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Thursday, October 12, 2017

Seven Ways Climate Change Makes Hurricanes Worse

As I’m writing this post, Hurricane Ophelia is forecast to hit Ireland, the first full-strength hurricane to hit the island since 1961, and the tenth consecutive storm this season to reach hurricane strength.  The year 2017 has already been a tragic and record-setting Atlantic hurricane season.  Hurricane Harvey hit Texas as a 1000-year rainstorm, dropping about 11 cubic miles of rain on Houston with enough weight to depress the earth’s crust by a measured 2 centimeters.  And within a period of two weeks, Hurricanes Irma and Maria struck the Caribbean Islands as category 5 hurricanes – the strongest measure on the Saffir-Simpson scale, setting a record for the duration of category 5 storms in one season. 

The obvious question is whether climate change is causing an increase in the frequency or intensity of these storms. 
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Hurricanes and Climate Change
A hurricane is a convection engine.  Warm tropical waters convey heat and humidity to the air over the water.  The warm, humid air is light, and begins to rise at random spots over the ocean.  The warm air cools as it rises, dropping below the dew point.  Water vapor condenses to form clouds and rain.  The condensation of water vapor reduces air pressure, further lowering the air density.  The low-pressure center draws warm air from the ocean surface toward itself, which feeds the rising convection column.

The converging air currents are affected by the Coriolis force, and begin to spin as they approach the growing low-pressure center.  As the storm develops structure, a downward current of air forms as the eye in the center of the storm, returning dry air from high altitude.  The hurricane eyewall of clouds, rain and ferocious winds spins around the central eye.  Surrounding the eye, spiral rain bands develop as subsidiary convection systems, with upward air flow in the rain bands and downward flow between those bands. 

The strength of a hurricane is often limited by high-level winds blowing across the top of the hurricane.  Strong high-level winds effectively decapitate a hurricane by blowing the top off of the convection column.  Hurricanes tend to drift westward in equatorial waters, as the globe spins eastward beneath them; and to drift toward the poles in temperate latitudes.  Areas of surrounding high and low pressure form steering currents, which modify the path of the hurricane as it drifts across the globe.

Climate change is expected to increase the intensity of hurricanes in a number of ways. Here are seven ways in which climate change is expected to make hurricanes worse.
Modified after image by Thompson Higher Education.

Temperature
1)  Average surface air temperature has risen around the globe by about one degree Celsius since 1980.  The particular warming is variable at different times and places, and may be greater over tropical waters at times.  Warmer surface air creates a greater tendency to form thermal convection currents.
Average annual global surface temperature, 1880 - 2016.  Image credit NASA.

2)  Average temperature in the upper 100 meters of the ocean has risen by ½ degree Celsius since 1980.  As with air, temperatures in the ocean vary seasonally and in complex patterns of time and space.  At times, tropical waters will be warmer by more than the average ½ degree Celsius global average.  A warmer ocean surface contributes more heat and humidity to a hurricane.

Average water temperature, 0-100 meters, 1955-2017.  Image credit NOAA.

3)  The average temperature of the ocean water at depth has also risen.  The average temperature of waters from the surface to 700 meters has risen by 1/10 of a degree Celsius since 1980.  Waters from 100 to 200 meters have warmed nearly as much as surface waters.  Hurricane waves churn up deeper water, bringing cooler water to the surface.  In the past, this stirring of deeper water cooled the ocean surface, and acted as a buffer on the intensity of a hurricane.  But now that deeper waters are also warmer, there is less tendency for wave action to moderate the strength of a hurricane. 
Average water temperature, 0 - 700 meters, 1955 - 2107.  Image credit NOAA.

Humidity
The principles of physics mean that higher air temperatures and higher water temperatures mean that more humidity is carried in tropical air before the formation of a tropical storm.  Warmer air raises the water-carrying capacity according to the principle of relative humidity, and higher water temperatures raise the humidity of the air according to the Clausius-Clapeyron equation.  Higher humidity acts in three ways to increase the intensity of a hurricane.

4)  Higher humidity lowers the density of the air, because the water molecule is lighter than the average molecular weight of air.  Intuitively, we tend to think that moist air is “heavy”, perhaps because liquid water seems heavy.  But the molecular weight of water is 20, while the molecular weight of nitrogen is 28, and oxygen is 32, giving dry air a molecular weight of about 29.  Molecules of water vapor occupy just as much space as gaseous molecules of nitrogen or oxygen, thus lowering the density of air.  [If we had a bucket of liquid water and a bucket of liquid air, the liquid air would be heavier.]   Lighter air contributes to stronger convection, which strengthens the hurricane.

5) Air with higher humidity has more moisture to condense, causing a stronger drop in air pressure.  This can lead to stronger winds and more rapid intensification of a hurricane.

6) Higher humidity raises the water-carrying capacity of the hurricane, and contributes to higher volumes of rainfall and flooding when a hurricane makes landfall.  The unusual volumes of rainfall associated with hurricanes Harvey and Maria probably reflect higher humidity caused by climate change.

Winds
7) Finally, it is possible that climate change has reduced the strength of high-level winds, reducing the tendency for these winds to blow the tops off of hurricanes.  Some scientists have observed a decline in the strength of high-level winds in recent years, and tentatively suggest that this may be a result of climate change.  However, the mechanisms by which climate change would affect these winds is unclear, and the proposal is still controversial. 

Quantification
I wrote to “Ask a Climate Scientist” on Facebook, and asked whether satellite data from NASA’s GOES satellites documented higher humidity over the Atlantic since the 1980s, either in actual hurricanes or in general background humidity.  I wanted to know if the data supported the idea that climate change is making hurricanes worse.  Here’s the answer I received:

"The GOES imager series involve technology upgrades and are not well calibrated, and so are not well suited for measuring changes in water vapour over time. 
However, the HIRS instrument aboard the NOAA polar orbiter series which began about the same time is fairly well calibrated, and does show increases in humidity. 
More recently the microwave radiometers on board the AMSU series of satellites also show the increases, as does the global radiosonde and surface-observing networks.  The increases are in line with expectations from thermodynamic principles (the Clausius-Clapeyron equation) and climate models. 
We are pretty confident that these increases are indeed causing a storm to dump more rain now than it would have a few decades ago, all other things being equal.”
Professor Steve Sherwood, Climate Change Research Center, UNSW Australia

I made a brief attempt to quantify changes in the hurricane system in the Gulf of Mexico that are due to climate change.  Assuming a 1.5 degree rise in sea surface temperature, humidity will rise by about 5 percent, from about 75% relative humidity to 80%, at an average daily temperature of 80 degrees F.  Along with rising humidity, air temperatures have risen by about 2 degrees F (global average).  Air density will fall, but not very much, only about ½ of one percent.  This will result in stronger convective activity, but I do not have the knowledge or modeling ability to translate that change into hurricane intensity. 

When water vapor in the air is converted to rain, air pressure drops.  Higher initial humidity will lower air pressure in the center of the hurricane.  This means stronger rotation and stronger winds.  Hurricanes are incredibly efficient at removing humidity from the air.  Almost all of the surface humidity in a hurricane is converted to rain, as convection drops the temperature of the air from 80 degrees Fahrenheit at the surface to minus 130 degrees F at the cloud tops.  But the initial saturation pressure of water in air is fairly small.  At 86 degrees F and 80 percent humidity, air contains only 3.3 percent water vapor.  Although climate change has raised the humidity by 5 percent, this means that the surface air in a hurricane now contains 3.5 percent water vapor.  When the vapor is converted to rain, the difference in air pressure is 0.2 percent. 

So, temperature and humidity reduce the air density in a hurricane by 0.5%; additional rain reduces the air pressure by another 0.2 %, for a total climate-change reduction in air pressure of 0.7%. 

Conclusion
Climate change produces higher temperature and humidity.  Those changes push the physical processes of a hurricane toward stronger convection, more rapid intensification, higher wind speeds, and greater rainfall. 

Quantifying those changes is difficult.  Without sophisticated modeling, it is impossible to say whether the small changes in air density and water vapor can result in a major change to a storm system.  But it is important to note that hurricanes are feedback systems.  Hurricanes start as a mild swirl of air over the water, or a rain squall no different than any other rain squall.  But like the proverbial “butterfly effect”, a small change in the initial conditions of the hurricane may result in profound changes in the ultimate intensity of the storm.  Feedback mechanisms in the convection system create the hurricane; it would not be surprising if those same feedback mechanisms amplify the small changes due to climate change to create monster storms. 

I am generally critical of strictly empirical reasoning in science.  Science is about providing explanations, identifying, observing and measuring processes which change the world.  But empirical evidence can support scientific reasoning, and give a clue that an explanation is on the right track.  Currently, the remarkable 2017 hurricane season is supporting the notion that Climate Change is producing stronger, more frequent storms, with more rapid intensification and heavier rain.
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References:
Temperature of cloud tops -90 degrees C.

Chart showing mass of water contained in air at 50% and 100% humidity, as a function of temperature.
Air with 80% humidity at 86 degrees Fahrenheit contains about 21 grams of water per kilogram of air.

Tells us there is a roughly 3 percent increase in average atmospheric moisture content for each 0.5 degrees Celsius of warming

Air density calculator

Average annual humidity for places in Texas.

Average annual humidity for places in Florida.

Temperature change for mid-Gulf surface waters, 1975 to the present.  Average temperatures have increased by 1.5 degrees F; high temperatures have increased by about 3 degrees F.

Average Gulf of Mexico air temperatures, by month.

Mass of water in air at 50% and 100% humidity, as a function of temperature.

Hurricane facts. 

Cloud top temperatures for hurricane Ingrid, 2013.

Partial pressure of water in saturated air, as a function of temperature.

Standard Air Pressure
14.70 psi
1013.25 millibars
Air Density @ 80 F & 75% humidity:  1.166 kg/m3
Air Density @ 82 F & 80% humidity:   1.16 kg/m3

Personal Communication from Profesoor Steve Sherwood, Climate Change Research Center, UNSW, Australia:

However, the HIRS instrument aboard the NOAA polar orbiter series which began about the same time is fairly well calibrated, and does show increases in humidity. 

More recently the microwave radiometers on board the AMSU series of satellites also show the increases, as does the global radiosonde and surface-observing networks.  The increases are in line with expectations from thermodynamic principles (the Clausius-Clapeyron equation) and climate models. 

We are pretty confident that these increases are indeed causing a storm to dump more rain now than it would have a few decades ago, all other things being equal."

Real time and archived statistics on global cyclone energy.

Friday, September 22, 2017

Flooding in Houston: Hurricane Harvey and Climate Change

September 2017 has been an active and violent hurricane season in the Atlantic tropical zone.  Most of this post was written following Hurricane Harvey, and before Hurricanes Irma and Maria.  Hurricane Harvey dumped record-setting volumes of rain on the south Texas coast.  Hurricanes Irma and Maria, occurring in a two-week period, were the strongest hurricanes on record in the open Atlantic Ocean.  For many years, researchers have warned that climate change would produce stronger hurricanes -- it seems that the future has arrived.

In the simplest analysis, I would ask the following question.  Is it more likely that Hurricane Harvey was a completely natural, unlikely event with a probability of 1:1000, or is it more likely that the storm was made worse by climate change, according to well-understood physical principles and predicted by scientists for over two decades? 

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In August 2017, Houston Texas became the face of climate change.   More than 20 inches of rain fell over an area of 28,949 square miles; > 30 inches over 11,492 square miles; and > 40 inches over 3643 square miles.  The maximum rainfall of 52 inches broke the record for rainfall from a single storm in the contiguous United States.  The intensely flooded area received 11 ½ cubic miles of water.  Within a few days, over 300,000 people had already filed claims for federal disaster assistance, and many more are likely to require assistance in the future.  The immediate death toll from the storm was 82, and illnesses relating to the storm are expected to persist for years.
Total Rainfall from Hurricane Harvey, August 30, 2017 

Climate change is often represented in terms of polar bear on melting ice; of changing migration patterns for wildlife; of seemingly trivial changes in long-term average temperatures; of higher sea level in the next century.  All of those are true and real.  But for many Americans, these issues do not impact their lives.  None of this matters in terms of day-to-day living. 

Hurricane Harvey is different.  It has been called a 1000-year flood by scientists.  This is an expression of the probability of an event of this magnitude in a given year, based on statistics of smaller events.  The storm dropped an awe-inspiring quantity of water on the earth, and America’s fourth-largest city was totally disrupted.  There was certainly no economic productivity from the city for a week, and the damages are considerable.  When floodwaters threatened to destroy Houston's earthen flood-control dams, emergency managers opened the floodgates, deliberately flooding neighborhoods downstream of the dams, to save other neighborhoods in a kind of triage.  An estimated 100,000 homes were flooded or damaged by the storm.  Many of them will be totally destroyed after sitting in flood waters for a month.  The human toll in lives and economic loss is huge. 

The damage is personal to me; two of my three former houses in Houston almost certainly flooded. Old friends and former neighbors are dealing with the loss of their homes, cars, and lifelong possessions. A number of deaths occurred in familiar neighborhoods
Image credit Joe Raedle/Getty Images
Image credit: David J. Phillip, AP
Image Credit: T.B. Shea, AFP/Getty
Image credit: AP

Climate Change: Prediction and Consequences

The Intergovernmental Panel on Climate Change (IPCC) 2014 Report includes this statement: “It is very likely that heat waves will occur more often and last longer, and that extreme precipitation events will become more intense and frequent in many regions.”

On August 27, 2017, the National Weather Service tweeted this statement regarding Hurricane Harvey: “This event is unprecedented & all impacts are unknown & beyond anything experienced. Follow orders from officials to ensure safety.  #Harvey”.   [Emphasis mine.] 

These statements are not unrelated. 

Scientific Analysis
The full scientific analysis of Hurricane Harvey will not be known for a long time, probably years.  And uncertainties will remain after the full analysis of all available data.  There are a number of known factors relating to climate change which will increase hurricane severity – it is simply physics.  These factors include higher temperatures at the ocean surface, higher temperatures in the upper 200 meters of the ocean, and higher humidity.  The factors are well-established -- the changing temperature of ocean waters have been observed by NOAA’s ARGO system of buoys since 2004, and by satellite since the 1980s.  Some scientists have also suggested that climate change is reducing the strength of upper level winds, though this proposal is not yet considered proved.

Higher temperatures at the ocean surface are believed to have caused a rapid, late intensification of the storm from category 2 to category 4 immediately before landfall. 

Higher temperatures in the water column are believed to have reduced the tendency of wave action to bring cooler water to the surface, weakening the hurricane.  

Higher humidity, relating to higher water temperatures and higher air temperatures, allowed the storm to carry more water than other storms.  Higher humidity and higher temperatures also lower air density, contributing to the strength of convection and wind speed.

At this time, it is unknown how much wind systems have changed due to climate change, or how much these changes might have affected Hurricane Harvey.  Hurricane Harvey stalled after moving onshore, caught between stationary high-pressure systems.   High-level winds, which sometimes reduce convection through wind shear, were also weak through the hurricane.  Quantifying these impacts using new observations and modeling is the job ahead for scientists. 

The specific magnitude of these changes is unknown, but the known factors contributing to the severity of the hurricane are clear.  According to one preliminary estimate, factors relating to climate change increased the volume of rainfall from Hurricane Harvey by 30%.   While this may seem to be only a moderate increment, thirty percent of extra water is what exceeded the capacity of flood-control reservoirs, caused neighborhoods to flood, and caused a number of deaths.

Conclusion
The earliest warning that climate change could result in more frequent and severe hurricanes was published in 1992, and incorporated into the IPCC Second Assessment Report.  At that time, statistical evidence that hurricanes were become worse was weak, and in 2017, statistical evidence is still weak.  However, science is not all about empiricism.  Explanations matter.  We understand the physical processes of hurricane convection, the Coriolis effect, and the importance of water temperature, air temperature, humidity and air density.  We have observed that these factors are changing due to accumulating greenhouse heat, and will increase the frequency and intensity of hurricanes.

At this time, we do not know the specific amount that climate change contributed to the Hurricane Harvey disaster.  But there is a simple, shortcut analysis that we can do now.  Simply consider which possibility is more likely: whether Hurricane Harvey was an extreme event with a probability of 1:1000, or whether climate change intensified an ordinary storm, as predicted by scientists for over twenty years?


 Graphical Representation of 1:1000 Probability Event
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References
http://www.npr.org/2017/09/01/547598676/at-least-100-000-homes-were-affected-by-harvey-moving-back-in-wont-be-easy

Long-term temperature is about one degree higher than a few decades ago.    Local conditions were 2.7 degrees to 7.2 degrees F higher than usual.   Humidity rises at about 3 percent per degree C., so humidity during Harvey was 3% to 5% higher than usual.
High level winds that typically steer tropical storms collapsed in 2010.  Although a meteorologist expects the winds to return in a few years, long-term climate modeling suggests that collapse of steering currents may become more common.    

Speech by Mike Pence


Immediately prior to landfall, and during the time of intensification to category 4, Harvey over water 4 degrees F warmer than average.

Waters off South Texas were 5 degrees warmer than usual during Hurricane Harvey.

Warm water extended deeper into the water column.

Atlantic Decadal Oscillation is trending to cooler temperatures, which may bring cooler waters to the tropics and weaken storms in coming years.   Another researcher suggests that GHG warming may keep the ADO positive for the coming decade.

Background conditions were about 2 degrees warmer than average, and then warmed further by an eddy of the Gulf Stream Loop Current.

“The human contribution can be up to 30 percent or so of the total rainfall coming out of the storm”.  Hurricane waves usually bring cooler water to the surface, which acts as a buffer to moderate the strength of the storm.  But Hurricane Harvey churned up water 100 m to 200 m below the ocean surface, but this water was still warm. 

Graphics representing 27 trillion gallons (about 25 cubic miles) of water. 

Probability of Hurricane Harvey, based on historical statistics, is 1:1000.


Immediate death toll from Hurricane Harvey was 82.  A number of serious health effects could persist for years. 

Daniel Huber, Jay Gulledge, Center for Climate and Energy Solutions, Extreme Weather and Climate Change, 2011.
“There is a physical basis for expecting hurricanes to have stronger winds and produce more rainfall due to global warming, and models with enhanced greenhouse gas levels show an increase in the number of such storms….However, observational evidence is insufficient to confirm that such a response has already begun.”

IPCC Second Assessment Report, 1995.
"Direct impacts on infrastructure would most likely occur as a result of changes in the frequency and intensity of extreme events. These include coastal storm surges, floods and landslides induced by local downpours, windstorms, rapid snowmelt, tropical cyclones and hurricanes, and forest and brush fires made possible in part by more intense or lengthier droughts."
“It is presently uncertain whether the frequency and severity of tropical cyclones will increase due to climate change.”

O'Brien, S.T., B.P Hayden, and H.H. Shugart, 1992: Global climatic change, hurricanes, and a tropical forest. Climatic Change , 22 , 1750-1790.

Monday, August 28, 2017

Where is the Dark Matter in the Earth's Core?

I just finished reading “Dark Matter and the Dinosaurs” by Harvard University physicist Lisa Randall.   Dr. Randall is an excellent popular science writer, as well as being a top-flight theoretical physicist.   Her exposition on dark matter gave me most of my exposure to this arcane topic in modern physics.  

My understanding of the topic is shallow, but I think some common-sense observations provide constraints on the distribution of dark matter, which need to be recognized in models of dark matter and experiments to find it.

My biggest question about dark matter is: Where is the dark matter in the earth's core?  

The Nature of Dark Matter
Dark matter is a form of matter that does not interact with ordinary matter or energy, except through the force of gravity.  A better name for dark matter might be “ghost matter”, as the lack of interaction with ordinary matter means the dark matter can occupy the same space as ordinary matter, or pass right through it, undetected. 


Lisa Randall writes: “Dark matter passes right through our bodies, and resides in the outside world as well…. Every cubic centimeter around you contains about a proton’s mass worth of [dark] matter….if those particles travel at the velocity we expect based on well-understood dynamics, billions of dark matter particles pass through each of us every second.  Yet no one notices that they are there.”

It is unknown whether dark matter can interact with ordinary matter at all, except through gravity.  Nevertheless, some theoretical results suggest there may be very weak interactions, and experiments are in progress seeking to detect dark matter, either directly or indirectly, through some kind of interaction with ordinary matter.  According to Dr. Randall, it is also unknown whether dark-matter interacts with itself. 

It seems to me that dark matter is necessarily self-interacting.  Evidence indicates that dark matter has a very low maximum density.   Dark matter must exclude other dark matter from occupying the same space.  The density of dark matter is much, much lower than regular matter.  Dr. Randall states that every cubic centimeter around you contains about 1 proton’s worth of dark matter.  Accordingly, the density of dark matter at the surface of the earth is only 1.7 x 10-24 gm/cc, or 1.7/1,000,000,000,000,000,000,000,000th of the density of water.

Evidence of Dark Matter
Dark matter is known through its influence on ordinary matter, via the force of gravity.  It is observed only on the scale of galaxies or larger structures.  Observed gravitational effects suggest that dark matter actually comprises 85% of the matter in the universe.  The evidence for dark matter is mostly derived from deep-space astronomy and cosmology, as follows.
  1. The orbital velocities of stars in galaxies are much too high, given the quantity of ordinary matter in the galaxy.  Additional mass, in the form of dark matter, is required to explain the cohesion of galaxies.
  2. The gravitational lensing of light around galaxies indicates a much greater mass in the galaxy than can be seen in ordinary matter.
  3. The background radiation of the universe which formed shortly after the Big Bang shows an irregular distribution, which can only be explained by gravitational accumulation, requiring more mass than is known to exist as ordinary matter.
  4. Modeling the development of the universe since the Big Bang shows that the gravitational influence of dark matter is necessary to create galaxies in the primordial universe. 
  5. Evidence of dark matter can also be seen in the gravitational lensing of distant objects near colliding galaxies, such as the spectacular Bullet Cluster.  In such events, dark matter becomes separated from ordinary matter, and is revealed by observations of separate patches of magnification by gravity lensing.


Bullet-Cluster Galaxy.  Image credit: NASA

Local Dark Matter and Self-Interaction of Dark Matter
Dark matter reveals itself through the force of gravity on a very large scale – the scale of galaxies or larger structures.  But what about smaller settings?  What can we deduce about dark matter by its small-scale behavior?

Black Holes of Dark Matter
If there was no exclusionary force to dark matter, particles of dark matter (which do interact through gravity) would fall together, presumably to a very high or infinite density.  Without an exclusionary force, dark matter would be particularly prone to forming black holes from small quantities of dark matter, collapsing to very high density.  [Despite the similarity in names and difficulty of observation, dark matter and black holes are quite different things, and should not be confused.]  But we don’t observe the gravitational influence of lots of small black holes, within the galaxy.  If they existed we would notice their presence by abnormalities in the velocities of stars in the Milky Way, by gravitational lensing of distant starlight, and by deflections in clouds of interstellar gas.  We don’t see those things, so there must be an exclusionary force prohibiting the close association of particles of dark matter. 

Dark Matter at the Center of the Earth
Also, since dark matter interacts with normal matter only through gravity, we might expect all of the dark matter in the neighborhood (that is traveling at less than escape velocity) to fall through the crust and mantle of the earth, and accumulate in the earth’s center.  Since dark matter comprises 80% of the matter in the universe, we ought to find a substantial gravity anomaly in the earth’s core, unexplained by the density of normal matter in the core. 

We don’t.

The Earth’s Structure and Core
We have a very good understanding of the structure and composition of the earth’s core.  The structure of the core is revealed by the behavior of earthquake seismic energy as it is transmitted through the earth.  Seismic waves generated by earthquakes travel through the earth, and can be recorded at most places around the earth following a major earthquake.  Compression waves and shear waves travel at different speeds, and behave differently depending on the nature of the transmitting media, whether solid or liquid.   The speed of the waves depends mostly on density, and interfaces between materials of different composition produce both reflections and refraction of the waves.  All of this information allows us to construct the specific solution of layers, mineral composition, and phase (liquid or solid) of the interior of the earth.
Image credit: Charles Sturt University, via Ethan on ScienceBlogs.com

The interior of the earth consists of a number of concentric shells of varying composition and consistency.  Below the atmosphere and oceans, there is the earth’s crust, which occurs as oceanic and continental components.   Below the crust, the upper mantle is divided into the lithosphere and asthenosphere.  The crust and mantle are composed of silicate minerals.  The crust and lithosphere are rigid, and move as plates on the ductile asthenosphere.  The lower mantle is also ductile, and deforms plastically to form convection cells, driving the motions of the shallower plates. 


Image Credit: Wonderopolis.org

The earth’s core is primarily composed of iron and nickel, with a small amount of lighter elements.  A huge clue to the composition of the core exists in form of iron-nickel meteorites, which are derived from some proto-planet in the early solar system.  Iron-nickel meteorites typically contain nickel in concentrations of about 6% to 10%.  Gravity shows that the density of the core is about 3% lighter than pure iron, implying about 10% of lighter constituents, probably silicon, oxygen and sulfur.   Nickel is slightly denser than iron, so higher nickel concentrations would imply correspondingly higher concentrations of light elements to compensate in overall density.

Seismic studies show that the inner core is solid, and the outer core is liquid.  Convection in the liquid outer core accounts for the earth’s magnetic field.  The mineral composition of the inner core can be replicated and studied using high-pressure tools in the laboratory.  The combination of seismic studies, gravity studies, mineral composition studies, meteorite studies, and magnetic studies yields a model that fully explains all observations about the earth’s core.  No dark matter is indicated by the observations; rather, the introduction of dark matter would require unreasonable changes to the most logical interpretation for the composition of the core.  

Model of the Earth's density from the center to the surface.  Image credit Wikipedia.
Model of the Earth's gravity from the center to outer space.  Image credit: Wikipedia

Dark Matter in the Cores of Stars
The sun is a delicately balanced fusion engine.  The heat generated by hydrogen fusion produces an expansion force, which is balanced by the gravity of the star.  When the balance is disrupted by the exhaustion of nuclear fuel, the star becomes unstable, exploding as a nova or supernova, or collapsing into a black dwarf, a neutron star or a black hole.  The processes of nuclear fusion are known and well-quantified as a result of nuclear weapons research and super-collider experiments. 

Any gravitational anomaly in the sun or in the theoretical models of other stars would surely be noticed, and would be glaringly apparent to scientists studying stars.  We have to conclude that there is no dark matter accumulated in the cores of stars.  

Evidence from Spacecraft
Dark matter may exist as a “soup” of uniform density larger than the solar system, so that there is equal gravitational attraction in all directions.  In this case, no anomaly could be detected, because the gravitational influence of dark matter would be the same in all directions.  By analogy, a point at the exact center of the earth would be weightless, subject to equal gravity in all directions.  But even in this case, as objects move in some direction through the soup, differential gravity should be detectable if there are heterogeneities or nearby limits to the dark matter soup. 

Our best experiments to find dark matter in the Solar System are the Pioneer 10, Pioneer 11, Voyager 1 and Voyager 2 spacecraft.  Data from the Pioneer spacecraft was last received in 2002 and 1995, respectively, but the probes lasted long enough to identify a potential gravity anomaly in the solar system, the Pioneer Anomaly.  As I read Lisa Randall’s book about Dark Matter, I initially thought that the Pioneer Anomaly might be the expression of dark matter in the solar system, but subsequent reading revealed that the anomaly was robustly explained in 2012 as a thermal recoil phenomenon relating to the spacecraft itself. 

The Voyager spacecraft are now the most distant man-made objects from earth.  Both are still transmitting, at distances of 12.9 billion miles (19 light-hours) away, and 10.7 billion miles (16 light-hours) away, respectively.  The craft are traveling at roughly a right angle to each other, providing two long baselines to measure any gravity anomalies in those directions, revealed by an unaccounted-for acceleration of the spacecraft.  None have been detected.

Conclusions

Any theory of dark matter must account for the lack of detectable dark matter in the cores of planets and stars.  The lack of a detectable dark matter core in these places is strong evidence that dark matter is self-interacting.  There must be a property of dark matter that prevents dark matter from accumulating at high density.  This exclusionary force must act on at least the scale of a planet, and probably on the scale of the solar system. 

The exclusionary force places a limit on the maximum density of dark matter.  To the best we can now recognize, that limit is the detectable limit of density anomalies in the sun or the earth.  It is a very small density compared to the density of ordinary matter.

The lack of acceleration anomalies in distant spacecraft shows the large-scale homogeneity of dark matter surrounding the solar system.  At this time, I don’t know the limits of velocity determinations of the Voyager spacecraft, but I think that these long-distance measurements would be quite sensitive to a local density anomaly.  It would be a worthwhile exercise to calculate the effect of a dark-matter accumulation (such as Dr. Randall’s posited dark-matter disk within the Milky Way) on the velocities of the spacecraft, and see if the results would be within the tolerance of the spacecraft velocity measurements. 

If not for the robust evidence of dark matter in cosmology, it would be tempting to dismiss the idea of dark matter entirely.  But perhaps the finding that there is an exclusionary force limiting the density of dark matter can be a clue to identifying the true nature of dark matter. 

Maybe dark matter doesn’t fit the particle model of matter at all.  At this time, all we know is that it is very sparsely dispersed gravity.  But the particle theory of matter has proven very useful at explaining most of our reality.  We shouldn’t give up on it too easily.  We should think for a moment about what constraints our observations put on a particle theory of dark matter.

Ordinary matter and dark matter are clearly different in scale.  A proton excludes other protons on the scale of 0.8414 x 10-15 linear meters.  If we assume a particle of dark matter has the same density as a proton, each dark matter particle must exclude other dark particles on the scale of 0.02 linear meters.  The ordinary proton occupies a volume of about 3 x 10-46 cubic meters, and a dark matter particle of the same mass would occupy a volume of 1 x 10-6 cubic meters, a difference of 40 orders of magnitude.

Perhaps the difference in size between ordinary matter and dark matter is the sole reason for the undetectability of dark matter.  It seems to me that electrical and other interactions between normal particles occur because the wave properties of the particles have similar wavelengths.  The waves can interfere, and therefore interact.  With wave properties of vastly different sizes, there is no interference, and therefore no interaction.

I am not optimistic about the present round of experiments looking for dark matter, as described by Dr. Randall.  If you are looking for an elephant with an electron microscope, you are likely to be unsuccessful. 

What kind of experiments could reveal particles which exist at a scale many orders of magnitude larger than ordinary matter?  I don’t know.  The electromagnetic spectrum is well-explored on that scale, and reveals nothing.  Perhaps other forces need to be synthesized, and examined at larger scales.  There may be practical technological benefits if instruments can be developed that directly detect dark matter.  Perhaps, such instruments could provide the ability to manipulate other forces, such as a way to generate, shape and manipulate artificial gravitational fields, in the way that artificial magnetic fields have been generated and used for almost 200 years.

That would be a real advance for mankind.


References
Lisa Randall, Dark Matter and the Dinosaurs; The Astonishing Interconnectedness of the Universe, 2015, 432p.

Density of the Outer Core (liquid):  9.9 to 12.2 gm/cc
Density of the Inner Core (solid):  12.6 to 13.0 gm/cc


Thickness (km)
Density (g/cm3)
Types of rock found
Top
Bottom
Crust
30
2.2
Silicic rocks
2.9
Andesite, basalt at base
Upper mantle
720
3.4
Peridotite, eclogite, olivine, spinel, garnet, pyroxene
4.4
Perovskite, oxides
Lower mantle
2,171
4.4
Magnesium and silicon oxides
5.6
Outer core
2,259
9.9
Iron + oxygen, sulfur, nickel alloy
12.2
Inner core
1,221
12.8
Iron + oxygen, sulfur, nickel alloy
13.1






Composition of the earth’s core.

The core is about 3% lighter than pure iron, implying about 10% of lighter constituents, probably silicon, oxygen, and sulfur.

Voyager 2: 10.7 billion miles (16 light-hours) away.

The last data received from Pioneer 10 was in 2002.  The last data received from Pioneer 11 was in 1995.


[The new] measurement measured [a proton] to be 0.8418±0.0007 fm.  A femtometer is 10-15 meters

Tuesday, July 4, 2017

The Global Heat Budget

Something odd is going on in the oceans. 

The oceans are apparently becoming more efficient at absorbing greenhouse gas heat from the atmosphere. Or, we are becoming better at quantifying the amount of man-made heat entering into earth systems.

According to published figures, the percentage of greenhouse gas heat entering the ocean has risen from less than 50% in the 1970s to over 92% today.  While the data from earlier years might be suspect, recent, high-quality data confirms the earlier pattern. The apparent change suggests we need to understand the cause of the higher rate of heat transfer to the oceans.  It may be that the rising rate of heat transfer is due to higher air temperatures, or there may be another explanation for the change.  If the rising rate of heat transfer is real, it begs the question of what was happening to the excess heat in earlier years.

In this post, I include other sources of man-made heat to calculate a world heat budget, comparing the quantity of man-made heat with the quantity of heat observed entering the oceans, atmosphere and melting ice.

Part I.  Anthropogenic Heat
Greenhouse Gases
In this post, we’ll look at the earth’s heat budget relating to greenhouse gases and other man-made sources of heat.   Greenhouse gases retain heat in the atmosphere, because they are transparent to much of the sun’s radiation spectrum, but opaque to most of the infrared thermal radiation.  Visible radiation from the sun penetrates the atmosphere and strikes the earth, but is trapped after being converted to infrared thermal radiation, instead of radiating back into space.  Increasing concentrations of greenhouse gases trap increasing amounts of the sun’s heat. 

We can calculate how much heat is retained, depending on the concentration of the gas.  We can look at the history of how greenhouse gas concentrations have changed, and calculate how much heat was retained a few years ago as compared to today.  And we can forecast how much heat will be retained in the future, and estimate what will happen because of the extra heat.

The amount of heat trapped by greenhouse gases was measured in laboratory measurements beginning in the late 1800s.  The Swedish chemist Arrhenius calculated the planetary warming that would result from doubling the amount of CO2 in the air, and published his results in 1896.   The high-altitude chemistry of the atmosphere was measured in high-altitude flights around the globe, improving the estimates of how much heat was trapped at varying concentrations of greenhouse gases.  Upcoming satellite missions will define the incoming and outgoing heat budget with even greater detail, unless cancelled by the current administration. 

The National Oceanic and Atmospheric Administration publishes a measure of the heat retained due to greenhouse gases, termed radiative forcing, reported in units of watts per meter squared (W/m2), equivalent to joules per second per meter squared (J/sm2).  NOAA's figures are global averages, with corrections for angle of incidence and cloudiness included.  From radiative forcing, we can calculate the global heat retained by each gas by multiplying by the cross-sectional area of the earth, and the time interval of interest.  Here is the annual global heat retained by various greenhouse gases, as reported by NOAA.  The heat retained by non-CO2 greenhouse gases was extrapolated for years that no data was available (1955 – 1978), assuming a constant ratio to CO2 for the earliest data available. 
Figure 1.  Annual Heat from Greenhouse Gases
The annual heat retained by greenhouse gases has increased by 75% since 1979, driven primarily by increases in atmospheric CO2 concentrations. 

The amount of heat retained by greenhouse gases is huge.  The annual heat retained by greenhouse gases in 2016 is 1.23 x 1022 joules, which is equivalent to about 533,000 Hiroshima-sized atomic bombs, every day, or one bomb a day on a grid of about 20 miles spacing, all over the world.  Fortunately, only a small fraction of that heat remains in the atmosphere, as we will see when we discuss heat sinks.

Primary Heat from Fossil Fuels, Nuclear Energy and Deforestation
For completeness and to give a sense of scale to the greenhouse effect, we can add other man-made additions to the earth’s heat budget.  Primary heat from fossil fuels and nuclear energy is published by the Energy Information Administration.  The volumes of biomass consumed by deforestation are uncertain, but I used published figures to estimate the primary heat derived from deforestation, through combustion or decay, using thermal values for combustion of wood, after corrections for moisture.

                Heat from Agriculture
Heat from agriculture is another source of anthropogenic heat, through the decay, combustion or consumption of agricultural biomass.  For the volume of agricultural biomass, I used data from the United Nations Environment Programme for the year 2009, and scaled that number in other years in proportion to world population.  Agricultural biomass displaces natural biomass, so I assumed an arbitrary 50% gain in productivity in agricultural biomass, as a result of irrigation and fertilizer.  I again used the thermal values for the combustion of wood, after corrections for moisture.

The sum of these inputs provides the history of human-caused heat to the earth’s heat budget, from 1955 – 2016.
Figure 2. Man-made Heat from All Sources, 1955 - 2015.

Part II.  Heat Sinks
To understand climate change, it is necessary to understand what happens to the heat retained by greenhouse gases.  Where is the heat going, and what is it doing to the planet?   The goal is to quantify the heat budget for the earth, and look at how it has changed over the past several decades. 

There are three main heat sinks we can measure: the ocean, the atmosphere, and volumes of melted ice.  I have used different methods to estimate the heat consumed by these processes, depending upon the data available.  It is quickly apparent that the ocean is the most significant heat sink.

Ocean Temperature
The temperature distribution of the ocean is seasonal, heterogeneous, and changing.  Maps of ocean temperatures at various depths can be seen on NOAA’s website.  Temperature data prior to 2000 was dependent on ship tracks, and variable methods of data collection.  These datasets had clusters of dense measurements in shipping lanes, and large gaps with no data.  The calculation of heat content is somewhat uncertain.  Beginning in 2000, NOAA launched ARGO, a system of free-floating buoys measuring ocean temperatures.  Today there are 3849 active buoys, which periodically dive to depths of 2000 meters, measuring temperatures within one-thousandth of a degree C, and returning to the surface every ten days to broadcast measurements to satellite receivers.  Here is a map of the current buoy locations. 

Figure 3.  Map of Current Location of ARGO Temperature Buoys, June 27, 2017.

The temperature rise noted over the past 50 years is relatively small, at 0.1 degrees C, but is two orders of magnitude more than the sensitivity of the ARGO instruments.  This temperature change represents an enormous amount of heat because of the huge volume of water and large heat capacity of the ocean.  As we might expect, the temperature of the shallow ocean is rising much faster than the temperature of the deep ocean.
Figure 4.  Ocean Temperature 0 -- 100 meters, NOAA
Figure 5.  Ocean Temperature 0 – 700 meters, NOAA
Figure 6.  Ocean Temperature 0 – 2000 meters, NOAA
The National Oceanographic and Atmospheric Administration (NOAA) published a chart of ocean heat content from 1957 through 2016.   NOAA did not publish the specific data used to generate the plot, so I have made an eye-ball fit to the data using a smooth function in Excel.   The NOAA chart, and my overlay of a smoothed function, is shown below.  The smoothed function allows me to calculate the annual change in ocean heat content.  As we will see when we compare heat sources to heat sinks, the heat absorbed by the ocean is very close to the amount of heat retained in the atmosphere by greenhouse gases.

Figure 7.  Cumulative Ocean Heat Content, NOAA, 1956 - 2016.
Figure 8.  Annual Change in Ocean Heat Content.
Figure 8 shows the annualized change in ocean heat content from my smoothed version of NOAA’s heat content chart, 0 – 2000 meters.  The change in heat content is constantly positive, but the rate of change dipped in the 1960s, before a continuous acceleration in heat content from the 1970s to the present.

Atmosphere
Similarly, the National Aeronautics and Space Administration (NASA) published a table of average global surface air temperatures. 
Figure 9.  Surface air temperature, 1880 – 2016, NOAA.

Figure 10.  Surface air temperature, 1955 – 2016, with polynomial regression to the data, NASA.
Assuming that the surface temperature change is representative of the atmosphere as a whole (or at least, the majority of the heat content), I calculated the annual change in atmospheric heat content.  The heat content of the atmosphere is significantly smaller than the heat content of the oceans despite a larger change in temperature, due to the very high heat capacity of water compared to air. 

Figure 11.  Annual change in Atmospheric Heat Content.
Melting Ice
NASA has used satellites and overflights to measure the changing volumes of ice on the Antarctic and Greenland ice sheets.  The determinations have been based on a combination of satellite gravity measurements, surface elevation measurements, and ice-penetrating radar measurements (from overflights).   Most of the high-quality surveys have occurred since the early 2000s, in particular NASA’s GRACE  (Gravity Recovery and Climate Experiment) gravity-measurement satellite, which was launched in 2002.
Figure 12.  Mass loss in Antarctica measured by NASA's GRACE satellite.

Figure 13.  Mass loss in Greenland measured by NASA's GRACE satellite.
Separately, the volumes of continental glaciers and ice sheets (other than Greenland and Antarctica) have been separately studied through an inventory of photos and elevation maps, and published as a time-series of melted ice volumes.  Volumes of melted ice from all three sources were reported in a single chart by Shuang Yi, et al, 2015.  As I did with NOAA data for oceans and temperatures, I fitted a smooth polynomial function to the published charts, and then extracted the annual change in melted ice for all three sources.  I should note that mass-loss data from NASA’s GRACE satellite shows about 10% smaller volume of ice loss than Shuang et al.  Shuang et al presumably incorporated other sources of information, including ice-penetrating radar, altimetry and flow measurements from other studies.

Figure 14.  Ice Loss from Antarctica, Greenland, and other Glaciers, Shuang Yi et al, 2015, with overlay of smoothed polynomial functions.
Image Credit:  Shuang Yi et al, 2015.

Finally, I calculated a volume of ice melted in the Arctic sea-ice, based on simple assumptions of new ice thickness (1 m) and multi-year ice (2 m), and the changing areas reported for new and multi-year ice.   The volume of melting Arctic sea-ice is relatively trivial compared to the volumes melting from glaciers and ice-sheets, but is still indicative of the overall melting trend in the Arctic.
Figure 15.  Diminishing Arctic Sea Ice, 1953 - 2010, image credit, National Snow and Ice Center.
I assumed that the volume of ice would need to be heated by 10 degrees Celsius before melting, based on temperature profiles from drilling on the Greenland and Antarctic ice sheets.  I then applied the heat of crystallization to the volumes of ice melted to obtain the total heat absorbed by melting ice. 
Figure 16.  Heat consumed by Global Melting Ice, 2004 – 2015.
The sum of oceanic, atmospheric, and melting ice heat sinks is shown below.  The ocean is by far the dominant heat sink. 
Figure 17.  Global Heat Absorbed by Heat Sinks.
Despite all of the news about rising atmospheric temperatures and volumes of melted ice, the ocean is absorbing the lion’s share of heat around the earth.  By comparison to the oceans, heat consumed by atmospheric heating and melting ice is negligible.  In 2016, the ocean absorbed about 96% of rising heat, compared to a little over 1% for the atmosphere, and 3% for melting ice. 

Part III.  The Heat Budget:  Retained Heat and Heat Consumed
Greenhouse Gas Heat and Ocean Heat Content
The simplest (but incomplete) expression of the planetary heat budget is the comparison between annual heat retained by greenhouse gases and the rising heat content of the oceans. 
Figure 18.  Annual Greenhouse Gas Heat and Annual Change in Ocean Heat Content
The first observation to make from this chart is that heat retained in the atmosphere by greenhouse gases and heat warming the oceans are very close to the same quantities. Greenhouse gases are retaining about 1,000,000,000,000,000,000,000 joules of excess heat in the atmosphere annually, and about 1,000,000,000,000,000,000,000 joules of unexplained heat is showing up annually in the oceans.  There is no known natural source of new heat warming the oceans.  It cannot be a coincidence.

The second, startling observation from this chart is that the rate of heat transfer from the atmosphere to the oceans is changing.  The amount of heat annually absorbed by the oceans apparently fell in the 1960s, although the data supporting this observation is weak.  However, continuing forward into the era of good data, the volume of heat absorbed by the oceans since 2000 has rapidly increased, and the fraction of  man-made heat absorbed by the oceans has also rapidly increased.  As temperatures rise, the oceans appear to be becoming more efficient at absorbing heat from the atmosphere.  

This raises a number of questions.  First, is the measurement of ocean heat content correct?  The quality and sensitivity of the ARGO system would suggest that the numbers are good since 2004. Second, are the calculations for man-made heat correct?  These numbers contain many assumptions, and are more likely to be uncertain than the ocean heat content.  Or third, was there another heat sink functioning in prior years, which is no longer working?  This seems unlikely.  Rather, the effort to align the sources and sinks in the earth's heat budget should focus on better quantifying of the sources of man-made heat.  

Full Heat Budget: Anthropogenic Heat and Heat Consumed in Heat Sinks
We can compare all sources of man-made heat from Part I to all heat sinks identified in Part II, to construct a heat budget for human-induced climate change.   Let’s look again at all sources of man-made heat, 1955 – 2015.
Figure 19.  Anthropogenic Heat, 1955 – 2015.
The red line indicates the sum of all man-made heat added to the earth’s atmosphere.

And let’s look again at the heat taken up by the oceans, atmosphere and ice.  As we noted before, the amount of heat entering the ocean overwhelmingly dominates heat entering the atmosphere or melting ice.
Figure 20.  Heat sinks, 1955 – 2015.
Let’s combine the two charts to see the full heat budget. 
Figure 21.  Man-made Heat and Heat Sinks, 1955 - 2016.
As we see, the gap between known sources of man-made heat and the observed amount of heat entering earth systems has narrowed, primarily due to an increasing amount of heat observed entering the oceans.  The amount of man-made heat that we can observe entering the oceans has risen from about 40% in the 1970s to nearly 90% today.  Much of that difference is probably due to the inadequacy of data prior to the deployment of the ARGO system of ocean buoys in the early 2000s.  However, there is a very real possibility that it also reflects a physical change in the process of transferring heat from the atmosphere to the ocean.

Let’s exclude questionable data, and only look at data from 2004.  First, man-made heat:  
Figure 22.  Anthropogenic Heat, 2004 – 2016.
And second, the heat budget including heat sinks in the oceans, atmosphere, and melting ice.
Figure 23.  Global Heat Budget, 2004 - 2016.
We are still left with the striking observation that between 2004 and 2016 the fraction of man-made heat absorbed by the ocean changed from 67 percent to 87 percent.  Given the high quality of recent observations, it seems that this is a real change in the rate of heat absorption by the ocean. 

Conclusions
People are the cause of rising temperatures and melting ice on earth.  Rising temperatures are observed in the atmosphere and oceans, and accelerating melting of ice in glaciers, arctic sea ice, and the Greenland and Antarctic ice sheets.  The heat responsible for these changes can be quantified and compared to the heat known to be retained in the atmosphere by human activities: CO2 and other greenhouse gases, primary heat from energy production, primary heat from deforestation, and heat produced from agricultural biomass.  There is a close correspondence between the quantity of man-made heat and observed heat appearing in natural systems; 89% of the heat generated by humans can be accounted for in known heat sinks.  Further, there is no known natural process which can be observed and measured presently warming the earth.

In more striking terms, people added 13,800,000,000,000,000,000,000 joules of heat to the atmosphere in 2016.   We observed 12,500,000,000,000,000,000,000 joules appearing in oceans, atmosphere, and melting ice with no known alternative cause.  This is not a coincidence.

The largest source of man-made heat is carbon dioxide from fossil fuels, which accounts for 57% of heat input into the Earth’s heat budget.  Greenhouse gases other than CO2 account for an additional 32%, bringing the total fraction due to greenhouse gases to 89% of man-made heat.  When I began this project, I focused on only greenhouse gases, but I found that heat appearing in heat sinks exceeded 100% of the heat retained by greenhouse gases.  I then added other sources of man-made heat that occurred to me: primary heat from fossil fuels and nuclear energy, primary heat from deforestation, and primary heat from the decay of agricultural biomass.  These other sources account for a smaller, but significant 11% fraction of man-made heat.

Large quantities of heat are retained in the atmosphere prior to absorption by other heat sinks.  Heat from human sources is put into the atmosphere before being transferred other to heat sinks.  Total man-made heat delivered to the atmosphere is equivalent to about 585,000 Hiroshima-sized bombs (533,000 from greenhouse gases alone) exploding in the atmosphere daily, or on a world-wide grid with a spacing of about 17 miles.  It seems likely that there will be increasingly severe consequences to weather systems, as such quantities of heat are moved from the atmosphere to their ultimate repository in heat sinks. 

Oceans are absorbing the bulk of heat produced by humans.  Oceans are absorbing 96% of the heat that can be observed going into heat sinks on earth; with atmospheric warming and melting ice accounting for only about 1% and 3% of the heat we can observe going into heat sinks.  If the oceans were not absorbing heat from the atmosphere, average temperatures would increase by 5 degrees C in about two years, equivalent to the temperature change between the current climate and the ice ages.   Without oceans on earth to absorb heat, man-made heat would quickly destroy civilization.  As climate scientist John Abraham said, “Global warming is really ocean warming.”

The fraction of man-made heat absorbed by the oceans is increasing rapidly, according to recent, high quality data.  The fraction of man-made heat absorbed by the ocean rose from 67% to 87% between 2004 and 2016.  The cause of the changing rate of heat absorption is not known.  Some likely possibilities include higher air temperatures and increased wave and wind activity at the ocean’s surface.

Explanations matter.  As I wrote in an earlier post, science is about delivering explanations in terms of physical processes.  Unfortunately, I could not find any physics describing the rate of heat transfer between the atmosphere and the ocean.  It is certain that higher surface air temperatures will mean a higher rate of heat transfer to the ocean, but it would be good to quantify that effect.  Other aspects of climate change, such has higher wind speeds and larger waves, may also play an important role.  Identifying the processes involved and quantifying the rate of heat transfer is unfinished work.

Future predictions of the consequences of anthropogenic heat depend on understanding the changing rate at which the ocean is absorbing heat from the atmosphere. 

--
Appendix I.
Other Potential Sources of Heat
When I began this study, the only source of anthropogenic heat I considered was from greenhouse gases.  I calculated the heat retained by greenhouse gases based on NOAA’s on-line publication of annual radiative forcing figures for each kind of greenhouse gas.  When I calculated the heat absorbed by heat sinks for 2015, I found that new heat showing up in earth systems exceeded the heat from greenhouse gases by a little bit.  So I went looking for other sources of man-made heat, and added primary heat from energy production, primary heat from deforestation, and net heat resulting from agriculture.  This brought the heat observed in earth systems to less than the sources of anthropogenic heat.

But the rapidly changing heat content of the ocean concerns me, and makes me think that I have not yet captured all of the sources of anthropogenic heat, or properly quantified heat from greenhouse gases.  In particular, it seems to me that published figures on radiative forcing are only considering the proportion of the sun’s radiant heat which is retained in the atmosphere due to the greenhouse effect.  However, greenhouse gases will affect all out-going thermal radiation from earth.  Thus, geothermal heat will also be retained in the earth’s atmosphere by greenhouse gases.  Geothermal heat is a steady-state process, adding about 1.4 x 10^21 joules to the earth’s heat budget (about 10% of total anthropogenic heat).  I ignored geothermal energy in this study because it is constant.  But if the amount of geothermal energy retained in the atmosphere is changing due to greenhouse gases, I may need to add additional heat to the analysis.  Similarly, natural biomass is a large part of the earth’s heat budget.  Like geothermal heat, I assumed that it is constant (except where displaced by agriculture).   Presumably, heat from all natural biomass is accounted for in the sun’s radiant heat, and therefore accounted for in the radiative forcing figures published by NOAA.  But it might be worth a question for clarification.  

Appendix II.

Assumptions for Global Heat Budget study: 
I made a number of simplifying assumptions in order to calculate the global heat budget.  Those assumptions are fairly sweeping in some cases, but all are based on reasoning, as explained below, and the results of the study are robust with respect to the simplifying assumptions made.

NOAA published data for radiative forcing for fifteen greenhouse gases other than CO2, for the years 1979 – 2016.   I extrapolated the radiative forcing for these gases for the years 1979 – 1974, scaling the radiative forcing for non-CO2 gases proportionally with CO2 for the years 1955 – 1978, for Figures 17, 18 and 20.

I assumed that volumes of deforestation biomass decayed or were burned, generating heat according to the heat content of wood fuel with an initial moisture content of 50%.  I used published figures for annual deforestation biomass through 2008, and assumed deforestation was constant at 2008 levels for all later years.

I assumed that net agricultural biomass was proportional to global population, and scaled reported biomass from 2009 for each year accordingly.  I also arbitrarily assumed that agricultural biomass productivity was 50% greater than the natural productivity it displaced, allowing me to calculate the net heat attributable to agricultural production, in excess of natural biomass production.   I assumed that the heat produced during decay is equal to the heat of combustion for wood, after corrections for moisture content.  Moisture was assumed to be 50%.  The heat of combustion for dry wood was used to calculate heat produced from dry agricultural biomass during decay.

I used a smooth 2nd order polynomial function for global ocean heat content (0 – 2000m).  I created the function as a visual overlay on NOAA’s plot of heat content, as I was unable to obtain digital data from NOAA for this parameter.

I used smooth 2nd order polynomial functions for ice melt from Greenland, Antarctica and other Glaciers and Ice Sheets, created by visual overlays on a figure from Shuang, 2015.  The functions for Antarctica and Greenland compare very well (about 10% higher) to digits available from NASA for the gravity-indicated mass losses from Greenland and Antarctica.  Numerous papers document ice loss from other glaciers and ice sheets, but without documented volumetric data.

I estimated the thickness of “multi-year [‘old’] ice” as 2 meters, and “new ice” as 1 meter, in calculating loss of Arctic sea ice.   Old ice diminished from 1,860,000 km2 in 1984 to only 110,00 km2 in 2016, a decline of 96%.   (Imster, E., in EarthSky, Nov. 8, 2016). Declining areas of multi-year ice and new ice were calculated separately, and attributed as a constant annual average, resulting in an estimate of 145 gigatonnes of Arctic sea ice melted annually. 

I assumed that all ice needed to be warmed 10 degrees C before melting.  This figure was an eyeball estimate based on temperature profiles of ice cores in Antarctica and Greenland.

I used a figure for the heat capacity for the atmosphere from college physics lecture notes available on-line.  I assumed that the temperature of the atmosphere changed proportionately with surface temperature when calculating atmospheric heat content.  I would note that recent satellite studies have shown that the troposphere is cooling, perhaps as the result of greater water content.  In any event, the bulk of the atmosphere’s mass, density and heat content are near the surface. 
--
References:

Heat Sources
Greenhouse Gases
James.H.Butler and Stephen.A.Montzka, 2017,  THE NOAA ANNUAL GREENHOUSE GAS INDEX (AGGI),
NOAA Earth System Research Laboratory
Source for radiative forcing data used in greenhouse gas heat calculations.

Fossil Fuels
Conti, J., et al, 2013.  International Energy Outlook, U. S. Energy Information Administration, Office of Energy Analysis, U.S. Department of Energy, Washington, D.C.    DOE/EIA-0484(2013)
http://www.eia.gov/forecasts/ieo/

U.S. Energy Information Agency, Data Tables, U.S. Energy Information Agency, Office of Energy Analysis, U.S. Department of Energy, Washington D.C. data tables, 2014
Source for data on primary heat from fossil fuel and nuclear energy.

Deforestation
Houghton, R.A. 2008. Carbon Flux to the Atmosphere from Land-Use Changes: 1850-2005. In TRENDS: A Compendium of Data on Global Change. Carbon Dioxide Information Analysis Center, Oak Ridge National Laboratory, U.S. Department of Energy, Oak Ridge, Tenn., U.S.A.
Source for biomass used in calculation of primary heat generated by deforestation.  Volumes of deforestation were assumed constant after 2008.

Adam Martin, 2011, A Reassessment of Carbon Content of Tropical Trees.  Average carbon content of dry wood is 47.2%.  This analysis includes volatiles, determined by the process of freeze-drying, instead of heat-drying the wood.

Agriculture
UNEP, 2009, Converting Waste Biomass Into a Resource

Heat Sinks
Atmosphere
NASA, 2017, Global Annual Mean Surface Air Temperature Change
Tabular data and chart

Dennis Hartman, 2017, University of Washington, Heat capacity of ocean, atmosphere and land.
Heat capacity of atmosphere used to calculate changing heat content.

Oceans
NOAA, National Centers for Environmental Information
Source of charts used with overlays to generate smoothed functions for ocean heat content and temperature.  Note: the charts showing average ocean temperature changes for 0 – 700 meters and 0 – 2000 meters were accessed on this site in June, 2017, but are no longer available.

The ARGO system and the Jason altimeter system allow separation of sea level rise into steric (heat & salinity; i.e. density) and mass components.   The steric component is dominant over the mass component in regional sea level variability and on a global basis it accounts for about 1/3 of total sea level increase in the past half century (Domingues et al 2008).

specific heat of seawater = 3.9 joules/g

John Abraham, 2017, New study confirms the oceans are warming rapidly, The Guardian.
Dr. John Abraham is a professor of thermal sciences. He researches in climate monitoring and renewable energy generation for the developing world. His energy development work has extended to Africa, South America and Asia.  “Global warming is really ocean warming.”

Ice Melt
Greenland, Iceland, and Antarctica
Shuang Yi, Wenke Sun, Kosuke Heki, and An Qia, 2015, An increase in the rate of global mean sea level rise since 2010, Geophysical Research Letters

NASA Global Climate Change, Vital Signs of the Planet
Source of charts and data tables for ice mass loss from Greenland and Antarctica.

Contribution of ice sheet and mountain glacier melt to recent sea level rise
J. L. Chen, C. R. Wilson, & B. D. Tapley

Arctic Sea Ice
Imster, E., in EarthSky, Nov. 8, 2016, Decline of Arctic’s thickest sea ice.  Multi-year ice grows up to 4 meters thick, while single-year ice is 2 meters thick at most.  The area covered by Arctic sea ice at least four years old has decreased from 1,860,000 square kilometres in September 1984 to 110,000 square kilometres in September 2016.