Viscous Dissipation in the NASA/GISS ModelE Code is Wrong …
it is not even viscous dissipation. There is also an energy imbalance in the NASA/GISS ModelE code due to the error in the ‘viscous dissipation’. The energy imbalance is about the same magnitude as the imbalance associated with increased concentrations of CO2 in the atmosphere.
Updated January 14, 2009, down near the end.
This is another continuation post about viscous dissipation. A followup on the discussions over at Luica’s and additional information related to both the general issue and some specific aspects of the NASA / GISS ModelE model / code.
I originally ran across a ModelE code segment that purported to be the viscous dissipation as mentioned in this post. There is a some discussion with NASA / GISS employee Gavin Schmidt there in the comments. That led to a more general discussion in this post, and given the estimated magnitude of the viscous dissipation, I posted this thread about the general issues of conservation of mass and energy in model equations.
At the time that this was ongoing, my preliminary conclusions were that the quantity purported to be viscous dissipation in the ModelE code was in fact not viscous dissipation. I also decided that the ModelE model / code had the potential to be missing a not-insignificant contribution to the thermal-energy form of the energy balance equation.
Based on additional reading, I will now say that both of my preliminary conclusions are correct. The ModelE model / code does not include accounting of the viscous dissipation, and the magnitude of the missing contribution is about the same value as the energy imbalance introduced by increased concentration of CO2 in the atmosphere. The papers by Becker and colleagues that I cited in this post, plus some others cited in these, provide important information. The papers that I find useful are Becker 2001, Becker 2003, and Burkhardt and Becker 2006. There is an earlier article by Fielder; Fiedler B. H., 2000: Dissipative Heating in Climate Models. Quart. J. Roy. Meteor. Soc., 126, 925–939, but I haven’t paid the bucks to get it.
Becker cites the following to conclude that the physical processes in the atmosphere correspond to viscous dissipation of about 2 W/m^2:
It is noteworthy that a simulated mean dissipation rate of about 1.9 W m–2 fits well within the range of existing estimates based on observational analyses of the general circulation. Oort (1964) finds a value of 2.3 W m–2 (see also Lorenz 1967, chapter V). Some modern textbooks suggest somewhat lower dissipation rates between 1.8 and 1.9 W m–2 (e.g., James 1994, chapter 5.3; Prandtl et al. 1989, chapter 8). Thus, with regard to the total kinetic energy typically contained in the atmosphere (~7 × 1020 J), neglecting dissipation in conventional GCMs means that within about 8 days an equivalent amount of kinetic energy is removed from the flow without being dissipated into heat (see also Pichler 1986, chapter 9).
Becker and colleagues base their developments on two fundamental aspects of fluid motions as follows; (1) conservation of angular momentum requires the stress tensor to be symmetrical, and (2) the viscous dissipation must always be a non-negative positive-definite contribution to the thermal-energy forms of the energy conservation equation.
I think Becker 2003 discusses explicitly the Total-Potential-Energy Change-Kinetic-Energy approach mentioned in passing by Gavin Schmidt:
It is worth mentioning that an attempt to include frictional heating in GCMs can also be found in Hamilton (1996) or Kiehl et al. (1996, sections 3b and 4d). With this alternative method, no knowledge of the stress tensor is required. One rather assumes that, locally, the rate of change of kinetic energy due to friction enters the thermodynamic equation of motion with negative sign. This assumption may be interpreted as the local counterpart of the fact that, in the global mean, the frictional loss of kinetic energy balances the frictional heating (e.g., Pichler 1986, chapter 9; Smagorinsky 1993; see also section 2b). However, assuming local equivalence implies that there is no turbulent stress at all acting at the resolved scales. Also, the local frictional heating rate has arbitrary signs.
I’ll have to track down the references he cites to see if this is what Schmidt is saying.
Based on the information in these papers and the coding in NASA / GISS ModelE code I now conclude
1. The quantity called ‘dissipation’ in the ModelE model / code is in fact not viscous dissipation. As I noted in the original post, I suspect the approach in the code implicitly contains numerical artifacts in addition to any ‘physical’ accounting.
2. The above error in the ‘dissipation’ leads to an energy imbalance of about 2 W/m^2 in the NASA / GISS ModelE calculations. This imbalance is about the same as that associated with increased concentrations of CO2 in the atmosphere.
3. The potential errors in the calculation of viscous dissipation seem to have been recognized only recently now early in the 21st Century. This situation leads me to conclude that the results of many GCM calculations used by the IPCC, especially in the first ‘Annual Reports’, very likely contain the same error as ModelE ( speculation on my part ).
4. The Total-Potential-Energy Change-in-Kinetic-Energy approach can indeed produce negative numbers for the ‘viscous dissipation’; a most unusual outcome, in my opinion.
In regard to 3. above, Boville and Bretherton 2003 state:
The spectral Eulerian core in CAM2 includes a biharmonic horizontal diffusion operator that cannot be represented by a symmetric stress tensor and therefore the kinetic energy dissipation cannot be correctly defined, as noted by Becker (2001).
Where Becker (2001) is the paper cited above. The authors note that the Change-in-Kinetic-Energy approach has the potential to give cooling of the fluid. These authors also give the viscous dissipation to be about 2 W/m^2.
Reverse engineering model equations from coding is a process that has very high probabilities for errors; especially when even the most rudimentary documentation is not available. I think there is more light than heat here, but under the present circumstances I can not be certain. Additionally, maybe I’m biased because these papers are in agreement with preliminary conclusion that I obtained 🙂
Here is the citation and abstract of the Fiedler (2000) paper:
B. H. Fiedler, “Dissipative Heating in Climate Models”, The Quarterly Journal of the Royal Meteorological Society, Volume 126 Issue 564, Pages 925 – 939, 2000.
The various strategies towards dissipative heating in meteorological models are reviewed. The strategies are implicit formulations, explicit formulations, and exclusion. A thermodynamic formulation based on dry static energy, which has long been used in hydrostatic models, implicitly includes dissipative heating. Many modem non-hydrostatic mesoscale models, being based on potential temperature, do not include any dissipative heating. The National Center for Atmospheric Research community climate model (CCM3) explicitly formulates the dissipative heating for the resolved motion, which is based upon potential temperature, but, uses the implicit formulation for the sub-grid motion.
Those who want to use the mesoscale-model formulation as a cloud-resolving climate model are advised to consider the effect of dissipative heating. We show a dry radiative-convective model, formulated without dissipative heating, in which climate equilibrium occurs with a net inward radiative flux of about 18 W m-2 at the top of the atmosphere, and a surface temperature that is 8 K too low.