Dissipation of Fluid Motions into Thermal Energy
I have been looking for information on this subject without much success. I have contributed these notes to a thread at Climate Science (the site appears too be down at this time, I’ll add a link later) and at Climate Audit. Viscous dissipation has also been the subject of this post. To date I have gotten very little feedback of any kind. Maybe all this is incorrect, but no one has said that either. All pointers to literature relating to the following will be appreciated.
To be clear I am referring to the conversion of fluid motions into thermal energy via the viscous shear-stress terms in momentum balance equations. These are momentum diffusion contributions to the momentum balance equations. The thermal energy due to viscous dissipation appears as a positive-definite contribution to the various forms of the thermal energy conservation equation. Viscous dissipation always acts to increase the thermal energy of the fluid. If a temperature representation is used as the thermal energy equation, it always acts to increase the temperature.
Viscous dissipation is a volumetric process occurring at all times so long as fluid motions are present. The process is constantly acting to increases the thermal energy content of the fluid and thus increase its temperature.
In contrast I am not referring to the explicit and implicit viscous-like terms that arise from, and sometimes added to, the discrete approximations to the continuous form of the momentum equations. Somewhat ironically, these terms seem to be frequently labeled as ‘momentum dissipation’. The label momentum dissipation seems to be used in the GCM world more than in other computational fluid dynamics applications. I think it is a good assumption that the viscosity-like coefficients that are used for momentum dissipation are not used to calculated the viscous dissipation contributions to thermal energy equations.
It is of course true that these momentum dissipation additions to the momentum balance equations have an indirect effect of the viscous dissipation to the extent that they modify the velocity distributions and gradients in the flow. These latter are the correct terms for calculating the viscous dissipation.
Modeling and calculation of the viscous dissipation and consequent thermal energy addition in GCM models has a somewhat checkered history. This is due in part to the evolutionary nature of the models and changing application areas. More nearly complete and comprehensive accounting of the components of, and physical phenomena and processes occurring in, the climate system have generally developed over decades of time. Applications to calculations of the thermal history of the planet over hundreds of years has required that the energy-conservation aspects of the modeling be fundamentally sound and theoretically correct. However, a large contribution to addressing the fundamentally sound and theoretically correct model has been the approximations made at the continuous-equation level of the modeling. The momentum balance equations used in the models are simplified versions of the complete equations. More specifically, the thin-atmosphere approximation on a spherical surface, the representation of surface drag, the no-slip condition at land-atmosphere interfaces, the corresponding boundary condition at ocean-atmosphere interfaces, and the decomposition of the velocity into horizontal and vertical fields has also contributed to the problem. While calculations and analyses with the GCM models/codes have been carried out over four or five decades, it seems that only late in the 20th century and early in the 21st century have the problems with the modeling and calculations of the viscous dissipation been corrected in some of the models/codes. Two somewhat recent discussion have been given by, Boville and Brethron and Becker.
Again this situation is most likely a reflection on the interests in carrying out calculations for 100s of years of time.
It is my understanding that the global-average volumetric viscous dissipation in the atmosphere is calculated to be equivalent to about 2W/m^2 of energy; and I have seen much higher values. I do not know if there are estimates available from measured data in the atmosphere. A very wide spread has appeared in the literature over the years. This conversion of fluid motions into thermal energy has occurred for as long as the present composition and motions in the atmosphere have been roughly equivalent to the present-day conditions.
The standard argument is that this is a small number relative to the other energy-addition contributions to an energy balance for the planet. However, the radiative-equilibrium argument means to me that, as equilibrium is approached few energy additions can be consider to be a small number and neglected. Almost all finite numbers will not satisfy attempts to make 0 = 0. As I understand the situation, the effect of doubling of CO2 in the atmosphere is equivalent to about 1.6 W/m^2 and that the effects of the consequent changes in the thermal-energy state for the planet will be easily measured and observed in time spans of only 100s of years. How can an equilibrium-based approach to descriptions of the thermal-energy state of the planet neglect the viscous dissipation if it is of the same order as the assigned imbalance?
I think another important issue is related to the development of the continuous equations used in CGM models/codes. Taken as a whole these equations are known to be incomplete. The basic-equation models for the fluid motions and thermal state are not the complete equations that describe the motions and energy conservation. The all-encompassing parameterizations, many of which deal with mass and energy sources and sinks and interchanges across sub-system interfaces, are ad hoc/heuristic, best-expert-approximations (EWAGs) and thus cannot be assured of complete accounting of the mass and energy balances for the processes that are parameterized. In summary, the continuous equations very likely do not accurately account for the mass and energy conservation that actually occur in the physical system. I suspect it is easily possible for the lack of completeness and complete understanding to be responsible for several W/m^2 difference between the model equations and physical reality.
Is it not possible that the differences between the model equations and the actual physical phenomena and processes incur errors on the order of a few W/m^2. Again this might be a small number relative to the macroscopic energy balance for the planet, but as equilibrium is approached, and the imbalance is accumulated over 100s years of time in a calculation, significant differences are very likely possible. It is an important issue that the level of incompleteness and imbalances in the modeling at the continuous-equation level, relative to physical reality, must be significantly less than the physical imbalances that are driving the planet toward a new equilibrium state.
Finally we come to, as we always do, the fact that the numbers are the results of numerical solution methods. In order to ensure that strict accounting and conservation of the energy distributions within the system requires extremely close attention to how the numerical methods are developed and implemented. An example of how easily it is to overlook important details is given by the usual practice of numerically integrating different parts of a model system using different time steps. A related issue is calculations using parallel-computing capabilities by various approaches to domain de-compositions. Exchanges of mass and energy at interfaces between subsystems presents another opportunity to overlook mass and energy conservation requirements. Generally these must be evaluated at the same time-step level in order to ensure strict conservation.
It is important to note that while there are many ways to account for the mass and energy conservation of a given calculation, this process in no ways ensures that the calculations are in accord with and reflect the actual mass and energy balances and conservation in the physical phenomena and processes. However it is an important issue that the numerical imbalances must be significantly less than the physical imbalances that are driving the planet toward a new equilibrium state.
As a stable equilibrium state, the radiative equilibrium state for example, is approached, no imbalance can be counted as small and dismissed. The imbalance between physically reality and the continuous model equations is almost certain to be a genuine problem. The effects of the viscous dissipation, constantly acting to increase the thermal energy of the atmosphere is physical reality. I am uncertain of the actual physical value. The imbalances introduced by numerical solution methods is very likely a problem in some GCM models/codes. This problem has been discussed as recently as 2003. Small imbalances acting constantly over long periods of time cannot be ignored as equilibrium states are approached.
Here is a curious side issue that was discussed in this olde paper by H. A. Dwyer from 1973 located here. The results given in the paper indicate that the effects of his estimate of the power generation activities by humans can easily be seen in the calculations with his model. He used 15.0 x 10^18 BTU/yr (1.58 x 10^22 Joules/yr) as the ‘heat generation’ by mankind over a period of 100 years. Energy conversion activities by humans is another one of those processes that is constantly occurring and adding energy into the climate system
Worldwide energy consumption by the human race is over 446 Quadrillion BTUs at the present time. This is equivalent to 131,400 TWhr or 471,000 PJ (= 10^15 J) per year. If we take an average efficiency to be 33%, the total energy conversion is about 3 times the consumption, or 1,413,000 PJ per year = 1.413 x 10^21 J/year. This is within a factor of 10 of the value used by Dwyer. (While we consumed about one-third of the total conversion, all the energy converted will always reside in the climate system until it is lost to space.) Can this be another source of internal energy conversion that cannot be ignored over long time scales as an equilibrium state is approached.
The thermal state of the planet, as measured by the temperature, is a strong function of the thermodynamic processes occurring within the climate system. The temperature distribution near the surface is determined by the transport and storage of the energy additions to the system.
Chaos and Butterflies
Dissipative systems (physical or mathematical)will have several attractors (assuming they exist). The typical Lorenz-like ODE systems are dissipative, and conserve energy in the dissipationless limit. Thus these systems, the original 1963 and the later 1984/1990 systems are examples, have more than a single attractor. The dissipative and energy-conserving-in-the-dissipationless-limit properties are generally considered necessary in order for systems of ODEs to be Lorenz-like.
The oscillatory/periodic-like/aperiodic response seen in calculational results from these systems remains bounded due primarily to the linear terms on the right-hand sides of the equations. The plots of the dependent variables from Lorenz-like ODEs ‘look’ like bounded numerical instabilities. These are damping terms in the equations; the resistance offered to fluid motions, for example. If the coefficients on these terms are increased slightly from the usual default values of unity, the system can be shown to become a little under-damped to massive over-damping with the trajectories smoothly approaching equilibrium states. The effect is very dramatic on graphical plots.
It is possible to find values of the coefficients that produce periodic responses having almost no change in frequency or amplitude. And some values will bring the initial trajectory motion away from the initial point to a screeching halt at a new equilibrium point. In general, the chaotic response properties are lost and deterministic predictability returns.
I suspect, but haven’t yet done any calculations, that coefficient values less than unity might lead to unbounded responses. The non-linear terms in the equations might also provide contributions that assist in maintaining bounded-ness. It would also be of interest to investigate the effects of modeling the momentum-equation resistance as a non-linear function such as in turbulent flows.
The numerical solution methods used in NWP and AOLGCM models/codes have both implicit and explicit numerical damping in addition to the physical damping contained in the basic-equation models for mass, momentum, and energy for the fluid motions. These systems are dissipative and might be energy-conserving in the dissipationless limit. I do not know that if the ‘momentum dissipation’ terms are not present the codes can even maintain bounded responses. The effects of the numerical damping relative to the ‘chaotic response’ properties of the continuous equations are not known, of course.
When performing an initial-value sensitivity analysis, specified initial conditions are varied and the response of the calculations are observed. It is not possible a priori to know to which attractor for a dissipative system a given trajectory will approach. It seems that this property means that even long-range calculations of responses cannot be assumed/hypothesized to be reliable. Again, all this assumes that attractor(s) exist.