Guest Post by Willis Eschenbach
Once again, the crazy idea that downwelling longwave radiation (DLR, also called infra-red or IR, or “greenhouse radiation”) can’t heat the ocean has raised its ugly head on one of my threads.
Figure 1. The question in question.
There are lots of good arguments against the AGW consensus, but this one is just silly. Here are four entirely separate and distinct lines of reasoning showing that DLR does in fact heat the oceans.
Argument 1. People claim that because the DLR is absorbed in the first mm of water, it can’t heat the mass of the ocean. But the same is true of the land. DLR is absorbed in the first mm of rock or soil. Yet the same people who claim that DLR can’t heat the ocean (because it’s absorbed in the first mm) still believe that DLR can heat the land (despite the fact that it’s absorbed in the first mm).
And this is in spite of the fact that the ocean can circulate the heat downwards through turbulence, while there is no such circulation in the land … but still people claim the ocean can’t heat from DLR but the land can. Logical contradiction, no cookies.
Argument 2. If the DLR isn’t heating the water, where is it going? It can’t be heating the air, because the atmosphere has far too little thermal mass. If DLR were heating the air we’d all be on fire.
Nor can it be going to evaporation as many claim, because the numbers are way too large. Evaporation is known to be on the order of 70 w/m2, while average downwelling longwave radiation is more than four times that amount … and some of the evaporation is surely coming from the heating from the visible light.
So if the DLR is not heating the ocean, and we know that a maximum of less than a quarter of the energy of the DLR might be going into evaporation, and the DLR is not heating the air … then where is it going?
Rumor has it that energy can’t be created or destroyed, so where is the energy from the DLR going after it is absorbed by the ocean, and what is it heating?
Argument 3. The claim is often made that warming the top millimetre can’t affect the heat of the bulk ocean. But in addition to the wind-driven turbulence of the topmost layer mixing the DLR energy downwards into lower layers, heating the surface affects the entire upper bulk temperature of the ocean every night when the ocean is overturning. At night the top layer of the ocean naturally overturns, driven by the temperature differences between surface and deeper waters (see the diagrams here). DLR heating of the top mm of the ocean reduces those differences and thus delays the onset of that oceanic overturning by slowing the night-time cooling of the topmost layer, and it also slows the speed of the overturning once it is established. This reduces the heat flow from the body of the upper ocean, and leaves the entire mass warmer than it would have been had the DLR not slowed the overturning.
Argument 4. Without the heating from the DLR, there’s not enough heating to explain the current liquid state of the ocean. The DLR is about two-thirds of the total downwelling radiation (solar plus DLR). Given the known heat losses of the ocean, it would be an ice-cube if it weren’t being warmed by the DLR. We know the radiative losses of the ocean, which depend only on its temperature, and are about 390 w/m2. In addition there are losses of sensible heat (~ 30 w/m2) and evaporative losses (~ 70 w/m2). That’s a total loss of 390 + 30 + 70 = 490 w/m2.
But the average solar input to the surface is only about 170 watts/square metre.
So if the DLR isn’t heating the ocean, with heat gains of only the solar 170 w/m2 and losses of 390 w/m2 … then why isn’t the ocean an ice-cube?
Note that each of these arguments against the idea that DLR can’t warm the ocean stands on its own. None of them depends on any of the others to be valid. So if you still think DLR can’t warm the ocean, you have to refute not one, but all four of those arguments.
Look, folks, there’s lot’s of good, valid scientific objections against the AGW claims, but the idea that DLR can’t heat the ocean is nonsense. Go buy an infrared lamp, put it over a pan of water, and see what happens. It only hurts the general skeptical arguments when people believe and espouse impossible things …
w.
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Tim Folkerts, Re: spherical radiation;
Another way of looking at it is to visualize the spines on a sea urchin as being radiation. (or maybe the dimples on a golf ball). The largest proportion of them tend towards the lateral. Nothing changes if you rotate the sphere through 90 degrees.
Let me clarify about the radiation direction Re: http://farm3.static.flickr.com/2522/3837627461_4fc91e7a03_z.jpg?zz=1 . I was a little hasty. I was concluding from symmetry that any given direction has as much radiation as any other given direction, which would be true.
An “elemental” parcel of air is presumably on that is small compared to other dimensions of interest — say a cubic meter of air (or a cubic cm of air). In this approximation,. there is no variation within the parcel. This will naturally lead to the radiation being isotropic. The same amount of radiation goes up as goes down as goes east as goes west as goes north as goes south.
You could indeed lump 4 of these together and call them “sideways”, which would make exactly 2x as much radiation leaving from the “4 sides” as from the “top & bottom”. The flip side is that there would be twice as much area on the sides to receive radiation.
The sides will be radiating to/from other parcels at the same altitude and same (roughly) temperature, so the net radiation would be zero sideways. In this sense, the sides are of no interest. Only the top and bottom are “aimed” are parcels of a different temperature and hence only they can lead to any net radiation.
To me, the logic comes down to this. Please point out any specific idea that you take issue with.
1) If some parcel of the ocean (say 1m x 1m x 100m deep but NOT counting the skin layer) is balanced in terms of energy, then the temperature will stay constant.
2) The balance for an “average parcel” is approximately + 170 W/m^2 net solar radiation and -170 W/m^2 conduction to the skin layer (with very close to 0 W/m^2 of energy flow sideways or downward).
3) To maintain the -170 W/m^2, the skin layer must maintain a temperature gradient of 0.2 – 0.3 K (as is indeed observed).
4) If the IR energy into the top layer increases, the top layer must respond.
5) The first change will be an increase in temperature right near the top of the skin (since the IR is absorbed there).
6) in response to this increased temperature, the IR up from the surface will increase, as will the evaporation. BUT NOTE: the surface temperature has to increase for either of these to happen. Evaporation will speed up IN RESPONSE TO the elevated temperatures. Evaporation can and does limit the temperature rise, but cannot 100% prevent it.)
7. Even a small rise in the surface temperature will lead to a decrease in the conduction from below.
8. Even a small decrease in conduction out of the bulk of the ocean will lead to warming of the bulk of the ocean over time.
At this point, we could discuss the EXTENT of the warming of the bulk due to an increase in the downward IR. IR might be inefficient at warming the oceans, but IR logically has SOME effect unless one or more steps above are incorrect. Perhaps 10 W/m^2 extra only warms the surface 0.01 K, leading to a 0.01 K increase in the bulk temperature to restore the gradient.
Or someone could point out a flaw in my logic.
Bob_FJ::
“The following depiction shows that radiation in the atmosphere is mostly lateral and not up and down as depicted in the cartoon. Furthermore, the initial emission from the surface must mean that many of the first absorptions are close to the surface, even if free path lengths are long. Furthermore that subsequent emissions in the air are spherically distributed so that it only partly returns towards the surface.”
Very interesting. Am studying. Thanks!
All I KNOW ALMOST FOR CERTAIN NOW is that the GHE doesn’t make any sense from an empirical evidence standpoint. Other planets prove it. Our planet proves it. Wish I was smart enough to explain it.
Bob_FJ,
please stop with the most of the radiation is lateral. ALL of the radiation is emitted spherically unless there are physical reasons that would distort it (such as being emitted from a flat surface making it hemispherical). Trying to say most is lateral or most is up and down or… is all equally incorrect.
Tim Folkerts,
“The sides will be radiating to/from other parcels at the same altitude and same (roughly) temperature, so the net radiation would be zero sideways. In this sense, the sides are of no interest”
Yes, this is the NET that is SUPPOSED to be computed when things emit at each other. Of course, for some reason, the Climate Types throw out what is real physics for pseudo physics in computing 390 and 324 when they look at up and down.
I would point out that we have convection in the atmosphere so we have substantial areas where we have substantial differences in the net!! 8>) We also have irregularities where the SB is reduced in accuracy as the object has a strutcture where it can absorb its own black body radiation. I would suggest that the ocean surface is this way most of the time as it is not perfectly smooth and waves radiate against each other and the trough… Brings a whole new aspect to this why the ocean doesn’t freeze bit. It heats itself?? And in reality, this cannot be ignored any more than “backradiation!”
kuhnkat @ur momisugly August 26, 2011 at 7:52 pm
So in the analogy I made about visualising the spines on a sea urchin as radiation from an elemental sphere of air, does not work for you? Can you not see that there are a lot more spines pointing laterally than up and down? (or if you prefer, the dimples on a golf ball). You appear to be very assertive in your opinion, but should explain why, and your next comment seems to support my point anyway. I did not discuss absorption because neither does S-B which is at the core of the issue.
kuhnkat @ur momisugly August 26, 2011 at 8:00 pm Re Tim Folkerts @ur momisugly August 26, 2011 at 6:07 pm
Further to my August 25, 2011 at 11:08 pm the issue is that S-B gives the TOTAL radiative energy from a flat surface, which is hemispherically distributed. As I understand it from others, (I’ve not had the fortitude to read in detail the Trenberth et al papers of 1997 and 2009), the 390 W/m^2 up is based on an S-B calculation on global average T. But, a great deal of the total emission, (which is not HEAT), is lateral, and the NET lateral is zero, and not part of upward. Thus the 390 up is seriously overstated
“Climate Types throw out what is real physics for pseudo physics in computing 390 and 324 when they look at up and down.”
NO! If I heat a 1 m^2 piece of ground to ~ 288 K it will emit ~ 390 W/m^2. If the surrounding area is 0K, then there will be no radiation coming back. If the surrounding are 288 K, then there will be the 390 W/m^2 coming back. If the surrounding are warmer, there could by 780 W/m^2 coming back. The temperature of the surrounds does not change the rate at with the photons are emitting.
It is not “pseudo physics” that at the opening between two chambers, molecules are moving both directions.
It is not “pseudo physics” that in a wire, electrons are moving both directions.
It is not “pseudo physics” that in the atmosphere, photons are moving both directions.
The net flow is certainly something interesting to calculate (in any of the cases above). But the actual movement of actual particles in interesting too, and certainly not “pseudo physics”!
Jae @ur momisugly August 26, 2011 at 7:04 pm
Yep, but I don’t think we can win the war by questioning if there is a GHE though. I prefer to go along for example with atmospheric physicist and elitist alarmist Andrew Dessler who has admitted that if sceptics like Roy Spencer can show that there are major negative feedbacks, then AGW is no problem.
Tim Folkerts said:
“7. Even a small rise in the surface temperature will lead to a decrease in the conduction from below.”
That is the exact issue that needs to be resolved. I have described and considered that point at length elsewhere.
That statement represents the application of Fourier’s Law which provides (I think) that the flow of energy across a boundary is proportionate to the temperature differential either side of it.
However I have come to the conclusion that the combined effect of increased evaporation plus upward convection radiation and conduction from the warmed molecules at the very TOP of the ocean skin removes the energy added by DLR fast enough to prevent any change in the temperature differential at the interface between the ocean bulk and the BOTTOM of the skin layer.
Thus the 1mm deep cooler ocrean skin acts as a buffer between the DLR and the ocean bulk.
Reasons for that conclusion:
i) That little ‘hook’ in the temperature profile from warm to cool of 1mm depth below the actual surface and above the bulk remains present day and night. If it is not removed by solar input on a diurnal basis I don’t see how it could be affected by a little more DLR.
ii) That coolness of 0.3C 1mm deep appears to be a fixture of the oceans averaged globally. There seems to be no evidence of significant variation on a globally averaged basis.
iii) If DLR were to reduce the rate of energy flow from the ocean bulk to the skin layer then the depth and intensity of the temperature change at that bulk/skin interface must change but there is no evidence that it does if averaged globally.
iv) There is a nice analogy available namely the concept of a tributary joining a river. The downstream VOLUME of flow may increase but there is no need for any change in the RATE of flow from UPSTREAM in either river or tributary except maybe in a small buffer zone around the actual junction. So it appears to be similar where the energy flow from the bulk meets the energy flow from the DLR within the ocean skin which is most likely a similar type of buffer zone. In the case of river and tributary gravity determines the flow rates. In the case of ocean and skin atmospheric pressure determines the flow rates.
v) Evaporation being an energy hungry process (5 times more energy taken up than is required to provoke it) and since the energy shortfall is taken from where it is most readily available it seems most likely that the shortfall will be taken from any remaining DLR once increased radiation conduction and convection have taken their slices.
So it seems to me to be unlikely that Tim’s point 7 is correct.
I saw something that led me to believe that those ‘in the know’ are aware of this issue and that experiments are in the design stage but not yet ready for deployment.
So it is not safe to say that DLR has ANY effect on the upward flow rate of energy from the oceans. Pronouncing to that effect is not adviseable. Confidence in AGW is misplaced until the issue has been resolved.
Bob_FJ says:
August 26, 2011 at 11:11 pm
“Yep, but I don’t think we can win the war by questioning if there is a GHE though. I prefer to go along for example with atmospheric physicist and elitist alarmist Andrew Dessler who has admitted that if sceptics like Roy Spencer can show that there are major negative feedbacks, then AGW is no problem.”
Battles can be won by employing more than one weapon.
And are sceptics to also support pseudo science because it might help to win this battle. Or promote real science
GHG backradiation is not based on real science. We can acknowledge this but also challenge the alarmists positive feedback theory and the hopelessness of solutions like ET schemes.
Tim Folkerts,
1) if it is balanced it simply means there is the same energy coming and going from that parcel. Does not say much about what is happening outside of that parcel, only that the flux is balanced within that parcel.
2) you are guessing. just because the input and output for the surface (not just the ocean surface) is 170 has little to do with the balance of any particular parcel which isn’t in balance anyway. Warming in the day, cooling at night in general.
3) again this is an average and will be changing due to instant flows.
4) OK, except what would increase this input?? Generally it is increased insolation that cycles through the ocean first meaning the surface temp goes up, evap and OLR is going up a fraction before the DLR that stabilizes it.
5) speculation
6) probably
7) maybe
8) not necessarily. convection is up although I believe Willis is trying to claim that a large enough delay in the flux up and out would allow conduction to move energy down. While certainly possible, I would need to see a lot of measurements to agree this is significant below the absorption layer without gross physical effects like currents carrying the energy.
Bob_FJ,
I look at half of a sea urchin and see ALL of the spines going AWAY from the surface. Since the bottom layer of the atmosphere is the most dense, the LATERAL will be just as likely to end up back at the ocean surface than what goes mostly up. It has more atmosphere to penetrate to leave the system.
You are misinterpreting the importance of the direction here. If you move up, say 100ft, this geometry will have slightly more than half of the energy going away from the earth due to the geometry. Of course, the energy ends up heating the rest of the atmosphere at least as much as it is being reradiated or I don’t think we would get nearly as much convection.
Bob_FJ,
You make some good points about radiation. Yes, I agree that there is more power radiating laterally than vertically from any chuck of air. The same could be be said for any small chunk of the surface . I don’t agree that this implies that the 390 W/m^2 is overstated.
A very similar (and more familiar to many people who have taken university physics) situation occurs in electrostatics. The electric field lines spreads outward from a point charge in all directions (like those spines on the sea urchin). Integrating the electric field gives the flux Φ = ∫ E·dA (mathematically identical to integrating the photons heading in various directions from the radiation surface). This is known as Gauss’ Law. Any calculus-based university physics text will show have a chapter discussing exactly the sort of question about flux passing thru spheres and boxes.
The key point for this discussion is that while the flux from a point charge spreads out radially (like the radiation from a point on the ground radially), the flux from an infinite plane does not spread out at all, heading out constantly and perpendicularly from the surface. Analogously, if the radiant flux is 390 W/m^2 at the surface of an infinite plane, the flux would be 390 W/m^2 at any distance from the plane.
And yes, the earth is not an infinite plane, but from an elevation of 1 meter or 100 m or even 10 km, the “infinite plane approximation” will be quite close (if you look out of the windows of a jet, the earth’s surface still covers pretty close to 50% of your entire field of view). The details would be easy to work out. My intuition says that even going up 1% of the earth’s radius (~ 64 km), the flux would only decrease about 2% (~ 8 W/m^2) due to the spherical shape of the earth.
Tim Folkerts,
IF we had been measuring this flux across the oceans to a high degree of accuracy for years and saw the DLR slowly increasing without the 390 increasing THAT MIGHT be a problem as that would seem to show the oceans were absorbing energy without warming the surface. Impossible eh?
If the 324 number were decreasing it would seem that the atmosphere was decreasing in temperature right? Of course, without this energy the ocean surface would seem to have to cool and the 390 drop. right?
We do not have enough measurements of the surface, upper 700 meters, atmosphere, the sun, or any other part of the system to have the vaguest idea of whether what we are seeing is good bad or indifferent, AND the accuracy of our measurements aren’t good enough to tell us what we are seeing within what we THINK we know. We have a range of measurements for the diurnal and annual changes that would seem to be much more informative, yet, we seem to get stuck talking about these silly averages that show us nothing about what happens when the fluxes change, which they are doing all the time.
I have no problem with the idea that the IR being emitted from the surface is absorbed and reemitted, partially downward, by things called GHG’s (and that they warm the local parcel through collisions or cool it if it is warmer). I have no problem with the idea that a surface warmer than the GHG’s appear to be absorbing the IR and in its turn, reemitting the IR (unless it is being reflected by a surface that can’t absorb it??). What I have a problem with is the idea that a surface emitting at a higher rate would be raised to an even higher level by this lower density energy. It simply turns the lower density energy around as it is already emitting at a higher density. This would appear to slow the FLUX within the material making up that surface.
What is interesting is that this is NOT taken back the next steps. The sun is irradiating the ocean in depth with an energy density of a particular level. If the flux is slowed in the ocean then the flux into the ocean from the sun would be slowed also as the ocean warms. Everyone seems to agree with the idea that two bodies will eventually come to an equilibrium radiating against each other. This happens throughout the system, not just at the ground air interface. The NET is what is actually being moved between and through the systems. As the system warms the rate of warming slows due to the decreased delta between.
None of this would necessarily preclude conduction warming the next layer down. What makes this idea a non-sequitur is that we have continuous downwelling in the arctic and antarctic replenishing the cold abyssal waters maintaining the general temperature profile. Even with all the undersea volcanoes, what our limited measurements appear to show is a very stable COLD mass of ocean that is NOT changing due to these small surface anomalies.
Then again, without better measurements I could be wrong!!
I simply am not interested in alarmists who called wolf dozens of times before they even knew what was happening and persist as their scam is almost daily shown to have been based on shoddy science and STILL have no solid evidence of a problem. Willis appeared to be worried about why the oceans don’t freeze. If he is worried, he should REALLY worry about what might happen with an effective method of reducing GHG’s in the atmosphere and overshoot!!!!! Frozen oceans and dead plants!!!
HAHAHAHAHAHAHAHAHAHAHAHAHAHAHAHA
kuhnkat says:
“What I have a problem with is the idea that a surface emitting at a higher rate would be raised to an even higher level by this lower density energy. ….
If he is worried, he should REALLY worry about what might happen with an effective method of reducing GHG’s in the atmosphere and overshoot!!!!! Frozen oceans and dead plants!!!”
These two statements of yours seem to be completely at odds with each other. First you are saying more GHGs couldn’t warm the earth by increasing their “low density energy”. Then you are saying that less GHGs would cool the earth, which would be due to the loss of that same “low density energy”.
Why should the earth be at such a unique point — where only changes in one direction have an affect?
>>Tim Folkerts said:
>>7. Even a small rise in the surface temperature will lead to a decrease
>>in the conduction from below.”
> Stephen Wilde said:
>That is the exact issue that needs to be resolved.
I am glad we are getting close to agreement. We seem to be focused in on the same issue. The question seems to be:
The “mainstream consensus seem to be that, with no feedback, a doubling of CO2 will lead to 3.7 W/m^2 increase in DLR and subsequently a 1 C increase in temperature. If this were the case, then X = 1/3.7 = 0.27 °C/(W/m^2)
You seem to to be saying that feedback reduces this value to X ≡ 0 °C
To me, this sounds like “I have a spring so stiff that it prevents any motion when I push on it”. The spring WILL more a bit no matter how stiff (even if that motion is incredibly small). The extra radiation would have to have SOME effect on the water before the water would react with more IR and more evaporation (even if that increase in temperature is incredibly small). I can accept that it may be the case that X << 0.27 W/m^2, but not exactly 0 W/m^2
====================================
I suspect we will not get any closer to an answer in this thread. Stephen presented legitimate arguments why X should be very close to zero — so close that it would represent an insignificant, immeasurable change in temperature. I presented very crude estimate earlier of changes in IR and evaporation that led me to conclude X ~ 0.1 C/(W/m^2). This is smaller than the "no feedback" number = 0.27, which would put this estimate in the "negative feedback" regime. On the other hand, the estimate is mostly just an order-of-magnitude estimate. If I am a factor of 5 high, then X = 0.02, which would be in line with Stephen's estimate. If I am a factor of 5 low, I in in line with "mainstream" estimates of 2C per doubling.
I don't have the time nor expertise nor funding to pursue this to get a more accurate estimate. So as I said a little earlier, we may have to leave it at that …
“To me, this sounds like “I have a spring so stiff that it prevents any motion when I push on it”.”
More like a spring that is bouncy at one end and which becomes progressively stiffer towards the other end.
Note that I am not denying a climate effect. Merely pointing out that the effect is limited to the surface molecules,a portion of the skin and the air.
The climate effect is a slight adjustment of surface pressure distribution but miniscule compared to natural variations forced by solar and oceanic variability. We could never be able to measure it.
Kuhnkat & Tim Folkerts, concerning your recent comments on spherical radiation.
I refer you back to your earlier exchange:
And, repeating, the S-B law gives the TOTAL emitted radiant energy for a single flat body, which of course is hemispherically distributed. It does NOT give radiation only normal to the surface.
As can be visualized in the sea urchin illustration, a very large proportion of its spines point laterally through 360 degrees, not upwards. (In conceptual terms, treat 0 to 45 degrees as lateral and 45 to 90 degrees as vertical). Translating these spines as representing radiation, it follows that adjacent elemental parcels in an elemental layer give the net result as you describe above. Also, adjacent ELEMENTAL layers contain lateral radiation that deviates slightly from the horizontal but still meet and result in a similar outcome between such layers. (up & down) Thus we have a lot of the original S-B calculated emission that does not comprise part of the vertical effect you (Tim) are looking for, or in other words, the vertical components are less than the total S-B emission. It might be interesting to someone to pursue the maths, and integrate the vectors etc, but in my old age, I’ve lost pleasure in such activity long ago.
Kuhnkat, I’m sorry but your most recent comment; me no understand
Tim, generally speaking, I don’t like analogies. The more complicated they are, the greater the tendency is for them to become problematic. I think your latest is inappropriate for this forum, and we don’t want to wander off into new territory do we?
Bob_FJ,
Gauss’s Law is not rally an analogy — it is exactly the same math applied to a different situation. In that sense it is much better than analogies about urchins or golf balls.
When you say :
quess what — this IS the integral I was pointing you to with gauss’s law. And the result is very clearly not what you suggest when you say
If 390.00 W/m^2 leave from everywhere on the surface, then 390.00 W/m^s WILL be going up thru a 1m x 1m square mounted 1 meter above the surface.
Dear moderator/webmaster, this is driving me absolutely nuts! My mobile (Oz-wireless) laptop (notebook) computer has almost become stationary on this very long myrrhed thread. Please can we have a page #2 continuation to speed things up?
Regards, Bob-FJ
REPLY: Sorry, no facility for page 2 on a single thread, I’d have to change the whole blog settings. More laptop memory or a bigger swap file or different browser might help – Anthony
Tim Folkerts,
Slow the cooling Tim. Yes that means it is warmer that it would be otherwise but that is confusing terminology and should not be used.
Tim Folkerts @ur momisugly August 27, 2011 at 9:12 pm
Quickly, returning to S-B and first principles:
The hemispherical emissions from the surface can be replicated by perpendicular and horizontal vectors. The integration of the latter will give a greater quantity than that normal to the surface. The horizontal is PART OF your 390 W/m^2 radiative energy leaving the surface, but does NOT go up. Thus from first principles, the Trenberth cartoon shows a vertical number that is far too high
A further difficulty with S-B is that it only describes single body emissions and does not embrace consequent absorptive effects for a large body immersed in air. On Earth, most of the initial emissions will be absorbed close to the surface, some very close, and this has consequences as the emission mode switches to spherical. And of course the atmosphere has myriad layering effects. Lots of stuff going on that disturbs the radiative considerations quite markedly.
“The horizontal is PART OF your 390 W/m^2 radiative energy leaving the surface, but does NOT go up. Thus from first principles, the Trenberth cartoon shows a vertical number that is far too high.
From first principles, consider a large grid of 1m x 1m squares on the ground, each radiating 390.0 W. (Or a sea urchin with 390 quills placed in every grid. Or 390 1W lasers pointing out randomly from each square). Then consider a 1m x 1m grid somewhere above the ground — perhaps 1 m up or 1000 m up.
I agree that the 390.0 W from one square will not all pass thru the 1m x 1m square in the grid directly above it. Some will go sideways to varying degrees and pass thru some OTHER part of the upper grid. (None will go absolutely horizontally because then it would not be emitted from the surface to begin with).
But the upper square that we are looking at will also GAIN some photons from OTHER parts of the lower grid that are radiating sideways (this is the part you seem to have neglected). The sideways radiation can’t just disappear. By symmetry, in a transparent atmosphere, the total thru the upper grid must be 390.0 W for each square meter.
Now the atmosphere is not transparent. This is where the “350 W/m^2” and “40 W/m^2” come into play. The calculations would get a bit involved to know how the “sideways” component factors into the absorption, but I will assume that the clever people who put together MODTRAN and similar calculations were aware of this and built it into the program. But if you think they are wrong, then you are welcome to perform a detailed analysis of there methods looking for errors.
And the earth is not flat. The 390 W from 1 square meter will indeed spread out SLIGHTLY as you go up. But this will result in a correction of no more than ~ 2% up to 60 km high and 0.02% up to 6 km high. Most of the action for absorption occurs in the first 6 km, so the errors from the curvature of the earth should be negligible. (And again, there is a good chance that this is indeed built into the models. If not, it could be a potential source of systematic error.)
TO WRAP UP: If people truly think they have a better grasp on radiation than Trenberth and/or the people who put together MODTRAN, then do the math in detail and find the errors.