## New Computer Fund ## Tuesday, February 19, 2013

### Why Root Mean Squared?

If you live in the United States and have a toaster, it is like rated for 120 volts and 15 amps or less.  Those values are RMS or Root Mean Squared values.  The actual alternating current provided by the wall receptacle that you plug the toaster into is supposed to be 340 Volts peak to peak and the breaker that serves the receptacle is likely rated for 20 or 15 amperes.  To figure out RMS, you have to consider the shape of the wave form providing the energy.

For just the basics, Wikipedia has a fair page on the subject.  Since the toaster is AC with a Peak to Peak of 340 Volts, the sine wave formula applies, RMS=0.707*a, where the 0.707 is the square root of two and a is amplitude or peak value, not the peak to peak value.  So the toaster gets 0.707*170=120.19 VAC, or just plain 120 or even 110 volts depending on the leg of the transformer your toaster's breaker is connected to at the panel.  What's is a few volts among friends.

So what has this got to do with climate science?  Probably a lot more than many think.

For the toaster, the RMS value is great because the AC voltage in goes positive and negative.  Since the toaster is likely grounded or at least insulated, in the middle of the sine wave current is zero.  It draws current on the positive and negative parts of the sine wave.   The current then would be in phase with the voltage.  Let's say that we have a floating neutral.  That would be where zero is not in the center of the sine wave but shifted up or down, the neutral of the transform is not grounded.  In this case, since we really don't know what the peak is, 0.707*340=240 volts might be the voltage applied to the toaster.    Since the toaster is a resistive load, now the current may be out of phase with the voltage, how much we don't know without checking things out.  Got to know your reference or ground state.

Since the Earth rotates, the energy provided by the sun is a sine wave effectively with a peak to peak value of 1360 Wm-2.  0.707*1360=961 Wm-2 would be the RMS value of the energy applied if the Earth were a flat disc.  Since Earth is spherical, that energy is at a peak near the equator and near zero at the poles so the cosine distribution would be 0.707*961=679 Wm-2.  This is twice the applied energy generally considered with the  TSI/4 average used in climate science.  This doesn't mean that climate science is wrong, just that there are other ways of looking at things.

The 679 Wm-2 is just the RMS value for the day surface.  Related to a fixed ground level of zero, the mean would be 1/2 or 339.5 Wm-2 or about the same as the 340 generally used.  But if the "neutral" is floating, i.e. there is something other than zero as a reference, the actually available energy would be higher.  One possible source of a floating "neutral" is the Greenhouse Effect.  Since it takes time for the night side to decay to "zero" or "neutral", Greenhouse Gases as well as other atmospheric effects would increase the delay.  Since there are two greenhouse effects in my opinion, I want to look at the other first.

The oceans also have their own delays in heat loss.  That residual energy left before the next day dawns, would effectively cause the floating "neutral".  Since the thermal mass of the oceans is much greater than the atmospheric, this residual energy would be the "source" of the atmospheric portion of the Greenhouse effect.  This is not disputed, but methods used to isolate the total impact of the atmospheric portion tend to assume away to impact of ocean thermal inertia or the lag required for the oceans to decay to a "zero" energy value.

The heat of fusion of water is just one of the inertia lags.  The latent heat of fusion, ~334 Joules per gram of water and the "energy" of fusion of 316 Wm-2 down to ~307 Wm-2, could provide a total energy of up to 334 + 316 = 650 Wm-2 is the area of surface and volume of ice happen to be equal.  So if the surface water can freeze quickly and deeply, it could release pretty much all or more of the energy gained in the day period, since the difference is only 29 Wm-2 and there are of course other factors.  I am going to use energy values instead of temperature for a reason that will hopefully explain itself later.

The question is how quickly.  For ice to form in fresh water it would first cool the water to maximum density, ~316 Wm-2.  The as the water freezes, releasing the 334 Wm-2 of energy, the density of the ice decreases to approximately 90% of the maximum.  This increasing density first then decrease, produces an counter flow that would delay heat loss.  Since fresh water has a maximum density at 4 C, the whole volume would have to cool before ice began to form.  Since the oceans are a huge volume, the rate of heat transfer would "define" a volume that could freeze.  With salt water, the maximum density is not limited to 316 Wm-2.  It would be limited by the energy and salinity.  The Fahrenheit temperature scale is based on the freezing point of saturated salt water or brine at 0 F and fresh water at 32F.  0F or -17,7 C has an equivalent energy of 241 Wm-2 (255.3K degrees).  Since salt is expelled in the freezing process, ocean water freezing would create a lag or delay from nearly 316 Wm-2 to 241 Wm-2.

Now to hopefully answer that question in the title, because of inertial limits.  In order to reach a perfect peak energy, things have to be perfect, the insulation of the capacitor, zero resistance in the transfer, perfect stability in the applied power.  Our world operates in okay ranges not perfect ranges.

Think of the oceans if we were to turn the sun off.  They would cool fairly rapidly until they reached about 316 Wm-2.  Then they would have to release about twice as much energy to continue cooling, so cooling would slow as they approach this inertia limit.  Since the oceans have salt that would be expelled in the freezing, not all but most, that expelled salt has to diffuse or mix away to allow less saline water to freeze.  That is another inertial limit.  Then once all the oceans have reached approximately 0F or -17.7 C, then the inertial limit is passed and more rapid cooling could begin again since counter flow mixing is no longer required.

It is like charging a battery, lots of current to start decaying to zero at full charge and when you discharge, there always seems to be a little energy left it you let is set a little while, the "floating neutral".  It is all in the mixing.