Modeling the Basics, A mathematical summary of condensation in climate models.
Posted by Jeff Id on October 21, 2010
An important point often made by critics of climate models is that they often represent our best guess at specific phenomena. One of the biggest uncertainties in climate models is in proper modeling of atmospheric moisture. Considering that H2O is widely accepted to be the strongest of all greenhouse gasses, water is fairly important component of climate models. As is often the case at tAV, I’m not the guy who figured this out but am the one who will attempt to translate the deficiency in models as I currently understand it.
Based on Makarieva et al. recent multi-author paper (M10) on the driving force behind winds several here at the Air Vent have discovered that the climate model CAM3.0 linked here doesn’t include precipitation condensation based pressure loss in its cloud parametrization. The model doesn’t even attempt a simulation for what I believe will soon be accepted as the primary driver of most winds including tornadoes and hurricanes, not just globally on earth but solar system wide. Jupiter’s red spot, bands in Saturn, all of it powered by condensation based pressure changes. Why is that so important? Because the additional energy stored in water vapor which translates into lower than modeled pressure and higher than modeled windspeeds in hadley cell updrafts.
Ask yourself to explain what powers a tornado, and soon you’ll find yourself describing strong temperature inversions where the hot surface air breaks through the cold upper air or something of that sort. In fact, that is what we’ve all been taught since junior high. M10 teaches that there is a component of basic gas physics missing from this explanatoin — condensation.
The water vapor component of saturated air at 30C has a Water vapor is about 0.6PSI where as standard air pressure at sea level is 14.7 PSI. When the vapor condenses, it no longer contributes to the gas pressure in the region of condensation much as it takes up nearly zero volume at that point. The air pressure in the saturated volume would drop by the vapor pressure amount when condensed. The pressure doesn’t sound like much but remember, we already have a model which creates some convection without it and 0.6PSI over a square yard is 780 lbs of additional upward force. That’s not all though, the heat release during condensation creates an additional pressure loss warming the surrounding air reducing the air pressure even further. The net effect is all toward powerful amplification of updrafts in a condensation environment.
I’m not intending to do the calculations any further here, because it is basic knowledge. None of what I’ve written above is in any way new and all of it has been known longer and more completely than we have known about CO2 absorbing infrared radiation. In fact, 20 years ago in my undergraduate thermodynamics class we were forced to calculate all of these factors for a variety of mixed gasses. So when I first read Anastassia’s paper on what powers hurricanes, it made perfect sense to me. Except for the part where she claimed it wasn’t part of mainstream literature.
Nick Stokes, who takes too much criticism some times, summed it up best, brackets are mine.
I have to say that it’s still not clear to me where condensation comes in in 3.3.6. However, I remain sure that they haven’t just forgotten about it. This stuff [models] has been around for thirty years, reviewed by thousands.
For some background reading Nick linked to the Zhang paper from which the cloud parametrization for CAM3.0 was adopted. Paper here and he also pointed out some very similar equations to M10 right from the CAM model there is a difference though.
From the CAM3.0 global climate model, chapter 3.3.2 paraphrased style, it’s not a direct quote and I’m not the original author 😉
The conservation of total air mass using as the prognostic variable can be written as
Similarly, the mass conservation law for tracer species (or water vapor) can be written as
There is no “law” of conservation of water vapor to my knowledge. Water vapor can condense, especially when V can be in the vertical direction toward lower pressure. This is clearly a simplification of the situation as mass of water vapor is obviously not conserved during condensation in an air volume.
q is the specific humidity of water vapor in this case. Pi is the pseudo density of the air. These conventions are different from molar density that M10 uses but they are interchangeable for the following points. M10 equation 34 is actually nearly identical but have an additional tweak to insure that water vapor is not conserved.
Calling your attention just to the right side of the equation for S ≡, w is velocity in the vertical z direction, Nv is the molar density of water vapor. So the first term w · dNv/dz is the change in water vapor density that is multiplied times vertical velocity resulting in a term for the change in water vapor density with respect to time, functionally identical to the first term in eq 3.366 which says change in total vapor mass in the volume vs time. The second term in eq34 w · Nv/N*dN/dz is the change in total mass with respect to z distance which when multiplied by w velocity (distance/time), it is the change in total mass vs time created by air flow. Again functionally nearly identical to the second term of eq 3.366 on which the model is based except that term 2 of 3.366 only addresses the water vapor while eq34 addresses total air mass. They are the same equation– almost.
The difference is that the model eq 2 above is set equal to zero representing conservaton of water vapor mass, whereas M10 has the equation set equal to S. Were it set to zero, the volume of vapor in term 1 would always equal term 2 and there would be no change in vapor density ratio – no condensation. IN eq 34 ‘S’ represents the condensation mass.
S (Eq. 34) is the sink term describing the non-conservation of the condensable component (water vapor).
See, Nick and I aren’t alone, the authors of M10 have never heard of conservation of water vapor either. Several of us scoured section after section of the model looking for where this effect was included. In the section on moisture, I found that while they used partial pressures to determine q, they didn’t recalculate pressure based on the new value maintaining conservation of water vapor mass. Surface pressure is held fixed as is the column’s molar gas mass even though they are calculating a condensation/evaporation process. The expansion and contraction processes are simply not calculated.
I commented to Nick, on the CAM model:
All they update is the q in 3.417 with no adjustment for the pressure created from the condensation. In fact they hold it to zero change during the update to maintain conservation of air mass (total gas mass and pressure) while changing q. By doing that in a condensation region, they’ve lost their conservation of mass in the form of condensate and increased the total energy per volume.
As further confirmation, Reader RuhRoh, discovered this little gem in a different model ‘CCM’ confirming quite clearly what we are discussing:
These conservation errors result in small imbalances ( << 1 W/m2) in the CCM. We note that there are also small inconsistencies present in conservation that are associated with the use of a moist mixing ratio, and moist surface pressure in the model. In principle, as any process removes water vapor from a cell, the surface pressure (PS), and the mass of air (dp) should change in a grid volume. This ought to also imply a change to any mass specific quantity affected by the parameterization. These changes are ignored in CCM parameterizations from one process to the next. We typically insist that processes conserve assuming a fixed mass of air (and hence a fixed surface pressure) within a parameterization.
Holding pressure fixed to conserve a mass of air ignoring the water vapor removal. Now that we’ve established that at least two climate models ignore this part, we caught some attention from modeler Gavin Schmidt of Real Climate who wrote.
Most GCMs use the hydrostatic approximation in which the pressure at any point is exactly equal to the weight of the air column above it. Very high resolution weather models sometimes use the proper non-hydrostatic equations, but this isn’t very important at coarse scale (I’m sure there is a paper that has demonstrated this somewhere).
While it sounds like a perfectly valid way to calculate pressure, the fact that delta P from condensation is not accounted for means that models will underrate the estimated flow to some degree in moist hadley cell regions. You can understand the thought process of the scientists when the models were developed. The mass of the vapor is small in comparison to the surrounding air, so as water condenses they work to capture only the energy of the condensation holding mass constant. This creates an imbalance in condensation where the mass in a volume after condensation is greater than the mass before. This imbalance shows up later in a minor energy imbalance corrected elsewhere in the model. Since the imbalance is small the effect is assumed to be small and everyone is happy.
None of this is unreasonable in my view, except that total energy in the volume is not the same as total gas pressure in the volume and you’ve now lost 0.6 psi from the flow in a hurricane model. But there is another detail we should talk about. The moisture is a continuing massive source of stored potential energy in the lower atmosphere. As global warming occurs from CO2 buildup, the lower atmosphere is expected to contain more vapor per unit volume. Basic physics right? The increased vapor would result in increased condensation and increased regions of updraft pumping more heat into the upper atmosphere where it can be radiated to space. Due to the warmer temperature profile in the tropics, heat from the surface could be carried quite a bit higher in the atmosphere before freezing occurs perhaps making the tropics a very effective.
So then the whole question comes down to, does this pressure change make much difference or is it just a fudge factor?
Well Makarieva et al. also proved the effect can drive hurricanes and tornadoes.
It is not small, and as my understanding grows seems correct that it isn’t standard in climate or weather science. It’s shocking, but this powerful and well known condensation effect appears to be missing from all of the climate models.