Strangely Attractive

Dr. Richard Muller of the Berkeley Earth Surface Temperatures (BEST) project has a new op/ed piece in the New York Times.  Dr. Muller is no longer skeptical of climate change.  Congratulations doctor, welcome to the world as we know it.

The real issue has not been if climate is changing or if man has an impact in climate change, but how much, how unusual and how serious is climate change.  From the article, "How definite is the attribution to humans? The carbon dioxide curve gives a better match than anything else we’ve tried. Its magnitude is consistent with the calculated greenhouse effect — extra warming from trapped heat radiation. These facts don’t prove causality and they shouldn’t end skepticism, but they raise the bar: to be considered seriously, an alternative explanation must match the data at least as well as carbon dioxide does. 


Yes, there are still reasons to be skeptical.

The BEST project is land surface temperature average.  The world is about 30% land and 70% ocean.  The term, "Global Warming" was once used instead of "Climate Change" possibly because the Globe is not Warming as much as the Land is warming.  Then the Oceans never have.  The plot above is the reconstruction of Lake Tanganyika temperatures of the past 60,000 years by Tierney et al.  The Green is the full temperature reconstruction with the mean value line also in green which on the left scale appears to be close to 23.4C degrees.  The Blue is what I think is the normal lower range which has a mean of about 22.7C degrees and the Red is the higher normal range with a mean of about 24.6C degrees.  So for the past 60,000 years the temperature range as determined by using Tierney et al, is 22.7 to 24.6 or about 1.9C degrees.    At the far left is another reconstruction of the temperature in Tasmania, but Cook et al.  The data for both of these are on the NCDC NOAA Paleo website if you care to double check.

So you can see the Tasmania part better, here is just the past 12,000 years.  It is hard to see but I have the average of the Tasmania lined up with the higher normal of the Lake Tanganyika.  The Yellow is a 150 year moving average of the Tasmania reconstruction.  The longer the period I average the less variation there is in the data.  Notice how the red plot has large spikes that settle down to around average.  While it is not as obvious in the Tasmania plots, you can see that it is also coasting into the same average.  Both are strangely attracted to that average for some odd reason. 

This plot has a number of reconstructions available on the NCDC NOAA Paleo site.  In this one the light blue is the Tasmania, the yellow is the Southern South America temperature reconstruction by Neukon Et al, the green is the Taymir Peninsular by Jacoby, the red this time is Greenland by Kobashi et al and the little burgundy series at the far right is the Sea surface temperature from the Hadley Center in the UK.   According to this plot, right now we are a little above average.  


Looking at those reconstruction, the global climate changes, it changes more on land and it appears to change the most near the poles if the Kobashi Greenland temperature reconstruction is to be believed.  Without a doubt, climate does change, but is the "Globe Warming"?


If you live in the higher northern latitudes, damn right it is warming.  You have industry and agriculture to thank or blame for that.  If there is a fix to that "Global Warming" that fix would be less warmth in the Northern high latitudes where the majority of the "Climate Change" has taken place.  


The title of this post is "Strangely Attractive".  That is because the Earth's climate system is strangely attracted to two "normal" climate "averages".  The high normal where we are now and the low normal where the glaciers grew on land.  There is a much larger range of "Global" temperature near the areas that the glaciers like to grow than there is near the liquid oceans.  Man, being the industrious devils we are, tend to take advantage of the retreating glaciers to expand territory.  That expansion is a part of what is called land use change.  That is one of the main causes of "Global Warming".  We are strangely attracted to expanding our territory and the Earth is strangely attracted to a range of temperatures.  There is a lot more to Greenhouse Gas Theory than CO2, though it is a strangely attractive theory.

Changing Climate - Land Versus Oceans

If you look at GISS LOTI, land and ocean temperature index versus HADSST2, the Hadley Center version 2 sea surface temperature record, the global average minus the sea surface temperature remains positive most of the time.  I did not change the baseline for either, all I want is to look at possible changes in the trend.

Since both have some possible issues early in the records, this plot from 1970 should be the more accurate portion of the record.  There is a trend where the land portion is warming greater than the oceans.  That is ancient history, no news there.

Moving into the satellite era, this chart compares the Northern extent, Southern extent and Tropical regions of the University of Alabama, Huntsville, MSU lower troposphere data.  This is a bit messy, but the southern extent has a negative trend while both the Northern extent and to a greater degree the Tropics have positive trends.

Looking at every 130 month linest trend for the land minus oceans there is nothing the really jumps out.  The Southern extent is mainly out of phase with the Northern extent, the tropics are generally in phase with the Northern extent.  Starting around 1998, the tropics and Northern extent ramp up to the Northern extent peak for the series.  The generally trend of the Southern extent is upward with more volatility.  Somewhat surprising, the tropics and Northern extent are fairly flat and less volatile.

This is the GISS minus SST2 with the 130 month linest.  Which of the UAH trends does this most resemble?  Yes, it looks like the southern extent where the majority of the energy is stored.  In case, you can't seen the correlation,

There is not a perfect correlation, but the GISS-SST2 tends to more closely match the SoExt.

What about variance?

While the HADSST2 accuracy is suspect going back in time, it is a portion of the data available.  The SST2 orange plot here is just the difference between one month and the next.  Not conclusive, but the monthly variance decreases as the SST warms at the end of the record, that could be indicating the approach to or arrival at, a heat capacity limit.

The 130 month LINEST plot in green is for the GISS minus SST.  That indicates they are becoming less disorganized, which may be real or a product of inaccuracies in either record.  Any either case, something worth looking into further.

Oop!  Missed the reason for all this,

Those two charts don't jive with GHE expectations.

Thermal Inertia

This is a new rendition of the LINEST (see description here and here) oceans using the UAH data.  There is a lot of noise in the ocean thermal energy.  Some of that noise is in phase and makes sense, some is out of phase and meaning has to be teased out.  Depending on the trend length, LINEST highlights different patterns.  This is for 130months or about 10.8 years, so you will notice the error in the chart title.  The period was selected to try and highlight the solar cycle.  The peaks are somewhat likely to be related to the solar variation, but not in every case.

The light blue tropics has a roughly 3.5 year pseudo cyclic oscillation. The main peak in 1999 for this chart shows that the Northern and Southern extents have a roughly 4 year lag and about 5 years later, 2008, the three sections of the oceans become more synchronized.   For this period, the most obvious non-synchronization appears to be 1994 to 1995 where the tropics shifted positive while both northern and southern continued negative.  From the peak of the 1995 non-synchronization it is about 4 years to the 1999 tropical peak and the synchronous shift of the northern and southern ocean expanses.

The 1999 peak is about twice the 1995 peak, which is similar to complementary wave response.  The inertia of the returning waves, the northern and southern extents, combined with the originating wave, the tropical oceans.  This appears to be a perfect example of thermal inertia in a fluid system.  With the periods not being perfectly timed, ~3.5 to ~4 years, the overall pseudo-clyclic period is hard to determine, but this appears to be about a half cycle or a full cycle.  With complimentary forcing, a weakening solar on a downward slope for example, this would be a half cycle.  With non-complimentary forcing, strengthening solar on a downward slope, it would appear to be a full cycle.  

My description is no doubt lacking, but there is evidence of thermal inertia in the ocean heat capacity which just like waves in a pond radiate to edges, the poles and return to the source, the tropics.  This is interesting, but it would greatly complicate attempting to assign a response to a particular force without full consideration of the natural internal oscillation frequencies.

The 1995 Climate Shift

Using approximation average temperatures of 290K for the northern and southern extent oceans and 300K for the tropical oceans, I have plotted the UAH lower troposphere oceans as three sections, No Ext, Tropics and So Ext using WM-2 instead of temperature anomaly.  This is far from being a valid approximation, it is just used as an illustration.  This perspective show the Southern extent being slightly higer energy moving to a neutral around 1995 and then into mode where the Northern extent is more energetic after 1995.    During the whole period, the tropical oceans remain in a fairly tight energy range.

 This chart I have replace the Tropics with Southern extent minus Northern extent.  Using the same approximation of Wm-2, there are "eyeballed" lines for the Northern extent dominate in red and the Southern extent dominate in blue.  The estimated range in energy is approximately 3.6Wm-2 over all or +/-1.8Wm-2 around the 1995 cross over.

Since I used 290K as an estimate for both, the approximate S-B perfect black body energy would be 401Wm-2.  For a +/-1.8 Wm-2 change, the temperature change would be approximately 0.32C degrees.

Can a minor shift of 0.32 degrees between the Northern and Southern oceans drive climate?

It looks like it By Golly!

There are no Steps! It is Constant Warming By Golly

As greenhouse gases change the chemical composition of the atmosphere, the stratosphere will cool

While the average global surface temperature rises.  There are no step changes in climate, only the constant warming due to CO2 that threatens to destroy the world as we know it.  Trust us, we are scientists.

http://www.realclimate.org/index.php/archives/2004/12/why-does-the-stratosphere-cool-when-the-troposphere-warms/

Thermal Wandering

In a Non-Equilibrium Thermodynamic system there is no true equilibrium or steady state, that's how it came by the name.  The approach to solving a non-equilibrium thermodynamic problem has to recognize the various possible causes of non-equilibria and impacts and the system changes.

One possible condition is a dissapative system that can reverse to a cumulative system.  That is a good description of the Earth liquid ocean and ice mass balance.  At some heat capacity, the oceans are dissapative.  Evaporation is transferred out of the liquid system into the ice system.  As long as the conditions for that dissipative state are met, the system "wanders" to ice.  Once a threshold is met, the ice "wanders" back into the system where the oceans begin to accumulate energy again.

The Plumber's Nightmare just shows the potential paths of mass and energy in the oceans system.  The thermal capacity of the system is limited by the freezing points of salt and fresh water.  Precipitation and ice formation can be in balance with evaporation and ice melt or not.  The system can wander from on state to the next.

The drawing just shows "an" equilibrium condition that "may" be possible, but is actually a better illustration of a change point.  Since the oceans have a huge thermal mass, they can "buffer" changes in the input and output energy values for a considerable time period.

From Wikipedia,"Glansdorff and Prigogine (1971)^[11] on page xv wrote "Dissipative structures have a quite different [from equilibrium structures] status: they are formed and maintained through the effect of exchange of energy and matter in non-equilibrium conditions."

With two more likely states, dissipative at higher energy and cumulative at lower energy (which corresponds to a dissipative ice state) the system is bi-stable.  


Greenhouse Effect Theory would have to consider the impact of these two potential states.  Assuming that albedo change is a sufficient consideration would not properly explain the various impacts that the location of the thermal masses, water and ice, would have on radiant energy conversion.  For example, loss of glacial mass at a high tropical location would have significantly less surface impact than the loss of ice mass at a higher latitude and lower elevation.

To help consider the thermal mass impact and moist air boundary layer that encompasses sea level liquid water and all altitudes with temperatures below the freezing point of water can be considered as a quasi-radiantless system with dissipative/cumulative properties.  Ice mass as well as energy can move into and out of that moist sir envelope. A second radiant envelope can be considered with quasi-conditiveless properties.

Since thermal mass can be dissipated and re-accumulated, there would be thermal inertia (heat capacity)of not just a theoretical nature but a mechanical nature that needs to be considered.

Reanalyse This

UPDATE:  Solar cycle added at end:

In Analyse This I described an anti-cherry picking method of using linear regressions on non-linear data.  You use every possible time period for a series, perform a linear regression for each period and plot the slope of each regression.  Open Source has a LINEST(a:b) function that works perfectly for that task.  There is nothing particularly note worthy about the process, other than it is easy to do.  In fact, it is not more informative than a good eyeball.

When comparing a number of time series, it does have a few advantages.  Above I compare all of the UAH ocean data sets.  At the end, the data set start synchronizing.  At the beginning the sets are somewhat synchronized and in the middle there is the least synchronization.  That is for the 60 month or five year lengths of the series.

In this plot, I use 180 month (15year) regressions and limit the chart to Global, Tropics, Northern and Southern extents.  The first arrow shows the peak slope for the tropics.  The other two arrows point out the peaks of the other series.  The warming due to the 1998 El Nino apears to have immediately impacted the tropics and to a lesser extent the rest of the ocean area, then there appears to be a delay of about eight years before that peak is seen in the rest of the series.  

The 1998 El Nino event is a natural event.  The magnitude of the event may be amplified by CO2 forcing, but then the following El Nino events would also have been amplified if that was the case.  Note that the yellow tropics curve becomes positive in 1995.  That may be due to CO2 amplification, then it returns to neutral in 2002, again just after 2008 and 2011.  The general slope of the tropics from 1998 is negative.  That is not in keeping with CO2 amplification.  It is not too unreasonable in my opinion to consider a good portion of the ocean temperature change as being due to a natural event with a delayed response.  Looking at just global temperatures, that would not appear to be the case.  

In this chart I converted the HADSST2 data set from temperature anomaly to approximate energy based on the average ocean surface temperature.  That conversion 5.67e-8*(294.25+Tanomaly)^4, amplifies the change by the T^4 power.  So the chart shows an approach to ~425.8Wm-2, an impulse perturbation around 1998 and a decay back to ~425.8Wm-2 following that perturbation.  This is not what I would expect to be a feature of CO2 enhancement.

Unfortunately, I don't have the new HADSST3 data set divided into the same regions.  If I did, I would suspect that I may be able to tease out a time constant or two from the longer time series, if they are accurate of course.  So far though, the oceans do not appear to be responding to CO2 forcing as much as "other" factors, some very natural in origin.

This plot is using 132 month (11year) regression lengths which should show some of the common solar cycle variation between sections of the oceans.  1998 still stands out as exceptional plus the 1995 bump in the tropical ocean regression is more prominent.

For the land fans.

Common Sense Versus Statistics

Recently a weather website meteorologist noticed that in the United States that for 13 straight months in a row, the average temperature was in the upper third of all temperatures in the historical record.  There was a heat wave.  Heat waves are news worthy, so the meteorologist determine the odds of that happening.

There is one chance in three of the temperature being in the upper third and 13 months of 1 in 3 chances would result in 1/3 raised to the 13th power.  That is a big number, like 1 in 1.6 million.  Oh my God! That is a huge long shot that just came in!  What can it mean?

Well, any month has a 50-50 change of being above average, 13 months of 50% chance is 1/2 raised to the 13th power or 1 in 8196, that is now were near as exceptional as 1 in 1.6million, but those are pretty long odds at first glance.

What are the odds that one year will be warmer than the average?  1 in 2, there is a 50-50 chance of one year being warmer than another.  What are the odds of one year being the warmest in 100 years?  1 in 100.

If the instrumental record is 100 years long and this past 13 months, which is just long than one year, the odds are about 1 in 100 that this year may be the warmest of the entire temperature record.

So how did a 1 in 100 shot get boosted to 1 in 1.6million? Think about the above average year.  Average is a set without any elements.  If you can't be average, you have to be above or below average.  The upper third is a set than can contain elements.  If you give average a range instead of a hard limit, you get a more realistic probability.  If average is a range of plus or minus one third of the range, then being in the upper third of the range would nearly the same as being above average.  You have one chance in three each of being below average, average or above average.  One chance in three but now the three is different.  The closer the value in the upper third is to the "average" boundary the less likely it is to be exceptional.  Since you have 1 in 100 chance of the year being exceptional, there are greater odds that each month of that year would be exceptional if the year is exceptional.  

So you can consider that in a warm year, the odds of any one month being in the lower 1 third are smaller that the odds of it being in the upper one third.  Once you remove the likelihood of record cold in a record warm year, you approach the month being either above the warm year average or below, 1 in 8196 if the year were a 1 in 100.  If not, for a 100 year record there are 12 months per year resulting in 1 chance in 1200 of that month being a record month and 1200/3 or 1 chance in 400 of any month being in the upper one third of the record.  1/100 time 1/400 results on 1 in 40,000 for any one year.  13 times 40,000 equals 520,000 or one in about a half million of any string of 13 months being in the upper third of a 100 year record if the year where not exceptional, but an exceptional year or an exception month in that year increases the odds of other months adjacent to that month also being exceptional.  So a simple approximation would be 520000 for a purely random occurance divided by 100 or 1 chance in 52,000, that 13 consecutive months in a record year would be in the upper one third of the entire record if the year where a record.

You can fine tune that, but one chance in 52,000 for 13 consecutive months being in the upper one third during a record year for a record 100 years long is close enough for government work.  The one in 100 is the more important statistic anyway.  Now if you where to pick the 13 months before hand, the odds of your winning ticket would be 1 in 1.6 million :)

Micro States are Not in Micronesia

Note:  I may come back to proof or expand this:

Since I got into an interesting discussion on what heat is, a kinda important subject in physics, I used the old standby joke about falling in an elevator, "it is not the fall, it is the sudden stop that kills ya."  Temperature is an indication of kinetic energy, but not all kinetic energy.

Kinetic energy is motion. Motion does generate heat.  It is resistance to motion that generates heat.   An elevator falling has potential energy even though it is moving, it doesn't release energy until it moves or attempts to move something else which is transferring energy.  So heat is related to energy transfer not energy proper.  Anytime energy is transferred there is a loss associated called Entropy so there will be heat.  

An Austria Physicist, Ludwig Boltzman figured that out and determine that there are tiny little energy transfers in substances that are important.  He worked out the Boltzmann constant which not surprisingly has the units, energy per absolute temperature, degrees Kelvin.  Energy per Kelvin are the units for Entropy.

Boltzmann's constant turned out to be the bridge between physics and quantum physics, classical mechanic and statistical mechanics or macro physics and micro physics if you prefer.  Those were grand old times in physics and Boltzmann was THE man for a while.

This lead to the discovery of the source of heat energy in an atom.  A monoatomic gas has three degrees of freedom, space or the Cartesian coordinates up, right and in which are allowed directions of motion.  The atom still doesn't contain measurable heat unless it changes direction as in a collision with something, another atom or a container.  So if the container is large enough and the number of atoms small enough, the atoms could move at any speed they like and there would be no measurable heat until the atom reached the wall of the container.

That concept is useful for most of physics, everything we can see or measure depends on the changes in the direction of something.  Only one problem, just because we can't see it or measure it does not mean it is not there.

Einstein figured that out and mentioned that something is missing, time.  No matter how big the container is, sooner or later that atom is going to hit something.  Einstein correctly figured out that that places a limit on how fast the atom can travel, the speed of light.  If the atom moved faster than the speed of light it would disintegrate.  That is a pretty hard limit.  If the atom disintegrated, every tiny building block of that atom would be measurable energy as it flew off into space.  So time, as in how fast the atom can travel, is another boundary of matter and energy.  Every part of the atom is a degree of freedom only measurable as heat energy when there is a change.   Since the speed of light is the limit, E=mc²   which means the smallest unit of mass is proportional to the smallest unit of energy by,  E/c².  If you knew what the smallest unit of energy or mass was and could prove it, you would be a hero in physics.  Since Einstein had to add another dimension for relativity, you probably will need another dimension or so for that exotic new discovery of say a graviton.

Going back to the larger micro world, if the molecule was diatomic, two atoms in a bond, the molecule has more degrees of freedom.  Atoms in a smaller molecule, like hydrogen may be able to move more while heavier atoms might be more tightly bond and could move less, but since they have more degrees of freedom to move, they can hold more energy which means they have a higher heat capacity.

The next level is more complex three and more atom molecules.  Each new atom in the bond adds more degrees of freedom to move which means there are more ways to stop moving or change direction of motion.

One atom three dimensional space, three degrees of freedom to move.

Two atoms, three dimensional space plus three directions of rotation in that space, 6 degrees of freedom.

Three atoms, three dimensional space plus three directions of rotation in that space, plus three vibrational directions, 9 degrees of freedom. 


Not all of these degrees of freedom matter though or at least to some don't matter.  A single atom can spin in any direction.  Since that spin direction doesn't tend to release energy, it doesn't tend to matter.  With a diatominc molecule, end over end rotation matters, but spinning like at top doesn't tend to release energy so it doesn't tend to matter.  If a molecule is not symmetrical in any orientation, then all of the possible degrees of freedom tend to release energy so they tend to matter.  


If you have ever played with a ball, you know that spin does matter sometimes, so it is not smart to just forget about that degree of freedom that generally doesn't tend to matter until it does :)


When it comes to Green House theory, note the small "t", the degrees of freedom that allow the release of a photon of energy are the only ones that appear to matter.


If you look at the Wikipedia link for the Boltzmann constant you will find that the number of microscopic degrees of freedom when considered with the macroscopic constraints, results in a formula for S, Entropy,


From Wikipedia, " 

In statistical mechanics, the entropy S of an isolated system at thermodynamic equilibrium is defined as the natural logarithm of W, the number of distinct microscopic states available to the system given the macroscopic constraints (such as a fixed total energy E):

This equation, which relates the microscopic details, or microstates, of the system (via W) to its macroscopic state (via the entropy S), is the central idea of statistical mechanics. Such is its importance that it is inscribed on Boltzmann's tombstone.

  
Now here is the fun part, the Boltzmann constant was actual never solved to the accuracy needed to be a true constant by Ludwig Boltzmann.  That was done years later by Max Planck.


So now think about what is being measured when you measure heat energy.  If you have a thermometer inserted or in contact with a fluid or solid, you would be measuring the collisional energy transferred to the material in the thermometer.  If you are using a non-contact thermometer, you would be measuring the energy transferred to the instrument by some other means than direct collisional contact.  If the instrument were in a vacuum, then the thermometer would only be measuring photons released by the object that struck the instrument.  If you are using a non contact thermometer inside of the object, you would have to compensate for the "contact" energy to determine the "non-contact" energy or the photons escaping from the object being measured.  The desired "non contact" energy would be Entropy or energy lost from the object being measured.  


What happens when we add more CO2 to the atmosphere?  We are adding more molecules with more micro states depending on their degrees of freedom.  The initial Entropy due to CO2 would be k*ln(Wco2), after doubling CO2 the final entropy due to CO2 would be k*ln(2Wco2)

So Ludwig Boltzmann with the help of Max Planck provided an alternate method of looking at global warming over a century ago.

Analyse This

UPDATE:  UAH oceans included at the end of the post:

Anyone can look at data.  If a chart is going up, it is going up.  Does it have to keep on going up?

You can draw a line on the chart to show it is going up.  You can start at different time to show it is going up less or more from that point.  Once you get near the end of the chart there is no more data. You can't predict the future, just look at the past, just see what happened in the past.  You can extend that line or curve you fit to the data in the chart, and extend those into the future as a guess of what may be coming.  If you pick a starting point that shows a future more like you want, you have "cherry picked" your starting point.  That's a no no.

So say you want to not fool yourself or others and just want the facts.  In the chart above I picked all possible 60 month trends for the UAH lower troposphere global temperature data.  If I wanted to show the most positive trend for a 60 month period, I would pick the 60 months ending in about 1989.  The most negative 60 month trend would end in about 1994.  From those two points the trends decrease.  For the whole period, the trend of the trends is upward a little bit.

If you do the same analysis for the Northern Hemisphere, the Southern Hemisphere and the Tropics, you see there are differences and similarities.  They don't do the same things at the same times.  The most negative 60 month trend for the tropics ended in about 2001, for the SH, the most negative land trend ended in 1994 but the oceans and total had their most negative trends en in about 2001.  For the NH, the most negative ocean trend was in about 2009.

If you were investing all your money in UAH Lower Troposphere temperature anomalies, would you buy or sell.

Bonus question

What about Sea Surface Temperature Anomalies?

 This is the same type of linear regression analysis done with the UAH ocean data.  Each series is 60 month linear regression ending at the date for each month of the series.  The beginning is obviously limited to 59 month after the beginning of the data.  Note that as the individual portions of the oceans synchronize something would be happening to the energy transfer inside the oceans.  With adjacent oceans sections synchronized, there would be less internal energy transfer or a more stable rate of internal energy transfer.  The oceans would be more likely to cool if they are in equilibrium or steady state.  This method is just a quick way to determine an estimate of the degree of synchronization.  

Back into the Ein=Eout Breach

One of the common assumptions made in climate science is that at the top of the atmosphere the energy in is equal to the energy out in equilibrium.  There is absolutely nothing wrong with that assumption, provided that the it is realistically considered.  Ein = Eout over some time period.  Ein never has to exactly equal Eout at any given moment in time.  In fact, Ein probably does not equal Eout very often.  It is a tool.  If measure Ein and it is greater than Eout then there is an imbalance, more energy is remaining in the system than being released.

I modified the standard Ein=Eout assumption to Ein=Eout +S-W, where S is entropy and W is work.  Depending on what you define as work or entropy, this is a more correct form to consider.  Work in my opinion is the atmospheric lapse rate, ocean temperature differential that powers the winds etc.  If the Earth were a billiard ball in space, Ein=Eout would be perfect.  Earth is not a billiard ball so it deserves a more realistic consideration.

A rather large reason that it is a little disrespectful to consider the Earth a billiard ball is because it has fairly large oceans.  Using the same logic as the Ein=Eout at the top of the atmosphere, I can consider that the oceans are a system that over some time period also meet the Ein=Eout requirements.  This is not received very well by many.  It is perfectly valid though since the Earth appears to have had oceans for a long time and not in any immediate danger of losing those oceans.  This assumption also should be treated with the same respect as the TOA Ein=Eout assumption.

So if both the oceans and the Top of the atmosphere have equilibrium conditions over some period of time and it is unlikely that the time periods are the same, what good is using the equilibrium assumption for both systems?  Well, in between the oceans and the top of the atmosphere are things inside the atmosphere and outside the oceans.  That is where the -W goes and the +S passes through.

Over a long enough time scale, the oceans have an equilibrium condition and the top of the atmosphere also has an equilibrium condition.  The in between the oceans and atmosphere may never and likely has not ever had an equilibrium condition.  Mountains erode, land erodes, new land forms, lakes form, rivers form they also fade.  The oceans rise and the oceans fall as ice builds and fades on land, but the oceans are a constant for the majority of the surface of the Earth.  By using the assumption of equilibrium for both the oceans and the top of the atmosphere, you can consider an equilibrium model that recognizes the chaos, but doesn't just dive right into it.

Now if the oceans gain more energy over some time period, the assumption implies they will get rid of the energy in the same amount over some time period.  The atmosphere can only get rid of energy by radiation according to Greenhouse Theory.  The oceans can release energy is a large number of ways, one is by losing mass.  That is right sports fans, evaporation does not have to return to the ocean immediately as condensation.  Ice formed near the poles can stack up on the land and not return for some time.  Ice formed can sublime to the atmosphere.  As long as that water and ice returns over some time period, the oceans will have some equilibrium.

That is not too hard for most to comprehend, but how is that possibly an equilibrium?  If the oceans derive most of their energy from the sun passing through the top of the atmosphere, and they transfer some of that energy to land, then the TOA Ein would not equal Eout while that was going on.

Just for grins, if say the oceans transferred billions and billions of tons of water to the polar regions as ice, Since each gram of that water would have to release 334 Joules of energy in that phase change, then the atmosphere would receive a large portion of billions and billions and billions of Joules of energy.  As the ice melted, it would have to absorb all that energy again either from the oceans of the atmosphere.  There would be no true equilibrium for either the oceans or the TOA while that was happening.  If you find the right time period where the oceans found equilibrium, you would likely find the time period that the TOA found equilibrium.

That is a temperature reconstruction for the past 60,000 years for data collected at Lake  Tanganyika in Africa near the equator.  That may be too short of time for there to have been either a true ocean or TOA equilibrium .  That 2.5C range though near the Equator where the majority of the energy is absorbed from the sun by the oceans, does provide a reasonable guess as to what equilibrium may be.  Comparing an idealized equilibrium at the top of the atmosphere to an idealized equilibrium in the oceans, is just a way to reduce the uncertainty in each.  See, there is no reason to assume that Ein=Eout at the top of the atmosphere over the past 120 years either is there, when the Earth has been changing for millions of years :)  If for the entire period the temperature was below average the oceans released energy, how long you reckon it would take for the oceans to regain that energy?

Simples

Most people just can't grasp the simplicity of this model.  The Earth is mainly ocean.  The ocean is water.  Water evaporates, freezes, flows, does water stuff.  Air with water in it, is different than air with no water in it.   So the first layer of the model is water, the ocean, and moist air.  Eaverage is the approximate radiant energy of the surface of the water at Taverage, the average temperature of the water.  The EMBL is the approximate radiant energy of a layer of air at TMBL, the temperature of that layer.  It is an imaginary layer.  If you look out your window you likely won't see it.  This imaginary layer is called a reference.  Why did I chose this as a reference?  Because by the time the moist air has cooled to that temperature is has likely started condensing into water droplets and maybe even ice.  There is also something else neat about that reference layer.  The total per gram of air at that temperature and 50% relative humidity is about 4.2 Joules per gram.  Oddly, the total energy of the water at the surface is about 4.2 Joules per gram.  Kinda funny huh?

Now here is a big word, THERMODYNAMICS.  The moist air in that Quonset hut looking shape has different THERMODYNAMIC properties than the dry air way outside the Quonset hut.

This is a bad drawing of our world without land.  The blue oval is a bad drawing of the moisture boundary layer (MBL).  The red oval is a bad drawing of the high temperature tropical water region.  The yellow oval is a bad drawing of the Radiant Boundary Layer (RBL).  The RBL is also imaginary.  I could pick any temperature and energy I like for my imaginary RBL.  Since I had 425Wm-2 for the water surface and 306.9 for the imaginary MBL I choose 188Wm-2.  Why?  Because 306.9 is 118.1 less than 425 and 188 is 118.1 less than 306.9.  You may have noticed that 188 is not exactly 118.1 less than 306.9.  That is because I ROUNDED.  That is a lazy way of doing things isn't it?

188Wm-2 would be equal to a temperature of 240K degrees.  That is a cold temperature.  So cold that there is very little moisture in air that cold.  That dry air would have different THERMODYNAMIC PROPERTIES than moist air.  Notice that my badly drawn yellow oval doesn't cover the entire badly drawn Earth with no land.  Those areas outside the yellow lines would have different THERMODYNAMIC RULES to follow.

If I am trying to figure out what impact THERMODYNAMIC CHANGES to the yellow RBL would have on other things, the impacts would be different inside the yellow oval than outside the yellow oval.  Those impacts would  be different than inside the blue oval and still more different inside the red oval.

If I were ANAL, a short but powerful word, I would worry about each and everything happening everywhere.  I am not ANAL, so I am only concerned with my imaginary boundaries on my imaginary world.

My imaginary world may have an AVERAGE GLOBAL SURFACE TEMPERATURE, I don't care.  My imaginary world may be 33C warmer because of the yellow oval.  I don't care.  If there were land on my imaginary world, it may have a different temperature or albedo.  I don't care.  All I care about is my imaginary oceans inside my imaginary MBL inside of my imaginary RBL.

Why would I be so uncaring?  It is called SIMPLIFICATION or simples if you like.  I start with the basic simple stuff and find out what hard stuff I need to figure out.  Once the imaginary ocean is balanced, then I can look at the imaginary land.

This is another funny little drawing.  It represents my imaginary ocean.  Notice that the light blue has the same 425 Wm-2 as the funny Quonset hut.  There is also a red layer with 495 Wm-2.  That is the hot red oval on the funny imaginary world with no land drawing.  There is a dark blue layer with 334.5 Wm-2.  That is a neat layer.  In the deep oceans there is a layer of water at about 4 degrees centigrade.  It seems to like that temperature.  On the sides are gray blocks with Ice 316Wm-2 and 307 Wm-2 Salt Ice.  I didn't type in the Wm-2 for the Salt Ice, because I didn't draw the block big enough and I am lazy.  Now here is something funny.  The Wm-2 stands for Watts per meter squared.  It is not correct to use Wm-2 in the water like that.

So why did I use Wm-2?  Because it SIMPLIFIES things and is informative.  From the 425 to the 307 is the same as in the cute Quonset hut except I ROUNDED.  From the 495 to the 307 would be 188Wm-2.  Does that sound familiar?  It should it is the same as the RBL from before, kind of odd huh?  It is almost like the system is in balance.  That is kinda the point of a static model.

Now the ice blocks.  The 9Wm-2 difference between the ice and salt ice is a joker.  These represent the energy of the heat sinks for the ocean and the atmosphere.  The temperature and the energy of the heat of fusion are nice reference values for an energy model.   The rate of energy transfer inside the ocean would vary between these two points as long as ice exists.  Sea ice that forms over the deeper ocean would insulate that portion of the ocean stabilizing temperatures.  If the deep ocean temperature is stable, then these references would make a very simple to use proxy for overall heat gain or loss.

This drawing shows why I think there is a need for that proxy.  Fresh Ice or ice forming in fresh water gains or loses 334 Joules per gram in phase change.  The gain or loss is neutral.  Salt Ice or ice that forms from salt water loses or gains 334 Joules per gram, but there is an extra 8 Joules per gram needed to cool the water below zero C degrees.  Since salt ice forms in salt water, there is energy lost to the water as the more dense salt water descends from the point of freezing.  There is a transition for the melt of ice that has to be considered.  Ice that melts away from the salt water, whether on land or on the ice, doesn't return the heat to the ocean immediately.  The melt water with the energy gained from the heat of fusion must flow back to the oceans for that heat to be returned.  So like real estate, thermal mass balance is all about location, location, location.

The ice difference is fairly easy for most to understand.  The more interesting though is the vaporization balance.  The heat of vaporization is non-linear.  At 100C and sea level pressure it varies with the salinity of the water.  It also varies with the temperature of the water and the barometric pressure.  Where evaporation takes place and where the condensate falls are extremely important.  Remember, there are the poorly drawn ovals on the poorly drawn world with no land to consider.

The First Day of Global Warming

The first day

All it would take is enough energy to melt some of the ice near the equator for the Earth to begin having a Green House Effect.  For day one, the 350 represents 1/2 of a faint sun insolation of 750Wm-2.  With no significant water vapor in the atmosphere there would be very little solar energy absorbed by or reflected by the atmosphere.  The energy values in the slab would represent near minimum values for a salt water open area with initial energy provided by a volcano or other natural event.  The 350Wm-2 would be equivalent to a temperature of 288.3K or 7 C degrees.  For twelve hours the water would absorb some solar energy.  With the minimum temperature for the water to remain liquid of -1.9C or the 307Wm-2 energy shown, there would be about 43Wm-2 transferred from the solar heated water to the water at the ice boundary.  While the solar energy may not be enough to melt the assume fresh water ice above this subzero liquid salt water, as the ice approached the melting point it would transfer up to 9 Wm-2 to the liquid.

Using Wm-2 inside the liquid is not a perfect illustration of energy transfer.  The energy transfer in water is typically in Joules per gram.  For the ice at 0C the transfer to -1.9C would be closer to 8.4 Joules per gram instead of 9 Wm-2.  The exact energy would depend on the salinity of the water and the purity of the ice.

For the first night, there is a blue bubble shown replacing the solar arrow.  As the 350Wm-2 warm pool cooled, the water on the surface would start to freeze.  Each gram of water that freezes would release 334 Joules per gram.  Again, Joules per gram is not a perfect conversion for Wm-2, but it is pretty close.

The water would receive 350 Wm-2 during the day at as the surface began to freeze, would be limited to approximately 334Wm-2 until it sufficiently froze.  That is only a gain of 16Wm-2, but because of the salinity of the water, could be as high as 25Wm-2, fresh ice is warmer than salt water at the freezing point.  This creates a little Mexican standoff. 


Rapidly cooling water has enough thermal inertia to freeze quickly.   Slowly cooling water actually takes longer to freeze since it has to deal with the point of fusion energy standoff, known as the Mpemba Effect.  The freezing of the surface tends to block convection from below reducing the rate of cooling.  Warmer water with more energy can diffuse more readily allowing it to overcome the convection blockage.  With a very small difference in energy, both the rate of convection and the rate of diffusion are minimized.  The temperature inversion and the ambitious density of the salt water makes things even more entertaining.


The maximum density of fresh water is at the temperature of 4 C degrees.  As water cools to that temperature is sinks slightly.  In salt water, the impurities cause that maximum density temperature to disappear.  The density is controlled by the salinity and as ice forms, some salt is expelled increasing the density causing the colder more dense water to sink and reverse of the more efficient convection path.  This allows more energy to be retained than one would expect at first glance.


Another consideration with the faint sun is energy absorbed in the atmosphere.  With little water vapor only ozone and oxygen would absorb any significant amount of incoming solar.  Since clouds current reflect nearly 24% of the incoming solar an absorb nearly the same percentage of the solar not reflected, the faint sun's energy would be more efficiently transferred to the surface water.  As long as the overall atmospheric composition is nearly the same, other than the water vapor content, energy would be transferred to the atmosphere both during 24 hour period by conduction.  That is a small but not negligible energy transfer over a long enough period of time.


In addition to the small solar energy gain, there is another small geothermal energy input into the oceans that likely caused the liquid water to exist to begin with.  Approximately 0.1 Wm-2 of energy are transferred to the oceans now.  A younger Earth was more likely a more geologically unstable Earth.  Even with a snowball Earth, there would likely be liquid water near the ocean floor with its huge pressures.  With that liquid water, the energy available would be on the order of 307Wm-2, the freezing point of salt water.  With the average solar energy available at the total surface area of 1/4 of the 700Wm-2 available at the top of the atmosphere and the weak absorbing atmosphere, there could be  175Wm-2  available at the surface with the 350Wm-2 available at the equatorial region.  The Sun and the Moon would still have a gravitational effect causing water to flow and thinner ice to fracture.  As long as there is a positive net gain in energy, the oceans would slowly warm, energy would be transferred to the young atmosphere and the globe would be warming.

The energy balance could end in a position like this.  We have a warmer ocean with less energy.  The solar in this drawing is based on 50% of 1361 Wm-2 available at the top of the atmosphere, 30% of that is reflected to space mainly by water vapor, 128Wm-2 is transferred to the atmosphere, absorbed by water and water vapor in the tropics and 345Wm-2 makes it to the surface.  This drawing is close to current reality, the solar available in the tropics drives the ocean heat engine that warms the global.  Global warming is why we are here to ponder global warming :)  For global averages that would be 345/2=172.5 absorbed by the surface, 64Wm-2 absorbed by the atmosphere.  There is only one major difference between this energy budget and the ones used in the Climate Change debate.  This one does not include the Antarctic an other regions where there is no significant water vapor.  Those regions are not in the moist air envelope that causes global warming and their climate never really changes.
  

More Stuff:

This is a plot of the specific enthaply of fresh water plus the 334 Joules required for the heat of fusion versus the ideal black body energy of an object of the same temperature.  There is no such thing as an ideal black body and our oceans are not pure water, so some adjusting would be required for these two energy measures to be comparable.  With most things though, it is not a bad idea to start with ideal relationships then work into the nitty gritty.  By adding the 334Wm-2 for the heat of fusion I am attempting to determine an absolute enthalpy.  Based on this ideal comparison, it would be easier for energy to leave the water, S-B than increase the absolute enthalpy above approximately 21 C degrees.  Below 21 C, the absolute enthalpy is greater than the S-B ideal energy.  According to S-B, the surface would still emit at its black body temperature but the emissivity of the surface is not fixed.  The object does not have to emit perfectly because it is not a perfect object.  Generally, the emissivity is assumed to be to be constant or at least the change assumed to be insignificant.  That may be a valid assumption, but the heat of vaporization is also non-linear.  Water evaporation at lower temperature due to partial pressure relationships have a higher energy per gram. That is still energy emitted, just not radiant energy.  If all energy transfer is considered then there would be a true emissivity or effective emissivity.


So think of a glass of fresh water.  Instead of thinking in temperature we think in terms of energy and density.

If the water in the glass contains more energy than 334.5Wm-2 it is less dense.  If it contains less energy than 334.5Wm-2 it is also less dense.  So in the glass water above the cute dark blue slice with the arrow would be colder, than light blue below the cute falling slice would have more energy than the bottom blue slice so the cute slice descends to meet up with its buddy with the same energy and density.  Using the Stefan-Boltzmann relationship, the table lists the energy, the fixed energy of fusion, the ratio of the S-B to the fixed energy of fusion and the final column is the total enthalpy difference from the maximum density if the world were perfect.  Even here, at 4C the enthalpy is not perfectly zero, there is 0.63 Watts or joules per some unit difference.  The difference to the ideal freezing point of water is -18.27 units of energy.  There is another factor though, the difference in density.  The main variable that controls density is the force of gravity.  On Earth, that force is determined by the gravitational acceleration constant of 9.81m/sec^2 at sea level.  It could be one or ten, it has to be some odd ball number like 9.81.  To simplify things I would change that sucker to 10, but then I would have to adjust all those numbers I slaved over a hot spread sheet to make.  It would be nice though to have a simple calculation to determine the specific enthalpy of at least water that directly relates to the S-B energy.

The dynamics of water though make that a little more complicated.  The speed of sound is a limit just like the speed of light is a limit for the Stefan-Bolzman equation which has this nifty little factor considered.

where:

• k_B is the Boltzmann constant;
• h is the Planck constant;
• ħ is the reduced Planck constant;
• c is the speed of light in vacuum.

A body of water is obviously not a vacuum.  So we would need to consider the real speed limit in water and the characteristics of heat flow in water which is a little different than photons shooting through a vacuum.


Because of the differences, in fluid dynamics you have different kinds of temperatures, Stagnation temperature, static temperature and there should be a dynamic temperature.  All based on the ability of the fluid to transfer heat.  


The stagnation temperature is the temperature at the zero velocity of a stream flow.  For our cute little glass, the stagnation temperature would be at the ice barrier, no water flows through that barrier.  Since I am working in energy, that point can be called the Stagnation Energy, which for pure water at our imperfect gravitational constant would be 316Wm-2 if the S-B number were perfect.  


Static temperature is related to Stagnation temperature, for air,

where:

 static air temperature, SAT (kelvin or degree Rankine)
 total air temperature, TAT (kelvin or degree Rankine)
 Mach number
 ratio of specific heats = approx 1.400 for dry air

The T(total) is the stagnation temperature, instead of the speed of light, the equation considers the Mach number or the speed of sound which varies with density and composition of the media, in this case air, of the path of the energy. 

So I could make my life simple and use every equation known to mankind, or tweak the S-B relationship to allow for changes in density that impact the rate of energy flow.  This is actually the basis for the Relativistic Heat Conduction equations.

"The most interesting observation about RHC is that it reduces the second law of thermodynamics to a statement of the form

which is the “no action at a distance” principle of special relativity. Essentially, the RHC asserts that relativity and the second law of thermodynamics are two alternative, but equal statements about the nature of time. Both physical principles are mutually derivable from each other and are complementary.^[1]"

Note: The C in RHC and the c in the S-B are not the same.  RHC use C as the speed of second sound similar to the Mach number in fluid dynamics.