Explanation:

Students don’t need to have this detail in their reports, but the reason this is happening is that the closer the basic environmental lapse rate gets to the dry adiabatic lapse rate, the less stable the atmosphere is and the easier it is for solar heating of the surface to set off convection that redistributes heat (basically bringing heat from the surface to the upper troposphere).

When the environmental lapse rate (ELR) is less than the moist adiabatic lapse rate, the surface temperature is close to \(T_{\text{surface}} = T_{\text{skin}} + h_{\text{skin}} \text{ELR}\), but when the ELR becomes greater than the moist adiabatic lapse rate, there is so much convection that it disrupts the simple picture I presented in class.

The picture I presented in class works well for an atmosphere like ours, which is marginally stable (i.e., the environmental lapse rate is roughly equal to the average moist adiabatic lapse rate). When the atmosphere is undergoing a large amount of constant convection, then heat flow behaves differently.

All of that convection transports lots of heat from the surface to the upper troposphere and brings lots of cold air down from the upper troposphere to the surface, which makes it hard for the surface to get warmer. This is why the ground temperature flattens out and stops changing very much after the lapse rate reaches about 6 K/km.

Preliminaries

Because I will be doing a lot of comparisons here, I want to automate some of the text using a compare function:

compare = function(x, y) {
   ifelse(x > y, "greater than", 
          ifelse(x < y, "less than", "the same as"))
}

# This function compares two values and prints the absolute value of 
# the difference followed by optional units (e.g., W/m^2^), and then the
# comparison string, so something like "3.2 W/m^2^ less than".
#
# If the two values are the same, we don't want to print 
# "0 W/m^2^ the same as", so we suppress printing the number and the units 
# in that case.
compare_val = function(x, y, digits, units = NULL) {
   delta = format_md(abs(x - y), digits = digits)
   comp_str = compare(x, y)
   ifelse(x == y, comp_str, str_c(delta, units, comp_str, sep = " "))
}

First, I recorded the albedo, surface temperature, and heat fluxes in and out of the earth under the default conditions of the earth’s average albedo.

default = run_rrtm()
albedo_default = default$albedo
T_surface_default = default$T_surface

I_in_default = default$i_in
I_out_default = default$i_out

flux_surf_default = default$fluxes %>% filter(altitude == 0)
flux_space_default = tail(default$fluxes, 1)

I_surf_long_default = flux_surf_default$lw_down
I_surf_short_default = flux_surf_default$sw_down
I_space_long_default = flux_space_default$lw_up
I_space_short_default = flux_space_default$sw_up

The albedo was 0.30, I_in = 244.8 W/m², and I_out = 244.8 W/m², so the heat budget is balanced.

Breaking down the heat flow into shortwave and longwave components, the heat absorbed by the ground consisted of 280.9 W/m² of longwave radiation and 266.3 W/m² of shortwwave radiation. The heat radiated out to space consisted of 244.8 W/m² of longwave radiation and 95.20 W/m² of shortwwave radiation.

2(a) Asphalt Surface

Next, I changed the surface type to “Asphalt” and recorded the changes in albedo, I_in, I_out, and heat balance.

asphalt = run_rrtm(surface_type = "Asphalt")
albedo_asphalt = asphalt$albedo

I_in_asphalt = asphalt$i_in
I_out_asphalt = asphalt$i_out

# This is the net heat flow into the earth.
# It should be zero when heat in = heat out
Q_asphalt = asphalt$Q

flux_surf_asphalt = asphalt$fluxes %>% filter(altitude == 0)
flux_space_asphalt = tail(asphalt$fluxes, 1)

I_surf_long_asphalt = flux_surf_asphalt$lw_down
I_surf_short_asphalt = flux_surf_asphalt$sw_down
I_space_long_asphalt = flux_space_asphalt$lw_up
I_space_short_asphalt = flux_space_asphalt$sw_up

Delta_I_in_asphalt = I_in_asphalt - I_in_default
Delta_I_out_asphalt = I_out_asphalt - I_out_default

Delta_I_surf_long_asphalt = I_surf_long_asphalt - I_surf_long_default
Delta_I_surf_short_asphalt = I_surf_short_asphalt - I_surf_short_default
Delta_I_space_long_asphalt = I_space_long_asphalt - I_space_long_default
Delta_I_space_short_asphalt = I_space_short_asphalt - I_space_short_default

The albedo for asphalt is 0.08, which is a good deal less than the default albedo. This caused I_in to increase from 244.8 W/m² to 296.8 W/m². I_out did not change and remained at 244.8 W/m².

Breaking down the heat flow into shortwave and longwave components, the heat absorbed by the ground consisted of 280.9 W/m² of longwave radiation, which was the same as in the default case, and 262.4 W/m² of shortwwave radiation, which was 3.9 W/m² less than in the default case.

The heat radiated out to space consisted of 244.8 W/m² of longwave radiation, which was the same as in the default case, and 43.2 W/m² of shortwwave radiation, which was 52.0 W/m² less than in the default case.

Then I opened the interactive RRTM model, set the surface type to “Asphalt,” and manually adjusted the surface temperature until the heat budget was balanced.

T_surface_asphalt = 325.6

# Check that the heat really does balance
asphalt_warming = run_rrtm(surface_type = "Asphalt", 
                           T_surface = T_surface_asphalt)
Q_asphalt_warmed = asphalt_warming$Q
Delta_T_asphalt = T_surface_asphalt - T_surface_default

The change in I_in caused a heat-budget imbalance of 52.0 W/m² going into the Earth. To put the heat budget back into balance, I had to raise the surface temperature from 284.4 K to 325.6 K, a change of 41.2 K.

2(b) Concrete Surface

Next, I did the same for a concrete surface:

concrete = run_rrtm(surface_type = "Concrete")
albedo_concrete = concrete$albedo

I_in_concrete = concrete$i_in
I_out_concrete = concrete$i_out

Q_concrete = concrete$Q

flux_surf_concrete = concrete$fluxes %>% filter(altitude == 0)
flux_space_concrete = tail(concrete$fluxes, 1)

I_surf_long_concrete = flux_surf_concrete$lw_down
I_surf_short_concrete = flux_surf_concrete$sw_down
I_space_long_concrete = flux_space_concrete$lw_up
I_space_short_concrete = flux_space_concrete$sw_up

Delta_I_in_concrete = I_in_concrete - I_in_default
Delta_I_out_concrete = I_out_concrete - I_out_default

Delta_I_surf_long_concrete = I_surf_long_concrete - I_surf_long_default
Delta_I_surf_short_concrete = I_surf_short_concrete - I_surf_short_default
Delta_I_space_long_concrete = I_space_long_concrete - I_space_long_default
Delta_I_space_short_concrete = I_space_short_concrete - I_space_short_default

Where asphalt had a much higher albedo albedo for concrete is 0.55 and this caused I_in to decrease to 183.8 W/m². I_out did not change and remained at 244.8 W/m².

Breaking down the heat flow into shortwave and longwave components, the heat absorbed by the ground consisted of 280.9 W/m² of longwave radiation, which was the same as in the default case, and 271.2 W/m² of shortwwave radiation, which was 4.9 W/m² greater than in the default case.

The heat radiated out to space consisted of 244.8 W/m² of longwave radiation, which was the same as in the default case, and 156.2 W/m² of shortwwave radiation, which was 61.0 W/m² greater than in the default case.

Then I manually adjusted the surface temperature in the interactive RRTM model until the heat out balanced the heat in.

T_surface_concrete = 247.0

# Check that the heat really does balance
concrete_warming = run_rrtm(surface_type = "Concrete", 
                           T_surface = T_surface_concrete)
Q_concrete_warmed = concrete_warming$Q
Delta_T_concrete = T_surface_concrete - T_surface_default

The change in I_in caused a heat-budget imbalance of -61.0 W/m² going out of the Earth. To put the heat budget back into balance, I had to lower the surface temperature to 247.0 K, a change of -37.4 K.

2(c) Discussion

Covering the earth in asphalt caused the planet to warm up by 41.2 K, so we could expect that covering a large part of a city in asphalt or similarly dark-colored material would cause the city’s temperature to rise. Conversely, covering the planet in concrete caused it to cool off by 37.4 K, so replacing asphalt with concrete in a city would cool the city off a good deal. This would provide cooler temperature for the people living in the city and would reduce the impact of global climate change.

3(a) Low Clouds

First, I changed the low-cloud fraction to 0.70.

low_clouds = run_rrtm(low_cloud_frac = 0.70)

I_in_low = low_clouds$i_in
I_out_low = low_clouds$i_out

flux_surf_low = low_clouds$fluxes %>% filter(altitude == 0)
flux_space_low = tail(low_clouds$fluxes, 1)

I_surf_long_low = flux_surf_low$lw_down
I_surf_short_low = flux_surf_low$sw_down
I_space_long_low = flux_space_low$lw_up
I_space_short_low = flux_space_low$sw_up

Delta_I_in_low = I_in_low - I_in_default
Delta_I_out_low = I_out_low - I_out_default

Delta_I_surf_long_low = I_surf_long_low - I_surf_long_default
Delta_I_surf_short_low = I_surf_short_low - I_surf_short_default
Delta_I_space_long_low = I_space_long_low - I_space_long_default
Delta_I_space_short_low = I_space_short_low - I_space_short_default

Q_low = low_clouds$Q

Adding low clouds caused I_in to decrease from 244.8 W/m² to 233.1 W/m², a change of -11.7 W/m². I_out increased from 244.8 W/m² to 240.2 W/m², a change of -4.6 W/m².

Breaking down the heat flow into shortwave and longwave components, the heat absorbed by the ground consisted of 320.8 W/m² of longwave radiation, which was 39.90 W/m² greater than in the default case, and 244.7 W/m² of shortwwave radiation, which was 21.60 W/m² less than in the default case.

The heat radiated out to space consisted of 240.2 W/m² of longwave radiation, which was 4.600 W/m² less than in the default case, and 106.9 W/m² of shortwwave radiation, which was 11.70 W/m² greater than in the default case.

Nwxt, I opened the interactive RRTM model, set the surface type to “low,” and manually adjusted the surface temperature until the heat budget was balanced.

T_surface_low = 280.45

# Check that the new temperature really does balance the heat flow.
low_warming = run_rrtm(low_cloud_frac = 0.70, T_surface = T_surface_low)
Q_low_warmed = low_warming$Q
Delta_T_low = T_surface_low - T_surface_default

The change in I_in caused a heat-budget imbalance of -7.1 W/m² going out of the Earth. To put the heat budget back into balance, I had to lower the surface temperature from 284.4 K to 280.4 K, a change of -4.0 K.

3(b) High Clouds

Next, I changed the high-cloud fraction to 0.70.

high_clouds = run_rrtm(high_cloud_frac = 0.20)
albedo_high = high_clouds$albedo

I_in_high = high_clouds$i_in
I_out_high = high_clouds$i_out

flux_surf_high = high_clouds$fluxes %>% filter(altitude == 0)
flux_space_high = tail(high_clouds$fluxes, 1)

I_surf_long_high = flux_surf_high$lw_down
I_surf_short_high = flux_surf_high$sw_down
I_space_long_high = flux_space_high$lw_up
I_space_short_high = flux_space_high$sw_up

Delta_I_in_high = I_in_high - I_in_default
Delta_I_out_high = I_out_high - I_out_default

Delta_I_surf_long_high = I_surf_long_high - I_surf_long_default
Delta_I_surf_short_high = I_surf_short_high - I_surf_short_default
Delta_I_space_long_high = I_space_long_high - I_space_long_default
Delta_I_space_short_high = I_space_short_high - I_space_short_default

Q_high = high_clouds$Q

Adding high clouds caused I_in to decrease from 244.8 W/m² to 234.7 W/m², a change of -10.1 W/m². I_out increased from 244.8 W/m² to 212.0 W/m², a change of -32.8 W/m².

Breaking down the heat flow into shortwave and longwave components, the heat absorbed by the ground consisted of 283.9 W/m² of longwave radiation, which was 3.000 W/m² greater than in the default case, and 255.4 W/m² of shortwwave radiation, which was 10.90 W/m² less than in the default case.

The heat radiated out to space consisted of 212.0 W/m² of longwave radiation, which was 32.80 W/m² less than in the default case, and 105.3 W/m² of shortwwave radiation, which was 10.10 W/m² greater than in the default case.

Next, I opened the interactive RRTM model, set the surface type to “high,” and manually adjusted the surface temperature until the heat budget was balanced.

T_surface_high = 298.05

# Check that the new temperature really does balance the heat flow.
high_warming = run_rrtm(high_cloud_frac = 0.20, T_surface = T_surface_high)
Q_high_warmed = high_warming$Q
Delta_T_high = T_surface_high - T_surface_default

The change in I_in caused a heat-budget imbalance of 22.7 W/m² going into the Earth. To put the heat budget back into balance, I had to raise the surface temperature from 284.4 K to 298.0 K, a change of 13.6 K.

3(c) Heat Flow

In the figure below, we can see how the upward and downward heat flows change with altitude for the case of low clouds.

plot_heat_flows(low_clouds)

Figure 2: Upward and downward fluxes of longwave and shortwave radiation with low clouds.

We can see that for the low clouds, the biggest changes are in the downward flow of radiation. Downward longwave radiation is a lot greater below the clouds and downward shortwave is smaller. Above the clouds, there is a small increase in upward longwave and shortwave, but the changes are much smaller than for the downward radiation.

plot_heat_flows(high_clouds)

Figure 3: Upward and downward fluxes of longwave and shortwave radiation with high clouds.

At the height of the clouds we can see big changes in the upward radiation: Outgoing longwave radiation is considerably smaller above the clouds and outgoing shortwave is larger. There are changes to the downward radiation at the height of the clouds, but these changes are much smaller than for the upward radiation.

As these figures show, the low clouds had the greatest effect on downward radiation and the high clouds had the greatest effect on upward radiation. In both cases, the clouds affected the longwave radiation much more than the shortwave.