An axisymmetric limit for the width of the Hadley cell on planets with large obliquity and long seasonality

TL;DR

This paper investigates the limits of Hadley cell width on planets with extreme obliquity and seasonality, using theoretical and numerical methods to identify the dominant factors influencing circulation patterns.

Contribution

It introduces an axisymmetric theoretical framework and GCM simulations to determine how the thermal Rossby number and rotation rate control Hadley cell characteristics on diverse planets.

Findings

Rotation rate critically influences Hadley cell width and position.

Thermal Rossby number governs the circulation's behavior.

Results explain the observed circulation features on Mars and Titan.

Abstract

Hadley cells dominate the meridional circulation of terrestrial atmospheres. The Solar System terrestrial atmospheres, Venus, Earth, Mars and Titan, exhibit a large variety in the strength, width and seasonality of their Hadley circulation. Despite the Hadley cell being thermally driven, in all planets, the ascending branch does not coincide with the warmest latitude, even in cases with very long seasonality (e.g., Titan) or very small thermal inertia (e.g., Mars). In order to understand the characteristics of the Hadley circulation in case of extreme planetary characteristics, we show both theoretically, using axisymmetric theory, and numerically, using a set of idealized GCM simulations, that the thermal Rossby number dictates the character of the circulation. Given the possible variation of thermal Rossby number parameters, the rotation rate is found to be the most critical factor controlling the circulation characteristics. The results also explain the location of the ascending branch on Mars and Titan.

Equations

$$u_M=\Omega a\frac{{\cos }^2{\varphi }_1-{\cos }^2\varphi }{\cos \varphi },$$

$$\frac{\theta (\varphi )-\theta ({\varphi }_1)}{{\theta }_0}=-\frac{{\Omega }^2{a}^2}{2gH}\frac{({\sin }^2\varphi -{\sin }^2{\varphi }_1{)}^2}{{\cos }^2\varphi },$$

$$\frac{\theta (\varphi )-\theta ({\varphi }_1)}{{\theta }_0}=-\frac{{\Delta }_H}{R_t}\frac{({\sin }^2\varphi -{\sin }^2{\varphi }_1{)}^2}{{\cos }^2\varphi },$$

$$\frac{{\theta }_e}{{\theta }_0}=1+\frac{{\Delta }_H}{3}(1-3(\sin \varphi -\sin {\varphi }_0{)}^2),$$

$${\int }_{{\varphi }_w}^{{\varphi }_1}(\theta -{\theta }_e)\cos \varphi d\varphi$$

$${\int }_{{\varphi }_1}^{{\varphi }_s}(\theta -{\theta }_e)\cos \varphi d\varphi$$

$$\theta ({\varphi }_1)$$

$$\theta ({\varphi }_w)$$

$$\theta ({\varphi }_s)$$

$$R_t=\frac{8\pi \rho GH{\Delta }_H}{3{\Omega }^{2}a}.$$

Full text

\journalname

Geophysical Research Letters

Weizmann Institute of Science, Department of Earth and Planetary Sciences, Rehovot, Israel

\correspondingauthor

Ilai Guendelmanilai.guendelman@weizmann.ac.il

{keypoints}

= The Hadley cell ascending branch position in planets with strong seasonal variation of temperature is mainly bounded by the rotation rate.

= A similar rotation rate dependence arises in the axisymmetric theory.

= This theory can explain the ascending branch position on Mars and Titan.

1 Introduction

Observations and models show that the Hadley circulation varies considerably between the Solar System terrestrial atmospheres of Venus, Earth, Mars and Titan. Venus’ lower atmosphere is composed of two hemispherically symmetric equator-to-pole Hadley cells (Read, 2013; Sánchez-Lavega et al., 2017). On Earth, similar to Venus, Hadley cells exist in both hemispheres, but due to Earth’s obliquity, the ascending branch latitude and the strength of the two cells vary seasonally, where during the solstice there is a strong and wide winter cell and a narrow and weak summer cell (e.g., Dima and Wallace, 2003).

Both Mars and Titan, exhibit stronger seasonality in the Hadley circulation compared to Earth, despite the fact that the obliquity of Mars and Titan is similar to Earth’s obliquity. The strong seasonality on Mars is due to its thin atmosphere and rocky surface resulting in a low thermal inertia and a short radiative timescale. Mars’ Hadley circulation transits from two hemispherically symmetric cells at equinox, to one solstice cell, with air rising at midlatitudes (Read et al., 2015). Thus, although at solstice, Mars’ maximum surface temperature is at the pole, the Hadley cell ascending branch does not reach the pole.

Titan’s tropospheric radiative timescale is considerably longer than its orbital period (Mitchell and Lora, 2016), which explains why Titan’s maximum surface temperature seem to stay near the equator during the seasonal cycle (Jennings et al., 2016; Lora et al., 2015). However, observations of Titan’s methane clouds, show a significant seasonal cycle as they shift from one pole to the other during Titan’s year (Brown et al., 2002; Turtle et al., 2011; Roe, 2012; Turtle et al., 2018). Different models associate the polar clouds to different phenomena. Schneider et al. (2012) associate the polar clouds to the meridional convergence (analogous to the inter tropical convergence zone, ITCZ, on Earth) indicating a pole-to-pole Hadley circulation (e.g, Roe, 2012). In contrast, Lora et al. (2015) and other studies (e.g., Mitchell et al., 2006, 2009; Mitchell, 2008; Newman et al., 2016) relate the polar clouds to intensive polar warming during solstice, while the meridional convergence occurs at midlatitudes.

The variability of the terrestrial atmospheric circulation within the Solar System, is a result of the variability in the planets’ orbit, rotation rate, atmospheric mass, radius etc. Different studies explored the effect of different planetary parameters on the atmospheric circulation, showing that the large scale circulation depends greatly on the planetary parameters and atmospheric characteristics (e.g., Ferreira et al., 2014; Kaspi and Showman, 2015; Linsenmeier et al., 2015; Faulk et al., 2017; Chemke and Kaspi, 2017).

More specifically, Faulk et al. (2017) studied the dependence of the meridional circulation seasonal cycle on the rotation rate using an idealized aquaplanet GCM, showing that in the case of an Earth-like rotation rate, the ITCZ and the ascending branch of the Hadley circulation do not reach the pole, even in an eternal solstice case, where the maximum temperature is at the pole. This result, together with Mars’ ascending branch not reaching the pole, even though its maximal surface temperature is at the pole (Forget et al., 1999; Read et al., 2015) and some Titan models predicting the ascending branch being poleward from the warmest latitude (e.g., Lora et al., 2015), is puzzling. Theoretical expectations are that the Hadley cell ascending branch, being a thermally driven circulation, will follow the warmest latitude (Neelin and Held, 1987), or the latitude of maximum low level moist static energy (Emanuel et al., 1994; Privé and Plumb, 2007), which is not the case for Mars and neither for the Faulk et al. (2017) simulations. This study shows that axisymmetric theory (Held and Hou, 1980; Lindzen and Hou, 1988; Caballero et al., 2008) has a similar rotation rate dependence as the modeling results and the observations on Mars and Titan show.

For Earth, there are several other theories regarding the Hadley circulation and the ITCZ position, that unlike the axisymmetric theory take into account the eddy contribution, and involve processes such as the flux of eddies across the equator (e.g., Kang et al., 2008; Bischoff and Schneider, 2014; Adam et al., 2016a, b; Wei and Bordoni, 2018), moist processes (e.g., Neelin and Held, 1987), baroclinicity (Held, 2000) and supercriticality (Korty and Schneider, 2008; Levine and Schneider, 2015). However, in this study which aims to understand the leading order effects over a wide range of conditions, we focus on the simpler, axisymmetric theory, as a leading order theory for the zonally symmetric climate balance. In section 2 we derive the axisymmetric theory for the solstice case following Lindzen and Hou (1988) (hereafter LH88), and solve it numerically to include a wide range of planetary parameters. In section 3 we briefly describe the numerical model and present the simulation results, relating them to the axisymmetric theory. In section 4 we discuss the results and their implication for the Solar System atmospheres.

2 Axisymmetric theory

The axisymmetric theory introduced by Held and Hou (1980) and further developed by LH88 to include the solstice case, is a theory for the Hadley circulation that neglects eddy contribution and diffusive processes. Despite the importance of eddies (e.g., Walker and Schneider, 2006), the axisymmetric theory has been found to overall give a good leading order estimate to the cell extent. Following LH88, angular momentum conservation at the top of the cell is assumed, and the angular momentum conserving wind, at latitude $\phi$, of an air parcel starting at rest from latitude $\phi_{1}$ (the ascending branch of the Hadley cell) is

[TABLE]

where $\Omega$ is the planetary rotation rate and $a$ is the planetary radius. Assuming that the flow is in cyclostrophic balance and that thermal wind balance holds to leading order, Eq. 1 together with hydrostatic balance, results in an expression for the angular momentum conserving potential temperature ($\theta$)

[TABLE]

where $\theta_{0}$ is some reference potential temperature and $H$ is the troposphere height. Equation 2 can be expressed using the thermal Rossby number $R_{t}=\frac{2gH\Delta_{H}}{\Omega^{2}a^{2}}$ (Held and Hou, 1980) to give

[TABLE]

where $\Delta_{H}$ is the meridional fractional change of the radiative equilibrium temperature (LH88 and Eq. 4). Taking a small angle approximation, the width of the circulation in the equinox case is $\propto R_{t}^{1/2}$ (Held and Hou, 1980) and for the solstice case $\propto R_{t}^{1/3}$ (Caballero et al., 2008).

In order to find the Hadley circulation edges, namely, the latitudes of the ascending and descending branches, we assume that the cells are energetically closed, that the temperature at the edge of the cells is continuous and that outside of the Hadley circulation the temperature is a radiative equilibrium temperature $\theta_{e}$

[TABLE]

where $\phi_{0}$ is the latitude of maximum $\theta_{e}$. The energetically closed cell and temperature continuity assumptions translate to the following set of equations

[TABLE]

where $\phi_{s}$ and $\phi_{w}$ are the latitudes of the Hadley cell descending branch in the summer and winter hemispheres (edges of the circulation), respectively.

The unknowns that equations (5)-(9) solve for are the ascending and descending branches latitudes $\phi_{1}$, $\phi_{s}$, $\phi_{w}$ and the temperature at the ascending branch $\theta(\phi_{1})$. Graphically, the energetically closed cell assumption translates to an equal area between the angular momentum conserving and the radiative equilibrium temperature curves inside each cell. Figure 1 depicts the angular momentum conserving (dashed) and the radiative equilibrium (solid) temperature curves, multiplied by $\cos\phi$ for different cases, depicting the closed cell argument. Figures 1a and 1c are similar to Figure 5 in LH88 with the difference that here $\theta$ and $\theta_{e}$ are multiplied by $\cos\phi$, as the small angle approximation is not appropriate in this case. Fig. 1a shows the hemispherically symmetric cell and Fig. 1c shows the $\phi_{0}=6^{\circ}$ case, representing an Earth-like scenario. Figures 1b and 1d are for different temperature gradients and different rotation rates, respectively, where the latitude of maximum radiative temperature is at latitude $45^{\circ}$. All plots in Fig. 1 show the position of the winter cell ascending (dotted line) and descending (dashed-dotted line) branches. Only the winter cell is shown, as for strong seasonal cases, which are the focus of this study, a summer cell barely exists. Comparing between Figures 1b and 1d, shows that slowing down the rotation rate is more efficient in widening the circulation than increasing the temperature gradient.

The axisymmetric theory solutions shown in Figure 1 together with equations (2) and (4), show that the latitudes of the ascending and descending branches depend on different parameters. Solving numerically equations (5)-(9) for a wide range of $\Omega,\Delta_{H},\phi_{0}$ and $H$ values (Fig. 2) shows a clear difference between cases where the rotation rate is slowed down (Fig. 2d-f), where the ascending branch easily reaches the pole, and cases where the rotation rate is kept with an Earth-like value (Fig. 2a-c). This demonstrates that an Earth-like rotation rate or faster, limits the expansion of the circulation, such that even if $\phi_{0}$ is at the pole, and, for example, $\Delta_{H}$ is increased (over a realistic range), it is unlikely for $\phi_{1}$ to reach the pole unless the rotation rate is slowed down. This result is consistent with the simulations of Faulk et al. (2017). The choice of parameter values is guided by the observed values in the Solar System. $\Delta_{H}$, the normalized horizontal temperature difference, gets its largest value for Mars ($\sim 0.4$, e.g., Read et al., 2015), and lowest value for Venus (with nearly zero temperature gradient, e.g., Read, 2013). The tropopause height, $H$, taken to be be the circulation height scale (Walker and Schneider, 2006), is highest on Titan and Mars reaching to $\sim 20$ km (e.g. Read et al., 2015; Lora et al., 2015).

In order to understand this rotation rate dependence, we plot the ascending branch latitude as a function of $R_{t}$ and $\phi_{0}$ (Fig. 3a), showing that for each value of $\phi_{0}$ the position of the ascending branch is strongly dependent on $R_{t}$ (Fig. 3a,b). For small values of $R_{t}$ there is a good agreement with the Caballero et al. (2008) scaling (Fig 3b). The values of $R_{t}$ for the Solar System terrestrial atmospheres vary from $0.06$ on Earth to $370$ on Venus, with the value for Mars being $0.2$ and for Titan $18$ (Read, 2011). Decomposing $R_{t}$ into $\frac{\Omega^{2}a^{2}}{2gH}$ and $\Delta_{H}$, which is a natural decomposition to a dynamical component ($\frac{\Omega^{2}a^{2}}{2gH}$) and a radiative one ($\Delta_{H}$), shows the range of possible values of $\frac{\Omega^{2}a^{2}}{2gH}$ is larger compared to that of $\Delta_{H}$ (Fig 3c). As a result, this factor will have a larger role in limiting the width of the circulation. Examining the elements in $\frac{\Omega^{2}a^{2}}{2gH}$ shows a strong dependence on the rotation rate and radius (Fig. 3d). Taking the Solar System terrestrial atmospheres as a proxy, and comparing between the range of the different parameters in $R_{t}$, shows that the rotation rate is the only parameter known to vary by two orders of magnitude (Earth and Venus), while all other parameters vary by one order of magnitude or less. This together with the strong dependence of $R_{t}$ on the rotation rate is what makes the rotation rate the limiting factor on the circulation extent. Also, taking a closer look at the radius dependence, shows that it is not as strong as the rotation rate dependence, considering the surface gravity dependence on the planetary radius $g=4\pi G\rho a/3$. Here $\rho$ is the planet’s mean density, which is a more fundamental characteristic of the planet than its surface gravity, and $G$ is the universal gravitational constant. Therefore, a more useful form to write the thermal Rossby number in this context, is

[TABLE]

Expressing $R_{t}$ in this form emphasizes the circulation dependence on the rotation rate.

3 Idealized GCM simulations

3.1 Model description

In order to test the theoretical framework presented in section 2 in a more complete model, we use an idealized moist aquaplanet GCM (Frierson et al., 2006), based on the GFDL dynamical core (Anderson et al., 2004), used in this context by Faulk et al. (2017). The model radiation scheme is augmented to include a diurnal mean seasonal insolation dependence on obliquity (Pierrehumbert, 2010). Similar to Faulk et al. (2017) the atmospheric optical depth is constant with latitude. The model solves the primitive equations with a horizontal spectral grid of $2.8^{\circ}\times 2.8^{\circ}$ (T42) and $25$ uneven vertical levels. To analyze the climate we use a $50$ year climatology after reaching a statistical steady state. Figure 4a shows the model results for Earth-like parameters. The model shows a generally similar climate as Earth’s (Kaspi and Showman, 2015).

3.2 Simulation results

Three simulations with different degree of seasonality: Earth-like, obliquity $90^{\circ}$ with an Earth-like orbital period and obliquity $90^{\circ}$ with four times Earth’s orbital period, are repeated with an Earth-like rotation rate and 1/4 of Earth’s rotation rate. Figures 4 and 5 show that shifting the maximal temperature poleward from a reference Earth-like state (Fig.  4a), does not result in a global pole-to-pole Hadley circulation for simulations with an Earth-like rotation rate (Fig. 4b), even when the temperature gradient is increased (Fig. 4c). However, slowing down the rotation rate, allows the ascending branch to reach the pole, similar to Faulk et al. (2017). These results coincide with the theoretical solution of equations (5)-(9) (Fig. 2), where the ascending branch, for a realistic range of $\Delta_{H}$, does not reach the pole for an Earth-like rotation rate.

Figure 5 shows that during the solstice of the strong seasonal cases, the meridional streamfunction follows the angular momentum contours (Fig. 5b-f), implying that the eddy contribution is small. This means that despite the importance of eddies in the more Earth-like cases for the extent of the Hadley circulation (e.g., Walker and Schneider, 2005, 2006; Korty and Schneider, 2008) eddies seem to play less of a role in these cases. This alignment, relates to a previously suggested regime transition between an eddy mediated circulation at equinox, to a thermally driven one at solstice, where eddies do not contribute, suggesting that the use of axisymmetric theory is appropriate (Bordoni and Schneider, 2008, 2010; Merlis et al., 2013; Geen et al., 2018). Aside from the seasonal regime transition, there is a rotation rate related transition, where by slowing down the rotation rate, the streamfunction follows angular momentum contours. This regime transition in both rotation rate and seasonality is a result of weaker eddy momentum flux convergence that in turn allow the streamfunction to follow the angular momentum contours (Faulk et al., 2017). Consistent with the angular momentum conserving cell these simulations do not exhibit superrotation. However, simulations with slower rotation rates (more similar to Titan and Venus) may exhibit superrotation (e.g., Kaspi and Showman, 2015), though the existence of superrotation together with a strong seasonal cycle is complex (Mitchell et al., 2014).

4 Discussion and conclusion

Previous studies showed that as a planet rotates faster, the Hadley circulation contracts, the streamfunction becomes multicellular and the number of jets increases (Navarra and Boccaletti, 2002; Walker and Schneider, 2006; Kaspi and Showman, 2015; Chemke and Kaspi, 2015a; Chemke and Kaspi, 2015b). Faulk et al. (2017), studying the effect of the rotation rate in a seasonal cycle, showed that for a planet with an Earth-like rotation rate the Hadley cell ascending branch and the latitude of the ITCZ do not reach the pole, even when the maximum surface temperature is at the pole and the seasonal cycle is very long.

Similar to Faulk et al. (2017), using an idealized GCM with different degrees of seasonality we show that for Earth-like rotation rate cases the Hadley cell ascending branch does not reach the pole (Figures 4 and 5). A similar rotation rate limitation arises from the axisymmetric theory, predicting that the Hadley cell ascending branch latitude is limited for Earth-like rotation rate cases (Figure 2). This rotation rate dependence is a result of the angular momentum conservation and thermal wind assumptions that makes the width of the circulation to be a function of the thermal Rossby number (Figure 3). The quadratic dependence of the thermal Rossby number on the rotation rate (Eq. 10), and the limited range the other thermal Rossby number parameters exhibit in the Solar System planetary atmospheres, imply that the strongest limiting factor in controlling the ascending branch of the Hadley circulation is the rotation rate.

Studying these extreme cases, and the climate dependence on different planetary parameters, gives insight to the expected climate on other planetary atmospheres. Our Solar System terrestrial atmospheres are a good example for a variety of circulations, due to their large variability in planetary characteristics. Of particular interest is the seasonality on Mars and Titan, both exhibiting a different circulation response to the seasonally varying surface temperature (Read et al., 2015; Mitchell and Lora, 2016). During the Martian solstice, maximum surface temperature is at the pole, however, the Hadley cell ascending branch is located at midlatitudes (Read et al., 2015), consistent with the axisymmetric theory using Mars’ $R_{t}$ (red dotted line in Fig. 3a).

On Titan, observational studies show that the maximum surface temperature stays around the equator during Titan’s year (Jennings et al., 2016); yet, cloud observations show a significant seasonal variation (e.g., Turtle et al., 2018). Models of Titan’s climate vary depending on their physical aspects (Hörst, 2017), with some models associating polar clouds with the Hadley cell ascending branch (e.g., Schneider et al., 2012) while others locate the ascending branch at midlatitudes (e.g., Lora et al., 2015). The warmest latitude also varies between models (e.g., the difference between dry and moist cases in Newman et al., 2016). Particularly, Lora et al. (2015) is an interesting case, where the peak surface temperature stays close to the equator while the ascending branch is located poleward, at midlatitudes. This variety of models can be explained using the axsymmetric theory. Following the blue dotted line in Fig. 3a, which represents the $R_{t}$ value of Titan, we indeed find that if $\phi_{0}$ is taken to be $\sim 10^{\circ}$ the position of the ascending branch is at $\sim 45^{\circ}$, in a general agreement with Lora et al. (2015). Also if $\phi_{0}\geq 30$ the ascending branch is predicted to be at the pole, similar to the dry case in Newman et al. (2016).

Acknowledgements.

We thank Rei Chemke for fruitful conversations and help with the model configuration. We also thank the reviewers that helped improve this manuscript. Datasets used in this manuscript is available on https://doi.org/10.5281/zenodo.1442928. The authors acknowledge support from the Minerva Foundation with funding from the Federal German Ministry of Education and Research, and from the Weizmann institute Helen Kimmel Center for Planetary Science.