Stability of Thermohaline circulation with respect to fresh water release
aa r X i v : . [ phy s i c s . a o - ph ] M a y Stability of Thermohaline Circulation with respect to fresh waterrelease
Ajay Patwardhan, Vivek Tewary
St. Xavier’s College, Mumbai
October 31, 2018
Abstract
The relatively warm climate found in the North-Western Europe is due to the Gulf Stream that cir-culates warm saline water from southern latitudes toEurope. In North Atlantic Ocean the stream givesout a large amount of heat, cools down and sinks tothe bottom to complete the Thermo-Haline Circula-tion. There is considerable debate on the stability ofthe stream to inputs of fresh water from the meltingice in Greenland. The circulation, being switched off,will have massive impact on the climate of Europe.Intergovernmental Panel on Climate Change (IPCC)has warned of this danger in its recent report. Ouraim is to model the Thermo-Haline Circulation at thepoint where it sinks in the North-Atlantic. We createa two- dimensional discrete map modeling the salin-ity gradient and vertical velocity of the stream. Welook for how a perturbation in the form of fresh waterrelease can destabilize the circulation by pushing thevelocity below a certain threshold.
In the ocean, water motion is not only generated bywind-forcing. The horizontal temperature and salin-ity differences caused by climatic influences at thesea surface, cause density differences which initiatecirculation. Such circulation requires that expansionshould take place at a higher pressure than contrac-tion does. In terms of ocean and water temperature,this means that the heat source must lie at a lowerlevel than the cold source. In the ocean, the heatand cold sources are located at the same level, at thesea surface. Water close to a heat source attains alower density than water close to a cold surface. Itbecomes lighter and spreads at the surface in the di-rection of the cold source. For continuity, water be-low the heat source will ascend and water below thecold source will descend and while spreading below, it will become warmer through heat conduction andmixing. In the layers down below, cold water movesfrom the higher latitudes to the lower ones and in theupper layer, warm water flows in the opposite direc-tion. This is not the entire story. At the sea sur-face globally, there are regions rich in salinity as wellas those deficient of salinity. The former has moreevaporation and ice formation (a cold source) and thelatter has more precipitation, continental run-offs andice-melting (a heat source). This causes a haline circu-lation. In places, the Thermal and the Haline circula-tions act in the same direction, reinforcing each otherto form what is called the Global Thermohaline Circu-lation (THC). It is often referred to as the Great Con-veyer Belt.In the North Atlantic, it manifests itself asthe meridional overturning current (MOC). This cur-rent, through the Gulf Stream and the North AtlanticCurrent, transports large quantities of warm water tothe northern latitudes. This has a strong effect onclimatic conditions. Compared to the correspondingparts of North America, Northern Europe has a muchmilder climate due to this heat transport. It has beenproposed by the Intercontinental Panel on ClimateChange (IPCC) that the challenge of global warmingis real and it is man- made. There are concerns [1] thatglobal warming can affect the stability of the MOC.During the last deglaciation (whose remnants are theGreat Lakes in the North America), a large amountof melt water entered the North Atlantic causing ashallower THC cell. North Atlantics ice cover has sig-nificantly changed over the past 20,000 years resultingin drastic climatic changes. However the ice cover overGreenland is much more massive than the polar seaice cover. Melt water from Greenland could trigger aswitchover of the MOC.
The aim here is to create a simple 2-D discrete modelof MOC. The amount of scaled fresh water intake isaken as a parameter. The impact of its variation isstudied. Our starting point is a modification of anMOC model by Timmerman, Lohmann and Mona-han [2, 3, 4]. ˙ x = x (1 − x ) + µ + yh (1)˙ y = hT x − g + αT y (2)Discretization yields x n+1 = ax n (1 − x n ) + µ + y n h (3) y n+1 = hT x n − g + αT y n (4)Now we shall fix some values for the parameters of themap. As the depth to which the meridional overturn-ing descends is approximately 3 km, we shall take h= 3000. We shall take the proportionality constant’ α ’ as 1. T, we take as 20. g, we keep as the accelera-tion due to gravity on Earth surface, 9.8. With thesevalues, the map much simplifies to x n+1 = ax n (1 − x n ) + µ + y n y n+1 = 7 . x n − . . y n (6) The variation of the map with a is not of much inter-est to us. µ is a scaled variable and we shall study itsvariation in the range (0 , µ = 0 . a = 1 .
1. The reason for the choice of ’ µ ’ is thatwe are talking of the North Atlantic, where freezingrather than melting predominates, and physically thatmeans a small, if any, fresh water intake for THC.Thefixed points of the system are¯ x = − . , ¯ y = − . x = 0 . , ¯ y = − . Now our intent is to find out what happens to the sys-tem if suddenly a large amount of melt water finds its way into the MOC. This would correspond to a largevalue of µ . µ = 0 . x = − . , ¯ y = − . x = 0 . , ¯ y = − . We are interested in how the stable fixed points of themap vary with change in µ . This can be visualised bythe Bifurcation Diagram for the range of (0, 1).Thebifurcation diagram shows a smooth slowing down ofthe stream velocity with respect to µ . Close to µ ∼ . µ B remains almost fixed at ∼ . The interpretation that we make of this is that a grad-ual change in would cause the stream to slow downonly gradually and only a sudden deglaciation lead-ing to release of a massive amount of fresh water cancause the Gulf Stream to slow down. Therefore, itcan be said that this is a low probability but disas-trous event. What we would like to further do withthe map is to standardise the values of the variablesusing real values and to see how the discrete time steptranslates into physical time because this would giveus some idea of the time in which the system can getdestabilised.2
10 20 30 40 50 60 70 80 90 100−10−8−6−4−20246810 timestep, i V e r t i c a l V e l o c i t y C o m ponen t, y V e r t i c a l V e l o c i t y C o m ponen t, y Figure 1: Time Series and Phase Plot showing convergence for low fresh water intake S c a l ed S a li n i t y G r ad i en t, x V e r t i c a l V e l o c i t y C o m ponen t, y Figure 2: Time Series and Phase Plot showing convergence for high fresh water intake3 .1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−10−9−8−7−6−5−4−3−2−10 Variation of µ V e r t i c a l V e l o c i t y C o m ponen t, y (a) T=20 µ V e r t i c a l V e l o c i t y C o m ponen t, y (b) T=100 µ V e r t i c a l V e l o c i t y C o m ponen t, y (c) T=500 µ V e r t i c a l V e l o c i t y C o m ponen t, y (d) T=1000 Figure 3: Bifurcation Plot showing variation of vertical velocity with change in µ for different values of parameter’T’ 4 eferenceseferences