Using an Iterative Adiabatic Model to study the Climate of Titan.

Guest Post by P Mulholland and Stephen Wilde. July 2019

“I would rather have questions that can’t be answered than answers that can’t be questioned.” Richard P. Feynman.

In this dual scene montage, we see on the left in natural colour the moon Titan orbiting above its parent body, the ringed planet Saturn. On the right is a near-infrared false-coloured composite image taken by the Cassini probe of Titan’s north pole. In this rare image of Titan the sunlight glints off the polar hydrocarbon seas, which are partially obscured by stratus clouds of condensed methane. The bright red arrow shape is from the anvil head of frozen methane cirrus clouds located above an actively raining convection storm (see also Schaller et al. 2006).

Abstract.

A mathematical model has been created based on meteorological principles, and intended to be applied as a correlative to the standard radiation balance equation used in current climate studies. The Dynamic-Atmosphere Energy-Transport climate model (DAET) is designed to account for the dual environmental nature of all terrestrial globes and moons, with sufficient mass and surface gravity to hold an atmosphere under a given solar radiation loading. The model consists of two distinct environments, a solar lit hemisphere dominated by surface radiative heating with an energy surplus, and a dark night-time hemisphere of energy deficit dominated by surface cooling, caused by direct through the atmosphere thermal radiative energy loss to space.

Energy exchange between the two hemispheres of energy surplus and energy deficit is mediated by a series of linked atmospheric mass movement processes. On the lit surface of energy surplus adiabatic convection and atmospheric overturning occurs. The lifted energy rich air then undergoes horizontal mass transport in the upper atmosphere. This process, characterized by air movement and energy transport towards the region of surface energy deficit, is representative of a thermal Hadley cell. The energy rich advected air subsequently descends onto the surface of the dark hemisphere which is under the influence of surface diabatic atmospheric cooling, and thermal radiant energy loss to space. This process is representative of the surface induced radiative cooling of a night time thermal environment. Near-surface advection of surface cooled dense air back to the energy rich sunlit environment, then completes the cyclical process of air mass transport and energy delivery.

1. Introduction.

Studies of the atmospheric dynamics of terrestrial solar system planets have a long and detailed history. The fundamental equation for the basis of this work is exemplified by the radiation balance equation used by Sagan and Chyba (1997): –

“The equilibrium temperature T_e of an airless, rapidly rotating planet (or moon) is: –

Equation 1: T_e ≡ [S π R²(1-A)/4 π R² ε σ]^1/4

where σ is the Stefan-Boltzmann Constant, ε the effective surface emissivity, A the wavelength-integrated Bond albedo, R the planet’s (or moon’s) radius (in metres), and S the solar constant (in Watts/m²) at the planet’s (or moon’s) average distance from the sun.”

Using Equation 1 for Titan: – T_e = 83.2 K, however the observed mean surface temperature for this moon is T_s = 94 K, therefore the difference Δ T between T_e and T_s = 10.8 K, this value is the atmospheric thermal enhancement effect for Titan. (Table 1).

2. Methods.

The Dynamic-Atmosphere Energy-Transport (DAET) climate model used here, is a 2-dimensional mathematical model that preserves the globular dual hemisphere components of daytime illumination and nighttime darkness. The forward model represents a globe with two environmentally distinct halves. The dayside is lit by a continuous incoming stream of solar radiation which creates an energy surplus, while the nightside 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, and then returns again is the fundamental requirement of this model. (Figure 2).

The climate model presented here is designed to represent the metrological processes of an illuminated globe. The model collects insolation at the surface boundary between the atmosphere and the solid surface of the globe, over the full face of its lit hemisphere. In this first instance of modelling analysis the illuminated surface partitions the captured power intensity flux equally. Half of the power intensity flux is converted to low frequency radiation. This radiant flux then exits the model, passing unimpeded through the overlying atmosphere and out to the vacuum of space.

The remaining half of the power intensity flux is captured by the air and advected to the unlit dark hemisphere. On this unlit side of energy deficit, the air is the only source of power intensity flux. Consequently, on contact with the cold unlit surface the air transfers half of its flux onto the solid ground, which then radiates this flux directly out to space. The now cold air returns back to the lit hemisphere, carrying its remaining part of the internally retained power intensity flux.

The two surfaces of the globe partition the power intensity flux equally (50% radiant flux ; 50% air thermal flux) , but because the absolute values are different the model contains two separate geometric series that tend to different limits, one limit for the lit hemisphere and one for the dark surface.

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

Equation 2: 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 3: 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 of course 1.

On the lit side of the globe the recycled air from the dark side supplies a second source of flux to the environment of the lit hemisphere. The equipartition of flux in the model, and the repetitive cycling of the air under conditions of constant solar radiation on the lit side, creates an infinitely repeating stable cycle of fluid mass motion and transported power intensity flux. The total energy recycling process in this model stores a power intensity flux in the atmosphere equal to that of the solar insolation flux, and so the model has a system gain of 2 (Figure 2).

We call this meteorological model the diabatic model because of the equipartition of power intensity flux at its critical surface boundary.

2.1. Modelling Slowly Rotating Titan.

The Saturnian moon Titan rotates only slowly on its axis, its rotational period is 382.7 hours (15.95 Earth days). This is the same length of time that Titan takes to orbit Saturn, and so the same face of this moon is always turned towards its parent planet. As a consequence of its slow axial rotation the sunlit day on Titan lasts for 191.35 hours, and the atmosphere experiences only a weak Coriolis effect. Like the planet Venus, Titan also has super-rotational winds in its upper atmosphere, and the daylit and nighttime surface temperatures are almost identical.

As with the previous study of the planet Venus, Modelling the Climate of Noonworld: A New Look at Venus, the application of a DAET model to analyse the climate of Titan will first be tested using the diabatic model described above. This meteorological model will be applied to slowly rotating Titan using the standard published atmospheric data (Table 2).

2.2.How the Presence of a Dynamic Atmosphere Distributes the Captured Solar Energy Across Titan.

To facilitate the modelling analysis, a number of simplifications have been made. The primary one is that the global atmosphere in the model world of Titan contains a fully radiatively transparent and free-flowing gas that connects the two hemispheres. Consequently, because the model atmosphere is fully transparent, it can only gain or lose heat from the solid surface at its base.

Next, a test is made of how the diabatic atmospheric model behaves when standard Titan Insolation parameters are applied. The solar irradiance that Titan experiences as a moon of Saturn is 14.82 W/m², and the Bond albedo of Titan is 0.265 (Table 2) so the average post-albedo power irradiance of the lit hemisphere is 5.45 W/m² (Table 3).

The diabatic equipartition energy budget for the lit hemisphere is 7.26 W/m² and 3.63 W/m² for the dark hemisphere, giving a total global energy budget for Titan of 10.89 W/m² (Figure 2). This quantity of flux is a doubling of the post-albedo irradiance experienced by the lit hemisphere, and so the diabatic model has generated a system gain of 2.

The result of applying the standard Vacuum Planet equation of astronomy to Titan is an Expected T_e of 83.2 Kelvin (Table 1). The meteorological diabatic forward model of Titan closely mirrors the results of this fundamental equation, it produces a modelled temperature for Titan of 82.3 Kelvin (Table 3), a difference of only 0.9 Kelvin (Figure 3).

2.3. Establishing the Global Energy Partition Ratio for Titan by Inverse Modelling.

The process of establishing the observed average surface temperature for slowly rotating Titan, in its customary orbit around Saturn, is achieved by applying the mathematical technique of inverse modelling to an equipartition diabatic forward climate model of the moon. The process of inversion adjusts the surface partition ratio in the diabatic model to create an adiabatic model of the moon’s climate, with an internal system gain that is greater than 2.

The following steps describe the logic flow of the modelling analysis: –

Step 1: That the repetitive air recycling process of a Hadley cell retains energy within the atmosphere, and that the quantity of energy retained by the air stabilises when the amount of outgoing radiant energy has the same value as the incoming solar flux (Table 3). This is the diabatic forward model.

Step 2: That on applying Titan’s insolation parameters to the diabatic forward model, an average global air temperature of 82.34 K is achieved. This value is a small underestimation of the Expected T_e for a vacuum Titan of 83.2 K that the standard radiative balance equation computes (Figure 3).

Step 3: That by applying the standard geoscience technique of inverse modelling to the basic diabatic atmospheric model of Titan, an adiabatic model can be created. This process has the ability to identify the surface energy partition ratio which determines the thermal enhancement observed in the atmosphere of slowly rotating Titan. It was established that 24 cycles of atmospheric overturn would produce a stable outcome for the DAET adiabatic model, and produce a global energy budget gain of 2.66 times the surface solar energy input (Table 4).

Step 4: Tests were made to try and establish the flux partition ratio for the unlit surface of Titan. However, in the absence of a suitable nighttime air temperature profile to constrain the modelling process, and because published sources report that the day and nighttime surface temperatures on Titan are almost equal (Courtin and Kim, 2002, Fig. 2), a pragmatic solution was adopted to this lack of control data. Consequently, for the purpose of this analysis it is assumed that the surface flux partition ratio is the same for both hemispheres of slowly rotating Titan. This approach of using a common flux partition ratio for both surfaces in the adiabatic model was also previously found to be suitable for the atmosphere of slowly rotating Venus.

On applying Titan’s insolation parameters to the adiabatic model, using the energy partition ratio identified by inverse modelling, an average global air temperature of 94 Kelvin (minus 179^oC) is achieved for this slowly rotating moon (Table 5).

The results of applying the inverse modelling run to Titan, using 25 cycles of internal planetary atmospheric overturn, are shown in Table 5. The surface energy partition ratio that achieved this result is 37.6% of the moon’s surface energy being directly lost to space, and 62.4% of the surface energy being retained by the atmosphere (Figure 4).

Details of the algorithms used in the diabatic and adiabatic models of Titan’s climate are recorded in the linked Excel Workbook (Mulholland, 2019).

3. Results of Applying the Adiabatic Meteorological Model to Titan.

The adiabatic model of Titan computes a thermal separation of 24.3^oC between the radiating surface and the air for the lit side surface (Table 5). In the model of slowly rotating Titan, the hemisphere of energy surplus is a proxy for the moon’s thermal Hadley cell. Using the troposphere 41.5 km gross thermal lapse rate of 0.53K/km for Titan (Figure 5), this temperature difference equates to a physical separation of 45.6 km. This value is the modelled estimate of the height of the radiant emitting surface for the lit hemisphere of Titan (Table 5).

The atmospheric profile for Titan shows that the minimum temperature in the moon’s atmosphere occurs at an altitude of 41.4 Km, where a temperature of 70.2 Kelvin (-202.8^oC) is recorded (Courtin and Kim, 2002: Table 1).

4. Discussion.

The numerical atmospheric model used here is based on the fundamental astronomical principle that all globes are sun lit on one side only. This fact applies to all solar system planetary bodies and moons of whatever form or type. For an atmospheric model to be valid, it must be capable of being applied to bodies that have all possible types of planetary rotation, including hypothetical bodies that are tidally locked, and always present the same face towards the Sun. To address this problem, a simple geometric model has been devised based on the “divide by 2” rule for a fully lit hemisphere surface illumination, coupled to the “divide by 4” rule of global thermal radiant emission.

Figure 3 demonstrates the close relationship between the standard Vacuum Planet radiative balance equation derived from astronomical principles, and our Dynamic-Atmosphere Energy-Transport climate model derived from metrological principles. The differences between these two analytical approaches and their appropriate use are listed in Table 6: –

The purpose of the diabatic meteorological model is to replicate the form of the standard radiation balance equation, that uses the illumination intensity divide by 4 rule of surface radiant energy distribution (Equation 1), and to apply this concept to a globe that is only lit on one side. For such a model, the sunlit energy is distributed over the surface of a single hemisphere, and so an illumination intensity divide by 2 rule of surface radiant energy capture is applied. In this model the transmission of energy from the lit hemisphere of energy surplus to the unlit side of energy deficit, is mediated by the meteorological atmospheric process of advection. The diabatic mathematical model shows that the system gain which stores this flux within the atmospheric reservoir has a value of 2.

Because an equipartition of energy between a radiatively heated (or cooled) solid surface and an overlying mobile fluid is characteristic of laminar flow, it is clear that this equipartition ratio cannot be used to describe the transmission of energy into (or from) a fluid that is undergoing turbulence at the critical boundary interface. For turbulent fluid motion, that is characteristic of forced radiative heating and adiabatic convection, a partition ratio weighted in favour of the air is the required metric.

The adiabatic model incorporates the numerical process of energy partition in favour of the turbulent air for the sunlit surface boundary. Because the required average surface air temperature for Titan is known a priori, the numerical technique of inverse modelling to establish the energy partition ratio can be applied. This algorithm creates the required thermal enhancement for an atmosphere of any opacity, and directly computes the gain that stores flux within the atmospheric reservoir. Because the model creates a thermal contrast between the surface and the air, this temperature difference can be used to calculate the height of the radiant emission surface, using the appropriate environmental lapse rate obtained from measured data.

The application of the Dynamic-Atmosphere Energy-Transport model to a planet or moon that is not tidally locked, introduces a new variable of surface daylength into the process of climate analysis. Both Titan and most especially Venus have a polar vortex of descending air in each hemisphere. It is apparent that the locus of the nighttime sector of descending air in the conceptual Noonworld model must shift from the antipodal zenith point (Figure 2) to the poles of rotation (Figure 4) for these real-world examples. Consequently, we expect that the polar vortices of Titan to be the primary atmospheric window for surface radiant flux exiting to space, and that the tropopause height will consequently be reduced in these regions. In the absence of polar atmospheric profile data for Titan to constrain our model, this remains a speculative prediction of our analysis.

5. Conclusions.

1. The mathematical model used in this study is designed to retain the critical dual surface element of a lit globe, namely night and day. The simple equipartition diabatic model, when applied to a fully transparent pure Nitrogen atmosphere, closely matches the results of the standard atmosphere equation, which is traditionally applied to an airless world (Sagan and Chyba, 1987).

2. By applying the inverse modelling process to the atmosphere of Titan, and accounting for the fact that there is little surface thermal contrast between day and night on this moon, then the modelling process can determine the global energy partition ratio that accounts for its thermally enhanced atmosphere.

3. By using the appropriate lapse rate for Titan, the inverse modelling process predicts the height of the radiant emission surface for a fully opaque atmosphere. Consequently, the computational dynamics of the adiabatic model with its fully transparent atmosphere demonstrates that the presence of a troposphere that is opaque to thermal radiation is not an a priori requirement for the retention of energy within an atmospheric system.

4. Titan has an atmosphere with a composition of 98.4% nitrogen gas, a surface pressure of 1.45 bar and a greenhouse effect of 10.8 Kelvin. The pure nitrogen model used here is fully valid for the composition of Titan’s atmosphere, and the adiabatic calculation achieves a surface temperature of 94 Kelvin with a much lower partition ratio than that used for the high-pressure environment of Venus.

5. The Bond albedo of Titan is 0.265. Titan is an optically veiled world with a uniform natural orange glow (ESA, 2004). This veil of photochemical smog results from the presence of Tholins in the upper atmosphere of the moon (Waite et al., 2007), and this hydrocarbon haze directly controls the intensity of the surface solar irradiance that drives the climate of Titan.

6. A slowly rotating moon, such as Titan, does not have a counter rotating mechanical Ferrel cell, therefore there is no dynamic restriction on the latitudinal reach of the Hadley cell on Titan, (as per Del Genio and Suozzo, 1987).

7. The atmosphere of Titan holds a number of environmental characteristics in common with the planet Venus:

a. Titan is a slow rotator.

b. Titan has two hemisphere encompassing Hadley cells which link directly into the moon’s two polar vortices.

c. Titan is an optically veiled world.

d. Titan has a super-rotational wind in its upper atmosphere.

e. Titan has similar day and night time surface temperatures.

8. The key insight gained from this analysis is that it is energy partition in favour of the air, at the lit surface boundary that achieves this thermal energy boost within a dynamic atmosphere. Therefore, the energy retention 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. Consequently, the retention of energy in the air by the process of convection is a critical feature of planetary atmospheric thermal cell dynamics.

6. References.

Courtin, R. and Kim, S.J., 2002. Mapping of Titan’s tropopause and surface temperatures from Voyager IRIS spectra. Planetary and Space Science, 50(3), pp.309-321.

Del Genio, A.D. and Suozzo, R.J., 1987. A Comparative Study of Rapidly and Slowly Rotating Dynamical Regimes in a Terrestrial General Circulation Model. Journal of the Atmospheric Sciences, Vol. 44 (6), pp. 973-984.

ESA, 2004. Titan’s True Colors. Astrobiology Magazine.

Li, L., Nixon, C.A., Achterberg, R.K., Smith, M.A., Gorius, N.J., Jiang, X., Conrath, B.J., Gierasch, P.J., Simon‐Miller, A.A., Flasar, F.M. and Baines, K.H., 2011. The global energy balance of Titan. NASA Reports Archive.

Limited Science, 2018. Possibility of life on Titan (Largest “Planet” of Saturn) Methane Sea of Titan. Limited Science Space & Universe.

Mulholland, P., 2019. Titan Climate Models 01Jun19 Excel Workbook. Research Gate Project Dynamic-Atmosphere Energy-Transport Climate Model.

NAIF JPL NASA 2019. Titan Occultations WebGeocalc Spice data and software.

Sagan, C. and Chyba, C., 1997. The early faint sun paradox: Organic shielding of ultraviolet-labile greenhouse gases. Science, 276 (5316), pp.1217-1221.

Schaller, E.L., Brown, M.E., Roe, H.G. and Bouchez, A.H., 2006. A large cloud outburst at Titan’s south pole. Icarus, 182(1), pp.224-229.

Waite, J.H., Young, D.T., Cravens, T.E., Coates, A.J., Crary, F.J., Magee, B. and Westlake, J., 2007. The process of tholin formation in Titan’s upper atmosphere. Science, 316(5826), pp.870-875.

Williams, D.R., 2016. Solar System Small Worlds Fact Sheet NASA NSSDCA, Mail Code 690.1, NASA Goddard Space Flight Center, Greenbelt, MD 20771.

Williams, D.R., 2018. Saturn Fact Sheet NASA NSSDCA, Mail Code 690.1, NASA Goddard Space Flight Center, Greenbelt, MD 20771.

Further Reading: –

Coustenis, A. and Taylor, F.W., 2008. Titan: exploring an earthlike world (Vol. 4). World Scientific.

Glossary:

Adiabatic: The process of air movement in which there is no energy exchange with the surroundings.

Advection: The process of horizontal transport of air by the mass motion of the atmosphere.

Albedo: An environmental property of a lit surface that acts as a radiant energy bypass filter. Defined as the ratio of reflected radiant energy to incident radiant energy.

A priori: Proceeding from a known value to deduce the consequential result.

Convection: The process of vertical transport of air by means of differential atmospheric heating and air density contrast.

Diabatic: The process of energy exchange by conduction between two adjacent bodies.

Forward Modelling: The technique of computing the result for an unknown parameter from a set of known measurements using a mathematical model.

Insolation: (Incoming Solar Radiation). The amount of direct sunlight energy received by the surface of a planet or moon.

Inverse Modelling: The mathematical process of determining the value of an unkown input parameter that creates a known measured result.

Lapse Rate: The change of atmospheric temperature with height in a given gravity field. The lapse rate is defined as positive when the temperature decreases with increasing elevation.

Laminar: An atmospheric layer in which air flow is smooth. This layer is usually associated with stable air mass formation and radiative surface boundary cooling.

Opacity: The capacity of a substance to impede the transmission of radiant energy.

Partition Ratio: The ratio of energy distribution at the boundary between two environments.

Terrestrial Planet: A solar system planetary body (or moon) that has a solid surface and is Earth-like in basic composition and form.

Tropopause: The upper limit of the troposphere marked by a transition to a zero or negative lapse rate in the atmospheric layer above.

Troposphere: The weather layer. The lowest layer of a terrestrial planet’s atmosphere dominated by surface daytime heating or nighttime cooling and energy transport by turbulent air motion.

Turbulence: The process of random mixing of air undergoing forced radiant thermal heating at the surface boundary.