Catherine Beltran
email: catherine.beltran@otago.ac.nz

Gabrielle Rousselle

Marc De Rafélis

Marie-Alexandrine Sicre

Nathalie Labourdette

Stefan Schouten

Marie- Alexandrine Sicre

Evolution of the zonal gradients across the equatorial Pacific during the Miocene-Pleistocene

Keywords: Equatorial Pacific, Miocene Pliocene climate, biomarker proxies, planktonic foraminifera, coccoliths

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The equatorial Pacific Ocean plays an important role in the global climate by influencing the heat transport from low to mid-and high latitudes. The modern structure of the equatorial Pacific is characterized by a pronounced east-west asymmetry of the thermocline depth and of sea-surface temperature (SSTs) with the warm pool (~ 29°C) in the west and the cold tongue (~ 23°C) in the east (Fig. 1). The thermocline in the West Equatorial Pacific is deep and controls the flux of atmospheric latent heat [START_REF] Tian | Heat balance in the Pacific warm pool atmosphere during TOGA COARE and CEPEX[END_REF] whereas in the east it is shallow and heat is absorbed [START_REF] Boccaletti | The thermal structure of the upper ocean[END_REF]. The SSTs across the equatorial Pacific are affected by the El Niño-Southern Oscillation [START_REF] Cane | A role for the tropical Pacific[END_REF], one of the most important component of the global climate system, which alternates at inter-annual time scale between warm "El Niño" and cold "La Niña" phases.

The El Niño-Southern Oscillation is sensitive to the SST distribution across the equatorial Pacific through the link between temperatures and the strength of the trade winds (Neelin et al., 1998). Changes in the meridional SST gradient in the equatorial Pacific have important consequences on the local and global climate.

The modern setting of the equatorial Pacific was established during the mid-to late Miocene [START_REF] Larivière | Late Miocene decoupling of oceanic warmth and atmospheric carbon dioxide forcing[END_REF] through the progressive closure of the Central American Seaway (CAS) and the Indonesian Seaway [START_REF] Keigwin | Isotope paleoceanography of the Caribbean and east Pacific: Role of Panama uplift in late Neogene time[END_REF][START_REF] Haug | Effect of the formation of the Isthmus of Panama on Atlantic Ocean thermohaline circulation[END_REF][START_REF] Srinivasan | Early Pliocene closing of the Indonesian Seaway: evidence from north-east Indian Ocean and Tropical Pacific deep sea cores[END_REF][START_REF] Schneider | Simulating the impact of the Panamanian seaway closure on ocean circulation, marine productivity and nutrient cycling[END_REF] together with Northern Hemisphere Glaciation [START_REF] Mudelsee | Slow dynamics of the Northern Hemisphere glaciation[END_REF] (Fig. 2). During this period, the onset of the modern thermohaline circulation and equatorial upwelling led to the progressive cooling of deepwater [START_REF] Woodruff | Miocene deep water oceanography[END_REF][START_REF] Ravelo | Regional climate shifts caused by gradual global cooling in the Pliocene Epoch[END_REF], upper-ocean stratification, and thermocline rise. [START_REF] Nathan | Early history of the Western Pacific Warm Pool during the middle to late Miocene (~13,2-5,8 Ma): Role of sea-level change and implications for equatorial circulation: Palaeogeography, Palaoclimatology[END_REF] dated the proto warm pool in the West Equatorial Pacific (WEP) between 11.6 and 10 Ma. The thermocline depth asymmetry set up around 4.8

Ma and induced a well-documented cold-water tongue in the Eastern Equatorial Pacific (EEP) between 4.4 and 3.6 Ma [START_REF] Chaisson | Pliocene development of the east-west hydrographic gradient in the equatorial Pacific[END_REF][START_REF] Wara | Permanent El Niño-Like Conditions During the Pliocene Warm Period[END_REF][START_REF] Steph | Early Pliocene increase in thermohaline overturning: A precondition for the development of the modern equatorial Pacific cold tongue[END_REF][START_REF] Rousselle | Sea-surface condition changes in the Equatorial pacific during the Mio-Pliocene as inferred from coccoliths geochemistry[END_REF].

Although numerous studies (e.g., [START_REF] Raymo | Mid-Pliocene warmth: stronger greenhouse and stronger conveyor[END_REF][START_REF] Haywood | Modelling middle Pliocene warmth: contribution of atmosphere, oceans and cryosphere[END_REF][START_REF] Ravelo | Regional climate shifts caused by gradual global cooling in the Pliocene Epoch[END_REF]Ravelo et al., , 2006;;[START_REF] Rickaby | Cool La Niña the warmth of the Pliocene[END_REF][START_REF] Wara | Permanent El Niño-Like Conditions During the Pliocene Warm Period[END_REF][START_REF] Barreiro | Simulations of warm tropical conditions with application to middle Pliocene atmospheres[END_REF][START_REF] Fedorov | The Pliocene paradox (mechanisms for a permanent El Niño)[END_REF][START_REF] Fedorov | Patterns and mechanisms of early Pliocene warmth[END_REF][START_REF] Dekens | A 5 million year comparison of Mg/Ca and alkenone paleothermometers[END_REF][START_REF] Brierley | Greatly expanded tropical warm pool and weakened Hadley circulation in the early Pliocene[END_REF] established that the equatorial Pacific zonal gradients were weaker during the early Pliocene warm period (El Niño-like, El Padre) and stronger during the mid-Holocene (La Niña-like; [START_REF] Koutavas | Mid-Holocene El Niño -Southern Oscillation (ENSO) attenuation revealed by individual foraminifera in eastern tropical Pacific sediments[END_REF], there is no consensus on the long-term evolution of the equatorial Pacific mean state since the Miocene (Fig. 2). This is a major limitation to predict El Niño-Southern Oscillation, in that its behavior is closely related to long-term changes in tropical Pacific oceanography that are still not fully understood. Some authors suggest that the equatorial thermal state remained unchanged over the last 12 Myr [START_REF] Zhang | A 12-million-year temperature history of the tropical Pacific Ocean[END_REF] whereas others identified alternating El Niño-like and La Niña-like conditions [START_REF] Wara | Permanent El Niño-Like Conditions During the Pliocene Warm Period[END_REF][START_REF] Ravelo | Evidence for El Niño-like conditions during the Pliocene[END_REF][START_REF] Nathan | Early history of the Western Pacific Warm Pool during the middle to late Miocene (~13,2-5,8 Ma): Role of sea-level change and implications for equatorial circulation: Palaeogeography, Palaoclimatology[END_REF][START_REF] Kamikuri | Evolution of Eastern Warm Pool and upwelling processes since the middle Miocene based on analysis of radiolarian assemblages: response to Indonesian and Central American Seaways: Palaogeography[END_REF][START_REF] Drury | Deciphering the State of the Late Miocene to Early Pliocene Equatorial Pacific: Paleoceanography and Paleoclimatology[END_REF] or a progressive shoaling of the thermocline over the last 13 Myr (LaRiviere et al., 2012).

Because the past variability of the equatorial Pacific mean state is also considered to have played a role in major climatic transitions [START_REF] Yin | The importance of tropical sea surface temperature patterns in simulations of Last Glacial Maximum climate[END_REF], reconstructing the longterm evolution of the thermal structure in this region is crucial.

While most published studies focus on very specific time intervals (e.g., [START_REF] Medina-Elizalde | Late Pliocene equatorial Pacific: Paleoceanography[END_REF][START_REF] Ford | The evolution of the equatorial thermocline and the early Pliocene El Padre mean state[END_REF], the purpose of this study is to examine the long-term evolution of equatorial Pacific surface conditions over the last 10 Myr. To do so, surface and subsurface temperature records from IODP Site U1338 (EEP) (Fig. 1) are generated from both alkenones ( ) produced by coccolithophores (marine unicellular haptophyte algae, [START_REF] Brassell | Molecular stratigraphy: a new tool for climatic assessment[END_REF] and glycero-dialkyl glycerol-tetraethers (GDGTs, TEX H 86) produced by Thaumarchaeota (Archaea, [START_REF] Schouten | Distributional variations in marine crenarchaeotal membrane lipids: a new tool for reconstructing ancient sea water temperatures?[END_REF][START_REF] Kim | New indices and calibrations derived from the distribution of crenarchaeal isoprenoid tetraether lipids: implications for past sea surface temperature reconstructions[END_REF]. Additional hydrological information is provided by the oxygen-isotope composition of calcareous nannofossils Noelaerhabdaceae spp. calcifying in the photic zone and the thermocline-dweller foraminifera Globorotalia menardii [START_REF] Martin | Taphonomy and temporal resolution of foraminiferal assemblages[END_REF][START_REF] Spero | Multispecies approach to reconstructing eastern equatorial Pacific thermocline hydrography during the past 360 kyr[END_REF] recovered from the ODP Site 806 (WEP) and IODP Site U1338 (Fig. 1).

OCEANOGRAPHIC SETTING

The equatorial Pacific surface ocean circulation, controlled by the Walker cell and Trade winds, results in two westward surface currents, the North and South Equatorial Currents, respectively (SEC and NEC; Fig. 1). The easterly winds pile warm waters in the WEP and create a subsurface eastward circulation known as the Equatorial Under Current (EUC), which brings cold and nutrient-rich waters to the surface in the EEP. The subsequent thinning of the mixed-layer and formation of a cold tongue in the EEP produce a strong W-E asymmetry of the equatorial thermocline depth, deeper in the West Pacific than in the East Pacific (Fig. 1). The modern equatorial SST gradient across the Pacific varies in response to

U k '
5 the inter-annual El Nino-Southern Oscillation. During El Niño years, the trade winds weaken, the thermocline deepens in the EEP, and the equatorial upwelling attenuates. As a consequence, the equatorial SST gradient is reduced, mean temperatures are warmer, and the extratropical heat distribution is impacted [START_REF] Molnar | El Niño's tropical climate and teleconnections as a blueprint for pre-Ice Age climate[END_REF][START_REF] Mcphaden | ENSO as an integrating concept in[END_REF].

Since the Miocene, the Pacific zonal temperature gradients and the thermocline tilt have varied. Extended periods of reduced equatorial SST gradient and deep thermocline that mimic the modern El Niño events are called permanent El Niño-like states. Similarly, periods of long-term strong equatorial Pacific zonal gradients are called La Niña-like states.

MATERIALS AND METHODS

IODP Site U1338 and ODP Site 806

The IODP Site U1338 (2° 30.469' N, 117° 58.178' W, 4200 meters water depth, leg 321) was drilled in the EEP, in the modern cold tongue [START_REF] Pälike | and the Expedition 320/321 Scientists[END_REF] (Fig. 1). Despite the northward drift of the Pacific plate, this site remained in the equatorial band for the past 10

Myr [START_REF] Pälike | and the Expedition 320/321 Scientists[END_REF]. Sediments at Site U1338 were deposited above the calcite compensation depth [START_REF] Pälike | A Cenozoic record of the equatorial Pacific carbonate compensation depth[END_REF] and are composed of well-preserved biogenic material, predominantly nannofossil oozes, with variable abundances of diatoms, radiolarians, and foraminifera [START_REF] Pälike | and the Expedition 320/321 Scientists[END_REF]. The age model of the core was constructed from biostratigraphic (nannofossils, foraminifers, diatoms, and radiolarians) and paleomagnetic data [START_REF] Pälike | and the Expedition 320/321 Scientists[END_REF][START_REF] Backman | Data report: a revised biomagnetostratigraphic age model for Site U1338[END_REF]. Average sedimentation rates are relatively high for the studied time interval (~ 27 m/Myr).

ODP Site 806 (0° 19,1' N, 159° 21,7' E, 2520 meters water depth, leg 130) is located on the northeast margin of the Ontong Java plateau in the area of the WEP warm pool (Fig. 1).

Sediments are composed mostly of well-preserved calcareous nannofossils and planktic foraminifera. The age model is based exclusively on biostratigraphic data of calcareous nannofossils, foraminifera, diatoms, and radiolarians established by [START_REF] Kroenke | Proceedings of the Ocean Drilling Program[END_REF], later refined by [START_REF] Takayama | Notes on Neogene calcareous nannofossil biostratigraphy of the Ontong Java Plateau and size variations of Reticulofenestra coccoliths[END_REF] and [START_REF] Chaisson | High-resolution Neogene planktonic foraminifer biostratigraphy of Site 806, Ontong Java Plateau (western equatorial Pacific)[END_REF]. Ages are corrected based on biostratigraphic data of [START_REF] Backman | Data report: a revised biomagnetostratigraphic age model for Site U1338[END_REF] (see Supplement). The average sedimentation rate at this site is higher than at U1338, ranging from ~ 45 m/Myr between 9

and 5 Ma to ~ 29 m/Myr during the Pliocene.

Proxy records at both sites are generated for the period between 10 and 0.18 Ma using samples from sections U1338B-22H7W through U1338B-1H2W and sections 806B-37X3W

through 806B-1H1W. The average sampling resolution of these records varies between 100 kyr and 400 kyr.

Oxygen Stable-Isotope Measurements

To determine changes in the thermocline depth in the EEP and WEP through time, we measured the oxygen-isotope ratios (d 18 O) in the calcite of calcareous-nannofossil-enriched fractions and of the upper-thermocline-dweller planktonic foraminifera Globorotalia menardii. Calcareous nannofossils calcify mainly in the photic zone and thus are used to derive surface-water properties [START_REF] Roth | Mesozoic paleoceanography of the North Atlantic and Tethys Oceans[END_REF] whereas G. menardii is selected to assess subsurface water changes.

d 18 O values are measured on calcareous nannofossils Noelaerhabdaceae spp. isolated from the bulk sediment following the procedure of [START_REF] Minoletti | Separation of sedimentary micron-sized particles for palaeoceanography and calcareous nannoplankton biogeochemistry[END_REF]. δ 18 O measurements are performed on the 2-5 µm fractions at both sites. For site U1338, we use the δ 18 O data previously published by [START_REF] Rousselle | Sea-surface condition changes in the Equatorial pacific during the Mio-Pliocene as inferred from coccoliths geochemistry[END_REF] based on the same protocol as in the present study.

δ 18 Ovalues are measured using a Finnigan Delta E mass spectrometer at 50°C, at a temporal resolution of 400 kyr. The quality of the granulometric separation based on 200 particles counting in smear slides is estimated to be at least 75% except between 8 and 9 Ma at Site 806, where Sphenolitus abies dominated. These measurements are thus not considered here.

In addition, five to seven specimens of G. menardii were hand-picked from the > 160 µm fractions for isotopic measurements. Analyses are performed on a VG ISOPRIME at 90°C with a precision of 0.1‰ and expressed with reference to the VPDB international standard. 45 samples at Site U1338 and 99 samples at Site 806 were analyzed. The average temporal resolution over the last 10 Myr is comprised between 120 and 150 ka.

Alkenone Analyses

Alkenone-derived SSTs of Site U1338 previously reported by [START_REF] Rousselle | Sea-surface condition changes in the Equatorial pacific during the Mio-Pliocene as inferred from coccoliths geochemistry[END_REF] are supplemented in the present study with 23 additional samples selected from the Pliocene interval (U1338B-8H5W to U1338B-1H2W) to reach a temporal resolution of 100 kyr.

Sedimentary alkenones are analyzed following the procedure described by [START_REF] Ternois | Climatic changes along the northwestern African continental margin over the last 30 kyrs[END_REF]. About 5 g of freeze-dried sediments are extracted in a mixture of CH2Cl2 / CH3OH.

Alkenones are isolated from the total lipid extract by silica-gel chromatography using solvents of increasing polarity. The fraction containing alkenones is concentrated, transferred into clean glass vials, and evaporated under a nitrogen stream. Gas-chromatography analyses are performed using a Varian 3400CX series equipped with a septum programmable injector (SPI) and a flame ionization detector (FID). We use a fused silica capillary column (Chrompack CP Sil5CB, 50 m long, 0.32 mm internal diameter, 0.25 µm film thickness) and helium as a carrier gas. The alkenone unsaturation ratio U K' 37 (C37:2/(C37:2 + C37:3) is converted into SSTs using the [START_REF] Conte | Global calibration of the alkenone unsaturation index (U K' 37) with surface water production temperature and a comparison of the core top integrated production temperatures recorded by U K' 37 with overlying sea surface temperatures[END_REF] calibration: T = -0.957 + 54.293 (U K' 37) -52.894 (U K' 37) 2 + 28.321 (U K' 37) 3 , which provides more accurate estimates in warm-temperature range than [START_REF] Prahl | Further evaluation of long-chain alkenones as indicators of paleoceanographic conditions[END_REF]. Average external precision of the SSTs using this calibration has been estimated at ~ 1.2°C [START_REF] Conte | Global calibration of the alkenone unsaturation index (U K' 37) with surface water production temperature and a comparison of the core top integrated production temperatures recorded by U K' 37 with overlying sea surface temperatures[END_REF] Forty samples from Site U1338 were analyzed for GDGTs at the Netherlands Institute for Sea

Research (NIOZ, Texel, The Netherlands). About 5 g of sediment are freeze dried and extracted in a mixture of CH2Cl2 and CH3OH. The total lipid extracts are separated following the protocol described in [START_REF] Schouten | Distributional variations in marine crenarchaeotal membrane lipids: a new tool for reconstructing ancient sea water temperatures?[END_REF], using Al2O3 column chromatography to separate non-polar compounds from the polar GDGTs. Polar fractions are then filtered through a 0.4 µm pore-size filter, evaporated under nitrogen, and diluted to a concentration of 2 mg/ml. GDGTs are analyzed using an Agilent HP1100 (HPLC) equipped with a Prevail Cyano column (3 µm, 150 mm x 2.1 mm) following [START_REF] Schouten | Improved analytical methodology and constraints on analysis of the TEX86 paleothermometer by high performance liquid chromatography/atmospheric pressure chemical ionization-mass spectrometry[END_REF]. The TEX86 index is calculated using peak areas of the following GDGTs as defined by [START_REF] Schouten | Distributional variations in marine crenarchaeotal membrane lipids: a new tool for reconstructing ancient sea water temperatures?[END_REF]:

TEX86 = [GDGT2]+[GDGT3]+[GDGT5'] / [GDGT1]+[GDGT2]+[GDGT3]+[GDGT5'].
The TEX H 86, defined as the log(TEX86), is used to calculate temperatures over the past 9 Myr.

Temperatures are calculated using the calibration established for temperatures above 15°C by EEP Temperature Reconstructions.

Alkenone-derived SSTs (SSTU K' 37) at Site U1338 are relatively stable and warm (~ 27°C) between 10 Ma and 6.8 Ma (Fig. 4A). They cool by 1°C between 6.8 Ma and 5.6 Ma, and show a broad warming until 3.7 Ma. From 3.7 Ma onwards, SSTs depict another cooling of about 4°C and superimposed cold episodes of 1 to 2°C amplitude around 2.9 Ma, 1.7 Ma, and 0.5 Ma.

From the Late Miocene to the Pleistocene, TEX86-derived temperatures (TTEX) indicate a long-term cooling. Between 8 and 5 Ma, TTEX and the SSTU K' 37 vary in the same range of values (between 24°C and 27°C; Fig. 4A), although they are slightly cooler. From approx. 4.5

Ma, the two temperature proxies show marked temperature difference. Note that TTEX increase around 7.5 Ma and 4.8 Ma, reaching values close to SSTU K' 37.

SSTU K' 37 at Site U1338 are warmer than TTEX by 2 to 3°C on average. Both progressively cool down between 9 Ma and 0.18 Ma but TTEX decreases by 5°C while SSTU K' 37 cools by only 3°C (Fig. 4A).

DISCUSSION

Surface and Subsurface Water Temperatures in the Equatorial Pacific

Warmer SSTU K' 37 than TTEX, as observed at Site U1338, is reported in several studies (e.g., [START_REF] Huguet | A study of the TEX86 paleothermometer in the water column and sediments of the Santa Barbara Basin[END_REF][START_REF] Lee | A study of the alkenone, TEX86, and planktonic foraminifera in the Benguela Upwelling System: Implications for past sea surface temperatures estimates[END_REF][START_REF] Leider | Core-top calibration of the lipid-based U K' 37 and TEX86 temperature proxies on the southern Italian shelf (SW Adriatic Sea, Gulf of Taranto)[END_REF][START_REF] Lopes Dos Santos | Glacialinterglacial variability in Atlantic meridional overturning circulation and thermocline adjustments in the tropical North Atlantic[END_REF][START_REF] Rommerskirchen | Miocene to Pliocene development of surface and subsurface temperatures in the Benguela Current system[END_REF][START_REF] Seki | Paleoceanographic changes in the Eastern Equatorial Pacific over the last 10 Myr: Paleoceanography[END_REF][START_REF] Zhang | A 12-million-year temperature history of the tropical Pacific Ocean[END_REF] and explained by different depth habitat or season of production. However, because seasonality is weak at low latitudes, it is unlikely to explain a temperature differences of up to 4°C, at Site U1338. Unlike haptophytes thriving in the photic zone, Thaumarchaeota, the sources of the GDGTs, occur throughout the water column (e.g., [START_REF] Karner | Archaeal dominance in the mesopelagic zone of the Pacific Ocean[END_REF][START_REF] Huguet | A study of the TEX86 paleothermometer in the water column and sediments of the Santa Barbara Basin[END_REF], and the export of GDGTs can be integrated for a larger part of the upper part of the ocean, likely including a part of the thermocline. [START_REF] Wuchter | Temporal and spatial variation in tetraether membrane lipids of marine Crenarchaeota in particulate organic matter: Implications for TEX86 paleothermometry[END_REF] reported maximum concentrations of GDGTs between 100 and 200 m depth in the Pacific Ocean, coinciding with high ammonium concentrations below the photic zone [START_REF] Murray | A time series assessment of planktonic archaeal variability in the Santa Barbara Channel[END_REF][START_REF] Karner | Archaeal dominance in the mesopelagic zone of the Pacific Ocean[END_REF][START_REF] Wuchter | Temporal and spatial variation in tetraether membrane lipids of marine Crenarchaeota in particulate organic matter: Implications for TEX86 paleothermometry[END_REF][START_REF] Huguet | A study of the TEX86 paleothermometer in the water column and sediments of the Santa Barbara Basin[END_REF][START_REF] Pearson | Stable warm tropical climate through the Eocene epoch[END_REF]. GDGT producers can thus preferentially develop in subsurface or deeper waters [START_REF] Huguet | A study of the TEX86 paleothermometer in the water column and sediments of the Santa Barbara Basin[END_REF][START_REF] Kim | Global sediment core-top calibration of the TEX86 paleothermometer in the ocean[END_REF][START_REF] Lee | A study of the alkenone, TEX86, and planktonic foraminifera in the Benguela Upwelling System: Implications for past sea surface temperatures estimates[END_REF][START_REF] Huguet | Coherent millennial scale patterns in U K' 37 and TEX H 86 temperature records during the penultimate interglacial-to-glacial cycle in the western Mediterranean[END_REF][START_REF] Kim | Pronounced subsurface cooling of North Atlantic waters off Northwest Africa during Dansgaard -Oeschger interstadials[END_REF]. The difference between TTEX and SSTU K' 37 in the EEP may then account for different production or export depth (Lopes dos [START_REF] Lopes Dos Santos | Glacialinterglacial variability in Atlantic meridional overturning circulation and thermocline adjustments in the tropical North Atlantic[END_REF][START_REF] Rommerskirchen | Miocene to Pliocene development of surface and subsurface temperatures in the Benguela Current system[END_REF][START_REF] Schouten | The organic geochemistry of glycerol dialkyl glycerol tetraether lipids: a review: Organic geochemistry[END_REF].

The interpretation of TTEX temperatures across the equatorial Pacific is complicated by both the upwelling in the EEP and the different trophic conditions between the EEP and the WEP that influence the export depth of GDGTs [START_REF] Seki | Paleoceanographic changes in the Eastern Equatorial Pacific over the last 10 Myr: Paleoceanography[END_REF]. Numerous studies indicate that upwelling impacts on the depth of production and export of GDGTs to the sediment [START_REF] Wuchter | Temporal and spatial variation in tetraether membrane lipids of marine Crenarchaeota in particulate organic matter: Implications for TEX86 paleothermometry[END_REF][START_REF] Huguet | A study of the TEX86 paleothermometer in the water column and sediments of the Santa Barbara Basin[END_REF][START_REF] Lee | A study of the alkenone, TEX86, and planktonic foraminifera in the Benguela Upwelling System: Implications for past sea surface temperatures estimates[END_REF][START_REF] Lopes Dos Santos | Glacialinterglacial variability in Atlantic meridional overturning circulation and thermocline adjustments in the tropical North Atlantic[END_REF][START_REF] Seki | Paleoceanographic changes in the Eastern Equatorial Pacific over the last 10 Myr: Paleoceanography[END_REF]. In the absence of upwelling, TTEX reflects the surface temperatures [START_REF] Sinninghe Damsté | Crenarchaeol: the characteristic core glycerol dibiphytanyl glycerol tetraether membrane lipid of cosmopolitan pelagic crenarchaeota[END_REF][START_REF] Wakeham | Archaeal lipids and anaerobic oxidation of methane in euxinic water columns: A comparative study of the Black Sea and Cariaco Basin[END_REF] whereas under upwelling conditions, TTEX represent thermocline temperatures [START_REF] Wakeham | Lipid biomarker fluxes in the Arabian Sea, with a comparison to the equatorial Pacific Ocean[END_REF][START_REF] Wuchter | Temporal and spatial variation in tetraether membrane lipids of marine Crenarchaeota in particulate organic matter: Implications for TEX86 paleothermometry[END_REF][START_REF] Huguet | A study of the TEX86 paleothermometer in the water column and sediments of the Santa Barbara Basin[END_REF].

Furthermore, the nitrate maximum across the modern equatorial Pacific exhibits a clear asymmetry in the depth. [START_REF] Könneke | Isolation of an autotrophic ammonia-oxidizing marine Archaeon[END_REF] found that the production of Thaumarchaeota occurs in the subsurface nitrate maximum, which thus is deeper in oligotrophic regions such as the WEP [START_REF] Brzezinski | Vertical distribution of ammonium in stratified oligotrophic waters[END_REF]. As a consequence, within the same basin TTEX may reflect surface and subsurface conditions in response to different environmental factors [START_REF] Hertzberg | Comparison of eastern tropical Pacific TEX86 and Globigerinoides ruber Mg/Ca derived sea surface temperatures: Insights from the Holocene and Last Glacial Maximum[END_REF].

In the EEP, the TTEX-SSTU K' 37 temperature gradient (DT) is proposed to monitor changes of the thermocline depth [START_REF] Seki | Paleoceanographic changes in the Eastern Equatorial Pacific over the last 10 Myr: Paleoceanography[END_REF] assuming that DT is driven by the depth migration of GDGT producers in response to upwelling activity. Therefore, at Site U1338, warming TTEX and reduced DT reflect a deep thermocline and surface export of GDGTs, whereas a more pronounced cooling of TTEX compared to SSTU K' 37, (increased DT) reflects a shallow thermocline, upwelling activity, and subsurface export of GDGTs.

In the WEP, the interpretation of TTEX-SSTU K' 37 is not straightforward either. At Site 806, SSTU K' 37 calculated from [START_REF] Pagani | High Earth-system climate sensitivity determined from Pliocene carbon dioxide concentrations[END_REF] are warm and stable, averaging 28 to 28.5°C over the last 5 Myr (Fig. 4C) except for a 2°C sharp cooling at ~ 1.5 Ma. High-resolution Mg/Ca-derived SSTs (SSTMg/Ca) corrected from Mg/Caseawater [START_REF] Medina-Elizalde | Implications of seawater Mg/Ca variability for Plio-Pleistocene tropical climate reconstruction[END_REF] from the same site show markedly warmer conditions in the surface of the WEP especially between 5 and 3.5 Ma, reaching up to 32°C. This difference with the alkenone-derived temperatures can be attributed to (1) the alkenone calibration being less accurate in the warmer temperature range [START_REF] Conte | Global calibration of the alkenone unsaturation index (U K' 37) with surface water production temperature and a comparison of the core top integrated production temperatures recorded by U K' 37 with overlying sea surface temperatures[END_REF], thus leading to the underestimation of SSTU K' 37 [START_REF] Tierney | BAYSPLINE: A new calibration for the alkenone paleothermometer[END_REF] or (2) the uncertainties on the seawater Mg/Ca correction. However, from 3 Ma onwards, SSTU K' 37 and SSTMg/Ca temperatures display similar trends and values, except between 1.8 Ma and 1.2 Ma, when they diverge. Since this discrepancy coincides with isotopic changes (Fig. 3) we can reasonably conclude that the difference between the two records during the late Pliocene reflects oceanographic changes.

During the early Pliocene, cooler SSTU K' 37 appear to be linked to a calibration limitation of the proxy at warm temperatures.

Comparison between the TTEX record [START_REF] Zhang | A 12-million-year temperature history of the tropical Pacific Ocean[END_REF] and subsurface temperatures derived from Mg/CaG.tumida [START_REF] Ford | The evolution of the equatorial thermocline and the early Pliocene El Padre mean state[END_REF] at ODP Site 806, highlights that from 5 to 3.5

Ma both reconstructions agree well within their uncertainties (Fig. 4C). From 3.5 to 2 Ma, TTEX match with SSTU K' 37 and Mg/Ca-derived SSTs from the surface dweller G. ruber (Mg/CaG.ruber [START_REF] Medina-Elizalde | Implications of seawater Mg/Ca variability for Plio-Pleistocene tropical climate reconstruction[END_REF], whereas Mg/CaG.tumida temperatures are systematically cooler than TTEX (Fig. 4C). After 1.3 Ma, TTEX values are intermediate between Mg/CaG.ruber and Mg/CaG.tumida temperatures, while SSTU K' 37 tend to agree with Mg/CaG.ruber SSTs. It is likely that in the WEP, the TTEX reflect production in different water depth in response to trophic changes and environmental conditions. We speculate that during the warm early Pliocene and after 1.3 Ma, TTEX reflect subsurface conditions and that during the Late Pliocene, the export of GDGTs derived mainly from surface waters. This interpretation nuances the interpretations of [START_REF] Zhang | A 12-million-year temperature history of the tropical Pacific Ocean[END_REF], who considered that TTEX reflect surface temperatures and could be used as an alternative to U K' 37 under warm conditions.

Changes in the Equatorial Pacific Mean State over the Last 10 Myr

To investigate the evolution of the surface stratification across the Equatorial Pacific, we calculate the oxygen-isotope gradients between Noelaerhabdaceae and G. menardii (Dd 18 O) (Fig. 4B) [START_REF] Nathan | Early history of the Western Pacific Warm Pool during the middle to late Miocene (~13,2-5,8 Ma): Role of sea-level change and implications for equatorial circulation: Palaeogeography, Palaoclimatology[END_REF][START_REF] Beltran | Paleoenvironmental trigger for the development of bloom forming calcareous nannofossils: an example from the Late Miocene Eastern Equatorial Pacific: Palaeoceanography[END_REF]. Note that in doing so, the sea-water thermocline [START_REF] Ravelo | Oxygen isotopic composition of multiple species of planktonic foraminifera: recorders of the modern photic zone temperature gradient[END_REF][START_REF] Ravelo | Evidence for surface-water circulation changes at Site 851 in the eastern tropical Pacific Ocean[END_REF][START_REF] Farell | Late Neogene sedimentation patterns in the eastern equatorial Pacific Ocean[END_REF][START_REF] Nathan | Early history of the Western Pacific Warm Pool during the middle to late Miocene (~13,2-5,8 Ma): Role of sea-level change and implications for equatorial circulation: Palaeogeography, Palaoclimatology[END_REF]Leckie, 2009, Beltran et al., 2014).

d
Variations in the gradient between TTEX and SSTU K' 37 also can be used to evaluate changes in the thermocline depth. TTEX and SSTU K' 37 values similar to those observed between 7.3 Ma and 6.8 Ma, and around 4.8 Ma at IODP Site U1338, indicate a deep thermocline (Fig. 4A).

Some intervals are characterized by an increase in DT (such as, for example, between 6.8 Ma and 5.3 Ma at IODP Site U1338) associated with a cooling trend of both proxies. These features indicate that GDGTs are produced deeper and that surface waters are concomitantly cooling, which is consistent with upwelling activity.

Based on temperatures and isotopic gradients between surface and subsurface waters from the EEP and WEP, the data reveal a pronounced oceanographic change at ca. 5 Ma with the thermocline progressively rising (Fig. 4A,4B).

Late Miocene [9 -5.3 Ma] Interval.---

During the late Miocene period, east and west Dd 18 O are reduced and stable, indicating a deep thermocline across the equatorial Pacific (Fig. 4B). From 10 Ma to 6.3 Ma, data indicate warm SSTs (~ 27 °C) in the EEP (Fig. 4A). This result coincides with the low abundances of upwelling radiolarian species reported between 9 and 7 Ma by [START_REF] Kamikuri | Evolution of Eastern Warm Pool and upwelling processes since the middle Miocene based on analysis of radiolarian assemblages: response to Indonesian and Central American Seaways: Palaogeography[END_REF]. Similar SSTU K' 37 and TTEX values at U1338 between 7.3 and 6.8 Ma also is consistent with a thick mixed layer in the EEP (Fig. 4A; [START_REF] Seki | Paleoceanographic changes in the Eastern Equatorial Pacific over the last 10 Myr: Paleoceanography[END_REF]. In the WEP, [START_REF] Nathan | Early history of the Western Pacific Warm Pool during the middle to late Miocene (~13,2-5,8 Ma): Role of sea-level change and implications for equatorial circulation: Palaeogeography, Palaoclimatology[END_REF] reported a similar decrease in the abundance of surface dwellers in favor of thermocline species, which suggests a deep thermocline. This reduced zonal gradient in SSTs and thermocline tilt across the equatorial Pacific corresponds to El Niño-like conditions, in agreement with earlier findings of [START_REF] Herbert | Late Miocene global cooling and the rise of modern ecosystems[END_REF], who indicate an expanded and weak

Hadley circulation before 8 Ma.

The long-term Late Miocene El Niño-like state was interrupted by a brief interval (6.5 -5.3

Ma) during which the thermocline shoaled in the EEP, as shown by the d 18 O gradient (Fig. 4B). The SST cooling and decrease of surface salinity recorded at Site U1338 [START_REF] Rousselle | Sea-surface condition changes in the Equatorial pacific during the Mio-Pliocene as inferred from coccoliths geochemistry[END_REF], supported by higher abundances of upwelling siliceous species during this interval [START_REF] Kamikuri | Evolution of Eastern Warm Pool and upwelling processes since the middle Miocene based on analysis of radiolarian assemblages: response to Indonesian and Central American Seaways: Palaogeography[END_REF], suggest the first establishment of the cold tongue and of upwelling in the EEP. Increasing abundances of surface-dweller species at Site 806 [START_REF] Nathan | Early history of the Western Pacific Warm Pool during the middle to late Miocene (~13,2-5,8 Ma): Role of sea-level change and implications for equatorial circulation: Palaeogeography, Palaoclimatology[END_REF]) (Fig. 3 and4), this interval further supports the idea of a weak and transient La Niña like period. [START_REF] Drury | Deciphering the State of the Late Miocene to Early Pliocene Equatorial Pacific: Paleoceanography and Paleoclimatology[END_REF] suggested a cooling of the thermocline waters related to the global late Miocene cooling (Federov et al., 2006;[START_REF] Drury | Late Miocene climate and time scale reconciliation: Accurate orbital calibration from a deep-sea perspective[END_REF]. Although the data do not rule out this hypothesis, the TTEX cooling is supportive of a change in depth of GDGT production or export in response to the onset of the EEP upwelling.

Plio-Pleistocene [from 5.3 Ma] Interval.---

The oxygen-isotope and temperature data indicate a long-term gradual thermocline shoaling and sea-surface cooling in the EEP (Fig. 4A,4B) throughout this interval. In the WEP, the thermocline shoaled between 5.3 and 4 Ma, returning to a deep position between 4 and 1.8

Ma and shoaling again after 1.8 Ma. This interval is characterized by the initiation of a progressive asymmetry in the thermocline depth across the equatorial Pacific.

After 5.3 Ma, SSTs at IODP Site U1338 were warm (~ 27 °C), in agreement with a deep thermocline until 4.5 Ma (Fig. 4A). This result is supported by a rapid increase of TTEX at U1338. Moreover, abundances of upwelling radiolarian species decreased again in the EEP [START_REF] Kamikuri | Evolution of Eastern Warm Pool and upwelling processes since the middle Miocene based on analysis of radiolarian assemblages: response to Indonesian and Central American Seaways: Palaogeography[END_REF]. In the WEP, the proto-warm pool period ended [START_REF] Nathan | Early history of the Western Pacific Warm Pool during the middle to late Miocene (~13,2-5,8 Ma): Role of sea-level change and implications for equatorial circulation: Palaeogeography, Palaoclimatology[END_REF]. This interval coincides with the Messinian salinity crisis [START_REF] Hilgen | Astronomical tuning as the basis for high resolution chronostratigraphy: the intricate history of the Messinian Salinity Crisis: Stratigraphy[END_REF] process would result in a deepening of the thermocline in the equator and El Niño-like conditions [START_REF] Rühlemann | Intermediate depth warming in the tropical Atlantic related to weakened thermohaline circulation: Combining paleoclimate data and modelling results for the last deglaciation[END_REF][START_REF] Timmermann | ENSO Suppression due to weakening of the North Atlantic thermohaline circulation[END_REF][START_REF] Barreiro | Abrupt climate changes: how freshening of the Northern Atlantic affects the thermohaline and winddriven oceanic circulations[END_REF], thus providing a possible explanation to the results herein.

Numerous studies have focused on the magnitude of Pliocene equatorial Pacific warmth (4.5 -3.5 Ma) and the factors responsible for it (e.g., [START_REF] Raymo | Mid-Pliocene warmth: stronger greenhouse and stronger conveyor[END_REF][START_REF] Haywood | Modelling middle Pliocene warmth: contribution of atmosphere, oceans and cryosphere[END_REF][START_REF] Ravelo | Regional climate shifts caused by gradual global cooling in the Pliocene Epoch[END_REF][START_REF] Barreiro | Abrupt climate changes: how freshening of the Northern Atlantic affects the thermohaline and winddriven oceanic circulations[END_REF][START_REF] Rickaby | Cool La Niña the warmth of the Pliocene[END_REF][START_REF] Brierley | Greatly expanded tropical warm pool and weakened Hadley circulation in the early Pliocene[END_REF][START_REF] Scroxton | Persistent El Niño-Southern Oscillation variation during the Pliocene Epoch: Paleoceanography[END_REF][START_REF] Ford | The evolution of the equatorial thermocline and the early Pliocene El Padre mean state[END_REF]. It is accepted that during the mid-Piacenzian Warm Period, persistent El-Niño like conditions prevailed. This interval, also known as El Padre mean state [START_REF] Ford | The evolution of the equatorial thermocline and the early Pliocene El Padre mean state[END_REF], is characterized by a deep and warm thermocline and warm surface conditions across the equatorial Pacific.

The low-resolution record presented here suggests a brief interval centered around 4 Ma when the thermocline deepened in the east and shoals in the west. SSTs in the EEP reach 30°C and around 32°C in the WEP (Fig. 4A, 4C, Mg/CaG.ruber SSTs; [START_REF] Medina-Elizalde | Implications of seawater Mg/Ca variability for Plio-Pleistocene tropical climate reconstruction[END_REF].

From 3.5 Ma, a pronounced zonal gradient of the thermocline depth and of the SSTs developed across the Pacific. The SST cooling trend and shoaling of the thermocline identified in the EEP coincides with the permanent establishment of the cold tongue [START_REF] Groeneveld | Pliocene mixed-layer oceanography for site 1241, using combined Mg/Ca and d 18 O analyses of Globigerinoides sacculifer[END_REF][START_REF] Dekens | Warm upwelling regions in the Pliocene warm period[END_REF][START_REF] Steph | Early Pliocene increase in thermohaline overturning: A precondition for the development of the modern equatorial Pacific cold tongue[END_REF][START_REF] Rousselle | Sea-surface condition changes in the Equatorial pacific during the Mio-Pliocene as inferred from coccoliths geochemistry[END_REF]. This result is in agreement with [START_REF] Chaisson | Pliocene development of the east-west hydrographic gradient in the equatorial Pacific[END_REF] and [START_REF] Kamikuri | Evolution of Eastern Warm Pool and upwelling processes since the middle Miocene based on analysis of radiolarian assemblages: response to Indonesian and Central American Seaways: Palaogeography[END_REF], who report higher abundances of subsurface-dweller foraminifera and radiolarian upwelling species in the EEP at that time. Simultaneous with the changes in the EEP, the warm pool became established and the thermocline deepened in the WEP, as suggested by warmer surface waters and reduced DT (Fig. 4C, [START_REF] Wara | Permanent El Niño-Like Conditions During the Pliocene Warm Period[END_REF][START_REF] Medina-Elizalde | Implications of seawater Mg/Ca variability for Plio-Pleistocene tropical climate reconstruction[END_REF] and planktonic foraminifera abundances [START_REF] Cannariato | Pliocene-Pleistocene evolution of eastern tropical Pacific surface water circulation and thermocline depth[END_REF][START_REF] Chaisson | Pliocene development of the east-west hydrographic gradient in the equatorial Pacific[END_REF][START_REF] Li | Late Miocene development of the western Pacific warm pool: planktonic foraminifer and oxygen isotopic evidence: Palaeogeography[END_REF]. This La Niña-like pattern is consistent with earlier studies [START_REF] Rickaby | Cool La Niña the warmth of the Pliocene[END_REF][START_REF] Kamikuri | Evolution of Eastern Warm Pool and upwelling processes since the middle Miocene based on analysis of radiolarian assemblages: response to Indonesian and Central American Seaways: Palaogeography[END_REF][START_REF] Steph | Early Pliocene increase in thermohaline overturning: A precondition for the development of the modern equatorial Pacific cold tongue[END_REF][START_REF] Ford | A deep Eastern Equatorial Pacific thermocline during the early Pliocene warm period[END_REF][START_REF] Rousselle | Sea-surface condition changes in the Equatorial pacific during the Mio-Pliocene as inferred from coccoliths geochemistry[END_REF] in favor of an active equatorial upwelling and a strong Walker cell.

Between 1.8 and 1.3 Ma, data indicate that the thermocline tilt and the SST gradient across the Pacific were reinforced (Fig. 4). This result is in agreement with the setting of the modern cold tongue identified by [START_REF] Martinez-Garcia | Southern Ocean dust-climate coupling over the past four million years[END_REF].

Evolution of the Pacific Mean State in a Regional and Global Context

Prior research has attributed changes in the thermocline tilt to local tectonic events such as the opening and closing of the Panama and Indonesian oceanic gateways (e.g., LaRiviere et al., 2012), and changes in sea-surface-temperature zonal gradients to global climate perturbations or changes in regional wind patterns and strength [START_REF] Hovan | Late Cenozoic atmospheric circulation intensity and climatic history recorded by eolian deposition in the eastern equatorial Pacific Ocean[END_REF][START_REF] Ford | A deep Eastern Equatorial Pacific thermocline during the early Pliocene warm period[END_REF].

However, there is still a strong debate about the timing of the closure of these oceanic gateways and their role in controlling the equatorial Pacific mean climate state. [START_REF] Montes | Evidence for middle Eocene and younger land emergence in Central Panama: Implications for Isthmus closure[END_REF][START_REF] Montes | Middle Miocene closure of the Central American Seaway[END_REF], [START_REF] Osborne | The seawater neodymium and lead isotope record of the final stages of Central American Seaway closure[END_REF][START_REF] Sepulchre | Consequences of shoaling of the Central American Seaway determined from modeling Nd isotopes[END_REF] suggested that the Central American Seaway was restricted by 10-11 Ma, which contradicts the commonly accepted late Pliocene timing of its closure [START_REF] Keigwin | Isotope paleoceanography of the Caribbean and east Pacific: Role of Panama uplift in late Neogene time[END_REF][START_REF] Duque-Caro | Neogene stratigraphy, paleoceanography and paleobiogeography in northwest South America and the evolution of the Panama Seaway: Palaeogeography[END_REF][START_REF] Haug | Effect of the formation of the Isthmus of Panama on Atlantic Ocean thermohaline circulation[END_REF][START_REF] Haug | Role of Panama uplift on oceanic freshwater balance[END_REF]. On the other hand, [START_REF] Molnar | Growth of the maritime continent and its possible contribution to recurring Ice Ages[END_REF] showed a marked restriction of the Indonesian Seaway from 5 Ma. This interpretation implies that the changes of the equatorial Pacific mean state during the Miocene would have been controlled mainly by global climate changes.

The brief La Niña-like interval between 6.5 and 5.3 Ma may be attributable to the combination of the restricted flow through the Central American Seaway [START_REF] Keigwin | Isotope paleoceanography of the Caribbean and east Pacific: Role of Panama uplift in late Neogene time[END_REF][START_REF] Keller | Late Neogene history of the Pacific -Caribbean gateway[END_REF] and the 25 m sea-level drop at ~ 6.5 Ma [START_REF] Haq | Chronology of fluctuating sea levels since the Triassic (250 million years ago to present[END_REF][START_REF] Hardenbol | Mesozoic and Cenozoic sequence chronostratigraphic framework of European basins[END_REF][START_REF] Haq | A chronology of Paleozoic sea-level changes[END_REF] due to the expansion of ice sheets in Greenland [START_REF] Larsen | Seven million years of glaciation in Greenland[END_REF], Alaska [START_REF] Bradshaw | The relative roles of CO2 and palaeogeography in determining late Miocene climate: Results from a terrestrial model-data comparison: Climate of the Past[END_REF] and Antarctica [START_REF] Billups | Late Miocene through early Pliocene deep water circulation and climate change viewed from the sub-Antarctic South Atlantic: Palaeogeography[END_REF][START_REF] Williams | Evidence for iceberg armadas from East Antarctica in the Southern Ocean during the late Miocene and early Pliocene[END_REF] and the 2 Myr-long Northern Hemisphere glaciation [START_REF] Larsen | Seven million years of glaciation in Greenland[END_REF]. This global cooling is thought to have contributed to the reinforcement of the equatorial upwelling and EUC.

Data from sediments younger than ~ 5 Ma indicate an important change in the equatorial Pacific mean state with the dominance of La Niña-like intervals. The onset of this long-term state coincides with a drastic restriction of the Indonesian Seaway [START_REF] Srinivasan | Early Pliocene closing of the Indonesian Seaway: evidence from north-east Indian Ocean and Tropical Pacific deep sea cores[END_REF][START_REF] Molnar | Growth of the maritime continent and its possible contribution to recurring Ice Ages[END_REF], and changes in the global climate. The Indonesian Seaway restriction is thought to have contributed to the cooling of the eastern tropical Pacific over the last 5 Myr through the strengthening of the Walker circulation, leading to the increase in the Pacific zonal temperature gradient (Fig. 3,4).

The La Niña-like event between 1.8 and 1.3 Ma coincides with pronounced ice growth in Antarctica [START_REF] Mckay | Antarctic and Southern Ocean influences on Late Pliocene global cooling[END_REF] and the development of the northern-hemisphere ice sheets.

The gradual increase of the pole-to-equator temperature gradient is thought to have reinforced the Walker cell, and affected the low-latitude zonal gradient [START_REF] Molnar | El Niño's tropical climate and teleconnections as a blueprint for pre-Ice Age climate[END_REF][START_REF] Mckay | Antarctic and Southern Ocean influences on Late Pliocene global cooling[END_REF]. Furthermore, modelling experiments show a positive feedback of the Pliocene low-latitude cooling on the global cooling [START_REF] Fedorov | The Pliocene paradox (mechanisms for a permanent El Niño)[END_REF]. The shoaling of the thermocline increased the climate sensitivity of this area by strengthening the temperature gradient through the tropics and subsequently the Walker and Hadley cells [START_REF] Brierley | Greatly expanded tropical warm pool and weakened Hadley circulation in the early Pliocene[END_REF][START_REF] Brierley | Relative importance of meridional and zonal sea surface temperature gradients for the onset of the ice ages and Pliocene-Pleistocene climate evolution: Paleoceanography[END_REF]. These processes led to the final setting of upwelling areas [START_REF] Philander | Role of tropics in changing the response to Milankovich forcing some three million years ago[END_REF][START_REF] Dekens | Warm upwelling regions in the Pliocene warm period[END_REF][START_REF] Brierley | Greatly expanded tropical warm pool and weakened Hadley circulation in the early Pliocene[END_REF][START_REF] Brierley | Relative importance of meridional and zonal sea surface temperature gradients for the onset of the ice ages and Pliocene-Pleistocene climate evolution: Paleoceanography[END_REF].

CONCLUSION

This study demonstrates the complexity of reconstructing the evolution of the equatorial Pacific mean climate state and oceanographic features and the need of a multi-proxy approach to get a comprehensive understanding of the mechanism at play over the past 10 Myr.

Combined isotopic zonal gradients between the surface (Noelaerhabdaceae) and subsurface (G. menardii) waters from the east and west equatorial Pacific reveals important steps towards the establishment of what is known today as El Niño-Southern Oscillation. The record suggests a general shoaling of the thermocline along the equator since the Late Miocene. The data show the appearance of the cold tongue at ~ 6.8 Ma before its proper establishment around 4.4 to 3.6 Ma [START_REF] Chaisson | Planctonic foraminiferal assemblages and paleoceanographic change in the trans-tropical Pacific Ocean: a comparison of West (leg 130) and East (leg 138), latest Miocene to Pleistocene[END_REF][START_REF] Cannariato | Pliocene-Pleistocene evolution of eastern tropical Pacific surface water circulation and thermocline depth[END_REF][START_REF] Chaisson | Pliocene development of the east-west hydrographic gradient in the equatorial Pacific[END_REF][START_REF] Molnar | El Niño's tropical climate and teleconnections as a blueprint for pre-Ice Age climate[END_REF][START_REF] Ravelo | Regional climate shifts caused by gradual global cooling in the Pliocene Epoch[END_REF][START_REF] Ravelo | Evidence for El Niño-like conditions during the Pliocene[END_REF][START_REF] Sato | Establishment of the western Pacific warm pool during the Pliocene: Evidence from planktic foraminifera, oxygen isotopes, and Mg/Ca ratios[END_REF][START_REF] Steph | Early Pliocene increase in thermohaline overturning: A precondition for the development of the modern equatorial Pacific cold tongue[END_REF][START_REF] Rousselle | Sea-surface condition changes in the Equatorial pacific during the Mio-Pliocene as inferred from coccoliths geochemistry[END_REF] [START_REF] Pagani | High Earth-system climate sensitivity determined from Pliocene carbon dioxide concentrations[END_REF] in green, TEX86-derived temperatures [START_REF] Zhang | A 12-million-year temperature history of the tropical Pacific Ocean[END_REF] in red, Mg/CaG.ruber-derived SST [START_REF] Medina-Elizalde | Implications of seawater Mg/Ca variability for Plio-Pleistocene tropical climate reconstruction[END_REF] in brown and Mg/CaG.tumida-derived temperature [START_REF] Ford | The evolution of the equatorial thermocline and the early Pliocene El Padre mean state[END_REF] in orange.

  . Replicate analyses indicate internal precision of ± 0.5°C. Because of the limited amount of sediment material at Site 806 (where alkenone concentrations are extremely low), previously published data from Pagani et al. (2010) provide data covering the past 5 Myr. Note that we recalculated SSTs from the U K' 37 values of Pagani et al. (2010) using the Conte et al. (2006) calibration to avoid calibration bias between sites (see Supplement).

d 18 O

 18 Record at Site 806.--d 18 ONoelaerhabdaceae at site 806 vary within a range similar (between -2.8 and 0.9‰) to site U1338 (Fig. 3B). d 18 ONoelaerhabdaceae values are relatively stable (around 0‰) between 7 Ma and 4.4 Ma and rapidly decline to more negative values between 4.5 Ma and 4 Ma, after which values remain stable until 0.38 Ma. The d 18 OG.menardii record at Site 806 shows a longterm decrease between 9.7 Ma and 7 Ma and becomes stable until 1.9 Ma. At 3.7 Ma oxygenisotope values decrease sharply by 1‰, and again between 2.5 Ma and 1.8 Ma. The d 18 OG.menardii values then increase to higher values (-0.4 ‰) until 0.5 Ma and further to 0.5‰ at 0.2 Ma. Overall, d 18 ONoelaerhabdaceae and d 18 OG.menardii reveal more positive values in the EEP than in the WEP and include larger-amplitude variations.

  18 O component is factored out. Higher (lower) Dd 18 O values reflect a shallower (deeper)

  , during which the isolation of the Mediterranean Sea would have triggered a reduction of the North Atlantic Deep Water formation (NADW) at ~ 6 Ma because of changing water density of the North Atlantic waters (Pérez-Asensio et al., 2012). According to model simulations, this

  and its modern configuration ~ 1.8 Ma. This result is in agreement with the findings of[START_REF] Larivière | Late Miocene decoupling of oceanic warmth and atmospheric carbon dioxide forcing[END_REF] and[START_REF] Zhang | A 12-million-year temperature history of the tropical Pacific Ocean[END_REF], who suggested progressive shoaling of the thermocline over the last 13 Myr.The data indicate two critical transitions at about 5.3 Ma, when the thermocline globally started shoaling across the equatorial Pacific and around 3.8 Ma from an El-Niño-likedominated mean state during the Late Miocene to a La-Niña-like-dominated state during the Plio-Pleistocene. These transitions coincide with marked regional and global changes.Locally, the restriction of the Indonesian Seaway and subsequent intensification of the Walker circulation increased the Pacific zonal temperature gradient over the last 5 Myr. From a global point of view, the growth of ice sheets in the northern hemisphere and in Antarctica most likely affected the Walker cell and the low-latitude zonal gradient.

  FIGURE CAPTIONS

Figure 2 .

 2 Figure 2. Comparison of El Niño-like (in red) and La Niña-like (in blue) time periods

Figure 3 .

 3 Figure 3. A) d 18 O records in the EEP. d 18 ONoelaerhabdaceae at IODP Site U1338 from Rousselle

Figure 4 .

 4 Figure 4. A) Temperature records at IODP Site U1338 (EEP): U K' 37-derived SST and TEX86-

  The oxygen stable-isotope signatures of the Noelaerhabdaceae-enriched fine fractions (d 18 ONoelaerhabdaceae) vary between -1.5‰ and 0.5‰ (Fig.3A). d 18 ONoelaerhabdaceae progressively increase between 8 Ma and 5 Ma, before declining to -1.3‰ until 2.7 Ma and rising again to 0.4‰. The d 18 ONoelaerhabdaceae record is similar to the higher resolution d 18 O record of the surface dweller G. sacculifer from the nearby Site 847 covering only the last 5 Myr(Wara et

	al., 2005) (Figs. 1, 3A).
	d 18 O values measured on the thermocline-dwelling planktonic foraminifera G. menardii
	(d 18 OG.menardii) over the last 8 Myr (Figure 3A) demonstrate that values vary between -1.4‰

[START_REF] Kim | New indices and calibrations derived from the distribution of crenarchaeal isoprenoid tetraether lipids: implications for past sea surface temperature reconstructions[END_REF]

(T = 68.4*TEX H 86 + 38.6

). The error on temperature estimates is ± 2.5°C.

Other calibrations have been proposed for TEX86 temperatures, in particular based on Bayesian statistics

[START_REF] Tierney | A TEX86 surface sediment database and extended Bayesian calibration[END_REF]

but these yield results similar to those shown here, whereas the

[START_REF] Kim | New indices and calibrations derived from the distribution of crenarchaeal isoprenoid tetraether lipids: implications for past sea surface temperature reconstructions[END_REF]

calibration is consistent with other studies from this area (e.g.,

[START_REF] Seki | Alkenone and boron-based Pliocene pCO2 records[END_REF][START_REF] Zhang | A 12-million-year temperature history of the tropical Pacific Ocean[END_REF]

. TEX H 86 -derived temperatures are obtained at a mean temporal resolution of 230 kyr. At Site 806, because of the very low GDGT concentrations, we use previously published data from

[START_REF] Zhang | A 12-million-year temperature history of the tropical Pacific Ocean[END_REF]

covering the past 12 Myr for comparison with the data from Site U1338.

RESULTS

Stable Isotopes of the Calcareous Nannofossil Fine Fractions and G.menardii

d 18 O Record at Site U1338.---and 1.25‰ and display trends similar to those of d 18 ONoelaerhabdaceae until 5 Ma. Then, d 18 OG.menardii become more positive than d 18 ONoelaerhabdaceae and still increase until 0.4 Ma, as also recorded in the deep thermocline dweller G. tumida d 18 O values (d 18 OG.tumida) at Site 847 (Figs. 1, 3A; Wara et al., 2005).

Pleistocene. Journal of Sedimentary Research, 2019, 89 (3), pp.242-252. 10.2110/jsr.2019.15. insu-

ACKNOWLEDGMENTS

We thank the European Consortium for Ocean Research Drilling (ECORD) for the ECORD Grant. We are also grateful to Anchelique Mets and Jort Ossebaar for support with the TEX86 measurements. We finally want to thank Fabrice Minoletti for providing us the separation technique development necessary to then perform the isotopic measurements on Noelaerhabdaceae calcite. We also thank the Ocean and International Ocean drilling Program (ODP and IODP) for providing samples from Sites 806 and U1338. SS and AS were supported by the Netherlands Earth System Science Center funded by the Dutch Ministry Science and Education (OCW). MAS was supported by CNRS (Centre National de la Recherche Scientifique) for salary.


Piotr Wolszczak
email: p.wolszczak@pollub.pl

Pawel Lonkwic
email: plonkwic@gmail.com

Americo Cunha
email: americo@ime.uerj.br

Grzegorz Litak
email: g.litak@pollub.pl

Szymon Molski
email: molski@agh.edu.pl

Robust optimization and uncertainty quantification in the nonlinear mechanics of an elevator brake system

Keywords: elevator brake system, nonlinear mechanics, nonlinear optimization, uncertainty quantification, parametric probabilistic approach

This paper deals with nonlinear mechanics of an elevator brake system subjected to uncertainties. A deterministic model that relates the braking force with uncertain parameters is deduced from mechanical equilibrium conditions. In order to take into account parameters variabilities, a parametric probabilistic approach is employed. In this stochastic formalism, the uncertain parameters are modeled as random variables, with distributions specified by the maximum entropy principle. The uncertainties are propagated by the Monte Carlo method, which provides a detailed statistical characterization of the response. This work still considers the optimum design of the brake system, formulating and solving nonlinear optimization problems, with and without the uncertainties effects.

Introduction

Considerations regarding the construction of lifting devices (design of cranes), and in particular brake systems, are not often discussed in the scientific literature, as the corresponding dynamical conditions are difficult to determine. The first person who addressed the issue of the impact of safety gears construction on the braking distance was Elisha Graves Otis, who in 1853 built the first safety gears and subjected them to experimental studies [START_REF] Lonkwic | Selected issues of the operating process of slide catches[END_REF][START_REF] Pater | Selected topics from the history of technology[END_REF]. Subsequent works on this subject, studying several aspects of cranes mechanics, appeared in the twentieth century and were published in journals and conference proceedings [START_REF] Yost | Configuring elevator systems[END_REF][START_REF] Lonkwic | Influence of friction drive lift gears construction on the length of braking distance[END_REF][START_REF] Lonkwic | Braking deceleration variability of progressive safety gears using statistical and wavelet analyses[END_REF][START_REF] Lonkwic | Nonlinear analysis of braking delay dynamics for the progressive gears in variable operating conditions[END_REF][START_REF] Kaczmarczyk | Dynamic response of an elevator car due to stochastic rail excitation[END_REF][START_REF] Kaczmarczyk | The dynamic behaviour of a non-stationary elevator compensating rope system under harmonic and stochastic excitations[END_REF][START_REF] Colón | On dynamic analysis and control of an elevator system using polynomial chaos and Karhunen-Loève approaches[END_REF].

For instance, Yost and Rothenfluth [START_REF] Yost | Configuring elevator systems[END_REF] describe how to configure a lifting device and how to select the correct components. These issues constituted a significant contribution to the development of the configuration of lifts, ensuring a trouble-free operation.

Lonkwic [START_REF] Lonkwic | Influence of friction drive lift gears construction on the length of braking distance[END_REF] presents a comparative analysis of the operation of slip safety gear of his own design study with the models by leading European manufacturers. Deceleration (braking time) values obtained in the physical experiment are analyzed. In [START_REF] Lonkwic | Braking deceleration variability of progressive safety gears using statistical and wavelet analyses[END_REF], the same author and collaborators address, by means of wavelet analysis, how certain variables influence on the operating conditions of deceleration. A similar analysis is presented in [START_REF] Lonkwic | Nonlinear analysis of braking delay dynamics for the progressive gears in variable operating conditions[END_REF], which concerns the selected braking parameters of CHP2000 and PP16 type chaters using the analysis of recursive patterns.

Regarding the study of elevator systems with uncertain operating conditions, the literature is not very comprehensive. The only works in this line known by the authors are [START_REF] Kaczmarczyk | Dynamic response of an elevator car due to stochastic rail excitation[END_REF][START_REF] Kaczmarczyk | The dynamic behaviour of a non-stationary elevator compensating rope system under harmonic and stochastic excitations[END_REF], developed by Kaczmarczyk et al., who attempt to analyze the behavior of balance ropes due to harmonic and stochastic excitations, and Colón et al. [START_REF] Colón | On dynamic analysis and control of an elevator system using polynomial chaos and Karhunen-Loève approaches[END_REF], who calculate the propagation of the rail profile uncertainties and study the effectiveness of a closed-loop control law.

Even with the scientific literature being rich in studies regarding the behavior of vehicles brake systems under changing operating conditions [START_REF] Renault | Experimental investigations for uncertainty quantification in brake squeal analysis[END_REF][START_REF] Dezi | Motorcycle brake squeal: experimental and numerical investigation on a case study[END_REF][START_REF] Knops | An optimum braking strategy[END_REF], it is surprising that, to the best of authors' knowledge, no similar research description on lift brakes has been reported up to the present date. Only general provisions contained in the British Standard Document BS EN 81 [START_REF]Safety rules for the construction and installation of lifts -lifts for the transport of persons and goodspart 20: Passenger and goods passenger lifts[END_REF][START_REF]Safety rules for the construction and installation of lifts -examinations and tests -part 50: Design rules, calculations, examinations and tests of lift components[END_REF] are to be found.

Thus, seeking to fill this gap, the present work aims to study the influence of some operating conditions on the efficiency of an elevator brake device, by analyzing how the operating parameters underlying uncertainties propagate through the mechanical system. In particular, the cam brake angle and the spring reaction force are of interest. In addition to quantifying the effects of uncertainties in operating conditions, this study also aims to achieve a robust design of a brake system by solving a nonlinear optimization problem, considering (or not) the uncertainty effects.

The remaining part of this paper is organized as follows: the deterministic modeling of the lift brake systems under study is presented in section 2. In section 3, the construction of a consistent stochastic model of uncertainties to deal with variabilities in the uncertain parameters, is presented. Two optimization problems, one classical and one robust, which seek to find an optimal design for the brake system are formulated in section 4. In section 5, numerical experiments are reported and discussed. Finally, in section 6, concluding remarks are presented.

Deterministic modeling

Elevator brake system

A schematic representation of the CHP 2000 safety gear, used by a typical friction crane brake system, is presented in Figure 1. It consists of a monolithic steel body [START_REF] Lonkwic | Selected issues of the operating process of slide catches[END_REF], in which a braking cam ( 5) is mounted on a bolt. The braking cam moves the brake roller [START_REF] Pater | Selected topics from the history of technology[END_REF], which has a knurled surface. This irregular surface is responsible for the braking process and for the cooperation with the guide roller surface. The brake roller moves over the braking cam surface until it contacts the lift guide [START_REF] Lonkwic | Nonlinear analysis of braking delay dynamics for the progressive gears in variable operating conditions[END_REF]. It is a free movement that does not cause any braking effect. The second part of the braking process is in constant contact with the lift guide surface.

An illustration of a typical friction crane brake system used by lifting devices is shown in Figure 2, which indicates the different components of the mechanism (see the caption). It consists of two safety gears, moving on the lift guides, connected to each other to ensure simultaneous operation when the brakes are activated by means of a trigger lever. A lift safety gear is placed in the frame, under a safety gear cabin. Its trigger is attached to the trigger lever, in which the end is connected to the rope speed limiter. In the upper part of the elevator shaft there is a speed limiter supervising the work of the safety gear, and in its lower part load responsible for causing the proper tension of the speed limiter rope is located. The speed limiter triggers the braking process when the nominal speed of the elevator car is increased by 0.3 m/s. After exceeding the nominal speed, the speed limiter is blocked, and the rope is also immobilized.

During the movement of an elevator car with locked components, the lever is moved in the opposite direction to the cabin, triggering the brake safety gear roll. In its turn, the roll is pressed against the guide causing elastic deformation towards the thrust plate located on the other side of the disk spring package, which induces the loss of energy in the accelerating mass. Therefore, the disc spring package is responsible for a variable force that presses the roller to the guide during the braking process.

Mathematical model

The design assumptions and safety gear structure shown in Figure 1 are taken into account to construct a mathematical model that relates the braking force with geometric parameters and other characteristics of the mechanical system. In this sense, equilibrium conditions for the system are deduced below.

A free-body diagram can be seen in Figure 3, which shows a schematic representation of the forces (in red) acting on the safety gear steel body, and the underlying geometric dimensions (in blue).

A balance of forces and the moments acting on the steel body gives rise to the equations

-N 4 + F s + R x = 0, (1)
T 4 -R y -(F g + F b )/2 = 0, (2)
-F s a -R y l + N 4 m -T 4 n = 0, (3)
where F s is the spring reaction force; F b is the inertial force from the cabin and lifting capacity; F g is the cabin and lifting capacity weight; T 4 is the friction force between the guide and brake retaining block, and N 4 is the corresponding normal force; R x and R y are the reaction forces in the braking cam rotation point; while a, l, m and n are geometric dimensions depicted in Figure 3.

The forces (in red) acting on the wedge during braking and immediately after stopping the cabin, until the safety gears are unlocked by technical maintenance of the lift, are shown in Figure 4, along with the relevant geometric dimensions (in blue).

A new balance of forces and moments provides

T 2 + N 1 cos α + T 1 sin α -F s -R x = 0, (4)
F s a + T 1 b -N 1 c -N 2 d -T 2 e = 0, (5)
where T 1 and T 2 are friction forces between brake elements (roller and cam), N 1 and N 2 are the corresponding normal forces; α is the braking cam angle; b, c, d and e are other geometric dimensions of the problem, shown in Figure 4.

In Figure 5 the reader can see characteristic dimensions (in blue) and forces (in red) acting on the brake roller inside the safety gear. Now the balance of forces gives where T 3 and N 3 respectively denotes the frictional and the normal forces between brake roller and the guide.

N 3 -T 2 -N 1 cos α -T 1 sin α = 0, (6)
T 3 -N 2 + T 1 cos α -N 1 sin α = 0, (7)
The frictional forces T 1 , T 2 and T 4 are, respectively, related to the normals N 1 , N 2 and N 4 through a Coulomb friction model, so that

T 1 = µ 1 N 1 , (8)
T 2 = µ 2 N 2 , (9)
T 4 = µ 4 N 4 , (10)
where µ 1 , µ 2 and µ 4 are friction coefficients.

On the other hand, the relationship between the frictional force T 3 and the normal N 3 takes into account the plastic deformation occurring in the contact between brake roller and the guide, so that

T 3 = f R N 3 , (11)
where f and R are geometric dimensions defined in Figure 5.

The vertical reaction force R y can be obtained from Eqs.( 2) and (3),

R y = -T 4 + (F g + F b ) 2 , (12)
R y = F s a -N 4 m + T 4 n l , (13)
which, when combined together with Eq.( 10), allows one to express N 4 as

N 4 = (F g + F b ) l/2 -F s a µ 4 (n + l) -m . (14)
Similarly, from suitable manipulations of Eqs.( 6), ( 8) and ( 9), it can be concluded that

N 3 = µ 2 N 2 + (µ 1 sin α + cos α) N 1 , (15)
as well as, from Eqs.( 5), ( 8) and ( 9), it is possible to obtain

N 2 = a F s + (b µ 1 -c) N 1 d + e µ 2 , (16)
which, in combination with Eqs.( 1), ( 4), ( 8) and ( 9), gives rise to

N 1 = N 4 - a µ 2 F s d + e µ 2 µ 1 sin α + cos α + µ 2 (b µ 1 -c) d + e µ 2 . ( 17
)
The braking force, resulting from the joint superposition of all frictional forces, is given by

F h = T 1 + T 2 + T 3 + T 4 , (18)
which, with aid of Eqs.( 8) to [START_REF] Dezi | Motorcycle brake squeal: experimental and numerical investigation on a case study[END_REF], can be rewritten as

F h = µ 1 N 1 + µ 2 N 2 + f R N 3 + µ 4 N 4 . ( 19
)
Note that, once normal forces N 1 , N 2 , N 3 and N 4 present explicit dependence on geometric dimensions, frictional coefficients, and non-frictional forces, the braking force F h is also a function of these parameters, i.e.,

F h = F h (α, F s , F g , F b , µ 1 , µ 2 , µ 4 , • • • • • • a, b, c, d, e, f, l, m, n, R). (20)

Stochastic modeling

The angle α and the spring reaction force F s are subjected to variabilities during the operation conditions of the brake system, so that their actual values may be very different from the nominal project values. Since they are the critical parameters for the brake system efficiency, studying the effect of such variabilities on the braking force is essential for a good design. In this way, a parametric probabilistic approach [START_REF] Cunha | Modeling and Quantification of Physical Systems Uncertainties in a Probabilistic Framework[END_REF][START_REF] Soize | Uncertainty Quantification: An Accelerated Course with Advanced Applications in Computational Engineering[END_REF] is employed here to construct a consistent stochastic model for uncertain parameters α and F s .

Probabilistic framework

Let (Θ, Σ, P) be the probability space used to describe the model parameters uncertainties [START_REF] Soize | Uncertainty Quantification: An Accelerated Course with Advanced Applications in Computational Engineering[END_REF][START_REF] Soize | Stochastic modeling of uncertainties in computational structural dynamics -recent theoretical advances[END_REF], where Θ is the sample space, Σ a σ-field over Θ, and P : Σ → [0, 1] a probability measure.

In this probabilistic setting, the parameters α and F s are respectively described by the random variables X 1 : Θ → R and X 2 : Θ → R, which are lumped into the random vector X : Θ → R 2 , which associates to each elementary event θ ∈ Θ a vector X = (X 1 , X 2 ).

The probability distribution of X is characterized by the map p X : R 2 → [0, +∞), dubbed the joint probability density function (PDF).

The mean value of X is defined in terms of the expected value operator

E {X} = R 2 x p X (x) dx , (21)
in which x = (x 1 , x 2 ) and dx = dx 1 dx 2 .

Maximum entropy principle

To perform a judicious process of uncertainty quantification, it is essential to construct a consistent stochastic model for the random vector X, that represents the uncertainties in α and F s in a rational way, trying to be unbiased as possible. In this sense, in order to avoid possible physical inconsistencies in the probabilistic model, only available information must be used in its construction [START_REF] Soize | Uncertainty Quantification: An Accelerated Course with Advanced Applications in Computational Engineering[END_REF][START_REF] Soize | Stochastic modeling of uncertainties in computational structural dynamics -recent theoretical advances[END_REF]. When this information materializes in the form of a large set of experimental data, the standard procedure is to use a nonparametric statistical estimator to infer the joint distribution of X [START_REF] Soize | Uncertainty Quantification: An Accelerated Course with Advanced Applications in Computational Engineering[END_REF][START_REF] Soize | Stochastic modeling of uncertainties in computational structural dynamics -recent theoretical advances[END_REF]. However, if little (or even no) experimental data for X 1 and X 2 is available, as is the case of this paper, such construction can be done based only on known theoretical information, with the aid of the maximum entropy principle [START_REF] Soize | Uncertainty Quantification: An Accelerated Course with Advanced Applications in Computational Engineering[END_REF][START_REF] Soize | Stochastic modeling of uncertainties in computational structural dynamics -recent theoretical advances[END_REF].

The available theoretical information about the random parameters X 1 and X 2 encompasses a range of possible values for each of then, i.e., [START_REF] Bonnans | Numerical Optimization: Theoretical and Practical Aspects[END_REF] as well as their nominal values µ X1 and µ X2 , that are assumed to be equal to their mean values, i.e.,

(X 1 , X 2 ) ∈ [α 1 , α 2 ] × [γ 1 , γ 2 ] ⊂ [0, 2π] × (0, +∞),
E {X 1 } = µ X1 ∈ [α 1 , α 2 ], (23)
E {X 2 } = µ X2 ∈ [γ 1 , γ 2 ]. ( 24
)
This information is translated into the statistical language through the normalization condition

γ2 γ1 α2 α1 p X (x) dx = 1, (25)
and the first order moment equation

γ2 γ1 α2 α1 x p X (x) dx = (µ X1 , µ X2 ) . (26)
From the information theory point of view, the most rational approach to specify the distribution of X in this scenario of reduced information is through the maximum entropy principle (MaxEnt) [START_REF] Soize | Uncertainty Quantification: An Accelerated Course with Advanced Applications in Computational Engineering[END_REF][START_REF] Soize | Stochastic modeling of uncertainties in computational structural dynamics -recent theoretical advances[END_REF][START_REF] Kapur | Entropy Optimization Principles with Applications[END_REF], which seeks the PDF that maximizes the entropy functional

S(p X ) = - γ2 γ1 α2 α1 p X (x) ln p X (x) dx , (27)
respecting the restrictions (information) defined by ( 25) and [START_REF] Cuellar | Non-intrusive polynomial chaos expansion for topology optimization using polygonal meshes[END_REF].

Using the Lagrange multipliers method it is possible to show that such joint PDF is given by

p X (x) = p X1 (x 1 ) × p X2 (x 2 ), (28)
with marginal densities

p X1 (x 1 ) = exp (-λ 10 -λ 11 x 1 ) 1 [α1,α2] (x 1 ), (29)
p X2 (x 2 ) = exp (-λ 20 -λ 21 x 2 ) 1 [γ1,γ2] (x 2 ), (30)
where λ 10 , λ 11 , λ 20 and λ 21 are parameters of the distribution of X, and

1 I (x) = 1, x ∈ I, 0, x ∈ I, (31)
denotes the indicator function of the interval I. Note that, since no information relative to the cross statistical moments between X 1 and X 2 has been provided, MaxEnt provides independent distributions. The parameters λ 10 , λ 11 , λ 20 and λ 21 depend on α 1 , α 2 , γ 1 , γ 2 , µ X1 and µ X2 . They are computed through the nonlinear system of equations obtained by replacing (28) in [START_REF] Cuellar | Non-intrusive polynomial chaos expansion for topology optimization using polygonal meshes[END_REF] and in the normalization conditions of the marginal PDFs (29) and (30).

In a scenario with little information, it is practically impossible not to be biased in choosing a probability distribution. The MaxEnt formalism provides the least biased distribution that is consistent with the known information, therefore constituting the most rational approach [START_REF] Kapur | Entropy Optimization Principles with Applications[END_REF].

Uncertainty propagation

The mathematical model relating the braking force F h with braking cam angle α and the spring reaction force F s , Eq.( 20), can be thought abstractly as a nonlinear deterministic functional M that maps a vector of input parameters x = (α, F s ) into a scalar quantity of interest y = F h , i.e.,

x → y = M(x).

(32) Thus, if the uncertain parameters α and F s are represented by the known random vector X, the braking force becomes the random variable Y = M(X), for which the distribution must be estimated.

The process of determining the distribution of Y , once the probabilistic law of X is known, is dubbed uncertainty propagation problem [START_REF] Cunha | Modeling and Quantification of Physical Systems Uncertainties in a Probabilistic Framework[END_REF][START_REF] Soize | Uncertainty Quantification: An Accelerated Course with Advanced Applications in Computational Engineering[END_REF], being addressed in this paper via the Monte Carlo simulation [START_REF] Kroese | Handbook of Monte Carlo Methods[END_REF][START_REF] Cunha | Uncertainty quantification through Monte Carlo method in a cloud computing setting[END_REF].

In this stochastic calculation technique, ν independent samples of X are drawn according to the density (28), giving rise to statistical realizations X (1) , X (2)

, • • • , X (ν) . ( 33
)
Each of these scenarios for X is given as input to the nonlinear deterministic map x → y = M(x), resulting in a set of possible realizations for the quantity of interest

Y (1) , Y (2) , • • • , Y (ν) , (34)
where Y (j) = M(X (j) ), j = 1, • • • , ν. These samples are used to estimate statistics of Y non-parametrically, i.e., without assuming the PDF shape known [START_REF] Wasserman | All of Nonparametric Statistics[END_REF].

Optimization framework

Regarding the improvement of brake system efficiency, an optimal design of its components is required. This work addresses this question by solving nonlinear optimization problems that seeks to maximize the braking force, using geometric dimensions of the system as design variables. Two optimization approaches are employed. The first one, named classical, is based on deterministic formalism of nonlinear programming [START_REF] Bonnans | Numerical Optimization: Theoretical and Practical Aspects[END_REF], while the latter, dubbed robust, takes into account the model parameters uncertainties, in order to reduce the optimum point sensitivity to small disturbances [START_REF] Beyer | Robust optimization -A comprehensive survey[END_REF][START_REF] Gorissen | A practical guide to robust optimization[END_REF].

In this framework, a set of two design variables (geometric dimensions) is denoted generically by the vector s. The other parameters of the model are denoted generically by x, and the model response is given by the nonlinear map (s, x) → y = M(s, x) The quantity of interest to be optimized (objective function) is denoted generically by J .

Classical optimization

In this classical optimization approach the s components are employed as design variables, while the braking force is adopted as objective function, i.e.,

J C (s) = y. ( 35
)
The admissible set for this optimization problem is defined by

A C = [s min 1 , s max 1 ] × [s min 2 , s max 2
], so that it can be formally stated as find an optimal design vector

s opt C = arg max s∈A C J C (s).
(36)

Robust optimization

In this robust optimization framework, which is based on those shown in [START_REF] Cunha | Computational modeling of the nonlinear stochastic dynamics of horizontal drillstrings[END_REF][START_REF] Cuellar | Non-intrusive polynomial chaos expansion for topology optimization using polygonal meshes[END_REF], the uncertainties are described according to the formalism of the section 3, where x becomes the random vector X, and, as a consequence, y = M(s, x) becomes the random variable Y = M(s, X). Thus, the robust objective function is not constructed directly from the model response, but with the aid of statistical measures of Y , which aims to guarantee greater stability to small disturbances (robustness) to an optimum point.

Specifically, the robust objective function is given by a convex combination between minimum, maximum, mean and standard deviation inverse, so that

J R (s) = β 1 min θ∈Θ {Y } + β 2 max θ∈Θ {Y } + β 3 E {Y } + β 4 1 E {Y 2 } -E {Y } 2 , (37)
where

β 1 + β 2 + β 3 + β 4 = 1.
Note that by maximizing this robust objective function, it is sought to raise both the lowest and the highest possible value, the mean, in addition to reducing the dispersion, by reducing the standard deviation.

Additionally, in order to avoid excessively small braking forces, the following probabilistic constraint is imposed

P |Y | > y * ≥ 1 -P r , (38)
where y * is a lower bound for the magnitude of Y , and P r is reference probability. Therefore, the admissible set for the robust optimization problem, denoted by A R , is defined as the subset of A C for which the probabilistic constraint (38) is respected. In this way, the robust optimization problem is formally defined as find an optimal design vector

s opt R = arg max s∈A R J R (s). ( 39
)

Results and discussion

The simulations reported below, conducted in Matlab, use the following numerical values for the deterministic parameters of the mechanical model:

Uncertainty quatification

The calculation of the propagation of uncertainties of X = (X 1 , X 2 ) through the mechanical-mathematical model (32) initially involves the generation of random samples according to the probabilistic model defined by Eq.(28). A set of 4096 random samples for X 1 (top) and X 2 (bottom) can be seen in Figure 6, which also shows some statistics (mean, standard deviation and 95% confidence interval) for this set of values. In Figure 7 the reader can see the statistics shown in Figure 6 compared to the analytical curves for the PDFs of X 1 and X 2 , and histograms constructed with the underlying random samples. It can be observed that the sampling process is well conducted, since the histograms and analytical curves present great similarity.

Note that the brake cam angle X 1 is modeled according to a probability density with a descending exponential behavior, which decays slowly between the ends of the support [α 1 , α 2 ] = [0, 18] o , whereas the spring reaction force X 2 is described by probabilistic law with an increasing exponential density, which grows rapidly from the left to the right extreme of [γ 1 , γ 2 ] = [0, 56] kN.

It is worth noting that, of course, the real system parameters do not follow these probability distributions. These are only approximations of the real distributions, constructed with the aid of the maximum entropy principle and the available information about these parameters. However, as in this paper the authors do not have experimental data to infer the real form of these distributions, in the light of information theory, the PDFs of Figure 7 are the best that can be inferred.

The next step involves the model evaluation in each pair (X 1 , X 2 ) previously generated, which gives rise to the set of possible values for the braking force F h , shown in the top part of fitting obtained by a smooth curve. Mean, standard deviation, and a 95% confidence interval can be seen in both, top and bottom figures. To prove that these estimates are reliable, the authors also show the convergence of the mean and standard deviation estimators, as a function of the number of samples, in Figure 9.

It may be noted that the PDFs of F h and F s have a very similar shape, suggesting that the mechanical model preserves the shape of the spring reaction force distribution. This result is at the least curious and unexpected, since the angular dependencies introduced in the mechanical model by Eqs.( 15) and ( 17) define a structure of multiplicative uncertainty between α and F s , what should make F h not invariant with respect to the input distribution.

This result suggests that the nonlinearity associated with the alpha parameter is very weak, which causes F h to behave as an affine map of F s , and thus to preserve the form of its distribution. This hypothesis is reinforced by analyzing the system response by keeping α distribution and F s support fixed, while the mean value of the latter parameter is varied, assuming the values equal to for 14 kN, 28 kN1 and 42 kN. The probability densities corresponding to these different inputs, and the corresponding outputs of the mechanical system can be seen in Figures 10 and11, respectively. In all cases the input and output PDFs have the same shape.

The results of this study allow one to conclude that uncertainties in α parameter does not have significant influence on the braking force behavior. Simulations propagating only α uncertainties, not included here because of space limitation, demonstrate such an assertion. However, the uncertainty propagation study also shows that the variability of F s cannot be ignored, since it has great influence on the statistical behavior of F h . tion (37). Although the objective function still maintains a smooth appearance, the problem gains a nonconvex status by the irregular forms of constraint.

Once the robust objective function takes into account other design criteria than in (35), its behavior is different from the classical objective function shown in Figure 12, thus having a different optimal point.

It is also worth noting that this second formulation of the optimization problem considers the effects of uncertainties, which in a realistic system are always present, thus offering a design option more suitable for projects that cannot ignore such variabilities.

Summary and conclusions

This work presents a study regarding the optimization and uncertainty quantification of an elevator brake system. The paper starts from an original construction of a safety gear for the brake, for which a mechanical-mathematical model is constructed. Studies involving the quantification of the braking force uncertainties due to the variability in the brake cam angle and the spring reaction force are presented, showing that spring force uncertainties are more significant. The paper also focuses on the optimal design of an elevator brake system, showing through the solution of a robust optimization problem that operating conditions uncertainties can significantly influence its efficiency.

Figure 1

 1 Figure 1 Schematic representation of the CHP 2000 safety gear used by a friction crane brake system, which consists of the following parts: 1 -steel body; 2 -brake roller; 3 -thrust plate; 4 -spring package; 5 -braking cam; 6 -lift guide.

Figure 2

 2 Figure 2 Illustration of a typical friction crane brake system used by lifting devices, which consists of the following parts: 1 -safety gear cabin; 2 -trigger lever; 3 -safety gears connector; 4 -rope speed limiter.

Figure 3

 3 Figure 3 Illustration of the forces (in red) acting on the safety gear steel body and the underlying geometric dimensions (in blue).

Figure 4

 4 Figure 4 Illustration of the forces (in red) acting on the wedge and the underlying geometric dimensions (in blue) and

Figure 5

 5 Figure 5 Illustration of the forces (in red) acting on the brake roller inside the safety gear and and the underlying geometric dimensions (in blue).

  F g = 50 kN; F b = 30 kN; µ 1 = 0.10; µ 2 = 0.10; µ 4 = 0.15; a = 55.0 mm; b = 16.6 mm; c = 52.7 mm; d = 34.5 mm; e = 60.7 mm; f = 0.005 mm; l = 49.0 mm; m = 40.0 mm; n = 17.5 mm; R = 29.0 mm. Regarding the two random parameters, the following information is assumed: [α 1 , α 2 ] = [0, 18] o ; [γ 1 , γ 2 ] = [0, 56] kN; µ X1 = 6 o ; and µ X2 = 42 kN.

Figure 6

 6 Figure 6 Generated samples and statistical measures for the cam angle (top) and the spring force reaction (bottom).

Figure 7

 7 Figure 7 Probability density functions and statistical measures for the cam angle (top) and the spring force reaction (bottom).

Figure 8 .

 8 In the bottom part of the same figure the reader can observe a histogram that estimates the F h PDF form, as well as a nonparametric

Figure 8

 8 Figure 8 Generated samples (top) and probability density function (bottom), with statistical measures, for the braking force.

Figure 9

 9 Figure 9 Statistical estimators convergence for the braking force mean (top) and standard deviation (bottom).

Figure 10

 10 Figure 10 Probability density functions and statistical measures for the spring force reaction with different mean values: 14 kN (top); 28 kN (middle); 42 kN (bottom).

Figure 11

 11 Figure 11 Probability density functions and statistical measures for the braking force with different input mean values for Fs: 14 kN (top); 28 kN (middle); 42 kN (bottom).

Figure 14

 14 Figure 14 Contour map for the robust objective function with the optimum point is indicated by a cross.

For this value, which corresponds to the midpoint of the support, the distribution degenerates into an uniform.

Funding

This research was financed in the framework of the project Lublin University of Technology-Regional Excellence Initiative, funded by the Polish Ministry of Science and Higher Education (contract no. 030/RID/2018/19). The fourth author acknowledge the financial support given by the Brazilian agencies Carlos Chagas Filho Research Foundation of Rio de Janeiro State (FAPERJ) under grants E-26/010.002.178/2015 and E-26/010.000.805/2018 and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001. Compliance with ethical standards

Optimization

In this section the problem of optimum design of the brake system is addressed. The geometric dimensions s = (a, c) are used as design variables, considering as admissible region 50 ≤ a ≤ 60 mm and 50 ≤ c ≤ 55 mm.

The optimization problem is solved using the standard sequential programming quadratic (SQP) algorithm obtained from Matlab (see chap. 18 of [START_REF] Nocedal | Numerical Optimization[END_REF]), being the contour map of the objective function (35) illustrated in Figure 12, which also highlights the optimum point found.

Despite the fact that this result offers a starting point for an optimal project for the brake system, it does not take into account the effect of the uncertainties underlying the operating conditions, which can considerably affect the system response, as shown in the previous section. In this way, robust optimization presents itself as a natural alternative.

For the robust optimization problem the design variables s = (a, c) are considered once more, with the same ranges of admissible values used above. The uncertainties in α and F s are modeled as in section 3, and the probabilistic constraint is characterized by the parameters y * = 0.5 kN and P r = 5%. The convex weights β 1 = β 2 = β 3 = 2/10 and β 4 = 4/10 are adopted in the robust objective function.

This second problem is much more complex because the constraint to be satisfied is nonconvex, offering additional challenges to the numerical solution procedure. But for the values described above the SPQ algorithm is able to find a solution.

The reader can see the contour map of the probabilistic constraint (38) in Figure 13, while Figure 14 presents the contour map for the robust objective func-

Conflict of Interest

The authors declare that they have no conflict of interest.