Hello, so I am trying to carry out a very simple and high-level approach for calculating the effect of building height on green roof cooling performance, and have been trying to use a certain building height-cooling decay rate I found in the literature, but am unsure if I am actually using it correctly and wanted to ask the research experts here for perhaps some guidance. I will further explain my project below with an example.

Let’s say I have 3 buildings, building A, B, and C in a city, each with a green roof. I also have a land surface temperature map of that city. I then want to find the cooling intensity/performance of each green roof. Here, this is defined as the difference in temperature between the LST of the green roof site itself, and the LST at the distance where the cooling effect ends. This means I need to calculate the cooling extent of each green roof. To accomplish this, I draw buffer doughnut rings around the green space site, and take the average LST value of each buffer ring. I will expect the ring averages to gradually increase moving away from the green roof site, as the cooling effect fades away. However, once I get to a ring distance where the cooling actually drops, then I will mark that as the point where the cooling extent ends. And so then, to find the cooling intensity/performance of each green roof, I take the absolute value of the difference between the LST at the green roof itself and the average LST at the distance where the cooling effect ends. For example, if the green roof itself has an LST of 25C and the distance where the cooling effect is marked as ending is 31C, then I take the difference and say that green roof has a cooling intensity of 31-25 = 6C. Ok great, done.

This buffer ring method to determine cooling extents is proposed in this paper “Quantifying the local cooling effects of urban green spaces: Evidence from Bengaluru, India” by Shah, Garg, and Mishra (2021):

Article Quantifying the local cooling effects of urban green spaces:...

Now I want to find, how is the cooling intensity of a green roof affected by the height of the building? I am basing this on the assumption that the higher a green roof is off the ground, the more the cooling performance will decay. I want to find what this building height to cooling performance decay relationship is. From this paper “Modeling the outdoor cooling impact of highly radiative “super cool” materials applied on roofs” by Sinsel et al. (2021):

Article Modeling the outdoor cooling impact of highly radiative "sup...

I see that for every increase in 1-meter height, the cooling performance decreases by .003C (the paper says .003 K, but just converting to C since C and K have the same magnitude). I am trying to apply this decay rate to my example to see if I can understand how cooling intensity will decrease with building height.

And so for my example: Building A is 15 meters tall, Building B is 25 meters tall, and Building C is 42 meters tall. I want to account for how severely the height of each building will damped/lessen its cooling intensity/performance. And so, for the cooling intensities I would have calculated using the previously mentioned technique, this would assume that the green roof was 0 meters off the ground. And so to actually find how that cooling intensity would be lowered, I will apply the .003C/meter rate. Calculating cooling intensity for each green roof I get: Building A is 6C, Building B is 4C, and Building C is 3.4C. And so I calculate:

Building A: .003C x (15 meters) = .045 C, and so 6C – 0.045C = 5.955 C cooling intensity

Building B: .003C x (25 meters) = .075C, and so 4C - .075C = 3.925 C cooling intensity

Building C: .003C x (42 meters) = 0.126C, and so 3.4C - 0.126C = 3.274C cooling intensity

However, this would assume that the relationship is linear, while the paper says the relationship is nonlinear. Since this paper does not say what “nonlinear” means here, I turn to this paper “The impact of building height on urban thermal environment in summer: A case study of Chinese megacities” by Wang and Xu (2021):

Article The impact of building height on urban thermal environment i...

which says the relationship between building height and LST is “negative logarithmic”, and so I use the natural log now and calculate:

Building A: .003C x ln(15 meters) = 0.00812C, and so 6C – 0.00812C = 5.99188 C cooling intensity

Building B: .003C x ln(25 meters) = 0.00966C, and so 4C – 0.00966C = 3.99034C cooling intensity

Building C: .003C x ln(42 meters) = 0.01121C, and so 3.4C - 0.01121C = 3.38879C cooling intensity

But wait, that’s barely anything! It’s almost as if the height of the building has no effect on cooling intensity at all now!

Just to sanity check this, let’s imagine a building D. Treating this green roof as if it were 0 meters above ground, I calculate a cooling intensity of 5.6C. Now let’s say this building is really tall, 200 meters tall. Now at that height, I think it would be reasonable to say that the green roof at the top would likely have a very, very small effect on cooling ground surface temperature for nearby pedestrians, if any effect at all. So let’s try this out with the linear approach and the nonlinear approach:

Building D: .003C x (200 meters) =  0.6C, and so 5.6C – 0.6C = 5C cooling intensity

I was expecting there to be no cooling intensity as all, given how high above the ground the green roof is, but OK.

Building D: .003C x ln(200 meters) =  0.01589 C, and so 5.6C – 0.01589 C = 5.58411C cooling intensity

And so a green roof being 200 meters above the ground have just about the same cooling intensity/performance as if it were 0 meters above ground?? That doesn’t make sense! How can a green roof 200 meters in the sky provide the same cooling relief to nearby pedestrians as a green roof (or technically a green space I suppose if it were 0 meters off the ground) 0 meters off the ground?? It just doesn’t make sense to me! And so while I recognize this building height-cooling decay rate is justified and credited in the literature, I am just not understanding how these results seem reasonable. And so I would like to ask, am I approaching the use of the rate correctly or is my math totally wrong here? Also, I completely recognize understanding green roof cooling performance is far more complex than what I have here, accounting for variables such as evapotranspiration rates and albedo, however, I am trying to keep this very simple and high level for the moment, just so I can understand the role of building height here. I would really appreciate any guidance and feedback on my approach here! Sorry for the very long post, but I wanted to fully explain my example problem! Thank you!