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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

0 0.00000000 0.00

Z1: Heating

Limit 86.43 3.57 0.06226276 37.25 160.10 110.000

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

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

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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 7and 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.