an Cana ia, i mo el
p esen e en Las Palmas de G an Cana ia, a ein iocho de oc ub e de
dos mil quince.
PÁGINA 1 / 1 ID. DOCUMENTO XwwN32p7awxT6 z34GMFFg$$
FIRMADO POR FECHA FIRMA ID. FIRMA
43646105V ISABEL PADILLA LEÓN 29/10/2015 13:05:48 NTI0NTY=
Documen o i mado digi almen e. Pa a e i ica la alidez de la i ma copie el ID del documen o y acceda a / Digi ally signed documen .
To e i y he alidi y o he signa u e copy he documen ID and access o
h ps://sede.ulpgc.es:8443/Ve i icado Fi mas/ulpgc/Ve i icacionAc ion.ac ion
Tesis doc o al p esen ada po
Ca men He e o Na a o
di igida po el D .
An onio Ga cía-Oli a es Rod íguez
pa a ob ene el g ado de Doc o a po la
Uni e sidad de Las Palmas de G an Cana ia,
Depa amen o de Física,
DOCTORADO EN OCEANOGRAFÍA
Bienio 2012-2014
En Ba celona, a oc ub e de 2015
RELAXATION MODELS
APPLIED TO
PALEOCLIMATE DYNAMICS:
SOUTHERN OCEAN
MECHANISMS CONTROLLING
GLACIAL-INTERGLACIAL CYCLES
MODELOS DE RELAJACIÓN APLICADOS A LA
DINÁMICA PALEOCLIMÁTICA: MECANISMOS
DEL OCÉANO AUSTRAL CONTROLANDO LOS
CICLOS GLACIALES-INTERGLACIALES
El di ec o La doc o anda

A la me a amília
Con en s
12 ............. P e ace
14 ............. Acknowledgemen s
22 ............. Glossa y
23 ........ INTRODUCTION
30 ....... Chap e 1. RELAXATION MODELS APPLIED TO LATE
PLEISTOCENE CLIMATIC OSCILLATIONS
33 .................1.1. INTRODUCTION
34 .................1.2. PP04-DERIVED MODELS
37 ................ 1.2.1. Biological expo p oduc ion model
43 ................ 1.2.2. Two esponse imes o CO2
44 ................ 1.2.3. Two esponse imes o ice olume
46 ................ 1.2.4. Oceanic pulse exponen ially
dependen on s a i ica ion
46 ................ 1.2.5. 3τ model
47 ................ 1.2.6. Local s a i ica ion model
48 .................1.3. MODELS PERFORMANCE
50.................1.4. DISCUSSION
56 ........ Chap e 2. ON THE PHYSICAL MECHANISMS BEHIND
GLACIAL-INTERGLACIAL DYNAMICS
59 ................. 2.1. INTRODUCTION
60................. 2.2. RELAXATION MODELS
63 ................. 2.3. NON-LINEAR ANALYSIS
63 ................ 2.3.1. Fou ie ans o m
16
Al se emb e del 2009, ja amb la ca e a p àc icamen acabada, aig p end e la
decisió de llui a pe un u u en el món de la ece ca. Va se alesho es quan aig
p esen a -me al despa x d’un dels meus p o esso s de la ca e a pe demana si po-
dia e p àc iques amb ell. Aquell p o esso no e a ni més ni menys que l’An onio
Ga cía-Oli a es, di ec o d’aques a esi, o i que ni ell ni jo no n’é em conscien s
del lla g camí que ens espe a a al da an .
No sab ia di si les opo uni a s apa eixen soles o són ui de busca -les, pe ò al
cap de mol pocs dies de comença a e les p àc iques, em an uca dien que hi
ha ia una acan pe la beca que ha ia demana pe ana uns mesos a la Uni e -
si a de les Illes Balea s, i que inicialmen m’ha ien denega . Vaig es a sis mesos
a Espo les, ap enen de g ans in es igado s com en Damià Gomis, en Biel Jo dà,
en F ancisco Cala a i sob e o , na Ma a Ma cos, amb qui aig eni el p i ilegi
de eballa ac i amen . Vaig conèixe gen me a ellosa amb els que enca a, o i
que espo àdicamen , compa im ucades i anècdo es: na Chus Alonso, en JuanJo
Ensenya , en GianPa Candini, en Miquel Gomila... Va se una expe iència genial i
un bon ap enen a ge.
Pe ò enia cla a on olia es a i aquell lloc e a, cla amen , l’ICM. Va se lla o s,
al se emb e del 2010, quan aig decidi emba ca -me en aques a g an a en u a.
Nada de es o hubie a sido posible sin i. G acias An onio po con ia siemp e en mi
y o ece me la opo unidad de ap ende , de c ece , de mejo a y de conoce nue os
aspec os de la ida; g acias po enseña me y po ansmi i me esa g an pasión po la
ciencia.
Sense cap mena de dub e l’al e g an supo d’aques a esi ha es a en Josep Lluís
Peleg í. G àcies pe dona -me l’opo uni a de eballa amb osal es, pe aju-
da -me dia e e dia a què madu és an com a in es igado a com a pe sona. G àcies
ambé pe con ia semp e en el meu c i e i i pe o e i -me g ans opo uni a s.
T eballa a l’ICM ha es a un g andíssim plae . No només pe què es ac a d’un
dels cen es de ece ca més po en s de o el Medi e ani, si no pe què la gen que el
o ma es me a ellosa. He compa i despa x amb g ans pe sones com l’E an Mason,

17
Ag aïmen s / Ag adecimien os / Acknowledgemen s
la Pa icia De La Fuen e, en Miquel Rossell, en Suso Peña, en Se gio Ramí ez... o s
ells g ans in es igado s i uns companys excel·len s. També hi ha en Ma c Gasse
amb el que, o i no ha e coincidi explíci amen a un despa x, hi ha una connexió
especial; g àcies Ma c pe o es les con e ses apassionan s que hem ingu i pe es a
semp e disposa a ajuda o a e una coca-cola.
L’o icialmen ex ingi DOF (Depa amen que O gani za més Fes es, Cla e e al.
2011) ha dona pas al DOFT (Depa amen O gani zado de Fes es i T aslla s)
pe ò pe so la gen que el o ma segueix sen an às ica. Començan pel «nos e»
passadís, amb els dos Jo dis (Fon i Sala ), i l’Emilio Ga cía, seguin ans e -
salmen amb en Kin xo Sal ado , en Pe e Fe nández, la Ma ibel Llo e i en Jose
Pozo, con inuan amb l’Ál a o Víudez, el mejo compañe o de despacho que alguien
pueda imagina (mil g acias po aguan a me es os úl imos meses), i més enda an
amb en Mikhail Emeliano i en Jo di Ise n. Una plan a més amun obem en Jose
An onio Jiménez, en Jo di Solé, en Jaume Pie a, la Ca ine Simon, en Miguel Ángel
Rod íguez i o el g up SMOS amb l’An onio Tu iel, en Quim Ballab e a, la Ca o
Gaba ó, en Ma cos Po abella, la Ma ia Piles, la Ve o González, la Ma ía Bel-
mon e, l’Es ella Olmedo, la Ma a Ramí ez, la Nina Hou eau, la Ma a Umbe ...
i pe descomp a , en Fe nando Pé ez i en Jus ino Ma ínez, semp e amb la solució
als nos es p oblemes. Alguns dels que ja han « ola del niu» són en Ped o Llanillo
o la Rocio Rod íguez, g andíssimes pe sones.
Hi ha qua e pe sones, pe ò, que es me eixen una menció especial, pe o d e d’apa i-
ció: Paola, Ma iona, Ma ia Rosa i Do le a. Paola, ha sido un au én ico place es a
a us espaldas; ienes un g an co azón y una ene gía aclapa ado a, es oy segu a que
llega as allí donde e p opongas. G acias, po supues o, po odos los co ees (aun-
que yo nunca he omado ca é) y po siemp e encon a un hueco pa a escucha me
y da me un buen consejo. La Ma iona, ai, Ma iona... aques s dies em balla pel cap
l’època del B36 i el meu amós «Dis u a, que això només es a un cop a la ida»;
deixam di - e que se ’ha oba a al a pe «aques os la es», pe ò e desi jo que
allà on e po i el en , ja sigui a una banda de l’A làn ic o a una al a, siguis mol
i mol eliç. Ma ia Rosa, ¿què puc di - e que no sàpigues?, ha es a un immens plae
pode compa i aques s úl ims anys al eu cos a , pa lan de ia ges i de no es ec-
18
nologies; ha es a un g an hono conèixe - e una mique a més a ons. Pe ce , e s
la me a millo alumna, sense cap mena de dub e! G àcies pe escol a -me i aconse-
lla -me amb la e a immensa sa iesa. Do le, ¿quién nos iba a deci que acaba íamos
como dos sin papeles en Sal ado de Bahía? Ha sido genial el habe e conocido, e es
an ás ica y es oy segu a que llega ás muy lejos. G acias po siemp e ene una pa-
lab a de ánimo a mano y po a on a la ida con una son isa. ¡Vi a la biod amina!
No, espe a, pe mí eme ememo a ... ¡Ojo, que olcamos!.
Pe so , l’ICM no es edueix a un únic depa amen , hi ha ida social més enllà
dels ísics i majo i à iamen la obem a Biologia. Als ICM- ood, pe o s els dina s
(amb o sense upe ), co ees, pels cines, pe les 4h d’espe a a la C êpe ie, pe les ba -
bacoes i ac i i a s a iades... g àcies! Mi eia, Elisa, Ma ía D. L. F., Xa i, Rami o,
Rachelle, Isabel, Denisse, Ca ol, Yaiza, Es ela, Sdena, Rosana, No ma, AnaMa i,
F an C., Laia, Da ne, Ca y po supues o, F an A. Habéis hecho que i a abaja
sea absolu amen e enomenal.
Es imado S . Don. Pe mí ame que me di ija a us ed en pa icula pa a ansmi i le
mi más sen ido ag adecimien o. En es e iempo hemos compa ido g andes pa idas
de DS, is o odos los capí ulos de Fu u ama (al menos yo, a us ed le ha al ado
alguno), comen ado g andes películas de la saga Ju assic Pa k (las buenas y las no
an buenas), ha cambiado us ed unas 13 eces de elé ono ( odas ellas jus i icadas),
pe o lo más impo an e es que siemp e ha es ado ahí pa a anima me cuando más
al a ha hecho. Muchos ánimos en es e úl imo sp in , ú puedes con es o y más!!! Sin
más, me despido. Suya, siemp e, una Nobel.
G àcies ambé a o el pe sonal d’adminis ació i se eis, amb la Nú ia Angos o,
l’E a López, en Césa Ga cía, en Jo di Es aña, en José Fo uño, en Jose Ma ía
Angui a, en Xa i Leal, la Rosa Cabanillas, la Ma i í Ma ínez de Albéniz i especial-
men , la Conchi a Bo uel, qui és capaç de esold e qualse ol p oblema. Mil g àcies
a o s i a o es pe ajuda -nos amb la bu oc àcia i pe aconsegui que l’ICM uncioni
«a las mil ma a illas». A l’ Eli B oglio, g àcies pe ajuda -nos amb la di ulgació i
pe o gani za les ac i i a s més di e ides i in e essan s que algú pugui imagina . A
Ani a Bonilla de Pelopan ón, g acias po ilus a la ciencia en colo y ae un poco
de diseño al abu ido mundo cien í ico, y po supues o, po se capaz de cap u a en
19
Ag aïmen s / Ag adecimien os / Acknowledgemen s
papel mis ideas. És més que possible que m’es igui oblidan més d’un i més d’una,
així que mil g àcies a o s i a o es pe aques s g andíssims 5 anys.
E iden men , no o es edueix a l’ICM. A odo el g upo PalMA, de la Uni e sidad
Complu ense de Mad id. G acias Ma isa po con es a al co eo de una es udian e
sin mucha idea diciendo que le gus aba es o de los modelos. G acias po acoge me,
po o ece me la opo unidad de abaja con oso os y po enseña me más de ce ca
el uni e so CLIMBER. G acias Jo ge y Alex po odas las discusiones apasionan es
y po ayuda me a en ende un poqui o más es o de los modelos, incluso los de hielo.
A Rubén, po no se solo un compañe o sino un g an amigo con el que pode comen-
a la jugada, ¡incluso la del moun ain bike! A E o , po odas las con e saciones
apasionan es ya sean de ísica o de o og a ía; a Lau a, po su en usiasmo y ene gía
posi i a; a Fidel, a Nú ia, a Edmundo, a Elena, a Ángela, a Jo ge N., a Jo ge C., y
segu o que me es oy dejando a algun@ po el camino. A odo PalMA, g acias, de
e dad, po hace me un huequi o semanal espo ádicamen e en ues as idas y po
odas esas palme as de chocola e que hemos compa ido (las mejo es del mundo
mundial, doy e) y po las que nos quedan po hace .
Thank you so much o he UCSB o an amazing opo uni y. My mos since e g a i-
ude o Lo aine Lisiecki, o welcoming me and o e ing me he chance o wo k wi h
he . A special hanks o Vi ian S opple and he adminis a ion g oup, o all he
help wi h he documen a ion and hei wa m welcome. Thanks o Rachel Sp a o
all he in e es ing alks, o showing me a ound and o all he lunches oge he ; o
my o ice-ma es, Debo ah Khide and Ced ic Twa dzik, o hei help and kindness;
o Will G ay, o all he able ennis games (I wan my e enge). Also hanks o
F ank Kinnaman, o being a g ea iend in Gole a and o sha ing my passion o
pho og aphy. A huge hanks o he Gans Family, o all he amazing imes and o
making us eel like home. On he o he hand, I’m deeply g a e ul o all he In ense
Family: S ebe elli, De me s, He ick, Nie meije , Bible, Palme ... Thanks so much
o being abso ackinglu ely ad #in ense o li e.
Can iem adicalmen d’ò bi a. G àcies als amics que he ana en en cadascuna de
les e apes de la me a ida. A la gen del Rose , pe compa i amb mi la in ància
i la jo en u ; especialmen a la Ma a, amb qui he compa i g ans momen s (com
20
una pizza amb gus de cas anya). A les nenes del ba xille a : l’Es e , la Ma , la C is,
l’Anna i la Elena; g àcies pe acolli -me quan aig a iba a una escola desconeguda
i e -me un lloc en les os es ides. A o a la gen que em a acompanya al lla g
de la ida uni e si à ia, especialmen als F eaksics: F an, Laly, To u, Tinoco, Gui-
llem, Palau, Pablo, Sebas, Se gi, Albe o, Jo di C., Jo di R., Ma io, Alba, Lau a,
Albe ... i segu que em deixo a algú; a o s, g àcies. No m’oblido d’aquells que an
compa i amb mi les aules pe p ime cop: en Daniel (jun amb la Julia i l’È ika),
qui ha passa de se una uni a indi idual a una molècula de 3 à oms, amb qui se-
guim compa in la ida enca a que ens sepa in 19300 km; sembla men ida que amb
els anys que han passa i la de coses que han can ia hi hagi ce s aspec es de la
ida que segueixen igual, com que hi ha con e ses que malau adamen només ú i jo
obem ascinan s. And o cou se, la Síl ia, qui a massa anys que és lluny pe ò amb
qui semp e ens queda à una màquina de ca è i un «sense go ». Als òp ics de la UAB
i de i a s, semp e disposa s a e un sopa , un ídeo o un club de ans: en Menxi
(qui é ulle es de pu illa) i la Lina, amics d’aquells que semp e e de gus eu e; en
Mompi, un dels millo s p o esso s que he ingu mai; en Da id, semp e a en u e i
som ien ; la Sònia, amb la se a ene gia posi i a i de qui sóc p esiden a en supe po-
sició del seu club de ans; i pe descomp a , dues pe sones clau: en Bensi i la Ca .
Albe , e s un c ack; deixa’m di - e que has es a el meu model a segui a la esi i
hem iscu mil i una a en u es, pe ò c ec que em quedo amb INVERNAAAAALIA-
AAA i la supe posició p esiden ís ica (amb el momen es el·la a l’Abacus, explican
que ha íem comp a 100 s ands de Quàn ic Lo e i no ens els ha ien en ia ). Ca ,
Ca Olid, Ca pe a, Ca ó i de i a s simila s... quan es coses hem iscu eh! Ca , si
algú ens hagués di en aquella p àc ica de col·lisions ela i is es o el que iu íem
jun es, no ens l’haguéssim c egu . G àcies pe o s els bons momen s, o es les idees
esboja ades, les pelis de Spide man (i Kill Bill) que hem compa i ... i pe o s els
bons momen s que enca a es an pe eni . A o s els amics bike s, especialmen a la
Vane i en Miki, qui semp e enen una es ona pe ajuda en o el que aci al a (pe
quan un sushi?); g àcies pe es a semp e al nos e cos a , sou mol g ans!
E iden men no hi ha p ou espai pe anomena a o hom ( o i que m’he es o ça
en mi a de e -ho), així que g àcies a o es aquelles pe sones que m’heu acompanya
al lla g de la me a pe i a his ò ia.
21
Ag aïmen s / Ag adecimien os / Acknowledgemen s
Un luga especial lo ese o a mis pad es, Lucía y Emilio, po que sin ellos absolu-
amen e nada de es o hubie a sido posible. G acias po c ee siemp e en mi, po
enseña me, po educa me, po da me ues o apoyo incondicional y ambién, po
malc ia me (y cuán o he dis u ado con ello). G acias po hace que me cues iona a
las g andes decisiones pa a es a segu a de que e a lo que ealmen e que ía (aunque
siemp e ue a nada más le an a me) y g acias po da me las he amien as necesa-
ias pa a se capaz de asumi las consecuencias de i adas. Nada de lo que pueda
esc ibi se á su icien e pa a ag adece os odo lo que habéis hecho po mi, pe o pe -
mi idme insis i : g acias po es a ahí, siemp e.
G acias, ambién, a la amilia que, pese a es a lejos, siemp e es án al o o lado
del elé ono dispues os a con a me las úl imas no edades o dispues os a compa i
los g andes momen os. A Ca mina, a los Pepes, a los íos, a los p imos y a los chicos
(Vic o ia, Pablo y Ana), g acias. Y, po supues o, g acias po odas esas eli as que
habéis encendido y que me han iluminado el camino.
A la amilia Guà dia - Pascual, pe acolli -me a casa se a i e -me pa del seu dia
a dia. A l’An onia i en Josep, a en Dídac i la Mi eia, a la Mon se i en Da id, a les
iaies Rosi a i Ma ía, als ie s i als cosins: g àcies.
Las bu no leas , a en Be na . To a comença a les aules de ísica, a a ja a uns
quan s anys, amb un «nosal es anem a seu e allà, ens?» i des d’alesho es la Te a
gi a a i me de pedal. G àcies pe compa i la ida amb mi i e que cada dia sigui
més especial que l’an e io . G àcies pe escol a -me, pe ajuda -me, pe anima -me,
pe e -me iu e cada ma í i pe e -me llui a pels meus somnis. G àcies pe ha e
c egu semp e en mi i pe e -me se cada dia millo pe sona. I don’ know how o
pu his, bu you’ e kind o a big deal.
Ba celona, 21 d’oc ub e de 2015

AABW An a c ic Bo om Wa e
ACC An a c ic Ci cumpola
Cu en
AMOC A lan ic Me idional
O e u ning Ci cula ion
AP A e P esen
BP Be o e P esen
CDW Ci cumpola Deep
Wa e
CO2 Ca bon dioxide
CRP C oss Recu ence Plo
DIC Dissol ed Ino ganic
Ca bon
EMIC Ea h Sys em Model o
In e me dia e Complexi y
HS1 Hein ich e en
Glossa y
ITCZ In e opical Con e gence Zone
LCDW Lowe Ci cumpola
Deep Wa e
LGM Las Glacial Maximum
MOC Me idional O e u ning
Ci cula ion
NADW No h A lan ic Deep Wa e
NH No he n Hemisphe e
NMOC No he n Me idional
O e u ning Ci cula ion
ppm Pa s pe million
SH Sou he n Hemisphe e
SMOC Sou he n Me idional O e u ning
Ci cula ion
SO Sou he n Ocean
YDS Younge D yas
23
Le me s a wi h a undamen al ques ion: why should we s udy clima e? Clima e
a ec s ou daily li e in many ways: on he ood we ea , on he houses we li e in, on
ou wo k, on how we a el... E en a ec s ou cul u e, ou spa e ime o ou heal h.
Bu all o hose a e only locally impo an . Wha we need o see he e is he big
pic u e: clima e affec s he way all li ing species ha e adap ed o he biosphe e, i is
key o ou su i al, bu we a e d as ically in luencing i and we canno p edic he
consequences o ou ac ions. I is, hen, undamen al o b oaden ou knowledge on
he clima e sys em o Ea h, he only home we ha e e e known, o s udy he pas
clima ic a iabili y on a ious ime scales o ob ain clues ha will help socie y ace
u u e clima e change. As Ca l Sagan o eseen on Cosmos, one o his mos known
books (Sagan [1980]):
Ou in elligence and ou echnology ha e gi en us he powe o
affec he clima e. How will we use his powe ? A e we willing o
ole a e igno ance and complacency in ma e s ha affec he en-
i e human amily? Do we alue sho - e m ad an ages abo e he
wel a e o he Ea h? O will we hink on longe ime scales, wi h
conce n o ou child en and ou g andchild en, o unde s and and
p o ec he complex li e-suppo sys ems o ou plane ? The Ea h
is a iny and agile wo ld. I needs o be che ished.
In oduc ion
Wha is a scien is a e all? I is a cu ious man looking h ough a keyhole, he keyhole
o na u e, ying o know wha ’s going on.
— Jacques-Y es Cous eau
24
Paleoclima ology is he science ha s udies hose changes in clima e on a eally
long ime scale; as long as he en i e his o y o Ea h, 4.54 billion yea s. I uses
da a p e iously p ese ed on a wide ep esen a ion o en i onmen s (i.e. ocks,
sedimen s, ice shee s, co als...) o econs uc he pas s a e o he Ea h clima e
and i s co esponding a iabili y. To ind de ailed da a as old as Ea h is almos
an impossible ask, bu we can ce ainly use a a ie y o p oxy me hods o ob ain
eliable ime-se ies wi h a ime scale o millions o yea s. In his wo k, we ha e used
p oxy eco ds wi h a maximum ime domain o 5 million yea s, and ou esul s a e
educed o wha we may conside he mos ecen ime, only 800,000 yea s, o in
paleoclima e uni s, 800 ky .
To unde s and clima e in de ail, we should look in o all ele an p ocesses o he
sys em, al hough his migh be sligh ly e i ying, as he numbe o hese p ocesses
which mus be unde s ood is as onishing. To make i mo e in e es ing, he amoun
o ue knowledge ha we ha e is limi ed and pa ial. Should we c y on despai ?
Ce ainly no . Luckily, paleoclima e, as all sciences, is based on he scien i ic me h-
od, which help us on de eloping an unde s anding o eali y using e ms as mech-
anism, model and heo y. We may desc ibe a heo y as a de ailed ma hema ical
explana ion o phenomena ha has sufficien ele ance o make p edic ions om
undamen al p inciples, o in o he wo ds, as s a ed by T uesdell and Toupin [1960],
a heo y is no mo e han a ma hema ical model o na u e. T u h is, he e is no clea
pa h o ob ain a easonable heo y o clima e in he nea u u e. Ra he , we all hope
o de elop models o clima e, in which gene al conse a ion p inciples and assump-
ions on local mechanisms o eedback allow us o de i e ma hema ical algo i hms
and p edic ions ha need o be compa ed wi h expe imen al obse a ions. As long
as he p edic ions and he expe imen s ma ch, we may conside he model alid o
desc ibe a eali y, a complex objec called clima e sys em.
His o ically, many in oads ha e been made o model he s uc u e o long- e m
clima e a iabili y. In he 19 h cen u y, Louis Agassiz (Agassiz [1838]) p oposed o
he i s ime ha clima e on Ea h could ha e been much colde in olde pe iods o
ime, a leas in he No he n Hemisphe e (NH), bu unde a ca as ophis pe spec-
i e. La e on, Joseph Adhéma (Adhéma [1842]) o mula ed he i s as onomical
RELAXATION MODELS APPLIED TO PALEOCLIMATE DYNAMICS:
SOUTHERN OCEAN MECHANISMS CONTROLLING GLACIAL-INTERGLACIAL CYCLES
25
In oduc ion
heo y o glacial ages, sugges ing ha as onomical pa ame e s should modi y
clima e, and explained he posi ion o ice shee s acco ding o he p ecession o equi-
noxes, and he subsequen change o loca ion o he pe ihelion (nea es poin o
he Sun on Ea h o bi ) in ela ion o seasons. Ne e heless, Adhéma o mula ion
was la gely c i icized, mos ly wi h un ounded a gumen s, al hough i had indeed
p oblems. James C oll (C oll [1867]) wo ked on a mo e elabo a ed e sion o his
heo y, conside ing he ole o eccen ici y as a modula ion o he p ecessional o c-
ing, and ollowing Adhéma ideas, insis ed on he ole o snow accumula ion du ing
win e s as lead o glacia ions. I was on he 20 h cen u y when Milu in Milanko i ch
(Milanko i ch [1941]) o mula ed a heo y ha is s ill alid nowadays. C oll’s main
p oblem was o conside win e as he c i ical season o he ice shee e olu ion,
while Milanko i ch demons a ed ha summe mel ing is a mo e impo an o
he ice mass balance, conside ing summe insola ion a key pa ame e on his heo y.
Fu he mo e, Milanko i ch conside ed a hi d as onomical pa ame e , he obliqui y
o he Ea h axis (axial il ), se ing he basis o he p esen as onomical heo y o
clima e.
On he o he hand, o he hypo hesis we e de eloped ollowing a comple ely di e en
poin o iew, he geochemical heo y. S an e A henius (A henius [1896]), inspi ed
by he p e ious wo k o Joseph Fou ie o John Tyndall, s a ed he ole o CO2 on
clima e. He co ec ly compu ed how a a ia ion o a mosphe ic CO2 le els affec s he
su ace empe a u e h ough he g eenhouse effec , conside ing he a mosphe e as
a ca bon ese oi able o con ol ice ages. Unde his pe spec i e, CO2 and global
clima e a e d i ing he ice shee changes, while Milanko i ch poin o iew s a es
ha ice shee s con olled by summe no he n insola ion a e he ones d i ing global
clima ic changes. Bo h heo ies ha e limi a ions and p oblems, and as he eade
may suspec , bo h a e essen ially alid and no exclusi e.
Hays e al. [1976] demons a ed ha a ia ions o glacial-in e glacials s a es du ing
he Qua e na y may be ul ima ely caused by changes in pa ame e s o he Ea h’s
o bi (Milanko i ch heo y), as he as onomical pe iodici ies a e ound in paleo-
clima ic eco ds. They also showed ha a he obliqui y (41 ky ) and p ecessional
bands (23 and 19 ky ) exis a di ec clima ic esponse o high la i ude summe
33 .................1.1. INTRODUCTION
34 .................1.2. PP04-DERIVED MODELS
37 ................ 1.2.1. Biological expo p oduc ion model
43 ................ 1.2.2. Two esponse imes o CO2
44 ................ 1.2.3. Two esponse imes o ice olume
46 ................ 1.2.4. Oceanic pulse exponen ially
dependen on s a i ica ion
46 ................ 1.2.5. 3τ model
47 ................ 1.2.6. Local s a i ica ion model
48 .................1.3. MODELS PERFORMANCE
50.................1.4. DISCUSSION
Con en s

33
1.1 INTRODUCTION
Some simple elaxa ion models ha e been p oposed o explain he glacial-in e gla-
cial cycles. Pa icula y, Pailla d and Pa enin [2004] p oposed a model (he ea e
PP04) ha inco po a es e y simple pa ame e iza ions a emp ing o ep esen
dense wa e o ma ion in he SO. Though he model is simple, he esul s a e en-
cou aging because hey co ec ly ep oduce he pace and e mina ion imes o all
he glacial cycles obse ed.
He e, he PP04 model has been gene alized and calib a ed o he δ18O and CO2
ime se ies a ailable o he las 800 ky Be o e P esen (BP) (Pe i e al. [1999];
Monnin e al. [2001]; Pepin e al. [2001]; Lisiecki and Raymo [2005]; Siegen hale
e al. [2005]; Lu hi e al. [2008]). The objec i es o his chap e a e:
Chap e 1
RELAXATION MODELS APPLIED TO LATE
PLEISTOCENE CLIMATIC OSCILLATIONS
We emba ked on ou jou ney o he s a s wi h a ques ion i s amed
in he childhood o ou species and in each gene a ion asked anew wi h
undiminished wonde : Wha a e he s a s? Explo a ion is in ou na u e.
We began as wande e s, and we a e wande e s s ill. We ha e linge ed
long enough on he sho es o he cosmic ocean. We a e eady a las o
se sail o he s a s.
‒ Ca l Sagan, COSMOS
RELAXATION MODELS APPLIED TO LATE PLEISTOCENE CLIMATIC OSCILLATIONS
34
The nex sec ion desc ibes he s uc u e o he ini ial PP04 model and i s bes cali-
b a ions o he δ18O and CO2 da a o he las 800 ky BP. A biological ca bon expo
a e is nex in oduced ins ead o he o iginal mechanism, and he modi ied model is
calib a ed and analyzed. An al e na i e model wi h wo diffe en esponse imes o
he CO2 emission and abso p ion is also conside ed. The model is hen modi ied by
wo diffe en esponse imes o accumula ion and abla ion o ice, and by adding an
exponen ial oceanic pulse. Nex , we compa e he pa e ns ound when using diffe en
model calib a ions, and analyze diffe en mechanisms con olling he oceanic pulse
ha igge s he deglacia ions. Finally, he esul s and conclusions a e summa ized.
1.2. PP04 DERIVED MODELS
The se o equa ions o he o iginal PP04 model a e he ollowing:
(i) o ob ain he se o pa ame e s ha bes i he la e Pleis o-
cene ime se ies in PP04’s o iginal model;
(ii) o in es iga e se e al gene aliza ions o Pailla d’s model by
in oducing some simple sub-models o he expo p oduc-
ion, he s eng h o deep s a i ica ion and o he mechanisms; and
(iii) o analyze how he diffe en sub-models affec he da a i .
1.1
dV = (V ‒ V ) ,
τV
d
1.3
dC = (C ‒ C),
τC
d
1.2
dA = (V ‒ A) ,
τA
d
Chap e 1
35
whe e V and A a e dimensionless indexes o non-An a c ic ice olume and ex en
o An a c ic ice shee espec i ely,C is a dimensionless index o a mosphe ic CO2,
I65 is he daily insola ion a 65° N on 21 June, P(F) is he deep ocean con ibu ion
o he e e ence CO2, which in he o iginal model is P(F) = H(F), whe e H is
he Hea iside unc ion (H = 1 i F < 0; H = 0 o he wise), and F is he sal y
bo om wa e o ma ion efficiency pa ame e . F inc eases wi h ice olume V and
dec eases when con inen al shel a eas a e educed ( h ough A) and when I60
(daily insola ion a 60 S on 21 Feb ua y) inc eases.
The main a iables V, A, and C end o decay exponen ially o a e e ence s a e
V , V, and C in cha ac e is ic imes τV , τA and τC , espec i ely. Re e ence ol-
ume (V ) dec eases when C inc eases, and when I65 is high. The CO2 o e e ence
(C ) inc eases when I65 is high h ough pa ame e α and when he oceanic pulse
o CO2 is on ( h ough pa ame e γ), and dec eases when he ice olume inc eases.
The β pa ame e ep esen s he eedback be ween ice olumeV and CO2.
Daily insola ions o he las 800 ky BP a e ob ained om he Be ge [1978a]
and Be ge and Lou e [1991] so wa e (Appendix A). We ha e used nega i e and
posi i e alues when espec i ely e e ing o imes be o e p esen (BP) and a e
p esen (AP), whe e yea ze o is aken as 1950 AD. Insola ions ha e been no -
malized wi h hei s anda d de ia ions o ob ain I65 and I60. No lowe bound is
imposed on he V, C and A a iables, which may ake posi i e o nega i e alues
because hey ep esen ela i e and no absolu e a ia ions. No e ha all esul s
o he model ha e been no malized by hei s anda d de ia ion be o e plo ing.
V
=
‒ xC ‒ yI65 + z,1.4
C
=
αI65 ‒ βV + γP ( ‒F ) + δ,1.5
F
=
aV ‒ bA ‒ cI60 + d, 1.6
RELAXATION MODELS APPLIED TO LATE PLEISTOCENE CLIMATIC OSCILLATIONS
36
In Pailla d and Pa enin [2004] no p ecise uning o he pa ame e s was pe o med
bu a able wi h cen al alue and ange o he pa ame e s was p esen ed. In o ig-
inal PP04, γ pa ame e had a alue o 0.5 due o a mis ake. No iced by Michael
C uci ix and Didie Pailla d in a la e e ision, he co ec alue is 0.7 (Table 1.1).
Th ee di e en in eg a ion me hods we e used o he diffe en ial equa ions (Ma h-
ema ica in e nal algo i hms, Ma lab in e nal algo i hms1 and P edic o -Co ec o
explici me hod wi h ime s ep o 50 yea s) and he esul s we e coinciden . Gi en
he s ong nonlinea i y in he a iable C dynamics, a leas 16000 s eps a e needed
in he la e algo i hm o g an ha esul s a e independen o he numbe o s eps.
A sys ema ic sea ch o he se o pa ame e s ha lead o maximum co ela ion was
implemen ed a ound his cen al case. Co ela ions o he V and C ime se ies wi h
he ( espec i ely) δ18O and CO2 expe imen al se ies a ailable o he las 800 ky
BP we e calcula ed o quan i y he maximum obse a ional explained a iance, gi -
ing us a i s app oxima ion o he simila i y be ween simula ed and obse a ional
ime se ies. The exp ession o he s a is ical co ela ion is
1Ma hema ica is a adema k o Wol am Resea ch and Ma lab is a adema k o The Ma hWo ks.
whe e co [s1, s2] is he co ela ion be ween he ime se ies s1 and s2, E[x] is he ex-
pec ed alue (mean) o x, he do symbol deno es scala p oduc , σ[x] is he s an-
da d de ia ion o x, and N is he numbe o componen s o a ays s1 and s2.
The co ela ions ob ained a e shown in Table 1.1. As can be obse ed in column 2
o he able, he co ela ions o he cen al case o PP04 can be imp o ed om 0.59
and 0.63 o 0.67 and 0.71 o CO2 and ice olume, espec i ely, when he α pa am-
e e akes he ze o alue (model PB). This pa ame e ep esen s a di ec o cing
co [s1, s2] = (s1 ‒ E [s1]) • (s2 ‒ E [s2]) ,
(N ‒ 1)σ[s1] σ[s2]
1.7
Chap e 1
37
o I65 on a mosphe ic CO2 and i s physical in e p e a ion is no clea , gi en ha
he CO2 esponse o ice olume is al eady included in he model. Thus, we ha e
elimina ed his pa ame e om he model he ea e . The second and hi d panels o
Figu es 1.1. and 1.2. show he ice olume and CO2 ime se ies simula ed wi h hese
wo pa ame e se s. The main diffe ence be ween he wo cases appea s in he CO2
ime se ies, whe e he oscilla ions o 41 and 21 ky in he case wi h α = 0 p esen a
lowe ampli ude han in he case wi h α = 0.15.
1.2.1. Biological expo p oduc ion model
Ma ínez-Ga cía e al. [2009] obse ed ha he expo p oduc i i y in he suban -
a c ic A lan ic du ing glacial s ages g ew exponen ially. This sugges s ha he ex-
po p oduc ion in he SO could be d i en by changes in he supply o i on by dus ,
which is known o Vl mechanism, is able o gene a e i s o simila co ela ion o
he ones ob ained wi h he o iginal model (PP04). Wi h his pu pose, he oceanic
pulse in exp ession C is eplaced by an exp ession ha ep esen s he biological
a e o ca bon expo a ion o he SO as
whe e
C = αI65 ‒ βV + B + δ,1.8
B
=
ɛ(ekV ‒ 1), V > 0 .
0, V < 0
1.9

RELAXATION MODELS APPLIED TO LATE PLEISTOCENE CLIMATIC OSCILLATIONS
38
PARAMETERS PP04 PB BIO 2τ4τEP 3τLS
τV15000 15000 3667 12000 17006 4600 16585 11325
τV2- - - - 3797 1100 3105.5 2325
τC5000 5000 100 800 6796 3040 13505 2793
τC2- - - 4500 17667 7600 -8414
τA12000 10500 -11333 8089 12000 9004 10266
x1.3 1.32 0.85 1.29 0.767 1.525 0.905 0.669
y0.5 0.45 10.5 0.442 0.2 0.489 0.527
z0.8 0.8 0.92 0.85 1.033 0.96 0.946 0.761
α0.15 - - - - - - 0.237
β0.5 0.496 0.76 0.496 0.406 0.476 0.336 0.793
γ0.7 0.506 0.49 0.513 1.642 0.46 2.044 1.955
δ0.4 0.4 0.4 0.434 0.407 0.445 0.228 0.146
a0.3 0.3 -0.3 0.395 0.3 0.54 -
b0.7 0.71 -0.71 0.8 0.71 1.205 0.936
c0.01 0.005 -0.005 ---0.533
d0.27 0.27 -0.27 0.27 0.255 0.483 0.069
ɛ- - 0.8 - - - - -
k- - 0.47 - - - - -
Vm- - 0.5 - - - - -
λ- - - - - 37 - -
RV0.63 0.71 0.58 0.82 0.89 0.85 0.88 0.87
RC0.59 0.67 0.45 0.75 0.79 0.77 0.79 0.76
Table
1.1
Cen al alues o he models s udied. Columns 2 o 9 co espond o: (2) o iginal Pailla d
and Pa enin [2004] model wi h γ = 0.7 (PP04); (3) bes i o p e ious model (PB); (4)
model wi h biological expo p oduc ion (BIO); (5) model wi h wo elaxa ion imes o
C (2τ); (6) model wi h wo elaxa ion imes o C and wo elaxa ion imes o V (4τ); (7)
model wi h oceanic pulse exponen ially dependen on s a i ica ion (EP); (8) model wi h
one elaxa ion ime o C and wo elaxa ion imes o V (3τ) and (9) model wi h local
s a i ica ion pa ame e s (LS). RV and RC ep esen he co ela ion be ween p oxy and
modeled da a o global ice olume, V, and a mosphe ic CO2 concen a ion, C, espec i ely.
Chap e 1
39
MODELS DESCRIPTION
NUMBER
OF
PARAMETERS
PP04 O iginal Pailla d and Pa enin [2004] model wi h γ = 0.7 14
PB Bes i o Pailla d and Pa enin [2004] model 13
BIO Biological a e o expo a ion included a he
CO2 e e ence s a e 11
2τ
Two diffe en esponse imes o he
accumula ion and abso p ion o CO2
τC = τC1 o he pe iods whe e C < C
and τC = τC2 when C > C
14
4τ
Two diffe en esponse imes o he accumula ion and
abso p ion o CO2 and wo diffe en esponse imes o
accumula ion and abla ion o ice:
τV = τV1 o pe iods when V < V and τV = τV2 when V > V
14
EP Oceanic pulse exponen ially dependen on s a i ica ion 14
3τModel wi h one elaxa ion ime o C and
wo elaxa ion imes o V13
LS Model wi h local s a i ica ion pa ame e s 15
Desc ip ion o he diffe en models.
Table
1.2
RELAXATION MODELS APPLIED TO LATE PLEISTOCENE CLIMATIC OSCILLATIONS
40
Figu e
1.1
Bes i s ob ained o he ice olume ime se ies. F om op o bo om: (1) o iginal Pailla d
and Pa enin [2004] model wi h γ = 0.7 (PP04); (2) bes i o p e ious model (PB); (3)
model wi h biological expo p oduc ion (BIO); (4) model wi h wo elaxa ion imes o C
(2τ); (5) model wi h wo elaxa ion imes o C and wo elaxa ion imes o V (4τ); (6)
model wi h oceanic pulse exponen ially dependen on s a i ica ion (EP); (7) model wi h
one elaxa ion ime o C and wo elaxa ion imes o V (3τ ) and (8) model wi h local
s a i ica ion pa ame e s (LS). P oxy eco ds o δ18O ime se ies (Lisiecki and Raymo [2005])
a e supe imposed in e e y panel. Blue bands ep esen in e glacial pe iods conside ing ben-
hic δ18O below 3.8 pe mil.
PP04
I
-2.5
0
2.5
PB
-2.5
0
2.5
ExP
-2.5
0
2.5
2τ
-2.5
0
2.5
4τ
-2.5
0
2.5
EP
-2.5
0
2.5
3τ
-2.5
0
2.5
LS
Time (ky )
-800 -700 -600 -500 -400 -300 -200 -100 0
-2.5
0
2.5
II
IIIIV
VVI
VIIVIIIIX
Chap e 1
41
Figu e
1.2
Bes i s ob ained o he CO2 ime se ies. F om op o bo om: (1) o iginal Pailla d and
Pa enin [2004] model wi h γ = 0.7 (PP04); (2) bes i o p e ious model (PB); (3)
model wi h biological expo p oduc ion (BIO); (4) model wi h wo elaxa ion imes o
C (2τ); (5) model wi h wo elaxa ion imes o C and wo elaxa ion imes o V (4τ);
(6) model wi h oceanic pulse exponen ially dependen on s a i ica ion (EP); (7) model
wi h one elaxa ion ime o C and wo elaxa ion imes o V (3τ ) and (8) model wi h
local s a i ica ion pa ame e s (LS). Reco ds o expe imen al CO2 ime se ies (Pe i e
al. [1999]; Monnin e al. [2001]; Pepin e al. [2001]; Siegen hale e al. [2005]; Lu hi e al.
[2008]) a e supe imposed in e e y panel. Blue bands ep esen in e glacial pe iods consid-
e ing expe imen al CO2 concen a ions abo e 250 ppm.
PP04
-2
0
3
PB
-2
0
3
ExP
-2
0
3
2τ
-2
0
3
4τ
-2
0
3
EP
-2
0
3
3τ
-2
0
3
LS
Time (ky )
-800 -700 -600 -500 -400 -300 -200 -100 0
-2
0
3
III
IIIIV
VVI
VIIVIIIIX
RELAXATION MODELS APPLIED TO LATE PLEISTOCENE CLIMATIC OSCILLATIONS
48
ob ained wi h a = 0. Fo his eason, we used he inal exp ession (Eq. 1.18) ha
uses one pa ame e less. S a i ica ion is hus conside ed dependen only on egional
a iables ( he ex en o An a c ic ice shee A and SO empe a u e, ela ed o C). In
pa allel wi h i , i was assumed ha SO empe a u e, as ep esen ed by a iable C,
led also he e olu ion o ice shee A (Eq. 1.15). This change has no effec on he
co ela ion ob ained o V bu i imp o es sligh ly he co ela ion ob ained o
C. The inal equa ions a e:
1.3. MODELS PERFORMANCE
The analysis o Table 1.1 gi es some hin s on he condi ions ha lead o he bes
da a i s. As can be obse ed in Figu e 1.1, all models ha e p oblems o accu a ely
simula e he minimum o V a –200 ky (cycle II), he iming o he V minimum a
–490 ky (Te mina ion VI) and he maximum ice olume a –750 ky (excep , o
some ex en , model BIO).
dA
=
(‒C ‒ A) ,
τA
d
1.15
V = ‒ xC ‒ yI65 + z,1.16
C = αI65 ‒ βV + γH ( ‒F ) + δ,1.17
F = ‒ bA ‒cC + d. 1.18
1.13
dV = (V ‒ V ) ,
τV
d
1.3
dC = (C ‒ C),
τC
d

Chap e 1
49
I may be obse ed ha all models ha e p oblems simula ing he glacial cycle be-
ween –600 and –500 ky , especially he iming o i s e mina ion. Inaccu a e i ing
o he iming o e mina ions p obably de i es om limi a ions o bo h model
and δ18O eco ds. A common way o es ima ing ime in paleoclima e eco ds is
o s e ch, squeeze and shi a eco d’s ch onology in o de o align i wi h a em-
pla e indica i e o changes in he Ea h’s o bi al and o a ional con igu a ion, a
p ocess gene ally e e ed o as o bi al uning (Huybe s [2011]). Fo his eason,
i is no clea ha he ime scale o he da a is be e o wo se han he ime scale
o he model (wi hin, le ’s say, a qua e pe iod o he as es o cing, he p eces-
sion, i.e. ±6 ky ). This is pa icula ly ue o he iming o e mina ions, which
appa en ly a e no linea ly co ela ed wi h he as onomical o cing in ensi y (see
e.g. Pa enin and Pailla d [2003]). Fo his eason, a “be e i ” o he iming o
e mina ions o he δ18O da a may no be an imp o emen , because he da a a e
no necessa ily e y accu a e in hei es ima ion o he p ecise da es when e mi-
na ions ake place. Time se ies o he i e o we e ob ained by sub ac ing he
simula ed and expe imen al ime se ies esul ing om di iding e e y ime se ies
by i s s anda d de ia ion.
The co ela ions ob ained wi h he EP model do no imp o e upon he ones ob-
ained wi h he 4τ model (Table 1.1) so, we can conclude ha he esul s ob ained
do no jus i y he in oduc ion o an addi ional pa ame e . Howe e , he ex a
con ibu ion o glacial CO2 p oduced in his model makes i possible o i he
expe imen al CO2 se ies wi h a lowe alue o he y pa ame e (sensi i i y o I65
o cing), which p oduces a V ime se ies wi h he high- equency oscilla ions mo e
damped. The “ ex u e” o his se ies, as can be obse ed in Figu es 1.1 and 1.2,
esembles ha obse ed in he expe imen al da a.
The pe o mance o 4τ and 3τ models in he simula ion o V is qui e simila ,
excep o Te mina ion V and he glacial cycle II, whe e 4τ pe o ms sligh ly
be e . Fo hese easons, 4τ co ela ion o V is sligh ly be e han 3τ (0.89 in-
s ead o 0.88). The use o a diffe en elaxa ion ime o emission and abso p ion
o CO2 allows 4τ o ma ch CO2 in glacial cycle V be e han 3τ. Howe e , he
ex eme simplici y o he ca bon model shows up in bo h models hough he lack
RELAXATION MODELS APPLIED TO LATE PLEISTOCENE CLIMATIC OSCILLATIONS
50
o high- equency oscilla ions in he e olu ion o C. The LS model has a con en o
high- equency a iance la ge han ha o 3τ because i pe mi s a di ec o cing
o I65 o C bu i s co ela ion is no be e because his a iance is equen ly ou o
phase wi h he obse a ional one.
1.4. DISCUSSION
S a ing om he Pailla d and Pa enin [2004] model, se e al box models inco po-
a ing simple pa ame e iza ions o he oceanic CO2 pumping and esponse imes o
ca bon and ice olume ha e been de eloped. The models’ pa ame e s we e calib a -
ed o p o ide he bes i o he δ18O and CO2 expe imen al ime se ies a ailable
o he las 800 ky BP. The PP04 model is insensi i e o he α pa ame e (di ec
o cing be ween I65 insola ion and CO2) and he Sun’s effec seems o eme ge only
h ough he il e o global ice olume. The i o his model o obse a ional da a
may be imp o ed i diffe en esponse imes a e assumed bo h o abso p ion/
emission o CO2 and o abla ion/accumula ion o ice. Co ela ions be ween simu-
la ed and expe imen al ime se ies inc ease om 0.59 and 0.63 o 0.79 and 0.89 o
CO2 and V, espec i ely.
Se e al modi ica ions o he PP04 model ha lead o he igh iming o he las
nine e mina ions we e es ed. In pa icula , he quali a i e beha io o he las
eigh glacial-in e glacial cycles may be oughly ep oduced wi h an expo p oduc-
ion model wi h expo dependen on ice olume (BIO). Howe e , in his model,
he bes i co esponded o a dependence be ween CO2 expo and V ha was no
exponen ial as p oposed by Ma ínez-Ga cía e al. [2009], bu a squa e unc ion. In
ou o mula ion, biological expo alone was no able o simula e he expe imen al
da a as accu a ely as he o he models, e en hough some kind o biological expo
o CO2 is e y p obably ac ing in syne gy wi h physical p ocesses in he glacial-in-
e glacial dynamics.
One limi a ion o he me hodology used in his s udy, common in paleoclima e mod-
els, is ha he e is no dis inc ion be ween calib a ion da a and alida ion da a. Fo
Chap e 1
51
his eason, i may be use ul o know whe he pa ame e op imiza ions ob ained
on hal o he da a in e al lead o nea ly he same bes es ima es. Ou models
seem o i be e he saw oo h oscilla ions ha clima e has shown in he las –400
ky han he mo e i egula oscilla ions occu ing be ween –800 ky and –400 ky .
In o de o include bo h kinds o dynamics, we used he in e al –600 o –200 ky
as calib a ion da a in he LS model. This in e al led o a bes i ha is coinciden
wi h he one shown in he las column o Table 1.1, and he co ela ions ob ained
we e 0.872 and 0.763, which a e sligh ly be e han hose ob ained o he comple e
in e al (0.866 and 0.757). We ha e ounded he la e esul s o wo decimal digi s
in he las column o Table 1.1. When a second and mo e exhaus i e sea ching is im-
plemen ed a ound he alues ob ained by he gene ic algo i hm, i is possible o ind
a diffe en pa ame e combina ion wi h e en be e co ela ions (0.884 and 0.774).
Howe e , he case is hen so igh ly i ed o he calib a ion in e al ha i ails o
ep oduce he pace o he whole in e al (and co ela ions d op o 0.43 and 0.22).
On he o he hand, he expe imen al da a used (δ18O: Lisiecki and Raymo [2005];
CO2: Monnin e al. [2001]; Pe i e al. [1999]; Siegen hale e al. [2005]; Lu hi e al.
[2008]) con ain much high- equency a iance, ep esen ing (among o he hings)
wea he noise, measu emen e o , and annual- o-cen ennial clima e a iabili y.
PP04-de i ed models do no con ain mechanisms able o p oduce signi ican a iance
in hese ime scales. Fo his eason, in one o ou uns we smoo hed he p oxy da a
wi h a mo ing a e age o 2 ky , and hen calib a ed he LS model wi h hese ime
se ies and he comple e in e al (–800 ky , 0 ky ). The co ela ions ob ained we e
0.869 and 0.761, espec i ely, whe eas he esul wi h un il e ed da a was 0.866 and
0.756, espec i ely. In conclusion, he smoo hing o he expe imen al ime se ies
p o ides an imp o emen o less han 1% in he i s o he wo a iables.
Fou models (4τ, EP, 3τ and LS) show a good pe o mance in he simula ion o he
ice olume obse ed in he las eigh glacial-in e glacial cycles and e en in many
de ails o ha signal belonging o he scale om 23 o 41 ky . The good i s ob ained
sugges ha densi y o deep wa e and i s a e o o ma ion may be impo an ac-
o s con olling he oceanic pulse ha igge s he deglacia ions. Schmi ne [2007]
has con i med, wi h a global clima e model, he sensi i i y o a mosphe ic CO2 o
p ocesses ha affec s a i ica ion in SO wa e s. In ou analysis, oceanic pulses ha
RELAXATION MODELS APPLIED TO LATE PLEISTOCENE CLIMATIC OSCILLATIONS
52
s ongly depend, in a non-linea way, on he deep ocean s a i ica ion a e necessa y
o igge deglacia ions. The pulses ob ained in he bes models a e always e y
close o he Hea iside unc ion p oposed by he o iginal wo k o PP04. Oceanic CO2
pulses wi h a du a ion o be ween 10 and 20 ky we e ound a he beginning o he
nine las deglacia ions acco ding o he bes models analyzed. In addi ion, diffe en
esponse imes o emission and abso p ion o CO2 and o accumula ion and abla-
ion o ice a e necessa y o ob ain he bes i o he a ailable da a. Accumula ion
and abla ion o ice a e diffe en physical mechanisms ha could ha e diffe en cha -
ac e is ic imes. Howe e , he wo esponse imes o emission and abso p ion o
CO2 a e mo e di icul o explain, as we ha e said be o e. To decide whe he hese
wo esponse imes co espond o eal physical mechanisms, a deepe unde s anding
o he long- e m ans e a es in he ca bon cycle should be achie ed.
Bjö ks öm [1979] iden i ied he wo slowes esponse imes o he emission o CO2
o he a mosphe e o be ela ed o he emine aliza ion o o ganic ca bon o dead
ma e ial in soils, and he long a el ime o he CO2 om he deep ocean com-
pa men s o he su ace. These au ho s used 1000 yea as an o de o magni ude
o bo h pa ame e s. B o kin e al. [2002] used he ange 400 - 1000 yea s o model
he slow soil esponse wi h he CLIMBER-2 paleoclima ic model. The cha ac e is-
ic ime o CO2 abso p ion in he long- e m scales is ela ed o he CO2 ad ec ion
om he su ace in o he deep ocean acco ding o Bjö ks öm [1979] and i should
be, again, o he o de o 1000 y .
Ou esul τC2 > τC could be meaning ul i he cha ac e is ic ime o deep wa e
ci cula ion was longe han he long- e m emission ime o soils. Mon eneg o e al.
[2007] ound ha 25% o an ins an aneous CO2 elease o he a mosphe e emains
he e a e 5000 yea s. A che e al. [2009b] e iewed he li e a u e on he ca bon
cycle, which ag ees ha 20% o 35% o an ins an aneous CO2 elease emains in he
a mosphe e in 200 o 2000 yea s, a e equilib a ion wi h oceans. Due o oceanic acid-
i ica ion, subsequen dissolu ion o CaCO3 inc eases he up ake capaci y o oceans
in he scale o 3 o 7 ky . These esul s a e appa en ly cohe en wi h he o de o
magni ude o he τC2 pa ame e ha we ha e ound. Howe e , mo e in es iga ion is
needed o ha e a p ecise unde s anding o he long- e m ca bon cycle.
Chap e 1
53
A diffe en p oblem is deciding wha con idence we can gi e o hose models wi h
highe co ela ions, gi en ha he eco d used o he pas ice olume p obably
suffe s diffe en biases. Indeed, he eco d is a s ack o δ18O da a om ben hic o-
amini e a, and he a io O18/O16 is known o depend on bo h he iso opic composi-
ion and he empe a u e o he wa e whe e he o amini e a de elops. Siddall e al.
[2010] ound ha ice olume becomes inc easingly sensi i e o empe a u e change
a low empe a u es. Waelb oeck e al. [2002] ound ha he ela ionship be ween
δ18O and ice olume is no linea , since δ18O dec eased as e han he inc ease
in ice olume a he beginning o he las glacia ions, and hen p og essi ely mo e
slowly un il he ice shee s eached hei maximum size. This may p oduce unce -
ain ies g ea e han 20% o he ice olume es ima ions in some pe iods. The e o e,
he models wi h he la ges co ela ions be ween p edic ed ice olume and δ18O a e
no necessa ily be e han o he models wi h sligh ly lowe co ela ion because o
he unce ain y in he obse a ional da a. This is a complex and impo an ques ion
equi ing an addi ional ma hema ical analysis ha we lea e o a u u e wo k. Fo -
una ely, he e is no such p oblem o CO2, which is di ec ly measu ed in ice co es.
I may be a gued ha i is no su p ising o ge good i s (Table 1.1) using models
wi h many pa ame e s. This is pa icula ly ue when expe imen al da a a e i ed
o gene ic ma hema ical unc ions ha a e dependen on an a bi a y numbe o
pa ame e s. Howe e , i is no so easy o ge he same good i wi h ma hema ical
exp essions ha imi a e geophysical mechanisms such as eedbacks and pumping
a es. An addi ional ou come om elaxa ion models o his kind is ha hey poin
o speci ic physical mechanisms ha a e po en ially d i e s o he obse ed changes.
Thus, when good i s a e ob ained wi h such models, he speci ic physical mechanisms
modeled should be in es iga ed in mo e de ail o con i m hem (and no an al e na-
i e mechanism p oducing a simila beha io ) as induce s o he obse ed dynamics.
To sum up, we ha e ob ained eigh diffe en i s wi h ice olume co ela ions be-
ween 0.58 and 0.89 which show a good quan i a i e and quali a i e ag eemen wi h
he empi ical ime se ies, especially 4τ, EP, 3τ and LS; al hough he wa m e en o
–500 ky is no p ope ly ep oduced by any model. The 4τ model imp o es he PP04
co ela ions (0.59 and 0.63) o 0.79 and 0.89 using he same numbe o pa ame e s

RELAXATION MODELS APPLIED TO LATE PLEISTOCENE CLIMATIC OSCILLATIONS
54
(namely, 14). The EP model uses 15 pa ame e s bu he addi ional pa ame e ,
which was aimed a imp o ing he o m o he oceanic pulse unc ion, did no im-
p o e he co ela ions and can be conside ed useless. The 3τ model ob ains almos
he same co ela ions as 4τ using only 13 pa ame e s. The LS model (15 pa ame-
e s) does no imp o e he co ela ions o 4τ (which a e sligh ly highe ) bu inco -
po a es a unc ional o m o s a i ica ion F ha seems mo e consis en wi h bo h
he mechanisms sugges ed by Pailla d and Pa enin [2004]. I ou aim we e o se-
lec he model wi h he bes explained a iance pe pa ame e , he choice would be
3τ . Howe e , a second objec i e o his wo k is o de e mine whe he models wi h
an explained a iance simila o he one ob ained by 3τ con ain pa ame e iza ions
ha can be ela ed as ealis ically as possible o obse ed mechanisms. LS can offe
a aluable insigh in o he eal meaning o he good pe o mance o PP04-de i ed
models. In he nex chap e , he dynamics and mechanisms inco po a ed in 3τ and
LS models will be analyzed and compa ed.
Chap e 2
ON THE PHYSICAL
MECHANISMS BEHIND
GLACIAL-INTERGLACIAL
DYNAMICS
Chap e based on he ollowing
published a icles:
Ga cía-Oli a es and He e o [2013],
Simula ion o glacial-in e glacial cycles
by simple elaxa ion models: consis ency
wi h obse a ional esul s. Clima e Dy-
namics 41:1307-1331
He e o and Ga cía-Oli a es [2014],
Non-linea analysis me hods applied o
obse a ional and simula ed clima ic ime
se ies. P oceedings In e na ional wo k-con-
e ence On Time Se ies. Volume 2, p1068-
1080. I.S.B.N: 978-84-15814-97-4
He e o and Ga cía-Oli a es [2015],
Non-linea analysis me hods applied o
obse a ional and simula ed clima ic ime
se ies. Bole ín Geológico Mine o, accep -
ed o an ITISE special olume.
64
ON THE PHYSICAL MECHANISMS BEHIND GLACIAL-INTERGLACIAL DYNAMICS
As shown in Figu e 2.3 op and Figu e 2.4, he 100-ky band is he dominan pe iod
in bo h he δ18O and V ime se ies o he 3τ and LS models, ollowed by he 41-ky
band, and powe in bo h equencies is simila ly dis ibu ed in obse a ional and
simula ed se ies. The δ18O ime se ies con ains pe iodici ies in he ange 1-10 ky ;
in con as , ou models do no gene a e any signi ican pe iodici y unde 10 ky . The
23-ky pe iod is weakly p esen in bo h expe imen al and 3τ ime se ies. The LS
and 3τ models show he same pa e ns in hei wa ele diag ams, which a e almos
indis inguishable om each o he .
Fo CO2 (Figu e 2.3 bo om and Figu e 2.5), he powe dis ibu ion in he simula ed
and obse a ional ime se ies a e simila o he 100-ky band. Powe in he 41-ky
band is somewha mo e homogeneously dis ibu ed in he obse a ional ime se ies
han in he simula ed ime se ies. The 23-ky con ibu ion is weake in bo h se ies
and is dis ibu ed in a diffe en way in he modeled and obse a ional ime se ies. The
diffe ence is g ea e in he i s h ee cycles, which a e ela i ely difficul o model.
The complex c oss-wa ele ans o m o wo ime se ies (Figu es 2.6 and 2.7) can
be in e p e ed as he sha ed powe in a gi en pe iodici y band (absolu e alue) and
he phase diffe ence be ween he wo se ies in ime equency space (G ins ed e al.
[2004]). Ligh e pu ple colo indica es g ea e sha ed powe in ha ime and pe io-
dici y band, and he a ow angle indica es he phase be ween he obse a ional and
simula ed se ies. As can be obse ed in Figu e 2.6, obse a ional and simula ed ice
olume sha e a g ea common powe in he 100-ky and 41-ky band. In he 100-ky
band, he wo pe iodici ies a e in phase, bu in he 41-ky band he modeled pe io-
dici y ends o lead he obse a ional one a some momen s, wi h a phase be ween 0
and π/4. In he 41 and 23-ky band, he wo pe iodici ies end o be in phase when
he common powe is high and ou o phase when he sha ed powe is low. The wo
models s udied (3τ and LS) show e y close pa e ns in hei c oss-wa ele ans-
o ms o V. Rega ding CO2 (Figu e 2.7), he la ges sha ed powe is in he 100-ky
band, especially a e ‒500 ky , when he i be ween simula ed and obse a ional
cycles is somewha be e (see Figu e 1.2 and 2.1). In he 41-ky band he sha ed
powe is lowe and has ela i e maxima a ound ‒600 ky , be ween ‒400 and ‒200
ky and be ween ‒150 and -50 ky . These in e als o good coincidence can be also

Chap e 2
65
0 20 40 60 80 100 120
0
500
1000
1500
2000
Spec al Powe V
0 20 40 60 80 100 120
0
500
1000
1500
2000
Spec al Powe C
Pe iod (Ky )
A op, spec al powe o no malized ice olume as p edic ed wi h 3τ model (blue),
LS model ( ed) and p oxy δ18O eco ds om Lisiecki and Raymo [2005] (g ey line); a
bo om, no malized CO2 as p edic ed wi h he 3τ (blue line) and LS ( ed line) models
and p oxy da a (Pe i e al. [1999]; Inde muhle e al. [2000]; Monnin e al. [2001]; Sie-
gen hale e al. [2005]; Lu hi e al. [2008]) (g ey line).
Figu e
2.2
obse ed in Figu es 1.1, 1.2 and 2.1 o he LS model. Low sha ed powe is obse ed
in he 23-ky band, wi h an ou -o -phase pa e n in some in e als.
Wa ele cohe ence o wo se ies can be in e p e ed as a localized co ela ion coefficien
in ime equency space. I is use ul o ind locally phase-locked beha io , ha is,
66
ON THE PHYSICAL MECHANISMS BEHIND GLACIAL-INTERGLACIAL DYNAMICS
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1/8
1/4
1/2
1
2
4
8
16
32
64
Time (Ky )
F equency (Ky )
−700 −600 −500 −400 −300 −200 −100 0
0.20
0.39
0.78
1.56
3.13
6.25
12.5
25
50
100
200
Wa ele diag am o p oxy δ18O eco ds om Lisiecki and Raymo [2005] ( op) and p oxy
CO2 da a om Pe i e al. [1999]; Inde muhle e al. [2000]; Monnin e al. [2001]; Siegen-
hale e al. [2005]; Lu hi e al. [2008] (bo om).
Figu e
2.3
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1/2
1
2
4
8
16
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Time (Ky )
F equency (Ky )
−700 −600 −500 −400 −300 −200 −100 0
0.20
0.39
0.78
1.56
3.13
6.25
12.5
25
50
100
200
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67
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2
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8
16
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Time (Ky )
F equency (Ky )
−700 −600 −500 −400 −300 −200 −100 0
0.20
0.39
0.78
1.56
3.13
6.25
12.5
25
50
100
200
Wa ele diag am o he simula ed V ime se ies wi h 3τ model ( op) and LS model
(bo om).
Figu e
2.4
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2
4
8
16
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64
Time (Ky )
F equency (Ky )
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0.20
0.39
0.78
1.56
3.13
6.25
12.5
25
50
100
200
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Time (Ky )
F equency (Ky )
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0.20
0.39
0.78
1.56
3.13
6.25
12.5
25
50
100
200
Wa ele diag am o he simula ed C ime se ies wi h 3τ model ( op) and LS
model (bo om).
Figu e
2.5
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2
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8
16
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64
Time (Ky )
F equency (Ky )
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0.20
0.39
0.78
1.56
3.13
6.25
12.5
25
50
100
200
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69
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1
2
4
8
16
32
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Time (Ky )
F equency (Ky )
−700 −600 −500 −400 −300 −200 −100 0
0.20
0.39
0.78
1.56
3.13
6.25
12.5
25
50
100
200
C oss-wa ele ans o m be ween p oxy δ18O eco ds om Lisiecki and Raymo [2005]
and simula ed V ime se ies wi h 3τ model ( op) and LS model (bo om)
.
Figu e
2.6
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2
4
8
16
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Time (Ky )
F equency (Ky )
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0.20
0.39
0.78
1.56
3.13
6.25
12.5
25
50
100
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4
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Time (Ky )
F equency (Ky )
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0.20
0.39
0.78
1.56
3.13
6.25
12.5
25
50
100
200
C oss-wa ele ans o m be ween p oxy CO2 da a om Pe i e al. [1999]; Inde muhle e
al. [2000]; Monnin e al. [2001]; Siegen hale e al. [2005]; Lu hi e al. [2008] and simu-
la ed C ime se ies wi h 3τ model ( op) and LS model (bo om).
Figu e
2.7
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8
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Time (Ky )
F equency (Ky )
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0.20
0.39
0.78
1.56
3.13
6.25
12.5
25
50
100
200
Chap e 2
71
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (Ky )
F equency (Ky )
−700 −600 −500 −400 −300 −200 −100 0
0.20
0.39
0.78
1.56
3.13
6.25
12.5
25
50
100
200
Wa ele Cohe ence be ween p oxy δ18O eco ds om Lisiecki and Raymo [2005] and
simula ed V ime se ies wi h 3τ model ( op) and LS model (bo om).
Figu e
2.8
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (Ky )
F equency (Ky )
−700 −600 −500 −400 −300 −200 −100 0
0.20
0.39
0.78
1.56
3.13
6.25
12.5
25
50
100
200
72
ON THE PHYSICAL MECHANISMS BEHIND GLACIAL-INTERGLACIAL DYNAMICS
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (Ky )
F equency (Ky )
−700 −600 −500 −400 −300 −200 −100 0
0.20
0.39
0.78
1.56
3.13
6.25
12.5
25
50
100
200
Wa ele Cohe ence be ween p oxy CO2 da a om Pe i e al. [1999]; Inde muhle e al.
[2000]; Monnin e al. [2001]; Siegen hale e al. [2005]; Lu hi e al. [2008] and simula ed
C ime se ies wi h 3τ model ( op) and LS model (bo om).
Figu e
2.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (Ky )
F equency (Ky )
−700 −600 −500 −400 −300 −200 −100 0
0.20
0.39
0.78
1.56
3.13
6.25
12.5
25
50
100
200
Chap e 2
73
momen s in which bo h se ies oscilla e wi h he same equency and a gi en phase
diffe ence. Figu e 2.8 shows he wa ele cohe ence o he δ18O and he simula ed V
se ies o 3τ and LS and Figu e 2.9 shows he wa ele cohe ence o he obse a ional
CO2 and he simula ed C se ies o he same models. I can be obse ed ha he
co ela ions o V o he wo models a e always in phase, a e la ge han 0.9 in he
100-ky band o all imes and a e especially high in he ou las cycles (‒400 ky
o 0). In he 41-ky band he simula ed and obse a ional se ies oscilla e oge he
wi h a co ela ion ha exceeds 0.9 a ound ‒600 ky , be ween ‒400 and ‒350 ky ,
and is always g ea e han 0.7 be ween ‒130 and ‒90 ky . The cohe ence is no sig-
ni ican be ween ‒750 and -630 ky and be ween ‒540 and -380 ky , co esponding
o in e als in which he models a e wo se a ma ching he obse a ional se ies (see
Figu es 1.1 and 2.1).
Rega ding CO2 (Figu e 2.9), in he wo models he cohe ence a he 100-ky band
is in phase and high a e ‒500 ky and weake and ou o phase be o e ‒500 ky
coinciding wi h he i s glacial cycles (see also Figu e 2.2), which a e mo e difficul
o simula e. In he 41-ky band he cohe ence is high in oughly he same in e als
as in he V se ies, and in he 23-ky band i is p ac ically inexis en . This can be
a ibu ed o he combined effec s o high- equency damping p oduced by pa am-
e e τC (and/o τC2 in LS) and poo ep esen a ion o he ca bon dynamics by
PP04-de i ed models.
2.3.3. Phase space po ai s
Phase space po ai s and embedding a ac o echniques can also be use ul o
quan i ying he pe o mance o a simula ed ime se ies in ma ching he dynamical
p ope ies o an obse a ional ime se ies. Time se ies o 16000 linea ly in e pola -
ed, equally spaced da a a e used o econs uc he a ac o s. To a oid au oco -
ela ed e ec s we use he me hod o mu ual in o ma ion (Ma wan e al. [2007]),
a well-es ablished measu e o de ec nonlinea dependencies wi hin a ime se ies,
which es ima es an op imal ime lag o 1433 o V and 1371 o C. The i s s ep
80
ON THE PHYSICAL MECHANISMS BEHIND GLACIAL-INTERGLACIAL DYNAMICS
Embedded a ac o o he simula ed C ime se ies wi h LS model.
Figu e
2.15

Chap e 2
81
2.3.4. C oss ecu ence plo s
We hen ob ained he c oss ecu ence plo (CRP) o obse a ional and simula ed
ajec o ies which, despi e i s name, does no ep esen ecu ences bu a he he
conjunc u es o s a es o he wo sys ems. The CRP e eals all he imes when
he phase space ajec o y o he i s sys em isi s oughly he same egion in he
phase space whe e he phase space ajec o y o he second sys em is (Ma wan e al.
[2007]). Figu es 2.16 o 2.19 show he CRP ob ained wi h he abo e pa ame e s o
he simula ed se ies o V and C ob ained by he models. The Ma lab ool c ea ed
by Ma wan e al. [2007] was used. Las glacial cycle does no show up because his
pe iod da a is smalle han D imes l, whe e D is he dimension (D = 3) and l is he
lag (l = 1433 o V and l = 1371 o C). We can see ha he pe sis en in e als o
good coincidence be ween he V and δ18O se ies (Figu e 2.1) coincide wi h in ense
black pa ches in Figu es 2.16 and 2.17, especially spo s ollowing Te mina ion IX
and Te mina ion VII, 5000 < N < 6000, 7000 < N < 9000, 9200 < N < 9500,
10500 < N < 11000 and 11500 < N < 12500 (Te mina ions a e shown in Figu es
1.1 and 1.2). Many pa e ns occu ing in he main diagonal appea de o med in
lines and columns, indica ing ha bo h se ies a e cyclical and he way in which V( )
coincides wi h δ18O( ) a ime shows simila i y o he way in which V( ) coincides
wi h δ18O( ‒T), whe e T is he 100-ky pe iod. Roughly 85% o he ime he p e-
dic ed and obse a ional V ajec o ies a e neighbo s a a dis ance o 0.46 imes he
s anda d de ia ion o he phase space. The C ajec o ies a e much mo e poo ly
simula ed, wi h he p edic ed and obse a ional ajec o ies coinciding less han
50% o he ime (Figu es 2.18 and 2.19). Howe e , he dynamics in he in e als
when he CO2 is maximum is no mally well simula ed, which seems o be sufficien
o allow he V dynamics o ob ain a good simila i y o he obse a ional dynamics.
2.4. ALTERNATIVE PROXY FOR ICE VOLUME
As poin ed ou in Chap e 1, he Lisiecki and Raymo [2005] eco d is a s ack o
δ18O da a om ben hic o amini e a, and he O18/O16 a io is known o depend on
82
ON THE PHYSICAL MECHANISMS BEHIND GLACIAL-INTERGLACIAL DYNAMICS
C oss- ecu ence plo s ob ained o smoo hed p oxy δ18O eco ds om Lisiecki and
Raymo [2005] and simula ed V ime se ies wi h 3τ model.
Figu e
2.16
Chap e 2
83
C oss- ecu ence plo s ob ained o smoo hed p oxy δ18O eco ds om Lisiecki and
Raymo [2005] and simula ed V ime se ies wi h LS model.
Figu e
2.17
84
ON THE PHYSICAL MECHANISMS BEHIND GLACIAL-INTERGLACIAL DYNAMICS
C oss- ecu ence plo s ob ained o smoo hed p oxy CO2 da a om Pe i e al. [1999];
Inde muhle e al. [2000]; Monnin e al. [2001]; Siegen hale e al. [2005]; Lu hi e al. [2008]
and simula ed C ime se ies wi h 3τ model.
Figu e
2.18
Chap e 2
85
C oss- ecu ence plo s ob ained o smoo hed p oxy CO2 da a om Pe i e al. [1999];
Inde muhle e al. [2000]; Monnin e al. [2001]; Siegen hale e al. [2005]; Lu hi e al.
[2008] and simula ed ime se ies wi h LS model.
Figu e
2.19

86
ON THE PHYSICAL MECHANISMS BEHIND GLACIAL-INTERGLACIAL DYNAMICS
bo h he iso opic composi ion and he empe a u e o he wa e whe e o amini e a
de eloped. Waelb oeck e al. [2002] ound ha he ela ionship be ween δ18O and
ice olume is no linea , since δ18O dec eases as e han he inc ease in ice olume
a he beginning o he las glacia ions, and hen p og essi ely mo e slowly un il
he ice shee s eached hei maximum size. This may p oduce unce ain ies o 20%
o he ice olume es ima ions a he beginning o glacia ions. Consequen ly, he e
is some isk o o e -in e p e a ion when model pa ame e s a e adjus ed o i ha
iso opic eco d igh ly. I ha e o was assumed o ake place du ing 20 ky a he
beginning o each glacial pe iod, i would cause a 7% loss o explained a iance o a
4% loss o co ela ion. This is sligh ly abo e he co ela ion di e ences in he models
ha we ha e s udied, so hei pe o mance mus be conside ed equi alen wi hin he
obse a ional unce ain y.
Bin anja e al. [2005] ied o elimina e he empe a u e e ec s om he δ18O se-
ies by using a model o ai and deep ocean empe a u es. The esul ing sea-le el
ime se ies ca be used as an al e na i e o δ18O. When we calcula e he co ela-
ions be ween his ime se ies and he V p edic ed by ou models, we ob ain he
su p ising esul ha hey inc ease om 0.88 o 0.90 o 3τ and om 0.87 o 0.89
o LS. This shows ha he calib a ion o ou selec ed models o an iso opic se ies
wi h expe imen al biases has p oduced a obus esul in spi e o he obse a ional
unce ain y, al hough a non-negligible componen is ha Bin anja’s ime se ies is
sligh ly smoo he han δ18O. In Figu e 2.20 he V p edic ed by he 3τ and LS mod-
els is compa ed wi h sea le el ime se ies (Bin anja e al. [2005]) and δ18O eco ds
(Lisiecki and Raymo [2005]). In he bo om panel, he powe spec um o all se ies
is shown. As can be obse ed, he beha io o he 3τ and LS models compa ed
wi h Bin anja e al. [2005] sea le el is good, ein o cing he obus ness o he i s
implemen ed. Al hough he co ela ion imp o es, he esul s do no offe any new
dynamical insigh s.
2.5. DYNAMICS OF THE MODELS
A common ea u e o PP04-de i ed models is hei lack o sensi i i y o he I60 o c-
Chap e 2
87
ing. This esul could be in e p e ed in h ee al e na i e ways:
(i) Ou unc ion F (V, A) is an app oxima e ep esen a ion o s a -
i ica ion bu I60 is no a good p oxy o Sou he n Ocean empe -
a u e.
(ii) S a i ica ion and/o CO2 elease in he SO is no con olled
by egional empe a u es.
(iii) Release o CO2 in he SO is no con olled by s a i ica ion and
he unc ion F (V, A) in ou models ep esen s a di e en p o-
cess.
A possible in e p e a ion belonging o g oup (iii) is ha A migh ep esen sea ice
ins ead o he ice-shee a ea and ha F (V, A) could be a p oxy o a a e o emis-
sion o CO2 ha is con olled di ec ly by sea ice a ea, A, and SO deep empe a u e.
SO deep empe a u e could be con olled by NH empe a u e ( h ough V) mo e
s ongly han by I60. One possible mechanism able o p oduce his effec was p o-
posed by Gildo and Tzipe man [2001]: s a i ica ion in he SO is composed o cold,
esh wa e abo e wa m, sal y wa e . Glacial condi ions in he NH cool he No h
A lan ic Deep Wa e (NADW) and, consequen ly, ia he sou hwa d low and up-
welling o NADW, lowe he deep empe a u e in he SO. Because o he pe manen
ice co e o e An a c ica, su ace ocean empe a u e in he SO nea An a c ica is
close o eezing poin du ing he en i e glacial cycle, so i canno cool e y much
e en du ing glacial condi ions. Glacial condi ions he e o e inc ease he densi y o
deep SO wa e bu no o su ace SO wa e and his s eng hens he e ical s a i-
ica ion he e. As a esul , e ical mixing in he SO is expec ed o be educed. This
mechanism implies ha empe a u e changes in he NH (supposedly co ela ed wi h
global ice olume V) lead o he CO2 changes in he SH. This mechanism is plausible
dA = (V ‒ A ‒ cI60) .
τA
d
2.7
88
ON THE PHYSICAL MECHANISMS BEHIND GLACIAL-INTERGLACIAL DYNAMICS
−800 −700 −600 −500 −400 −300 −200 −100 0
−2
−1
0
1
2
Time (Ky )
Global Ice Volume
0 20 40 60 80 100 120
0
500
1000
1500
2000
Pe iod (Ky )
Spec al Powe
Top panel: no malized ice olume as p edic ed wi h he 3τ (blue line), LS ( ed line)
models, p oxy δ18O eco ds om Lisiecki and Raymo [2005] (g ey line) and Bin anja e
al. [2005] sea-le el ime se ies (g een line). Bo om panel: powe spec um o no malized
ice olume as p edic ed wi h 3τ model (blue), LS model ( ed), p oxy δ18O eco ds om
Lisiecki and Raymo [2005] (g ey line) and Bin anja e al. [2005 sea-le el ime se ies
(g een line).
Figu e
2.20
Chap e 2
89
bu in e p e ing F as a model o such mechanism is unce ain, because co ela ion
be ween V and No h A lan ic empe a u e ime se ies (Ba d [2003]) is low, only
0.64 o he las 800 ky .
On he o he hand, o de e mine whe he A is sensi i e o I60, Equa ion 2.2 o he
model 3τ was modi ied as
Howe e , he bes i s ob ained a e his change did no imp o e upon hose ob-
ained by ou 3τ model, showing ha Equa ion 2.2 is also insensi i e o I60.
These esul s a e consis en wi h he ollowing conclusions: a a iable A ha ol-
lows he e olu ion o V wi h a delay o 5 o 7 ky seems necessa y o ob ain good
i s. This a iable can be in e p e ed, as Pailla d and Pa enin [2004] do, as he ice
shee ex en on he An a c ic shel , bu i can also be in e p e ed as he whole SO
ice ex en since he capping e ec (Keeling and S ephens [2001]) and he con ol o
A on he esidual ci cula ion ac in he same di ec ion as he deep s a i ica ion.
CO2 elease o he a mosphe e in he SO may be con olled by ei he s a i ica ion
o sea ice ex en , o by bo h. Howe e , nei he o he wo co esponding a iables
in ou models (F and A) seems o be sensi i e o I60 insola ion. The conclusion is
ha ei he some o hese mechanisms a e no sensi i e o SO empe a u e, which is
difficul o concei e, o I60 is no a good p oxy o SO empe a u e.
Huybe s and Den on [2008] poin ed ou an impo an ac ha had gone unpe -
cei ed and can b ing new ligh o hese conclusions: An a c ic summe du a ion is
highly co ela ed wi h I65 insola ion. In addi ion, i is plausible ha he An a c ic
clima e emains in nea adia i e equilib ium wi h local hea accumula ion, which
is con olled by summe ime du a ion. In con as , no he n changes a e media ed
h ough he esponse o he no he n ice shee s, which a e much mo e sensi i e o
insola ion a he sols ice (I65) han o summe ime du a ion. Thus, SO empe a u e
could be a c ucial a iable ha has in luence on he sou he n sea ice o he densi y
o deep wa e , o bo h; bu his sou he n empe a u e does no need o be con olled
ia eleconnec ion wi h he no h, i may espond o some egional as onomical
o cing diffe en o I60.
96
ON THE PHYSICAL MECHANISMS BEHIND GLACIAL-INTERGLACIAL DYNAMICS
he wes e lies bel (Fische e al. [2010]). Howe e , obse a ions o wind and ain
pa e ns in HS1 and YDS sugges ha he In e - opical Con e gence Zone (ITCZ)
and ade winds shi ed sou hwa d (Pe e son e al. [2000]; Leduc e al. [2009]; Saikku
e al. [2009]; Wang e al. [2004]; G i hs e al. [2009]). Two wa ming pulses in he
SH coincided wi h HS1 and YDS, espec i ely, sugges ing a no h-sou h connec ion
mechanism such as he bipola seesaw sugges ed by B oecke [1998]. The 231Pa/230Th
a ios om co e GGC5 o he Be muda Rise (McManus e al. [2004]) e lec a s ong
educ ion in A lan ic o e u ning ci cula ion du ing HS1 and a mode a e educ ion
du ing YDS. In con as , biogenic opal lux in he SO, in e p e ed as a p oxy o
changes in upwelling sou h o he An a c ic Pola F on (Ande son e al. [2009])
shows an inc ease in upwelling du ing HS1, a dec ease du ing he Bolling-Alle od
pe iod and a u he inc ease in o e u ning du ing YDS. This sugges s ha he shu -
down in he A lan ic MOC (AMOC) emo ed a sou ce o dense wa e o he ocean
in e io and ha his “densi y acuum” (B oecke [1998]) p ecipi a ed an inc ease in
AABW o ma ion o ill ha acuum (Sigman e al. [2010]).
These pieces o e idence ha e been in e p e ed simila ly by diffe en au ho s (see
Cheng e al. [2009]; Den on e al. [2010]; Shakun e al. [2012]). The sequence o
e en s would s a wi h a ising bo eal summe insola ion con olled mainly by in-
clina ion and p ecession cycles (Huybe s [2011]). This insola ion p oduces a g adual
no he n wa ming in he p esence o a massi e, isos a ically dep essed NH ice shee
which has g ow h g adually o 80-100 ky . In hese condi ions, an ice ins abili y
mechanism makes he Lau en ide, G eenland and Eu opean shee s abla e and col-
lapse locally, p oducing a s ong mel wa e pulse in o he No h A lan ic, which in-
i ia es a Hein ich e en . This slows he AMOC and lowe s he sou h-no h A lan ic
hea lux, wi h s adial condi ions in he no h and sou he n wa ming and sou hwa d
mo emen o clima ic zones in he SH. The abo e h ee s udies ag ee ha his se-
quence is he deglacial igge .
A numbe o mechanisms a e po en ially able o connec his igge wi h an in-
c eased elease o CO2 in he SO and subsequen a mosphe ic CO2 ise:
(i) Sou hwa d mo emen o clima ic zones could include a sou h-
wa d shi in he wes e lies, esul ing in enhanced wind-d i en

Chap e 2
97
upwelling on he An a c ic di e gence (Ma chi o e al. [2007]).
This would p omo e en ila ion o CDW and he obse ed p o-
duc i i y peak (Ande son e al. [2009]), would e ode he An -
a c ic salini y-d i en s a i ica ion and would hus acili a e he
o ma ion o deep wa e in An a c ica (Toggweile e al. [2006]).
The impo ance o his mechanism dese e u he in es iga-
ion, as ecen s udies ha e shown an oscilla o y beha io o
AMOC be ween weak and s ong s a es due o a ia ions on
he densi y g adien in he A lan ic Ocean, de i ed om a ia-
ions o CO2 and s eng h o SO winds (Bande as e al. [2015]).
(ii) Wa ming o he SO could mel sea ice, leading o an inc ease
in en ila ion as a esul o enhanced esidual no hwa d ans-
po and also o degassing e ec s. As sugges ed by Fische
e al. [2010], he inc ease and dec ease in sea ice co e age
could ha e modula ed he annual ne hea gain o he SO su -
ace, and he educed buoyancy gain would ha e weakened he
MOC du ing glacial pe iods h ough i s con ol o he esidual
ci cula ion. This would ha e limi ed he upwelling o CO2-en-
iched deep wa e , ac ing in he same di ec ion as he capping
effec ha win e ice ex en is assumed o ha e on deep-wa e
ou gassing o CO2 (S ephens and Keeling [2000]).
(iii) Wa ming associa ed wi h sou he ly shi s could educe Pa-
agonian glacia ion, lowe ing he lux o dus and i on om
Pa agonia o he SO. This would educe he iciency o he
biological pump (Ma in and Fi zwa e [1988]).
(i ) Inc eased CO2 elease om An a c ic di e gence would de-
c ease he concen a ion o dissol ed ino ganic ca bon (DIC)
and hus inc ease he ca bona e ion concen a ion in he deep
ocean. This would inc ease he bu ial a e o calcium ca -
bona e, o cing a dec ease in whole-ocean alkalini y, dec easing
solubili y and pu ing an addi ional ac ion o CO2 in o he
98
ON THE PHYSICAL MECHANISMS BEHIND GLACIAL-INTERGLACIAL DYNAMICS
a mosphe e. Acco ding o Sigman e al. [2010], his ca bona e
compensa ion in coope a ion wi h swi ch on/off o deep wa e
o ma ion o a degassing e ec is able o accoun o 40 ppm
o pCO2 diffe ence be ween in e glacial and glacial condi ions.
I is p obable ha nei he An a c ic o e u ning no gas ex-
change we e comple ely shu o /on. Howe e , mo e comple e
nu ien consump ion and lowe expo o o ganic ma e ha e
been obse ed in he An a c ic du ing glacial pe iods, sugges -
ing a educed supply o nu ien s and wa e om he deep o
he su ace ocean (Sigman e al. [2010]). Thus, o e u ning de-
c eased in he glacial SO while p oduc i i y did no dec ease
as much as he nu ien supply om below did, making pCO2
dec ease as a esul . This mechanism would ha e comple-
men ed he p e ious ones, gene a ing he a mosphe ic pCO2
diffe ence o 40 ppm.
( ) In he sub-An a c ic zone, he sinking lux o o ganic ma -
e was g ea e in glacial pe iods han oday, wi h some e i-
dence o mo e comple e nu ien consump ion (Sigman e al.
[2010]). The effec o his zone on alkalini y and a ailabili y
o nu ien s a low-la i udes is g ea , eaching 40 ppm o he
pCO2 glacial-in e glacial diffe ence (Hain e al. [2010]).
As obse ed by Sigman e al. [2010] he declines in a mosphe ic pCO2 o hei minima
du ing peak ice ages end o occu o e ens o housands o yea s and/o in s eps. This
may esul om he p og essi e ac i a ion o diffe en pCO2 educing p ocesses (Hain
e al. [2010]). In pa icula , he An a c ic cooling ea ly in he las ice age, 115 ky BP,
has been in e p e ed as educed An a c ic o e u ning o inc eased sea-ice supp ession
o gas exchange (Peacock e al. [2006]) wi h an addi ional ac ion o CO2 up ake
due o ocean cooling. In con as , he second majo decline in a mosphe ic pCO2,
70 ky BP, coincides wi h a majo dus lux inc ease in An a c ic and sub-An a c ic
zones (Ma ínez-Ga cía e al. [2009]), which may ha e caused i on e iliza ion o he
SO (Wa son e al. [2000]). Addi ionally, his pe iod may ha e suffe ed he sha pes
Chap e 2
99
ansi ion om NADW o GNAIW o ma ion (Hodell e al. [2003]), which may ha e
imp o ed he abili y o he deep SO o lowe a mosphe ic pCO2 (Sigman e al. [2010]).
2.7. DISCUSSION
To wha ex en a e PP04-de i ed models consis en wi h he obse a ional dy-
namics summa ized in las sec ion? Fi s , he sequence o ac i a ion o p ocesses
ha elease CO2 du ing he e mina ions, sugges ed by he igge ing sequence
men ioned abo e, is e y diffe en o he sequence o s eps ha educe CO2 du -
ing glacial imes, and his diffe ence gene a es he ypical saw- oo h shape o he
CO2 glacial-in e glacial oscilla ions. The lack o a comple e model o he biological
and ca bona e pumps in he PP04-de i ed models makes i impossible o p ecisely
ep oduce he se ies o e en s ha cha ac e ize glacial pe iods, and using diffe en
elaxa ion imes o momen s o inc easing and dec easing CO2 is he easies way o
ake in o accoun his asymme y and o ep oduce he double slope cycle. When
a single elaxa ion ime τC is used (τC2 = 0) in he LS calib a ion, he co ela ion
ob ained o V dec eases by less han 1% (0.85 ins ead o 0.86) bu he co ela ion
o C dec eases by 8% (0.68 ins ead o 0.74) because o he a mo e ab up ups and
downs o he simula ed C du ing he glacial pe iods. The 3τ model is much less sen-
si i e o his τC2 pa ame e , and main ains a e y high co ela ion when i is elim-
ina ed (0.88 and 0.79, o V and C, espec i ely, in he 3τ model). We hus ob ain
a model wi h almos he same co ela ions as 4τ and wi h one pa ame e less han
PP04. The good pe o mance in ela ion o PP04 mus be a ibu ed only o he
in oduc ion o a di e en ime o accumula ion (τV) and o abla ion o ice (τV2).
Though he glacial decay is only oughly simula ed by PP04-de i ed models, he
p ecise i ing o pace and in ensi y o in e glacial CO2 ha hese models gene a es
is su icien o ob ain a good ma ch o he ice olume e olu ion.
The e m cI60 has been emo ed om Equa ion 2.6 in 3τ model as one conclusion o
his s udy is ha PP04-de i ed models a e no sensi i e o I60 sou he n insola ion. In
hese models, C seems o be a much be e p oxy o An a c ic empe a u e han I60
100
ON THE PHYSICAL MECHANISMS BEHIND GLACIAL-INTERGLACIAL DYNAMICS
is. This indi ec ly ein o ces he claim o Huybe s and Den on [2008] ha An a c ic
empe a u e is no sensi i e o insola ion du ing a pa icula summe day and ha i
could be sensi i e o ano he kind o a iable, such as aus al summe du a ion.
The s a i ica ion unc ion, F, is a c ucial a iable o PP04-de i ed models, which
mus be in e p e ed as s a i ica ion p oduced by densi y diffe ence be ween AABW
and NADW wi h a possible mino con ibu ion om s a i ica ion p oduced by den-
si y diffe ence be ween NADW and su ace wa e . F con ols he upwelling o CO2
in a nonlinea way: an ab up oceanic elease o CO2 simila o a ec angula pulse
o 10 o 20 ky , e y sensi i e o F, is necessa y in hese models o gene a e he e -
mina ions. Mixing a es may dec ease as densi y diffe ences inc ease (Wa son and
Ga aba o [2006]) i we use he in e nal wa e pa ame e iza ion o he e ical mixing
coefficien sugges ed by Ga ge [1984]. I he densi y diffe ence be ween AABW and
NADW is small, he blending p ocess o he wo wa e masses may be mo e efficien
and a la ge ac ion o he AABW wells up wi h he mixed Lowe Ci cumpola Deep
Wa e (LCDW) in o he wo loops o he me idional o e u ning ci cula ion (NMOC
and SMOC). Bou es e al. [2012] ha e shown he plausibili y o his s a i ica ion-de-
penden mechanism wi h a model o in e media e complexi y.
The changes in ocean ci cula ion associa ed wi h wes e lies, deep ocean s a i ica-
ion, expansion o sea ice, a ia ion o AMOC e iciency, oge he wi h a ia ions
o biological up ake and ca bona e compensa ion seems o be he main d i e s o
he CO2 a mosphe ic diffe ence be ween glacial and in e glacial s a es. Fe a i e al.
[2014] has ecen ly p oposed a new mechanism able o link An a c ica sea ice expan-
sion wi h he empe a u e d op, he ea angemen o deep wa e masses and he
change o ci cula ion in LGM, p o ing ha hey a e no independen mechanisms,
bu eedbacks. The p ecise mechanism is lead by he isopycnals o he SO, which
con ol he upwelling and mixing o AABW and NADW, sepa a ing he NMOC
and SMOC cells. The slope and posi ion o his isopycnal change acco ding wi h he
expansion o summe sea ice, mixing ac i ely AABW and NADW in he in e glacial
pe iod, bu con ining he mixing-d i en upwelling o abyssal wa e s a he LGM,
closing he SMOC and p e en ing he mixing o NADW and AABW. As a esul ,
AABW illed oceans basins up o 2 km dep h, ins ead o he p esen 4 km dep h.
Chap e 2
101
This closed abyssal o e u ning cell would ac as a ocean s o age o ca bon. This
app oach, conside s he expansion o quasi-pe manen sea ice in he SH as he d i e
o he e mina ion. This sugges ha sea ice ex en may ha e an effec on he CO2
elease in ensi y which wo ks in he same di ec ion han s a i ica ion du ing glacial
and in e glacial pe iods; he e o e, we canno disca d a possible pa allel con ibu ion
o sea ice a ea in he in ensi y o he CO2 elease.
PP04-de i ed models need a quick esponse o egional SO empe a u e ( ep esen -
ed by C in LS and by –V in 3τ) o I65 ising, as quick as τV2 = 3 ky in 3τ and τC
= 3 ky in LS. The eal clima e sys em could p oduce a quick esponse o his kind
du ing deglacia ions by he bipola seesaw mechanism (B oecke [1998]).
A is ano he c ucial a iable o his kind o models and ollows he e olu ion o V
and C wi h a delay o abou 5-7 ky . This dependence shows ha A is sensi i e o
An a c ic empe a u e, and he delay sugges s ha i ep esen s he ex en o ice
shee on An a c ic shel es and no on SO sea ice ex en . Polynyas a e he places
whe e mos o he b ine is p oduced and a e loca ed be ween he ice shee edges and
sea ice. Thus, he posi ion o ice- shee bounda ies is c ucial o he a e and densi y
o he deep wa e o med. Acco ding o Ande son e al. [2002], he An a c ic ice-
shee edge was close o he con inen al slope in he Las Glacial Maximum (LGM)
and i s a ed o e ea on o he con inen some housand yea s a e he LGM
on mos o he sho eline. This delay seems o be cohe en wi h he delay ob ained
o he A a iable in ou calib a ions. The e o e, PP04-de i ed models ein o ce an
in e p e a ion o he oceanic CO2 elease du ing deglacia ions as con olled by he
deep s a i ica ion a he han by a capping e ec .
As can be obse ed in Figu e 2.21, A akes abou 80 o 100 ky o each a le el
ha makes s a i ica ion become c i ical (F < 0). The e o e, he 100-ky pe iod
(main equency o glacial cycles) mus be conside ed as in e nally gene a ed by he
clima e sys em. Mo e speci ically, i mus be in e p e ed as he cha ac e is ic ime
needed o he An a c ic ice shee o each he con inen al slope. The obse a ions
o Ande son e al. [2002] on he ad ance and e ea o ice shel es a e no in disa-
g eemen wi h his ime scale, bu addi ional obse a ions a e needed o con as
his p edic ion o PP04-de i ed models.

102
ON THE PHYSICAL MECHANISMS BEHIND GLACIAL-INTERGLACIAL DYNAMICS
Acco ding o Tzipe man e al. [2006], many o he models ha bes i glacial oscil-
la ions ha e a “phase locking” be ween a Milanko i ch equency ωm and an in e nal
cyclici y ωi. F equen ly, a good i can be ob ained when he quo ien ωm/ωi is a
a ional numbe . Howe e , his good i does no imply ha he mechanisms ep-
esen ed by he model a e he co ec ones. Phase locking is a necessa y condi ion
bu no a su icien one. Howe e , Tzipe man e al. [2006] conclude ha “ he ac ual
glacial cycles may also be simila ly phase locked o he Milanko i ch o cing”. Tha
is, his p ope y is p obably also one o he clima e sys em i sel . I his is so, show-
ing nonlinea phase locking is a good ‒ hough no su icien ‒ ea u e o a co ec
paleoclima e model.
Wha should be su icien ? The e is no clea ma hema ical c i e ion o deciding.
Howe e , i he igh mechanisms we e o be inco po a ed in a model wi h nonlinea
locking, we would expec he model o display he ollowing ea u es:
(i) I s p edic ions should ha e high co ela ions wi h he expe i-
men al ime se ies.
(ii) I should ep oduce he ups and downs o he expe imen al
ime se ies in he scales be ween housand o million yea s
be e han o he models o simila complexi y.
(iii) I should gene a e ime se ies wi h equency composi ion and
embedding dynamics simila o he obse a ional ones.
Howe e , ul ima ely, only he expe imen al con i ma ion ha he p oposed mecha-
nisms a e eally ac ing would be a de ini i e sufficien condi ion.
The isual inspec ion o Figu e 2.1 as well as he good co ela ions ob ained show
ha ou models o e g ea imp o emen s in c i e ia (i) and (ii) in compa ison wi h
o he simple models s udied in he li e a u e. The wa ele analysis and phase space
po ai s show ha he models ha we ha e discussed he e sa is y he hi d c i e ia,
a leas o a iable V. On o he hand, he pace o o bi al imescale is well simula ed
Chap e 2
103
o V and C, bu in he sub-o bi al imescales he phasing o C is poo ly ma ched in
many pe iods ha include he deglacia ions (Figu e 2.9). In addi ion, he c oss- e-
cu ence analysis shows ha he sho e m cohe ence be ween simula ed and ob-
se a ional CO2 is only spo adic, indica ing ha bo h ime se ies do no ollow he
same dynamical beha io (Figu es 2.18 and 2.19). Howe e , in he deglacial pe iods
he wo ca bon se ies become dynamically close. Finally, nei he simula ed C no V
ma ch he obse a ions on imescales 1-10 ky , which a e he imescales whe e he
i s e en s o he deglacial igge ake place.
The mos impo an conclusion o his s udy is ha a de ailed dynamics o CO2
is o li le ele ance o ob ain e y good ag eemen be ween simula ed ice olume
and δ18O. All ha is needed is a nonlinea ins abili y ha is eached a e 80-100
ky and ha enables ha e en a modes sub-o bi al a ia ion may igge a sudden
elease o a mosphe ic CO2. This nonlinea ins abili y seems o be con olled by V
o C (he e in e p e ed as An a c ic empe a u e) and by A (in e p e ed as An a c ic
ice shee ) in PP04-de i ed models. Mo e speci ically, An a c ic cooling would p ess
owa d g ea e s a i ica ion because a coole egional empe a u e would p econdi-
ion he su ace laye empe a u e making easie he wa e column on he shel o
ge he ‒1.9 o ‒2°C needed o eezing and b ine o ma ion. On he o he hand, ice
shee ex en would end o block he ac i e polynyas, p oducing an opposi e e ec .
Wi h he p og ess o glacia ion he la e effec would become dominan and would
place he sys em close o ins abili y, due o he dec ease in s a i ica ion.
A p ecise simula ion o he de ailed se o e en s cons i u ing he deglacial igge
would equi e adding some model o he No h A lan ic ice shee ins abili y o ou
models as well as mo e complex ca bon models. PP04-de i ed models p oduce good
i s in spi e o hei simplici y in he modeling o oceanic CO2 elease, which is made
o depend only on s a i ica ion. The o he con ibu ions (ca bona e compensa ion,
shoaling o deepening o NADW o ma ion, change o p oduc i i y in SO and slow-
ing down o SO o e u ning) a e no included in he model, bu should be included
in u u e imp o emen s o ob ain a close ep esen a ion o all he physical p ocesses
ha a e appa en ly in ol ed.
Hein ich e en s seem o belong o a cyclic p ocess o cal ing wi h a ecu ence pe i-
104
ON THE PHYSICAL MECHANISMS BEHIND GLACIAL-INTERGLACIAL DYNAMICS
od much sho e han deglacia ions (Bond and Lo i [1995]). Howe e , acco ding o
he in e p e a ions o Cheng e al. [2009]; Den on e al. [2010]; Shakun e al. [2012],
Hein ich e en s a e able o p oduce la ge icebe g discha ges and long s adials only
when ice shee s become la ge enough. Long s adials a e needed o p oduce a s ong
CO2 oceanic elease able o s a he e mina ion acco ding o Den on e al. [2010].
Howe e , la ge oceanic CO2 emissions can be p oduced only i s a i ica ion has
weakened a he end o he glacial pe iod (Bu ke and Robinson [2012]; Ma inez-Bo i
e al. [2015]) o he lack o b ine o ma ion (Pailla d and Pa enin [2004]; Bou es e
al. [2012]) o may be weakened by he expec ed SO upwelling ac i a ion. Ul ima ely,
s a i ica ion change plausibly con ols a la ge ac ion o he oceanic CO2 elease
and is synch onous wi h o he mechanisms which con ibu e an addi ional ac ion
in wa m pe iods, such as biological and ca bona e pumps in he sub-An a c ic zone
and sea ice ex en which con ols he esidual ci cula ion (Hain e al. [2010]; Sigman
e al. [2010]) and he dep h o upwelled wa e (Fe a i e al. [2014]). In his ega d,
s a i ica ion may play he same ole as an “o de pa ame e ” (Haken [1987]) plays
in a syne ge ic nonlinea change.
IMPACT OF ANTHROPOGENIC CO2 ON THE NEXT GLACIAL CYCLE
112
The In e go e nmen al Panel on Clima e Change (IPCC) published a Special Re-
po on Emission Scena ios (SRES) ha con ains 40 scena ios o u u e ossil uel
p oduc ion o assess u u e clima e change (SRES [2000]). Fo example, he SRES
A1 se uses an a e age accumula ed ca bon emission be ween 1750 and 2100 AD o
app oxima ely 1995 G ce, and he A1F1 ex eme scena io implici ly assumes 4100
G ce o URR, acco ding o he Hubbe linea iza ion done by Be g and Boland [2013].
Howe e , hese scena ios a e based on simple ex apola ions o p esen emission a es
wi h no conside a ion on ac ual URR alues. Höök e al. [2010] and Be g and Boland
[2013] show ha his epo is based on un easonably op imis ic expec a ions abou
u u e ossil uel p oduc ion which do no ake in o accoun ealis ic es ima es o
p o en and p obable ossil uels ese es. In o de o ob ain a ealis ic es ima e o
U we ha e used he analysis o bo h Lahe e e [2006] and he Ene gy Wa ch G oup
[2007] on he URR o he h ee main ossil uels (coal, oil and gas).
The Ene gy Wa ch G oup [2007] es ima es 479 G o bi uminous coal and an h a-
ci e, 272 G o sub-bi uminous coal, and 158 G o ligni e. Assuming 92%, 40%
and 66% o ca bon espec i ely o hese h ee mine als, we ob ain app oxima ely
558 G ce and he peak o coal-de i ed CO2 emissions on yea 2042 AD. The analysis
o Lahe e e [2006] gi es an es ima e o 3000 Gb o all kind o oils (Gb = Gigaba -
el, one oil ba el is e y close o 159 li e s o oil), which amoun s o 353 G ce i we
use he con e sion ac o ecommended by EPA2: 0.118 ons o ca bon pe ba el.
The In e na ional Ene gy Agency (IEA [2010, 2012]) ecognizes ha con en ional
oil p oduc ion is a o e y close o a maximum since 2006 AD and ha his p o-
duc ion will ha dly inc ease much mo e, which amoun s o an implici ecogni ion
o he oil peak a i al (Lahe e e [2012a]). Rega ding gas, Lahe e e [2006] p edic s
an URR o 9100 Tc , which amoun s o abou 120-132 G o ca bon. Recen ly, La-
he e e [2012b] has upda ed he URR coal es ima e o 750 giga ons o oil equi alen
(G oe). Using a ca bon con en o 25.7 ce/TJ o a e age coal, in he uppe ange
o epo ed alues (Gassan-zade [2004]), his amoun s o 808 G ce. Gi en hese con-
side a ions, he o al ossil uels URR amoun s o app oxima ely 1300 G ce, being
his ou bes es ima e o he in eg a ed an h opogenic emission o ca bon.
2
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Chap e 3
113
Figu e 3.1 shows he his o ical da a on ossil uels emissions du ing he 1800 o
2010 AD pe iod and he co esponding i using a Lo en z unc ion (Equa ion 3.1),
ob ained se ing U = 1300 G , which co esponds o a peak emissions on yea 2037
AD. This i has been done such ha he in eg a ed ca bon emission a 2010 AD
has exac ly he same a ea han he his o ical ime-se ies un il 2010 AD (dashed
lines in Figu es 3.1 and 3.2). This modi ied Lo en z unc ion is he one we use o
p oduce he ull ime se ies o he accumula ed an h opogenic emission o ossil
uels (Figu e 3.2). The peak o emissions occu s in yea 2037 AD, coinciden wi h
he in lec ion poin in he accumula ed cu e, and ends o ze o nea yea 2324 AD,
when
he a
ccu
m
u
l
a
ed
emissions each
h
ei
m
a
x
i
m
u
m
a
l
u
es
.
1800 1850 1900 1950 2000
0
2
4
6
8
10
Time !Yea AD"
G #y
His o ical da a o ossil uels emissions du ing he 1800-2010 pe iod (con inuous line)
and sligh ly modi ied Lo en z unc ion such ha he a ea un il 2010 equals he accumu-
la ed his o ical emissions (dashed line).
Figu e
3.1
IMPACT OF ANTHROPOGENIC CO2 ON THE NEXT GLACIAL CYCLE
114
3.3.2. CO2 a mosphe ic esponse
The ela i ely as , a he 10 o 100 y scale, an h opogenic CO2 emission will lead
o a ise in a mosphe ic concen a ion bu i will also p oduce an inc eased s o age
a e in o he ocean, g ound ege a ion, ees, de i us and soil (Kheshgi and Jain
[2003]). Ou models do no include his ype o ela i ely apid eedback clima ic
mechanisms. To ake hem in o accoun we equi e a ans e unc ion o accumu-
la ed ca bon emissions (Ac, in G ) o a mosphe ic CO2 concen a ion (pCO2, in
ppm ). Wi h his objec i e we u n o he global compa men al model ISAM (Jain
e al. [1994]; Kheshgi and Jain [2003]), which may be in e ac i ely un online3.
1800 1900 2000 2100 2200 2300
0
2
4
6
8
10
12
1800 1900 2000 2100 2200 2300
0
2
4
6
8
10
12
Time !Yea AD"
G #y
10
2
G
Modi ied Lo en z unc ion illus a ing he an h opogenic ca bon elease pe yea (dashed
line) and he accumula ed ca bon elease (solid line).
Figu e
3.2
Chap e 3
115
In his model, he a es o ans e om he a mosphe e o he abo e i e com-
pa men s ha e been calib a ed o ma ch he a e age p ojec ions (by six dynamic
global- ege a ion models and 10 coupled ocean-a mosphe e models) o CO2 up ake
be ween yea s 2000 and 2200 AD.
In o de o ob ain he Ac o pCO2 ans e unc ion, we use he ISAM model wi h
he A1T scena io pa ame e s, as p oposed by he IPCC (SRES [2000]). We un he
ISAM model o alues o URR be ween 0 and 3000 G , a inc easing URR in e als
o 100 G , eplacing he ossil uels inpu o A1T scena io wi h he modi ied Lo en z
unc ion. Fo each URR alue we ob ain he co esponding a mosphe ic esponse;
hese inal esponse poin s a e hen linea ly in e pola ed in o de o ha e a con in-
uous cu e (Figu e 3.3). The co esponding ans e unc ion should be adequa e
o model he sho - e m (a decadal and cen u y ime scales) esponse in he ange
om 0 o 3000 G ; in pa icula , i may be used o model he esul s wi h ou URR
bes es ima e, U = 1300 G (Figu es 3.1 and 3.2). No e ha con ibu ions om
u u e de o es a ion a e conside ed by he ISAM model, he e o e hey a e e lec ed
in he ans e unc ion o Figu e 3.3.
We may now inally es ima e he ime e olu ion o he an h opogenic a mosphe ic
CO2 pulse. The his o ical da a o a mosphe ic concen a ions is used o he pe iod
be ween 1750 and 2010 AD. The ea e , and un il he end o he an h opogenic
emission in yea 2324 AD, he a mosphe ic CO2 is es ima ed using he p edic ed ac-
cumula ed emissions om he modi ied Lo en z unc ion (Figu es 3.1 and 3.2) and
he ans e unc ion as deduced om he ISAM model (Figu e 3.3). In his way he
accumula ed ca bon emission is 361 G in yea 2010 AD, 642 G in yea 2037 AD,
and eaches a maximum alue o 1300 G by yea 2324 AD, and he a mosphe ic
concen a ions a hose da es can be ob ained om he co esponding o dina es in
Figu e 3.3.
3h p://clima e.a mos.uiuc.edu/isam2
IMPACT OF ANTHROPOGENIC CO2 ON THE NEXT GLACIAL CYCLE
116
3.4. PROJECTIONS FOR THE NEXT GLACIAL CYCLE
Be o e making a p ojec ion o he nex 300 ky we mus ake in o conside a ion
i he e a e o he plausible mechanisms which a e no inco po a ed in ou mod-
els. In his wo k we conside wo po en ial con ibu ions: he e y well es ablished
wea he ing compensa ion mechanism and he less known emission o me hane om
cla h a es. We ha e chosen hese wo mechanisms no only because hey a e indeed
ele an e ec s bu also because hey a e illus a i e o he so o modi ica ions
ha eedback mechanisms may cause in he Pleis ocene dynamics.
Acco ding o A che and Ganopolski [2005], he silica e wea he ing cycle will cause
ha 7% o he an h opogenic CO2 will s ill emain in he a mosphe e 100 ky a e
an an h opogenic pe u ba ion. This implies ha , a e ending he an h opogenic
emission, abou 9% o he CO2 concen a ion will decay wi h a ime cons an o ap-
p oxima ely 400 ky . We ake in o conside a ion his non-linea bu e (ocean ca bon
chemis y) e ec by simply assuming ha 91% o he an h opogenic pe u ba ion will
ollow he dynamics gi en by Equa ions 2.1 o 2.6, while he emaining 9% will ha e
a long- e m decay, wi h a 400 ky ime decay cons an . A simila app oach was used
by Pailla d [2006].
A che e al. [2009a] ha e s udied he g eenhouse e ec expec ed o occu in he
nex 10 ky as he ising empe a u es lead o he emission o a ac ion o he
me hane cla h a es om con inen s and shel es. Thei p ojec ion has a la ge unce -
ain y ela ed o de e mining he c i ical ac ion o me hane bubbles ha a e able
o each he ocean su ace. Taking his ac ion as 2.5% and conside ing a scena io
o 1000 Pg o ca bon emission, su cien ly close o he one used in ou models, he
p edic s an inc ease o 0.4 – 0.5°C ela ed o he escape o me hane du ing some
10 ky . Taking a clima e sensi i i y o 3°C o a doubling o CO2, he e ec o he
p edic ed me hane emission, Cme , is equi alen o 40.5 ppm o CO2 concen-
a ion du ing a pe iod o abou 10 ky . We ha e in oduced his con ibu ion in
ou p ojec ions by assuming ha Cme (addi ional equi alen CO2 concen a ion
de i ed om me hane) g ows linea ly om 0 o 40.5 ppm be ween 0 and 2 ky ,
emains cons an be ween 2 and 10 ky , and dec eases linea ly om 40.5 ppm o
0 be ween 10 and 20 ky .
Chap e 3
117
Following hese conside a ions, we a e now eady o s a he p ojec ion. The model
equa ions a e in eg a ed o wa d, wi h he an h opogenic ca bon emission scena io
in Figu e 3.2 as he ini ial condi ion and p ope ly o ced by he insola ion a 65°N
(Be ge [1978b]), o p edic he e olu ion o he in e glacial-glacial ansi ions du ing
he nex 300 ky . In o de o ob ain a CO2 p ojec ion in dimensional uni s (ppm ),
a he han he model non-dimensional uni s, a concen a ion o 280 ppm is assumed
o yea 1750 AD and a dimensional scale is ob ained by conside ing he ange o
a mosphe ic concen a ions in he Taylo Dome Ice Co e be ween he las glacial
maximum (181 ppm ) and yea 1750 AD (280 ppm ) (Inde muhle e al. [2000]).
Simila ly, in o de o ans o m om model ice olume o ben hic δ18O (pe mil),
he con e sion used is
V = (Vno malized + s d(δ18O)) + mean(δ18O),3.2
whe e Vno malized is he modeled ice olume no malized wi h i s own s anda d
de ia ion and mean. All o he a iables emains non-dimensional as he e is no
dimensional scale a ailable o hei ans o ma ion.
The p ojec ion o he nex 300 ky is compu ed o bo h unpe u bed and pe -
u bed condi ions using ei he he 3τ o LS models; Figu e 3.4 illus a es, as an
example, he p edic ed e olu ion o a mosphe ic CO2 ollowing he an h opogenic
CO2 pulse. Bo h models display an exponen ial decay in a mosphe ic CO2, wi h
cha ac e is ic ime decay somewha longe o 3τ han o LS. The wo models,
despi e ha ing di e en pa ame e iza ions o ocean s a i ica ion and ice shee e o-
lu ion, do p oduce a e y simila ou come o he ollowing glacial cycle. He ea e ,
we will ocus on he u u e p ojec ion ob ained wi h he 3τ model, a compa ison o
bo h models shows analogous esul s.
Figu e 3.5 shows he CO2 e olu ion p edic ed by model 3τ ollowing he an h opo-
genic sho - e m pulse, a e adding he long- e m wea he ing compensa ion mech-
anism and u he including me hane emissions om cla h a es. I may be obse ed
ha in all cases he iming o he nex in e glacials does no subs an ially change.
The main e ec is caused by he wea he ing compensa ion which causes somewha

IMPACT OF ANTHROPOGENIC CO2 ON THE NEXT GLACIAL CYCLE
118
highe concen a ions and a sligh ly longe du a ion o he p esen in e glacial. The
e ec o he me hane emission is much educed excep o a ela i ely sho pla eau
in CO2 alues a e ending he an h opogenic pulse.
The in oduc ion o he an h opogenic CO2 pulse clea ly pe u bs he na u al cycle o
all ou model a iables du ing he o hcoming 300 ky (Figu e 3.6). No ice ha he
alues o a iables I65, V, A and F in Figu e 3.6 a e ela i e a ia ions, only C has
been ansla ed o physical uni s (ppm ); u he no e ha a minimum bound o V
has been se o an addi ional 10 me e s sea le el ise wi h espec o yea ze o, which
co esponds o a mel ing abou 30% highe han he G eenland ice shee (IPCC [2001])
and, o his eason, V doesn’ ake alues unde ‒ 0.93/ ‒1.94 o 3τ/LS models.
0500 1000 1500 2000 2500 3000
0
100
200
300
400
500
600
700
Ac !G "
pCO
2
!ppm "
Rela ion be ween he accumula ed an h opogenic ca bon emissions (Ac) and he a -
mosphe ic CO2 concen a ion (pCO2). The da a is ob ained om an agg ega e o A1T
scena ios (SRES [2000]) as p edic ed by model ISAM (Jain e al. [1994]), a ailable a
h p://clima e.a mos.uiuc.edu/isam2/, o e e y possible alue o URR.
Figu e
3.3
Chap e 3
119
A compa ison be ween he di e en panels in Figu e 3.6 illus a es he in luence o
I65 on he sequence o e en s, as discussed wi h de ail in Chap e 2 and Ga cía-Ol-
i a es and He e o [2013]. In bo h he pe u bed and unpe u bed simula ions,
a V minimum (maximum) is p oduced a ew housand yea s a e e e y posi i e
(nega i e) peak in I65. Du ing a glacia ion, he long- e m inc ease in V p oduces
an inc ease in A which ends o dec ease he F alue (due o he in e se depen-
dence o F on A, Equa ion 2.6). The ansi o he in e glacial elies hea ily on
F u ning nega i e ( h ough he Hea iside unc ion in Equa ion 2.5): F eaches
nega i e alues only du ing pe iods o ad anced glacia ion because a hese imes
A is also la ge. In hese si ua ions, peaks in I65 lead o ela i e minima in V and
a subsequen dec ease in bo h A and F, which may induce nega i e F alues ha
igge he oceanic CO2 emission. The condi ion o gene a e a deglacia ion is,
hus, he coincidence o a maximum o I65 in a pe iod when V is la ge, as we ha e
seen p e iously in Chap e 2.
The model p edic s a maximum CO2 concen a ion o 519 ppm in 2300 AD
ollowed by an exponen ial decay. To de ine in e glacial condi ions we use wo
simple c i e ia (C uci ix [2011]): CO2 concen a ions abo e 250 ppm and ben hic
δ18O below 3.8 pe mil; he CO2 c i e ion has been used also by IPCC [2007].
Acco ding o he CO2 c i e ion he cu en in e glacial would end 40 ky AP (35
ky AP) o 3τ (LS), and acco ding o he δ18O c i e ion i would end 46 ky AP
(40 ky AP) o 3τ (LS). The e o e, he leng h o his in e glacial would be 54 ‒
47 ky (acco ding o he CO2 alues) and 58 ‒ 50 ky (acco ding o he δ18O)
as p edic ed by he 3τ and LS models. This in e glacial would be ollowed by a
glacial pe iod las ing un il 113 ky AP.
The model also p edic s he disappea ance o he nex in e glacial unpe u bed
age, which should s a a 64 ky AP. This is due o he abno mally educed ice
olume and ice shee a ea p edic ed o he p esen in e glacial, which akes long
o eco e . The ice shee , A, ollows he e olu ion o V wi h a delay o abou 7
ky (Equa ion 2.2 and τA alue in Table 1.1). The F unc ion, howe e , u ns
nega i e only when A is la ge enough (due o he high alue b = 1.205). A 64
ky AP, V dec eases (C ises) in esponse o he posi i e o cing o he I65 insola-
ion bu A is no high enough o F o become nega i e. I V and A we e la ge,
IMPACT OF ANTHROPOGENIC CO2 ON THE NEXT GLACIAL CYCLE
120
as occu s in he unpe u bed case, F would be close o he h eshold alue (F = 0)
and he dec ease o V ( ise o C ) a ha ime would d i e F o nega i e alues,
igge ing he oceanic pulse. This delayed eco e y o he ice olume V (and A) is
caused by he an h opogenic pulse in oduced in he model, which p oduces 20 ky
o abno mally high g eenhouse e ec . Unde hese condi ions V needs a longe ime
o each he glacial maximum and he nex glacial cycle mo es 44 ky o wa d in
ime. A 235 ky AP pe u bed and unpe u bed in e glacials coincide again and
he u he e olu ion o all a iables emain in phase, sugges ing he eco e y o he
na u al pe iodici y.
0 50 100 150 200 250 300
200
250
300
350
400
450
500
Time (ky )
C (ppm )
P edic ion o he dimensional a mosphe ic CO2 concen a ion o he nex 300 ky
using, as an ini ial condi ion, he an h opogenic ca bon elease scena io shown in Fig-
u e 3.2. The blue and ed lines co espond o he p ojec ion by he 3τ and LS models,
espec i ely.
Figu e
3.4
Chap e 3
121
We ha e also explo ed how sensi i e a e he model esul s, in pa icula he s a -
ing ime o he nex in e glacial cycle, o di e en URR alues. One main esul
is ha a p og essi e inc ease in he an h opogenic pulse leads o smalle , wi h
lowe CO2 alues, in e glacial cycles. A qui e ele an esul is ha he iming o
he nex in e glacial will change disc e ely as U exceeds di e en h eshold alues
(inse in Figu e 3.5). Fo example U = 475 G ep esen s he URR h eshold be-
yond which he nex in e glacial will expe ience a delay o 44 ky ; he ollowing
h esholds will occu a 1725 and 2100 G , wi h co esponding addi ional delays
o 42 and 72 ky (inse in Figu e 3.5).
0 50 100 150 200 250 300
200
250
300
350
400
450
500
550
Time (ky )
C (ppm )
0 500 1000 1500 2000 2500 3000
50
100
150
200
250
300
To al emission (G C)
Time o nex in e glacial (ky )
Dimensional a mosphe ic CO2 concen a ion o he nex 300 ky as p edic ed by model
3τ ollowing he an h opogenic sho - e m pulse ( ed line), a e adding he long- e m
wea he ing compensa ion (g een line), and a e u he inco po a ing he me hane
emissions om cla h a es (blue line). The inse shows how he s a o he nex in e -
glacial shi s depending on he o al emissions o ca bon.
Figu e
3.5

Chap e 4
INSIGHT TO MARINE
ISOTOPIC STAGE 13
USING LATE PLEISTOCENE
RELAXATION MODELS
AND SEA LEVEL STACK
Ch
a
p
e
based on
he
ollowing
published
a
icl
es
:
He e o, Lisiecki and Ga cía-Oli a es [2015],
Insigh o Ma ine Iso opic S age 13 using
la e Pleis ocene elaxa ion models and sea
le el s ack. In p epa a ion.
131 .............. 4.1. INTRODUCTION
132 .............. 4.2. RELAXATION MODELS
134 .............. 4.3. NON-LINEAR ANALYSIS OF BOTH SIMULATED AND
PROXY TIME SERIES
137 .............. 4.4. DYNAMICS OF SEA LEVEL RE-CALIBRATED MODELS
137 .............. 4.5. DISCUSSION
Con en s
131
4.1. INTRODUCTION
As we ha e seen in p e ious chap e , glacial-in e glacial oscilla ions o la e
Pleis ocene clima e (las 800 ky ) e eal a cha ac e is ic 100-ky ice-age cy-
cle, assumed o be mainly de i ed om o bi al pa ame e s and om in e nal
eedbacks o he clima e sys em (Hays e al. [1976]; A che e al. [2000];
Pailla d [2010, 2015]).
Following Pailla d and Pa enin [2004], se e al elaxa ion models, based on
simple pa ame e iza ions o deep ocean s a i ica ion, ha e been de eloped
(Ga cía-Oli a es and He e o [2012, 2013]). Two o hese models, 3τ and
LS, ha e good skills ep oducing he condi ions du ing he las eigh glacial
cycles (Chap e 1). The 100-ky glacial-in e glacial pe iodici y is in e nally
gene a ed h ough h ee coupled a iables: a mosphe ic CO2 concen a ion,
global ice olume and he ex ension o he An a c ic ice shel ( o a de ailed
dynamics, see Chap e 2)
Chap e 4
INSIGHT TO MARINE ISOTOPIC STAGE 13
USING LATE PLEISTOCENE RELAXATION
MODELS AND SEA LEVEL STACK
I seems o me ha he na u al wo ld is he g ea es sou ce o ex-
ci emen ; he g ea es sou ce o isual beau y; he g ea es sou ce o
in ellec ual in e es . I is he g ea es sou ce o so much in li e ha
makes li e wo h li ing.
- Da id A enbo ough
131 .............. 4.1. INTRODUCTION
132 .............. 4.2. RELAXATION MODELS
134 .............. 4.3. NON-LINEAR ANALYSIS OF BOTH SIMULATED AND
PROXY TIME SERIES
137 .............. 4.4. DYNAMICS OF SEA LEVEL RE-CALIBRATED MODELS
137 .............. 4.5. DISCUSSION
INSIGHT TO MARINE ISOTOPIC STAGE 13 USING LATE PLEISTOCENE RELAXATION MODELS AND SEA LEVEL STACK
132
δ18O da a om ben hic o amini e a (Lisiecki and Raymo [2005]) may be used as
a p oxy o ice olume (Shackle on e al. [2000]; Shakun e al. [2015]; Waelb oeck
e al. [2002]), al hough he O18/O16 a io is known o depend on bo h he iso opic
composi ion and empe a u e o he wa e whe e o amini e a de elop. Waelb oeck
e al. [2002] ound ha he ela ionship be ween δ18O and ice olume is no linea ,
causing some unce ain ies in he ice olume a ia ions.
Global ice olume changes de i ed om g ow h and e ea o con inen al ice shee s
may be associa ed wi h sea le el a ia ions in he glacial-in e glacial cycles (Chap-
pell and Shackle on [1986]; Waelb oeck e al. [2002]; Lambeck e al. [2014]), a p oxy
wi h less unce ain ies. Some econs uc ions o sea le el om ocean sedimen co e
da a ha e been pe o med using di e en p oxies and models, each o hem limi ed
by measu emen e o , local a ia ions in salini y and empe a u e, and assump ions
pa icula o each echnique. Sp a and Lisiecki [2015] ha e compiled a wide ep e-
sen a ion o hese econs uc ions, de eloping a sea le el s ack, which ep esen s he
eus a ic sea le el eco d mo e accu a ely han each o he indi idual econs uc ions.
In his wo k, we use Sp a and Lisiecki [2015] sea le el s ack o analyze and iden-
i y di e ences be ween δ18O and sea le el as p oxy da a o he ice olume. In
nex sec ion, he models a e p esen ed and some non-linea me hods a e applied in
Sec ion 4.3 o ou line he di e ences be ween δ18O and sea le el s ack. Sec ion 4.4
shows he ecalib a ion and op imiza ion o 3τ and LS models o he new se o sea
le el da a, and he discussion o he esul s, as well as some b ie conclusions, a e
p esen ed in las sec ion.
4.2. RELAXATION MODELS
The e a e se e al di e ences be ween models 3τ and LS (Figu e 4.1, bo om panel).
One di e ence is he alue used o he e e ence An a c ic ice shee , ei he ‒C ( ep-
esen ing he in e se e ec o An a c ic empe a u e in An a c ic ice shee ex en )
o model LS o V o model 3τ. Ano he mino di e ence is he inclusion o I65 in
model LS when speci ying he e e ence a mosphe ic CO2 concen a ion alue, C .
Chap e 4
133
Howe e , he main di e ence be ween bo h models, is hei pa ame e iza ion o he
ocean s a e, wi h F = F (V, A) in model 3τ and F = F (C, A) in model LS. Bo h V
and C a e good p oxies o he Sou he n Ocean (SO) egional empe a u e and ei-
he F = F (V, A) o F = F (C, A) a e plausibly ways o model he local o ma ion
o b ines. The ac ha he dependence F = F (V, A) pe o ms somewha be e
han F = F (C, A) may be due o he non-negligible ole ha V has on s a i ica ion
h ough eleconnec ions (Ga cía-Oli a es and He e o [2013]; Gildo and Tzipe man
[2001]) o o a possible la ge e ec o sea le el han An a c ic empe a u e on b ine
o ma ion (Chap e 2).
−3
−2
−1
0
1
2
3
P oxy
−800 −700 −600 −500 −400 −300 −200 −100
0
−3
−2
−1
0
1
Time (Ky )
Models
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Top panel: p oxy δ18O eco ds om Lisiecki and Raymo [2005] (g ey line) and sea le el
s ack om Sp a and Lisiecki [2015] (g een line); numbe s ep esen Ma ine Iso opic
S age labels o he ea ly Pliocene om Lisiecki and Raymo [2005]. Bo om panel: no mal-
ized ice olume as p edic ed wi h he 3τ (blue line) and LS ( ed line models.
Figu e
4.1

INSIGHT TO MARINE ISOTOPIC STAGE 13 USING LATE PLEISTOCENE RELAXATION MODELS AND SEA LEVEL STACK
134
The bes - i co ela ions a e 0.87/0.88 (be ween δ18O and modeled V ) and 0.76/0.79
(be ween he econs uc ed a mosphe ic CO2 concen a ion and modeled C ) o
models LS/3τ. The pa ame e alues ha lead o he bes da a i o ei he model
a e shown in Table 4.1.
The wo di e en p oxies, δ18O eco ds om Lisiecki and Raymo [2005] and sea le el
s ack om Sp a and Lisiecki [2015], ha e simila dynamics bu he iming o de-
glacia ions is sligh ly shi ed in all cases, as well as dep h o glacial cycles, specially
in he i s ou cycles ( om ‒800 o ‒400 ky ) (Figu e 4.1, op). The di e ence is,
somehow, mo e e iden a ound ‒500 ky whe e he sea le el da a econs uc he
cycle wi h less a iance. A compa ison be ween sea le el s ack and ben hic δ18O has
been pe o med in Sp a and Lisiecki [2015], showing ha he ela ionship be ween
ben hic δ18O and sea le el is well-desc ibed by a linea unc ion in he i s ou cy-
cles ( om ‒800 o ‒400 ky ) and a quad a ic unc ion in he las cycles ( om ‒400
o 0 ky ).
4.3. NON-LINEAR ANALYSIS OF BOTH
SIMULATED AND PROXY TIME SERIES
Some non-linea analysis ha e been pe o med o analyze in e nal equencies o
each ime se ies. Spec al powe o obse a ional and simula ed ime se ies a e
ob ained wi h Fas Fou ie T ans o m (Figu e 4.2). The sea le el s ack has mo e
powe han any o he se ies in he 100-ky band, bu he powe in he 41-ky band
is isibly lowe han any o he ime se ies. As i seems, he sea le el s ack ends o
o e p edic he powe o long wa e equencies and o unde p edic i o sho
wa e equencies when compa ed wi h δ18O da a.
The complex c oss-wa ele ans o m o wo ime se ies can be in e p e ed as he
sha ed powe in a gi en pe iodici y band (absolu e alue) and he phase be ween
he wo se ies in ime equency space (G ins ed e al. [2004]). Ligh e pu ple colo
indica es g ea e sha ed powe in ha ime and pe iodici y band, and he a ow
angle indica es he phase be ween he obse a ional and simula ed se ies. I is in-
Chap e 4
135
e es ing o see ha in he main 100-ky band he sha ed powe is maximum (as
expec ed) and he phase is eally simila o all h ee cases (Figu e 4.3), bu he
mos in e es ing ea u es a e he blue spo s in he 41-ky band a he sea le el and
δ18O case. Appa en ly, he e a e some signi ican di e ences be ween bo h p oxies
a ound ‒500 and ‒250 ky , in e ms o obliqui y con en , which made bo h p oxies
dynamically di e en a hose imes.
0 20 40 60 80 100 120
0
500
1000
1500
2000
2500
Spec al Powe
Pe iod (Ky )
Spec al powe o no malized ice olume as p edic ed wi h 3τ model (blue), LS model
( ed), p oxy δ18O eco ds om Lisiecki and Raymo [2005] (g ey line) and Sp a and
Lisiecki [2015] sea-le el s ack (g een line).
Figu e
4.2
INSIGHT TO MARINE ISOTOPIC STAGE 13 USING LATE PLEISTOCENE RELAXATION MODELS AND SEA LEVEL STACK
136
1/64
1/32
1/16
1/8
1/4
1/2
1
2
4
8
16
32
64
Time (Ky )
F equency (Ky )
−700 −600 −500 −400 −300 −200 −100 0
0.20
0.39
0.78
1.56
3.13
6.25
12.5
25
50
100
200
1/64
1/32
1/16
1/8
1/4
1/2
1
2
4
8
16
32
64
Time (Ky )
F equency (Ky )
−700 −600 −500 −400 −300 −200 −100 0
0.20
0.39
0.78
1.56
3.13
6.25
12.5
25
50
100
200
1/64
1/32
1/16
1/8
1/4
1/2
1
2
4
8
16
32
64
Time (Ky )
F equency (Ky )
−700 −600 −500 −400 −300 −200 −100 0
0.20
0.39
0.78
1.56
3.13
6.25
12.5
25
50
100
200
C oss-wa ele ans o m be ween p oxy sea le el s ack om Sp a and Lisiecki [2015]
and δ18O eco ds om Lisiecki and Raymo [2005] ( op), simula ed V ime se ies wi h 3τ
model (middle) and LS model (bo om).
Figu e
4.3
Chap e 4
137
4.4. DYNAMICS OF SEA LEVEL RE-CALIBRATED MODELS
Conside ing he exis ing dynamic di e ences be ween he sea le el s ack and he
ben hic δ18O, a new calib a ion has been done o bo h 3τ and LS models using
he sea le el (SL) s ack om Sp a and Lisiecki [2015]. This has de i ed in new
3τ and LS models, called 3τSL and LSSL he ea e .
The compa ison be ween pa ame e alues o all models is shown in Table 4.1.
The co ela ions be ween he o iginal models and he new ecalib a e models
do no change much; o V, he co ela ion has inc eased om 0.88 (3τ) o 0.89
(3τSL), jus 1%, while he e is no change o he LS/LSSL case. Fo C, howe e ,
he co ela ion has sligh ly dec eased in bo h cases om 0.79 (3τ) o 0.77 (3τSL)
and om 0.76 (LS) o 0.73 (LSSL), a 2% and a 3% espec i ely. This change in he
co ela ion should no be impo an , as he unce ain ies in he δ18O may p o-
duce an o e in e p e a ion o he da a, meaning ha hese co ela ion di e ences
o less han 4% a e included unde he obse a ional unce ain y. Ne e heless,
he mos ema kable ea u e o hese new op imiza ions is a double peak shown
in he 3τSL ice olume a ound ‒500 ky . This double peak was no p esen in he
o iginal 3τ models, and i is no shown in any LS model, sugges ing ha he e a e
a signi ican dynamic di e ence be ween bo h p oxies and/o bo h models.
4.5. DISCUSSION
Ma ine Iso opic S age 13 (MIS-13) in e glacial occu ed app oxima ely 500,000
yea s ago (‒500 ky ). I is o pa icula in e es , as i su e ed se e e summe
monsoons simul aneously wi h inc easing ma ine oxygen iso ope and dec eas-
ing An a c ic ice co e eco ds in empe a u e compa ed o o he in e glacials
(Yin and Guo [2008]; Lang and Wolff [2011]; Mu i e al. [2012, 2013]). All o
hese anomalies indica e a wa m No he n Hemisphe e (NH) and a cool Sou he n
Hemisphe e (SH), and consequen ly a s ong asymme y o hemisphe ic clima es
du ing MIS-13 (Guo e al. [2009]).

145
I
n his inal chap e I will b ie ly e ise he esul s p esen ed in his hesis. In
Chap e 1 se e al elaxa ion models inco po a ing a wide ep esen a ion o physical
mechanisms, like he oceanic CO2 pumping o he esponse imes o ca bon and
ice olume, ha e been de eloped. The models’ pa ame e s ha e been calib a ed o
p o ide he bes i o he δ18O and CO2 expe imen al ime se ies a ailable o he
las 800 ky BP. We ha e desc ibed eigh diffe en sub-models, all de i ed om he
o iginal Pailla d and Pa enin [2004], o e alua e how hose diffe en pa ame e iza-
ions affec he da a i . Some o hose mechanisms like he biological expo a ion
p oduc ion o an exponen ial CO2 oceanic pulse does no offe new insigh s. On he
o he hand, he sub-models wi h diffe en esponse imes o accumula ion and ab-
la ion o ice, and/o emission and abso p ion o CO2, show a good quan i a i e and
quali a i e ag eemen wi h he empi ical ime se ies, especially 4τ , EP, 3τ and LS,
sugges ing ha diffe en elaxa ion imes a e a possible way o ep oduce he asym-
me y in he glacial cycles and ha he mechanisms ha hese models inco po a e
may be impo an ac o s con olling glacial-in e glacial oscilla ions.
Conclusions
We on Ea h ha e jus awakened o he g ea oceans o space and ime
om which we ha e eme ged. We a e he legacy o 15 billion yea s o
cosmic e olu ion. We ha e a choice: We can enhance li e and come o
know he uni e se ha made us, o we can squande ou 15 billion-yea
he i age in meaningless sel -des uc ion. Wha happens in he i s sec-
ond o he nex cosmic yea depends on wha we do, he e and now, wi h
ou in elligence and ou knowledge o he cosmos.
‒ Ca l Sagan, COSMOS
146
The 4τ model imp o es he PP04 co ela ions (0.59 and 0.63) o 0.79 and 0.89 using
14 pa ame e s. The EP model uses 15 pa ame e s bu he addi ional pa ame e ,
which was aimed a imp o ing he o m o he oceanic pulse unc ion, did no im-
p o e he co ela ions and can be conside ed useless. The 3τ model ob ains almos
he same co ela ions as 4τ using only 13 pa ame e s. The LS model (15 pa ame-
e s) does no imp o e he co ela ions o 4τ (which a e sligh ly highe ) bu inco -
po a es a unc ional o m o s a i ica ion F ha seems mo e consis en wi h he
mechanisms sugges ed by Pailla d and Pa enin [2004]. 3τ is, hen, he model wi h
he bes explained a iance pe pa ame e , and LS, ha ing a explained simila a -
iance, con ain pa ame e iza ions ha can be ela ed o he obse ed mechanisms.
Bo h models offe a aluable insigh in o he eal meaning o he good pe o mance
o PP04-de i ed models, and dese e u he analysis.
In Chap e 2, he dynamics and mechanisms inco po a ed in 3τ and LS models ha e
been analyzed and compa ed, o explo e he mos plausible physical in e p e a ions
o he ma hema ical exp essions. Fi s , we ha e e alua ed hei non-linea dynamics
using wa ele ans o m, c oss-wa ele ans o m, wa ele cohe ence, Fou ie analy-
sis, a ac o s and c oss- ecu ence plo s. We ha e, hen, iden i ied he mechanisms
and dynamics ha lead o good da a i and we ha e compa ed he esul s wi h he
obse a ional dynamics ha supposedly cause glacial-in e glacial oscilla ions.
We ha e demons a ed ha , in ac , elaxa ion models a e a use ul ool o analyze
and s udy he physical mechanisms o he clima e sys em, as well as o iden i y eed-
backs and he in ol ed a iables. We ha e shown ha he models a e no sensi i e
o he sou he n insola ion; a he , hey espond o a diffe en a iable, C, which
seems o be a much be e p oxy o An a c ic empe a u e. One impo an conclu-
sion i ha he de ailed dynamics o CO2 a e no impo an o ob ain a good ma ch
be ween he models and he p oxy da a. Wha is eally impo an is a nonlinea in-
s abili y, eached a e 80 - 100 ky (allowing V o each a maximum le el), allowing
any modes inc ease o he insola ion o cause a elease o a mosphe ic CO2.
Va iable F is c ucial in he models. We ha e poin ed ou ha an ab up oceanic
elease o CO2, simila o a ec angula pulse o 10 o 20 ky , is necessa y o igge
he e mina ion, and F is he non-linea con ol o his upwelling o CO2. I can be
Conclusions
147
desc ibed as a deep ocean con ol pa ame e . We belie e ha F can be physically
conside ed he s a i ica ion synch onously combined wi h he biological and ca -
bona e pumps in he sub-An a c ic zone, and wi h he sea ice ex en which con ols
he esidual ci cula ion and he dep h o upwelled wa e .
A p ecise simula ion o he de ailed se o e en s cons i u ing he deglacial igge
would equi e adding some pa ame e iza ion o he No h A lan ic ice shee ins a-
bili y o ou models as well as mo e complex ca bon dynamics. PP04-de i ed models
p oduce good esul s in spi e o hei simplici y in he modeling o oceanic CO2 e-
lease, and o ob ain a close ep esen a ion o all he physical p ocesses in ol ed in
he clima e sys em, mo e mechanisms should be included in u u e e sions.
In Chap e 3 we ha e illus a ed an applica ion o he elaxa ion models, use ul
oo o answe mo e di ec dynamic ques ions. We ha e used 3τ and LS models o
p edic he u u e e olu ion o global Ea h a iables du ing he o hcoming 300
ky , wi h and wi hou he a mosphe ic CO2 pe u ba ion caused by an h opogenic
ossil uels emissions.
The an h opogenic CO2 pulse p oduces 20 ky o abno mally high g eenhouse e ec ,
in ol ing a delay in he u u e ad ance o he ice shee o e he An a c ic shel . As
a esul , he co esponding peak o no he n insola ion, causing a e mina ion in he
unpe u bed scena io, will ha e no e ec on he s abili y o he de eloping glacia-
ion. Howe e , he ollowing insola ion peak will ake place in an app op ia e s a e
o he clima e sys em and will be sufficien o induce he new deglacia ion, mo ing,
acco dingly, he nex glacial cycle 44 ky o wa d in ime. A e h ee cycles, pe -
u bed and unpe u bed in e glacials coincide again and he u he e olu ion o all
a iables emain in phase, sugges ing he eco e y o he na u al pe iodici y. On he
o he hand, a p og essi e inc ease in he an h opogenic pulse leads o smalle , wi h
lowe CO2 alues, in e glacial cycles, and he iming o he nex in e glacial will
change disc e ely as U exceeds diffe en h eshold alues.
In Chap e 4 he models ha e been e-calib a ed using a new sea le el da a s ack
and a e used o unde s and he dynamics o a e y pa icula in e glacial e en , he
MIS-13. We ha e compa ed 3τ and LS models wi h Sp a and Lisiecki [2015] sea
148
le el s ack o analyze and iden i y diffe ences be ween δ18O (Lisiecki and Raymo
[2005]) and sea le el as p oxy da a o he ice olume.
The compa ison o he deep ocean pa ame e , F , wi h a p oxy o deep s a i ica-
ion (ben hic δ13C da a, Lisiecki [2010]) shows a misma ch in he i s cycles ( om
‒800 o ‒400 ky ), sugges ing ha F migh be ep esen ing no only he deep wa e
o ma ion bu also o he a iables, like he An a c ic ice shee . On he las cycles
( om ‒400 o 0 ky ), F seems o be in good pace wi h he s a i ica ion, al hough
sligh ly shi ed, sugges ing a ep esen a ion o a mixed s a e o he deep ocean. This
ein o ces he esul s o Chap e 2, whe e F ep esen s no only he s a i ica ion
bu also he sea ice ex en .
The appea ance (o no appea ance) o a double CO2 peak a MIS-13 in ou mod-
els, sugges ha he pa ame iza ion F = F (V, A), affec ed by bo h NH and SH
dynamics, is mo e app op ia e o simula e he clima e.
We may hen assess ha elaxa ion models can be ex emely use ul ools o cha -
ac e ize he complex clima e sys em, and help ul o examine speci ic ques ions as
he u u e e olu ion o clima e unde diffe en scena ios o an h opogenic ossil u-
els emissions. Relaxa ion models may con ibu e o b oaden he knowledge o he
mechanisms con olling he a iabili y o he la e Pleis ocene glacial cycles; ne e -
heless, hei esul s should be compa ed wi h hose ob ained om an in e media e
complexi y model, which should include conse a ion equa ions, ealis ic geome y
and long- e m global ca bon dynamics. The analysis o he p esen esul s unde
he complemen a y pe spec i e gi en by hese complex models would be a na u al
con inua ion o his hesis.
Appendix
150 ............
Appendix
A. On insola ion o cing
152 ...........
Appendix
B. Gene ic algo i hms o op imiza ion
155 ............
Appendix
C. Resumen en cas ellano

150
Appendix A
We ha e used Be ge [1978a] and Be ge and Lou e [1991] so wa e o calcula e in-
sola ion ime se ies a diffe en la i udes, co e ing a ime domain om las 800 ky
o he nex 300 ky in ime s eps o 100 yea s. We ha e used nega i e and posi i e
alues when espec i ely e e ing o imes be o e p esen (BP) and a e p esen
(AP), whe e yea ze o is aken as 1950 AD. In Chap e s 1, 2 and 4 we ha e aken
in o accoun only he pas ime (las 800 ky ) bu in Chap e 3 we ha e made some
p ojec ions, using he nex 300 ky domain.
Two se s o insola ion da a has been de eloped o he pas ime. On one hand,
insola ion ega ding one single day o he yea a one speci ied la i ude, his is
No he n Hemisphe e summe insola ion (65°N) on 21s June, named I65, and on
he o he hand, la e Aus al summe insola ion (60°S) on 21s Feb ua y, named I60
(Figu e A.1).
Fo Chap e 3, a new se o No he n Hemisphe e summe insola ion (65°N) on 21s
June has been calcula ed, his ime conside ing he whole ime domain ( om ‒800
o +300 ky ), shown in Figu e A.2.
A. ON INSOLATION FORCING
151
Appendix A
Figu e
A.1
No he n Hemisphe e summe insola ion ( op) and la e Aus al summe insola ion
(bo om) o he pas 800 ky
−800 −700 −600 −500 −400 −300 −200 −100 0
−3
−2
−1
0
1
2
3
i
65
−800 −700 −600 −500 −400 −300 −200 −100 0
−3
−2
−1
0
1
2
i
60
Time (ky )
Figu e
A.2
No he n Hemisphe e summe insola ion (65°N) om he las 800 ky o he nex
300 ky .
−800 −700 −600 −500 −400 −300 −200 −100 0 100 200 300
−3
−2
−1
0
1
2
3
i65
Time (ky )
152
To pe o m op imiza ions o ma hema ical exp essions, we ha e implemen ed a
mul i-objec i e gene ic algo i hm wi h he aim o maximize co ela ion be ween
expe imen al and modeled da a. This has been done wi h Global Op imiza ion
Toolbox om Ma lab1.
Fi s o all, we mus answe a key ques ion: wha exac ly is a gene ic algo i hm?
Acco ding o Cha bonneau [2002], gene ic algo i hms a e, undamen ally, a class o
sea ch echniques ha use simpli ied o ms o he biological p ocesses o selec ion/
inhe i ance/ a ia ion.
This op imize s a e based on na u al selec ion, which s a es ha indi iduals be e
adap ed o hei en i onmen , i.e., o wha e e eason be e a ob aining ood,
a oiding becoming lunch, and inding/a ac ing/compe ing o ma es, will, on a -
e age, lea e behind mo e offsp ing han hei less ap colleagues.
B. GENETIC ALGORITHMS FOR OPTIMIZATION
1Ma lab is a adema k o The Ma hWo ks.
Appendix B
153
Fo na u al selec ion o lead o e olu ion, wo mo e essen ial ing edien s a e equi ed:
1- inhe i ance: o sp ing mus e ain a leas some o he
ea u es ha made hei pa en s i e han a e age, o he wise
e olu ion is e ec i ely ese a e e y gene a ion.
2- a iabili y: a any gi en ime indi iduals o a ying i -
nesses mus coexis in he popula ion, o he wise na u al selec ion
has no hing o ope a e on.
To be e unde s and he mechanism, he e we show a gene ic op imiza ion p oblem
based in ou model. One is gi en a model ha depends on a se o pa ame e s u
(like 3τ), and a unc ional ela ion (u) ha e u ns a measu e o quali y, o i ness,
associa ed wi h he co esponding model; in ou case, his unc ion is he co ela ion
be ween he modeled ou pu and he expe imen al da a. The op imiza ion ask usu-
ally consis s in inding he “poin ” u* in pa ame e space co esponding o he model
ha maximizes he i ness unc ion (u); in ou case, we sea ch in a pa ame e space
o 11 o 15 coo dina es, depending on he model. We now de ine a popula ion as a
se o Np ealiza ions o he pa ame e s u.
A op-le el iew o a basic gene ic algo i hm is hen as ollows:
1- Randomly ini ialize popula ion and e alua e i ness o i s
membe s (pa en s).
2- B eed selec ed membe s o cu en popula ion o p o-
duce offsp ing (child) popula ion (selec ion based on i -
ness).
3- Replace cu en popula ion by offsp ing popula ion.
4- E alua e i ness o new popula ion membe s.
5- Repea s eps (2) h ough (4) un il he i es membe
o he cu en popula ion is deemed i enough.
Appendix B