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Abstract and Figures

The application of a lapse rate controlled pressure profile for the atmosphere of Venus as a calibration check for the DAET climate model.
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A Modelled Atmospheric Pressure Profile of Venus.
Philip Mulholland and Stephen P.R. Wilde.
1st March 2021
Abstract:
In this paper we present the results of a calibration check on the validity of our previous application
of the Dynamic-Atmosphere Energy-Transport (DAET) model used to study the climate of the planet
Venus [1]. To achieve this, we have created an eXcel spreadsheet application designed to calculate
the pressure profile for the Venus atmosphere at 1 metre increments. We have applied this
calculation from the surface to the lower stratosphere, a modelled vertical height of 100 kilometres
[2].
Two equations of state are used to achieve this objective, these are the pressure, volume,
, and the application of 
shells, used to calculate the reduction in strength of the gravity field as the height above the surface
of Venus increases. For the purpose of this study, we have created a set of four linked predictive
lapse rate equations based on published data [3]. These equations are used as the fundamental
temperature control of the atmospheric pressure profile. The temperature data that controls these
equations is calibrated to a surface datum global average temperature for Venus of 699 Kelvin [4].
By using the troposphere model lapse rate profile as the constraint on cooling by vertically
convecting air, we have established that the height of the tropopause convection limit in our model
is a close match to the level of the observed static atmosphere height for the 250 Kelvin freezing
point level of 75% by weight of concentrated sulphuric acid [5]. Sulphuric acid is the primary
condensing volatile in the Venusian atmosphere. We hypothesise that the impact on planetary
albedo by the solidification of this ggests that the
observed albedo of Venus is a response to and not a cause of planetary atmospheric solar radiant
forcing.
By using the thermal lapse rate for the troposphere of Venus in its top-down mode of application,
we calculate the depth below the tro
effective heating of the Venusian atmosphere. We have established that this radiant quenching
depth delineates a pool of upper tropospheric air that both captures and responds to solar radiant
forcing.
This radiatively forced thermal imbalance sets up an initial density variation within the upper
atmosphere that causes convection to begin. However once started at the solar zenith, the point of
maximum solar radiant forcing, the atmospheric convection responds and develops to include the
entirety of the Venusian atmosphere from surface to space.
We note that the observed cloud patterns on Venus recorded in 1974 by the Mariner 10 NASA probe
do appear to support the development of the type of insolation induced convective disturbance that
our DAET hypothesis requires and indeed predicts (Figure 1). It is the solar induced disruption of the
mass/gravity lapse rate slope at upper levels that forces convection to begin. Once started the
motion of rotating planetary dynamics ensures that this convection develops throughout the entire
depth of the Venusian atmosphere, this occurs even when no effective solar energy reaches the
surface of Venus.
As a direct consequence of this top of the troposphere solar radiant forcing, a process of full
troposphere convective overturn occurs and delivers solar heated air to the ground via the action of
forced air descent in the twin polar vortices of Venus. This forced descent of the topside heated air
means that it undergoes adiabatic heating as it falls in the gravity field of Venus. The descending

to circulate vertically in a pair of giant hemispheres encompassing Hadley cells. By this means the
compressed air is heated as it falls, therefore the thermal limit to radiative forcing set by the
insolation of Venus is easily surpassed.
From this analysis we conclude that the high surface temperatures observed at the surface of Venus
are a direct consequence of, and maintained by, a process of topside thermal radiant capture by the
air followed by mass motion energy delivery to the surface. Our DAET model demonstrates
therefore that the observed high air temperatures of Venus are due to the mass motion adiabatic
heating of descending air in a gravity field, and not due to a process of thermal radiant energy flux
impediment by atmospheric thermal radiant opacity.
Introduction:
Close proximity observations of the planet Venus by the NASA Mariner 10 space probe in 1974 have
shown that its upper atmosphere displays a set of cloud bands that are part of a global atmospheric
circulation system, which connects the solar zenith point of maximum solar radiant forcing to both
polar vortices of the planet. (Figure 1).
Figure 1: NASA 1974 Mariner 10's Portrait of Venus.
In 2020 we presented the results of our first application of the Dynamic-Atmosphere Energy-
Transport ((DAET) mathematical model to a study of the climate of Venus (Mulholland and Wilde,
2020) [1]. Two subsequent criticisms made of the DAET model and its application to the planet
Venus are:
1. That the intensity of the dim sunlight is too weak to fully energise the surface of the planet
at the base of a 63.4 km thick troposphere.
2. Consequently, the temperature at the base of the atmosphere of 699 Kelvin (426OC) has a
value that far exceeds the effective solar radiative thermodynamic temperature of the
surface insolation received by Venus (Table 1).
Table 1 : Venus Lit Hemisphere Illumination Interception Geometry.
At its base Venus has a dense atmosphere with a value of 69.69 kg/m3 (Table 2), while this is far less
than the density of liquid water (1,000 kg/m3), the oceans of Earth do provide a model as to how the
topside of a planetary atmosphere can be heated. On Earth the photic zone is that shallow part of
the ocean (typically less than 200 m water depth) where sunlight energy is absorbed and the water is
heated.
On Venus the average post-albedo insolation received by the lit hemisphere is 299 W/m2, which
equates to a thermodynamic temperature of 269.5 Kelvin (-3.6OC) (Figure 2). This average intensity
will apparently provide heating for only the upper 5 km of the troposphere at heights above 58.4
Km, where the lapse rate reduced air temperature is below the 269.5 Kelvin value (Figure 3).
Location
Latitude
(Degrees)
Elevation
Angle
(Degrees)
Sine Angle
Solar Power
Intensity
(W/m2)
Effective Flame
Temperature
(Kelvin)
Quenching
Limit Height
(Km)
Air Pressure
(hPa)
North Pole 90.00
Z1: 85 5 0.08715574 52.15 174.14 88.092 0.37
Z1: 80 10 0.17364818 103.89 206.89 76.617 5.84
Z1: 75 15 0.25881905 154.85 228.60 70.502 20.89
Z1: 70 20 0.34202014 204.63 245.10 64.669 64.11
Z1: 65 25 0.42261826 252.85 258.41 61.065 123.05
Z5: Hemisphere
Average Flux
60 30 0.50000000 299.15 269.51 58.461 192.90
Z1: 55 35 0.57357644 343.17 278.92 56.454 269.32
Z1: 50 40 0.64278761 384.58 286.97 54.850 348.82
Z1: 45 45 0.70710678 423.06 293.90 53.540 428.56
Z1: 40 50 0.76604444 458.32 299.84 52.461 505.92
Z1: 35 55 0.81915204 490.10 304.91 51.570 578.81
Z1: 30 60 0.86602540 518.14 309.18 50.839 645.33
Z1: 25 65 0.90630779 542.24 312.71 50.247 703.99
Z1: 20 70 0.93969262 562.22 315.55 49.798 751.53
Z1: 15 75 0.96592583 577.91 317.73 49.474 787.52
Z1: 10 80 0.98480775 589.21 319.27 49.244 813.95
Z1: 5 85 0.99619470 596.02 320.19 49.107 830.05
Z1: Equator 0 90 1 598.30 320.50 49.061 835.52
Illumination Intensity versus Solar Elevation
Figure 2: Venus Lit Hemisphere Illumination Interception Geometry
At depths in the atmosphere below this average insolation level the average energy contained in the
sunlight is less than the ambient temperature of the surrounding air, so no heating is apparently
possible. However, and perhaps more importantly the local intensity of the insolation at solar zenith
has sufficient power to heat the Venus atmosphere down to a level of 49 km, in a column that is 14.4
km thick (Figure 3).
Figure 3: Venus Atmospheric Solar Radiant Thermal Heating Pool
It is this concentration of energy at the solar zenith, heating the upper air that creates the bow-wave
disruptor observed in the centre of the blue disk, as the thermal impact of the solar zenith travels
around the planet (Figure 1).
The energy imparted by the sun into that zenith induced bow-wave powers the circulatory system of
the upper atmosphere. So, sun clearly heats
the top of the Venus atmosphere. However, ns the heated topside
atmosphere of Venus is a compressible gas held at high elevation in a gravity field. This has clear
implications for the process of surface heating by full troposphere mass-motion solar forced
convection overturn of a compressible gas in the presence of a gravity field.
In order to study this circulation process using the DAET climate model we have created a pressure
profile model for the Venus troposphere at 1 metre increments. We have applied this calculation
from the surface to the lower stratosphere, a modelled vertical height of 100 kilometres (Mulholland
and Wilde, 2021) [2].
Two equations of state are used to achieve this objective, these are the pressure, volume,

shells, used to calculate the reduction in strength of the gravity field as the height above the surface
of Venus increases. For the purpose of this study, we have created a set of four linked predictive
lapse rate equations based on published data (Justus and Braun, 2007) [3]. These equations are
used as the fundamental temperature control of the tropospheric pressure profile. The temperature
data that controls these equations is calibrated to a surface datum global average temperature for
Venus of 699 Kelvin (Singh, 2019) [4].
Method:
Our spreadsheet analysis of the pressure profile of the Venusian atmosphere is built on the following
Baseline Parameters:
1. The surface pressure measured in Pascal.
2. The surface temperature measured in Kelvin.
3. The Molecular Weight of the Venus atmosphere measured in g/mole.
4. The surface gravity of Venus measured in m/s2.
5. The planetary mass of Venus in kg.
6. The mean radius of Venus in metres.
Figure 4: Venusian Atmosphere: Temperature versus Altitude
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7. A set of four predictive temperature Lapse Rate equations for the atmosphere of Venus
based on published data [3] measured in K/m and corrected to a surface datum value of 699
Kelvin [4] (Figure 4).
8. 
law, a pressure profile is created for the atmosphere, applying the predictive lapse rate
equations as temperature control over the relevant height intervals (Figure 5).
The predictive lapse rate equations used in the pressure profile model are listed in Table 2.
Table 2: Predictive Lapse Rate Equations.
The datum parameters used for the pressure profile analysis are listed in Table 3.
Table 3: Datum Values
Item He ight (m) Predictive Temperature Lapse Rate Equation
Temperature
(Kelvin)
Pressure
(hPa)

(kg/m3)
LR1: Lapse Rate: Start of
Troposphere Equation
0
=-7.967*I2/1000+699 699.000 93,219.00 69.69140601
LR1: Lapse Rate Tie Point: End of
Troposphere Equation
39,220
=-7.967*I39222/1000+699 386.534 3,001.63 4.058083903
LR2: Lapse Rate Tie Point: Start of
Solar Heating Convection Equation
39,221
= -6.71*I39223/1000 + 649.7 386.527 3,001.27 4.057677378
LR2: Lapse Rate Tie Point: End of
Solar Heating Convection Equation
50,000
= -6.71*I50002/1000 + 649.7 314.200 729.81 1.213822584
LR2: Lapse Rate Tie Point: Start of
Tropopause Convection Overshoot
Equation
50,000
= -6.71*I50002/1000 + 649.7 314.200 729.81 1.213822584
LR2: Lapse Rate: End of Tropopause
Convection Overshoot Equation
70,000
= -6.71*C20032/1000 + 649.7 180.000 16.67 0.048388192
LR3: Lapse Rate Tie Point: Start of
Tropopause Breaking Equation
50,001
= 0.092*(I50003/1000)^2 - 15.26*I50003/1000 +
847.2
314.194 729.70 1.213669421
LR3: Lapse Rate Tie Point: End of
Tropopause Breaking Equation
70,477
= 0.092*(I70479/1000)^2 - 15.26*I70479/1000 +
847.2
228.686 20.99 0.04796827
LR4: Lapse Rate Tie Point: Start of
Stratosphere Equation
70,478
=( 0.0027*(I70480/1000)^3 - 0.5809*(I70480/1000)^2
+ 38.611* (I70480/1000)- 537.31)+(-
0.0146*(I70480/1000)^2 + 1.594*(I70480/1000) -
54.84)
228.683 20.99 0.047959369
LR4: Lapse Rate: End of
Stratosphere Equation
100,000
=( 0.0027*(I100002/1000)^3 -
0.5809*(I100002/1000)^2 + 38.611* (I100002/1000)-
537.31)+(-0.0146*(I100002/1000)^2 +
1.594*(I100002/1000) - 54.84)
173.350 0.01588 4.78577E-05
Results:
In order to verify our DAET climate model of Venus, we first re-calibrated the model against the new
surface datum temperature of 699 Kelvin [4]. This process was achieved by reducing the energy
intensity flux partition ratio to an atmosphere retained percentage of 98.071%, down from the
previously published value of 99.1138% [1]. This adjustment is in line with our modelling concept
that the average global surface temperature of a planet is a function of the energy flux partition
ratio between the retained atmospheric energy in the troposphere, and the radiant energy loss to
space from the stratosphere (Table 4).
Table 4: Adiabatic Model of Venus showing Internal Energy Recycling for Both Hemispheres
The key results for the atmosphere of Venus are listed in
Table 5 and displayed in Figure 5.
These results include the following:
1. The average post-albedo irradiance for the lit hemisphere of Venus is 299 W/m2, this
intensity (Table 1, Z5) converts to a thermodynamic temperature of 269.5 Kelvin (-3.6OC).
This temperature occurs at an altitude of 58.46 Km and a pressure of 192.9 hPa. By
geometry the average intensity value of 299 W/m2 also occurs at a solar elevation angle of
30o (Figure 2).
2. The post-albedo solar zenith irradiance for Venus is 598.3 W/m2, this maximum possible
intensity (Figure 3, Z1) converts to a thermodynamic temperature of 320.5 Kelvin (47.3OC).
This temperature occurs at an altitude of 49.06 Km and a pressure of 835.5 hPa (Table 5).
Table 5: Predicted Pressures
Figure 5: Venusian Tropopause: Temperature versus Altitude
Item Hei ght (m) Predictive Temperature Lapse Rate Equation
Temperature
(Kelvin)
Pressure
(hPa)

(kg/m3)
PE =mgh (Joules)
Z6: Mean Air Temperature (699
Kelvin)
0
=-7.967*I2/1000+699 699.000 93,219 69.69140601 0
Z1: Solar Zenith (albedo applied) 49,061 = -6.71*I49063/1000 + 649.7 320.50 1 835.52 1.3623 26292 5 83,372
Z5: Space Incoming Captured
Radiation (W/m2)
58,461
= 0.092*(I59315/1000)^2 - 15.26*I59315/1000 +
847.2
269.512 192.90 0.374026892 190,285
Z8: Night Tropopause 63,240 = -6.71*I63242/1000 + 649.7 225.360 76.31 0.17 6958051 97,234
Z9: Tropopause Ceiling 100 hPa,
235.84 Kelvin, 62.2 32 Km
62,232
= 0.092*(I62234/1000)^2 - 15.26*I62234/1000 +
847.2
253.839 100 0.205873161 111,356
Z10: Mean Tropopause 63,337 = -6.71*C13369/1000 + 649.7 224.709 74.83 0.174025191 95,766
Z11: Lit Tropopause 63,433 = -6.71*C13465/1000 + 649.7 224.065 73.39 0.171162378 94,331
Z12: Dark Side Radiant Partition is
1.0929% (W/m2)
71,156
=( 0.0027*(I71158/1000)^3 - 0.5809*(I71158/1000)^2
+ 38.611* (I71158/1000)- 537.31)+(-
0.0146*(I71158/1000)^2 + 1.594*(I71158/1000) -
54.84)
226.299 18.33 0.042334801 26,106
Z13: Mean Radiant Exit
Temperature (Kelvin)
71,062
=( 0.0027*(I71064/1000)^3 - 0.5809*(I71064/1000)^2
+ 38.611* (I71064/1000)- 537.31)+(-
0.0146*(I71064/1000)^2 + 1.594*(I71064/1000) -
54.84)
226.631 18.68 0.043076419 26,529
Z14: Space Outgoing Radiation
Balance (W/m2)
71,062
=( 0.0027*(I71064/1000)^3 - 0.5809*(I71064/1000)^2
+ 38.611* (I71064/1000)- 537.31)+(-
0.0146*(I71064/1000)^2 + 1.594*(I71064/1000) -
54.84)
226.631 18.68 0.043076419 26,529
Z15: Vacuum Planet Equation
Expected Te of Venus (226.627
Kelvin)
71,063
=( 0.0027*(I71065/1000)^3 - 0.5809*(I71065/1000)^2
+ 38.611* (I71065/1000)- 537.31)+(-
0.0146*(I71065/1000)^2 + 1.594*(I71065/1000) -
54.84)
226.627 18.68 0.043068467 26,525
Z16: Lit Side Radiant Partition is
1.0929% (W/m2)
70,985
=( 0.0027*(I70987/1000)^3 - 0.5809*(I70987/1000)^2
+ 38.611* (I70987/1000)- 537.31)+(-
0.0146*(I70987/1000)^2 + 1.594*(I70987/1000) -
54.84)
226.902 18.97 0.043692829 26,880
Z17:r Rising Air Droplet Cloud Tops
(260 Kelvin)
58,077
= -6.71*C9600/1000 + 649.7 260.003 201.35 0.404689223 204,558
Z17:s Stable Air Droplet Cloud Tops
(260 Kelvin)
60,673
= 0.092*(I60675/1000)^2 - 15.26*I60675/1000 +
847.2
260.002 131.82 0.264950232 139,792
Z18:r Rising Air Latent Heat
Freezing Point of 75%wt H2SO4
(250.0 Kelvin)
59,568
= -6.71*C9600/1000 + 649.7 249.999 154.26 0.322448865 167,091
Z18:s Stable Air Free zing Point of
75%wt H2SO4 (250.0 Kelvin)
63,266
= 0.092*(I63268/1000)^2 - 15.26*I63268/1000 +
847.2
249.999 82.97 0.173424014 95,331
3. The DAET adiabatic climate model of Venus predicts for the lit hemisphere a thermal
emission intensity (Table 4, Z16) of 150.4 W/m2 and a thermodynamic temperature of 226.9
Kelvin (minus 46.3OC). This temperature occurs at an elevation of 70.99 Km and at a
pressure of 19 hPa (Table 5).
4. The DAET adiabatic climate model of Venus also predicts for the dark hemisphere a thermal
emission intensity (Table 4, Z12) of 148.75 W/m2 and a thermodynamic temperature of
226.3 Kelvin (minus 46.8OC). This temperature occurs at an elevation of 71.15 Km and at a
pressure of 18.33 hPa (Table 5).
5. The modelled height of the Venusian droplet cloud planetary veil (Z17:s) occurs at an
elevation of 60.67 Km and a temperature of 260 Kelvin [5] with an associated pressure of
18.33 hPa (Table 5).
6. The measured freezing point of 75% wt H2SO4 (Z18:s) s 250 Kelvin (-23OC) [5]. This
temperature is found at a model altitude of 63.27 Km, and a pressure of 83 hPa (Table5).
This near association between the stable air freezing point of concentrated sulphuric acid,
the main condensing volatile in the Venus atmosphere, and the DAET modelled height of the
convection tropopause warrants further study. Solid aerosol particles are efficient thermal
emitters and can enhance atmospheric thermal radiation loss to space through the
transparent lower Stratosphere (Figure 4).
The Energy Consequence of Air Convection in a Gravity Field.
At the modelled convection tropopause of Venus, over 63.3 km  (Table 5,
Z10), a cubic metre of Venusian air has a mass of 174 g and possesses a potential energy of 95.7
Kilojoules. All air mass held aloft in a gravity field contains a considerable quantity of potential
energy. On descent to the surface this air will undergo adiabatic heating and consequent air
temperature rise as it falls . In doing so it loses potential energy by the
process of conversion to kinetic energy (Figure 6).
Figure 6: Scaled Comparison Chart of Pressure, Gravity, Discrete Mass, Discrete Potential Energy (PE)
and Cumulative PE Curves for Venus
Discussion - The Rationale:
On Venus the solar forced radiant heating of the upper troposphere at the zenith creates a process
of pole-ward advection of heated air. Figure 2 shows how the upper atmosphere of the lit
hemisphere of Venus intercepts the energy of the sunlight in a pattern of concentric rings of
intensity centered around the zenith, the point at which the overhead sun provides the maximum
flux that heats the atmosphere. When the sun heats the cold upper part of the Venusian
atmosphere it will distort the lapse rate slope to the warm side. That forces the lapse rate profile
downward. That compression then steepens the lapse rate slope lower down which causes
convection to accelerate as a negative compensation mechanism.
As Venus slowly turns from east to west, the locus of the solar zenith tracks along the equator
towards the east creating a point of disturbance in the upper air. This forms a bow shockwave
disruptor dividing the equatorial flow of the zonal circulating winds which are forced apart and made
to track towards higher latitudes (Figure 1)
Due to the conservation of angular momentum associated with the slow planetary rotation of
Venus, these winds travel faster than the ground surface below them and are called super-rotation
the rim of the lit
hemisphere. Here the illumination intensity of the low angle sun within 5O of the terminator does
not have sufficient power to heat the tropospheric air. At the poles of Venus, the low power of the
sunlight, combined with the angular momentum of the super-rotational winds creates a cyclonic
vortex which drives the air down into the deep atmosphere below (Ignatiev, et al, 2009) [6].
Figure 7: Planetary Rotation and the Conservation of Angular Momentum
This forced descent of the topside heated air, means that the compressible air undergoes adiabatic
heating as it falls in the gravity field of Venus. The descending mass flow within the polar vortex

hemisphere encompassing Hadley cell (Figure 3). By this means the compressed air is heated as it
falls and the apparent thermal limit set by insolation at the top of the atmosphere is easily
surpassed.
Figure 3 shows the impact of upper atmosphere heating, the circulation system powered by the
solar zenith constantly replenishes the forced descent vortex over both poles which heats the
surfaces beneath. That energy then flows across the entire Venusian surface so that it can reach
temperatures much higher than predicted by the Stefan-Boltzmann (S-B) radiation equation. The
greater the mass of the Venus atmosphere the greater the systems efficiency, and the more heat
that will be delivered to the surface by the air descent at the poles.
The piston-like hydrostatic circulation is fuelled by whatever energy is available from any source, but
can never exceed the amount of energy required to balance the upward pressure gradient force with
the downward force of gravity. The pattern of differing lapse rate slopes within the vertical plane is
infinitely variable, but must always average out to the slope dictated by mass and gravity.
The Utility of the DAET climate model
The key physical process that the DAET climate model describes is that mobile compressible fluids
circulating within a gravity field over and above the surface of a rotating terrestrial planet, will at the
same time capture, store and transport energy in various guises. Not all of these are thermal and so
not all are subject to radiative loss. While energy can flow from cold to hot (e.g., the meteorological
process of cooling rain falling onto the surface of a hot desert below), however heat being a directed
dynamic process cannot flow from cold to hot (e.g., Unconfined rivers of water cannot flow uphill).
Mass motion is a process that generates a system lag because it is inherently slower than radiative
processes. Convection is also a process that deals with albedo variations because convection just
shifts to equalise these perturbations. There is still enough room for internal climate variability as
the system lags somewhat in response to destabilising influences, but it always gets there quickly
enough to retain the atmosphere in a dynamically stable state.
The Venus surface is at the temperature it is simply because that is the temperature needed to
balance the mass of the atmospheric gases against gravity. It makes no difference what the source
of that energy is. It is the same for stars in the cold of space and the gas planets far from the sun.
Convection always settles at a level that keeps the gases suspended against the downward force of
gravity. Until, in the case of stars, a fusion reaction starts whereupon convection adopts a new
equilibrium.
It is by this mechanism of circulating mass motion of a compressible gas acted on by a gravity field,
within the context of a rotating spherical planet that surface thermal enhancement is created, and
which we now propose to call the Maxwell Mass Effect after the work of James Clerk Maxwell
(Maxwell, 1868) [7].
Conclusion - The Venus Heating Paradox Explained:
In conclusion we will now address the matter of the high surface temperature of the planet Venus,
and the paradox of the dim surface sunlight not being able to create this 699 Kelvin global average
temperature.
The process of deep atmospheric convection throughout the whole 62.2 km (100 mbar limit) of the
Venus troposphere means that sunlight heated air at the top of the atmosphere can and does
ace. Instead of solar radiation, this process of energy delivery to the
surface occurs by the mechanisms of full troposphere planetary rotation-forced mass-motion, the
circulation of polar vortex descending air and heating by adiabatic auto compression.
The warming at the surface of Venus is from the mechanical process of convection, and any
potential warming effect from downward radiation is neutralised by convective adjustments.
Instead, descending air heats both itself and the surface beneath via reconversion of potential
energy (PE) to kinetic energy (KE). The atmosphere is held aloft by potential energy which is not
thermal energy. Heat cannot be amplified, but it can be stored in a non-kinetic form as potential
energy so that it is not then sensed as temperature.
Potential energy is in effect a form of Latent Heat. This store of energy within mass is then returned
again as temperature at a later time and critically at a lower elevation. So, as long as there is
constant mass motion recycling to and fro between PE and KE as the air moves vertically within a
gravity field, then the surface will receive kinetic energy from the descending air and be warmed.
To maintain long term hydrostatic equilibrium the total energy retained at the surface must be a
dynamic equilibrium that is just right to support the weight of atmospheric gases against the
downward force of gravity. It makes no difference whether the source of the necessary energy is
from the sun, the surface, volcanic outbreaks, atmospheric opacity, particulate aerosols or anything
else.
We also know that atmospheres vary hugely in composition, and we aver that the way the
composition of an atmosphere is sorted into differing compositional layers will affect the vertical
boundaries between those layers. Thus, a tropopause can vary in height somewhat depending on
the various compositionally induced stratifications within a  atmosphere. However, if an
atmosphere is to be retained by a planet, then the average lapse rate slope between surface and
space must always net out to the slope specified by mass and gravity.
Convection always adjusts in order to balance energy into the system from space with energy out to
space derived from the net combination of all energy transfer mechanisms between surface and
atmosphere. If it were not so then the tiniest radiative imbalance would prevent the formation and
retention of an atmosphere. We know that atmospheres are ubiquitous and last for geological eons
in the absence of catastrophe, so it must be that convection neutralises all 'normal' radiative
imbalances.
However, our DAET concept needs to apply to every scenario, whatever the density or opacity of an
atmosphere the final outcome must be the same, and the cause will be a combination of heating of
upper levels and heating of the surface with the proportions related to atmospheric opacity to
radiation. It is a universal rule of meteorology that any temperature induced density variations in
the vertical plane will lead to convective overturning in the entire depth of an atmosphere, with
consequent heating of the surface.
When the energy present in a mass aloft is converted from PE (not thermal) to KE (thermal) by
forced descent then that solves the problem of the apparent radiative limit. Radiation does not limit
the mechanical transformation of energy between PE and KE in convection. Climate theorists who
have been fixated on radiation miss this point. Potential energy is a form of latent heat. Kinetic
energy (motive heat) is the direct consequence of the conversion of potential energy during the
descending phase of convective overturning.
As the case of Venus proves insolation does not need to reach the surface to provoke planetwide
deep convection. It is sufficient if insolation beneath the solar zenith creates density differentials at
any point within the mass of an atmosphere. Convection can achieve that because radiative
imbalances alter the lapse rate slope and the rate of convection changes in a negative response to
this forcing (Wilde, 2012) [8].
References
[1] Mulholland, P. and Wilde, S.P.R., 2020. Inverse Climate Modelling Study of the Planet Venus.
International Journal of Atmospheric and Oceanic Sciences, 4(1), pp.20-35.
[2] Mulholland, P. and Wilde, S.P.R. 2021 A Modelled Atmospheric Pressure Profile of Venus:
Venus Gravity Profile 01Mar21.xlsx Research Gate Project Update
[3] Justus, C.G. and Braun, R.D., 2007. Atmospheric Environments for Entry, Descent, and
Landing (EDL) NASA Natural Environments Branch (EV13)
[4] Singh, D., 2019. Venus nightside surface temperature. Scientific reports, 9(1), pp.1-5.
[5] Young, A.T. 1973. Are the clouds of venus sulfuric acid? Icarus 18(4), pp. 564-582.
[6] Ignatiev, N.I., Titov, D.V., Piccioni, G., Drossart, P., Markiewicz, W.J., Cottini, V., Roatsch, T.,
Almeida, M. and Manoel, N., 2009. Altimetry of the Venus cloud tops from the Venus Express
observations. Journal of Geophysical Research: Planets, 114(E9).
[7] Maxwell, J.C., 1868. XXII. On the dynamical theory of gases. The London, Edinburgh, and
Dublin Philosophical Magazine and Journal of Science, 35(236), pp.185-217.
[8] Wilde, S.P.R., 2012. The ignoring of Adiabatic Processes Big Mistake. 
14/12/2012 WordPress.
[9] Jenkins, J.M., Steffes, P.G., Hinson, D.P., Twicken, J.D. and Tyler, G.L., 1994. Radio occultation
studies of the Venus atmosphere with the Magellan spacecraft: 2. Results from the October 1991
experiments. Icarus, 110(1), pp.79-94.
Appendix:
The Pressure Profile Calculation Method:
Starting with the Gas Equation:
P1.V1/T1 = P2 V2/T2 Equation 1
Where:
P1 Is the Initial Gas Pressure measured in Pascal.
V1 Is the Initial Gas Volume measured in Litres.
T1 Is the Initial Gas Temperature measured in Kelvin.
P2 Is the Final Gas Pressure measured in Pascal.
V2 Is the Final Gas Volume measured in Litres.
T2 Is the Final Gas Temperature measured in Kelvin.
In the Troposphere as height above the surface increases, air pressure and air temperature both
decrease. In the open atmosphere the air pressure decrease is a function of air mass and gravity,
while the temperature decrease, the Lapse Rate, is a measured known parameter (Jenkins et al. Fig.
4. 1994) [9]. This then leaves the change in volume with height as the one variable to be calculated.
Equation 1 can be rearranged to give Volume in terms of Pressure and Temperature
V2 = P1.V1.T2/T1 /P2 Equation 2
The next issue to be resolved is to determine the rate of pressure reduction with height.
In a column of air, the pressure is a function of the overlying mass, so if we imagine that the
atmosphere is a stack of one metre cubes of air, then for each one metre rise in height the mass of
the overlying column will be less, and so this mass reduction will cause a pressure reduction which
can be calculated.
Pressure is a force; it is defined as the product of mass times acceleration. In the atmosphere the
accelerat at that level, and this
can be determined gravity law of spherical shells. The value of the surface gravity of a
planet can be calculated by using the Universal Gravity Equatio
its average radius.
But we also need a standard measured quantity of gas.
To do this we have adopted the process used by chemists to find the relationship between the mass
in grams and the volume in litres (dm3) at standard temperature and pressure (STP) for one mole of
gas.
At 273.15 Kelvin (0oC) and 1013.25 hPa (mbar) the volume is 22.414 litres (dm3) and so for air with a
molecular weight of 43.45 g/mol (standard Venus atmospheric composition) the mass contained in
molecular volume (22.414 dm3) will be 43.45g.
Phase 1: Building the Pressure Ladder for the Venus Atmosphere.
Step 1: From knowledge of the surface pressure of the Venusian atmosphere and the value of the
surface gravity of Venus, compute the total atmospheric mass in a column bearing down on 1 square

Using the equation of force F = m.a we find that Pressure/Gravity = Mass
For Venus the equation of state is:
9,321,900/8.87039 = 1,050,990.969 kg (1,051 tonnes / sq metre).
Step 2
temperature and pressure conditions on the surface of Venus.
Using the constant Pressure Volume Temperature relationship of P1.V1/T1 = P2.V2/T2 we can establish
the unknown V2 (the volume of 1 mole of gas at the surface of Venus).
V2 = P1.V1.T2/T1 /P2
V2 = 101,325 * 22.414 * 699 / 273.15 / 9,321,900 = 0.623 Dm3 (Litres)
Step 3: Compute the density of the unit mole of compressed gas at the surface of Venus under
ambient surface conditions.
Using the standard formula: Density = Molecular Weight / Volume
Surface Density = 43.45 / 0.623 = 69.691 Kg/m3.
Step 4: Convert the Gas Density to Discrete Mass of Gas per Unit Metre Cube.
Discrete Mass = 69.691 Kg
Step 5: Establish the Mass of Gas in the Atmospheric Column lying above this Unit Cube.
Mass Bearing Down = Column Mass minus Unit Mass
Mass Bearing down at 1 metre elevation = 1,050,900.969 69.691 = 1,050,831.277 Kg.
We now have sufficient information to begin climbing the Pressure Ladder of the Venus Atmospheric
Profile at Unit Steps of 1 metre increment.
Phase 2: Climbing the Pressure Ladder of the Venus Atmosphere.
Step 1: Compute the new P2, The Base Pressure of the Overlying Column of Gas.
Using the standard equation of Force: F = m.a where m is the mass of the overlying column and a is
the value of gravity at the surface of Venus.
P2 = 1,050,831.277 * 8.87039 = 9,321,282 Pascal
Step 2: Compute the new value of T2 one metre above the base surface temperature of Venus using
the relevant predictive Tropospheric Lapse Rate equation in K/m and inputting the height value h
where h is the full distance above the surface in metres.
T2 = 699-7.967*h/1000= 698.9920 Kelvin
Step 3: Compute the new value of V2 at one metre elevation using the standard Pressure Volume
Temperature relationship V2 = P1.V1.T2/T1 /P2
V2 = 9,321,900 * 623.458 * 698.9920 / 699.0000 / 9,321,282 = 623.492 cm3
N.B. The Volume increase for V2 is due to the reduction in Pressure P2 (which increases volume)
dominating over the reduction in Temperature T2 (which decreases volume) in the equation as we
step up the ladder.
Step 4: Compute the Density of the New Unit Cube of Gas.
Using the standard formula: Density = Molecular Weight / Volume
New Density = 43.45 / 623.492* 1,000 = 69.68758 Kg/m3.
Step 5: Convert the New Gas Density to Discrete Mass of Gas per Unit Cube.
Discrete Mass = 69.688 Kg
Step 6: Subtract the Discrete Unit Cube Mass from the Column Mass at this level to give the
overlying Column Mass Bearing down on this Unit Cube.
Overlying Mass = Current Column Mass minus Unit Mass
Bearing Down Mass = 1,050,831.277 69.688 = 1,050,761.590 Kg
Step 7and set M2 to be the unit mass to compute the
reduced value of planetary gravity at this new increment of 1 metre elevation.
We now have the required information to climb one step of the ladder and continue the calculation
cycle. The bearing down Mass at the top of the unit cube defines the Pressure at the base of the
next unit cube above.
Step 8: Return to Step 1 and continue climbing the Pressure Ladder by 1 metre increments.
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