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In this paper we use the inverse modelling technique, first applied to the atmosphere of the planet Venus, to demonstrate that the process of convective atmospheric mass motion can be invoked to explain the greenhouse effect of the Earth's climate. We propose that the atmospheric cell is the fundamental element of climate, and have developed an alternative climate model based on this process of atmospheric circulation for a hypothetical tidally locked world. The concept of climate derives from studies by the Greek philosopher Aristotle, who identified the three main climatic zones known to the ancient world; the equatorial torrid zone, the polar frigid zone and in between the favoured temperate zone of the Mediterranean world. Aristotle's three climatic zones can be directly linked to the three main atmospheric circulation cells that we now recognise within the Earth's atmosphere. These three cells are the Hadley cell, the Polar cell and the Ferrel cell. Based on the clear association between the traditional Greek concept of climate and the modern meteorological concept of atmospheric circulation cells, we propose that climate be defined as the presence and action of a particular circulation cell type within a given planetary latitudinal zone. We discuss how with knowledge of three simple meteorological parameters of tropopause elevation, tropopause temperature and lapse rate for each atmospheric cell, combined with the measurement of the area of that cell, the average global surface temperature can be calculated. By means of a mathematical model, the Dynamic-Atmosphere Energy-Transport (DAET) climate model we apply an individual climate analysis to each of the three atmospheric cells, and next generate a parallel composite model of the Earth's planetary climate using these data. We apply the concepts and techniques of the adiabatic version of the DAET climate model, and show how this model can be compared with the published NASA image of the Earth's outgoing long-wave radiation recorded by the CERES (Clouds and the Earth's Radiant Energy System) Instrument onboard the NASA Aqua Satellite. Our analysis of the CERES image suggests that the Tibetan plateau forms a permanent geological thermal radiant leak point in the Earth's atmosphere. We also compare the observed temperature found at the maximum elevation of the Antarctic ice cap with the freezing point of super-cooled water, and suggest that there is therefore a temperature controlled and latent heat related upper limit to the vertical development of a continental icecap.
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International Journal of Atmospheric and Oceanic Sciences
2020; 4(2): 36-53
http://www.sciencepublishinggroup.com/j/ijaos
doi: 10.11648/j.ijaos.20200402.11
ISSN: 2640-1142 (Print); ISSN: 2640-1150 (Online)
Return to Earth: A New Mathematical Model of the Earth’s
Climate
Stephen Paul Rathbone Wilde, Philip Mulholland*
Mulholland Geoscience, Weybridge, Surrey, UK
Email address:
*Corresponding author
To cite this article:
Stephen Paul Rathbone Wilde, Philip Mulholland. Return to Earth: A New Mathematical Model of the Earth’s Climate. International Journal
of Atmospheric and Oceanic Sciences. Vol. 4, No. 2, 2020, pp. 36-53. doi: 10.11648/j.ijaos.20200402.11
Received: March 21, 2020; Accepted: May 18, 2020; Published: June 8, 2020
Abstract: In this paper we use the inverse modelling technique, first applied to the atmosphere of the planet Venus, to
demonstrate that the process of convective atmospheric mass motion can be invoked to explain the greenhouse effect of the
Earth’s climate. We propose that the atmospheric cell is the fundamental element of climate, and have developed an alternative
climate model based on this process of atmospheric circulation for a hypothetical tidally locked world. The concept of climate
derives from studies by the Greek philosopher Aristotle, who identified the three main climatic zones known to the ancient
world; the equatorial torrid zone, the polar frigid zone and in between the favoured temperate zone of the Mediterranean world.
Aristotle’s three climatic zones can be directly linked to the three main atmospheric circulation cells that we now recognise
within the Earth’s atmosphere. These three cells are the Hadley cell, the Polar cell and the Ferrel cell. Based on the clear
association between the traditional Greek concept of climate and the modern meteorological concept of atmospheric circulation
cells, we propose that climate be defined as the presence and action of a particular circulation cell type within a given planetary
latitudinal zone. We discuss how with knowledge of three simple meteorological parameters of tropopause elevation,
tropopause temperature and lapse rate for each atmospheric cell, combined with the measurement of the area of that cell, the
average global surface temperature can be calculated. By means of a mathematical model, the Dynamic-Atmosphere Energy-
Transport (DAET) climate model we apply an individual climate analysis to each of the three atmospheric cells, and next
generate a parallel composite model of the Earth’s planetary climate using these data. We apply the concepts and techniques of
the adiabatic version of the DAET climate model, and show how this model can be compared with the published NASA image
of the Earth’s outgoing long-wave radiation recorded by the CERES (Clouds and the Earth’s Radiant Energy System)
Instrument onboard the NASA Aqua Satellite. Our analysis of the CERES image suggests that the Tibetan plateau forms a
permanent geological thermal radiant leak point in the Earth’s atmosphere. We also compare the observed temperature found at
the maximum elevation of the Antarctic ice cap with the freezing point of super-cooled water, and suggest that there is
therefore a temperature controlled and latent heat related upper limit to the vertical development of a continental icecap.
Keywords: Atmospheric Cell, Climatology, CERES, Climate Model
1. Introduction
The history of Climatology is long and honourable, indeed
the very concept of climate goes back to the Greek philosopher
Aristotle, who identified the three main types of climatic zone
known to the ancient world [1]. These zones are:
A. The Torrid Zone - located to the south of Greece in Africa.
B. The Frigid Zone – located to the far north of Greece
where Boreas, the god of the north wind and winter lived.
C. The Temperate Zone of Europe, where the four annual
seasons occur, and Greece is most favourably located.
Aristotle’s three climatic zones can be directly linked to the
three main atmospheric circulation cells that we now recognise
within the Earth’s atmosphere. These three cells are:
A. The Hadley cell [2], which is a thermal cell, driven by
solar radiation from space heating the planet’s surface.
Two zones of Hadley cells exist in our atmosphere,
these are both found in the Tropics and are generally
located between the Equator and the Tropic of Cancer in
the northern hemisphere and the Tropic of Capricorn in
37 Stephen Paul Rathbone Wilde and Philip Mulholland: Return to Earth: A New Mathematical
Model of the Earth’s Climate
the south. The Hadley cell’s poleward limit is located in
the Horse Latitudes; where a zone of descending air
exists forming surface high pressure anticyclones. It is
the Hadley cell that is the defining atmospheric feature
of Aristotle’s Torrid Zone.
B. The Polar cell [3], which is also a thermal cell, but it is
driven by atmospheric circulation caused by radiation
cooling from the ground surface directly to space. This
radiative cooling produces an atmospheric surface
inversion, that is most noticeable in winter. The Polar
cell’s equatorward limit is marked by the Polar Front,
an oscillating band with an associated strong horizontal
surface temperature gradient; above which is found the
jet stream of the upper troposphere. The Polar cell is
responsible for the formation and surface export
towards the equator of cold dense airmasses. It is the
Polar cell that is the defining atmospheric feature of
Aristotle’s Frigid Zone.
C. The Ferrel cell [4], which is a mechanical cell, located
between the Hadley and Polar cells. It acts as a buffer or
cog between the latitudinal limits of the two thermal
cells, and has a circulation pattern that abuts and links
these two opposing cells. The Ferrel cell forms a zone
of mixing and ascending air that is associated with cold
cored cyclones. It is the Northern Hemisphere Ferrel
cell that accounts for Aristotle’s Temperate Zone, with
its annual seasonal changes and varied weather patterns.
In this paper we adopt the climate modelling techniques first
applied to the planet Venus [5] where we developed an
alternative climate model, using the process of atmospheric
circulation on a hypothetical tidally locked planet. We use this
model to demonstrate that convective atmospheric mass
motion recycling can be invoked to explain the planetary
greenhouse effect. We use here the modelling strategy of
Noonworld and by creating a three-element parallel model
constrained to atmospheric data, apply this concept of
convective atmospheric mass-motion energy recycling to study
the climate of the Earth.
In building a three-element parallel model the primary
distinction being studied is between slowly rotating Venus,
with its hemisphere encompassing pair of Hadley cells, and
rapidly rotating Earth, with its triple cell per hemisphere
configuration of Hadley, Polar and Ferrel cells [2-4].
The use of a parallel configuration for the model design
addresses the issue that the primary motion of the Earth’s
atmosphere is predominantly zonal and not meridional flow,
and that tropospheric air motion is constrained by the
Coriolis effect [6]. This design also addresses a feature of
tropical climate best summed up as “Nighttime is the winter
of the tropics.”
The issue of planetary axial tilt, leading to polar winters
dominated by little or no solar flux and summers with no
nighttime, was addressed by splitting the polar cell sub-unit
in the climate model into two distinctly separate modes of
operation. For the summer all convective activity is assumed
to recycle the descending air directly back onto a lit arctic
surface. By this means the low intensity solar flux inherent at
high latitudes is amplified by a process of atmospheric short-
circuiting during the arctic summer. By contrast during the
arctic winter the polar sub-cell unit in the model is assumed
to have zero solar influx, and all of the power intensity used
to drive the atmospheric circulation arrives by advection
from the adjacent Ferrel cell.
Finally, we have adopted in the model a nuanced approach
to the issue of lapse rate, using a wet adiabatic lapse rate for
the Hadley cell, an intermediate environmental lapse rate for
the Ferrel cell, and a dry adiabatic lapse rate for the Polar
cell. This approach is in contrast to previous work into the
study of terrestrial radiation, where a uniform planetary lapse
rate of 6°C/km was used across all atmospheric cells [7].
2. Methodology
Our modern understanding of the dynamics of the Earth’s
planetary climate, and the role that radiant energy has in
defining the features of the atmospheric circulation system, has
progressed with the formulation of the energy budget diagram
used to quantify and rank the importance of the constituent
elements of the climate system [8]. A key metric of the energy
budget is the standard Vacuum Planet or rapidly rotating airless
planet equation [9]. This equation is used in Climate Science to
calculate the expected thermal emission temperature Te of an
illuminated globe under the average solar irradiance that
pertains for a planet’s average orbital distance from the sun,
and for that planet’s specific Bond albedo.
“The equilibrium temperature Te of an airless, rapidly
rotating planet is:
Equation 1: Te ≡ [S π R2 (1-A)/4 π R2ε σ]1/4
where σ is the Stefan-Boltzmann Constant, ε the effective
surface emissivity, A the wavelength-integrated Bond albedo,
R the planet's radius (in metres), and S the solar constant (in
Watts/m2) at the planet's average distance from the sun.”
The results of applying this Vacuum Planet equation to the
Earth are shown in Table 1:
Table 1. The Expected Surface Temperature for an Airless Earth compared
with its actual Atmospheric Temperature.
Parameter Symbol
Earth Units Dimensions
Solar Constant at distance a
S 1361 W/m2 MT-3
Radius of Body R 6,367,445 m L
Bond Albedo A 0.306 Constant
Constant A
Stefan-Boltzmann Constant
σ 5.67E-08 W/m2/K4
MT-3K-4
Effective surface emissivity
ε 1 Constant
Constant ε
Expected Te T
e
254 Kelvin K
Greenhouse Effect GE 34 Kelvin K
Actual Ts T
s
288 Kelvin K
Distance from the Sun a 1.4960E+11
m L
2.1. Applying the Dynamic-Atmosphere Energy-Transport
(DAET) Model to the Study of Earth’s Climate
We are attempting here to simultaneously analyse the
energy flows for the Earth’s three atmospheric circulation
cells, using the adiabatic form of the Dynamic-Atmosphere
Energy-Transport (DAET) model, previously used for the
study of the climates of Venus and Titan [5, 10]. The Earth is
International Journal of Atmospheric and Oceanic Sciences 2020; 4(2): 36-53 38
modelled as a spherical globe that cuts a circular silhouette,
or disk shadow from the beam of the solar irradiance at the
planet’s average orbital distance from the Sun. The average
fraction of the illumination beam, that the silhouette for each
of the three circulation cells intercepts during the course of a
year, is latitude dependent.
For the purpose of this analysis it is assumed that the
latitudinal reach of the Hadley cell for each hemisphere is
from the equator to latitude 30°. The Ferrel cell extends from
latitude 30° to the (ant) arctic circle and the Polar cell
occupies the remaining latitudes around the pole of rotation.
The tropical Hadley cell of net energy surplus intercepts
60.90% of the illumination, the temperate mechanical Ferrel
cell of energy transport intercepts 36.29%, and the frigid
thermal Polar cell of net energy deficit intercepts the
remaining 2.81% of the Sun’s energy cut out by the disk
silhouette (Figure 1).
Figure 1. Earth’s Planetary Disk Silhouette for the average annual axial tilt of the globe.
The Earth has the form of a globe, and because of this
spherical shape the surface of the Earth is unevenly
illuminated. The location of the maximum possible power
intensity at the surface of the Earth occurs at the solar zenith,
the unique point on the Earth’s surface where the Sun is
directly overhead. At all other locations the slant of the Earth’s
surface to the sun’s beam of light lowers the interception
intensity. This is a feature of illumination that we observe at
both dawn and dusk when our shadows reach their maximum
length as the sunlight grazes the surface of the Earth.
The average power intensity at the Earth’s surface is
different for each of the three atmospheric cells. The tropical
Hadley cells, which occupy 50% of the surface of the Earth,
intercept 60.90% of the beam silhouette, and therefore receive
the highest radiant beam intensity. The Ferrel cells, which
occupy 36.29% of the surface of the Earth, intercept 41.75% of
the beam silhouette and therefore receive a lower radiant beam
intensity; while the Polar cells, which occupy 8.25% of the
surface of the Earth, intercept only 2.81% of the disk
silhouette, and therefore receive the lowest average radiant
beam power intensity. This quantity of radiant power intensity
is defined by the illumination power intensity dilution divisor
or “divide by rule” that is specific for each of the three
atmospheric cells. This metric is used to constrain the value of
the insolation flux used in the modelling process (Figure 2).
Figure 2. Globular Earth’s Lit Hemisphere Illumination Intensity for the average annual surface attitude of the planet [11].
39 Stephen Paul Rathbone Wilde and Philip Mulholland: Return to Earth: A New Mathematical
Model of the Earth’s Climate
The power intensity of the Earth’s average annual
irradiance is 1361 W/m2 [12]. This flux which arrives at the
Top of the Atmosphere (TOA) is then reduced by the Earth’s
planetary Bond albedo of 0.306 that acts as a bypass filter
diverting radiant solar energy back out to space [12]. It is
only the remaining 69.4% of the radiant flux which is
absorbed by the planet, and consequently the value of the
solar irradiance that drives the Earth’s climate is reduced to a
power intensity of 944.53 W/m2 (Figure 3).
It is fundamental to what comes next that the following
aspect of power intensity distribution within the Earth’s
climate system is appreciated in its full subtlety and
implications. In figure 1 we observed that the planet
intercepts sunlight as if it is a planar disk. However, because
of the attitude (slant) of the surface of a sphere with respect
to the parallel rays within the solar beam, the strength of the
beam striking the surface decreases from its maximum
possible post-albedo value of 944.53 W/m2 at the solar
zenith, down to a value of zero at the terminator, the great
circle line of dawn and dusk (Figure 3).
Figure 3. The Scaled Model Globular Earth’s Lit Hemisphere Illumination Interception Geometry.
Perhaps the most fundamental issue at the heart of climate
modelling is the use of the power intensity illumination
divisor of integer 4, that is present in the vacuum planet
equation (Equation 1). In this special case divisor 4 is used to
dilute the insolation to one quarter of the radiant beam
intensity. The original, valid and only purpose of the vacuum
planet equation is to establish the radiant exhaust temperature
of an illuminated planetary body. Planetary bodies of
whatever shape or form are only ever illuminated over the
surface of a single hemisphere. So, the appropriate divisor
required to calculate the average insolation power intensity
for the climate system, supplied by the fully lit face of a
planet, is integer 2.
Furthermore, there will exist on every lit planet a unique
location, the solar zenith, at which the radiant power intensity
at the base of the atmosphere is exactly equal to the value of
the solar irradiance at that planet’s orbital distance from the
sun (Figure 3). It should again be appreciated that the power
intensity illuminating the planets atmosphere at the solar
zenith is scaled down by the Bond albedo which acts as a
bypass filter. The albedo filter acts by removing insolation
from the climate system, and returning this discarded portion
of the high-frequency radiant flux directly back to space.
2.2. Calculating the Average Surface Temperature of the
Earth
The Earth’s atmosphere is a dynamic system composed of
three zonally separated and interlocking cells that are
symmetrically distributed in each hemisphere (Figure 1). The
global areal distribution of each atmospheric cell is shown in
Figure 2. The higher percentage of insolation intercepted by
the tropical Hadley cells, compared to the low value
intercepted by the Polar cells, means that less surface power
intensity is delivered to the boreal regions of the planet
(Figure 3). The weakened strength of the sunlight in the
arctic, caused by the sun’s lower angle of elevation in the
sky, is a fundamental reason for the lower average surface
temperatures found at the poles.
For each atmospheric cell we can compute the average
surface temperature if we know the average annual values of
the following meteorological parameters:
1. The average height of the tropopause.
2. The average temperature of the tropopause.
3. The average environmental lapse rate of each cell.
Using these three metrics we can then calculate the
average surface temperature of each atmospheric circulation
cell (Table 2).
Table 2. Atmospheric Cell Parameters.
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
Average Annual Surface Temperature (Celsius) 27.9 6.5 -20.0
International Journal of Atmospheric and Oceanic Sciences 2020; 4(2): 36-53 40
From our knowledge of the average surface temperature
for each atmospheric cell (Table 2) combined with our
calculation of the global areal footprint for each cell (Figure
2) we can now calculate the global average surface
temperature of the Earth. The key parameters are the global
extent of each of the three meteorological cells of Hadley,
Ferrel and Polar, and their respective average annual
temperatures. By combining these three temperature values
and using an area weighted average, the average annual
temperature of the whole planet can be derived (Table 3).
Table 3. Calculating the Global Average Temperature of the Earth.
Cell Type Hadley
Ferrel Polar
Percentage of Global Area 50.00%
41.75%
8.25%
Cell Average Temperature (Celsius) 27.9 6.5 -20.0
Area Weighted Proportion AW% (Celsius) 13.94 2.71 -1.65
Average Annual Global Temperature
(Celsius) AW% of Hadley+AW% of
Ferrel+AW% of Polar
15.0
2.3. Applying the Dynamic-Atmosphere Energy-Transport
(DAET) Climate Model to the Earth
The Dynamic-Atmosphere Energy-Transport (DAET)
climate model contains a mechanism for energy flux
recycling using the meteorological process of atmospheric
circulation. This model was used to demonstrate that
convective atmospheric mass motion can be invoked to
explain the planetary greenhouse effect for both Venus and
also Titan, the tidally locked moon of Saturn [5, 10].
Atmospheric data for these two bodies show that there is
little or no thermal contrast between the lit daytime and the
dark nighttime hemispheres on these slowly rotating worlds.
Our studies indicate that when applied to slowly rotating
Venus, or the moon Titan, the adiabatic model required only a
single energy flux partition ratio, common to both the lit and
dark sides of each globe, to achieve an appropriate thermal
boost within these atmospheres.
However, when a single common energy partition ratio
was applied to the process of creating an adiabatic model for
the Hadley cell on rapidly rotating Earth, the model failed
and created an atmosphere in which the tropopause is higher
on the unlit dark side of the globe (Table 6, Attempt 0).
Clearly this result is in direct contrast to observed
atmospheric data, where we find that the convective process
on the lit hemisphere produces a tropopause with a higher
elevation during the hours of daylight compared to the
nighttime. The solution to this failure of the analysis is to
apply two distinct and separate energy partition ratios during
the process of inverse modelling, one for each side. On the lit
side of energy collection, the partition ratio should be biased
in favour of the air. However, on the dark side of energy loss,
the partition ratio should be biased in favour of the radiant
surface of energy loss to space.
The justification for using two distinct energy partition
ratios, for the atmospheric circulation cells on rapidly
rotating Earth, is based on observation and deduction. The
primary observation is that for the Earth atmospheric
convection is predominantly a sunlight driven phenomenon.
It creates turbulent air motion at the lit surface boundary of
the planet, and in the presence of a gravity field turbulent
mixing favours energy retention by the air over direct surface
radiant energy loss to space.
Contrastingly at night, in the absence of solar radiant
loading, the process of surface radiant cooling predominates
as the atmosphere stabilises and develops a surface inversion
of cold dense air. This near surface vertical thermal profile
results in lateral movement of dense air downslope, away
from land surface high-elevation points of radiantly efficient
emission to space. Consequently, at these locations the
overlying air preferentially delivers retained and advected
daytime acquired thermal energy down onto the now cooler
nighttime surface.
The parameters listed in Table 4 have been used to
constrain the adiabatic climate modelling process:
Table 4. Earth Climate Metrics used to constrain the three-element parallel
cell DAET climate model.
Earth Climate Metrics
Earth's TOA Solar Irradiance W/m2 1361.0
Earth Bond Albedo 0.306
Dimmed Intercepted Beam at Solar Zenith W/m2 944.53
Earth Hemisphere Average Distribution W/m2 472.27
Disk Silhouette of Tropical Hadley Cell 60.90%
Disk Silhouette of Temperate Ferrel Cell 36.29%
Disk Silhouette of Frigid Polar Cell 2.81%
Global Surface Area of Tropical Hadley Cell 50.00%
Global Surface Area of Temperate Ferrel Cell 41.75%
Global Surface Area of Frigid Polar Cell 8.25%
Hadley Cell Power Intensity Dilution Divisor 1.642042
Ferrel Cell Power Intensity Dilution Divisor 2.300839
Polar Cell Power Intensity Dilution Divisor 5.873730
Average Daily Hadley Cell Illumination W/m2 575.22
Average Daily Ferrel Cell Illumination W/m2 410.52
Average Daily Polar Cell Illumination W/m2 160.81
Hadley Annual Surface Temperature (Celsius) 27.9
Ferrel Annual Surface Temperature (Celsius) 6.5
Annual Polar Surface Temperature (Celsius) -20.0
Tropical Wet Lapse Rate (K/km) 4.6
Temperate Lapse Rate (K/km) 6.5
Frigid Dry Lapse Rate (K/km) 8.8
2.3.1. Modelling the Earth’s Hadley Cell
The two planetary Hadley cells, present in the tropics of
the northern and southern hemispheres, together occupy 50%
of the surface area of the Earth (Table 4), and in total
intercept 60.90% of the light that creates the disk silhouette
of the planetary beam shadow (Table 4). Because the surface
area of the globe’s lit hemisphere is twice the cross-sectional
area of the total disk silhouette, it follows that the power
intensity illumination divisor for the Hadley cells has a value
of (0.5*2)/0.609=1.642 (Table 4). This divisor is then applied
to the post-albedo dimmed irradiance to create the Hadley
cell specific power intensity flux of 575.22 W/m2. This flux
is then in turn used to analyse the process of energy recycling
within the Earth’s Hadley cell by atmospheric mass motion
using the adiabatic climate model of the captured solar
energy (Table 5).
41 Stephen Paul Rathbone Wilde and Philip Mulholland: Return to Earth: A New Mathematical
Model of the Earth’s Climate
Table 5. The inverse modelling process used to determine the dual power intensity flux partition ratios for the Earth's Hadley cell.
Cycle
Number
Hadley
Incoming
Captured
Radiation
Heating the
Hadley Cell Lit
side
Lit side Hadley Cell
Radiant Loss to Space
Lit side Hadley Cell
Thermal Export to
Dark Side
Darkside
Hadley Cell
Radiant Loss to
Space
Darkside
Hadley Cell
Thermal Return
to Litside
Radiant Energy
Exiting Tropical
Zone to Space
Hadley Partition Ratio: Target
Annual Temperature 300.9 Kelvin
(27.9oC)
24.1541% 75.8459% 53.8273% 46.1727%
0 575.22
0
1 575.2191101 575.2191101 138.9391858 436.2799243 234.8378307 201.4420936 373.7770165
2 575.2191101 776.6612037 187.5957759 589.0654278 317.0781865 271.9872413 504.6739624
3 575.2191101 847.2063514 204.6353443 642.5710071 345.8788108 296.6921963 550.5141551
4 575.2191101 871.9113065 210.6026119 661.3086946 355.9648075 305.3438871 566.5674194
696 575.2191101 885.2257096 213.8185904 671.4071191 361.4005197 310.0065994 575.219110121
697 575.2191101 885.2257096 213.8185904 671.4071191 361.4005197 310.0065994 575.219110121
698 575.2191101 885.2257096 213.8185904 671.4071191 361.4005197 310.0065994 575.219110121
699 575.2191101 885.2257096 213.8185904 671.4071191 361.4005197 310.0065994 575.219110121
700 575.2191101 885.2257096 213.8185904 671.4071191 361.4005197 310.0065994 575.219110121
Cycle
Number
Proportionate
Hadley Insolation
Heating the
Hadley Cell
Litside
Lit side Hadley Cell
Radiant Loss to Space
Lit side Hadley Cell
Thermal Export to
Dark Side
Darkside Hadley
Cell Radiant
Loss to Space
Darkside Hadley
Cell Thermal
Return to Litside
Radiant Energy
Exiting Tropical
Zone to Space
Infinity 575.22 885.23 213.82 671.41 361.40 310.01 575.22
S-B 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08
Kelvin 317.4 353.5 247.8 329.9 282.6 271.9 317.4
Celsius 44.4 80.5 -25.2 56.9 9.6 -1.1 44.4
Statistic Mean Exit Temp
Mean Air Temp Lit-side Dark-side Hadley Average
Kelvin 265.18 300.90 W/m2 W/m2 W/m2
Celsius -7.82 27.9 885.226 671.407 778.316
Atmospheric Response Thermal Enhancement
(Kelvin)
Lapse rate Tropopause Height (km)
K/Km Delta K Km
Daytime Hadley Cell 36.1 4.6 82.1 18.0
Nighttime Hadley Cell
4.6 47.3 10.4
The objective of the inverse modelling process used in Table
5 is to establish the daytime convection and nighttime
advection pair of energy partition ratios for both of the
atmospheric circulation cells. The inverse modelling process is
constrained by the two known parameters of annual average
temperature and also the average tropopause height for the
energy collection (lit side) of each cell. The process of
establishing these partition ratios (daytime and nighttime) for
the Earth’s Hadley cell involved a sequence of tuning that
required a “see-saw” approach of iterative “nudges” (Table 6).
Starting with a neutral nighttime energy partition ratio of
50% radiant loss to space and 50% thermal retention by the
air, the inverse modelling process was run with the objective
of establishing the lit surface energy partition ratio that
creates a daytime tropopause height of 18 km (Table 6,
Attempt 1). This first attempt resulted in an adiabatic model
of the Hadley cell with an average annual temperature of
33.75°C, which is warmer than the required average
temperature of 27.9°C.
In order to reduce the model temperature to the required
value of 27.9°C the inverse modelling process was then
repeated, but this time adjusting the nighttime energy
partition ratio to achieve an increased energy loss to space
from the dark side, thereby reducing the average temperature
to the required value (Table 6, Attempt 2). This second
attempt produced a modelling result in which the daytime
tropopause height of 17.8 km is too low.
This undershoot was then corrected by repeating the search
for the lit side energy partition ratio that creates a tropopause
height of 18 km (Table 6, Attempt 3). This third attempt to
tune the model by increasing the retention of flux into the air
on the lit side produces an average annual temperature of
28.25°C, which is still too warm.
The fourth attempt, with its increased nighttime radiant
loss to space, cools the return flow of air to the lit side
sufficiently to successfully achieve both targets of a lit
hemisphere tropopause height of 18 km, and an average
annual temperature of 27.9°C (Table 6, Attempt 4).
Table 6. Establishing the dual set of energy partition ratios for the Earth's Hadley cell.
Adiabatic
Nudge Lit side of Energy Surplus
Dark Side of Energy
Deficit Hadley Average
Air
Temperature °C
Hadley Average
Radiant
Temperature °C
Tropopause
Height (km) Comments
Attempt Loss to
Space
Air
Retention
Loss to
Space
Air
Retention
Lit Side
Dark
Side
0 37.7351% 62.2649% 37.7351%
62.2649%
27.90 -7.51 8.2 15.1
Single Ratio Adiabatic
Model. Lit side Tropopause is
lower than dark side
1 24.3854% 75.6146% 50.0000%
50.0000%
33.75 -7.31 18.0 11.6 Lit side Tropopause Target 18
International Journal of Atmospheric and Oceanic Sciences 2020; 4(2): 36-53 42
Adiabatic
Nudge Lit side of Energy Surplus
Dark Side of Energy
Deficit Hadley Average
Air
Temperature °C
Hadley Average
Radiant
Temperature °C
Tropopause
Height (km) Comments
Attempt Loss to
Space
Air
Retention
Loss to
Space
Air
Retention
Lit Side
Dark
Side
km: Hadley is too warm
2 24.3854% 75.6146% 53.6077%
46.3923%
27.90 -7.71 17.8 10.4 Target 27.9°C: Lit side
Tropopause is too low
3 24.1541% 75.8459% 53.6077%
46.3923%
28.25 -7.79 18.0 10.4 Target 18 km: Hadley is too
warm
4 24.1541% 75.8459% 53.8273%
46.1727%
27.90 -7.82 18.0 10.4 Both Targets Reached
Successfully
2.3.2. Modelling the Earth’s Ferrel Cell
The process of establishing the dual component flux partition
ratio for the Ferrel cell adopts the same strategy as that
established for the Hadley cell described in Section 2.3.1.
The two planetary Ferrel cells, present in the temperate
zones of the northern and southern hemispheres, together
occupy 41.75% of the surface area of the Earth (Table 4) and
in total intercept 36.29% of the light that creates the disk
silhouette of the planetary beam shadow (Table 4). Because
the surface area of the globe’s lit hemisphere is twice the
cross-sectional area of the total disk silhouette, it follows that
the power intensity illumination divisor for the Ferrel cells
has a value of (0.4175*2)/0.3629=2.3008 (Table 4). This
divisor is then applied to the post-albedo dimmed irradiance
to create the Ferrel cell specific power intensity flux of
410.52 W/m2. This flux is then in turn used to analyse the
process of recycling of the captured solar energy by
atmospheric mass motion, within the Earth’s Ferrel cell using
the adiabatic climate model (Table 7).
Table 7. The inverse modelling process used to determine the dual power intensity flux partition ratios for the Earth's Ferrel cell.
Cycle
Number
Ferrel Incoming
Captured
Radiation
Heating the Ferrel
Cell Lit side
Ferrel Thermal
Radiation Loss
to Space
Ferrel Cell
Export to Dark
Side
Darkside Ferrel
Cell Radiant
Loss to Space
Darkside Ferrel
Cell Thermal
Return to Litside
Radiant Energy
Exiting Temperate
Zone to Space
Ferrel Partition Ratio: Target Annual
Temperature 279.5 Kelvin (6.5°C) 21.6206% 78.3794% 54.2258% 45.7742%
0 410.52
0
1 410.5171466 410.5171466 88.7562776 321.7608690 174.4772691 147.2835999 263.2335467
2 410.5171466 557.8007465 120.5998782 437.2008683 237.0754834 200.1253849 357.6753616
3 410.5171466 610.6425315 132.0245901 478.6179414 259.5342051 219.0837363 391.5587952
4 410.5171466 629.6008829 136.1234998 493.4773831 267.5918499 225.8855332 403.7153497
696 410.5171466 640.2083996 138.4169087 501.7914909 272.1002379 229.6912530 410.5171466
697 410.5171466 640.2083996 138.4169087 501.7914909 272.1002379 229.6912530 410.5171466
698 410.5171466 640.2083996 138.4169087 501.7914909 272.1002379 229.6912530 410.5171466
699 410.5171466 640.2083996 138.4169087 501.7914909 272.1002379 229.6912530 410.5171466
700 410.5171466 640.2083996 138.4169087 501.7914909 272.1002379 229.6912530 410.5171466
Cycle
Number
Proportionate
Ferrel Insolation
Boosted Ferrel
Temperature
Ferrel Thermal
Radiation Loss to
Space
Ferrel Cell
Export to Dark
Side
Darkside Ferrel
Cell Radiant Loss
to Space
Darkside Ferrel
Cell Thermal
Return to Litside
Radiant Energy
Exiting Temperate
Zone to Space
Infinity 410.517 640.208 138.417 501.791 272.100 229.691 410.517
S-B 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08
Kelvin 291.7 326.0 222.3 306.7 263.2 252.3 291.7
Celsius 18.7 53.0 -50.7 33.7 -9.8 -20.7 18.7
Statistic Mean Exit Temp Mean Air Temp Lit-side Dark-side Ferrel Average
Kelvin 242.74 279.50 W/m2 W/m2 W/m2
Celsius -30.26 6.50 640.208 501.791 571.000
Atmospheric Response
Thermal
Enhancement
(Kelvin)
Lapse rate Tropopause Height (km)
K/Km Delta K Km
Daytime Ferrel Cell 34.3 6.5 84.4 13.0
Nighttime Ferrel Cell
6.5 43.5 6.7
As with the Hadley cell model the determination of the parameters for the Ferrel cell starts with a neutral nighttime
energy partition ratio of 50% radiant loss to space and 50% thermal retention by the air. The inverse modelling process is
then run with the objective of establishing the lit surface energy partition ratio that creates a daytime tropopause height of
13 km, for an average annual cell temperature of 6.5°C. As with the analysis of the Hadley cell, a process of “see-saw”
iterations were used to achieve the final pair of partition ratios that satisfy both of these data constraints for the Ferrel cell
(Table 8).
43 Stephen Paul Rathbone Wilde and Philip Mulholland: Return to Earth: A New Mathematical
Model of the Earth’s Climate
Table 8. Establishing the dual set of energy partition ratios for the Earth's Ferrel cell.
Adiabatic
Nudge
Lit side of Energy
Surplus
Dark Side of Energy
Deficit Ferrel Average
Air
Temperature °C
Ferrel Average
Radiant
Temperature °C
Tropopause
Height (km) Comments
Attempt Loss to
Space
Air
Retention
Loss to
Space
Air
Retention
Lit
Side
Dark
Side
0 36.7784%
63.2216% 36.7784%
63.2216% 6.50 -28.90 5.8 10.1
Single Ratio Adiabatic Model.
Lit side Tropopause is lower
than dark side
1 21.9039%
78.0961% 50.0000%
50.0000% 12.66 -29.59 13.0 7.6 Lit side Tropopause Target 13
km: Ferrel is too warm
2 21.9039%
78.0961% 53.9690%
46.0310% 6.50 -30.11 12.8 6.7 Target 6.5°C: Lit side
Tropopause is too low
3 21.6206%
78.3794% 53.9690%
46.0310% 6.90 -30.22 13.0 6.7 Target 13 km: Ferrel is too warm
4 21.6206%
78.3794% 54.2258%
45.7742% 6.50 -30.26 13.0 6.7 Both Targets Reached
Successfully
2.3.3. Modelling the Earth’s Polar Cell
The two planetary polar cells together occupy 8.25% of the surface area of the Earth (Table 4) and in total intercept only
2.81% of the light that creates the disk silhouette of the planetary beam shadow (Table 4). As before, because the surface area
of the globe’s lit hemisphere is twice the cross-sectional area of the total disk silhouette, it follows that the power intensity
illumination divisor for the Polar cells has a value of (0.0825*2)/0.0281=5.874 (Table 4). When this divisor is applied to the
silhouette of the post-albedo dimmed irradiance it creates the Polar cell specific power intensity flux of 160.81 W/m2.
Modelling tests established that this power intensity can be used to create an average annual Polar cell temperature of minus
20°C (Table 9).
Table 9. Testing the model of energy partition ratios for the Earth's Polar cells.
Adiabatic
Nudge
Lit side of Energy
Surplus
Dark Side of Energy
Deficit Polar Average Air
Temperature °C
Polar Average
Radiant
Temperature °C
Tropopause
Height (km) Comments
Attempt Loss to
Space
Air
Retention
Loss to
Space
Air
Retention
Lit
Side
Dark
Side
0 25.5567%
74.4433%
25.5567%
74.4433% -20.00 -79.34 7.0 8.6
Single Ratio Adiabatic
Model. Lit side Tropopause
is lower than dark side
1 17.9095%
82.0905%
50.0000%
50.0000% -42.26 -81.94 9.0 4.5 Lit side Tropopause Target
9 km: Polar is too cold
2 17.9095%
82.0905%
34.5138%
65.4862% -20.00 -79.89 9.6 7.1 Target -20°C: Lit side
Tropopause is too high
3 19.3021%
80.6979%
34.5138%
65.4862% -22.31 -79.55 9.0 7.0 Lit side Tropopause Target 9
km: Polar is too cold
4 19.3021%
80.6979%
33.0183%
66.9817% -20.00 -79.42 9.1 7.3 Target -20°C: Lit side
Tropopause is too high
5 19.4497%
80.5503%
33.0183%
66.9817% -20.25 -79.39 9.0 7.3 Lit side Tropopause Target
9 km: Polar is too cold
6 19.4497%
80.5503%
32.8566%
67.1434% -20.00 -79.37 9.0 7.3 Both Targets Reached
Successfully
The stable value that results from this initial test, and presented in Table 10 achieves an average annual temperature of minus
20°C for the Polar cell. However, the range of minimum average air temperature from minus 7.4°C for the summer to minus
32.6°C for the winter is actually too small to account for the known winter extrema air temperatures observed in polar regions.
For example, air temperatures of lower than minus 50°C for July were recorded during advected katabatic drainage storms at
the Little America exploration base, on the ice edge of the Ross Sea in Antarctica [13].
Table 10. The inverse modelling process used to test the interlocked dual power intensity flux partition ratios for the Earth's Polar cells.
Cycle
Number
Polar Incoming
Captured
Radiation
Heating the
Polar Cell Lit
side
Lit side Polar
Cell Radiant
Loss to Space
Lit side Polar Cell
Thermal Export to
Dark Side
Dark side Polar
Cell Radiant
Loss to Space
Dark side Polar
Cell Thermal
Return to Lit side
Radiant Energy
Exiting Polar
Zone to Space
Polar Partition Ratio: Target Annual
Temperature 253 Kelvin (-20
°
C) 19.4497% 80.5503% 32.8566% 67.1434%
0 160.81
0
1 160.8064999 160.8064999 31.27644873 129.5300512 42.55911054 86.97094068 73.83555927
2 160.8064999 247.7774406 48.19207196 199.5853687 65.57687337 134.0084953 113.7689453
3 160.8064999 294.8149952 57.34075479 237.4742404 78.02585079 159.4483896 135.3666056
4 160.8064999 320.2548896 62.28874851 257.9661411 84.75878308 173.2073580 147.0475316
696 160.8064999 350.2205534 68.11699269 282.1035607 92.68950726 189.4140534 160.8064999
International Journal of Atmospheric and Oceanic Sciences 2020; 4(2): 36-53 44
Cycle
Number
Polar Incoming
Captured
Radiation
Heating the
Polar Cell Lit
side
Lit side Polar
Cell Radiant
Loss to Space
Lit side Polar Cell
Thermal Export to
Dark Side
Dark side Polar
Cell Radiant
Loss to Space
Dark side Polar
Cell Thermal
Return to Lit side
Radiant Energy
Exiting Polar
Zone to Space
697 160.8064999 350.2205534 68.11699269 282.1035607 92.68950726 189.4140534 160.8064999
698 160.8064999 350.2205534 68.11699269 282.1035607 92.68950726 189.4140534 160.8064999
699 160.8064999 350.2205534 68.11699269 282.1035607 92.68950726 189.4140534 160.8064999
700 160.8064999 350.2205534 68.11699269 282.1035607 92.68950726 189.4140534 160.8064999
Cycle
Number
Proportionate Polar
Insolation
Heating the Polar
Cell Lit side
Lit side Polar
Cell Radiant
Loss to Space
Lit side Polar Cell
Thermal Export to
Dark Side
Dark side Polar
Cell Radiant Loss
to Space
Dark side Polar
Cell Thermal
Return to Lit side
Radiant Energy
Exiting Polar
Zone to Space
Infinity 160.81 350.22 68.12 282.10 92.69 189.41 160.81
S-B 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08
Kelvin 230.8 280.3 186.2 265.6 201.1 240.4 230.8
Celsius -42.2 7.3 -86.8 -7.4 -71.9 -32.6 -42.2
Statistic Mean Exit Temp
Mean Air Temp Lit side Dark side Polar Average
Kelvin 193.63 253.0 W/m2 W/m2 W/m2
Celsius -79.37 -20.0 350.221 282.104 316.162
Atmospheric Response
Thermal
Enhancement
(Kelvin)
Lapse rate Tropopause Height (km)
K/Km Delta K Km
Daytime Polar Cell 49.6 8.8 79.4 9.0
Night time Polar Cell
8.8 64.5 7.3
Modern icecap temperature data recorded for Antarctica regularly reach values of minus 70°C in winter (Figure 4), and so an
alternative modelling strategy was devised to account for these extreme temperature values recorded for winter in polar regions.
Figure 4. Australian Antarctic Division [14]: Dome Argus Temperature Profile: 12th – 19th August 2008.
The key difference between the polar cells and the two
other atmospheric cells present in the Earth’s atmosphere, is
that in summer the high latitude polar regions experience
months of continuous daylight. The effect of continuous
daylight is that any atmospheric convective activity, that
results in vertical overturning in the Polar cell, returns air
back onto a lit surface. This return of air onto the illuminated
surface effectively short circuits the surface energy partition
process, and delivers an energy flux boost directly back to the
lit summer Polar cell environment.
By contrast, during their respective winter season, each
Polar cell experiences months of continuous darkness and
there is no direct input of radiant solar energy. Consequently,
all of the energy flux experienced by the cells throughout the
months of continuous darkness is a direct result of advected
air transported into the polar environment from the abutting
Ferrel cell.
In order to address the dichotomy of continuous summer
illumination and continuous winter darkness, the design of the
adiabatic model of the Polar cell was altered to incorporate the
convective feedback process of summer, and also the advected
process of winter into two separate modelling streams. For the
purposes of this analysis, and as merely a scoping proposal, the
average Polar cell summer temperature is assumed to be plus
5°C, and the average winter temperature is assumed to be
minus 45°C. These two separate seasonal values combine to
create the required average annual temperature for the Polar
cell of minus 20°C (Table 11).
45 Stephen Paul Rathbone Wilde and Philip Mulholland: Return to Earth: A New Mathematical
Model of the Earth’s Climate
Table 11. The inverse modelling process used to determine the seasonally separated dual power intensity flux partition ratios for the Earth's Polar cells.
Cycle
Number
Polar
Incoming
Captured
Radiation
Heating the
Summer
Polar Cell
Summer Polar
Thermal
Radiant Loss to
Space
Summer Polar Cell
Return to Surface
Winter Air
Advected from
Ferrel Cell
Retained in
the Winter
Polar Cell
Winter Polar
Cell Radiant
Loss to Space
Winter Polar
Cell Cold Air
Returned to
Ferrel Cell
Polar Partition Ratio: Target
Summer Temperature 278
Kelvin (5°C)
32.1957% 67.8043%
Polar Partition Ratio: Target
Winter Temperature 228 Kelvin (-
455°C))
64.0791% 35.9209%
0 160.81
273.33
0
1 160.8064999 160.8064999 51.7728363 109.0336636 273.3325458 273.3325458 175.1490034 98.1835424
2 160.8064999 269.8401635 86.8770269 182.9631366 273.3325458 371.5160883 238.0641223 133.4519660
3 160.8064999 343.7696366 110.6791650 233.0904716 273.3325458 406.7845119 260.6638065 146.1207053
4 160.8064999 393.8969716 126.8180294 267.0789421 273.3325458 419.4532512 268.7818192 150.6714320
696 160.8064999 499.4652071 160.8064999 338.6587071 273.3325458 426.5549856 273.3325458 153.2224397
697 160.8064999 499.4652071 160.8064999 338.6587071 273.3325458 426.5549856 273.3325458 153.2224397
698 160.8064999 499.4652071 160.8064999 338.6587071 273.3325458 426.5549856 273.3325458 153.2224397
699 160.8064999 499.4652071 160.8064999 338.6587071 273.3325458 426.5549856 273.3325458 153.2224397
700 160.8064999 499.4652071 160.8064999 338.6587071 273.3325458 426.5549856 273.3325458 153.2224397
Cycle
Number
Proportionate
Polar
Insolation
Heating the
Summer Polar
Cell
Summer Polar
Thermal Radiant
Loss to Space
Summer Polar Cell
Return to Surface
Winter Air
Advected from
Ferrel Cell
Retained in
the Winter
Polar Cell
Winter Polar
Cell Radiant
Loss to Space
Winter Polar
Cell Cold Air
Returned to
Ferrel Cell
Infinity
160.81 499.47 160.81 338.66 273.33 426.55 273.33 153.22
S-B 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08
Kelvin 230.8 306.4 230.8 278.0 263.5 294.5 263.5 228.0
Celsius
-42.2 33.4 -42.2 5.0 -9.5 21.5 -9.5 -45.0
Statistic Mean Exit Temp
Mean Air Temp Summer Winter Polar Average
Kelvin 247.13 253.00 W/m2 W/m2 W/m2
Celsius -43.61 -20.00 499.465 338.659 419.062
Atmospheric Response Thermal Enhancement
(Kelvin)
Lapse rate Tropopause Height (km)
K/Km Delta K Km
Summer Polar Cell 75.6 8.8 75.6 8.6
Winter Polar Cell 31.0 8.8 66.5 7.5
We have now completed the individual modelling process
for each of the Earth’s three atmospheric cells [15, 16].
3. Discussion of the Modelling Results
The triple-cell parallel adiabatic model of Earth’s climate
is tuned to produce the expected value of the average annual
atmospheric temperature of 288 Kelvin (15°C) using the
previously established method of weighted area to determine
the average annual temperature of the Earth (Table 3).
The results of the inverse modelling process demonstrate
that to achieve a stable average air temperature and also an
appropriate cell specific tropopause height, solar energy must
be preferentially retained in the climate system by the air that
is located over the lit portion of the Earth’s surface (Table
12). Retention in favour of the air occurs because convection
at the solar heated surface boundary is a turbulent process. In
the presence of a gravity field solar heated air ascends by
buoyancy displacement which removes it from contact with
the ground. Because the solid ground surface of a planet is
the primary low-frequency radiator, ascending air becomes
decoupled from this surface and so retains its energy
internally as it rises.
Thermal radiant exhaust of energy to space is the primary
control on the ambient atmospheric temperature. Even under
conditions of reduced atmospheric opacity, the ground
surface radiator of the Earth continues to operate through the
Infrared Window, first identified in 1928 as a critical
component of atmospheric radiant energy transmission [7].
Under conditions of zero solar radiant loading, either at
night or during the polar winter, the ground surface radiator
continues to operate through the atmospheric infrared
window. The nighttime is an environment of energy deficit,
gasses are poor absorbers and emitters of radiant thermal
energy, so they heat most effectively by contact with the
sunlit warmed surface during the day, and cool most
effectively by contact with the radiatively cooled ground
surface by night.
The Antarctic winter temperature inversion profile (Figure
4) is a direct consequence of thermal equilibrium being
established and maintained by the process of surface
radiative cooling. This cooling is caused by direct radiative
energy loss to space through the dry transparent atmosphere
above the high elevation Antarctic icecap. The radiative
process results in the development and maintenance of a
surface air temperature inversion. Under these conditions the
atmosphere delivers energy to the ground surface radiator,
and consequently the energy partition ratio for the winter
polar cell is heavily weighted in favour of radiant energy loss
to space (Table 12).
International Journal of Atmospheric and Oceanic Sciences 2020; 4(2): 36-53 46
Table 12. Results of the inverse modelling process used to establish power intensity flux partition ratios for the Earth's atmospheric cells.
Daily Atmospheric Cell
Lit side Partition Ratio Lit side Dark side Partition Ratio Dark side
Thermal Radiant
Loss to Space
Thermal Export
to Dark Side
Tropopause
Height (km)
Thermal Radiant
Loss to Space
Thermal Return to
Lit side
Tropopause
Height (km)
Hadley cell: Target Annual
Temperature 300.9 Kelvin (27.9°C)
24.1541% 75.8459% 18.0 53.8273% 46.1727% 10.4
Ferrel cell: Target Annual
Temperature 279.5 Kelvin (6.5°C) 21.6206% 78.3794% 13.0 54.2258% 45.7742% 6.7
Seasonal Atmospheric Cell
Summer Polar cell
Thermal Radiant
Loss to Space
Summer Polar cell
Thermal Return to
Surface
Tropopause
Height (km)
Winter Polar Cell
Thermal Radiant
Loss to Space
Winter Polar Cell
Cold Air Returned
to Ferrel Cell
Tropopause
Height (km)
Polar cell: Target Summer
Temperature 278 Kelvin (5°C) 32.1957% 67.8043% 8.6
Polar cell: Target Winter
Temperature 228 Kelvin (-45°C) 64.0791% 35.9209% 7.5
3.1. Studying the Effects of Energy Flux Variations Within
the Adiabatic DAET Model
In conducting the modelling analysis presented here the
key question that must be addressed is this. What is the
justification for using energy partition ratio as the basis for
determining the average annual temperature of the Earth?
There are three fundamental physical parameters that
underpin our DAET modelling process that relate directly to
planetary climate, these are: -
1. Global Atmospheric Temperature.
2. Global Atmospheric Pressure.
3. Global Atmospheric Volume.
We have already demonstrated that if we know the areal
weighting of the three atmospheric cells, their respective
tropopause heights, their TOA temperatures and also
respective lapse rates, then the global average temperature of
the planetary atmosphere can be calculated. We also know
that the average pressure of the atmosphere can be
determined by measurement and is common across all three
cells, so the remaining issue is the determination of the
planetary atmospheric volume.
If we assume that the tropopause is a pressure related
phenomenon, and that the 100 mb pressure marks the upper
limit of the troposphere [17], then the question of applying
Boyle’s Law to the total planetary atmosphere potentially has
merit and requires investigation. The key objection that the
Boyle’s Law relationship relates only to a confined volume
of gas assumes that planetary atmospheres are completely
unconfined. Clearly this is not strictly true, the total surface
area of the Earth does not change, the total mass of the
atmosphere, and therefore its pressure is also a fixed
quantity.
So, in the presence of a gravity field that binds the
atmosphere to the planet it follows that the volume change
we observe associated with a change in tropospheric height
for each atmospheric cell must be related to the temperature
of that cell. Consequently, we can study the planetary
atmosphere in total by treating it as a single gravity confined
entity with measurable parameters of temperature, pressure
and volume.
In order to test the relationship between atmospheric
temperature, pressure and volume, a simple single
hemisphere adiabatic model was created with an illumination
intensity dilution divisor of integer 2. This model is assumed
to have simple diabatic radiative cooling from the dark unlit
hemisphere, and so a constant partition ratio of 50% radiant
energy loss to space and 50% retention by the air was applied
to this part of the model (Table 13).
Table 13. Testing the Whole Earth PVT Adiabatic Model.
Cycle
Number
Incoming Captured
Radiation
Heating the Lit
side
Lit side Radiant
Loss to Space
Lit side Thermal
Export to Dark
Side
Darkside
Radiant Loss
to Space
Darkside Thermal
Return to Litside
Radiant Energy
Exiting to Space
Litside Variable Partition Ratio: Target
Annual Temperature 288 Kelvin (15
°
C) 26.9553% 73.0447% 50.0000% 50.0000%
0 472.267
0
1 472.267 472.267 127.3008921 344.9661079 172.483054 172.483054 299.783946
2 472.267 644.750054 173.7941822 470.9558718 235.4779359 235.4779359 409.2721181
3 472.267 707.7449359 190.774629 516.9703069 258.4851535 258.4851535 449.2597824
4 472.267 730.7521535 196.9762889 533.7758646 266.8879323 266.8879323 463.8642212
2996 472.267 743.990 200.5444627 543.4450745 271.7225373 271.7225373 472.267
2997 472.267 743.9895373 200.5444627 543.4450745 271.7225373 271.7225373 472.267
2998 472.267 743.9895373 200.5444627 543.4450745 271.7225373 271.7225373 472.267
2999 472.267 743.9895373 200.5444627 543.4450745 271.7225373 271.7225373 472.267
3000 472.267 743.990 200.5444627 543.4450745 271.7225373 271.7225373 472.267
Cycle
Number
Incoming Captured
Radiation
Heating the Lit
side
Lit side Radiant Loss
to Space
Lit side Thermal
Export to Dark
Side
Darkside
Radiant Loss to
Space
Darkside Thermal
Return to Litside
Radiant Energy
Exiting to Space
47 Stephen Paul Rathbone Wilde and Philip Mulholland: Return to Earth: A New Mathematical
Model of the Earth’s Climate
Cycle
Number
Incoming Captured
Radiation
Heating the Lit
side
Lit side Radiant
Loss to Space
Lit side Thermal
Export to Dark
Side
Darkside
Radiant Loss
to Space
Darkside Thermal
Return to Litside
Radiant Energy
Exiting to Space
Infinity 472.27 743.99 200.54 543.45 271.72 271.72 472.27
S-B 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08
Kelvin 302.1 338.5 243.9 312.9 263.1 263.1 302.1
Celsius 29.1 65.5 -29.1 39.9 -9.9 -9.9 29.1
Statistic Mean Exit Temp
Mean Air Temp Lit-side Dark-side Hadley Average
Kelvin 253.49 288.00 W/m2 W/m2 W/m2
Celsius -19.51 15.0 743.990 543.445 643.717
Atmospheric Response Thermal Enhancement
(Kelvin)
Lapse rate Tropopause Height (km)
K/Km Delta K Km
Lit Hemisphere 36.4 6.0 94.6 15.8
Dark Hemisphere
6.0 49.8 8.3
The energy flux within the model was then adjusted by
varying the Bond albedo. For each increment of Albedo
related radiant power intensity, the inverse modelling process
was run to determine the lit surface energy partition ratio that
restored the global atmospheric temperature back to a
constant value of 15°C.
Because we are now adjusting the Bond albedo, the power
intensity flux in our simple model varies from a maximum
case of 680.5 W/m2 [1361/2*(1-0.0)] for a totally absorptive
Earth (albedo=0.0), down to a lower limit of 272.2 W/m2,
[1361/2*(1-0.60)] for a bright reflective Earth (albedo=0.60).
The power intensity flux lower limit of 272.2 W/m2 occurs
because below this value it is impossible for the model Earth
to maintain an average annual temperature of 15°C if its
albedo becomes any brighter.
The results of these tests are shown in Figure 5.
Figure 5. The Variation of Energy Partition Ratio with Power Intensity Influx for a Single Lit Hemisphere Adiabatic Model.
Using data from the American Vacuum Society (AVS) the
temperature and pressure profiles for the average atmosphere
are shown in Figures 6 and 7 [18]. These data show that for a
standard Earth atmosphere and a tropopause defined as
occurring at a pressure of 100 mbar (Figure 6) then the
average elevation of this pressure is at a height of 16 km
(Figure 7).
Figure 6. Earth’s Average Atmosphere Temperature Profile (AVS data).
International Journal of Atmospheric and Oceanic Sciences 2020; 4(2): 36-53 48
Figure 7. Earth’s Average Atmosphere Pressure Profile (AVS data).
Starting with a biased surface datum of minus 50 km, the
calculated pressure versus height relationship for the Earth’s
standard atmosphere (Figure 7) was extended downwards to
create a model high pressure atmosphere using an
exponential pressure altitude equation (Km versus mbars):-
Equation 2: Pressure=1060.9*EXP (-0.146*C2) mbar
Where C2 is the Datumed Biased Altitude in kilometres.
Equation 2 is constructed to create the standard atmospheric
pressure of 1013 mbar at the reference zero altitude of the
Earth’s surface under current atmospheric conditions. For
Equation 2 negative altitudes relate to higher than ambient
surface pressure, while positive altitudes relate to lower than
ambient pressure. The calculated pressures range from a high
pressure state for a model atmosphere thickness of 68 km
(equation biased altitude of minus 50 km), down to a low
pressure state for a model atmosphere thickness of 5.66 km
(equation biased altitude of plus 13 km) (Figure 8).
Using a model specific wet adiabatic lapse rate of 3.8
K/km for the lit side of the single cell model, the atmosphere
“thickness” records a low of 5.66 km for the high solar
energy input case, with a commensurate balancing high
radiant energy loss to space. The maximum value of 68 km
of atmospheric thickness is achieved for the low solar energy
input case, and commensurate balancing low radiant energy
loss to space (Figure 8). There is therefore a clear
relationship between solar energy input and immediate
energy shedding to space by the lit surface. This energy
shedding is required to maintain the constant modelled
average global temperature of 15°C, and is a pressure
dependent effect (Figure 8).
Figure 8. Lit Ground % Energy Partition vs Surface Atmospheric Pressure for a Constant Earth 15°C.
In Figure 8 we see the effective pressure dependent limits
under which an Earth with an average planetary temperature
of 15°C can exist for a given range of radiant energy loadings
at its current orbital distance from the Sun. With the high
albedo, (low energy capture) thick atmosphere end-member
of the model we are effectively simulating a high pressure,
low temperature version of the atmosphere of Venus.
3.2. The CERES Image of the Earth’s Radiant Emission to
Space
Each of the three atmospheric cells that constitute the
circulation system of the Earth’s atmosphere has a distinct set
of meteorological parameters of areal extent, average
insolation power intensity flux, average annual temperature
49 Stephen Paul Rathbone Wilde and Philip Mulholland: Return to Earth: A New Mathematical
Model of the Earth’s Climate
and adiabatic lapse rate. The following image (Figure 9)
shows the Earths outgoing long-wave radiation recorded by
the CERES (Clouds and the Earth’s Radiant Energy System)
Instrument onboard the NASA Aqua Satellite [19].
Figure 9. The Earth’s outgoing long-wave radiation recorded by the CERES (Clouds and the Earth’s Radiant Energy System) Instrument onboard the NASA
Aqua Satellite [19, 20].
The colour table legend records the energy flux of the
outgoing thermal radiation. This flux ranges from a minimum
value of 150 W/m2 displayed as white, to a maximum flux of
350 W/m2, 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 3: 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 3 we can determine that the emission
temperatures recorded by the CERES instrument range from
a minimum value of 226.8 Kelvin (-46.2°C) for the 150
W/m2 low-end flux, to a maximum value of 280.3 Kelvin
(7.3°C) for the 350 W/m2 high-end flux.
3.3. Calibrating the CERES Image
The CERES image (Figure 9) 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.
Visual inspection of the CERES image (Figure 9) 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/m2
and have an emission temperature of 227 Kelvin (-46.2°C).
In order to determine the elevation of this emission, we need
to use atmospheric parameters for the Hadley, Ferrel and
Polar cells established in Section 2.3. (Table 2), which are:
1. The height of the tropopause.
2. The temperature of the tropopause.
3. The environmental lapse rate for each atmospheric cell.
Using the values established in Table 2 we can determine
the top down temperature profile for each of the three
atmospheric cells.
3.3.1. Calibrating the Hadley cell
The calculations for the Hadley cell show that to maintain
a 17 km tropopause with a temperature of 190 Kelvin (-
83°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°C)
(Table 14).
Table 14. Hadley Cell - CERES Image Emissions Calibration.
Energy Flux
W/m2 Kelvin
Celsius Hadley Cell
Profile Km
Radiant
Depth Km Hadley Cell Measurements
465 301 27.9 0.00
Average Tropical Surface Temperature
426 294 21.4 1.00
389 288 14.9 2.00
350 280 7.3 3.16 13.84 Maximum observed radiant emission depth of the Hadley cell seen from space
325 275 2.1 3.96 13.04
309 272 -1.4 4.50 12.50 Tibetan Plateau 4500m
300 270 -3.3 4.80 12.20
275 264 -9.1 5.69 11.31
270 263 -10.4 5.89 11.11 Height of Kilimanjaro 5892m
250 258 -15.3 6.64 10.36 Midpoint observed radiant emission depth of the Hadley Cell seen from space
225 251 -22.0 7.68 9.32
International Journal of Atmospheric and Oceanic Sciences 2020; 4(2): 36-53 50
Energy Flux
W/m2 Kelvin
Celsius Hadley Cell
Profile Km
Radiant
Depth Km Hadley Cell Measurements
200 244 -29.3 8.80 8.20 Height of Mount Everest 8850m
175 236 -37.3 10.03 6.97
150 227 -46.2 11.40 5.60 Maximum supercooled liquid water atmosphere elevation in the Hadley cell
140 223 -50.1 12.00 5.00
124 216 -56.6 13.00 4.00
110 210 -63.1 14.00 3.00
97 203 -69.6 15.00 2.00
85 197 -76.1 16.00 1.00
74 190 -82.6 17.00 0.00 Hadley Cell Tropopause Height
Converting this average surface temperature of ~28°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/m2. This value is 115 W/m2
higher than the maximum observed flux of 350 W/m2 in the
Ceres image (Figure 9), 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°S to 30°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.
3.3.2. Calibrating the Ferrel cell
The calculations for the Ferrel cell show that to maintain a
13 km tropopause with a temperature of 195 Kelvin (-78°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°C)
(Table 15).
Table 15. Ferrel Cell - CERES Image Emissions Calibration.
Energy Flux
W/m2 Kelvin
Celsius
Ferrel Cell
Profile Km
Radiant
Depth Km Ferrel Cell Measurements
346 280 6.5 0.00
Average Temperate Surface Temperature
325 275 2.1 0.67
300 270 -3.3 1.51 11.49 Maximum observed radiant emission depth of the Ferrel cell seen from space
275 264 -9.1 2.40 10.60
250 258 -15.3 3.36 9.64
225 251 -22.0 4.39 8.61
200 244 -29.3 5.51 7.49
190 241 -32.5 6.00 7.00 Height of Denali (Alaska) 6190m
175 236 -37.3 6.74 6.26
150 227 -46.2 8.11 4.89 Maximum supercooled liquid water elevation in the Ferrel cell
135 221 -52.0 9.00 4.00
120 215 -58.5 10.00 3.00
106 208 -65.0 11.00 2.00
93 202 -71.5 12.00 1.00
82 195 -78.0 13.00 0.00 Ferrel Cell Tropopause Height
Converting this average surface temperature of 6.5°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/m2. This value is 46 W/m2
higher than the maximum observed flux in the CERES image
of 300 W/m2 for the temperate zone as seen from space
(Figure 9). 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.
3.3.3. Calibrating the Polar cell
The calculations for the Polar cell show that to maintain a
9 km tropopause with a temperature of 194.5 Kelvin (-
78.5°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°C) (Table 16).
Table 16. Polar Cell - CERES Image Emissions Calibration.
Energy Flux
W/m2 Kelvin
Celsius Polar Cell
Profile Km
Radiant Depth
Km Polar Cell Measurements
232 253 -20.0 0.00
Average Polar Surface Temperature
225 251 -22.0 0.31 8.69 Maximum observed radiant emission depth of the Polar cell seen from space
200 244 -29.3 1.43 7.57
175 236 -37.3 2.66 6.34
169 234 -39.5 3.00 6.00 Elevation of North Dome, Greenland 3000m
150 227 -46.2 4.03 4.97 Maximum supercooled liquid water elevation in the Polar cell
51 Stephen Paul Rathbone Wilde and Philip Mulholland: Return to Earth: A New Mathematical
Model of the Earth’s Climate
Energy Flux
W/m2 Kelvin
Celsius Polar Cell
Profile Km
Radiant Depth
Km Polar Cell Measurements
149 226 -46.6 4.09 4.91 Elevation of Dome Argus, Antarctica 4093m
134 221 -52.5 5.00 4.00
119 214 -59.0 6.00 3.00
105 208 -65.5 7.00 2.00
93 201 -72.0 8.00 1.00
81 194.5 -78.5 9.00 0.00 Polar Cell Tropopause Height
Converting this average surface temperature of -20°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/m2. This value is just 7
W/m2 higher than the maximum observed flux in the CERES
image of 225 W/m2 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.
4. Conclusions and Observations
1. By creating a dual surface climate model, with one day
lit surface of energy surplus and a second dark night
surface of energy deficit, we can apply two separate
energy partition ratios to these two distinct
environments, and study the impacts of these ratios on
energy retention and distribution within the DAET
climate model.
2. By assuming that the daytime environment on Earth is
dominated by adiabatic convection and has an energy
partition ratio weighted in favour of the air, we can
account for the process of atmospheric uplift and energy
retention by the air.
3. By assuming that the night-time environment on Earth
is dominated by radiative cooling, and has an energy
partition ratio weighted in favour of radiant loss to
space, we can account for the standard nighttime air
temperature profile, and the development of surface
temperature inversions in air.
4. By applying a process of inverse modelling, we can
establish the values of the energy partition ratio for the
Earth’s lit daytime and dark night-time environments. It
is this daytime energy retention in favour of the air that
creates the climatic thermal enhancement observed on
Earth.
5. By using the appropriate adiabatic lapse rate for each
cell, our inverse modelling process can be tuned to
replicate the expected tropopause height for the Earth’s
tropical Hadley Cell of energy surplus, that of the
temperate Ferrel cell, and also the height for the Earths
Polar Cell of energy deficit.
6. By constructing a simple single lit hemisphere adiabatic
model, the range of energy partition ratios required to
maintain a constant whole Earth temperature under
various solar radiation loadings can be explored. Using
an extrapolated pressure altitude equation, the
relationship between the energy partition ratio for the lit
surface of energy collection and confining atmospheric
pressure can be established.
7. Convection efficiency is a pressure related
phenomenon. High pressure gaseous environments are
more efficient at removing energy from a solar heated
surface in the presence of a confining gravity field.
8. Our modelling studies suggest that the opacity of the
atmosphere fundamentally controls the height of the
radiant emission surface that vents energy to space (as
per [17]). However, there is no requirement for opacity
to be an atmospheric energy amplifier via radiative
feed-back contra [8].
In our analysis of the CERES image (Figure 9) we are
looking at climate from a geological perspective. Our
motivation for this approach was to try and determine if high
elevation solid surfaces, such as the Tibetan Plateau, are
"thermally visible" from space, and if the contrast in global
land surface elevation between the Cretaceous and the
Tertiary Periods can provide a physical explanation for the
long-term planetary cooling of the Earth over the last 65
million years.
This analysis is based on the following points: -
1. All planets shed energy to space via thermal radiation.
2. Flexure of solid materials is the fundamental process
that interlinks vibrating matter with thermal radiation.
The coincidence of the lowest thermal radiant
temperature of -46.2°C in the CERES image with the
lowest observed temperature of super-cooled water
suggests that there is a relationship between planetary
Bond albedo and atmospheric thickness (Tables 14, 15,
16). Albedo therefore could be an emergent
consequence rather than a cause of planetary climate.
3. Solid surfaces (either rock or ice) held at high elevation
are efficient radiators precisely because they are
composed of flexible materials.
4. Icecaps can melt, therefore the ice age high elevation
solid ice surfaces of Canada and Scandinavia can
rapidly disappear with major implications for surface
radiation loss, whereas the Tibetan plateau with an
elevation of 4,500m remains as a long-term geological
high elevation radiant leak point (Table 14).
5. The apparent coincidence between the maximum
elevation of the Antarctic Icecap at Dome Argus
(4093m) and the maximum elevation at which super-
cooled liquid water can exist in the atmosphere of the
Polar cell (4030m) requires further study (Table 16). A
possible explanation is that once the ice surface rises
above the point where it is no longer possible to have
International Journal of Atmospheric and Oceanic Sciences 2020; 4(2): 36-53 52
supercooled liquid water droplets in the atmosphere,
then there is no further possibility for latent heat of
crystallisation to be released. Therefore, there is no
energy left in the meteorological system to power moist
convection, and so vertical ice accumulation stops.
6. On Earth the rapid daily rotation limits the latitudinal
reach of the Hadley cell [21, 22], it creates forced
descent of upper atmospheric air in the mid-latitudes
and directly accounts for the existence of the Ferrel cell.
The Ferrel cell is a mechanical cell that acts as a cog
between the tropical thermal Hadley cell of solar
heating and energy surplus, and the thermal Polar cell of
surface radiant cooling and energy deficit.
7. The impact of the forced descent of Hadley cell air in
the mid-latitudes can clearly be seen in the CERES
image (Figure 9). This descent creates a zone of high
surface pressure and reduced moist convection that
allows for significant planetary thermal loss to space
over continental land areas, such as the Sahara Desert of
North Africa. Over the adjacent mid-Atlantic and
southern Indian Oceans, the same latitudinal zone of
forced air descent allows for clear skies that lead to
solar energy capture and retention by the marine waters
below.
Our fundamental criticisms of the standard radiative
climate model currently used by climate science are as
follows: -
First, all materials heat and cool diabatically (laminar
exchange of energy through the surface interface), solids do
not significantly change their position when they are heated.
Gaseous atmospheres not only heat and cool diabatically, but
in addition air also heats adiabatically, which is a turbulent
motion process of energy acquisition, and is a critical part of
daytime surface heating.
Second, it is physically impossible to lose potential energy
by radiant thermal emission. Atmospheric adiabatic energy
transport is a meteorological process that delivers energy,
without any transport loss, to a distant surface that is itself
undergoing diabatic cooling by radiant thermal emission to
space.
We have designed our climate model to retain the critical
dual surface element of a lit globe, namely night and day.
The standard climate model is a single surface model that
does not include adiabatic energy transfer, because diabatic
thermal equilibrium is assumed at all times (both night and
day). When in our model we apply the missing element of
adiabatic energy transfer from the lit side, by using distinct
and separate energy partition ratios for night and day, then
the requirement for back radiation greenhouse gas heating is
no longer necessary.
We are able to quantify the degree of adiabatic lit surface
energy partition in favour of the air by using the process of
inverse modelling, a standard geoscience mathematical
technique. The issue of atmospheric opacity then becomes a
passive process, and the purported atmospheric action of
greenhouse heating by back-radiation can be discounted. We
believe that our modelling work presented here should lead
to a fundamental reassessment of the atmospheric processes
relating to energy partition, retention and flow within the
Earth’s climate system.
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... Both of these diagrams when combined provide detailed energy budget information for the Earth's climate; however, their parameters are recorded as percentages of solar illumination at the top of the atmosphere (TOA). Neither diagram published by OK-First records the actual values of solar power intensity, nor is it demonstrated how they can be used to estimate the global average temperature for the surface of the Earth, something that has been shown by us to be achievable using basic climate budget data [3]. ...
... It is therefore impossible for the Earth to experience a runaway greenhouse gas effect due to changes in the atmospheric thermal radiant opacity if the total mass of the atmosphere does not increase [3]. ...
... For a resolution of this paradox we propose the adoption of a new climate model, the Dynamic Atmosphere Energy Transport (DAET) model, that is based on meteorological principles and is applicable to all solar illuminated terrestrial type astronomic bodies that possess a dense semi-opaque thermally radiant atmosphere [3,17,18]. ...
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