Updated Spencer Ocean Model
Posted by Jeff Id on July 2, 2011
Roy Spencer pointed out that his simple ocean model had an error of a factor of ten in the heat capacity of water. It is a simple spreadsheet linked below that anyone can work with.
FOLLOWUP NOTE: The above spreadsheet has an error in the equations, which does not change the conclusions, but affects the physical consistency of the calculations. The heat capacity used for water is 10 times too low, and the diffusion coefficients are also 10x too low. Those errors cancel out. I will post a new spreadsheet when I get back to the office, as I am on travel now.
Paul K also noted the error claiming to have found an energy conservation error in the missing heat thread.
Jeff et al,
I pulled down Dr Spencer’s spreadsheet with a view to testing higher order integration, and discovered that there are two major errors in the spreadsheet, which probably make further conversation on his findings a bit useless, at least until he has had a chance to review and correct the errors, and modify his conclusions accordingly.
1) Dr Spencer noted (in an update) an error of a factor of about 10 in the heat capacity term, but argued that this was compensated for by a change in the heat diffusion term of the same order. In fact, the argument for compensatory errors is only valid for the calculations below the first layer, where the calculation involves only terms from interlayer heat-flow. The argument is not valid for the calculation of the temperature of the first layer. This critically includes the integration (over time) of the total heat flux due to radiative imbalance – expressed in the model in the form of : F(t) – lamda * DeltaT [F is the forcing, lamda is the feedback parameter and T is temperature] There is no compensatory mechanism for the error in heat capacity, and this introduces a substantial error in the first-layer temperature calculation.
2) The heat capacity term in the model for each layer is given by 50*418000/86400. It is not clear where these values come from, but it is easily confirmed that the final value from this expression is too small by a factor of about 10. I calculated it should be 2555 on the back of an envelope. However, the heat flow term out of layer 1 into layer 2 includes a factor of 41,800 for the layer 1 calculation and a factor of 418,000 for the layer 2 calculation of heat flow from layer 1 into layer 2, which causes the model to bust conservation of energy.
These two errors are sufficient for me to throw the towel in. Pity. I’m going to bed.
I fell asleep early and woke up at 2am so the house is quiet and I began reading the equations carefully. It turns out that they are both right. The spreadsheet cells from the air/water boundary contained the following equation:
I couldn’t figure out PaulK’s envelope calculation of 2555 but I did figure out that Roy’s number 86400 is the number of seconds in a day and $AJ$10 is the number of days in a timestep - 30 in this case. The other factor 418000 is supposed to be the number of joules to change the temperature of 1 cubic meter of sea water by 1 degree, in fact it should be about 4,180,000 although I found slightly lower references on the internet it is close enough. Note the middle term though has 41,800 which is actually a factor of 100 off and definitely violates conservation of energy as Paul K states.
The diffusion factor does dominate this model though and that is what leaves me wondering how realistic any of this is in comparison to measurements of ocean energy exchange. I’m an engineer not a climatologist after all. I updated the above numbers, and modified the diffusion by a factor of 10 as Doc. Spencer suggested and calculated the following answer. simple-forcing-feedback-ocean-heat-diffusion-model-v1.0-1 revised Jeff
You can see that the red line ends up matching the blue line very closely. It would only take a slight tweak of the diffusion coefficients to place them directly on top of each other. In other words, correcting the problems results in not much change to the result. What is interesting is that there is meaning in the diffusion coefficients and my diffusion coefficients are 10X higher than the original. It seems to me that these energy diffusion numbers would be nailed down a little better than that, but I don’t really know. That is the point of Roy’s demonstration though, nobody really knows and a good match to PCM1 only requires tweaks to the top diffusion layers above the thermocline to get a good match to observation and a completely different sensitivity.
Anyway, the diffusion per layer is shown in the following table:
|Meters Depth||Diffusion Coefficient Observations||Diffusion Coeficient PCM1 Model|
The diffusion coefficient is the number of times that the full energy capacity of the 50Meter thick layer exchanges with the above and below layers each month. A value of 4 would mean that the exchange occurs once per week. A value of 30 is once per day – interesting no? I didn’t mess with the values at all, except to multiply by 10. For Dr. Spencer to complete his model now, he’s going to have to mess with the values a little more to get the quality of match he used to have, but these are not far off.