By Philip Mulholland

1. Introduction.

The following image shows the Earth’s outgoing longwave radiation recorded by the CERES (**C**louds and the **E**arth’s **R**adiant **E**nergy **S**ystem) Instrument onboard the NASA Aqua Satellite (Damadeo and Hanson, 2017). This image is compiled from measurements made on March 18 2011, near the time of the Vernal Equinox.

Image credit: NASA

The colour table legend records the energy flux of the outgoing radiation. This flux ranges from a minimum value of 150 W/m^{2} displayed as white, to a maximum flux of 350 W/m^{2}, displayed as yellow. Using the Stefan-Boltzmann law of radiative emission these energy flux values can be converted to emission temperatures using the following equation: –

Equation 1:** T** = (** j***/

**σ**

**)**

^{0.25}Where **T** is the thermodynamic temperature in Kelvin.

*j****** is the black body radiant emittance in Watts per square metre.

**σ** is the Stefan-Boltzmann constant of proportionality.

(Sigma has a value of 5.670373 * 10^{-8} W m^{-2} K^{-4})

Using equation 1 we can determine that the emission temperatures recorded by the Ceres instrument range from a minimum value of 226.8 Kelvin (-46.2^{o}C) for the 150 W/m^{2} low-end flux, to a maximum value of 280.3 Kelvin (7.3^{o}C) for the 350 W/m^{2} high-end flux.

2. Calibrating the CERES image

The CERES image is a single snapshot of the Earth’s thermal radiant emission to space. This image contains a significant amount of information, however to understand this in its global context we must first calibrate the image against known measurements of the major components of the Earth’s atmospheric system.

The Earth’s atmosphere is a dynamic system composed of three separate types of interlocking cells, symmetrically distributed in each hemisphere. These cells consist of two thermal cells and one mechanical cell, they are: –

1. The tropical thermal Hadley cell located between the equator and latitude 30 degrees.

2. The temperate mechanical Ferrel cell located between latitude 30 degrees and latitude 66.56 degrees.

3. The frigid thermal Polar cell located around each pole and defined by the Arctic and Antarctic circles.

The global areal distribution of each cell is as follows: –

1. The Hadley cells occupy 50% of the surface area of the globe, and in total intercept 60.9% of the sun’s insolation.

2. The Ferrel cells occupy 41.75% of the surface area of the globe, and in total intercept 36.29% of the sun’s insolation.

3. The Polar cells occupy 8.25% of the surface area of the globe, and in total intercept only 2.81% of the sun’s insolation.

The high concentration of insolation intercepted by the tropical Hadley cells, compared to the low insolation intercepted by the Polar cells, is the fundamental reason for the low surface temperatures found in the polar regions of our planet.

Visual inspection of the CERES image shows the presence of cloud tops associated with the convective storms of the equatorial intertropical convergence zone (ITCZ) or doldrums. These storms are radiating at 150 W/m^{2} and have an emission temperature of 227 Kelvin (-46.2^{o}C). In order to determine the elevation of this emission, we need to establish three atmospheric parameters for the Hadley, Ferrel and Polar cells, which are: –

1. The height of the tropopause.

2. The temperature of the tropopause.

3. The environmental lapse rate of the atmospheric cell.

Using these three metrics we can then calculate the temperature elevation profile that relates to the given emission rate for each of the three atmospheric circulation cells. Published information for the temperature of the tropopause is not easy to establish, however using various sources the values used in this analysis were obtained and are recorded in Table 1.

Cells | Hadley | Ferrel | Polar |

Tropopause Height (km) | 17 | 13 | 9 |

Tropopause Temperature (Celsius) | -83 | -78 | -78.5 |

Environmental Lapse Rate (K/km) | -6.5 | -6.5 | -6.5 |

Information Source | Environmental Lapse Rate |

Table 1: Atmospheric Cell Parameters.

Using the values established in Table 1 we can now determine the top down temperature profile for each of the three atmospheric cells. The calculations for the Hadley cell show that to maintain a 17 km tropopause with a temperature of 190 Kelvin (-83^{o}C) and a lapse rate of -6.5 K/km, then the average surface temperature of the tropical zone must be 301 Kelvin (27.9^{o}C).

Table 2: Hadley Cell – CERES Image Emissions Calibration Table.

Converting this average surface temperature of ~28^{o}C into a radiant energy emission flux, by using the Stefan-Boltzmann equation, we can establish that the tropical surface energy flux is 465 W/m^{2}. This value is 115 W/m^{2} higher than the maximum observed flux of 350 W/m^{2} in the Ceres image, and so we have established that this image does not record direct sea level surface radiant emission. Rather, with this image we are observing the atmospheric temperatures at elevations of 3,160 m (10,370 ft) and above. Consequently, all high elevation land surfaces in the latitude zone of 30^{o}S to 30^{o}N, such as the Tibetan plateau at 4,500m (14,750 ft), will be capable of directly emitting thermal radiant energy to space through the overlying atmosphere.

The calculations for the Ferrel cell show that to maintain a 13 km tropopause with a temperature of 195 Kelvin (-78^{o}C) and a lapse rate of -6.5 K/km, then the average annual surface temperature of the temperate zone will be 280 Kelvin (6.5^{o}C).

Table 3: Ferrel Cell – CERES Image Emissions Calibration Table.

Converting this average surface temperature of 6.5^{o}C into a radiant energy emission flux, by using the Stefan-Boltzmann equation, we can now establish that the temperate zone surface energy flux is 346 W/m^{2}. This value is 46 W/m^{2} higher than the maximum observed flux in the Ceres image of 300 W/m^{2} for the temperate zone as seen from space. Once again, although this image does not record direct sea level surface radiant emission, all land surfaces with an elevation above 1,500 m (4,920 ft) will be capable of directly emitting thermal radiant energy to space through the overlying atmosphere.

The calculations for the Polar cell show that to maintain an 9 km tropopause with a temperature of 194.5 Kelvin (-78.5^{o}C) and a lapse rate of -6.5 K/km, then the average annual surface temperature of the polar zone will be 253 Kelvin (-20^{o}C).

Table 4: Polar Cell – CERES Image Emissions Calibration Table.

Converting this average surface temperature of -20^{o}C into a radiant energy emission flux, by using the Stefan-Boltzmann equation, we can now establish that the polar zone surface energy flux is 232 W/m^{2}. This value is just 7 W/m^{2} higher than the maximum observed flux in the Ceres image of 225 W/m^{2} for the region of the Southern Ocean, south of the Antarctic circle. This calculation demonstrates that all parts of the polar regions above 310 m (1,020 ft) elevation, and in particular the high elevation ice domes, will be capable of directly emitting thermal radiant energy to space through the overlying atmosphere.

3. Conclusion

Using the average surface temperature values calculated for each cell (Tables 2, 3 & 4), combined with the percentage of the global areal distribution of each cell (Table 5)

Table 5: Calculating the Global Areal Distribution of the Atmospheric Cells.

we can now compute the average annual surface temperature of the Earth (Table 6).

Table 6: Calculating the Global Average Temperature of the Earth.

The average temperature of the Earth determined by this calculation method is 288 Kelvin, which is the currently accepted value of 15^{o}C used by climate science.

4. Discussion

The process of establishing the average temperature of the surface of the Earth presented here, relies on the following atmospheric measurements and fundamental planetary parameters.

These are: –

1. The height of the tropopause for each atmospheric cell.

2. The temperature of its tropopause.

3. The environmental lapse rate of each atmospheric cell.

4. The relative proportion of the Earth’s surface occupied by each cell.

With these measurements established, the global average temperature of the Earth is simply the arithmetical sum of the relative proportion surface temperatures for the three atmospheric cells.

Perhaps the most interesting part of this analysis is the apparent coincidence between the maximum surface elevation of the Antarctic Icecap at Dome A (4093m) and the maximum elevation of supercooled water (Moore and Molinero, 2011) in the atmospheric profile of the Polar Cell (Rubin, 1953). It would appear that the vertical elevation of continental ice caps is limited by atmospheric processes, however it is equally clear that no such vertical constraint occurs with solid rock land surface elevation. Because mountain ranges can reach vertical elevations that lie within the radiant transmission zone to space for each atmospheric cell, it appears that these topographic features can form leak zones that emit radiant energy to space independently of the transmission properties of the overlying atmosphere.

References

Beal, A. 2011. The Surface Area of a Sphere Between Parallel Planes. Online Blog

Damadeo, K. and Hanson, H. 2017. CERES Clouds and the Earth’s Radiant Energy System. NASA 9pp.

Moore, E.B. and Molinero, V. 2011. Structural transformation in supercooled water controls the crystallization rate of ice. Nature, 479, 506-508.

Rubin, M.J., 1953. Seasonal variations of the Antarctic tropopause. Journal of Meteorology, 10(2), pp.127-134.

Excel spread sheet containing the source tables for the essay.

via Watts Up With That?

May 19, 2019 at 10:19AM