The Venusian Insolation Atmospheric Topside Thermal Heating Pool

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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
This paper develops the results of the application of the Dynamic-Atmosphere Energy-Transport ((DAET) mathematical model to a study of the climate of Venus (Mulholland & Wilde, 2020) and addresses the following two issues: 1) That the intensity of the dim sunlight is too weak to fully energise the surface of the planet Venus at the base of a 63.4 km thick troposphere.
2) Consequently, the temperature at the base of the atmosphere of 699 Kelvin (426 O C) (Singh, 2019, pp.1-5) has a value that far exceeds the effective solar radiative thermodynamic temperature of the surface insolation received by Venus (Table 1).At its base Venus has a dense atmosphere with a value of 69.69 kg/m 3 (Table 2), while this is far less than the density of liquid water (1,000 kg/m 3 ), 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/m 2 , which equates to a thermodynamic temperature of 269.5 Kelvin (-3.6 O C) (Figure 2).This average intensity will apparently provide heating for only the upper 5 km of the troposphere at heights above 58.4Km, where the lapse rate reduced air temperature is below the 269.5 Kelvin value (Figure 3).

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
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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 (Limaye, 2010).So, just like the sun heats the top of the Earth's oceans, the sun clearly heats the top of the Venus atmosphere.However, unlike water in the Earth's oceans 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 a pressure profile model for the Venus troposphere at 1 metre increments has been created.This calculation has been applied from the surface to the lower stratosphere, a modelled vertical height of 100 kilometres (Mulholland & Wilde, 2021).
Two equations of state are used to achieve this objective, these are the Pressure, Volume, Temperature (PVT) version of Boyle's law, and the application of Newton's gravity law of spherical 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 a set of four linked predictive lapse rate equations based on published data has been created (Justus & Braun, 2007).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, pp. 1-5).

Method
The spreadsheet analysis of the pressure profile of the Venusian atmosphere presented here is built on the following Baseline Parameters (Williams, 2023): 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/s 2 .
5) The planetary mass of Venus in kg.
6) The mean radius of Venus in metres.8) Using the two physical relationships of Boyle's gas law, and Newton's spherical shell gravity 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.The datum parameters used for the pressure profile analysis are listed in Table 3.

Result
In order to verify the DAET climate model of Venus, the model was first calibrated against the new surface datum temperature of 699 Kelvin (Singh, 2019, pp. 1-5).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% (Mulholland & Wilde, 2020, pp. 20-35).This adjustment is in line with the 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 from the Boyle's Law Pressure Model Analysis for the atmosphere of Venus (Mulholland & Wilde, 2021) 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/m 2 , this intensity (Table 1, Z5) converts to a thermodynamic temperature of 269.5 Kelvin (-3.6 O C).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/m 2 also occurs at a solar elevation angle of 30 o (Figure 2).
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2) The post-albedo solar zenith irradiance for Venus is 598.3W/m 2 , this maximum possible intensity (Figure 3, Z1) converts to a thermodynamic temperature of 320.5 Kelvin (47.3 O C).This temperature occurs at an altitude of 49.06 Km and a pressure of 835.5 hPa (Table 5).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/m 2 and a thermodynamic temperature of 226.3 Kelvin (minus 46.8 O C).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 (Young, 1973, pp. 564-582) with an associated pressure of 18.33 hPa (Table 5).
6) The measured freezing point of 75% wt H 2 SO 4 (Z18:s) is 250 Kelvin (-23 O C) (Young, 1973, pp. 564-582).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).
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The Energy Consequence of Air Convection in a Gravity Field
At the modelled convection tropopause of Venus, over 63.3 km above the planet's surface (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 towards the planet's surface.In doing so it loses potential energy by the process of conversion to kinetic energy (Figure 6).

Discussion
On Venus the solar forced radiant heating of the upper troposphere at the zenith creates a process of pole-ward advection of heated air that feed the planet's polar vortices (Luz et al., 2011).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.

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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 winds (Zasova et al., 2007).Eventually these circulating winds reach the planet's poles at a point on the rim of the lit hemisphere.Here the illumination intensity of the low angle sun within 5 O 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).

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 provides a hydrodynamic piston drive that causes the planet's air to circulate vertically in a giant 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 (Lacis & Hansen, 1974).
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 system's efficiency, and the more heat that will be delivered to the surface by the air descent at the poles.
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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 it is proposed here to call the Maxwell Mass Effect after the work of James Clerk Maxwell (Maxwell, 1868).

Conclusion: The Venus Heating Paradox Explained
In conclusion 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 will now be addressed.
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 deliver heat to the planet's surface.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.
It is known that planetary atmospheres vary hugely in composition, and 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 planet's 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.It is known 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.
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V 2 = P 1 .V 1 .T 2 /T 1 /P 2 Equation 2The 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 that the atmosphere is modelled as 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 acceleration acting on the air parcel at rest in the column is the planet's gravity at that level, and this can be determined by Newton's gravity law of spherical shells.The value of the surface gravity of a planet can be calculated by using the Universal Gravity Equation, and knowing the planet's mass and its average radius.
But a standard measured quantity of gas is also required.
To do this the process used by chemists to find the relationship between the mass in grams and the volume in litres (dm 3 ) at Standard Temperature and Pressure (STP) for one mole of gas has been adopted here.
At 273.15 Kelvin (0 o C) and 1013.25 hPa (mbar) the volume is 22.414 litres (dm 3 ) 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 dm 3 ) 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 metre of the planet's surface.
Using the equation of force F = m.a this equation can be restated as Pressure/Gravity = Mass For Venus the equation of state is: 9,321,900/8.87039= 1,050,990.969kg (1,051 tonnes/sq metre).
Step 2: Compute the volume change for I mole of gas from STP at the Earth's surface to the ambient temperature and pressure conditions on the surface of Venus.
Using the constant Pressure Volume Temperature relationship of P 1 .V 1 /T 1 = P 2 .V 2 /T 2 this establishes the unknown V 2 (the volume of 1 mole of gas at the surface of Venus).
Step 4: Convert the Gas Density to Discrete Mass of Gas per Unit Metre Cube.

Figure 6 .
Figure 6.Scaled Comparison Chart of Pressure, Gravity, Discrete Mass, Discrete Potential Energy (PE) and Cumulative PE Curves for Venus