Wednesday, February 27, 2013

RMS versus Average - the Radiant Shell Game

This is the problem.  Assuming "average" instead of RMS you have figure (a)  With RMS you have figure (b) as a peak and (a) as zero.  As the total heat content of the surface with average energy value (a) increases, the range of (b) minus (a) changes.  For a "ballpark" estimate, (a) is fine, but since the variation from (a) to (b) can be 17 Wm-2 equivalent or more compared to a total CO2 forcing of 4Wm-2, the "ballpark" estimate is not adequate.

The link to the article containing that graphic is, Science Focus: Sea Surface Temperature Measurements MODIS and AIRS Instruments Onboard of Aqua Satellite

If you are looking for "average" and get something else, you might start blaming the satellite guys.  So let me try to compare the "average" and RMS values of the energy applied to the "surface" again.

Surface is in quotes because there are various surfaces in a complex system.  The ocean atmosphere interface or boundary layer is one of the more important surfaces.  The actual solar energy that can reach this surface varies with cloud cover, season, time of day, and there are longer term "minor" solar cycle variations.  The energy provided is always positive, ranging from zero to roughly 1410 Wm-2.

A simple average for a sphere with 1410 Wm-2 available would 1410/2=705 Wm-2 for the active or day face and 705/3=352.5 Wm-2 for the diurnal average.  An RMS estimate based on the sinusoidal shape of the applied energy would 1410/1.414=997 Wm-2 and due to spherical shape 997/1.414=705.  Since there is not abrupt fall to zero with day to night transition, for diurnal change the RMS value would be 705/1.414=498 Wm-2.  Cloud cover is generally non-stationary, so local variations change with some unknowable time period, amounting to roughly a pulsing energy wave shape with some average duty cycle.  Since most tropical clouds form after peak solar from energy gained up to peak solar, the cloud cover effectively shifts the "surface" upwards.

If you neglect the clouds for the moment, the 498Wm-2 is the RMS energy at some "surface".  As a radiant "shell" that surface would emit energy up and down, producing 259 Wm-2 up and 259 Wm-2 down.  Then the RMS value would be equal to the "Average" value, but that requires an ideal "surface" with precise depth and negligible thermal mass.  To maintain the energy in that radiant "shell" there must be 259Wm-2 provided constantly from some source, i.e. an ideal black body.

Clouds plus changes in atmospheric density and composition would vary the depth and distribution of the "shell", but the "shell" energy is solely dependent on the two black body sources, the Sun and the thermal mass of the core.  Since the core must store energy gained from the Sun, the nocturnal energy available depends on the core's ability to transfer energy to the radiant "shell".

I am going to stop here to let everyone ponder this concept with a little food for thought,
E=Eo*e^(-t/RC),  what is "t" for a planet?