Download Exploring the Possibility of Altered Ocean Circulation Patterns Using

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Adrienne Propp
ES 112 Semester Project
9 May 2016
The Day After Tomorrow: Exploring the Possibility of
Altered Ocean Circulation Patterns Using the Second Law
of Thermodynamics
Climate change, one of the most urgent and universal issues of our time, involves a
complicated web of interrelated processes, many of which are quite complicated, themselves. As
a result, although there is general consensus that climate change is occurring, it is difficult to
determine what its effects will ultimately be or when they will be realized. As this has
potentially critical consequences for society, a deeper understanding of how human actions affect
the climate system must be pursued.
Ocean circulation, one of the major processes affecting the climate system, is believed by
some to be undergoing a fundamental change. Specifically, warmer temperatures and higher
levels of atmospheric CO2 are believed to be contributing to the weakening of what is
colloquially called the “great ocean conveyor belt,” or Earth’s network of ocean currents
(Marshall, 2012; Weijer, Maltrud, Hecht, Dijkstra, & Kliphuis, 2012). If true, this could be
indicative of the level of severity of anthropogenic climate change, and have potentially
disastrous consequences.
In this paper I will first provide a brief overview of ocean circulation’s role in the global
climate system. I will then discuss the nature of the reports being made. Finally, I will present
ocean circulation in the context of the Second Law of Thermodynamics, and discuss an
investigation of the validity of these claims using this thermodynamic perspective. This
approach is valuable because “the sequence of all natural processes is determined by the
principle of entropy increase” (Fenn, 1982, p. 242) – in other words, the Second Law of
Thermodynamics determines the course that a system will actually follow when multiple courses
satisfy the First Law of Thermodynamics, or conservation of energy.
Ocean Circulation and the Global Climate System
Earth’s climate is influenced by many factors, including solar radiation, wind, ocean
currents, and the interactions between them. As the oceans cover about 71% of the Earth, they
are unsurprisingly a major component of the global climate system. Indeed, the oceans are both
responsible for and responsive to many changes in environmental conditions. (Pidwirny, 2007)
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More than their sheer size and volume, the influence exerted by oceans comes largely
from their circulation patterns. Ocean currents transport enormous amounts of heat around the
world and absorb gases in the atmosphere. In fact, oceans are estimated to transport a maximum
of heat just under 3 petawatts, and to have absorbed up to half of all of the CO2 produced by the
burning of fossil fuels since the beginning of the industrial revolution (Bollmann et al., 2010).
Therefore, whether the climate will change in the future, and by how much, is strongly linked to
ocean circulation.
Ocean circulation is both mechanically and thermally driven. It is mechanically affected
by wind stresses, waves, and the like. Thermally, ocean circulation is affected by the sun, via
radiative heating, and the core of the Earth, via geothermal heating. (Bollmann et al., 2010)
(Rahmstorf S. , Thermohaline Ocean Circulation , 2006) Overall, most of the large-scale
circulation is driven by density, which depends on salinity and temperature. Thus, this
characterization of ocean circulation is often called thermohaline circulation. Figure (1) gives a
broad overview of what is
colloquially called the “great
ocean conveyor belt”, or the
manifestation
of
this
thermohaline
circulation.
Density, a measure of how
tightly
packed
together
molecules are, governs the
direction, location, and depth
of currents (Ocean and
Climate - The Odd Couple).
Warm water and fresh water
rise due to low density, while cold water and salty water sink due to high density. Furthermore,
water in regions with high concentrations of heat and salt diffuses into regions with low
concentrations, dissipating the unequal distributions of heat and salinity, and thus diminishing
the density gradients.
These unequal distributions of heat and salinity are caused by precipitation and
evaporation, as well as differential heating between the polar and equatorial regions. Overall,
there are net gains of heat and salt in the equatorial regions, and net losses of heat and salt in the
polar regions (Shimokawa & Ozawa, 2002). This flux imbalance results in the inhomogeneous
distribution of temperature and salinity at the ocean surface that is often considered to be the
driving force behind global-scale thermohaline circulation.
One manifestation of this is the water mass produced by these convective processes in the
Arctic, termed the North Atlantic Deep Water (NADW). Warm, salty surface water spreads
from the tropics into the North Atlantic. In the Labrador and Greenland seas, the cold
Figure 1: The "Great Ocean Conveyor Belt"
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temperature and formation of ice increases the region’s water density, pulling it down to rest on a
layer of even denser, deeper water produced by convection in the Antarctic, the Antarctic Bottom
Water (AABW), that extends up the entire length of the Atlantic Ocean. (Bollmann et al., 2010)
(Gordon, 2004)
Approximately one third of the world’s ocean water is involved in thermohaline
circulation, or about 400,000 cubic kilometers of water (Bollmann et al., 2010). Although it
transports about 20 million cubic meters of water per second, there may be hundreds of years
between sinking and returning to the surface (Conkling, Alley, Broecker, & Denton, 2011).
Indeed, oceans react very gradually to change. As a result, though ocean circulation exerts a
huge influence on, and is a powerful indicator of, the state of the global climate, the impacts of
climate change evident in the oceans today do not yet reflect the total extent of climate change
already caused by human activity (Bollmann et al., 2010). Thus, the decisions made today may
have consequences that extend far into the future.
Are Humans Altering the Course of Ocean Circulation?
In Florida and much of the southeastern United States, citizens anxiously await
predictions of the severity of the approaching hurricane season each spring. This year, there is
growing concern that the presence of a “cold blob,” or a region of unusually cool water, in the
North Atlantic will affect Atlantic Ocean currents. This could potentially initiate a transition
from El Niño to El Niña and increase the likelihood of a severe hurricane season (MacMath,
2016). Scientists’ concern reaches even farther – some climate models have predicted that the
Atlantic turnover process will weaken by about 25% by the end of this century (Bollmann et al.,
2010). The 2004 film, The Day After Tomorrow, may have been inspired by, and probably
aggravated, public concern over the issue.
Indeed, there is evidence that ice sheets in the North Atlantic are melting at an increasing
rate, discharging large amounts of cold fresh water into the ocean (e.g. Frauenfeld,
Knappenberger, & Michaels, 2011; Marshall, 2012). Furthermore, rising Arctic temperatures
inhibit the formation of sea ice, diminishing the level by which salinity is increased in this
region. As a result, the increase in water density that usually occurs in this region is diminished,
weakening the convective forces that pull the water down, driving circulation. As a result, many
believe the Atlantic Meridonal Overturning Circulation (AMOC) will be weakened, affecting
global climate patterns as well as the ocean’s uptake of CO2, likely contributing to a positive
feedback cycle. (Bollmann et al., 2010)
These changes are significantly tied to human activity, such as the burning of fossil fuels.
However, information about past environmental conditions, drawn from ocean floor sediment
and paleo-data, indicates that shifts in oceanic circulation patterns have occurred in the past, and
corresponded to shifts in overall climatic conditions. Scientists believe that certain cold climate
episodes, occurring over a few decades or even less, were caused by abrupt disturbances in the
ocean currents of the North Atlantic. (Bond et al., 1993 as cited in Rahmstorf, 1997) The
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significance of this is multifaceted. On one hand, it is concerning that a major shift in ocean
circulation patterns is indeed a possible scenario. On the other hand, changes in ocean
circulation are a naturally occurring phenomenon that may be out of our control. That said, there
is evidence that this naturally occurring phenomenon is encouraged by the very changes we are
exerting on our environment today. For this reason, a holistic and thorough investigation of the
likelihood of this occurring is critical to ensuring that we are prepared for the potentially
inevitable consequences of our actions.
One Method of Investigation: The Second Law of Thermodynamics
Ocean Circulation in the Context of the Second Law
As it turns out, thermohaline circulation is a fantastic example of the second law of
thermodynamics, manifested in the dissipation of heat and salt gradients. In this context, the
second law of thermodynamics will be considered with the ocean as an open dissipative system –
it is open because the system exchanges heat and salt with the surroundings, and dissipative
because the system is not at equilibrium. Fluxes of heat and salt between the ocean and its
surroundings (most notably the atmosphere) produce gradients of temperature and salt
concentration, which processes like thermohaline circulation act to diminish. In effect, these
processes occur in an attempt to bring the system closer to equilibrium, towards a state of higher
entropy. The rate at which the ocean system approaches equilibrium is described as the rate of
entropy production. (Ozawa & Shimokawa, 2000)
From the Second Law of Thermodynamics, we know that the overall entropy of the
universe must increase.
𝑑𝑆!"# ≡ 𝑑𝑆!"## + 𝑑𝑆!"! ≧ 0 1, where
𝛿𝑄!"# 𝑑𝑈 + 𝑝𝑑𝑉 − ! 𝜇! 𝑑𝑁!
=
𝑇
𝑇
Therefore, for any spontaneous process to occur, it must result in an overall increase of entropy.
We also know that any spontaneous process in any isolated system always results in an increase
in the entropy of that system (Fenn, 1982).
𝑑𝑆 =
𝑑𝑆!"#$%&'( ≧ 0 1
Furthermore, entropy is additive, and the total change in entropy of a system must be the sum of
internal changes (i) and exchanges (e) with the surroundings.
𝑑𝑆!"! = 𝑑! 𝑆 + 𝑑! 𝑆 1
Internal entropy changes are the result of irreversible processes, and entropy exchanges are the
result of heat exchanges in a closed system, or heat and matter exchanges in an open system.
1Obtainedfromlecturenotesandacompilationofarticles,books&onlinesources
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Thus, for an open system,
𝑑𝑆!"# =
!!!"##
!!"##
+
!"
!
!!!
!"##
𝑑𝑁!,!"## + 𝑑𝑆!"! ≥ 0 1
or,
𝑑𝑆!"# = −
!!!"!
!!"##
−
!
!"
!!!
!"##
𝑑𝑁! + 𝑑𝑆!"! ≥ 0 1
since the fluxes from the perspective of the system and the perspective of the surroundings are
equal and opposite (𝛿𝑄!"## = −𝛿𝑄!"! and 𝑑𝑁! = 𝑑𝑁!,!"! = −𝑑𝑁!,!"## ). Combining the above
expressions,
𝑑𝑆!"# = 𝑑𝑆!"## + 𝑑! 𝑆!"! + 𝑑! 𝑆!"! ≥ 0
!!!"!
𝑑𝑆!"# = −
𝑑𝑆!"# = −
!!!"!
!!"##
−
!!"##
!
−
!
!"
!!!
!"##
!"
!!!
!"##
𝑑𝑁! +
𝑑𝑁! + 𝑑! 𝑆 + 𝑑! 𝑆 ≥ 0
!!!"!
!!"!
+
!
!"
!!!
!"!
𝑑𝑁! + 𝑑! 𝑆 ≥ 0
Here, the entropy change in the surroundings is a result of the interaction with the system. This
is a very simplified case with many assumptions, such as reversible heat transfer, but the main
idea is that there are separate components of entropy that contribute to ensuring that the overall
entropy is nonnegative.
Shimokawa and Ozawa (2000) formulated expressions of entropy specifically relating to
ocean circulation. In their expressions, they describe the rate of entropy increase of the system –
the ocean, in this case – as depending on heat and salt transports. They do not separate dSsys into
deS and diS. However, they do express dSnet as the sum of dSsys and dSsurr. According to them,
where, 𝑆!"# = 𝑆!"! ,
𝑆!"# =
! ! !"#
!
!"
+ div 𝜌𝑐𝑇𝒗 + 𝑝 div 𝒗 𝑑𝑉 − 𝛼𝑘
!"
!"
+ div 𝐶𝒗 ln 𝐶 𝑑𝑉
2
In the above expression, the first integral term represents the entropy increase rate due to heat
transport, and the second integral term represents the entropy increase rate due to salt transport.
They integrate over volume to account for the transport throughout the entire ocean. They also
describe the rate of entropy change of the surroundings.
𝑆!"## =
!!
!
𝑑𝐴 − 𝛼𝑘 𝐹! ln 𝐶 𝑑𝐴 2
In the above expression, the first integral term represents the entropy increase rate due to heat
flux through the boundary surface, and the second integral term represents the entropy increase
rate due to salt flux through the boundary surface. They use a surface integral because the
2ObtainedfromShimokawaandOzawa,2000
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entropy increase of the surroundings is due to the interaction between the ocean and
surroundings across the boundary of the ocean surface.
Together, the rate of entropy increase of the ocean and the rate of entropy increase of the
surroundings comprise the overall rate of entropy increase due to ocean circulation.
𝑆!!!"# = 𝑆!"# + 𝑆!"##
2
However, before combining the expressions, we can simplify them by making a few
assumptions. When we assume that seawater is incompressible, div v = 0. When we assume that
volumetric heat capacity is constant, 𝜌𝑐 = constant. This removes the divergence terms from the
equation for 𝑆!"# .
𝑆!!!"# =
!" !"
! !"
𝑑𝑉 +
!!
!
𝑑𝐴 − 𝛼𝑘
!"
!"
ln 𝐶 𝑑𝑉 𝛼𝑘 − 𝛼𝑘 𝐹! ln 𝐶 𝑑𝐴 2
Through mathematical manipulation, this expression can be rewritten as the following:
𝑆!!!"# =
𝐹! ∙ grad
!
!
!
𝑑𝑉 +
!
𝑑𝑉 − 𝛼𝑘
!! ∙ !"#$ !
!
𝑑𝑉 2
The first term on the right side of the equation represents the rate of entropy increase due to
thermal dissipation, the second represents the rate of entropy increase due to viscous dissipation
(or the dissipation of kinetic energy as heat), and the third represents the rate of entropy increase
due to the diffusion of salt ions. The gradient function gives directional information about the
gradients of temperature and salinity. Heat flows from hot to cold, the dissipation function is
always nonnegative, and molecular diffusion takes place from high to low concentration –
therefore, this sum should always be positive, resulting in a positive rate of change of entropy,
consistent with the Second Law. (Ozawa & Shimokawa, 2000)
Before continuing with this model, it is important to confirm that these are reasonable
expressions for the rate of entropy increase due to thermohaline circulation. The separation into
surface and volume integrals, as well as the elements of temperature and concentration, indicate
that it likely is reasonable. Indeed, another group, Yan et al. (2004), independently arrived at a
rather similar expression for the rate of entropy increase due to the boundary interactions.
!! !
!" !
!
= −!
!
!
!
∙
!!"
!!"#
!! !
!" !
where
!! !
!"
!
+ !!" +
!!"
=
!
!!
! ! !
!!!
!!!"
𝑑Σ
!
+ ! !!
!!"
𝑑Σ
3
3
is the rate of entropy exchange across the boundary of the ocean system, h refers to
heat, m refers to mass, Σ represents the area of the global sea surface, dΣ represents the area of
each cell, Tsun represents the temperature of the sun, Tsst represents the temperature of the sea
surface, 𝐹!" and 𝐹!" are the shortwave and longwave radiant energy fluxes, respectively, 𝐹!! and
𝐹!! are the latent heat flux and sensible heat flux, respectively, I0 is freshwater flux, and S is
3ObtainedfromYanetal.(2004)
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salinity. This version exhibits similarities to and differences from the expressions proposed by
Shimokawa and Ozawa (2000). We still integrate over the surface to describe interactions across
the boundary. The heat transfer expression still exhibits flux divided by temperature, while the
salt transfer expression still exhibits division by salt concentration. This indicates that
Shimokawa and Ozawa’s (2000) thermodynamic approach to analyzing shifts in ocean
circulation is likely based on sound reasoning.
Shimokawa and Ozawa (2000) found that the entropy increase rate in the steady state is
zero for the ocean system, but positive for the surroundings in both heat and salt transport. If the
entropy change for the system is zero,
𝑆!"#,!"#$%"&' = −𝑆!"#,!"#!!"#$%
Because the entropy change of a system is the sum of its entropy production and the entropy
flux, and we know that the ocean produces entropy through the dissipation of temperature and
salt gradients, there must be entropy lost to the surroundings through boundary fluxes of heat and
salt. In other words, the entropy exchange is negative from the perspective of the ocean system.
This may be partly due to the fact that the gradients are constantly replenished by precipitation,
evaporation, glacial melting, and differential heating of the globe – in effect, the ocean’s
interaction with its surroundings is actually the cause of the gradients that keep the system away
from equilibrium. Overall entropy does increase, however, which makes sense because there are
irreversible dissipative processes taking place, though the system itself stays in a state of
constant entropy.
Using the Second Law to Predict the Feasibility of Various Scenarios
Several methods have been investigated in the attempt to determine whether or not a
significant change in ocean circulation is imminent. Shimokawa and Ozawa (2002) explore the
issue as an initial-value-boundary-condition-type problem, investigating irreversible transition to
a state with a higher rate of entropy production. In this type of analysis, the initial values and
boundary conditions determine the outcome that is specific to the situation in question. In the
case of ocean circulation, these initial values and boundary conditions would describe properties
such as air temperature, water temperature, salinity, wind strength, etc. Under a chosen set of
initial values and boundary conditions, one can then examine the resulting steady states and their
responses to perturbation.
For their investigation, Shimokawa and Ozawa (2002) chose restoring boundary
conditions that were symmetric about the equator, with the initial temperature distribution as a
function of depth and latitude, initial salinity assumed to be constant at 34.9%, and initial
velocity field set to zero. After the system reaches a steady state with northern sinking
(approximately 4000 years), they switch to mixed boundary conditions (restoring condition for
temperature and a fixed flux condition for salinity), perturb it with a salinity flux in high latitude,
and integrate the model described in Shimokawa and Ozawa (2000) for 500 years so that the
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system moves to a state regulated by the perturbation. They then remove the perturbation and
integrate for another 1000 years. If the system finds a new steady state, they repeat this
procedure with the same perturbation, using the new state as the initial state. If it returns to the
original steady state, they repeat this procedure with a new perturbation. They end up finding a
series of steady states of thermohaline circulation that all satisfy the same set of wind forcing and
mixed boundary conditions, as well as the rate of entropy increase for each.
The results of their study are shown in Figure (2) in the Appendix. In general, a positive
salinity perturbation added to a high-latitude region in the northern hemisphere intensifies
northern sinking and weakens southern sinking, while a negative salinity perturbation weakens
northern sinking and intensifies southern sinking. When transitions to new states occurred, the
rate of entropy production was always higher in the final state than in the initial state.
Transitions in the opposite direction did not occur, indicating that these were irreversible
transitions. For these reasons, it appears that a thermodynamic approach to the issue of changing
ocean circulation is extremely useful, as the rate of entropy production seems to govern the
behavior of the circulation patterns.
One concerning result is that, starting from the initial steady state after spin up with no
perturbation, the system transitioned to S1, a southern sinking state. This situation, called the
“halocline catastrophe,” or the collapse of the NADW, has been observed in previous models
(e.g. McPhaden et al., 1992) and is appropriately named due to its potentially disastrous
consequences. Furthermore, in the cases where the initial state was northern sinking, and the
perturbation was a decrease in salinity (paralleling what is actually occurring in the North
Atlantic, with the influx of fresh water), the system always transitioned to southern sinking.
These results are extremely concerning in their implications that the increased melting of polar
ice sheets, such as the Greenland ice sheet, may have extremely drastic effects on global ocean
circulation. If the ocean were to transition to a state of southern sinking, the entire pattern of
global climates to which we have become accustomed would shift – for example, as Western
Europe’s current climate is quite temperate due to the Gulf Stream, this region might become
more like Canada with the onset of these shifts. Even if the current pattern of thermohaline
circulation were to remain intact, but merely weaken, landmasses would likely experience
increased temperatures due to the diminished transport of heat around the globe (Bollmann et al.,
2010). In other words, these results are not to be taken likely.
This approach has limitations, however, as the boundary and initial conditions are very
simplified, and in reality, the perturbation would be fresh water flux rather than salinity flux.
The equation for calculating the rate of entropy production also involves assumptions about
incompressibility and constant volumetric heat capacity, as mentioned earlier. In addition, their
model assumes the entropy increase rate due to wind and tidal dissipation to be negligible. They
are indeed small – about two orders of magnitude smaller than the total rate – but not zero.
Despite these limitations, their approach gives a reasonable indication of how a similar situation
might play out, and supports the hypothesis made by Shimokawa and Ozawa (2002), as well as
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Sawada (1981) and others, that a nonlinear system is likely to move to a state with maximum
entropy production by perturbation.
Conclusion
In analyzing the dependence of ocean circulation on the Second Law of
Thermodynamics, it is clear that significant and prolonged changes in a property affecting
temperature or salinity gradients can cause extreme changes in ocean circulation. Given the
evidence that the North Atlantic is becoming cooler and fresher due to increased melting of
glaciers, the results of Shimokawa and Ozawa’s (2002) experiment are extremely foreboding.
Decreasing the salinity of the North Atlantic seawater directly affects its ability to sink and drive
circulation in that region. Furthermore, the layout of salinity and temperature gradients will be
significantly altered, and thus the most entropically favorable course of ocean circulation may
change, as well. Based on past evidence that such transitions are indeed possible, and
Shimokawa and Ozawa’s (2002) evidence that in many realistic circumstances, they are indeed
favorable, I am unfortunately convinced that these transitions may be in our future.
Thankfully, the time scale over which this will occur is likely many generations, if not
lifetimes. However, this is not a reason to delay action – as discussed earlier, the actions of
today will impact ocean circulation many years from now due to its gradual nature.
Furthermore, it is possible that immediate action to stop anthropogenic climate change could
prevent the perturbation we have caused from becoming large enough or prolonged enough to
cause a complete transition. Conversely, if the North Atlantic ice sheets continue to melt at this
rate, we may reach a critical threshold beyond which a transition is unavoidable (Bollmann et al.,
2010). Unfortunately, the exact location of this threshold is unclear. Either way, it is likely that
the “cold blob” in the North Atlantic will affect weather patterns in the coming years to some
extent, such as the severity of next year’s hurricane season.
The Second Law of Thermodynamics has provided a valuable and nuanced perspective
on this topic. Deeper understanding must obtained yet, however, as it remains unknown where
the current state of the climate system stands in relation to these models. Confirmation of
Shimokawa and Ozawa’s (2002) results by a separate group, as well as an extension of their
method to more initial values and boundary conditions, could provide even more insight into
how imminent the threat of altered ocean circulation patterns really is. Hopefully, with a
combination of robust scientific evidence and effective policy measures, we will be able to
ensure that future generations do not have to cope with the consequences of our era’s actions.
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Appendix
Figure 2: Results of Shimokawa and Ozawa's 2002 experiment
Figure 3: Visualization of the results from Shimokawa and Ozawa's 2002 experiment
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