Coupling convectively driven atmospheric circulation to surface rotation: Evidence for active methane weather in the observed spin rate drift of Titan

Models of the interior of Titan for plausible ammonia concentrations predict the existence of a water ocean under a thin ice I crust of perhaps 100 km thickness (Sohl et al., 2003). A subsurface ocean would dynamically isolate the surface from the more massive core, effectively lowering the moment of inertia of the surface. For the expected magnitude of seasonal angular momentum change in the atmosphere, a thin ice crust experiences measurable changes in length-of-day, or equivalently spin rate, in a matter of a few years (Tokano & Neubauer, 2005).

Cassini RADAR spin rate measurements (Stiles et al., 2008), seasonal winds from an atmospheric model prediction (Tokano & Neubauer, 2005) and Huygens wind observations (Bird et al., 2005) were used to infer the moment-of-inertia (MOI) of Titan’s surface (Lorenz et al., 2008). The surface MOI is well below that allowed for any solid internal configuration, but is consistent with models of Titan’s interior that with a thin surface shell overlaying a subsurface ocean (Sohl et al., 2003). Seasonally changing zonal winds in the atmosphere torque the freewheeling surface, introducing a forced libration about synchronous rotation. The forced libration of the surface, however, unexpectedly lags seasonal forcing by a few years. Here we suggest this time lag signals an active methane weather cycle in the atmosphere.

Titan’s atmosphere is known to superrotate, with angular momentum exceeding that of solid body rotation. Stratospheric zonal winds exceed 100 m/s in the prograde direction (Hubbard et al., 1993; Achterberg et al., 2008). Between 10 and 60 km, the troposphere weakly superrotates with winds limited to 10-20 m/s (Bird et al., 2005). The lowest $\sim$10 km at low latitudes show slight retrograde motion, which is consistent with a seasonal, cross-equatorial circulation, or Hadley cell, driven by convection (Tokano, 2007). Winds in the lowest kilometer turn prograde again (Karkoschka et al., 2007), which is corroborated by the directionality of equatorial dune features (Lorenz et al., 2006).^***The presence of prograde winds at the equatorial surface is mysterious given frictional coupling with the surface. Explanation of this phenomenon is beyond the scope of this work. Atmospheric models suggest angular momentum is seasonally redistributed in the atmosphere by a cross-equatorial circulation that changes direction with seasons (Hourdin et al., 1995; Tokano et al., 1999; Tokano, 2005; Rannou et al., 2006; Mitchell et al., 2006; Tokano, 2007; Mitchell, 2008).

Though stratospheric zonal winds dwarf those in the troposphere, very little mass is being advected in the stratosphere. Zonal winds in more than 90% of the atmospheric mass are below 20 m/s. The lowest one-third of the atmospheric mass is subject to the largest seasonal change in angular momentum due to frictional torque from the surface acting on a seasonally reversing Hadley cell. In the following, we develop a theory for the magnitude of seasonal change in atmospheric angular momentum (AAM) and its effect on the spin rate. We then develop a full time-dependent model of this process and show by comparison with numerical simulations that a seasonal time lag of a few years is a natural consequence of methane thermodynamics in Titan’s atmosphere. We conclude by offering predictions for current and future Cassini observations.

Here and throughout we assume axisymmetry of the atmosphere; by doing so we preclude the development of superrotation. Asymmetric surface heating by sunlight drives a convective updraft at a certain latitude. Surface friction imparts angular momentum to converging, surface-level air, which is then lofted into the upper troposphere by the convective updraft. At the top of the convective layer, air diverges along meridians while conserving angular momentum. The Hadley cell homogenizes the angular momentum in this upper, divergent branch of the flow. The triple requirements of angular momentum and energy conservation and cyclostrophic balance (Vallis, 2006) allow prediction of the width and strength of the Hadley circulation (Held & Hou, 1980); this model predicts a global Hadley cell on Titan. Air sinks at the poles and returns along the surface where friction resets the angular momentum to the surface value.

The updraft of the Hadley cell is located where convection is favorable, which on Earth is nearly steadily located at warm equatorial oceans. On Titan, the surface warms quickly relative to a season, and one therefore might expect the convective updraft to follow the latitude of maximum solar forcing. However, the atmosphere could not quickly respond to such warming since the radiative cooling time exceeds the length of a Titan year (Flasar et al., 1981). For reasonable estimates of surface heat capacity (Tokano, 2005), the radiative cooling time of the surface is perhaps a month; it is clear that the atmosphere provides the dominant source of heat capacity in the climate system.

Dry (i.e. neglecting methane condensation and evaporation) GCM simulations suggest the large heat capacity of the atmosphere does not significantly attenuate the seasonal cycle of the Hadley cell; the convective updraft of the Hadley cell travels from one pole to the other each Titan year, roughly in phase with the seasonal cycle of solar radiation (Tokano, 2005). However, increasing the amount of methane evaporation at the surface introduces a time lag of the response of the Hadley circulation to seasonal forcing (Mitchell et al., 2006). Moist, precipitating convection penetrates deeper into the cold, dense atmosphere than dry convection; one can think of the increase in convective depth arising from a reduction in the vertical temperature gradient resulting from moist convection versus dry convection (the lapse rate). By doing so, moist convection communicates surface warming into the large heat capacity reservoir in the upper troposphere that is not accessed by dry convection alone. The Hadley cell efficiently communicates convective warming throughout the latitudinal domain, which increases the thermal inertia of the system. The net result is an increase in the mass of the atmosphere through which the warming of the subsolar surface is communicated, which introduces a time lag in the response of the Hadley cell to seasonal forcing.

Figure 3 shows the predicted vertical pressure velocity with the scale truncated to highlight regions of large-scale updraft (colored contours) as a function of latitude and date over the last half Titan year of three model simulations^†††This model is similar to our earlier one (Mitchell et al., 2006) except a more realistic radiation scheme was implemented (Mitchell, 2008).; the insolation pattern is overplotted for reference (black contours). The position of the most intense updraft features in Figure 3 relative to the solar forcing gives a measure of the time lag of the Hadley cell. The “dry” limit resembles other dry simulations (Tokano, 2005), and very little time lag is predicted (Lorenz et al., 2008). In the “moist” case, in which deep, precipitating convection is most favorable, the main convective updraft is in quadrature with the seasonal forcing. The “intermediate” case, in which methane evaporation was restricted to produce $\sim$50% relative humidities near the surface, has sporadic precipitating convection, and as a result has a smaller time lag.

Atmospheric angular momentum (with the minimum subtracted) and the induced spin rate drift (for an assumed 70 km ice I shell with the average drift subtracted) over one Titan year for our three simulated cases are shown in Figures 1 and 2, respectively. The time lag of the Hadley cell updraft evident in Figure 3 translates directly into a time lag in these quantities. For comparison, the measured spin rate drift and the acceleration are shown as a solid, bold line (Lorenz et al., 2008; Stiles et al., 2008). The “intermediate” case fits the data very well, indicating methane thermodynamics plays a role in establishing the observed time lag in the measured spin rate drift. While our dry model reproduces the change in atmospheric angular momentum in the previous model estimate (Tokano & Neubauer, 2005), the change is somewhat smaller in the intermediate and moist cases. Our model predicts the spin rate drift should currently be changing directions; Cassini RADAR observations through 2010 should unambiguously determine whether our prediction holds.

Work must be done to change the spin rate of the ice shell. The torque exerted by Hadley cell oscillations is driven by seasonal changes in insolation. The thermodynamic efficiency of this process can be estimated given the change in rotational kinetic energy and the integrated sunlight over a quarter Titan season.

The kinetic energy of the surface shell is

The change in kinetic energy resulting from a change in surface angular momentum, $\Delta L=I_{s}\Delta\Omega$, is approximately

For the numbers listed previously, $\Delta T_{s}\sim 10^{20}$ Joules. The time to go from maximum to minimum spin is roughly a single season, or $\sim$7 years; in this time, Titan receives 7 $\times 10^{22}$ J of radiant energy from the Sun. The thermodynamic efficiency of seasonal angular momentum exchange is $\epsilon\sim 0.1$%. Sohl et al. (2003) suggest tidal dissipation in Titan could be 10¹⁰ W which over a Titan season amounts to $\sim 10^{18}$ Joules, two orders of magnitude smaller than the surface kinetic energy change.

Earth experiences seasonal changes in length of day of a millisecond due to atmospheric torques. Earth’s moment of inertia about the spin axis is 8 $\times 10^{37}$ kg m², and the kinetic energy associated with the spin rate change is $\sim 5\times 10^{21}$ Joules (see Rosen (1993) for a review of the topic). Earth receives $\sim 10^{24}$ Joules of radiant energy from the Sun in a season, implying an efficiency of $\epsilon\sim 0.5$%, the same magnitude as that of Titan. Titan receives 1/15th of the solar energy the Earth receives over the time of their respective seasons (7 years for Titan, 3 months for Earth). Taking the change in AAM as a measured quantity, the difference in solar energy nearly cancels the factor of 16 due to the difference in rotation rate in Equation 8 when estimating the efficiency. The change in Titan’s relative AAM, $L_{max}$, is $\sim$1/5th of Earth’s, which accounts for Titan’s lower efficiency. However, since Titan’s stratosphere either absorbs or reflects more than half of the incoming solar radiation (McKay et al., 1991), Titan’s tropospheric Hadley cell is driven by a reduced level of solar radiation relative to the top-of-atmosphere. If this is taken into account, the efficiency increases to near the level of Earth.

Rosen (1993) attributes Earth’s seasonal changes in length-of-day to variability in the subtropical jets, which mark the poleward flanks of Earth’s angular-momentum-transporting Hadley cell. Figure 4 compares changes in relative AAM on Earth from NCEP/NCAR reanalysis data for 2007 (Kalnay et al., 1996) to the our “intermediate” Titan simulation; the top two panels display the vertical mass-weighted average of $ua\cos(\varphi)$, which is the specific angular momentum of the atmosphere relative to rotation (measured positive in the direction of rotation), and the bottom two panels show the corresponding relative AAM. Earth’s subtropical jets contribute bands of positive AAM between 30-40 N/S latitude in Figure 4(a). The Northern hemisphere jet has stronger seasonality, with a minimum during Northern hemisphere summer corresponding to a global minimum of relative AAM in Figure 4(c). Titan’s subtropical jets are spread further poleward because of the increased width of the Hadley cell (Figure 4(b)), but the pattern with season shows similarity to the Earth’s. In particular, minima in relative AAM occur when either the Northern or Southern subtropical jet is at minimum.

There are a few physical processes which could additionally contribute a time lag, and we presently estimate their effects. First, a sluggish atmospheric overturning circulation might contribute a time lag; we assumed in our analytic model that the circulation is vigorous and able to redistribute the angular momentum of the convective updraft instantaneously. The overturning timescale varies by an order of magnitude between the “moist” case and the “dry” limit. Since our analytic model assumes the atmosphere instantaneously acquires the angular momentum of the surface at the updraft, the effect of a sluggish circulation is likely positively contributing to the time lag in AAM. Since the atmosphere is most sluggish in the “moist” case, the effect goes in the same direction as the thermal inertia. The weaker Hadley circulations in the “intermediate” and “moist” cases reduce the amplitude of AAM changes relative to the “dry” limit, as seen in Figure 1 in the main text; as a result, we infer a somewhat shallower ice shell depth of $\sim$70 km.

Second, frictional (Ekman) drag at the interfaces of the subsurface ocean could dynamically couple the surface to the deep core. Though the drag is only communicated to a thin layer of depth $\delta\sim(2\nu/f)$, where $\nu$ is the viscosity of water and $f\sim 2\Omega$ is the coriolis parameter, mass transport in this layer converges (diverges) to form downwelling (upwelling) through the depth. The characteristic timescale of this overturning is $\tau_{Ek}\sim H(\frac{1}{2}f\nu)^{-1/2}$, where $H$ is the ocean depth (Vallis, 2006). Sohl et al. (2003) estimate the ocean depth to be 224 km for an assumed 15 wt.% NH₃. If we assume $H\sim$ 200 km and $f\sim 2\Omega$, then $\tau_{Ek}\sim$ 3000 years. Ekman drag in a laminar boundary layer cannot contribute a time lag of a few years. However, the Richardson number, $Ri$, in an assumed laminar boundary layer may be order unity^‡‡‡We estimate the interior heat flux from the model of Sohl et al. (2003) and calculate the temperature difference over the laminar Ekman layer due to thermal conduction. We then use this temperature difference along with an estimate of the velocity in the boundary layer, $u\sim\Delta\Omega a$, to estimate $Ri\sim 0.7$‘. In this case the molecular viscosity, $\nu$, should be replaced with an effective eddy viscosity, $\kappa$. We estimate $\kappa\sim L_{d}f/\sqrt{Ri}$, where $L_{d}\sim u/f$ is the radius of deformation (Vallis, 2006) and $u\sim\Delta\Omega a$ is the velocity difference in the boundary layer induced by libration of the crust. For previously listed values, $\kappa\sim 2-3\times 10^{-2}$ m²/s and the Ekman spin-down due to the turbulent boundary layer is $\tau_{Ek}\sim$ 20 years. This rough estimate suggests that if the boundary layer is turbulent, dynamic coupling of the surface and core by Ekman drag could introduce a time lag in the drift rate.

Third, we have implicitly assumed torques on a permanent quadrupole moment in the crust cannot overcome frictional torques provided by the atmosphere, which may not be the case. The full equation of motion (EOM) of the surface crust including these terms reads

where $C_{s}=I_{s}$ is the polar moment of inertia of the crust and $\ddot{\Theta}=$ 0.1825 deg/yr² is the measured rotational acceleration of the surface. The $T$’s represent torques of various mechanisms averaged over an orbital period. $T_{atm}$ is the seasonal torque of the atmosphere; our analysis has assumed this term dominates the others, $C_{s}\ddot{\Theta}\simeq T_{atm}$. $T_{Sat}$ is the torque of Saturn’s gravitational field acting on permanent deformations contributing a quadurpole moment to the crustal layer. $T_{coup}$ represents gravitational coupling between a permanent surface quadrupole and a permanent quadrupole in the core. Gravity measurements from the Cassini spacecraft can constrain the quadrupole moments, but none have yet been reported in the literature. By assuming they do not contribute significantly to Equation 9, we have implicitly placed limits on the surface and core quadrupoles contributing to $T_{Sat}$ and $T_{coup}$, which we presently quantify. We will express limits in terms of the ratio $(B-A)/C$, where $C$ is the polar moment of inertia and $B$ and $A$ are the moments orthogonal to the spin axis. The subscripts “$s$” and “$c$” will respectively refer to moments of the surface and core.

Saturn’s gravity would apply a torque to a permanent quadrupole in the surface. Neglecting this effect in the EOM implies $\ddot{\Theta}>>T_{Sat}/C_{s}$. The torque is proportional to $(B-A)_{s}$, the quadrupole of the surface, and $n\simeq\Omega$, the orbital mean motion (Murray & Dermott, 1999). Together with the Cassini measurement of rotational acceleration, the constraint implies $((B-A)/C)_{s}<<10^{-7}$. Assuming the surface is a dynamically isolated ice shell of 70 km thickness, $(B-A)_{s}<<2\times 10^{27}$ kg m². The maximum $(B-A)_{s}$ could, for instance, be due to an isostatically uncompensated disk of ice of depth 1 km and diameter smaller than 40 km sitting on the equatorial surface; a 100 m thick disk would need to be smaller than 125 km.

A surface quadrupole could also gravitationally couple to a quadrupole in the core, represented by $(B-A)_{c}$. As expressed in the EOM, we have assumed $\ddot{\Theta}>>T_{coup}/C_{s}$. We derive a constraint on $((B-A)/C)_{c}$ by making the following assumptions: 1) the quadrupole of the surface layer relative to its polar moment takes the maximum value estimated in the previous paragraph, $((B-A)/C)_{s}\sim 10^{-7}$; 2) the core remains fixed in the rotating frame; and 3) the core radius is 2250 km and has the composition shown in Figure 6 of Sohl et al. (2003). Under these assumptions, $((B-A)/C)_{c}<<5\times 10^{-5}$. This minimum value is intermediate between the moon (which is roughly the size of Titan’s rock and iron core), $((B-A)/C)_{m}\sim 2\times 10^{-4}$, and the Earth, $((B-A)/C)_{\earth}\sim 2\times 10^{-5}$.

We have developed an analytical model for the time-dependent atmospheric angular momentum in an atmosphere with an oscillating Hadley cell which agrees with a simulation of Titan’s atmosphere (Tokano & Neubauer, 2005); while both models predict the observed magnitude of spin rate drift, neither can produce the observed time lag of the seasonal drift. Numerical simulations with methane thermodynamics included show an increase in the thermal inertia of the system, thus introducing a time lag with the seasonal forcing. The time lag is sensitive to the supply of methane evaporation at the surface. If methane is allowed to efficiently evaporate from the model surface, deep precipitating convection persists at all seasons and warms the high-heat-capacity atmosphere which increases the thermal inertia of the system. In this “moist” case, the convective updraft of the Hadley circulation is in quadrature with solar forcing, which translates to a $\sim$ 7 year time lag. If instead surface evaporation is uniformly restricted, the numerical model has the appropriate amount of time lag, matching the observed lag in the spin rate drift (see Figure 2). Our model predicts the spin rate drift has changed directions; current Cassini RADAR observations of Titan may show this trend.

In conclusion, the observed time lag of Titan’s drift in rotation rate is consistent with the lag introduced by thermodynamic-dynamic coupling resulting from the methane cycle in the atmosphere of Titan. We have shown the change in total atmospheric angular momentum is primarily controlled by the latitude of the Hadley cell updraft, and this updraft marks the location of the most persistent clouds features. The recent disappearance of persistent south polar clouds (Schaller et al., 2006) suggests the Hadley cell updraft is moving equatorward in response to the changing season. The resulting transfer of angular momentum to the atmosphere from the surface should currently be decelerating the spin rate, as shown in Figure 2. Other sources of torque might measurably contribute to the observed spin rate drift; however better knowledge of Titan’s gravitational moments and interior structure is required to estimate their effects.