In Part One we saw how the ocean absorbed different wavelengths of radiation:
- 50% of solar radiation is absorbed in the first meter, and 80% within 10 meters
- 50% of “back radiation” (atmospheric radiation) is absorbed in the first few microns (μm).
This is because absorption is a strong function of wavelength and atmospheric radiation is centered around 10μm, while solar radiation is centered around 0.5μm.
In Part Two we considered what would happen if back radiation only caused evaporation and removal of energy from the ocean surface via the latent heat. The ocean surface would become much colder than it obviously is. That is a very simple “first law of thermodynamics” problem. Then we looked at another model with only conductive heat transfer between different “layers” in the ocean. This caused various levels below the surface to heat to unphysical values. It is clear that turbulent heat transport takes place from lower in the ocean. Solar energy reaches down many meters heating the ocean from within – hotter water expands and so rises – moving heat by convection.
In Part Three we reviewed various experimental results showing how the temperature profile (vs depth) changes during the diurnal cycle (day-night-day) and with wind speed. This demonstrates very clearly how much mixing goes on in the ocean.
The Different Theories
This series of articles was inspired by the many people who think that increases in back radiation from the atmosphere will have no effect (or an unnoticeable effect) on the temperature of the ocean depths.
So far, no evidence has so far been brought forward for the idea that back radiation can’t “heat” the ocean (see note 1 at the end), other than the “it’s obvious” evidence. At least, I am unaware of any stronger arguments. Hopefully as a result of this article advocates can put forward their ideas in more detail in response.
I’ll summarize the different theories as I’ve understood them. Apologies to anyone who feels misrepresented – it’s quite possible I just haven’t heard your particular theory or your excellent way of explaining it.
Hypothesis A – Because the atmospheric radiation is completely absorbed in the first few microns it will cause evaporation of the surface layer, which takes away the energy from the back radiation as latent heat into the atmosphere. Therefore, more back-radiation will have zero effect on the ocean temperature.
Hypothesis B – Because the atmospheric radiation is completely absorbed in the first few microns it will be immediately radiated or convected back out to the atmosphere. Heat can’t flow downwards due to the buoyancy of hotter water. Therefore, if an increase in back radiation occurs (perhaps due to increases in inappropriately-named “greenhouse” gases) it will not “heat” the ocean = increase the temperature of the ocean below the surface.
For other, more basic objections about back radiation, see Note 2 (at the end).
I believe that Part Two showed that Hypothesis A was flawed.
I would like to propose a different hypothesis:
Hypothesis C – Heat transfer is driven by temperature differences. For example, conduction of heat is proportional to the temperature difference across the body that the heat is conducted through.
Solar radiation is absorbed from the surface through many meters of the ocean. This heats the ocean below the surface which causes “natural convection” – heated bodies expand and therefore rise. So solar energy has a tendency to be moved back to the surface (this was demonstrated in Part Two).
The more the surface temperature increases, the less temperature difference there will be to drive this natural convection. And, therefore, increases in surface temperature can affect the amount of heat stored in the ocean.
Clarification from St.Google: Hypothesis = A supposition or proposed explanation made on the basis of limited evidence as a starting point for further investigation
An Excellent Question
In Part Three, one commenter asked an excellent question:
Some questions from an interested amateur.
Back radiation causes more immediate evaporation and quicker reemission of LWR than does a similar amount of solar radiation.
Does that mean that the earth’s temperature should be more sensitive to a given solar forcing than it would be to an equal CO2 forcing?
What percentage CO2 forcing transfers energy to the oceans compared to space and the atmosphere?
How does this compare with solar forcing?
Is there a difference between the effect of the sun and the back radiation when they are of equal magnitude? This, of course, pre-supposes that Hypothesis C is correct and that back radiation has any effect at all on the temperature of the ocean below the surface.
So the point is this - even if Hypothesis C is correct, there may still be a difference between the response of the ocean temperatures below the surface – for back radiation compared with solar radiation.
So I set out to try and evaluate these two questions:
- Can increases in back radiation affect the temperature of the ocean below the surface? I.e., is Hypothesis C supported against B?
- For a given amount of energy, is there a difference between solar forcing and back radiation forcing?
And my approach was to use a model:
Oh no, a model! Clearly wrong then, and a result that can’t fool anyone..
For a bit of background generally on models, take a look at the Introduction in Models On – and Off – the Catwalk.
Here is one way to think about a model
The idea of a model is to carry out some calculations when doing them in your head is too difficult
A model helps us see the world a bit more clearly. At these point I’m not claiming anything other than they help us see the effect of the well-known and undisputed laws of heat transfer on the ocean a little bit more clearly.
The ocean model under consideration is about a billion times less complex than a GCM. It is a 1-d model with heat flows by radiation, conduction and, in a very limited form, convection.
Here is a schematic of the model. I thought it would be good to show the layers to scale but that means the thicker layers can’t be shown (not without taking up a ridiculous amount of blank screen space) – so the full model, to scale, is 100x deeper than this:
To clarify – the top layer is at temperature, T1, the second layer at T2, even though these values aren’t shown.
The red arrows show conducted or convected heat. They could be in either direction, but the upwards is positive (just as a convention). Obviously, only a few of these are shown in the schematic – there is a heat flux between each layer.
1. Solar and back radiation are modeled as sine waves with the peak at midday. See the graph “Solar and Back Radiation” in Part Two for an example.
2. Convected heat is modeled with a simple formula:
H=h(T1-Tair), where Tair = air temperature, T1 = “surface” temperature, h = convection coefficient = 25 W/m².K.
Convected heat can be in either direction, depending on the surface and air temperature. The air temperature is assumed constant at 300K, something we will return to.
3. Radiation from the surface:
E = εσT4 – the well-known Stefan-Boltzmann equation, and ε = emissivity
For the purposes of this simple model ε = 1. So is absorptivity for back radiation, and for solar radiation. More on these assumptions later.
4. Heat flux between layers (e.g. H54 in the schematic) is calculated using the temperature values for the previous time step for the two adjacent layers then using the conducted heat formula: q” = k.(T5-T4)/d54, where k= conductivity, and d54 = distance between center of each layer 5 to the center of layer 4.
For still water, k = 0.6 W/m.K – a very low value as water is a poor conductor of heat.
In this model at the end of each time step, the program checks the temperature of each layer. If T5 > T4 for example, then the conductivity between these layers for the next time step is set to a much higher value to simulate convection. I used a value for stirred water that I found in a textbook: kt = 2 x 105 W/m.K. What actually happens in practice is the hotter water rises taking the heat with it (convection). Using a high value of conductivity produces a similar result without any actual water motion.
For interest I did try lower values like 2 x 10³ W/m.K and the 1m layer, for example, ended up at a higher temperature than the layers above it. See the more detailed explanation in Part Two.
5. In Part Three I showed results from a number of field experiments which demonstrated that the ocean experiences mixing due to surface cooling at night, and due to high winds. The mixing due to surface cooling is automatically taken account of in this model (and we can see it in the results), but the mixing due to the winds “stirring” the ocean is not included. So we can consider the model as being “under light winds”. If we had a model which evaluated stronger winds it would only make any specific effects of back radiation less noticeable. So this is the “worst case” – or the “highlighting back radiation’s special nature” model.
Problems of Modeling
Some people will already know about the many issues with numerical models. A very common one is resolving small distances and short timescales.
If we want to know the result over many years we don’t really want to have the iterate the model through time steps of fractions of a second. In this model I do have to use very small time steps because the distance scales being considered range from extremely small to quite large – the ocean is divided into thin slabs of 5mm, 15mm.. through to a 70m slab.
If I use a time step which is too long then too much heat gets transferred from the layers below the surface to the 5mm surface layer in the one time step, the model starts oscillating - and finally “loses the plot”. This is easy to see, but painful to deal with.
But I thought it might be interesting for people to see the results of the model over five days with different time steps. Instead of having the model totally “lose the plot” (=surface temperature goes to infinity), I put a cap on the amount of heat that could move in each time step for the purposes of this demonstration.
You can see four results with these time steps (tstep = time step, is marked on the top left of each graph):
- 3 secs
- 1 sec
- 0.2 sec
- 0.05 sec
Figure 2 – Click for a larger image
I played around with many other variables in the model to see what problems they caused..
The model is written in Matlab and runs on a normal PC (Dell Vostro 1320 laptop).
To begin with there were 5 layers in the model (values are depth from the surface to the bottom edge of each layer):
- 5 mm
- 50 mm
- 1 m
- 10 m
- 100 m
I ran this with a time step of 0.2 secs and ended up doing up to 15-year runs.
In the model runs I wanted to ensure that I had found a steady-state value, and also that the model conserved energy (first law of thermodynamics) once steady state was reached. So the model included a number of “house-keeping” tests so I could satisfy myself that the model didn’t have any obvious errors and that equilibrium temperatures were reached for each layer.
For 15 year runs, 5 layers and 0.2s time step the run would take about two and a half hours on the laptop.
I find that quite amazing – showing how good Matlab is. There are 31 million seconds in a year, so 15 years at 0.2 secs per step = 2.4 billion iterations. And each iteration involves looking up the solar and DLR value, calculating 7 heat flow calculations and 5 new temperatures. All in a couple of hours on a laptop.
Well, as we will see, because of the results I got I thought I would check for any changes if there were more layers in my model. So that’s why the 9-layer model (see the first diagram) was created. For this model I need an even shorter time step – 0.1 secs and so long model runs start to get painfully long..
Case 1: The standard case was a peak solar radiation, S, of 600 W/m² and back radiation, DLR of 340 with a 50 W/m² variation day to night (i.e., max of 390 W/m², min of 290 W/m²).
Case 2a: Add 10 W/m² to the peak solar radiation, keep DLR the same. Case 2b – Add 31.41 W/m² to solar.
Case 3a: Keep solar radiation the same, add 3.14 W/m² to DLR. This is an equivalent amount of energy per day to case 2, see note 3. Case 3b – Add 10 W/m² to DLR.
Many people are probably asking, “Why isn’t case 3a – Add 10 W/m² to DLR?”
Solar radiation only occurs for 12 out of the 24 hours, while DLR occurs 24 hours of the day. And the solar value is the peak, while the DLR value is the average. It is a mathematical reason explained further in Note 3.
The important point is that for total energy absorbed in a day, case 2a and 3a are the same, and case 2b and 3b are the same.
Let’s compare the average daily temperature in the top layer, 1m, 10m and 100m layer for the three cases (note: depths are from the surface to the bottom of each layer; and only 4 layers of the 5 were recorded):
The time step (tstep) = 0.2s.
The starting temperatures for each layer were the same in all cases.
Now because the 4 year runs recorded almost identical values for solar vs DLR forcing, and because the results had not quite stabilized, I then did the 15 year run and also recorded the temperature to the 4 decimal places shown. This isn’t because the results are this accurate – this is to see what differences, if any, exist between the two different scenarios.
The important results are:
- DLR increases cause temperature increases at all levels in the ocean
- Equivalent amounts of daily energy into the ocean from solar and DLR cause almost exactly the same temperature increase at each level of the ocean – even though the DLR is absorbed in the first few microns and the solar energy in the first few meters
- The slight difference in temperature may be a result of “real physics” or may be an artifact of the model
And perhaps 5 layers is not enough?
Therefore, I generated the 9-layer model, as shown in the first diagram in this article. The 15-year model runs on the 9-layer model produced these results:
The general results are similar to the 5-layer model.
The temperature changes have clearly stabilized, as the heat unaccounted for (inputs – outputs) on the last day = 41 J/m². Note that this is Joules, not Watts, and is over a 24 hour period. This small “unaccounted” heat is going into temperature increases of the top 100m of the ocean. (“Inputs – outputs” includes the heat being transferred from the model layers down into the ocean depths below 100m).
If we examine the difference in temperature for the bottom 30-100m deep level for case 2b vs 3b we see that the temperature difference after 15 years = 0.011°C. For a 70m thick layer, this equates to an energy difference = 3.2 x 106 J, which, over 15 years, = 591 J/m².day = 0.0068 W/m². This is spectacularly tiny. It might be a model issue, or it might be a real “physics difference”.
In any case, the model has demonstrated that DLR increases vs solar increases cause almost exactly the same temperature changes in each layer being considered.
For interest here are the last 5 days of the model (average hourly temperatures for each level) for case 3b:
and for case 2b:
Results – Convection and Air Temperature
In the model results up until now the air temperature has been at 300K (27°C) and the surface temperature of the ocean has been only a few degrees higher.
The model doesn’t attempt to change the air temperature. And in the real world the atmosphere at the ocean surface and the surface temperature are usually within a few degrees.
But what happens in our model if real world situations cool the ocean surface more? For example, higher temperatures locally create large convective currents of rising hot air which “sucks in” cooler air from another area.
What would be the result? A higher “instantaneous” surface temperature from higher back radiation might be “swept away” into the atmosphere and “lost” from the model.. This might create a different final answer for back radiation compared with solar radiation.
It seemed to be worth checking out, so I reduced the air temperature to 285K (from 300K) and ran the model for one year from the original starting temperatures (just over 300K). The result was that the ocean temperature dropped significantly, demonstrating how closely the ocean surface and the atmosphere (at the ocean surface) are coupled.
Using the end of the first year as a starting temperature, I ran the model for 5 years for case 1, 2a and 3a (each with the same starting temperature):
Once again we see that back radiation increases do change the temperatures of the ocean depths – and at almost identical values to the solar radiation changes.
Here is a set of graphs for one of the 5-year model runs for this lower air temperature, also demonstrating how the lower air temperature pulls down the ocean surface temperature:
Figure 8 – Click for a larger image
The first graph shows how the average daily temperature changes over the full time period – making it easy to see equilibrium being reached. The second graph shows the hourly average temperature change for the last 5 days. The last graph shows the heat which is either absorbed or released within the ocean in temperature changes. As zero is reached it means the ocean is not heating up or cooling down.
Inaccuracies in the Model
We can write a lot on the all the inaccuracies in the model. It’s a very rudimentary model. In the real world the hotter tropical / sub-tropical oceans transfer heat to higher latitudes and to the poles. So does the atmosphere. A 1-d model is very unrealistic.
The emissivity and absorptivity of the ocean are set to 1, there are no ocean currents, the atmosphere doesn’t heat up and cool down with the ocean surface, the solar radiation value doesn’t change through the year, the top layer was 5mm not 1μm, the cooler skin layer was not modeled, a number of isothermal layers is unphysical compared with the real ocean of continuously varying temperatures..
However, what a nice simple model tells us is how energy only absorbed in the top few microns of the ocean can affect the temperature of the ocean much lower down.
“It’s obvious“, I could say.
My model could be wrong – for example, just a mistake which means it doesn’t operate how I have described it. The many simplifications of the model might hide some real world physics effect which means that Hypothesis C is actually less likely than Hypothesis B.
However, if the model doesn’t contain mistakes, at least I have provided more support for Hypothesis C – that the back radiation absorbed in the very surface of the ocean can change the temperature of the ocean below, and demonstrated that Hypothesis B is less likely.
I look forward to advocates of Hypothesis B putting forward their best arguments.
Update - Code files saved here
Note 1 – To avoid upsetting the purists, when we say “does back-radiation heat the ocean?” what we mean is, “does back-radiation affect the temperature of the ocean?”
Some people get upset if we use the term heat, and object that heat is the net of the two way process of energy exchange. It’s not too important for most of us. I only mention it to make it clear that if the colder atmosphere transfers energy to the ocean then more energy goes in the reverse direction.
It is a dull point.
Note 2 - Some people think that back radiation can’t occur at all, and others think that it can’t affect the temperature of the surface for reasons that are a confused mangle of the second law of thermodynamics. See Science Roads Less Travelled and especially Amazing Things we Find in Textbooks – The Real Second Law , The Real Second Law of Thermodynamics and The Three Body Problem. And for real measurements of back radiation, see The Amazing Case of “Back Radiation” -Part One.
Note 3 – If we change the peak solar radiation from 600 to 610, this is the peak value and only provides an increase for 12 out of 24 hours. By contrast, back radiation is a 24 hour a day value. How much do we have to change the average DLR value to provide an equivalent amount of energy over 24 hours?
If we integrate the solar radiation for the before and after cases we find the relationship between the value for the peak of the solar radiation and the average of the back radiation = π (3.14159). So if the DLR increase = 10, the peak solar increase to match = 10 x π = 31.4159; and if the solar peak increase = 10, the DLR increase to match = 10/π = 3.1831.
If anyone would like this demonstrated further please ask and I will update in the comments. I’m sure I could have made this easier to understand than I actually have (haven’t).