2. Climate Forward Modelling.

The process of Forward Modelling creates a numerical prediction, that must be matched and verified against external data for the model to be both valid and useful. The modelling process starts with the identification of the set of fundamental principles, that contain the irreducible minimum set of axioms, from which the actions of a system are designed and constructed. With the set of first principles established and measured, then the mathematical algorithm that combines these elements can be created.

With forward modelling studies of a planet’s energy budget, the first and overarching assumption is that the only way that a planet can lose energy is by thermal radiation from the planetary body to space. This assumption is not in dispute, and it leads to the adoption of the Stefan-Boltzmann (S-B) equation of thermal radiation, which is used to establish the direct relationship between power intensity flux in Watts per square metre (W/m²) and the absolute thermal temperature of the emission surface in Kelvin (K).

The second critical assumption made in the analysis of a planet’s energy budget, is that it receives incoming thermal energy in the form of insolation from a single central star. Solar system planets orbit around this central source of light, and consequently all planets have both a lit (day) and a dark (night) hemisphere.

A technique for establishing the energy budget of a planet, and hence how the power being consumed is distributed within its climate system, is a technical challenge that has already been addressed by astronomy. An equation was required that could be used to compute the average surface temperature of any planet, by establishing its thermal emission temperature under a given insolation loading. To solve this problem, a set of modelling assumptions were made that include the following simplifications: –

1. That the planet being observed maintained a constant average surface temperature over a suitably long period of time.

2. To make this assumption valid, the total quantity of solar energy intercepted by the planet is averaged out over its annual orbital year.

3. This annual averaging therefore removes the effect of distance variation from the Sun, inherent for the trajectory of any planet’s elliptical orbit.

Next the complex problem of how a planetary orb intercepts solar energy, and how this sunlight energy is distributed over the planet’s surface, was addressed. Planets contain the following geometric features in common:

1. They are near-spherical globes.

2. They are only lit on one side from a sun that is located at a focus of their orbit’s ellipse.

3. They often (but not always) have a daily rotation rate that is significantly faster than their annual orbital period.

4. They commonly have an obliquity or axial tilt, although each planet’s angle of tilt is unique.

Given the above list of distinct features, it is clear that the computation for the surface capture of solar energy on an orbiting, rotating, axially tilted planet is a complex mathematical calculation. To address this complexity the following simplification was applied: –

That all planets intercept solar energy at their orbital distance, as if they are a disk with a cross-sectional area that is equal to the planet’s radius (i.e. π R²). However, due to daily rotation and seasonal tilt, planets emit radiation from all parts of their surface over the course of each year.

Therefore, the total surface area of the planet that emits thermal radiation to space is four times the surface area of its intercepting disk (i.e. 4π R²). It is this geometric fact that is responsible for the “divide by 4” rule that is contained within the calculation of planetary radiative thermal balance.

Having devised a simplified way of calculating the amount of energy that the total surface of an orbiting, rotating, axially tilted planet would receive during the course of its year, we can now move to the next stage of the calculation. Namely, the computation of the annual average surface temperature associated with this energy flux.

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 distance from the sun.”

However, when we apply this logic to calculate the average surface temperature of the planet with a gaseous atmosphere, such as the planet Venus, then the parameters appropriate for Venus at its average orbital distance from the Sun, do not produced the known surface temperature of 464^oC (737 Kelvin) (Williams, 2018). Instead the equation produces a value of -46.4^oC (226.6 Kelvin), some 510^oC too low. (Table 1).

Table 1: Venus Atmosphere Parameters.

The Dynamic-Atmosphere Energy-Transport Model (DAET) of Planetary Climate, presented here, is a 2-dimensional forward model that preserves the dual hemisphere component of planetary illumination (Fig. 2). The forward model represents a planetary globe with two environmentally distinct halves. A dayside lit by a continuous incoming stream of solar energy which creates an energy surplus, and a nightside that is dark and has an ongoing energy deficit, due to the continuous exit to space of thermal radiant energy. Consequently, a mobile fluid atmosphere that transports energy from the day to the night side is the fundamental requirement of this climate model.

In order to study the process of atmospheric energy transmission within the model climate system of Noonworld, a number of simplifications have been made. The primary one is that the planetary atmosphere in the model has total clarity to incoming solar radiation, it also contains no greenhouse gases and therefore has no opacity to outgoing thermal radiation. The model has a free-flowing atmosphere of pure Nitrogen gas that connects the two hemispheres. Consequently, because the model atmosphere is fully transparent to all wavelengths, it can only gain or lose heat from the solid surface at its base.

Because Noonworld has only one hemisphere that is permanently lit, we need to invoke a “Divide by 2” rule that relates the cross-sectional area of the Noonworld disc’s interception of solar irradiance to the surface area of its single illuminated hemisphere. This divide by 2 relationship is valid for any planet with captured-rotation illuminated by a single sun.

Figure 2: Basic Noonworld Globe with Initial Static Model: Showing Energy Vectors and Start-Up Energy Partitions

3.2. Starting the Dynamic-Atmosphere Energy-Transport (DAET) Engine from Cold.

On Noonworld the atmospheric process of energy transmission begins on the sunlit side (Fig. 2). Here the solid surface is illuminated and warms as it receives radiant energy from the sun. As it warms it also warms the air above it by conduction. This warmed air then rises by convection, and because it is fully transparent, and also because it is no longer in contact with the ground, it retains all of its energy internally.

The lit ground surface however does not retain all its energy. It cools in two separate ways; it both loses energy to the air above it by surface conduction, and also transmits radiant energy, through the transparent atmosphere, directly out into space. In the forward modelling process, we assign a partition ratio of 50% to conduction and 50% to radiation to study this dual process of energy loss. This assignment is chosen to permit a first assessment to be made of the impact the energy partition process has on the energy budget of the planet.

On the dark side of the planet the ground surface is continuously emitting thermal radiation directly out to space. As this solid surface cools, it also cools the air above it, creating a surface pool of cold dense air. It is a critical feature of this model that as the air cools it retains its mobility, and does not freeze onto the solid surface below. Consequently, the cold dense gaseous lower atmosphere is able to advect back across the planet’s surface to the sunlit side, where it can again be warmed.

As the cold air moves away across the surface of the planet towards the lit hemisphere, more air from above descends onto the dark cold surface, delivering energy to the ground which is also then lost to space by direct thermal radiation. As with the lit surface, we assign an energy partition ratio of 50% to be retained by the advecting air, and 50% to the ground to study this dual process of energy transfer to, and subsequent radiant loss of energy to space from the dark surface.

The process of energy collection on the lit side; energy delivery to the dark side; energy loss by the unlit surface, and then cold dense air return to the source of heat on the lit side, forms a closed loop of energy transport that can then begin to endlessly cycle (Table 2).

Table 2: Starting the Dynamic-Atmosphere Energy-Transport Engine from Cold.

The cycling of air driven by thermal imbalance is a characteristic feature of a Hadley Cell. Because for the cycle to be maintained it must retain energy internally, the Hadley Cell therefore has the capacity to form an energy transmission system, capturing and delivering energy across the planet.

Because the priming stage of the process completed above retains energy within the atmosphere, the next overturning cycle starts with 1 unit of insolation plus ¼ unit of thermal energy left over from the first cycle. Clearly the retention of energy within the atmospheric system by this first cycle overturn means that the radiant energy loss to space does not balance at this point. However, the endless mass movement recycling by the air and the progressive energy retention by the developing Hadley Cell does not grow indefinitely. Our model has two separate geometric series that both tend to different limits, one for the lit and one for the dark surface.

The geometric series for the lit side energy loss to space is: –

Equation 3: ¹/₂ +¹/₈ + ¹/₃₂ + ¹/₁₂₈ ^…. + 2^{-n (odd)} = 2/3

While the geometric series for the dark side energy loss to space is: –

Equation 4: ¹/₄ +¹/₁₆ + ¹/₆₄ + ¹/₂₅₆ ^…. + 2^{-n (even)} = 1/3

Note that the aggregate sum for the limits of both series is: –

2/3 + 1/3 = 1

and so, the total energy recycling system will now be in radiative balance (Table 3).

Table 3: Building the Dynamic-Atmosphere Energy-Transport Forward Model.

We can consider that the consequence of this process of infinite recycling is the formation and maintenance of a dynamic machine made of air (Fig. 3).

Figure 3: Basic Globe with a Stable Diabatic Advection Forward Model: Showing Energy Vectors and Unitary Energy Distributions.

This machine is Noonworld’s single global Hadley Cell, a thermal and mass motion entity formed as the result of diabatic movement of air. The Hadley Cell machine transports air and energy across the planet from a region of energy surplus to a region of energy deficit, and then returns to endlessly repeat the cyclical process of energy transport. (Table 4).

Table 4: Running the Dynamic-Atmosphere Energy-Transport Engine “Warmed Up”.

3.4. Testing the Computational Algorithm within the Diabatic Model of Noonworld.

Using an Excel spreadsheet, a simple repetitive cyclical computation sum can be created in which the descending series of fractions in the geometric series shown in Equations 3 & 4 can be cascaded to any required degree of precision. The degree of precision in the computational algorithm is controlled by the number of repetitive cycles of addition of the declining fractional elements contained within the geometric series. The cascade algorithm requires 14 cycles of repetitive summation to achieve 8 decimals of precision (Table 5).

Table 5: Testing the Cascade Algorithm of the Diabatic Model of Noonworld

3.5. How the Presence of an Atmosphere Distributes the Captured Solar Energy Across a Planet.

Having established the required degree of precision, we now need to test how the Noonworld climate model behaves when standard Venus Insolation parameters are applied. The Venusian annual average solar irradiance is 2601.3 W/m² and the planet’s Bond Albedo is 0.770 (Williams, 2018) which means that the Annual Average Planetary Energy flow that the lit Venusian globe receives is 149.575 W/m² (Table 1). However, for our hypothetical captured-rotation planet Noonworld, because it only ever receives insolation over one hemisphere, the radiation loading will be double this value (Table 6).

Table 6: Internal Energy Recycling on Venus with Equipartition of Energy for Both Hemispheres.

It is this energy flux of 299.15 W/m² (post albedo), that determines the quantity of energy available to drive the Venusian climate system, and this is the insolation energy value that will be used in the Noonworld modelling analysis of Venus, where the “Divide by 2” rule applies.

3.6. Results of Applying the Noonworld Diabatic Model to Venus.

Converting the stable system (Cycle 14) energy values recorded in Table 6 into temperatures in Kelvin by using the S-B equation, shows that the Lit side power intensity flux converts into a day time air temperature of -29.5^oC, while the Dark side power intensity flux converts into a night time air temperature of -62.8^oC (Table 6). The average of these two temperature values produces a global average air temperature of -48.8^oC (Table 6). This temperature is slightly lower than the Vacuum planet temperature for Venus of -46.4^oC (Table 1). The discrepancy arises because we have unevenly distributed the energy flux between the two hemispheres, if we sum these two fluxes then the aggregate value is 299.1495 W/m², which produces a global surface area average of 149.575 W/m², and the Vacuum Planet relationship is satisfied (Table 1).

The forward modelling study shows that the global atmospheric recycling system of Noonworld, while redistributing energy from the lit to the dark hemisphere (Fig. 3), also stores and retains an additional 100% of the solar influx within the atmosphere to give a global energy budget which is 2 times the intercepted insolation flux (Table 7).

The diabatic recycling system has created a global average air temperature of -48.8^oC, however while closely matching the Vacuum Planet relationship (Table 1) the diabatic model has obviously not retained sufficient energy within the atmospheric reservoir to raise the surface Global Air Temperature to the observed Venusian value of 464^oC. (Table 7).

Table 7: Stable Energy Values for Noonworld achieved by Global Air recycling using a 50%A:50%S flux partition at the Ground to Air interface.

Two important facts have now been established about planetary climate on terrestrial globes: –

1. That the presence of a fully transparent mobile-fluid atmosphere can both retain and recycle solar energy within the atmospheric reservoir, and that this recycling achieves a stable energy flow across the planet’s surface.

2. The stable limit of the energy flow within the system is set by the partition ratio of energy between the radiant loss to space of the emitting surface, and the quantity of energy retained and recycled by the air.

We have also established that by using forward modelling techniques to apply an energy partition ratio of 50% surface radiant loss to space, and 50% thermal retention by the air; (hereafter 50_S : 50_A); the average global air temperature of the Noonworld model of Venus is approximately minus 48.8^oC, a value slightly below that achieved by the vacuum planet equation (Equation 2).

Convection is a fluid movement buoyancy process that takes place in the presence of a gravity field. When heated at its base air becomes less dense and more buoyant; because of gravity the warmed air rises away from the source of heat at the surface, to be replaced by cooler air, either arriving from the side (an advection cold front) or from above (convection overturning). The more energy put in to heating the surface the faster the mobile fluid system cycles between hot and cold, in effect the process of convection “steals” energy from the ground. In a dynamic mobile convecting atmosphere a 50_S : 50_A thermal equilibrium energy partition ratio is only rarely ever achieved; so, the partition of energy on the lit side must always be in favour of the air (conduction loss) and not the ground (radiation loss). Consequently, a lit surface thermal equilibrium ratio of 50_S : 50_A should not as a general rule be expected or applied.

4.1. Establishing the Energy Partition Ratio for Noonworld by Inverse Modelling using Venusian Climate System Parameters.

Inverse modelling is the process of establishing the value of a given variable within a modelling algorithm, that can be adjusted to achieve a known target result. Put more simply: inverse modelling is used when we already know the answer but are not sure what the question was. The process of inverse modelling was applied to the Noonworld forward climate model. By constructing a cascade algorithm, the initial unknown energy partition ratio of the lit hemisphere of Venus that creates the planet’s average surface temperature of 464^oC, can be found.

The value of the unknown surface partition ratio can be determined using the Excel Inverse Modelling Tool called “Goal Seek”, when applied to a suitably designed cascade algorithm with sufficient repetitive length. Initial tests were undertaken to establish the number of iterative cycles that are required to create a stable thermal outcome for a given partition ratio. It was established that the more highly asymmetric the partition ratio, the greater the number of cycles required to achieve stability.

For the example of Venus, where a TOA insolation flux of ~300 W/m² supports a surface thermal flux of ~17,000 W/m² (a gain of 56.67), then a partition ratio of 0.8862% radiant loss versus 99.1138% retention by the air is required. The inverse modelling process needed a cascade of 1203 cycles of atmospheric recycling to produce the stable outcome, by which the 737 Kelvin (464⁰C) target global average surface air temperature of Venus could be achieved (Table 8).

Table 8: Testing the Cascade Algorithm for the Adiabatic Model of Noonworld

The total global energy budget for the adiabatic model of Noonworld, using Venusian insolation parameters and a power intensity flux tuned to achieve the Venusian global average temperature of 737 Kelvin (464^oC) is 112.840577 units (Fig. 4).

Figure 4: Inverse Climate Model of Noonworld (Venus Target Temperature): showing Energy Vectors and Final Energy Distributions.

Figure 5 shows the final global energy distribution that is achieved, by applying the NASA values for the Venusian sunlit hemisphere post albedo solar energy interception flux of 299 W/m² (Williams, 2018) to the final adiabatic convection model of Noonworld

Figure 5: Inverse Climate Model of Venus: showing Energy Vectors and Final Energy Distributions.

The total global energy budget is now 33,756 W/m² (Fig. 5). Table 9 records the thermal effects of this energy partition, and shows that the Venusian global average air temperature has now been achieved.

Table 9: Stable Energy Values for Noonworld achieved by Global Air Recycling using a 0.8662%A: 99.1138%S Flux Partition.

4.2. Exploring the Results of the Adiabatic Convection Model that Creates Greenhouse Noonworld.

The results of the inverse modelling process have demonstrated that it is eminently feasible to achieve energy retention, and thermal enhancement within a climate system by repetitive thermal air recycling.

The key insight gained from this analysis is that it is the energy partition in favour of the air, at the surface boundary that achieves this energy boost within a dynamic atmosphere; and that the greenhouse effect is a direct result of the standard meteorological process of convection. Put simply energy retention by surface conduction and buoyancy driven convection wins over energy loss by radiation, and that the retention of energy by the air is a critical feature of planetary atmospheric thermal cell dynamics.

The DAET Model has its limitations, as does every model. The most critical limitation with the adiabatic model of Noonworld is that the model was populated by a fully radiatively transparent, non-greenhouse gas atmosphere. Consequently, in the model, all radiative loss to space takes place from the ground surface at the base of the atmosphere. If we now apply to the model an opaque atmosphere that can only emit radiation to space from its upper boundary, or Top of Atmosphere (TOA) altitude (as per Robinson & Catling, 2014), in general understanding this would be a greenhouse gas atmosphere. However, we do not need to invoke any back-radiation energy retention process for such an atmosphere. Its radiant opacity merely acts as a delaying mechanism to the transmission of radiant energy, rather than a feed-back amplifier.

By applying a troposphere lapse rate of 6.7 K/Km to the Venusian atmosphere (Justus and Braun, 2007, Table 3.1.2) we can now estimate the thickness of this opaque atmosphere at its TOA altitude. Its topside surface will be emitting energy to space at a point where the lapse rate achieves the same temperature in air, as the model radiant ground surface maintained under the original fully-transparent atmosphere. The thermal separation between the surface air temperature, and the temperature of the radiant emitting surface can be achieved for an opaque atmosphere at an altitude of ~76 Km (Table 10).