New Computer Fund

Wednesday, December 4, 2013

Simple Circuit Models

Simple RC circuits are great for modeling a bunch of processes.  I haven't touched on this much because I think you need to be careful of what you are actually trying to model before you start going nuts.  But, an example of using a "pull up" resistor RC circuit was mentioned.  A "pull up" or "pull down" resistor is used in logic circuits to make sure that the output voltage is easily recognized as a "yes" or "no".  If you have a 3 volt circuit a "no" could be less than 1 volt and a "yes" greater than 2 volts with between 1 and 2 volts something that is not recognized.  That is just an example, but you need to make sure the signal is not confused due to power supply drift or changes in load which effect the voltage.

A "pull up" would not be my first modeling choice but it has some advantages.  Since R' and R are in series between V+ and V- (the right hand diagram), the current through R' and R would be equal once the capacitors charge.  Then the current through R' and R would be (V+-V-)/(R'+R).  If you happen to know R' then the voltage across R', V(R')=R'V/(R'+R).  The drawback is that if the capacitors are not charged, you need more information.  This is the kind of circuit model you would use if you are sure that there is a change in V+ that is causing most everything.

If you assume that 3.7Wm-2 of "current" will produce 1.5C additional "Voltage" the 1.5/3.7 = 0.40 which is approximately (R'+R).  If you know or think you know the values of the capacitance, then you can work out some time constant for the entire circuit to settle out or reach steady state.

The voltage divider circuit on the left is more my idea of what should be used.  Vref is the temperature at some surface with R'C' a portion of the atmosphere and RC a portion of the ocean.  You don't really know anything other than if the system is in a steady state the V- upper will equal V- lower.  Then you can assume a value for V-, like say 4C degrees which I use in my static models.  Same principle just there are resistors and capacitors.  The with 4C which is 277.15K degrees with an effective S-B energy of 334.5 Wm-2, I can vary Vref and find a range of values.  For example if the "average" ocean surface temperature is 18.5C (291.65K @ 410.2Wm-2) I would have R'=R=~0.19 K/Wm-2, which if the average surface temperature of the ocean was actually 18.5 C and I can neglect that pesky ~0.926 factor in the S-B equation, would mean that 3.7Wm-2 times 0.19K/Wm-2 equals 0.70 C would be the "transient" sensitivity.  If nothing else changes then the "forced" surface at 4C, 334.5Wm-2 would increase to 334.5+3.7=338.2Wm-2 which would be 4.75C requiring the lower surface to respond over however long it takes to charge the lower capacitance to 4.75C degrees with the same caveat about the pesky ~0.926 in the S-B equation.

18.5C is a fair estimate for the "average" SST unless you try to include Sea Ice Surface (SIS) in you estimate.  Also I noted in the static model that the SH ice free SST is closer to 17C and the NH SST is closer to 20C meaning there are two lower capacitors to consider and the largest of the two will win.  There is a small difference in the R-value using the 17C temperature as Vref, but the big difference is time.

 My thoughts on this stays with the ice free oceans which have a moist atmosphere with an average dew point temperature of approximately 4C degree meaning that the marine atmosphere would be likely to have clouds, saturated to super-saturated water vapor and a higher moisture content in general.  With the 18.5 C temperature (~410Wm-2) and the "sink" temperatures at ~4C (334.5Wm-2) that difference is 75.5 Wm-2.  The oceans obviously are not absorbing 75.5Wm-2 to any depth, but the latent energy released is in that ballpark.  This to me implies that simple models are great for ballparks, but with R-Values so low and time constants likely extremely long, that they are not going to prove much to anyone without some time to verify which approach is the better approach.  Still, the inverse of 0.19K/Wm-2 is 5.26 which is close to the saturated moist adiabatic lapse rate and that magic 5.4 multiplier for the Arrhenius CO2 forcing equation.  The 0.4 is more than twice as much, pretty much like the climate models which seem to be over-estimating "sensitivity". 

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