1. BIFURCATIONS AND TIPPING POINTS IN CLIMATE MODELS
Thermohaline Circulation
and Ice Sheets

2. What is a bifurcation?
Differential equations often contain parameters (constants), which can change values depending on the specifics of an equation.
Often times, a small change in a parameter can drastically change the long-term behavior of solutions to a differential equation. When a change in a parameter drastically effects the long-term behavior of a differential equation, we say that a bifurcation has occurred.
Other Terms:
Equilibrium Solutions – A solution to a differential equation at which the rate of change is 0 (i.e. dy/dt = 0). Other solutions tend to diverge or converge to equilibrium solutions as time increases.
Stability of Solutions:
Stable – If solutions to a differential equation approach the equilibrium solution, we say the equilibrium solution is stable.
Unstable – If solutions to a differential equation diverge from the equilibrium solution, we say the equilibrium solution is unstable.

3. An Example of a Saddle Node Bifurcation for the Differential Equation dy/dt= y2 – 2y + a
Saddle Node Bifurcations:
Saddle node bifurcations occur when a pair of stable and unstable equilibrium solutions collide.
The two equilibrium solutions eventually disappear as the parameter is varied.
Let us consider equilibrium solutions, dy/dt = 0.
We can see that the equation 0 = y2 -2y + a has different numbers of solutions depending on the parameter a. Thus, we have different numbers of equilibrium solutions depending upon a.
We can see that the discriminate for the equation 0 = y2 -2y + a is 4 – 4a. If a > 1, we have no equilibrium solutions. If a = 1, we have one equilibrium solution. If a <1, we have two equilibrium solutions.
Thus, we can conclude a saddle node bifurcation occurs at a = 1 which changes the number of equilibrium solutions to the differential equation.

4. Thermohaline Circulation
Thermohaline Circulation is driven by density differences in ocean water.
Dense water sinks.
Density of water is affected by temperature and salinity.
Cooler water is denser than warmer water.
Saltier water is denser than less salty water.
Warm water near the equator travels at the surface of the ocean to higher latitudes were it becomes cooler. The cooler water is less dense and sinks, traveling deep in the ocean. The sinking of cold water in the high latitudes is called North Atlantic Deep Water Formation (NADWF).
The thermohaline circulation carries about 1.2 x 1015 Watts of heat northwards and carries about 17 x 106 Sverdrups(1 Sverdrup = 1 x 106 m3/second) of water!

5. The Stommel Model
The Stommel Model is one of the simplest models for thermohaline circulation.
It consists of two boxes representing the low latitudes and high latitudes respectively.
The main differential equation for the Stommel model is:
ds/dt = Π - |1-s|s – Ks
s - Represents the difference in salinity between the low latitude and high latitude boxes
t – Represents the time
Π (parameter) – Represents the amount of freshwater addition
K (parameter) – A ratio for measuring the transport of water due to wind gyres to the transport of water due to the thermohaline circulation
Note: Wind gyres are caused by the Coriolis effect.

6. Equilibrium Solutions for s < 1
Equation: ds/dt = Π - |1-s|s – Ks
Let us consider equilibrium solutions (ds/dt = 0).
Let us consider s < 1 (this allows us to disregard the absolute value and solve the equation).
For s < 1, we can solve the expression 0 = Π – (1-s)s – Ks for s using algebra and the quadratic formula.
We get the two equilibrium solutions:
s(1) = ½[(1+K) − 𝐾 2 − 4Π ]
s(2) = ½[(1+K) 𝐾 2 − 4Π ]
Because s < 1, the difference in salinity between the two boxes is negligible, and the temperature difference between the high latitude and low latitude boxes creates the density difference in the ocean water. Because the temperature difference dominates the density, water travels to the high latitudes were it sinks because of cooler temperatures.
Thus, s < 1 represents the way the current thermohaline circulation functions: we have North Atlantic Deep Water Formation (NADWF), and the deep water travels the ocean.
Note: It can also be shown that s(1) = ½[(1+K) − 𝐾 2 − 4Π ] is a stable equilibrium solution, where as s(2) = ½[(1+K) 𝐾 2 − 4Π ] is an unstable equilibrium solution.
If solutions to the differential equation converge to s1, the thermohaline circulation will run as normal.

7. Equilibrium Solutions for s > 1
Equation: ds/dt = Π - |1-s|s – Ks
Let us consider equilibrium solutions (ds/dt = 0).
Let us consider s > 1 (this allows us to disregard the absolute value and solve the equation).
For s >1, we can solve the expression 0 = Π + (1-s)s – Ks for s using the quadratic formula.
We get the expression s(3) = ½[(1-K) −𝐾 Π ].
Note: We would also get a fourth solution, but this would contradict our assertion that s >1.
Because s > 1, the difference in salinity between the two boxes creates the density difference in the ocean water. It can be shown that tropical water is saltier than polar water; thus, for s >1, we have water traveling to the equator where it sinks due to the salinity of the tropical waters.
In this case, the thermohaline circulation is reversed. Thus, if solutions converge to s3, we have a reversal of the thermohaline circulation!

8. Stommel Model Bifurcations
Equilibrium solutions:
s(1) = ½[(1+K) − 𝐾 2 − 4Π ],
s(2) = ½[(1+K) 𝐾 2 − 4Π ],
s(3) = ½[(1-K) −𝐾 Π ]
Like we saw in the first example, different values of K (a parameter) and Π (freshwater addition) can effect our equilibrium solutions.
Consider K = 0:
If Π > ¼, the expression under the square root 1+𝐾 2 − 4Π becomes negative for s(1) and s (2). In other words, the equilibrium solutions s1 and s2 disappear when Π > ¼ for K =0.
Thus, we conclude a saddle-node bifurcation occurs at Π = ¼ for K = 0 . The solutions s1 and s2 are annihilating each other, making both solutions disappear. What we are left with is s3, a stable solution. Thus, solutions to our equation will converge to s3 for Π > ¼. In other words, our thermohaline circulation will be reversed!
Other cases of K:
Note: As we increase the parameter K, the movement of the ocean is affected more by wind gyre rather than density differences.
Because the thermohaline circulation causes the saddle-node bifurcation to occur, the bifurcation becomes less pronounced for increasing K.

9. Graphs of Bifurcation
Main features: The x-axis represents the amount of freshwater added (Π), and the y-axis represents the maximum flow strength of the thermohaline circulation or (1-s).
When K =0, we can see that at Π = 0.25 our saddle-node bifurcation occurs. A slight increase in freshwater over 0.25 destroys the solutions s1 and s2. We are left with s3, a stable solution representing the reversal of the thermohaline circulation.
Also, notice increasing freshwater always reduces the flow strength of the thermohaline circulation because it decreases the density of the water.

10. Conclusions about the Stommel Bifurcation
Increasing the amount of freshwater by a small amount can drastically affect the flow strength of the thermohaline circulation.
The closer K is to 0, the more prone the thermohaline circulation is to the saddle node bifurcation. Increasing the freshwater input for a small K value can reverse the thermohaline circulation.
Upsetting the flow of the thermohaline circulation would drastically change the climate. The thermohaline circulation carries so much heat northward.
Note: The Stommel Model is an extremely simplified model for the thermohaline circulation. There are much more complex models out there; however, even the complex models behave similar to the Stommel Model.

11. Less Math, Overview of Ice Sheets
Why do we want to discuss ice sheets? Ice sheets can have a huge impact on the thermohaline circulation.
How do ice sheets form? Ice sheets form as snow falls in the winter and does not melt completely in the summer. Over millions of years, the snow accumulates into packs of ice and compresses the old layers.
Each winter snow and ice accumulate on the ice sheet. In the summer, the ice melts. If the snow accumulation in the winter does not replenish the melting of the summer, the ice sheet shrinks.

12. Young Dryas Event
Younger Dryas event – An unexpected cooling just as the Earth was warming up after the last ice age.
As the earth was warming, the ice sheet over North America began to melt, cutting a channel into the North Atlantic, increasing the input of freshwater.
As we have seen with the Stommel Model, a small increase of freshwater input can have disastrous results.
Because of the increase of freshwater input into the ocean, the thermohaline circulation shut down, cutting of warm surface water traveling to the north.
This started the Younger Dryas cooling.
Eventually, the re-start of the thermohaline circulation ended the Dryas cooling, and the earth went into a warmer era.
Main point: Increasing temperature has caused massive ice sheet melting in the past. This melting ice has caused bifurcations in the thermohaline circulation which cooled our climate.

13. Arctic Ice Loss Today
Today, the minimum summer area cover of the Arctic ice sheet is declining.
We had the second lowest areal coverage in 2008 and fourth lowest areal coverage in 2009
Similarly, the winter area cover of Arctic ice sheet is declining.
From 1997 to 2007, we lost 1.5 million km2 of ice.
Melting ice causes other factors that in turn lead to more melting (positive feedback).
Example: Melting ice exposes more dark ocean water which increases absorption of solar radiation.
Many models are projecting an ice-free Arctic starting around summer
It remains to be seen how this increasing freshwater input into the ocean will change our future climate.
We have shown how increasing freshwater input into the Atlantic has upset the thermohaline circulation in the past.

14. Greenland Ice Sheet Shrinkage and Tipping Points
The Greenland ice sheet is losing mass at an increasing rate.
Surface mass balance - The accumulation vs. melting at the surface of an ice sheet.
If the warming of the earth continues, the Greenland ice sheet will be committed to irreversible (bifurcation) meltdown if the surface mass balance goes negative.
As the surface mass balance of the ice sheet declines, the altitude of the ice sheet shrinks.
As the altitude of the ice sheet decreases, the ice sheet increases in temperature.
Thus, with decreasing altitude, the ice sheet continues to get warmer and warmer until the bifurcation becomes irreversible (positive feedback).
In models, just an increase in three to six degrees of the earth will cause the Greenland ice sheet to reach an irreversible tipping point.

15. Consequences of Melting Ice Sheets
One consequence of melting ice is massive sea level rise affecting any land mass around water.
Most importantly, melting ice sheets will greatly affect the thermohaline circulation, causing a much cooler earth.
We have seen in the Younger Dryas event how this process has taken place.
Melting freshwater ice will decrease the density of ocean water.
Decreasing the density upsets the North Atlantic Deep Water Formation.
The thermohaline circulation will be slowed, reducing the amount of warm surface water traveling to the north, making the earth much cooler.

16. Conclusions
Bifurcations or tipping points are very important in climate models: changing a small parameter such as temperature or freshwater input can cause drastic changes for our climate.
With respect to thermohaline circulation, we saw how a bifurcation caused by increasing freshwater input can drastically change the flow rate of the thermohaline circulation. The shut down of the thermohaline circulation would greatly change our climate.
Similarly, bifurcations can occur in ice sheets in which melting becomes irreversible. We saw this with the Greenland ice sheet.
Melting of ice sheets could in turn lead to bifurcations in the thermohaline circulation.
In conclusion, bifurcations are very important with respect to climate models. Knowledge of a tipping point in a climate system is extremely important to understanding how a small change in our climate can drastically change a climate system such as thermohaline circulation or various ice sheets.

17. References
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Willeit, Matteo. "Klimageschichte Ubung 9 Thermohaline Circulation - the Stommel Box Model." Accessed October 14,
Rahmstorf, Stefan. "Bifurcations of the Atlantic Thermohaline Circulation in Response to Changes in the Hydrological Cycle." Nature (1995): Web. 19 Oct. 2014 
 "National Snow and Ice Data Center." Quick Facts on Ice Sheets. Accessed November 2,
 Thompson, J. Michael T., and Jan Sieber. "Predicting Climate Tipping As A Noisy Bifurcation: A Review." International Journal of Bifurcation and Chaos 21 (2010):
Lenton, Timothy M. "Arctic Climate Tipping Points." Ambio 41.1 (2012): Web. 19 Oct
Lenton, Timothy M. "Earth System Tipping Points." Accessed October 14, pdf.
Bergman, Jennifer. "Thermohaline Circulation: The Global Ocean Conveyor." - Windows to the Universe. January 26, Accessed October 14,
"OCEANS & SEA LEVEL RISE Consequences of Climate Change on the Oceans RISE." Climate Change and Sea Level Rise. Accessed October 14,
Kloeppel, James. "News Bureau | University of Illinois." Global Warming Could Halt Ocean Circulation, with Harmful Results. Accessed December 6,