## New Computer Fund ## Tuesday, June 19, 2012

### Isothermal Boundary Layers

The Moisture Boundary Layer model uses the latent heat of fusion as a reference boundary.  The drawing above shows the MBL in light blue roughly oriented to the actual Earth.  Since the Earth has predominately salt water available in liquid form, the temperature of the MBL would approximately 271.25K degrees (-1.9C) slightly below the freezing point of fresh water.  To simplify calculations, instead of actual temperature, it is easier to use energy units based on the Stefan-Boltzmann relationship E=5.67e-8*(Tkevlin)^4.  So the value of the MBL would be 306.9Wm-2.

Below 271.25K water would turn to ice and water vapor would decrease to nearly negligible amounts.  Above water would be mainly in liquid phase with water vapor in the atmosphere based on the temperature and pressure.  Using the sea level barometric pressure simplifies calculations, so the liquid surface temperature would determine the profile of the MBL.  As shown, the equatorial temperatures are warmer which would increase moisture in the atmosphere expanding the MBL vertically.  Temperature decreases with altitude AND heat content.  Moist air contains much more energy which takes longer to dissipate, so the altitude of the MBL increases with the energy content of the moist air.  The width of the MBL also varies with the total amount of moist energy which means the total volume of the MBL varies with total moist heat content.

By using an "average" temperature of the ocean surface  and relating that to the fixed value of the MBL a simple energy flow relationship can be obtained.  With the "average" sea surface temperature (SST) being approximately 294.25K degrees with an equivalent energy of 425Wm-2 and the MBL energy of 306.9Wm-2, there would be a steady state energy flow of 425-306.9=118.1Wm-2.  This is THE average total energy flow from the sea surface to the MBL.  This is way too simple a concept for most geniuses to grasp.

So by defining the Moisture Boundary Layer one can start at the most relevant thermodynamic frame of reference in the complex climate system of Earth.  It does help knowing what TOTAL energy means in terms of a steady state flux.  By multiplying this steady state flux times the total surface area of the MBL, one can get the big ass number they are looking for to impress their friends.  Then if someone just wants to solve the problem, they can stick with the stead state energy flux between boundary layers and not mess with the big numbers until the end :)

On the drawing above is a red band with a small oval noted as CBL, the Conductive Boundary Layer.  With the average SST energy of 425Wm-2 and the flow of 118.1Wm-2 to the MBL, the maximum energy of the surface at the CBL would be 425+118.1=543.1Wm-2.  For the degree K fans, that would be a temperature of 312.9K or 39.07C degrees.  This is well below the boiling point of water.  If the MBL was the only limit of energy flow, that would be the maximum SST.  With the Earth in some conditional state of equilibrium, meaning we are here to discuss this subject, there would be another limit or the oceans would continue to gain energy until there is no ice to provide a sink of the heat internally.  Kaboom is a word that comes to mind.

Since the energy of the MBL is 306.9Wm-2 and the steady state energy flow to this boundary is 118.1Wm-2, there would be a second Radiant Boundary Layer (RBL) that would have an energy of 306.9-118.1=188.8Wm-2.  Why would this layer exist?  If the MBL is stable as in conditional equilibrium and receiving 118.1Wm-2 of energy it would also have to rejected 118.1 Wm-2 of energy to maintain the 306.9Wm-2 of energy fixed by the freezing point of salt water.  That gives us a conditional equilibrium value for the RBL of 188.8Wm-2 with a temperature for the degree K fans of 240.2K or -33C degrees.  Conversely, the average maximum SST energy would be 306.9+188.8=405.7Wm-2 which the degree K fans would figure out is 305.8K or 32.6C degrees at equilibrium.

Looking at the opening figure, one should notice that the MBL does not enclose the entire surface of the Earth.  If it did, the kaboom word would be useful.  To not go Kaboom, the internal flows of energy would have to be in a steady state as well.  Electically, this would be a parallel circuit internal to the system.

With the flow to the Radiant Boundary Layer at 188.8Wm-2 and the flow to the MBL at 118.1Wm-2 for this conditional equilibrium condition, the internal flow from equator to ice the same over some period of time.  That caveat needs to be added because there is a 1.9C difference between the freezing point of salt water and fresh water.  For now it is easy to stick with the salt water temperature and energy reference so let's stick with easy and say the moist air flow is the same to the poles as it is to the MBL, which is kinda the point of defining the MBL, simplicity.

There will be heat flow in the liquid water of the oceans though that does not have the luxury of picking the path of least resistance.  In order to estimate this internal flow, the enthalpy of water has to be considered. In order to change the temperature of one gram of water one degree C, it required 4.2Joules of energy be transferred.  To be consistent, Wm-2 would be nice.  Joules per second and Watts per second are conveniently identical.  A cubic meter of water contains 1000 kilograms of water and there are 1000 grams per kilogram.  That means there are 1,000,000 grams of water per cubic meter.  If we think of a cubic meter as a stack of 1,000,000 grams of water then we can have 1 gram equivalent of water per face of the cubic meter.  In other words, 4.2Wm-2 per second in water is equal to 4.2 Joules per second, tah dah!  The average temperature of the SST is 294.25K and the freezing point of salt water is 271.25K which is a temperature difference of 23K degrees.  To change water 23 K degrees requires 4.2Wm-2 per degree equaling 96.6Wm-2 of internal average energy flux for a salt water sink.  The freezing point of fresh ice is 1.9 C warmer, which would be 4.2*21.1=88.6Wm-2.  This energy flux does not include phase change and would likely not include any significant radiant heat transfer.  This is energy flux is most likely simple thermal diffusion, aka sensible heat transfer.  The  use of the term "most likely" is intentional.  There would be some phase change from ice to water and back at the MBL and the amount of salt and other impurities in the water would change the thermal conductivity of the water.  This will provide an interesting challenge.

In this drawing the 425Wm-2 average is used with the optional sink energies for the range of salt to fresh dominate conditions.  The deep ocean sink would be lower, 2.75C degrees for salt dominate and 4.4C for fresh dominate, for this conditional equilibrium state.  Under salt dominate conditions, more energy would be absorbed in the deep oceans which would increase the energy of the system causing the MBL envelope to expand.  That expansion would include more fresh ice which negatively feeds back on the internal heat content by raising the freezing temperature at the MBL.  So with this range of potential feed backs, it would likely be better to build the model based on the average.  Interestingly, this guarantees the model will be incorrect, but would make it more useful for determining the impact of the feed backs.

By using (306.9+315.6)/2=311.25Wm-2 for the MBL reference, the model would look like this:

For the specified average ocean energy emission of 425Wm-2, the average MBL atmospheric energy emission would be 113.75Wm-2, the average ocean energy flux internally to the sink temperature would be 92.6Wm-2, which would produce an absymal depth temperature of 3.6C with an inward energy flux of 92.6Wm-2.  which gives us a new flux to complete the balance.  331.5-311.25=20.5Wm-2 from the deep ocean depths to the polar ice regions, with, 113.75-92.9=21.15Wm-2 from the surface atmosphere to the polar ice region.  The difference, 21.15-20.5=0.65Wm-2 indicating either some error or there is another source of energy not considered, possibly, geothermal :)

The range possible from fresh dominate to salt dominate regimes, should span the glacial and inter glacial climates.  That story will be left for another day.