Models Methods Software

Dan Hughes

More on Mass and Energy Conservation

The focus on this previous post was the fact that approximations made at the continuous-equation level mean that the model’s mass and energy budgets are different from the mass and energy budgets of the physical system. Note that there are very significant additional issues associated with the discrete approximations applied to the continuous equations and the numerical solutions of these. These issues in the discrete domain, in my opinion, have the potential to far outweigh issues in the continuous domain. Accurate integration of the discrete approximations over the enormous times scales of interest is a very tough problem.

The present post looks at some more issues associated with energy conservation in the discrete domain.

I find it very interesting, and more than a little disconcerting, that the issue of conservation of energy seems to have come to the fore-front only recently. The paper mentioned below was published in 2007, for example. Additionally, I have not yet found papers or reports in which the convergence of the solutions of the discrete equations has been investigated. If convergence is not tested and attained, the ‘Fixers’ used to force energy conservation are functions of the numerical solution methods and will change as the temporal and spatial discrete increments are changed. That is, for a given spatial discretization, the mass and energy inventory at the initial time will have values M01 and E01, say. If the spatial discretization is changed, refined say, the new values will be M02 and E02. The ‘Fixers’ applied in the two cases will not be the same. Energy-Conservation Fixers are Fixers for numerical artifacts.

An interesting Section from a review article about the evolution of of GCMs is given below. The Section that I have included here addresses energy conservation. This sentence from that Section is an interesting summary of the energy-conservation problem:

For coupled model climate change applications, the dynamical atmospheric component of climate system models must conserve energy to tenths of W m^-2 or better ( Boville 2000); a greater imbalance could produce drift of the deep ocean in a coupled system which is large enough to imply a non-equilibrium solution.

My bolding.

Also note that I have included farther below the abstract of one of the literature citations from this Section. It is also very interesting in that one of the subjects of the paper are Energy Fixers. We met those in this post. The paper addresses directly this problem noted in the extracted Section:

Care must be taken in defining such fixers to ensure they do not affect the flow in adverse ways. An example of where an energy fixer did just that is illustrated in Williamson et al. (2007).

That is, the Energy Fixer itself led to incorrect calculated results. This example is for a version of the NCAR Community Atmosphere Model (CAM3.1) code. While I have always advocated Verification of the code, numerical methods, and calculations, I find it very unsettling that applications of Verification procedures was apparently delayed until after several version of the code had been released and used in production calculations. This approach is just simply wrong.

I think the Energy Fixers discussed in this paper have been developed from the same concept as used in the NASA / GISS ModelE model and code.

Here is the citation for the first paper mentioned above and then the extracted Section.

David L. Williamson, “The Evolution of Dynamical Cores for Global Atmospheric Models”, Journal of the Meteorological Society of Japan, Vol. 85B, pp. 241– 269, 2007. Abstract here:

4.4 Energy conservation
An area that has not received enough attention in the past is conservation of total energy in baroclinic dynamical cores in practical applications such as climate change simulations. While of less importance for NWP, it is becoming critical for coupled climate system models. Thuburn (2006) discusses many issues associated with conservation in dynamical cores and trade-offs associated with different choices. Here we consider just one aspect which has become important in coupled climate system modeling. For coupled model climate change applications, the dynamical atmospheric component of climate system models must conserve energy to tenths of W m^-2 or better ( Boville 2000); a greater imbalance could produce drift of the deep ocean in a coupled system which is large enough to imply a non-equilibrium solution. Since the models have finite resolution, a means must be included to control the build up of kinetic energy at the smallest resolved scales in order to maintain a reasonable kinetic energy spectrum during long simulations. Many models including most based on spectral transform approximations add a horizontal diffusion term, often of [ del^4 form ], to control the energy at the smallest resolved scales. Other cores with shape preserving approximations may be able to control the smallest scales via monotonicity constraints in the numerical scheme and do not require an explicit diffusion term. In semi-Lagrangian schemes theinterpolants control the energy at the smallest resolved scales to some extent, but such models often include explicit diffusion as well. The energy loss associated with these damping mechanisms is around 2 W/m^2 at resolutions normally applied to climate simulation. This is clearly not negligible. As a result some models include a frictional heating term that corresponds to the momentum diffusion to make the momentum damping process conservative, e.g., the CAM (Collins et al. 2004). Dissipative heating associated with vertical momentum diffusion is also an issue for energy conservation, although this is usually considered as part of the parameterization suite rather than as part of the dynamical component. Boville and Bretherton (2003) present a consistent formulation for the heating due to kinetic energy dissipation associated with the vertical diffusion of momentum.

Although the frictional heating associated with an explicit horizontal momentum diffusion seems a reasonable approach, it is somewhat arbitrary and does not capture the true energetics of the system. Such heating provides for a greater degree of energy conservation than described above, but terms such as diffusion on temperature lead to lack of conservation in the tenths of W/m^2 range. In formulations in which monotonicity constraints might control the smallest scales the energy loss due to the inherent damping is not explicitly known and cannot be compensated by a frictional heating term such as that associated with explicit diffusion. Thus to compensate for these unknown sinks an a posterior energy fixer has been included in some models to ensure conservation. An example is the CAM in which one is applied every time step (Collins et al. 2004). Care must be taken in defining such fixers to ensure they do not affect the flow in adverse ways. An example of where an energy fixer did just that is illustrated in Williamson et al. (2007). Tomita and Satoh (2004) avoid this problem by formulating their approximations to guarantee conservation of mass and total energy. It remains to be seen if the kinetic energy and temperature variance spectra in very long runs are consistent with atmospheric estimates.

Thuburn, J., 2006: Some conservation issues for the dynamical cores of NWP and climate models. J. Comput. Phys., doi:10.1016/

Boville, B.A., 2000: Toward a complete model of the climate system. In P. Mote and A. O’Neill (eds.), Numerical Modeling of the Global Atmosphere in the Climate System, Kluwer Academic Publishers, 419–442.

Collins, W.D., P.J. Rasch, B.A. Boville, J.J. Hack, J.R. McCaa, D.L. Williamson, J.T. Kiehl, B. Briegleb, C. Bitz, S.-J. Lin, M. Zhang, and Y. Dai, 2004: Description of the NCAR Community Atmosphere Model (CAM3.0). NCAR Technical Note NCAR / TN-464+STR, xii+214 pp.

Boville, B.A. and C.S. Bretherton, 2003: Heating and Kinetic Energy Dissipation in the NCAR Community Atmosphere Model. J. Climate, 16, 3877–3887.

Williamson, D.L., J.G. Olson, and C. Jablonowski, 2007: Two Dynamical Core Formulation Flaws Exposed by a Baroclinic Instability Test Case. To be submitted to Mon. Wea. Rev.

Two flaws in the semi-Lagrangian algorithm originally implemented as an optional dynamical core in the NCAR Community Atmosphere Model (CAM3.1) are exposed by steady-state and baroclinic instability test cases. Remedies are demonstrated and have been incorporated in the dynamical core. One consequence of the first flaw is an erroneous damping of the speed of a zonally uniform zonal wind undergoing advection by a zonally uniform zonal flow field. It results from projecting the transported vector wind expressed in unit vectors at the arrival point to the surface of the sphere and is eliminated by rotating the vector to be parallel to the surface. The second flaw is the formulation of an a posteriori energy fixer which, although small, systematically affects the temperature field and leads to an incorrect evolution of the growing baroclinic wave. That fixer restores the total energy each time step by changing the provisional forecasted temperature proportionally to the magnitude of the temperature change that time step. Two other fixers are introduced that do not exhibit the flaw. One changes the provisional temperature everywhere by an additive constant, and the other changes it proportionally by a multiplicative constant.
Received: March 6, 2008. DOI: 10.1175/2008MWR2587.1

Tomita, H. and M. Satoh, 2004: A new dynamical framework of nonhydrostatic global model using the icosahedral grid. Fluid Dyn. Res., 34, 357–400.


January 14, 2009 - Posted by | Verification | , ,


  1. Hi Dan,

    This is very interesting stuff. My take on the “fixers” is that they are included as a way of dealing with the build up of numerical errors as the simulation proceeds. If your goal is to simulate a century’s worth of climate, controlling temporal discretization error is the name of the game!

    What I’ve always wondered is why the climate models appear to have little skill over the 1 – 5 year range. You would think this is a long enough period for “weather noise” to not be a factor but short enough for the temporal errors (requiring the “fixers”) to be relatively small.

    I guess we shall see soon, as the NASA GISS folks are saying either this year or next will be the warmest on record…


    PS it’s -3 F here in tropical NH – brrrrr

    Comment by Frank K. | January 15, 2009 | Reply

  2. A short summary about Energy Conservation.

    I have cited peer-reviewed papers in which it is stated that numerical solution methods must not have extraneous energy source greater than a few tenths of W/m^2. The numerical solution methods must maintain an energy balance within a few tenths of W/m^2.

    How does this relate to my comments relative to the mass and energy model equations at the continuous-equation level?

    (1) I maintain that the continuous equations must represent the physical system of interest with the same degree of accuracy in accounting; a few tenths of W/m^2.

    (2) Additionally, the discrete approximations must in turn represent the continuous equations with the same degree of accuracy. All discrete approximations contain implicitly errors in representations of the continuous equations; they are after all correctly characterized to be discrete ‘approximations’. Discrete approximations having conservative properties relative to continuous equations also in conservative form are still none-the-less truncations of the series said to represent completely the continuous equations.

    (3) And the solution method applied to the discrete approximations must also conserve the energy balance. The truncation errors can be reduced only by reducing the size of the temporal and spatial discrete increments used in the application of the numerical solution methods. If convergence is not tested, it is not known if an application calculation is in fact operating at the theoretical truncation limit. Process models such as GCMs that include huge numbers of algebraic parameterizations are notorious for operating well above theoretical truncation estimates. Well above meaning the truncation errors are nowhere close to the theoretical limit. Truncation errors for systems of ‘pure’ PDEs do decrease as the discrete increments are refined and the theoretical limit can be attained. In contrast, systems of PDEs plus algebraic parameterizations and switches, even when these latter are continuous in their first derivatives, can easily unknowingly introduce properties that make the truncation errors much larger that theoretical values. This is not very clear and based primarily on personal experiences. I’ll see if I can find some examples. Consider the case for which the timing of an event changes as the spatial resolution is increased. Or even the case for which the location of a physically important aspect of the problem changes as the spatial resolution changes.

    The rather longish notes for (3) correspond to the importance that I place on the numerical solution methods as sources of real-world problems in codes and Verification of methods and calculations.

    I’ll state that given the complexity of the physical phenomena and processes of interest plus the spatial and temporal extent of the application of interest, it is highly unlikely that GCM applications and calculations have yet attained correspondence to the physical world to within tenths of W/m^2.

    Comment by Dan Hughes | January 15, 2009 | Reply

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