Global climate by Rossby number in the Solar system planets

TL;DR

This study investigates the relationship between planetary Rossby number and a scaled height parameter across Solar system planets, revealing a positive correlation that could inform the search for life on exoplanets.

Contribution

It introduces a simplified model linking Rossby number to a scaled height parameter and explores its correlation across Solar system planets using observational wind data.

Findings

Found a power-law correlation between Rossby number and height parameter with an exponent around 0.5.

Identified Earth's height parameter as mainly influenced by lunar tides.

Suggests exoplanet-moon systems as promising targets for life detection.

Abstract

On the largest scales, planetary climates can be described by their Rossby number (\textit{Ro}). \textit{Ro} is in response to $Gr/Re^2$ , where Gr is the Grashof number and Re is the Reynolds number. We here simplify $Gr/Re^2$ as h, where $h=H/H_{Earth}$ with $H=gP/(2\pi V_e)$ for a planet with surface gravity \textit{g}, rotation period \textit{P} and equatorial velocity V\textsubscript{e}. Unlike \textit{h}, \textit{Ro} is difficult to obtain because of a large diversity in observation. We perform on an in-depth literature search on average (av) and maximum (mx) wind velocity for each planet in the Solar system by various observational methods and by altitude. We explore a correlation between \textit{Ro} and \textit{h} expressed as a power law with index $\alpha$ based on wind velocities of planets in the Solar system. We obtain a correlation between \textit{Ro} and \textit{h} with $\alpha=0.56$ (av) and $\alpha=0.52$ (mx). Earth's $H=H_{Earth}$ ($h=1$) is primarily due to lunar tidal interaction, given our relatively distant habitable zone (HZ) to the Sun. Our positive correlation, therefore, suggests exoplanet-moon systems as the `go-to-place' in our searches for potentially advanced life in exosolar system.

Tables

Table 1. Table 1.1 : Average wind velocity data for each planet in the Solar system by various observational methods.

Methods		Earth	Mars	Jupiter	Saturn	Uranus	Neptune	Pluto
Image Analysis	Cloud					${84}^b$	${325}^{de}$
	Movement
	Atmospheric			${100}^f$	${400}^f$
	Marking
	Surface		55					${10}^h$
	Topography		$\pm {21}^{g\star }$
Radio Observation	Doppler			90
	shift			$\pm {10}^l$
	Occultation					110
						$\pm {40}^n$
	Rotation					${200}^p$	${400}^p$
	period
In-situ Probes	Descent	50
	Probes	$\pm {10}^i$
	Landing		23
			$\pm 4^j$
Spectra	Thermal
	Emission
∗unspecified				${100}^p$	${500}^r$
Average		50	39	97	450	97	∙325	10
values		$\pm 10$	$\pm 23$	$\pm 6$	$\pm 71$	$\pm 18$

Table 2. Table 1.2 : Maximum wind velocity data for each planet in the Solar system by various observational methods.

Methods		Earth	Mars	Jupiter	Saturn	Uranus	Neptune	Pluto
Image Analysis	Cloud	${50}^a$	${40}^a$	${170}^a$	${450}^a$	240	${560}^e$
	Movement					$\pm {50}^c$
	Atmospheric							${10}^h$
	Marking
	Surface
	Topography
Radio Observation	Doppler			175
	shift			$\pm {25}^l$
	Occul		${170}^m$					${10}^o$
	tation
	Rotation
	period
In-situ Probes	Descent	${70}^i$
	Probes
	Landing
Spectra	Thermal		85
	Emission		$\pm 7^{k\circ }$
unspecified				${172}^q$	${500}^q$	${250}^{qs}$	${556}^s$
Maximum		60	98	172	475	235	505	10
value		$\pm 14$	$\pm 66$	$\pm 3$	$\pm 35$	$\pm 24$	$\pm 91$

Table 3. Table 2 : Wind velocities of the planets in the Solar system in the two temperature-negative gradient regions by altitude. See the notes of Table 1.1 for references.

Altitude(km)	Earth	Mars	Pluto
$50\sim 80$	$50\pm {10}^{ac}$	$85\pm 5^r$
$0\sim 10$		${40}^a$	${10}^g$

Table 4. Table 3.1 : Average values of Ro and h for all planets in the Solar system, where R ^ ​ o = R ​ o / R ​ o E ​ a ​ r ​ t ​ h ^ 𝑅 𝑜 𝑅 𝑜 𝑅 subscript 𝑜 𝐸 𝑎 𝑟 𝑡 ℎ \hat{R}o=Ro/Ro_{Earth} .

	Earth	Mars	Jupiter	Saturn	Uranus	Neptune	Pluto
Ro	95.2	144	6.8	38.9	33.2	106	651
$\hat{R}o$	1	1.51	0.07	0.41	0.35	1.11	6.84
h	1.00	0.76	0.04	0.02	0.12	0.14	13

Table 5. Table 3.2 : Maximum values of Ro and h for all planets in the Solar system, where R ^ ​ o = R ​ o / R ​ o E ​ a ​ r ​ t ​ h ^ 𝑅 𝑜 𝑅 𝑜 𝑅 subscript 𝑜 𝐸 𝑎 𝑟 𝑡 ℎ \hat{R}o=Ro/Ro_{Earth} .

	Earth	Mars	Jupiter	Saturn	Uranus	Neptune	Pluto
Ro	114	362	12	41.1	80.4	164	651
$\hat{R}o$	1	3.17	0.1	0.36	0.70	1.44	5.70
h	1.00	0.76	0.04	0.02	0.12	0.14	13

Equations

$$Ro=\frac{U}{2\Omega R},$$

$$Gr=\frac{\beta \Delta T{R}^{3}g}{{\nu }^2}$$

$$Re=\frac{UR}{\nu },$$

$$\frac{Gr}{R{e}^2}\approx \frac{gP}{2\pi V_e}\equiv H$$

$$h=\frac{H}{H_{Earth}}.$$

$$Ro=c_0{\left(\frac{Gr}{R{e}^2}\right)}^{\alpha }\propto h^{\alpha },$$

Full text

††journal: Journal of New astronomy

1 introduction

Currently, there is a vigorous quest for potentially habitable exoplanets, following the pioneering discoveries by Sagittarius Window Eclipsing Extrasolar Planet Search (SWEEPS) in HST [24], Convection, Rotation and planetary Transits (CoRoT) [3], Kepler [19], Kepler*′* Second light (K2) [13], Gaia [16], Arcsecond Space Telescope Enabling Research in Astrophysics (ASTERIA) [18] and, today, Transiting Exoplanet Survey Satellite (TESS) [8].

At present, conditions of habitability remain somewhat speculate, though general conditions favoring liquid water and, perhaps, an oxygen-rich atmosphere are essential [5]. Of interest also, and perhaps even more speculate, are conditions favoring advanced life. In particular, these conditions may not be the same as those for abiogenies, suggesting the need for co-evolution of life and planetary conditions as a whole. Here, we explore some conditions for a global clement climate relevant to the development of advanced life on Earth.

Planetary climates in our Solar system can be roughly divided into two groups: clement climates on Earth, Mars and Pluto and extreme climates on Jupiter, Saturn, Uranus and Neptune. We shall omit Mercury and Venus. Mercury is omitted for its essentially complete lack of atmosphere; Venus, tidally locked to the Sun, is heated one-sided driving a global climate different from those of the other planets in the Solar system. These kind of different wind patterns are commonly characterized by their Rossby number,

[TABLE]

where U is the wind velocity of a planetary atmosphere, $\Omega$ is the angular frequency of planetary rotation. A global clement climate is proposed as a natural condition conducive for advanced life [31].

Quite generally, the global climate of a planet is a buoyancy-driven atmospheric flow on a rotating sphere subject to Coriolis forces. The driving force of buoyancy is governed by the Grashof number

[TABLE]

and the induced large scales flows are characterized by the Reynolds number,

[TABLE]

where g is surface gravity, $\beta$ is the coefficient of thermal expansion $\beta=1/T$, $\Delta T$ is a characteristic scale of the driving temperature difference (e.g. polar-to-equator) induced by exposure to the Sun, R is the radius of a planet, $\nu$ is the kinematic viscosity, and $U=V_{e}$ is the equatorial velocity, $V_{e}=\Omega R$.

On the largest scales, the resulting buoyancy-driven flows are effectively described by $Gr/Re^{2}$. Here, ignoring variants of $\beta\Delta T$ across the different planets, we simplify

[TABLE]

and normalize

[TABLE]

h was recently proposed as a habitability index for its general correlation to global climate [31]. Here, we seek to quantify the correlation of Ro to h in the planetary atmospheres of our Solar system (§3). Ro is not so easy to infer because of a large diversity in observational measurements and associated uncertainties while h is well defined for all Solar system planets. For this reason, we focused on an in-depth literature search on wind velocity data of all planets in the Solar system.

In §2, we organize the wind velocities inferred for planets in the Solar system by observing methods and altitudes. In §3, we report on a correlation of Ro to h using available data on average and maximum wind velocities of planets in the Solar system. In §4, we summarize our exploration with a future outlook.

2 Data

Wind velocities of planets in the Solar system are observed across different observing atmospheric heights (altitudes) by different methods ranging from ground-based, space-based and in-situ measurements. This poses a challenge in our effort to derive a homogeneous set of data.

$*$: Unspecified: cited papers did not mention methods.

$\star$: 55 m/s is the average of 70 m/s, the wind velocity required for activation of Meridiani

Planum observed by Opportunity, and 40 m/s, estimated from image of reversing dust

streaks (Mars Orbiter).

$\bullet$: The mean wind velocity of Neptune, 325 m/s, is measured also by Voyager 2 with no

uncertainty reported.

$\circ$: $85\pm 7$ is westward jet velocity in both hemispheres, though a further 40 m/s eastward

in the Southern hemisphere exists.

a) Limaye, S. S., 2010 [20]; b) Hammel, H. B., 2001 [10]; c) Sromovsky, L. A. et al, 2005

[27]; d) Hammel, H. B. 1989 [9]; e) Smith, B. A., et al, 1989 [26]; f) Hide, R. 1984 [12];

g) Jerolmack, D. J. et al., 2006 [15]; h) Telfer, M. W. et al., 2018 [28]; i) Keegan, T. J.

1961 [17]; j) Pollack, J. B. et al., 1976 [23]; k) Santee, M. L. et al, 1995 [25]; l) Atkinson,

D. H. et al, 1998 [2]; m) Tellmann, S. et al, 2013 [29]; n) Lindal, G. F., 1987 [21]; o)

Zalucha, A.M. 2016 [33]; p) Helled, R. et al., 2010 [11]; q) Guide to Space - Universe

today [7]; r) Ingersoll, A. P., 1990 [14]; s) Planets, Moons, and Dwarf Planets — NASA

[22]

Occasionally, we compromise, e.g., taking wind velocities from the mesosphere of Earth - with negative temperature gradient similar to that at low altitudes - alongside space-based observation of the other planets.

Observational techniques to measure wind velocities of a planet are mostly by image analysis, radio observations, in-situ probes and spectral methods (Tables 1.1-2). For instance, wind velocities can be obtained from images of cloud movement, atmospheric markings, e. g., Jupiter*′s Giant Red Spot, and surface topography, e.g., Pluto′*s dunes. Also, wind velocities can be obtained via radio observation: radio Doppler shift, radio occultation, and radio rotation period. Additionally, wind velocities have been measured by in-situ probes: descent probes and Landers. Furthermore, wind velocities can be obtained through spectral methods such as thermal emission. Tables 1.1-2 summarize our literature search on wind velocities for each planet in the Solar system by various methods. In this table, uncertainties refer to scatter in data from multiple measurements reported in the literature. For this broad range of measurement approaches, our references are listed in the notes of Table 1.1-2.

Pluto is an interesting special case with wind velocities estimated for the first time by the New Horizons mission [4]. In its extremely tenuous atmosphere, wind velocities are now inferred from atmospheric temperature structure and also from dunes on its surface. The first shows that winds concentrate about polar regions, from greater than 10 m/s down to a few m/s at intermediate latitudes [33]. This is corroborated by wind velocities of about 10 m/s inferred from dunes at similar intermediate latitudes [28]. Thus, we use Pluto*′*s wind velocity of 10 m/s in Table 1.1-2. With no known uncertainties, uncertainty is left unspecified in Table 1.1-2.

Quite generally, wind velocities vary with altitude. As a compromise in deriving a reasonably homogeneous data set, we would focus on velocities in layers with negative temperature gradients. On Earth, there are two such layers, at low altitudes 0-10 km and at high altitudes 50-80 km. Most other planets in the Solar system have a similar structure, excluding perhaps Uranus and Mars [1]. In Table 2, we list results for Earth, Mars and Pluto by altitude. These data are indeed consistent with the mean of the average (av) and maximum (ma) values of Table 1.1-2. On this admittedly limited basis, we shall proceed with the data for all planets in Table 1.1-2 despite the absence of such explicit data by altitude for the other four planets.

3 A Ro-h correlation

We calculate Ro from the wind velocities of Table 1.1-2 and h from the physical parameters of planets in the Solar system. Table 3.1-2 summarizes our estimates of Ro and h for all planets in the Solar system excluding Mercury and Venus.

We consider the correlation of Ro to h be of the form of a power law:

[TABLE]

To estimate the power law index $\alpha$, we draw a loglog plot of our Ro to h and calculate $\alpha$ by using the MatLab fitnlm (h, Ro, y, $B_{0}$). Here, y is the linear function $y=ax+b$ with $y=\log_{10}Ro$ and $x=\log_{10}\textit{h}$ ,and $B_{0}$ are initial data of the unknown coefficients a and b.

Figure 1 shows our result for Ro using average wind velocities and our estimate $\alpha=0.56$ with standard error of 0.15, giving equal weights to all planets in the Solar system. The same figure shows our result for Ro using maximum wind velocities and our estimate $\alpha=0.52$ with standard error of 0.14 obtained similarly.

4 Conclusion and Outlook

Here, we have reviewed wind velocity data of the planets in the solar system by observational methods (Table 1.1) and altitudes (Table 1.2). We used this review to explore a novel correlation of Ro and h.

Ro characterizes the response to $Gr/Re^{2}$ measured by wind velocity. Jupiter, Saturn, Uranus and Neptune have low Ro and indeed show global extreme climate. On the other hand, Mars, Earth and Pluto have high Ro and indeed show global clement climate.

h is our reduced expression for $Gr/Re^{2}$ and, relatively easy to observe for planets of our Solar system, may indeed indicate potentially favorable conditions to advanced life. Jupiter, Saturn, Uranus and Neptune have small h-values because they are fast spinning whereas Mars, Earth and Pluto are slow spinning, indicated by the respective h-values.

Quantitatively, Ro and h satisfy a power law with index of $\alpha=0.56$ and $\alpha=0.52$ based on average (av) and, respectively, maximum (mx) wind velocities in Table 1.1-2, upon attributing equal weights to all planets.

While the observed positive correlation of Ro and h appears to be robust, there is clearly a need for data with a much more uniform uncertainty than which is available today, especially by what would be an anomalous tilt by Jupiter in estimates of $\alpha$ weighted by present uncertainties.

However, h will be difficult to observe for relatively small potentially Earth-like exoplanets. Quite generally, h of order unity corresponds to slow spin. Slow spin, in turn, can be attributed to lunar tidal interactions in relatively distant habitable zone‘s such as ours around the Sun. It appears, therefore, that exoplanet-moon systems are possibly the $“$go-to-places$”$ for advanced life in exosolar systems [30].

The upcoming James Webb Space Telescope (JWST) [6] and the Extremely Large Telescope (ELT) [32] may guide us to such systems in the near future.

Acknowledgment

The authors thank the reviewer for a detailed reading of the manuscript and constructive comments. Support is acknowledged from the National Research Foundation of Korea under grants 2015R1D1A1A01059793, 2016R1A5A1013277 and 2018044640.