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Titan, the giant moon of the planet Saturn, is recognized to have meteorological processes involving liquid methane that are analogous to the water generated atmospheric dynamics of planet Earth. We propose here that the climatic features of Titan by contrast are more akin to those of the planet Venus, and that this structural similarity is a direct result of the slow daily rotation rate of these two terrestrial bodies. We present here a simple mathematical climate model based on meteorological principles, and intended to be a replacement for the standard radiation balance equation used in current studies of planetary climate. The Dynamic-Atmosphere Energy-Transport climate model (DAET) is designed to be applied to terrestrial bodies that have sufficient mass and surface gravity to be able to retain a dense atmosphere under a given solar radiation loading. All solar orbiting bodies have both an illuminated hemisphere of net energy collection and a dark hemisphere of net energy loss. The DAET model acknowledges the existence of these dual day and nighttime radiation environments and uses a fully transparent non-condensing atmosphere as the primary mechanism of energy storage and transport in a metrological process that links the two hemispheres. The DAET model has the following distinct advantages as a founding model of climate: It can be applied to all terrestrial planets, including those that are tidally locked. It is an atmospheric mass motion and energy circulation process, and so is fully representative of a Hadley cell; the observed fundamental meteorological process of a terrestrial planet's climate. The diabatic form of the DAET model fully replicates the traditional vacuum planet equation, and as it applies to a totally transparent atmosphere it therefore demonstrates that thermal radiant opacity, due to the presence of polyatomic molecular gases, is not a fundamental requirement for atmospheric energy retention. For the adiabatic form of the DAET model, where the turbulent asymmetric daytime process of forced radiant convection applies, the intercepted solar energy is preferentially retained by the ascending air. The adiabatic DAET climate model shows that the atmospheric greenhouse effect of surface thermal enhancement is a mass motion process, and that it is completely independent of an atmosphere's thermal radiant opacity.
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Journal of Water Resources and Ocean Science
2020; 9(1): 15-28
http://www.sciencepublishinggroup.com/j/wros
doi: 10.11648/j.wros.20200901.13
ISSN: 2328-7969 (Print); ISSN: 2328-7993 (Online)
An Iterative Mathematical Climate Model of the Atmosphere
of Titan
Philip Mulholland
*
, Stephen Paul Rathbone Wilde
Mulholland Geoscience, Weybridge, Surrey, UK
Email address:
*
Corresponding author
To cite this article:
Philip Mulholland, Stephen Paul Rathbone Wilde. An Iterative Mathematical Climate Model of the Atmosphere of Titan. Journal of Water
Resources and Ocean Science. Vol. 9, No. 1, 2020, pp. 15-28. doi: 10.11648/j.wros.20200901.13
Received: October 26, 2019; Accepted: January 4, 2020; Published: January 31, 2020
Abstract:
Titan, the giant moon of the planet Saturn, is recognized to have meteorological processes involving liquid
methane that are analogous to the water generated atmospheric dynamics of planet Earth. We propose here that the climatic
features of Titan by contrast are more akin to those of the planet Venus, and that this structural similarity is a direct result of
the slow daily rotation rate of these two terrestrial bodies. We present here a simple mathematical climate model based on
meteorological principles, and intended to be a replacement for the standard radiation balance equation used in current studies
of planetary climate. The Dynamic-Atmosphere Energy-Transport climate model (DAET) is designed to be applied to
terrestrial bodies that have sufficient mass and surface gravity to be able to retain a dense atmosphere under a given solar
radiation loading. All solar orbiting bodies have both an illuminated hemisphere of net energy collection and a dark
hemisphere of net energy loss. The DAET model acknowledges the existence of these dual day and nighttime radiation
environments and uses a fully transparent non-condensing atmosphere as the primary mechanism of energy storage and
transport in a metrological process that links the two hemispheres. The DAET model has the following distinct advantages as a
founding model of climate: It can be applied to all terrestrial planets, including those that are tidally locked. It is an
atmospheric mass motion and energy circulation process, and so is fully representative of a Hadley cell; the observed
fundamental meteorological process of a terrestrial planet’s climate. The diabatic form of the DAET model fully replicates the
traditional vacuum planet equation, and as it applies to a totally transparent atmosphere it therefore demonstrates that thermal
radiant opacity, due to the presence of polyatomic molecular gases, is not a fundamental requirement for atmospheric energy
retention. For the adiabatic form of the DAET model, where the turbulent asymmetric daytime process of forced radiant
convection applies, the intercepted solar energy is preferentially retained by the ascending air. The adiabatic DAET climate
model shows that the atmospheric greenhouse effect of surface thermal enhancement is a mass motion process, and that it is
completely independent of an atmosphere’s thermal radiant opacity.
Keywords:
Climate Model, Titan Atmosphere, Atmospheric Dynamics, Terrestrial Planets
1. Introduction
Titan is the largest moon of the giant planet Saturn, and the
second largest moon in the solar system [1]. Titan is a unique
moon in that it possesses a thick atmosphere of nitrogen gas
with a surface pressure of 1.43 Bar [2}. There are convective
meteorological processes occurring in the atmosphere of
Titan that generate liquid methane rain [3]. These low
temperature processes can be compared with the standard
convective rainfall on planet Earth that involves the
evaporation and condensation of water. In addition to the
direct weather analogy that can be made between the
terrestrial bodies of Titan and Earth there is another analogy
that can be considered, but this time between the climates of
Titan and the planet Venus. These two terrestrial bodies have
the following features in common.
1. Titan and Venus are slow rotators and therefore both
bodies possess a single atmospheric system of
hemispheric encompassing Hadley cells [4].
2. Titan and Venus are both veiled worlds that have an
optically translucent upper atmosphere [5]. In the case
of Titan, this is caused by the presence of tholins [6].
16 Philip Mulholland and Stephen Paul Rathbone Wilde: An Iterative Mathematical Climate Model of the Atmosphere of Titan
Tholins are complex hydrocarbons generated by the UV
ionisation of methane to create methyl radicles, which
then polymerise and condense to form the particles of
the upper atmospheric veil.
3. Titan, like the planet Venus, has a solar zenith
generated, bow-shockwave structured, super-rotational
wind in the upper atmosphere [7].
4. Like Venus, Titan also has uniform day and night time
temperatures, with no significant surface diurnal effect
seen in the data points corresponding to the dayside and
those of the nightside of this slowly rotating moon [2].
The objective of this study is to generate a dynamic
atmospheric model that accounts for the climatic features of
Titan, specifically the observed uniform diurnal surface
temperatures and the presence of a hemispheric
encompassing Hadley cell.
Under the influence of intercepted solar radiation, the
atmosphere of a rotating terrestrial body self-organises into a
series of latitudinally defined closed loops, or cells, that
transport mass and energy across the surface of a globe.
Climate is the presence and action at the surface of a
terrestrial planet (or moon) of this series of solar energy
driven mechanically coupled atmospheric cells. The
atmosphere of a terrestrial body under solar radiation loading
can be considered to act as a dynamic mobile-fluid mass-
transport and energy delivery system.
A terrestrial body’s mobile-fluid mass transport system
collects high-frequency energy from a region of incoming
surplus around the solar zenith, and delivers it to a region of
energy deficit towards the poles of rotation. At the location
of energy deficit, the energy imported by the atmospheric
system is then lost to space by low-frequency thermal
radiation from the planet. As with any mass transport system
it must form a closed loop, otherwise all of the heat necessary
for the dynamic flow will be dissipated and the system will
run down. Indeed, if too much energy is lost from the cell at
the region of energy deficit, then the transport mechanism
will cease, as the mobile fluid carrying the heat freezes solid,
and the planetary body will no longer possess a gaseous
atmosphere.
Consequently, it is a fundamental requirement that
sufficient energy is retained by the mobile fluid for it to
return to the original location of energy surplus. On its return
to this origin, the mobile fluid is then able to collect
additional energy, and the mass transport system becomes
recharged. This interception of additional solar energy by the
globe’s surface reheats the mobile-fluid, and so the cycle is
able to continue and repeat indefinitely, and be maintained as
a sustainable system.
Modern climate science starts with the vacuum planet
equation, a conceptual model devised by astronomy, and
used to determine the average thermal emission temperature
of a rapidly rotating solar illuminated planetary body. The
basic form of this equation is used by [8] to calculate the
equilibrium temperature T
e
of an airless, rapidly rotating
planet (or moon).
Equation 1 (corrected from the published error pers comm):
-
T
e
≡ [S π R
2
(1-A)/4 π R
2
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/m
2
) 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 uniform surface temperature for this moon is T
s
=
94 K, therefore the difference T between T
e
and T
s
= 10.8
K, and this value is the atmospheric thermal enhancement
effect for Titan. (Table 1).
Table 1. The Expected Surface Temperature for an Airless Titan compared with its actual Surface Atmospheric Temperature [8].
Parameter Symbol Titan Units Dimensions
Solar Constant at distance a S 14.82 W/m
2
MT
-3
Radius of Body R 2,576,000 m L
Bond Albedo A 0.265 Constant Constant A
Stefan-Boltzmann Constant σ 5.67E-08 W/m
2
/K
4
MT
-3
K
-4
Effective surface emissivity ε 1 Constant Constant ε
Expected T
e
T
e
83.2 Kelvin K
Greenhouse Effect GE 10.8 Kelvin K
Expected T
s
T
s
94 Kelvin K
Distance from the Sun a 1.4270E+12 m L
All models are abstractions based on assumptions, in order
to understand modern climate science, the elements of the
model and the fundamental assumptions need to be
recognised and understood. The primary assumption that
directly arises from the use of the vacuum planet equation is
this: -
It is the daily rotation of the globe that distributes the
intercepted solar energy from the lit to the unlit hemisphere
of the terrestrial body.
We can demonstrate that this assumption is paramount
because it is correctly deduced that if the vacuum planet was
tidally locked in its orbit around the sun, then only one
hemisphere the sun-facing side would ever receive solar
energy. Consequently, the unlit far side would have a never-
ending night, and experience vacuum surface temperatures of
only a few degrees above absolute zero.
Data show that the surface day and night time
temperatures on slowly rotating Titan are almost identical [2].
It appears therefore that it is not rotation that distributes the
solar energy between the lit and unlit hemispheres of this
Journal of Water Resources and Ocean Science 2020; 9(1): 15-28 17
terrestrial body, instead it is the presence of the thick and
mobile non-condensing atmosphere of Titan that performs
this function.
For the purposes of this study, we define climate as the
presence and action of a solar energy driven atmospheric cell
over the surface of a terrestrial body. Modelling studies of
planetary atmospheric dynamics have shown that the
latitudinal reach of a Hadley cell for a given terrestrial body,
is determined by its daily rotation rate [4]. Because all
planets and moons are rotating globes (even tidally locked
ones) it follows that the latitudinal reach of the primary
atmospheric cell determines the number of climate zones on
a given terrestrial body. Slowly rotating Titan has a Hadley
cell that extends from the moon’s equator to its pole of
rotation, and therefore it has a single climatic zone with a
uniform surface temperature.
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. In order to
study the climate of a terrestrial body with a thick
atmosphere, we need to formulate a model that can be
applied in all possible scenarios. The vacuum planet equation
[8] by its very definition cannot be applied to a planet with an
atmosphere, nor can it be applied to a planet that is tidally
locked.
To address both of these restrictions a new planetary
climate model, the Dynamic-Atmosphere Energy-Transport
(DAET) Climate Model, was devised using a tidally locked
planet as the fundamental element. We call this model planet
“Noonworld” and populate it with a transparent atmosphere
of pure nitrogen gas, so that all the processes of energy
conversion can only take place at the basal surface boundary
of the model.
2. Methods
With forward modelling studies of a terrestrial body’s
energy budget, the first and overarching assumption is that
the only way that it can lose energy is by thermal radiation to
space. This assumption is not in dispute, and it correctly
leads to the adoption of the Stefan-Boltzmann (S-B) equation
of thermal radiation, which is used to establish the direct
relationship between energy flux in Watts per square metre
(W/m
2
) 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 solar radiation or insolation from a
central star or sun, and so is only ever lit on one side. Solar
system bodies orbit around our Sun, and consequently they
all have both a lit (day) and an unlit (night) hemisphere, and
are never illuminated on both sides simultaneously.
2.1. Building the Dynamic-Atmosphere Energy-Transport
(DAET) Climate Forward Model
The Dynamic-Atmosphere Energy-Transport Model of
planetary climate presented here is a 2-dimensional
atmospheric model, that preserves the globular dual
hemisphere component of solar illumination (Figure 1). This
forward model represents a 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 our atmospheric model.
Figure 1. Basic Thick Atmosphere Globe with Initial Static Model: Showing Energy Vectors and Start-Up Energy Partitions.
On all rotating terrestrial planets and moons, the solid
ground cools by thermal radiation to space all of the time
(both day and night), but the surface only gains radiant heat
during the hours of sunlight throughout the day. The current
18 Philip Mulholland and Stephen Paul Rathbone Wilde: An Iterative Mathematical Climate Model of the Atmosphere of Titan
accepted climate paradigm assumes that it is the effect of
daily rotation, and seasonal axial tilt that distributes the
energy intercepted from the Sun across the full surface area
of the globe. In order to remove the complications associated
with axial rotation, and the impact that rapid daily rotation
has on global atmospheric cell circulation patterns; we will
assume that the model world presented here is tidally locked
in its orbit around the Sun, and so the Coriolis Effect on air
motion [9] is minimised.
2.2. Starting the Dynamic-Atmosphere Energy-Transport
Engine from Cold
On all illuminated planets and moons the atmospheric
process of energy transmission begins on the sunlit side
(Figure 1). Here the solid surface is illuminated and warms as
it receives radiant energy from the sun, and as it warms it
also warms the air above it by conduction. This buoyant
warmed air then rises by convection, and because in our
model example the atmosphere is fully transparent, and also
because the air is no longer in contact with the ground, it
retains all of its energy internally.
However, the lit ground surface below our model’s
transparent atmosphere does not retain all of its energy. It
cools in two separate ways; it both loses energy to the air
above it, and also transmits low frequency radiant energy
directly out into space. In the forward modelling process, we
assign an initial partition ratio of 50% to conduction and 50%
to radiation to study this dual process of energy loss. This
assignment was chosen to permit a first assessment to be
made of the impact that the partition process has on the
energy budget of the atmospheric model.
On the dark side of the planet, the ground surface is also
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 the DAET model that as the air cools it retains its
mobility, and does not freeze on to the solid surface below.
Consequently, the gaseous atmosphere is able to advect back
to the sunlit side, where it can again be warmed and
replenished.
As the cold air moves away across the surface of the
ground towards the lit hemisphere, more warm air from
above descends on to the dark cold surface. This descent of
air delivers heat 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. This partition allows us
to study the dual cooling process of energy transfer to, and
subsequent radiant loss of energy from the dark surface to
space, and the associated role of horizontal cold air advection
in the transport of atmospheric energy.
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 Vacuum Empty.
Step Process Energy Flow
1 Interception of solar energy by the lit surface 1 unit.
2 No flow of frozen air from the dark side 0 Unit
3 Total energy available to drive the system 1 unit.
4 50:50 partition of the intercepted energy between the ground and the air leading to: -
5 Direct radiant loss to space from the warm surface ½ unit
6 Retention of energy by the lit air, followed by transport and delivery of this warm air to the dark side ½ unit
7 50:50 partition of the delivered energy between the ground and the air leading to: -
8 Radiant loss to space from the ground on the dark side ¼ unit
9 Return flow of cold surface air from the dark side to the lit side ¼ unit
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 amplifier,
capturing and retaining solar energy within the reservoir of
the atmospheric system.
2.3. Warming up the Dynamic-Atmosphere Energy-
Transport Engine
Because the priming stage of the process retains energy
within the atmosphere, the next overturning cycle starts with
1 unit of insolation plus ¼ unit of thermal atmospheric
energy left over from the first cycle. Clearly the retention of
energy within the atmospheric system by this first cycle
overturn means that the global 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. We have here two separate geometric series that
both tend to different finite limits, one for the lit and one for
the dark surface.
The geometric series limit 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 limit 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:
-
2/3 + 1/3 = 1
and so, the total energy recycling system is now in radiative
balance (Table 3).
Journal of Water Resources and Ocean Science 2020; 9(1): 15-28 19
Table 3. Developing the Dynamic-Atmosphere Energy-Transport Forward Model.
Building the Dynamic-Atmosphere Energy-Transport Model
Cycle
Number
Space
Incoming
Captured
Radiation
(Units)
Heating the
Lit
Hemisphere
(Units)
Lit Hemisphere
Thermal
Radiation Loss to
Space (50%
Lost)
Thermal Cell
Export to Dark
Side (50%
Retained)
Dark Hemisphere
Thermal Radiation
Loss to Space (50%
of 50% lost)
Surface Return
Loop from Dark to
Lit Hemisphere
(50% of 50%
Returned)
Radiant
Energy
Exiting to
Space (Units)
0 1 0
1
1 1
0.5 0.5
0.25 0.25 0.75
2
1 1.25
0.625 0.625
0.3125 0.3125 0.9375
3
1 1.3125
0.65625 0.65625
0.328125 0.328125 0.984375
4
1 1.328125
0.6640625 0.6640625
0.33203125 0.33203125 0.99609375
5
1 1.33203125
0.666015625 0.666015625
0.333007813 0.333007813 0.999023438
6
1 1.333007813
0.666503906 0.666503906
0.333251953 0.333251953 0.999755859
7
1 1.333251953
0.666625977 0.666625977
0.333312988 0.333312988 0.999938965
8
1 1.333312988
0.666656494 0.666656494
0.333328247 0.333328247 0.999984741
9
1 1.333328247
0.666664124 0.666664124
0.333332062 0.333332062 0.999996185
Infinite
Series
Limit
1 1.333333333 Energy Surplus Thermal Cell Energy Deficit Thermal Cell
Lit Hemisphere Budget Dark Hemisphere Budget
0.666666667 0.666666667 0.333333333 0.333333333 1
Process Insolation
Insolation plus
Thermal
Recycled Air
Final Lit Side
Thermal Radiant
Energy Lost to
Space
Atmospheric
Energy
Transported to
the Dark Side
Final Dark Side
Thermal Radiant
Energy Lost to Space
Atmospheric
Energy Returned
to the Lit Side
Space
Outgoing
Radiation
Balance
Hemisphere Energy Budget 1.3333 0.6667 Units
Total Global Energy Budget 2.0000
We can consider that the consequence of this process of infinite recycling by the Hadley cell is the formation and
maintenance of a dynamic machine made of air (Figure 2).
Figure 2. Basic Globe with a Stable Diabatic Advection Forward Model: Showing Energy Vectors and Unitary Energy Distributions.
20 Philip Mulholland and Stephen Paul Rathbone Wilde: An Iterative Mathematical Climate Model of the Atmosphere of Titan
If we couple the planet’s thermal Hadley cell of energy
surplus and link it to a thermal Polar cell of energy deficit,
we create a global atmospheric mass motion entity. This
entity is the global climate system that transports energy
from a region of surface energy surplus and delivers it to a
region of surface energy deficit, and then returns to repeat the
cycle endlessly (Table 4).
Table 4. Running the Dynamic-Atmosphere Energy-Transport Engine “Warmed Up”.
Step Process Energy Flow
1 Interception of solar energy by the lit surface 1 unit.
2 Return flow of cold air from the dark side 1/3 unit
3 Total energy available to drive the system 4/3 unit
4 50
S
:50
A
partition of the intercepted energy between the ground and the air leading to: -
5 Direct radiant loss to space from the warm surface 2/3 unit
6 Retention of energy by the lit air, followed by transport and delivery of this warm air to the dark side 2/3 unit
7 50
S
:50
A
partition of the delivered energy between the ground and the air leading to: -
8 Radiant loss to space from the ground on the dark side 1/3 unit
9 Return flow of cold surface air from the dark side to the lit side 1/3 unit
2.4. Modelling Slowly Rotating Titan
In order to study the process of energy transmission within
a globular modelled atmospheric system, we will first apply
our diabatic model to Titan, the slowly rotating Saturnian
satellite, using standard published metrics for this moon
(Table 5).
Table 5. The Metrics of Titan (Various Sources).
Parameter Value Units Source
Diameter of Titan 5150 Km [10]
Radius of Titan 2576 Km [11]
Average Surface Atmospheric Pressure 146.7 Kpa [11]
Average Surface Temperature 94 Kelvin [11]
Average Surface Temperature -179 Celsius [11]
Expected T
e
83.2 Kelvin [8]
Greenhouse Effect 10.8 Kelvin [8]
Eclipse duration Minimal % time in eclipse [12]
Surface gravity 1.352 m/s
2
[10]
Tropopause height 45.3 km [2]
Tropospheric lapse rate 0.533 K/km [2]
Average Solar insolation per solar orbit 14.82 W/m
2
[13]
Bond Albedo 0.265 A (Constant) [14]
Average Surface Solar Insolation 5.45 W/m
2
[13] [14]
Average Orbital Distance (Saturn) 1,427,000,000 Km [13]
If for the purpose of analysis, we remove the presence of
the giant planet Saturn (Titan’s parent body), from the solar
system and replace it with a hypothetical solar orbiting
tidally-locked minor planet Titan. Then for a vacuum Titan
with no atmosphere, the solar lit hemisphere of the now
minor planet will receive 5.45 W/m
2
, for a surface Bond
albedo of 0.265 at the average orbital distance of Saturn. The
Stefan-Boltzmann (S-B) equation gives us a temperature of
99.0K for the lit surface of a vacuum Titan (Table 6). The
unlit dark side of solar tidally locked vacuum Titan therefore
receives no radiant heat, and so will be at a nominal
temperature of zero Kelvin, therefore the tidally locked
vacuum Titan’s surface average temperature will be 49.5K
(Table 6).
Table 6. The Impact on Global Surface Temperature of Adding an Atmosphere to a Vacuum Titan.
Hypothetical
Circulation Type
Incoming
Solar
Radiant
Energy
(W/m
2
)
Lit
Hemisphere
Outgoing
Thermal
Radiant
Energy (W/m
2
)
Lit
Hemisphere
Surface
Temperature
(Kelvin)
Dark
Hemisphere
Outgoing
Thermal
Radiant Energy
(W/m
2
)
Dark
Hemisphere
Surface
Temperature
(Kelvin)
Global
Average
Surface
Temperatur
e (Kelvin)
Hemisphere
Average
Outgoing
Thermal
Radiant
Energy (W/m
2
)
No Atmosphere, No
Solar Day, Vacuum Titan
5.446 5.446 99.0 0.000 0 49.5 N/A
No Circulation Static
Atmosphere Titan 5.446 2.723 83.2 2.723 83.2 83.2 2.723
If we now add a fully optically transparent atmosphere of
Nitrogen gas to vacuum Titan, and let this atmosphere
distribute the heat from the Lit to the Dark Hemisphere, then
on achieving thermal equilibrium (and assuming no ongoing
Journal of Water Resources and Ocean Science 2020; 9(1): 15-28 21
cycling of mass and energy) the energy on the lit side will be
halved to 2.72 W/m
2
, and the remaining half (also 2.72 W/m
2
)
will be distributed to the dark hemisphere, and raise its
temperature from zero Kelvin to 83.2 Kelvin (Table 6).
This temperature is identical to the Expected T
e
value of
83.2K previously established for Titan (Table 1) using
Equation 1 of [8], and this demonstrates that daily rotation is
not an a priori requirement to distribute captured solar
energy across the surface of a globe. Fluid atmospheric
circulation that links the two separate hemispheres achieves
exactly the same thermal effect.
2.5. How the Presence of an Atmosphere Distributes the
Captured Solar Energy Across a Planet
We now need to test how our DAET model behaves when
we apply standard Titan insolation parameters to the diabatic
form of our model. The annual average solar irradiance for
Titan is 14.82 W/m
2
, and its Bond Albedo is 0.265 [14],
which means that the Annual Average Planetary Energy flow
that the lit hemisphere of Titan receives is 5.446 W/m
2
. This
insolation flux equates to 2.723 W/m
2
per hemisphere after
internal atmospheric redistribution (Table 7).
Table 7. The Titan energy budget and atmospheric system energy deficit.
Metric Thermal Radiation (W/m
2
) S-B Sigma Kelvin Celsius
Titan Average Annual Solar Insolation (Expected T
e
) 2.723 5.67E-08 83.2 -189.8
Target Global Average Annual Air Temperature (Actual Ts) 4.427 5.67E-08 94.0 -179.0
Titan Thermal Energy Deficit 1.704 10.8
The total global atmospheric energy budget that the diabatic equipartition model achieves for Titan is 10.89 W/m
2
(Figure 3).
This value results from applying the solar interception flux for the sunlit hemisphere of 5.45 W/m
2
(Table 5), to the
equipartition diabatic advection model (Table 3).
Figure 3. Stable Diabatic Advection Model of Titan: Showing Energy Vectors and Total Energy Distributions.
This equipartition energy ratio, when applied to the
modelled atmosphere of Titan, creates an air temperature of
82.3 K (-190.7°C), which is slightly below the T
e
= 83.2 K of
the standard vacuum planet equation for the moon (Table 1).
This lower temperature value for Titan (Table 8) is as result
of the larger surface radiant energy loss from the lit surface
to space (Figure 3), because of the higher insolation applied
to our diabatic model under the divide by 2 rule of lit
hemisphere radiation loading.
Table 8. Diabatic Model of Titan showing Internal Energy Recycling with Equipartition of Energy for Both Hemispheres.
Diabatic Model Partition Test 50
S
:50
A
Titan Insolation Parameters
Cycle
Number
Space
Incoming
Captured
Radiation
(W/m
2
)
Lit Ground
Received Energy
(W/m
2
)
Lit Hemisphere
50% Thermal
Radiation Loss
to Space (W/m
2
)
Lit Hemisphere
50% Export to
Dark Side
(W/m
2
)
Dark
Hemisphere
50% Thermal
Radiation Loss
to Space (W/m
2
)
Dark
Hemisphere
50% Surface
Return Loop to
Lit Side (W/m
2
)
Space
Outgoing
Radiation
Balance
(W/m
2
)
Diabatic Equipartition Ratio 50.0000% 50.0000% 50.0000% 50.0000%
0 5.4464
1 5.4464 5.4464 2.7232 2.7232 1.3616 1.3616 4.085
2 5.4464 6.8079 3.4040 3.4040 1.7020 1.7020 5.106
3 5.4464 7.1483 3.5742 3.5742 1.7871 1.7871 5.361
4 5.4464 7.2334 3.6167 3.6167 1.8084 1.8084 5.425
10 5.4464 7.2618 3.6309 3.6309 1.8154 1.8154 5.446
11 5.4464 7.2618 3.6309 3.6309 1.8154 1.8154 5.446
22 Philip Mulholland and Stephen Paul Rathbone Wilde: An Iterative Mathematical Climate Model of the Atmosphere of Titan
Diabatic Model Partition Test 50
S
:50
A
Titan Insolation Parameters
Cycle
Number
Space
Incoming
Captured
Radiation
(W/m
2
)
Lit Ground
Received Energy
(W/m
2
)
Lit Hemisphere
50% Thermal
Radiation Loss
to Space (W/m
2
)
Lit Hemisphere
50% Export to
Dark Side
(W/m
2
)
Dark
Hemisphere
50% Thermal
Radiation Loss
to Space (W/m
2
)
Dark
Hemisphere
50% Surface
Return Loop to
Lit Side (W/m
2
)
Space
Outgoing
Radiation
Balance
(W/m
2
)
12 5.4464 7.2618 3.6309 3.6309 1.8154 1.8154 5.446
13 5.4464 7.2618 3.6309 3.6309 1.8154 1.8154 5.446
14 5.45 7.26 3.63 3.63 1.82 1.82 5.45
S-B 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08
Kelvin 99.0 106.4 89.5 89.5 75.2 75.2 99.0
Celsius -174.0 -166.6 -183.5 -183.5 -197.8 -197.8 -174.0
Statistic Mean Exit Temp Mean Air Temp Lit-side Dark-side Global
Kelvin 82.3 82.3 W/m
2
W/m
2
W/m
2
Celsius -190.7 -190.7 7.26 3.63 10.89
Thermal
Enhancement
(Celsius)
Atmospheric Response Lapse rate Top of Atmosphere (TOA)
K/Km Delta K Km
Lit Hemipshere 0.533 16.9 31.8
0.0 Dark Hemisphere 0.533 14.2 26.7
The most critical feature of the diabatic climate model of Titan is that it fully and accurately replicates the computation of
the vacuum planet equation of astronomy (Figure 4).
Figure 4. The Direct Equivalence of the Vacuum Planet Equation Top of Atmosphere Radiant Exhaust Temperature (Astronomy) with the Diabatic Climate
Model Surface Atmospheric Temperature (Meteorology).
2.6. Establishing the Global Energy Partition Ratio for Titan by Inverse Modelling
The process of establishing the average surface
temperature that Titan experiences as a consequence of the
adiabatic meteorological actions of turbulent air motion,
energy partition and internal atmospheric pressure induced
energy retention, is achieved by applying the mathematical
technique of inverse modelling to our standard diabatic
forward model.
The following steps describe the logic flow of our inverse
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 (Figure 3). This is the diabatic forward
model.
Step 2: That on applying Titan insolation parameters to the
diabatic forward model we have achieved an average global
air temperature of 82.3 K. This temperature is a small
underestimation of the Expected T
e
for a vacuum Titan of
83.2 K that the standard radiative balance equation computes.
The difference arises because in our model we apply a divide
by 2 rule to the distribution of incoming solar energy, rather
than the divide by 4 rule of the standard vacuum equation
model. (Table 8).
Journal of Water Resources and Ocean Science 2020; 9(1): 15-28 23
Step 3: That by applying the standard geoscience
technique of inverse modelling to our basic diabatic
atmospheric model of Titan, we can create an adiabatic
model that has the flexibility to allow exploration of the
surface energy partition ratios that determines the thermal
enhancement observed in the atmosphere of this slowly
rotating moon. (Table 9).
Table 9. Adiabatic Energy Partition Test for Titan (~37.6% Thermal Radiant Loss to Space: ~62.4% Atmospheric Energy Retention).
Adiabatic Partition Test of Slowly Rotating Titan
Cycle
Number
Space
Incoming
Captured
Radiation
(Units)
Lit Ground
Received Energy
(Units)
Lit Ground
Energy
Partition is
~37.6% Lost to
Space (Units)
Sun Lit Air
Energy Partition
is ~62.4%
Retained and
Exported (Units)
Dark Ground
Energy Partition
is ~37.6% Lost
to Space (Units)
Dark Air
Energy
Partition is
~62.4%
Retained and
Exported
(Units)
Radiant
Energy
Exiting to
Space (Units)
Partition Ratio Target Temperature
94 Kelvin (-179°C) 37.6017% 62.3983% 37.6017% 62.3983%
0 1 0
1
1 1
0.376017256 0.623982744
0.234628279 0.389354465 0.610645535
2
1 1.389354465
0.522421253 0.866933212
0.325981847 0.540951365 0.8484031
3
1 1.540951365
0.579424303 0.961527062
0.361550767 0.599976295 0.94097507
20
1 1.637611244
0.615770086 1.021841158
0.384229908 0.63761125 0.999999994
21
1 1.63761125
0.615770088 1.021841162
0.384229909 0.637611253 0.999999998
22
1 1.637611253
0.615770089 1.021841164
0.38422991 0.637611254 0.999999999
23
1 1.637611254
0.615770089 1.021841164
0.38422991 0.637611254 1.000000000
24
1 1.637611254
0.615770089 1.021841165
0.38422991 0.637611254 1.000000000
Infinite
Series
Limit
1 1.64 Energy Surplus Thermal Cell Energy Deficit Thermal Cell
Lit Hemisphere Budget Dark Hemisphere Budget
0.62 1.02 0.38 0.64 1.00
Process Insolation
Insolation plus
Thermal
Recycled Air
Final Lit Side
Thermal Radiant
Energy Lost to
Space
Atmospheric
Energy
Transported to
the Dark Side
Final Dark Side
Thermal Radiant
Energy Lost to
Space
Atmospheric
Energy Returned
to the Lit Side
Space
Outgoing
Radiation
Balance
Hemisphere Energy Budget 1.64 1.02 Units
Total Global Energy Budget 2.66
Step 4: That on applying Titan insolation parameters to the adiabatic model, using the energy partition ratio identified by
inverse modelling, we achieve the known average global air temperature of 94 K (-179°C) for this slowly rotating moon (Table
10).
Table 10. Adiabatic Model of Titan showing Internal Energy Recycling for Both Hemispheres
Titan Adiabatic Model
Cycle
Number
Space Incoming
Captured
Radiation (W/m
2
)
Lit Ground
Received
Energy (W/m
2
)
Lit Ground
Partition is
~37.6% (W/m
2
)
Sun Lit Air
Partition is
~62.4%
(W/m
2
)
Dark Ground
Partition is
~37.6%
(W/m
2
)
Dark Air
Partition is
~62.4%
(W/m
2
)
Space Outgoing
Radiation
Balance (W/m
2
)
Partition Ratio Target Temperature 94
Kelvin (-179°C) 37.6017% 62.3983% 37.6017% 62.3983%
0 5.446
1 5.446 5.446350000 2.047921580 3.398428420 1.277867728 2.120560692 3.325789308
2 5.446 7.566910692 2.845288992 4.721621700 1.775411234 2.946210467 4.620700226
24 Philip Mulholland and Stephen Paul Rathbone Wilde: An Iterative Mathematical Climate Model of the Atmosphere of Titan
Titan Adiabatic Model
Cycle
Number
Space Incoming
Captured
Radiation (W/m
2
)
Lit Ground
Received
Energy (W/m
2
)
Lit Ground
Partition is
~37.6% (W/m
2
)
Sun Lit Air
Partition is
~62.4%
(W/m
2
)
Dark Ground
Partition is
~37.6%
(W/m
2
)
Dark Air
Partition is
~62.4%
(W/m
2
)
Space Outgoing
Radiation
Balance (W/m
2
)
3 5.446 8.392560467 3.155747554 5.236812912 1.969132019 3.267680893 5.124879574
20 5.446 8.919003998 3.353699406 5.565304592 2.092650559 3.472654033 5.446349965
21 5.446 8.919004033 3.353699419 5.565304614 2.092650567 3.472654046 5.446349986
22 5.446 8.919004046 3.353699424 5.565304622 2.092650571 3.472654052 5.446349995
23 5.446 8.919004052 3.353699426 5.565304625 2.092650572 3.472654054 5.446349998
24 5.446 8.919004054 3.353699427 5.565304627 2.092650572 3.472654054 5.446349999
25 5.45 8.92 3.35 5.57 2.09 3.47 5.45
S-B 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08 5.67E-08
Kelvin 99.0 112.0 87.7 99.5 77.9 88.5 99.0
Celsius -174.0 -161.0 -185.3 -173.5 -195.1 -184.5 -174.0
Statistic Mean Exit
Temp Mean Air Temp
Lit-side Dark-side Global
Kelvin 82.82 94.00 W/m
2
W/m
2
W/m
2
Celsius -190.18 -179.0 8.92 5.57 14.48
Thermal
Enhancement
(Celsius)
Atmospheric Response Lapse rate Top of Atmosphere
(TOA)
K/Km Delta K Km
Lit Hemisphere 0.533 24.3 45.6
11.2 Dark Hemisphere 0.533 21.6 40.5
The final adiabatic global energy budget for Titan is 14.48
W/m
2
(Figure 5). This value results from applying the solar
interception flux for the sunlit hemisphere of 5.45 W/m
2
(Table 5), to the Titan adiabatic convection model (Table 9)
with a surface partition of 62.4% of the intercepted solar
energy being retained by the air. By this means of solar
energy capture and retention, as a result of atmospheric
adiabatic convection on the lit hemisphere, the stable average
air temperature of 94 K (-179°C) is thereby achieved for
Titan (Table 10).
Figure 5. Stable Adiabatic Model of Titan: Showing Energy Vectors and Total Energy Distributions.
3. Results of Applying the Adiabatic
Modelling Process to Titan
We have now established three important facts about a
planetary atmosphere on terrestrial globes:
1. That the presence of even a fully transparent mobile-
fluid atmosphere raises the global average surface
temperature above that of a non-rotating vacuum world.
2. That this atmosphere both retains and recycles solar
energy, and achieves a stable energy flow across the
globe’s surface.
3. The stable limit of the energy flow within the system is
set by the partition ratio of energy between the radiant
Journal of Water Resources and Ocean Science 2020; 9(1): 15-28 25
loss to space of the emitting surface of both
hemispheres, and the quantity of energy retained and
recycled by the air.
The process of inverse modelling was applied to the
DAET forward model of atmosphere, by constructing a
cascade algorithm that allowed the initial unknown energy
partition ratio of the lit hemisphere to be determined. This
value was established by means of the Excel Inverse
Modelling Tool called “Goal Seek”. Initial forward
modelling tests where undertaken to establish the number of
iterative cycles that are required by Goal Seek to create a
stable thermal outcome.
It was established that 25 cycles of atmospheric overturn
would produce a stable adiabatic outcome for the atmosphere
of Titan. The global average air temperature of this giant
moon was achieved using the DAET model with a global
atmospheric reservoir energy budget of 2.66 times surface
solar energy input (Table 9).
The results of applying the inverse modelling run to Titan
are shown in Table 10. 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 its surface energy
being retained by the atmosphere (Figure 5).
The adiabatic model of Titan computes a dark side thermal
separation between the surface and the air of 21.6°C (Table
10). In our model of Titan, the hemisphere of energy deficit
is a proxy for the moon’s thermal polar cell. Using the
troposphere 45 km gross thermal lapse rate of 0.533 K/km
for Titan (Figure 6), this temperature difference equates to a
physical separation of 40.5 km (Table 10). This value is our
modelled estimate of the tropopause height for the polar
regions of Titan.
Figure 6. Titan Atmosphere Temperature Profile [2].
4. Discussion
Our numerical atmospheric model 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 and type. For a
climate model to be valid, it must be capable of being applied
to all bodies, and include all possible types of planetary
rotation, including those bodies that are tidally locked.
To address the problem that the standard vacuum planet
equation of astronomy cannot be applied to a tidally locked
planet, we created a geometric climate model based on
meteorological principles. The fundamental distinction of our
model is that it honours the reality of a single illuminated
hemisphere, and therefore uses the divide by 2 rule as the
power intensity divisor for the intercepted surface
illumination flux.
The purpose of the diabatic meteorological model is to
replicate the form of the standard radiation balance equation
of astronomy. The standard model uses the divide by 4 rule
of surface radiant energy distribution (Equation 1), and we
apply this concept of flux intensity dilution 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 a
divide by 2 rule of surface radiant energy distribution is
applied. For our meteorological model the transmission of
energy from the lit hemisphere of energy surplus to the unlit
side of energy deficit, is mediated by the atmospheric process
of advection.
A comparison of the approach to climate analysis used by
the two different scientific disciplines of astronomy, which
26 Philip Mulholland and Stephen Paul Rathbone Wilde: An Iterative Mathematical Climate Model of the Atmosphere of Titan
uses the vacuum planet equation as its founding concept, and
our meteorological approach, using a model of the Hadley
cell as the fundamental element, is given in Table 11.
Table 11. A comparison of the approach to climate analysis used by the two different scientific disciplines of astronomy and meteorology.
Source Discipline Astronomy Meteorology
Analytics Vacuum Planet Equation Dynamic-Atmosphere Energy-Transport
Illumination Intensity Divisor 4 2
Power Intensity Distribution Mode Rapid Planetary Rotation Atmospheric Mass Motion
Surface Environment Uniform whole globe Two distinct hemispheres
Analytical Approach Descriptive Physical Equation Explanatory Mathematical Model
Initial Conditions Starts by assuming that there is no atmosphere Starts by assuming that the planet does not rotate
Physical Process Low frequency Radiant Flux recycling Air Motion Thermal recycling
Application Treats the atmosphere as an opaque system Treats the atmosphere as a compressible mobile fluid
Appropriate Use Measures the external low frequency radiant flux
exhaust temperature of a planet
Studies the internal process of solar energy capture
and atmospheric distribution
Critical Distinction Applies the concept of energy balance to the top of the
atmosphere
Applies the concept of energy balance to the base of
the atmosphere
In order to replicate as closely as possible, the form of the
standard equilibrium temperature equation of astronomy, the
energy partition ratio between the surface and the air in our
DAET model is maintained at a diabatic 50%:50% ratio for
both the lit and unlit hemispheres. Our model therefore
performs, by means of an analogous numerical series with a
finite limit (Equations 2 and 3), the same function as, and it
sums to the same result as that observed for the standard
radiation balance equation of astronomy (Equation 1).
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 turbulent motion at the boundary interface.
Turbulent fluid motion under daytime solar illumination is
characteristic of forced radiative heating and adiabatic
convection, consequently a partition ratio weighted in favour
of the air is the required metric.
Because the DAET atmosphere model has two distinct
surfaces, both representative of the separate environments of
energy surplus and energy deficit, we have the opportunity to
explore the effects on energy flow within the atmospheric
model by applying the process of inverse modelling. Using
the known atmosphere parameters, we can explore how two
distinct energy partition ratios, one for day and a separate one
for the night, impact on internal energy retention within the
modelling system. Inspection of the day time temperature
profile for Titan (Figure 6) suggests that the energy partition
ratio should be weighted in favour of the air for the
convective environment on this slowly rotating moon.
An important feature of our DAET model is its predictive
capability, specifically the ability of the adiabatic model of
slowly-rotating Titan to predict the tropospheric altitude of
the Hadley cell of energy surplus, the Polar cell of energy
deficit, and also the parameters and dynamics of the Titan
atmosphere.
Our adiabatic model incorporates the numerical process of
energy partition in favour of the turbulent air for the sunlit
surface boundary. Because we know a priori the required
average surface air temperature for Titan, we can apply the
numerical technique of inverse modelling to establish the
energy partition ratio that creates the required thermal
enhancement for an atmosphere of any opacity. Because our
model creates a thermal contrast between the surface and the
air, we can also use this temperature difference as a measure
of the tropospheric height by applying the appropriate
environmental lapse rate for Titan from measured data.
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; and that the energy retention 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.
5. Conclusions
We have designed our mathematical model to retain the
critical dual surface element of a lit globe, namely night and
day. Figure 4 shows that our simple process diabatic model,
when applied to a fully transparent pure Nitrogen atmosphere,
matches the results of the standard atmosphere equation
which is traditionally applied to an airless world [8].
The following key points arise from the application of our
model to Titan:
1. By applying a diabatic forward model to the atmosphere
of Titan we have demonstrated that the expected
temperature of an airless planet can be replicated on a
moon that has a thick (but fully transparent) atmosphere,
which can transport air from a lit region of net energy
surplus to a dark region of net energy deficit.
2. 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 [4], and consequently this slowly
rotating moon experiences a single climatic surface
environment with a common energy partition ratio for
both hemispheres.
Journal of Water Resources and Ocean Science 2020; 9(1): 15-28 27
3. By applying the inverse atmosphere modelling process
to the atmosphere of Titan, and accounting for the fact
that there is no surface thermal contrast between day
and night on the slowly rotating moon; we can
determine the global energy partition ratio on Titan that
accounts for its thermally enhanced atmospheric energy
retention, and explain the presence of super-rotational
winds.
4. By using the appropriate planetary lapse rate for Titan,
our inverse modelling process predicts the global
atmospheric thickness for this Saturnian moon.
5. Our modelling study of Titan confirms that the opacity
of an atmosphere fundamentally controls the height of
the radiant emission zone that vents energy to space
[15].
6. Consequently, the computational dynamics of the
adiabatic model demonstrate 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.
Our adiabatic model can be tuned to replicate the known
conditions of surface atmospheric temperature and pressure
of Titan. The issue of atmospheric opacity, due to the
presence of polyatomic molecular gases, then becomes a
passive atmospheric process, and the concept of thermal
radiative feedback as an explanation for the greenhouse
effect can be abandoned. This assessment agrees with the
recent analysis using balloon profile data that the Greenhouse
Gas hypothesis as an explanation for the thermal structure of
the Earth’s tropopause is flawed [16].
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: The amount of direct sunlight energy received
by the surface of a planet or moon.
Inverse Modelling: The mathematical process of determining
the variable 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 its 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 lowest layer of a terrestrial planet’s
atmosphere dominated by surface heating and cooling, and
turbulent air motion.
Turbulence: The process of random mixing of air
undergoing forced radiant thermal heating at the surface
boundary.
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... We are attempting here to simultaneously analyse the energy flows for the Earth's three atmospheric circulation cells, using the adiabatic form of the Dynamic-Atmosphere Energy-Transport (DAET) model, previously used for the study of the climates of Venus and Titan [5,10]. The Earth is modelled as a spherical globe that cuts a circular silhouette, or disk shadow from the beam of the solar irradiance at the planet's average orbital distance from the Sun. ...
... The Dynamic-Atmosphere Energy-Transport (DAET) climate model contains a mechanism for energy flux recycling using the meteorological process of atmospheric circulation. This model was used to demonstrate that convective atmospheric mass motion can be invoked to explain the planetary greenhouse effect for both Venus and also Titan, the tidally locked moon of Saturn [5,10]. Atmospheric data for these two bodies show that there is little or no thermal contrast between the lit daytime and the dark nighttime hemispheres on these slowly rotating worlds. ...
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... For a resolution of this paradox we propose the adoption of a new climate model, the Dynamic Atmosphere Energy Transport (DAET) model, that is based on meteorological principles and is applicable to all solar illuminated terrestrial type astronomic bodies that possess a dense semi-opaque thermally radiant atmosphere [3,17,18]. ...
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Images of Titan acquired over five nights in October 2004 using the adaptive optics system at the Keck Observatory show dramatic increases in tropospheric cloud activity at the south pole compared with all other images of Titan clouds to date. During this time, Titan's south polar clouds brightened to more than 18 times their typical values. The Cassini Ta flyby of Titan occurred as this storm was rapidly dissipating. We find that the brightness of this cloud outburst is consistent with the dramatic transient brightening of Titan observed in atmospheric windows on two nights in 1995 by Griffith et al. [Griffith, C.A., Owen, T., Miller, G.A., Geballe, T., 1998. Nature 395 (6702) 575–578] if we scale the brightness of the cloud by projecting it onto the equator. While apparently infrequent, the fact that large cloud events have been observed in different seasons of Titan's year indicates that these large storms might be a year-round phenomenon on Titan. We propose possible mechanisms to explain these occasional short-term increases in Titan's cloud activity.
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Atmospheric mixing ratios of ∼10−5 ± 1 for ammonia on the early Earth would have been sufficient, through the resulting greenhouse warming, to counteract the temperature effects of the faint early sun. One argument against such model atmospheres has been the short time scale for ammonia photodissociation by solar ultraviolet light. Here it is shown that ultraviolet absorption by steady-state amounts of high-altitude organic solids produced from methane photolysis may have shielded ammonia sufficiently that ammonia resupply rates were able to maintain surface temperatures above freezing.
Titan: Facts About Saturn's Largest Moon
  • N T Redd
Redd, N. T. 2018. Titan: Facts About Saturn's Largest Moon. https://www.space.com/15257-titan-saturn-largest-moon-factsdiscovery-sdcmp.html.
The Coriolis Effect: Four centuries of conflict between common sense and mathematics, Part I: A history to 1885
  • A O Persson
Persson, A. O., 2005. The Coriolis Effect: Four centuries of conflict between common sense and mathematics, Part I: A history to 1885. International Commission on the History of Meteorology 2, 24pp.