Why Pi?

When I mention that the subsurface temperature tends to the energy equivalent of TSI/Pi(), eyes roll.  I use it as a convenient approximation.  If I were to get into more detail, I would use the integrated incident irradiation  which would consider the azimuth of the sun by latitude and time of day with a day isolation factor that included seasonal variations.  Since I am more concerned with the ocean subsurface energy, I would also have to consider land mass and sea ice.

I find it easy to just assume that liquid ocean is most likely between 65S and 65N, then Io*Cos(theta) from -65 to 65 would be about 450 Wm-2 average while TSI/Pi() would be about 430 Wm-2 excluding the land mass.  Those would be average insolation values for the day time portion of a rotation.  If the insulation were close to perfect, these would be the more likely values of the subsurface energy actually stored.

Another reason I like this approximation is that if the oceans were never frozen at the poles, the angle of incidence over 65 degree would mean close to 100% reflection off a liquid surface anyway.  The only energy that would likely penetrate to the subsurface directly would be between roughly 65S and 65N except for a month or so in northern hemisphere summer.  It is a lot easier to say the subsurface energy will tend toward TSI/Pi(), than to carry out a lot of calculations which are about useless without knowing the actual cloud albedo at each and every location on Earth.  Roughly though, if you consider a noon band with a zero tilt and the average 1361 Wm-2 TSI, from 65S-65N the insulation could be as high as 910 Wm-2 versus 665 Wm-2 for 90S-90N for the oceans.  That should give you an idea of the difference between subsurface and surface solar forcing using average ocean land distribution.  There are of course central Atlantic and Pacific noon bands with nearly zero land mass between 65S and 65N that can allow much more rapid subsurface energy uptake.  So until I either take the time or determine a need for a more accurate estimate, TSI/Pi() is convenient.

Purist won't like that, but hey.

There are quite a few things that can bung up that estimate.  One is how well the subsurface energy is transferred pole ward.  If the transfer is slow, like before the Panama closure, the equatorial temperatures would tend higher which could increase the average or the Drake Passage could open which would decrease the average.  When the average changes the cloud cover percentage and extent would change making the puzzle a bit more challenging.  With a baseline, "surface tends tend toward TSI/Pi()", you can at least estimate the impact of changes in ocean circulation on average ocean energy.  It is far from perfect, but a convenient approximation.  It is also convenient to assume that albedo is fixed.  Subsurface versus surface energy provides a reasonable explanation why albedo may be some what fixed.

If you have a more elegant estimate, break it out.


Note:  This post is just an explanation in case some wants more detail about the TSI/Pi() approximation.  It may be revised or expanded as required.