Models Methods Software

Dan Hughes

Radiative-Equilibrium Models and the Weather Temperature

I started working on my own toy 0-D and 1-D models of the combined radiation-convection-conduction heat transfer problem aspect of energy balance approaches. I got to the point of assigning symbols to physical quantities and ran into a problem with some basic concepts. Details are discussed in the following paragraphs.

There are many examples of 0-D and 1-D radiative-equilibrium modeling approach both in the peer-reviewed literature, all in respectable journals of course, and in Web space. Professor Pielke Sr. has posted on some aspects of the problem and lucia has carried out some parameter-estimation exercises based on such a model. Raypierrie has an article published by Woods Hole, and Schwartz and Tung have also presented radiative-equilibrium models. You’ll find many such models in the archives of the journals of the AMS with the articles available for download. I’m not going to list these, but let me know if you would like to have additional information regarding them.

The following discussion has focused my thoughts and I’m now inclined to think that Paltridge, and Bejan and colleagues have presented the better way to address simple models. The key to success is to focus on the hydrodynamic and thermodynamic processes occurring on the surface of the planet. Radiative energy transport, of course will contribute to the model equations.

The following two guidelines are part of what got me thinking about the 0-D radiative-equilibrium model energy equation approach.

1. The quantities used in model equations must conform to the basic fundamentals of thermodynamics. I mention thermodynamics here because we’re talking about temperature and energy.

2. The symbols used in model equations must correspond exactly to specific quantities in the physical system of interest. You don’t call A the exit temperature from a turbine, say, and then measure something at the inlet as data for A.

For the 0-D radiative-equilibrium approach these mean that (a) whatever the quantity called temperature, T, is, it must have a thermodynamic connection with the quantity called energy, E, in the usual nomenclature. For a simple thermodynamic system comprised of a single pure substance, for example, temperature and energy are related through basic and fundamental thermodynamic relationships that cannot be violated. And that (b) measured data for the temperature must correspond to the radiative-equilibrium temperature. It is not clear to me that the radiative-equilibrium model energy balance equation approach satisfies either of these requirements. Consider the following arguments.

I think that the fundamentals of thermodynamics require that whenever energy is added into a material the temperature increases, and when energy is removed the temperature decreases. The subsystems present in the climate system, and the complete climate system itself are not closed systems comprised of a single simple material, of course.

The Measured Data
My interpretation of the situation is as follows. The temperature reported in some of the plots of the Surface temperature Record is some kind of (very) rough approximation of the temperature within the atmosphere near the surface of Earth. I’ll try to get back to those plots that have some kind of average for the Land + Ice + Ocean. But the temperature being measured is not an accurate reflection of the radiative-equilibrium balance approach. The 0-D radiative-equilibrium model energy balance equation cannot capture the physical phenomena and processes that dominate and control the quantity measured as T. The question of equilibrium is a whole nother open issue imo.

The plots of the quantity T do not show a monotonic dependence of the temperature with energy; taking the energy to be monotonically increasing as time increases. The smoothing and/or neglecting of the oscillatory nature of the temperature does give a more or less monotonic increasing of temperature. But maybe the time periods for which the temperature is decreasing are trying to tell us something. Plus something deep in the recesses keep saying to me that the specific heats must be positive numbers. The variability in the plotted temperature, when decreasing, means that energy has been removed from the system, if the temperature is the physical quantity associated with the energy. My understanding is that there is seldom an actual net reduction of energy in the Earth system. More is incoming than outgoing; let me know if that is not correct. So when a decrease in the temperature is measured, that actually means that the phenomena and processes down here have taken control of the temperature. The energy already added into the system has caused/been-a-part-of some transport/storage processes that result in changes in the temperature. Neglecting for a moment the cases for which energy does in fact get blocked/reflected back. By the same token, when the temperature is increasing that very likely is not an indication that an excess energy addition has occurred in contrast to that processes here that control the temperature have simply changed to other processes.

Whatever the case, the radiative transport problem/model in no ways reflects the actual physical system. The media through which the radiation, in both directions, passes is an interacting media, as you well know. Plus after the energy gets to the surface the surface is not a purely radiative body. All the radiative transport properties of the interacting surface vary all over the map (you might say). Covering the full ranges for about 0.0 to about 1.0. The energy is stored and transported in all the stuff here on the surface in addition to a part acting in a radiative-energy-transport way.

Thermo-and Hydrodynamical Physical Phenomena and Processes Determine the Surface Temperature
The Thermodynamic phenomena and processes present/undergoing down here are a heat engine in which the energy additions to the atmosphere provide the driving potentials required to move fluids from regions of higher temperature (energy) to regions of lower temperature (energy). Typically from the tropics toward the poles. The poles, being at lower temperature level, cannot reject all the energy transported to those areas. It would seem that as the driving potential for thermally induced motions decreases, the motions themselves will decrease and thus change the energy-transport mechanisms.

And here I’ll guess that the greater temperature increases at the North compared to the South is a reflection of the larger amounts of liquid and solid phases of water in the South. The liquid form, of course, has a high specific heat (and there’s tons of it around) and the solid form can absorbed energy at constant temperature after it reaches the melting temperature. Getting it up to the melting temperature might also require significant amounts of energy; I haven’t made an approximation.

I think the temperature measured in the atmosphere is more likely a function of the thermodynamic states of this heat engine at the locations where the measurements are made than a function of the energy additions to the system. There are of course simple systems for which energy additions act solely to increase the temperature. A closed system comprised of a pure homogeneous material initially at an equilibrium state internally is an example; the Earth is not such a system. Phase-change and energy storage and transport processes within the Earth system dominate the temperature here, I think.

The new operating states of the topics-to-polar heat engine will be a function of the relative temperature changes at the source and sink ends as excess energy accumulates. If both these increase the engine will operate at a higher temperature level, although I think the tropics end of the engine is controlled more by latent and sensible heat transfer thermodynamic processes rather than by excess energy additions. I don’t know if the power and dissipation will increase or decrease?

Plus, the various motions, large scale bulk motions in the atmosphere and oceans can, and do, affect the numbers reported to be the temperature of the day at all locations. All those Pacific Ocean hot and cold things, and all those shifts and oscillatory things. But, again, it is the motions and not the energy additions that have caused the variations in the temperature.

Equilibrium and Not Equilibrium
The lack of equilibrium both within and between the climate subsystems, causing interactions that effect the measured temperature are especially not functions of energy additions to the total system. The subsystems within the climate system have never been, and will never be in equilibrium either within a given subsystem and most certainly between subsystems. A thermally and mechanically static Earth is not going to happen. The daily and seasonal and yearly variations in the measured temperature are not stationary cyclical variations. The number of potential thermodynamic states internal to the total system seems to me to be quite large.

Some of the motions are of course a result of the energy additions. Many large-scale motions in both the atmosphere and oceans, however, are inherent in the basic properties of fluid motions on a rotating sphere. Additionally the exchanges of momentum at the interfaces between the subsystems can also induce motions in the respective subsystems. And finally, none of the subsystems is ever in equilibrium internally with respect to any of the driving potentials for either motion or energy transport/storage. Upwelling of deep liquid from deep within the oceans subsystems generally brings cooler material nearer the surface and thus affects the temperature on a large scale. Cool (sometimes really cold) air sometimes moves from the polar regions and causes significant changes in the temperature.

All the physical phenomena and processes active here at the surface are indications that attempts to tie the excess energy to the surface temperature record does not work in a straight-forward manner. The introduction of positive and negative feedbacks and forcings (my nomenclature here is not necessarily correct) is necessary in order to accommodate basing a technical analysis with an incomplete (or not the best approach, incorrect) basis.

Additionally, as Professor Pielke Sr. constantly reminds us, the effects of human and ‘natural’ processes are constantly making changes that significantly impact both the measured temperature and physical quantities that impact the radiative energy balance approach.

Isn’t it even possible that the lack of equilibrium and the available phenomena and processes internal to the system might even allow states of lower temperature level to be attained even as excess energy additions are occurring? Shouldn’t this possibility be investigated and eliminated before we assume that the Weather temperature will respond in only an increasing way to the excess energy additions?

In summary, the temperature here on the surface is a function of which way the wind is blowing, jet stream, macro (meso?)-scale motions in the oceans, as mentioned above, etc. And the macro-scale conditions near the measuring stations will significantly affect the reported values of the temperature (ocean-side vs. desert, for example); not to even begin consideration of micro-scale level issues. Again, all of these are functions of things other than energy additions.

The Weather Temperature
The quantity being measured near the surface is the Weather temperature. It has always been the Weather temperature. It will always be the Weather temperature. I think to take it to be the radiative-equilibrium temperature is not the right thing to do. I guess the assumption is that long-term averages of the Weather temperature are in fact the radiative-equilibrium temperature. Is that assumption sound? If the Weather temperature continues to decrease as well as increase while all the time the energy content is increasing, I think the assumption needs to be examined in a little more depth.

Short Partial Summary
I suspect that since part of the excess energy addition goes into changing the properties and characteristics of the Earth heat engine, the Weather temperature will not ever reflect the complete extent of the increase in the energy content of the system.

The typical 0-D model energy balance equations do not account for the heat engine processes, constant-temperature phase-change processes, dissipation, work. I certainly understand the concepts on which such an approach is based. And maybe everyone is happy even when they see a temperature decreasing, for whatever the time period, while the energy content of a system is increasing.

I suggest improved 0-D modeling based on

1. A radiative-equilibrium balance written for its more appropriate physical system in the overall scheme of things. I don’t know what this equation might include nor where in physical space it should be applied.

2. 0-D model energy balance equations that account for the heat-engine processes occurring down here on the surface; storage, phase change, transport, work, dissipation, and radiative energy additions.

I’m certain that this is in fact the approach taken in the early modeling days. However I think a good argument can be made that deeper understanding of the system and its responses might in fact be more readily available through study of these more simple approaches. Additionally there is software that will fit parameter values appearing in ODEs to data. So as the number of ODEs increases to be beyond hand/analytical work, the software can save the day.

That’s as far as I’ve gone on this, so it’s a very rough draft.

Raymond T. Pierrehumbert, “Lecture 6: energy Balance Models”, Woods Hole Oceanographic Institution, 2001.

G. W. Paltridge, “Global dynamics and Climate – A System of Minimum Entropy Exchange”, Quart. J. R. Met. Soc., Vol. 101, pp. 475-484, 1975.

G. W. Paltridge, “The Steady-State Format of Global Climate”, Quart. J. R. Met. Soc., Vol. 104, pp. 927-945, 1978.

Stephen E. Schwartz, “Heat Capacity, Time Constant, And Sensitivity Of Earth’s
Climate System”, Accepted for publication in Journal of Geophysical Research, Brookhaven National laboratory, June 2007.

Ka Kit Tung, “Simple Climate Modeling“, Discrete And Continuous Dynamical Systems–Series B, Vol. 7, No. 3, pp. 651–660, 2007.

Adrian Bejan and A. Heitor Reis, “Thermodynamic Optimization of Global Circulation and Climate”, Int. J. energy Res., Vol. 29, pp. 303–316, 2005. DOI: 10.1002/er.1058.

A. Heitor Reis and Adrian Bejan, “Constructal Theory of Global Circulation and Climate”, Int. J. Heat Mass Trans., Vol. xxx, pp. yyy-www, 2006.


February 11, 2008 - Posted by | Uncategorized |


  1. It is not easy to comment because you don’t define what you understand under “Earth system” like in “My understanding is that there is seldom an actual net reduction of energy in the Earth system. ” .
    You have local considerations expressed by differential equations and then “global” considerations expressed by integrals of some functions over some integration space which has to be carefully defined and justified .
    Then you have “averages” which are just a variation of an integral of something over some domain too .

    Now as the only thing you have are the laws of nature that are local you can always write locally the relevant differential equations and try to solve them .
    In this process you will obviously always have functions of t (f.ex T(x,y,z,t)) and nothing will stay constant .
    Indeed there is never equilibrium , everything is transient .
    During the day the local incoming energy will gradually increase then decrease . The matter will react on that by gradually warming then cooling with a certain lag . During the night the flow of incoming energy is brutally cut off and the matter goes in a cooling phase .
    Then the energy transport is added .
    You may very well have a piece of surface heating up during the night because the sky is covered and a hot wind is blowing . It may be rare but there is nothing preventing that . The energy given up by the hot wind has its source somewhere else somewhere in the past .
    T(x,t) depends in a complex higly non linear way on some f(y,t-t0) where f is not the same function all the time and neither y nor t0 stays constant .

    You will notice that every cartoon and presentation of the models always says somewhere : “The absorbed radiative energy must be IN AVERAGE radiated to the space” .
    That is the alpha and omega of everything .
    At first degree that statement is trivially wrong because it expresses a state of radiative equilibrium of the WHOLE system and the system as we have seen is never in radiative equilibrium .
    The key is in the word average .
    Common sense says us that with a time long enough (let’s take the most obvious period of the system , a year) ir should be true because otherwise the internal energy of the system (thus its temperatures – notice the plural because there is an infinity of temperatures) would vary .
    Well sure it would but why shouldn’t it ?
    There is no law saying that the system must have a constant internal energy for states that are separated by 1 year .
    OK then what about 10 years ?
    Again you see that there is nothing in the nature saying that 2 states separated by 10 years should have a constant internal energy .
    And that is true even if the energy input in the system stays RIGOROUSLY constant .
    By changing its internal states (f.ex producing more or less clouds and ice and dissipating more or less) the system would oscilate among different internal energies and infinities of temperature distributions without ever needing to match the incoming energy over any time interval .
    There would appear all the time random time intervals where the match would be true but it would never have the same length .

    Having said that , I don’t know if it helps but given the above I can’t even see what a 0D model (which would be necessarily some sort of averaging) could say .
    On the contrary a 0D model because it averages would necessarily postulate some kind of equilibrium unless it “parametrises”/”constrains” somehow the infinity of temperature distributions and the corresponding internal energy states .
    Perhaps I misunderstood your intent but here you are .

    Comment by Tom Vonk | February 12, 2008 | Reply

  2. Thanks for your, as usual, clear and concise comment, Tom.

    For me the Earth Climate System is comprised of everything related to energy and CO2, because these are the main focus of the global warming issues. So a partial inventory includes the land, ocean, atmosphere, ice, and other major bits of materials, the biological systems that interact with these, and all the chemical processes, both organic and inorganic, related to energy and CO2. And as Professor Pielke Sr. reminds us all the activities of humankind that affect these relative to the thermal response of the systems. At some point, of course a condition of diminishing returns is encountered and it doesn’t make sense to continue to try to keep track of all possible materials, processes, and interactions.

    My post was an attempt to focus on three main issues as follows.

    (1) If a model equation is not in agreement with the fundamental equations, thermodynamics in this case, the model is incorrect on a level that is beyond repair and should be replaced.

    (2) If a model equation cannot, in contrast to does not, predict the response shown in measured data, the model is incorrect on a level that is beyond repair and should be replaced.

    (3) Whenever a quantity is measured and purported to be the quantity represented by a symbol in a model equation, the measured quantity must be associated with the physical phenomena and processes represented by the model equation.

    For me the 0-D radiative-equilibrium model equation is not in agreement with thermodynamic fundamentals and cannot predict the response shown in measured data. It is not a correct statement relative to an energy balance for planet Earth. And it is not at all clear to me that the surface temperature record correspond to the radiative temperature of the planet.

    That such fundamental problems can exist in a field that has received so much investigation over 20 to 30 years of time is a mystery. Maybe I’m wrong.

    Comment by Dan Hughes | February 12, 2008 | Reply

  3. Hi Dan,

    An interesting, but difficult to understand post. Just to make sure I do understand…

    A 0D model of the climate will have a single equilibrium temperature variable, say T0. A 1D model of the climate will have a range of equilibrium temperatures that vary with height, with the surface temperature being, say T1. A 3D model of the climate will have a persistently dynamical range of temperatures that also vary both with surface location as well as height. Surface temperatures (only) may be averaged (somehow) to arrive at a mean equilibrium temperature, say T3.

    Now the question is: do T0, T1, and T3 represent the same thing? You seem to be saying no. And I agree.

    Although T0 and T1 are model data points, there is no point in the real climate that they can represent. And T3 isn’t even a model data point. It is an average taken to best bring out a ‘signal’ about the climate model. T3 is climate model metadata, useful for talking about the real climate, for sure, but not a real climate property itself.

    Now we can also average (somehow) actual temperature instrument recordings (and various proxies) globally and over time, say Tg. Is Tg a property of the real climate or metadata? Metadata.

    Does Tg have a relationship to T3, T1, or T0? You would think, but who knows to what extent?

    So I don’t find it surprising that, for instance, the global temperature Tg has been going slightly down over the past few years while the models say T3 is going way up and Nature says let’s reduce the minimal extent of the Arctic ice pack.

    Comment by George Crews | February 13, 2008 | Reply

  4. Hey, next time link and drop me a trackback– then I’ll see the post more quickly. I’m just starting to find the various blogs. 🙂

    There are difficulties in relating the various quantities in a zero – d model to the the full problem. I don’t think it’s a real problem, but it does require someone to write down the various equations and talk about how each term relates to the thermodynamic quantity it purports to describe.

    I actually did a little on the radiative imbalance terms and posted yesterday. So, I have a new term, which Tom Vonk will hate because he’ll see: second order moments! Yes, it will look like the horror of RANS.

    Still, if you think of the terms as describing a phenomanon rather than something one eventually wishes to solve, the terms will tell us something about the way in which the o-d model is deficient vis this particular issue.

    I have done nothing on the ‘true meaning’ of dH/dt nor ‘f’. So, yes, you’ll notice the analysis doesn’t address that. But, there are obvious issues, and you’ve touched on them: the time dependence, internal motion of heat etc.all affect things.

    Still, one can always learn from simple models– but it IS important to know what’s missing, and which missing phenomena are important to leading order.

    Comment by lucia | February 15, 2008 | Reply

  5. re: #4 lucia, I have started a comment to point George to additional information and in which I include links to your discussions and Eli and Atomz, too. Didn’t get back to it yet. After I left the short comment at your blog, I started one of those stream-of things and decided to make a post here.

    As I mentioned in the post, I started a small toy model with coupled radiation, convection, and conduction. The objectives included looking into how lack of convergence in the conduction might affect the response and accuracy and to look at conservation of energy in such a system having energy exchange across interfaces. Didn’t get back to that either. Way too many interesting things to check into and way too little time.

    Comment by Dan Hughes | February 16, 2008 | Reply

  6. I wouldn’t like to fall in chit-chatting but I do not “hate” the Reynolds stress tensor .

    It is a perfectly legitimate change of variables (assuming some regularity conditions) and it is no really big deal mathematically .
    There where it becomes doubtfull is that it fabricates a whole set of new unknown functions that don’t represent anything physical .
    It is for that reason that I consider that this change of variables mathematically legitimate makes physically no sense .
    It makes even less sense that it is clear that there are no additionnal physical laws that would ever enable to close the system .
    So it remains undetermined and as such useless .

    Of course if one uses a hammer and a crowbar long and hard enough in order to put the tensor in a given shape then it may give a numerical result that for certain conditions (like gaussian noise) and for steady states or at least for short times isn’t completely absurd .
    That is a justification a posteriori for a very restricted catalogue of conditions and makes me think about those broken watches that give the exact hour 2 times a day .
    But if one looks at such a watch only at these hours , it is perfectly adapted to the task 🙂

    And what bothers me is not so much the unphysicality of the thing , it is that it wastes time of people by forcing them to learn tons of uninteresting and irrelevant stuff .
    The above doesn’t apply for the cases where the method obviously serves practical needs like epicycles served practical needs despite being generally wrong .

    Comment by Tom Vonk | February 19, 2008 | Reply

  7. Dan , radiation is like those expensive restaurants in Paris where only saying “Good evening” has already cost you 20 € .
    So only saying “radiation” has cost you 2D .
    The simplest model I could think of would be an initialy isothermal non rotating sphere with only surface conductivity submitted to a constant parallel monochromatic radiation .
    Already such a model is a 2D problem and you will get a set of rather ugly differential equations defining T(theta,psi) .
    You can’t get anything less like 0D or 1D because there is simply no way for the temperature field to depend on less than 2 variables .
    If the sphere rotates it becomes uglier .
    And here you have only a rather unearthy system coupling radiation and surface conduction a bit like the Moon .
    You add anything more to that (like atmosphere and/or hydrosphere) and you are in for a full scale 3D + time model .

    But perhaps there is a way to construct some purely abstract system (makes me again think of the constructal theory) that doesn’t involve geometrical things like spheres but only systems composed of subsystems with equations governing them and boundary conditions separating them .
    I haven’t really thought of such an abstraction so I have only gut feelings .
    The gut feeling is that conduction is a problem because it is defined by a gradient .
    And in 0D the gradient is 0 so there is no conduction .
    So the minimum would be 1D in steady state – a kind of radiating string with some exotic 1D atmosphere .

    Comment by Tom Vonk | February 19, 2008 | Reply

  8. Tom V: The term I was discussing would share all the features you describe for the Reynolds stress tensor. 🙂

    Yes, I know you don’t hate the Reynolds stress tensor. I was sort of joking. 🙂

    I have the same issues with it myself. I just know that you periodically post comments bewailing the closure problem– which does exist and is a problem, generally.

    I usually resort to averaging not to actually do modeling, but to do order of magnitude estimates of certain effects, or to assess whether or not a particular modeling assumption would obviously introduce HUGE errors, moderate errors or relatively small errors. (I’m also aware that relatively small can be deceptive.)

    Comment by lucia | February 20, 2008 | Reply

  9. Lucia , I knew that you would say that you knew because I knew that you knew .
    From that I inferred that you were joking .

    Now if your thing shares the same features , you got the same problems .
    Unless you define with excrutiating acurateness the conditions where some empirical closure is valid and why , you won’t be able to trust what it says .

    Actually I must admit that somewhere I do hate it a bit because I find it frustrating that people spend energy to manoeuver themselves in a dead end when nobody is compelling them to do so .
    On the other hand I also admit that I find SOMETIMES a virtue to the saying “Better any number than no number at all .”
    It depends then if your model stands in one of those sometimes 🙂

    And I know that you know that I know why averaging as such might be useful especially if one wants to check gross orders of magnitude …. and that’s why this phrase is actually useless .

    Comment by Tom Vonk | February 21, 2008 | Reply

  10. Tom–
    The higher order moment terms I came up with in a zero-order model for a lumped parameter energy balance are simply to illustrate that an effect Pielke says would need to be included in an IPCC equation is really there.

    The purpose of finding the terms is to show they exist, and then later, if someone does an empirical analysis, they can estimate how much noise of bias might be introduced in a computation. So, in that sense, they terms needed to be unveiled because otherwise, the assumption was that the term was “zero”.

    But as for coming up with a model that one then progressively improves by developing more and more equations to predict more and more moments… no. Not going to do that. Thankfully, no one will!

    Comment by lucia | February 25, 2008 | Reply

  11. Dan

    You might consider reading :
    I have read it only once so have no precise opinion sofar .
    But the model looks very easy to implement (well there is that LBL bit that could complicate) and interesting enough to look at it .


    Increasing the freedom degrees of a system is a 2 edged sword .
    Either they may correspond to something and lead to farther insights or they may correspond to nothing and lead to confusion .
    So there is no absolute answer for all cases – very simple can be too simple and it gains with additionnal degrees while very complex can be too complex and it becomes unphysical with additionnal degrees .
    is for me definitely the latter .

    Comment by Tom Vonk | March 7, 2008 | Reply

  12. I didn’t know that the blog didn’t accept the brackets . The last phrase above should read : [ViVj]is for me definitely the latter .

    Comment by Tom Vonk | March 7, 2008 | Reply

  13. Btw while we are at the subject of freedom degrees .
    I don’t know if some of you did some linear programming .
    I did a lot of that many years ago and there are really strange things happening when one goes down from an undetermined system and begins to impose more and more constraints (= suppressing freedom degrees) .
    That teaches among other an extreme distrust to additionnals degrees of freedom unless one has a very good (aka physicaly justified) reason for it .

    Comment by Tom Vonk | March 7, 2008 | Reply

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