In the series so far we have seen how radiation interacts with the atmosphere for a given surface/atmospheric condition.
That is, if the temperature is say 288K (15°C) and the atmospheric temperature decreases at 6.5 K/km (the “lapse rate”) and the concentration of water vapor is this, and the concentration of CO2 is that.. then:
- what is the outgoing radiation, OLR, at the top of the atmosphere (TOA)?
- what is the surface downward radiation, DLR?
- what do the spectra look like and why?
- how do surface temperatures, water vapor concentrations and CO2 concentrations change these values?
These are all important questions, and the necessary first step. Because if we don’t understand these points then it is impossible to work out how the atmosphere reaches a steady state under those conditions, and of course, impossible to work out how a new steady state will be reached if something changes.
The earlier model had some ability to step forward in time and calculate temperature change (but with many limitations and some flaws). Specifically I wanted to be able to track the change of energy in each layer and also account for convective heat flow.
A recent commenter asked (about the effect of doubling CO2):
All what you’re saying would show is *if* the surface temperature were to increase by 1.1C (independent of mechanism) it would restore radiative balance for both +3.7 W/m^2 of post albedo solar power and +3.7 W/m^2 of GHG absorption (which I agree with). In no way does this prove or demonstrate that +1.1C at the surface (for no feedback) is itself a requirement to restore balance at the TOA from +3.7 W/m^2 of GHG absorption. This is because you’ve made no accounting for cause and effect – you’ve only shown that the outcome of one potential effect (i.e. +1.1C at the surface) would restore balance at the TOA.
This is an interesting question, and this kind of question is part of the reason for this series.
Let’s first look at the simple question of how any steady state is reached. I add more specifics about the model (v.0.10.1) at the end of the article and have added the code to Part Five – The Code.
This update of the model now includes an “ocean” and some very simple solar heating of the atmosphere:
This is shown in °C/day but is easily converted to W/m² by dividing by 86,400 (number of seconds in a day) and multiplying by the heat capacity of that layer of the atmosphere. Each of the atmospheric layers in the model have roughly the same number of molecules so the graph of W/m² absorbed in a given layer looks quite similar in shape.
I introduced this solar heating partly because the old model had bad accounting at the top of atmosphere, where solar absorption just (magically) kept the stratosphere isothermal (the same temperature). That constraint is now gone in this update of the model.
Here is a model run:
Figure 2 – Click to expand
What’s going on here? Well, let’s first take a look at the energy balance at the surface and for the whole planet (TOA):
Figure 3 – Positive downward (so positive flux imbalance at TOA means the planet is heating up)
The starting point for this model was an ocean temperature of 288K and a lapse rate of 6.5 K/km, up to a tropopause of 200 hPa with an isothermal stratosphere. The solar radiation absorbed by the climate was 242 W/m², with about 100 W/m² absorbed in the atmosphere and the balance absorbed by the surface.
Each time step was 2 hours, and the model was run for 800 days (10,000 time steps).
Why is the model out of balance to begin with?
There’s no reason it should be in balance. I’ve simply prescribed a surface temperature and an atmospheric temperature profile and humidity and CO2 concentration. Why should that happen to be the steady state condition for the solar absorbed radiation of 242 W/m² with 100 W/m² absorbed in the atmosphere?
[Update Jan 22nd - A good point added from Pekka: "It’s perhaps not clear enough to every reader that this thread is describing what happens when the starting point is an initial state that you happened to choose rather than a state that the system could have reached at different external conditions. Thus the Figure 4 tells what happens initially for this specific case rather than values somehow applicable to the real atmosphere."]
So the model works by energy accounting in each time step for each layer in the model. Energy cannot be created or destroyed. Radiation emitted and absorbed is calculated by the relevant equations (already explained). Convection moves heat if the atmosphere above is too cold (see Potential Temperature and Density, Stability and Motion in Fluids). Energy retained increases the temperature of the layer. Energy lost reduces the temperature of the layer.
Let’s consider the surface. On timestep 1 the surface is radiating 376 W/m² (note 1). All of the surface fluxes are shown (in W/m²) in the diagram:
The net is 18 W/m² and so the “ocean” absorbs this energy which means it heats up. In 2 hours (one timestep) this comes to 129 kJ/m² and as the “ocean” in this model is just 10m deep (to allow quicker progress to any equilibrium) this equates to a temperature increase of 0.0031 °C.
If the net heat absorbed is 18 W/m² why doesn’t figure 3 show that? Ok, let’s zoom into the first month of figure 3 and we can see it clearly:
How did the convective flux get calculated? Why isn’t it higher?
The model calculates all the radiative fluxes up and down through each layer, works out the absorbed energy and the resulting temperature increase. Then it checks between each layer to see if the lapse rate is exceeded (see Temperature Profile in the Atmosphere – The Lapse Rate). This means the atmosphere would be unstable, resulting in convection.
The model then calculates the transfer of heat which would satisfy the lapse rate – the layer below loses X Joules, the layer above gains X Joules and new temperatures are calculated based on their respective heat capacities. This is what the model calculates and then adjusts temperatures, logs the convective heat moved and adjusts the energy change in each layer for that timestep.
The graph below zooms in on the first 30 days of the bottom right graph in figure 2:
So convection in this particular instance isn’t any higher at the start simply because of the respective temperatures. Then the first atmospheric layer starts cooling via radiation (it loses more heat via radiation than it gains via solar heating) and this means that convection increases from the surface with each timestep – until a more steady condition is reached.
Now a key point is that the surface imbalance changes over time - which we see in figure 3.
Now there’s no magic “model driver” that makes this happen. It’s just basic heat transfer laws. The model just reflects, in a simplistic way, how heat is transferred between an ocean, an atmosphere and space.
Now let’s look at the TOA balance – look back at figures 3 and 5. This is the balance for the whole climate. What might be interesting is to see that the climate is initially out of balance – losing heat.
But why doesn’t the cooling of the climate mean that the imbalance just reduces until a steady state is reached? How is it possible for the climate to start heating at day 10 and peak somewhere around day 50 and then gradually reduce?
This is very typical of complex dynamic scenarios. Readers familiar with dynamic heat transfer (and any kind of dynamic physics/chemistry/engineering problems) will have seen these kind of graphs before – overshoot, decay to equilibrium.
What is completely unsurprising though is that the ocean and atmosphere end up in a steady state where cooling to space matches solar absorption – that is, the balance at TOA is ultimately zero.
Here’s a summary of the energy change, in kJ per timestep of 2 hours, of ocean, energy and TOA:
In the first few days the ocean and atmosphere are very much out of balance and so a big “reshuffle” of energy takes place where the ocean absorbs energy and the atmosphere loses energy until they are in much closer balance. Then there is a gradual cooling of the system (primarily via ocean cooling) which eventually leads to an overall TOA balance – which can be seen in figure 2.
In a subsequent article we will take this steady state condition, then increase the CO2 concentration and see what happens.
We’ve seen via one specific example how heat transfer, via radiation and convection, lead to a new equilibrium condition. This can include some oscillation on the way to equilibrium.
This particular case has no claim to be the “definitive median atmospheric condition”. It’s just a sample atmosphere that wasn’t in perfect balance for its conditions.
Many people have conceptual models of how heat moves in the atmosphere and often these mental models are wrong. The purpose of this article is to illustrate how the basic heat transfer mechanisms work. As we can see with this simple example, it would be surprising to get the right answer about dynamic and final temperatures from some hand-waving arguments.
If you have questions please ask. We can examine the energy transfer from many different perspectives.
Some Model Specifics
This update to the model has removed the constraint of keeping the stratosphere isothermal (see note 2).
Instead solar radiation is absorbed in the atmosphere according to the standard heating curves, for example, those found in Petty 2006 p.315:
The stratosphere is not well modeled because the higher levels of the stratosphere are not included. These absorb most of the solar radiation via O2 & O3 and consequently keep the lower levels warmer than the equilibrium reached in this model.
This model had 12 layers, with the TOA at 20 hPa (most previous models had 10 going up to 50 hPa) – the reason for going higher was just curiosity about the resulting temperature profile.
Clearly the solar absorption in the atmosphere should be calculated via the absorption characteristics of the various molecules but this will take some work, and the current model is just an interesting starting point (or resting place depending on my interest level in this aspect of the model). The main flaw in the current approach is that increasing water vapor in the lower atmosphere should increase heating via solar radiation but the model has a static absorption profile shown in figure 1. The main advantage is that it is a lot more accurate than having zero atmospheric absorption.
The convective accounting is also a little challenging. The problem is first that to calculate a convective adjustment we can’t just change layer 2 temperature to make it layer 1 temperature + lapse rate x height. Because after we work that out we have to move the right amount of heat from layer 1 to layer 2 to make layer 2 heat up enough. This reduces layer 1 heat by an equal amount and reduces layer 1 temperature dependent on its heat capacity and the temperature difference is now incorrect. This first problem is easily solved with a formula (see note 3).
Quick people unlike myself will immediately realize that we have not solved the problem at all because when we now consider layer 2- layer 3, the result will move layer 2 temperature and now layer 1-layer 2 is incorrect.
A bigger simultaneous equation might do the trick, but I’m pretty sure that it would come unstuck without some careful thinking about layers where the actual lapse rate in a given time step is less than the prescribed lapse rate. A quick solution was to do multiple loops (iterate towards a solution) and check the lapse rates via a graph and some printing out. Matlab is truly the friend of the mathematically lazy.
There are some checks and balances in my coding. Each time step the model calculates the difference between the TOA balance and the energy absorbed in all layers of the model. This should be zero otherwise I have not implemented the first law of thermodynamics. In this model run the maximum absolute error in any time step was 3 nJ/m² – or less than 4 pW/m². This is just rounding errors in the maths.
The model currently is tasked with printing an error message if ever more than 1J/m² goes missing in a time step.
The Matlab function returns the temperature profiles vs time, energy changes vs time for each layer, convective energy vs time for each layer, along with surface balance, TOA balance and lots of other parameters. The energy vs time graphs in figure 2 are shown as W/m² so they can be related to other fluxes, but they are stored as Joules per m² – it’s just difficult to consider whether 1.67 x 105 J/m² is the kind of value we are expecting or not – and of course the (Joules) numbers change as the time step is adjusted.
Part One - some background and basics
Part Two - some early results from a model with absorption and emission from basic physics and the HITRAN database
Part Three – Average Height of Emission - the complex subject of where the TOA radiation originated from, what is the “Average Height of Emission” and other questions
Part Four – Water Vapor - results of surface (downward) radiation and upward radiation at TOA as water vapor is changed
Part Five – The Code - code can be downloaded, includes some notes on each release
Part Six – Technical on Line Shapes - absorption lines get thineer as we move up through the atmosphere..
Part Seven – CO2 increases - changes to TOA in flux and spectrum as CO2 concentration is increased
Part Eight – CO2 Under Pressure - how the line width reduces (as we go up through the atmosphere) and what impact that has on CO2 increases
Part Ten – “Back Radiation” - calculations and expectations for surface radiation as CO2 is increased
Part Eleven – Stratospheric Cooling - why the stratosphere is expected to cool as CO2 increases
Part Twelve – Heating Rates - heating rate (‘C/day) for various levels in the atmosphere – especially useful for comparisons with other models.
The data used to create these graphs comes from the HITRAN database.
The HITRAN 2008 molecular spectroscopic database, by L.S. Rothman et al, Journal of Quantitative Spectroscopy & Radiative Transfer (2009)
The HITRAN 2004 molecular spectroscopic database, by L.S. Rothman et al., Journal of Quantitative Spectroscopy & Radiative Transfer (2005)
Note 1: The emission of thermal radiation by a surface at 288K with an emissivity of 1.0 is 390 W/m². This is across all wavelengths. The model looks at the range of wavenumbers 200 – 2500 cm-1 (equates to 4-50 μm) to ease up the calculation effort required. Across this range the emission is 376 W/m².
Note 2: This was partly to avoid what would look like confusing energy accounting, where solar absorption = the amount prescribed in the model + what we find necessary to keep the stratosphere isothermal.
Note 3: If T1 and T2 are the unadjusted temperatures (found via radiative energy movement), and T1′ and T2′ are the temperatures that should result from lapse rate Γ and height difference z, then:
Convective heat, CE = [(T1-T2) - zΓ]/(1/Cp1 + 1/Cp2)
and then T2′ = T2 + CE/Cp2, T1′ = T1 – CE/Cp1