Seasonal Evolution of Titan's Stratospheric Tilt and Temperature Field at High Resolution from Cassini/CIRS

Figure 10 shows zonal cross sections of retrieved temperature, averaged over each Earth year during the Cassini mission. This is the highest meridional resolution temperature field observed in this pressure region to date. The effects of the stratospheric tilt have been removed by mapping the temperature field into the tilted frame, such that given latitudes are in the atmospheric frame rather than Titan's solid-body frame. We use the mission-average optimal tilt axis determined in Section 5.1 in the inertial-fixed frame (tilt β = 4$\mathop{.}\limits^{\unicode{x000b0}}$0, directed ψ = 120° west of the subsolar point at northern spring equinox). The uncertainty on the retrieved temperature is typically 5 K at pressures greater than 0.1 mbar level, around 5–10 K in the 0.1–0.01 mbar region, and around 10–20 K at pressures less than 0.01 mbar, which approaches the pressure limits probed by the CIRS nadir observations used in this study (Figure 2).

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Our retrieved temperature structures are in broad agreement with those found by limb (R. K. Achterberg et al. 2008a, 2011; S. Vinatier et al. 2015, 2020; N. A. Teanby et al. 2017, 2019) and nadir observations (J. Sharkey et al. 2021; R. K. Achterberg 2023). Early in the mission, in mid- to late northern winter, we observe a high and hot stratopause at the north pole. An elevated winter polar stratopause is observed in previous studies (R. K. Achterberg et al. 2008a, 2011; S. Vinatier et al. 2015; R. K. Achterberg 2023) and in a Titan GCM (N. A. Lombardo & J. M. Lora 2023b). Pre-equinox (2004–2009, L_s < 360°), we observe the north polar stratopause to be approximately 205 K and located at around 0.01 mbar. The stratopause at lower latitudes, south of ∼60° north, is around 15 K cooler and about a decade deeper in pressure, in agreement with R. K. Achterberg et al. (2008a).

The stratopause is observed to be hot at the winter pole, most likely due to adiabatic heating of the descending branch of the meridional circulation cell (N. A. Teanby et al. 2017). Lower in the stratosphere, north polar temperatures are much colder than the stratopause, with temperatures reaching almost as low as 115 K at the 5 mbar level. These temperatures are lower than the TAM temperatures (∼140 K) but similar to the ∼120 K temperatures observed by P. Rannou et al. (2004) and S. Lebonnois et al. (2012), who include microphysics and photochemistry in their models. These extremely cold temperatures can be explained by the presence of trace molecules, acting as long-wave radiative coolers, which have been observed to be enriched at the winter pole (N. A. Teanby et al. 2017, 2019). Trace gases produced in the upper atmosphere descend with the circulation at the winter pole into the polar vortex, which acts as a mixing barrier, effectively trapping the trace gases at the winter pole (N. A. Teanby et al. 2008).

In the 2 yr leading up to the equinox (Figure 10, panels 2008 and 2009), the north polar stratopause cools slightly by ∼15 K, consistent with values found by R. K. Achterberg et al. (2011), S. Vinatier et al. (2015), N. A. Teanby et al. (2017), and R. K. Achterberg (2023), but remains warm and elevated compared to the stratopause at lower latitudes. This is most likely due to weakening of the meridional circulation cell (R. K. Achterberg et al. 2011), where the velocity of the descending branch decreases, leading to reduced adiabatic heating at the north pole. We observe a global decrease in stratospheric temperatures from 2006 to 2013, consistent with the reduced solar flux received by Titan during this period, due to Saturn's orbital eccentricity (S. Vinatier et al. 2015).

Post-equinox (after 2009 August, L_s > 0°), the north polar stratopause is observed to gradually cool by ∼20 K and descends to the same level as the lower-latitude stratopause (∼0.01 mbar). However, the north polar stratopause remains around 10 K warmer than lower latitudes throughout the remainder of the mission, consistent with CIRS limb observations (S. Vinatier et al. 2015, 2020; R. K. Achterberg 2023). At the south pole, we observe an elevated stratopause from at least 2012 (L_s ≈ 30°), which cools during 2012–2014, before warming again in early 2016 (L_s ≈ 75°). This behavior is consistent with that found by R. K. Achterberg (2023). From at least 2012, the lower south polar stratosphere (pressures greater than ∼0.1 mbar) rapidly cools and continues to do so until the end of the mission, with temperatures dropping to approximately 120 K at pressures greater than 1 mbar. Rapid cooling of the lower stratosphere is consistent with other nadir observations (N. A. Teanby et al. 2017; J. Sharkey et al. 2021) and with limb observations (S. Vinatier et al. 2020; R. K. Achterberg 2023) and is similar to what we observe in the lower north polar stratosphere during late northern winter. This unexpected cooling is, again, likely caused by long-wave-emitting trace species (N. A. Teanby et al. 2017), which become enriched at the south pole after the global circulation reversal.

By around mid-2016 (L_s ≈ 80°), a high and hot stratopause has formed over the south pole, at around 0.002 mbar, consistent with N. A. Teanby et al. (2017), J. Sharkey et al. (2021), and S. Vinatier et al. (2020). This is higher than the hot stratopause observed in the north during northern winter, but Cassini observations are not seasonally symmetric, as the mission did not capture the beginning of northern winter. The hot south polar stratopause then warms and descends to ∼0.01 mbar, extending away from the pole (Figure 10, panel 2017)—still a factor of 10 deeper in the atmosphere than the stratopause at ∼0.1 mbar, consistent with limb observations (S. Vinatier et al. 2020). This is most likely due to strengthening of the descending branch of the meridional circulation, leading to increased adiabatic heating at high southern latitudes (N. A. Teanby et al. 2017).

Figure 10 shows our retrieved zonal mean temperature and also the zonal mean temperature field produced by TAM (N. A. Lombardo & J. M. Lora 2023b, 2023a). TAM is a three-dimensional GCM that has been used extensively to study Titan's lower atmosphere and climate. Recently, TAM has been updated to improve the vertical extent and resolution of the model, as well as to include the radiative impacts of seasonally varying radiative species. The three-dimensional nature of TAM allows for the direct simulation of three-dimensional eddies, which have been shown to be crucial in the transportation of momentum through the middle atmosphere. The simulations presented here are based on the configuration of the model presented in N. A. Lombardo & J. M. Lora (2023b) and include an approximate 5° meridional resolution and 50 atmospheric layers distributed from the surface to an altitude of approximately 500 km (10⁻² Pa; N. T. Lewis et al. 2023; J. M. Lora et al. 2015). TAM temperatures are shown for the period of Titan's seasonal cycle covered by our observations and have been averaged over the same Earth-year bins to allow for a direct comparison.

The stratopause in the TAM temperature field is roughly a factor of 5 deeper in pressure compared to our observed temperature in all time bins (Figure 10). A ubiquitous challenge in comparing observations of Titan's middle atmosphere with GCM simulations is accounting for the decreasing strength of the gravitational acceleration with altitude. The dynamical cores of many Titan GCMs assume a uniform gravity, which, while accurate for Earth's very thin atmosphere, introduces discrepancies at the high altitudes of Titan's middle atmosphere. This leads to the effective altitude of pressure levels in Titan GCMs being lower than in reality. Along the same lines, the high strength of Titan's zonal winds leads to an atmospheric equatorial bulge, which is not included in most Titan GCMs (N. A. Lombardo & J. M. Lora 2023a). For the most robust insight into Titan's dynamics, it is best to compare observations and simulations not by direct altitude or pressure comparisons but by the structure of the atmospheric layers themselves, e.g., compare the temperature and winds of the observed stratosphere with those of the simulated stratosphere, regardless of exact altitude. We plot the TAM temperature field in Figure 10 over a reduced pressure range to facilitate a more realistic comparison to our observations. The evolution of our observed mean zonal temperature is broadly consistent with the average zonal temperatures produced by TAM.

In northern winter, we observe a hot, elevated stratopause at the north pole. A similarly hot stratopause (∼205 K) is obtained in TAM (N. A. Lombardo & J. M. Lora 2023b, 2023a) temperature cross sections, but the hot spot is much more confined to polar latitudes compared to the TAM temperature (Figure 10). The TAM temperature also exhibits a hot, although not elevated, stratopause (∼195 K) at the south summer pole that is much less pronounced. During mid- to late northern winter, the TAM temperature field is almost symmetric about the equator, whereas our observations show a clear asymmetry, with an elevated hot spot over the north pole that is constrained to high latitudes, but we do not observe the same feature at the summer south pole.

In the TAM temperature field, the south polar hot spot persists, becoming hotter and more elevated post-equinox (after 2009, L_s > 0°), suggesting that the south polar vortex develops more quickly post-equinox in the TAM simulation compared to observations. We only observe a south polar hot spot in 2016 (L_s ≈ 75°–86°). During 2012–2015, and most noticeably in 2013 (L_s ≈  40°–52°), we observe a cold spot at the south pole, around 0.1–0.01 mbar, which is not reproduced with the TAM. Instead, TAM produces a hot spot at the south pole immediately post-equinox (Figure 10, right 2009 panel). This may be due to the absence of highly enriched trace gas coolers in the model, as suggested by N. A. Teanby et al. (2017). Broadly, the observed and simulated temperature fields are in good agreement, especially during times when the atmosphere is more stable and the polar vortex is established, i.e., 2004–2009 and 2016–2018, although the observed south polar stratosphere is much colder than the simulation (Figure 10). The observed and TAM temperatures exhibit some common features, including a hot, elevated stratopause and a cold, lower stratosphere at the winter pole. At the northern winter pole (2005–2010, L_s < 0°), we observe temperatures around 110–150 K at pressures greater than ∼1 mbar, whereas the TAM produces temperatures around 160–180 K. Similarly, post-equinox, the cold lower-stratospheric temperatures at the south pole are around 110–120 K in our observations, compared to around 140–150 K in TAM. During the transitional period post-equinox, the observed and simulated temperatures are somewhat similar, but perhaps with an offset in the transition timing.

Titan's atmosphere is generally in cyclostrophic balance, such that the meridional pressure gradient is balanced with the total horizontal component of the centrifugal force. For such an atmosphere, the vertical gradient of the zonal wind is related to the meridional temperature gradient through the TWE. This is expected to hold for Titan (F. M. Flasar et al. 2005). The TWE can be written as (F. M. Flasar et al. 2005)

$$\begin{eqnarray}&&{\left.\displaystyle \frac{\partial }{\partial {z}_{| | }}\left(2{\rm{\Omega }}u+\displaystyle \frac{{u}^{2}}{r\cos \phi }\right)=-\displaystyle \frac{g}{T}\displaystyle \frac{1}{r}\displaystyle \frac{\partial T}{\partial \phi }\right|}_{p},\end{eqnarray} \tag{ 2 }$$

where Ω is the solid-body rotation rate of Titan, u is the (eastward) zonal wind velocity, ϕ is the latitude, g is the gravitational acceleration, T is the temperature, p is the pressure, r is the radial polar coordinate (radius of Titan + altitude: r = R_T + h), and z_∣∣ is the axis parallel to Titan's atmospheric rotation axis. To calculate zonal wind speed, we integrate the TWE over cylinders concentric to Titan's rotation axis (along the z_∣∣ axis; see, e.g., F. M. Flasar et al. 2005, supplementary material for integration). Consequently, the zonal wind speed is unconstrained by the TWE at low latitudes, although P. S. Marcus et al. (2019) show that an equatorial TWE can be formulated for some planetary atmospheres.

We use a finite-difference method to approximate the differential in Equation (2):

$$\begin{eqnarray}&&{\left({\left.\frac{\partial T}{\partial \phi }\right|}_{p}\right)}_{ij}=\frac{1}{{\phi }_{i+1}-{\phi }_{i-1}}\left[{T}_{i+1,j}-{T}_{i-1,j}\right],\end{eqnarray} \tag{ 3 }$$

for the ith latitude and jth pressure level in Titan's stratosphere, and T_ij = T(p_j, ϕ_i). Calculating the zonal wind speed field requires a boundary condition. We apply the boundary condition setting the zonal wind speed at 10 mbar as 4 times Titan's solid-body rotation rate (u(p = 10 mbar) = 4Ω), as used in R. K. Achterberg et al. (2008a), which is consistent with the Doppler wind experiment on board the Huygens descent probe (M. K. Bird et al. 2005; W. M. Folkner et al. 2006).

Since the zonal wind is dependent on the temperature meridional gradient, the temperature field must be smoothed before the TWE is applied, to limit the amplification of noise. We smooth the Earth-year average temperature field by fitting cubic B-splines (N. A. Teanby 2007) with a knot spacing of 10° and then calculate the mean zonal wind. This local fitting procedure preserves local temperature gradients better than the smoothing applied in previous studies (R. K. Achterberg et al. 2011; J. Sharkey et al. 2021). This, in addition to the higher meridional resolution of the underlying temperature field, preserves finer features in the zonal wind field. The resulting resolution of our zonal wind field is approximately 15°. The low spectral resolution nadir CIRS observations used in this study probe a narrower vertical range, and hence achieve a smoother vertical temperature gradient, than the higher spectral resolution nadir observations used by J. Sharkey et al. (2021). Additionally, our observations have a more consistent coverage of latitudes: while the S/N in J. Sharkey et al. (2021) becomes poor near the pole, due to their binning procedure, our S/N is more consistent over all latitudes. Thus, at high latitudes, we achieve a better S/N, and hence a more robust zonal wind field, than J. Sharkey et al. (2021). Thus, despite the effects of smoothing producing a wind field with a comparable meridional resolution, our observations can better preserve meridional structure in the zonal wind field.

Figure 11 shows the zonal mean wind field velocities derived from the mean zonal temperature. Determining zonal wind speed, u, from the TWE requires solving a quadratic in u, for which the discriminant can sometimes be negative, most typically due to noisy data. Consequently, the zonal wind speed cannot be calculated in some regions, typically where the meridional temperature gradient is small, in addition to being unconstrained near the equator (Figure 11). The unconstrained equatorial region has been determined previously through stellar occultations in northern midwinter (B. Sicardy et al. 2006) and with ground-based millimeter-wave observations (E. Lellouch et al. 2019; M. A. Cordiner et al. 2020; S. Light et al. 2024). We estimate the uncertainty in the zonal wind using the retrieved temperature error (outlined in the Appendix). The uncertainty in the zonal wind field is typically 2–15 m s⁻¹. In Figure 11, the regions where the wind speed error exceeds the contour interval (>10 m s⁻¹) are not plotted.

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Figure 11 also shows the zonal mean wind speeds produced by TAM (N. A. Lombardo & J. M. Lora 2023b, 2023a). Note that the TAM wind field is plotted over a reduced pressure range to compensate for Titan GCMs having a slightly depressed atmosphere compared to observations. The zonal winds have been averaged over the same Earth-year bins as our observations to allow for a direct comparison. The structure and evolution of our mean zonal wind field are overall in good agreement with previous studies using CIRS limb (S. Vinatier et al. 2020) and CIRS nadir (J. Sharkey et al. 2021) observations. However, our improved spatial resolution allows finer structure to be resolved. The observed wind field is also in good agreement with the TAM wind (N. A. Lombardo & J. M. Lora 2023b, 2023a), although TAM produces overall lower wind speeds.

The TAM winds also exhibit a second, weaker zonal wind maximum at ∼0.002 mbar (outside the range plotted in Figure 11), which we do not observe, perhaps due to the vertical shift that occurs in the GCM results and the limited sensitivity at these pressures. We observe the southern hemisphere circumpolar jet to reach higher wind speeds than the northern jet, in agreement with S. Vinatier et al. (2020) and J. Sharkey et al. (2021). Peak wind speeds are observed to be around 210 m s⁻¹ in the north and 230 m s⁻¹ in the south, with an uncertainty within 10 m s⁻¹. These values are comparable to the peak wind speeds observed by J. Sharkey et al. (2021; close to 200 m s⁻¹ in the north and 220 m s⁻¹ in the south) and R. K. Achterberg (2023; close to 200 m s⁻¹ in the north and 250 m s⁻¹ in the south). S. Vinatier et al. (2020) observe wind speeds of 240–280 m s⁻¹ in the south. Our observed winds are slightly slower than those of J. Sharkey et al. (2021), likely due to the higher meridional resolution of our observations, which better preserve the horizontal temperature gradient. On the other hand, our wind speeds are slightly slower than those observed by S. Vinatier et al. (2020), perhaps due to the viewing geometry of our observations. Nadir observations give a smoother temperature profile, and hence less extreme winds, whereas limb observations, used by S. Vinatier et al. (2020), better resolve vertical temperature structure and hence better preserve high wind speeds. Our observed wind speeds are significantly faster than those simulated by TAM (Figure 11), which produces wind speeds around 130 m s⁻¹ in the north and up to 160 m s⁻¹ in the south. This discrepancy is a known issue, likely related at least in part to the model's resolution and lack of trace gas and haze coupling with the circulation.

The structure of our mean zonal wind field is consistent with TAM, but with additional structure due to the higher meridional resolution we achieve. TAM predicts a single zonal jet core in the winter hemisphere. We now describe the observed evolution of the mean zonal wind structure in detail, and our interpretation is summarized in Figure 12.

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We observe a circumpolar jet at around 40°–50° north in northern winter (2004–09, L_s ≈ 290°–5°) with peak speed ∼210 m s⁻¹, reducing to ∼190 m s⁻¹ by 2009 (L_s ≈ 355°–5°). This is consistent with the predicted weakening of the meridional overturning circulation (R. K. Achterberg et al. 2011) as Titan approaches northern spring equinox in 2009. Less angular momentum is transported to high northern latitudes in late northern winter, leading to weakening of the jet. The lower-latitude jet migrates equatorward and gradually reduces in speed from ∼190 m s⁻¹ in 2010 (L_s ≈ 10°) to ∼120 m s⁻¹ in 2017 (L_s ≈ 90°), consistent with J. Sharkey et al. (2021). We observe the northern zonal jet to persist until at least 2013 (L_s ≈ 45°), although wind speeds only reach ∼135 m s⁻¹.

Pre-equinox, we observe a weak zonal jet at around 20° south with peak wind speed ∼110 m s⁻¹. This has some symmetry, with the northern jet observed to persist in the northern hemisphere well after the equinox, which sits at around 25° north in 2017 (L_s ≈ 90°). The weak spring/summer-hemisphere jet is not resolved in GCMs (C. E. Newman et al. 2011; S. Lebonnois et al. 2012; N. A. Lombardo & J. M. Lora 2023b, 2023a). Instead, GCMs predict the main winter-hemisphere jet to extend into the summer hemisphere, for example, in the TAM model (N. A. Lombardo & J. M. Lora 2023b, 2023a; Figure 11). Previous observational studies have also predicted an extension of the northern jet into the south (e.g., R. K. Achterberg et al. 2008a; J. Sharkey et al. 2021). Our observed wind contours are subject to uncertainties and agree, within unknowns, with the previous results and simulations that there is a single jet that crosses the equator.

Immediately after the equinox (from 2009 onward, L_s ≲ 0°), the weak southern jet rapidly strengthens and gradually migrates poleward, from ∼110 m s⁻¹ at 25° south in 2009 to ∼230 m s⁻¹ at 40° south in 2013 (Figure 11, left). This is consistent with the predicted reversal of the meridional overturning circulation that transports momentum to the southern hemisphere (S. Vinatier et al. 2020). Meanwhile, there is a suggestion that a second circumpolar jet might form at higher southern latitudes, around 70° north, first evident in 2012 (L_s ≈ 35°). However, the robustness of this result is low given the observational noise and given that, dynamically, this configuration would be extremely unstable. Due to the instability of this configuration, we would not expect to observe multiple jets in the simulated wind field. This apparent second jet appears to descend from altitudes above the sensitive region of our observations (pressures ≳0.001 mbar) and then gradually extend to lower altitudes, reaching 0.5 mbar in 2016 (L_s ≈ 80°), strengthening until it reaches a peak wind speed of ∼230 m s⁻¹ in 2015 (L_s ≈ 70°). However, it is unclear whether this jet is increasing in speed or the fast jet core is simply descending into our probed region. The jet core simulated by the TAM GCM follows a similar descent, from around 0.1 mbar in 2011 (L_s ≈ 25°) to 2 mbar by 2017 (L_s ≈ 90°). Two jets in the south are observed to persist in individual observations during 2016. After 2015 (L_s ≳ 65°), we observe weakening of the southern jet, in agreement with S. Vinatier et al. (2020). The two jets appear to merge in 2017 (L_s ≈ 90°), resulting in a single jet of speed ∼225 m s⁻¹.

PV is a measure of rotation within large-scale atmospheric flow and combines absolute vorticity and the vertical stability of the atmosphere. It is particularly useful because it is conserved through time in the absence of frictional or diabatic forces acting on it. As such, it gives us an indication of the rotation and intensity of the polar vortices as they evolve throughout the season. The edge of the polar vortex acts as a mixing barrier, leading to confinement of trace gases at the pole (N. Teanby et al. 2006). A strong horizontal PV gradient can be evidence of the mixing barrier (D. G. Andrews 1987), as is the case at the edge of Earth's antarctic polar vortex (M. E. McIntyre & T. N. Palmer 1983; M. McIntyre 1989), so it can be used to probe a vortex edge. A steep PV gradient may also indicate the existence of atmospheric waves such as Rossby waves (B. J. Hoskins et al. 1985; E. R. Nash et al. 1996); however, Rossby waves are not expected to propagate in Titan's stratosphere (J. Sharkey et al. 2020). The redistribution of PV can be closely related to the propagation of Rossby waves, which may be important in the breakup of the polar vortex. PV distributions are therefore important in describing the structure and stability of Titan's polar vortices.

Previous studies have investigated Titan's stratospheric PV distribution (R. K. Achterberg et al. 2008b, 2011; N. A. Teanby et al. 2008; D. M. Mitchell et al. 2021; J. Sharkey et al. 2021). PV has been used previously to probe the weakening of the north polar vortex (N. A. Teanby et al. 2008; R. K. Achterberg et al. 2011; J. Sharkey et al. 2021) and the strengthening of Titan's south polar vortex (J. Sharkey et al. 2021). In these studies, there is a suggestion of an annular PV structure encircling Titan's winter pole, where a PV maximum sits at high latitudes with a local PV minimum at the winter pole. Opposing PV gradients have been shown to be barotropically unstable (D. G. Dritschel 1986; L. Rayleigh 1879), so an annular PV structure should not persist without some external restoring force. However, J. Sharkey et al. (2020) find that the north polar vortex is zonally uniform, suggesting that the vortex is largely barotropically stable. We determine the zonal mean PV distribution in Titan's stratosphere from the zonal wind field and use it to diagnose the shape and strength of the north and south winter polar vortices.

In a rotating fluid, the PV, q, describes a combination of the absolute vorticity (the local rotation of the fluid) and stratification (the vertical gradient of the potential temperature, Θ):

$$\begin{eqnarray}&&q=-g(f+{\xi }_{{\rm{\Theta }}})\frac{{\rm{\partial }}{\rm{\Theta }}}{{\rm{\partial }}p}\end{eqnarray} \tag{ 4 }$$

(P. L. Read et al. 2006), where the potential temperature, Θ, is the temperature of an air parcel if it were moved adiabatically—that is, without loss or gain of heat—from some pressure level, p, to some reference pressure, p₀ (J. Houghton 2002),

$$\begin{eqnarray}&&{\rm{\Theta }}=T{\left(\frac{{p}_{0}}{p}\right)}^{\kappa }.\end{eqnarray} \tag{ 5 }$$

The ratio of specific heat capacities $\kappa =\frac{{c}_{p}-{c}_{v}}{{c}_{p}}=0.281$ for Titan (N. A. Teanby et al. 2008), and we use reference pressure p₀ = 10 mbar. The vertical component of the relative vorticity on surfaces of constant potential temperature is

$$\begin{eqnarray}&&{\xi }_{\Theta }=-\displaystyle \frac{1}{{r}^{2}\cos (\phi )}\displaystyle \frac{\partial }{\partial \phi }(ru\cos \phi ),\end{eqnarray} \tag{ 6 }$$

and f = 2Ωsinϕ is the Coriolis parameter. We determine the PV field from the potential temperature, following Equation (4). We use a finite-difference method to approximate the derivatives:

$$\begin{eqnarray}&&{\left(\frac{{\rm{\partial }}{\rm{\Theta }}}{{\rm{\partial }}p}\right)}_{{ij}}\approx \frac{{{\rm{\Theta }}}_{i,j+1}-{{\rm{\Theta }}}_{i,j-1}}{{p}_{j+1}-{p}_{j-1}}\end{eqnarray} \tag{ 7 }$$

and

$$\begin{eqnarray}&&{\xi }_{{\rm{\Theta }},{ij}}\approx -\displaystyle \frac{1}{{r}_{j}^{2}\cos {\phi }_{i}}\displaystyle \frac{({r}_{j}{u}_{i+1,j}\cos {\phi }_{i+1})-({r}_{j}{u}_{i-1,j}\cos {\phi }_{i-1})}{{\phi }_{i+1}-{\phi }_{i-1}}\end{eqnarray} \tag{ 8 }$$

for the ith latitude and jth pressure level, and r_j = r(p_j) and u_ij = u(p_j, ϕ_i). The radial polar coordinate, r = R_T + h, must remain a variable in Equation (6), as Titan's deep atmosphere means that it has significant atmospheric curvature compared to its radius.

PV has a large vertical variation, due its inverse dependence on the pressure, p_j+1 − p_j−1, so we scale the PV by factor (Θ/100)ⁿ to reduce this variation while preserving the horizontal structure and conservative properties of PV. This scaling was first suggested by L. R. Lait (1994), who use n = −9/2. We instead use n = −7/2, which is more appropriate for Titan's atmospheric composition (N. A. Teanby et al. 2008). Cross sections of the scaled PV magnitude are plotted against the potential temperature in Figure 13 for each Earth year. Regions where the scaled PV error exceeds the contour interval (>5 × 10⁻⁵ m² K kg⁻¹ s⁻¹) have been removed. The Appendix outlines how we determine the uncertainty in Lait PV. The radiative timescales in Titan's stratosphere range in length from a Titan day to a Titan season (B. Bézard et al. 2018), so PV features can persist in the stratosphere over multiple Earth years. Since PV effectively depends on the second derivative of temperature, it can be significantly influenced by fine structure in the temperature field, and although smoothing has been applied, only large-scale structure in PV should be considered reliable. The PV field is calculated from the temperature field smoothed with cubic B-splines with a knot spacing of 10°. This should preserve finer meridional structure than achieved by previous studies of the PV structure of Titan's stratosphere (R. K. Achterberg et al. 2008a, 2011; J. Sharkey et al. 2021). Our interpretation of the observed Lait PV structure in relation to Titan's polar vortices is summarized in Figure 12, and we now describe the broad structure in the determined zonal PV.

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Throughout mid- to late northern winter, we observe a suggestion of an annulus of PV encircling the north pole, with a PV maximum around 60°–80° north and a local PV minimum at the north pole (Figure 13). This is consistent with N. A. Teanby et al. (2008), R. K. Achterberg et al. (2011), and J. Sharkey et al. (2021), who observe a tentative annulus of PV at around 65° north in northern midwinter. This annulus of PV persists from at least 2005 (L_s ≈ 305°) through to the northern spring equinox in 2009 (L_s ≈ 0°), suggesting that Titan's northern polar vortex has an annular shape for at least 5 yr leading up to equinox. The annular PV structure does not seem to be an artifact due to increasing emission angles toward the pole: local PV minima are observed in multiple individual observations with emission angles of less than 40°. Due to the limited coverage of our observations, it is unclear whether the north polar vortex has an annular shape immediately post-equinox (2010–2011). However, a weak PV gradient persists at high northern latitudes at least during 2012–2014 (L_s ≈  30°–65°), and there are hints of this in 2015 and 2016 (L_s ≈ 65°–85°), although it is unclear whether this takes an annular shape. This is consistent with evidence for the confinement of trace species in the north polar region, which is observed to persist until 2016 (N. A. Teanby et al. 2019).

In Figure 13, there is a hint of a secondary PV annulus at higher altitudes, at pressures ≲0.1 mbar (Θ ∼ 1000 K), sitting persistently at around 40° north. PV is effectively the second derivative of the temperature field and is therefore very sensitive to noise in the temperature. Subtler features in the observed PV structure could be an artificial product of noise amplification in the temperature field, despite our efforts to limits this to PV through smoothing the temperature field. However, the northern secondary PV annulus is evident in almost all observation sequences that have sufficient latitude coverage in 2006 (L_s ≈ 330°), 2008 (L_s ≈ 345°), 2009 (L_s ≈ 0°), and 2012–2016 (L_s ≈  30°–85°).

There is a suggestive detached PV maximum low in stratosphere, around 1 mbar (Θ ≈ 300 K). We observe this at approximately 70° north during 2006–2009 (L_s ≈ 315°–5°) and 2012–2014 (L_s ≈ 30°–65°), although it significantly weakens after Titan's equinox. This feature is seen in the majority of individual observation sequences acquired during this time and is also evident in the PV structure determined from CIRS limb observations by N. A. Teanby et al. (2008) and CIRS nadir observations (J. Sharkey et al. 2021) in the northern hemisphere during northern winter. This package of PV may have been deposited by breaking gravity waves. Breaking waves can deposit energy into the zonal flow, producing an enhanced PV gradient that prevents mixing across it. This would prevent air at high northern latitudes from mixing with warmer air from the south, and thus we observe this feature as a cold spot low in the stratosphere near the north pole (Figure 10). Gravity waves have been observed in the equatorial region to propagate vertically into Titan's lower stratosphere up to at least the 3 mbar level (R. D. Lorenz et al. 2014), if not higher. Perhaps gravity waves also exist in the north polar region: topographic variations are weak on Titan (R. D. Lorenz et al. 2013), but waves could alternatively be generated by differences in surface-lake temperatures owing to the longitudinal asymmetry in lake distribution at high northern latitudes.

Regions of high meridional PV gradient can be evidence of a mixing barrier and so can diagnose the location of the vortex edge. The right panels of Figure 13 show the Lait PV on isentropic surfaces with potential temperature Θ = 350, 700, and 1000 K, which correspond to approximately 1, 0.1, and 0.01 mbar in Titan's atmosphere, respectively. The north polar annulus has a width of approximately 20° at ∼1 mbar (Θ = 350 K; Figure 13, pink lines), with the outer edge sitting at around 60° north. At higher altitudes, ∼0.1 and 0.01 mbar (Θ = 700 and 1000 K; Figure 13, blue and green lines), the vortex seems to be wider, around 30°, with the outer edge sitting at ∼50° north. The width of the vortex does not have a noticeable time dependence during this period, but it appears to be moving generally poleward.

From at least 2012 (L_s ≈ 35°), we observe a large PV at high southern latitudes, which persists until at least 2016 (L_s ≈ 80°). The PV maximum appears to sit at the south pole, although we cannot say for certain, as our observations do not extend to the pole. Near the end of the mission (L_s ≈ 80°), the PV is observed to be annular, particularly clear in the right 2016 panel of Figure 13. It is therefore suggestive that Titan's polar vortex begins its life as a conventional vortex, a stable configuration, persisting for at least 6 yr (approximately a quarter of a Titan year) after the equinox, and then begins to transition to an annular shape, with maximum PV at ∼75° south in 2016 (L_s ≈ 80°) at 1–0.001 mbar and ∼70° south in 2017 (L_s ≈ 90°) at 0.1–0.001 mbar. This is in agreement with J. Sharkey et al. (2021), who find the PV to be maximum at the south pole in 2013 (L_s = 42°) and then at ∼60° south in 2017 (L_s = 86°) at the 0.1 mbar level. The PV in the south during formation of the south polar vortex is around 4 times stronger than the maximum PV observed in the northern hemisphere during breakup of the north polar vortex, consistent with J. Sharkey et al. (2021). We would not expect to observe symmetry between the north/south vortices, as Cassini observed different parts of Titan's seasonal cycle for the northern/southern hemispheres.

As in the northern hemisphere, there is also a suggestion of a secondary PV annulus in the south. During 2007–2016 (L_s ≈ 330°–85°), there appears to be a PV annulus in the southern midlatitudes, around 60° south: in general, migrating equatorward leading up to equinox, then poleward post-equinox. This is observed in the majority of individual observation sequences during these time frames. However, as with secondary structures in the zonal wind field, the robustness of this feature is low given the considerations of observational noise and dynamical stability. Furthermore, in the left 2017 panel of Figure 13, we see a detached PV maximum lower in the stratosphere, around 1 mbar (Θ ≈ 300 K), mirroring the similar PV feature observed in the northern hemisphere earlier in the mission.

Again, we examine the Lait PV on some isentropic surfaces to diagnose the vortex edge. At Θ = 700 K (∼0.1 mbar), the south polar vortex outer edge varies between around 65° and 75° north during its formation (2012–2017, L_s ≈ 30°–90°). The outer edge of the vortex appears to migrate equatorward during this period, suggesting that the vortex is growing in size, or perhaps becoming annular if an inner edge is forming. Later, during 2016 (L_s ≈ 80°), there is evidence of an inner edge of the vortex, as we observe a negative meridional PV gradient at latitudes south of 80° south, suggesting an annular shape, which appears to continue to migrate equatorward during 2016–2017. The gradient of the PV in the south is significantly larger than in the north, suggesting that the (forming) south polar vortex edge acts as a stronger mixing barrier than the edges of the (decaying) north polar vortex, in agreement with J. Sharkey et al. (2021) and the observed isolation of species at the winter pole (e.g., N. A. Teanby et al. 2019; S. Vinatier et al. 2020). Composition mapping in the stratosphere suggests that there are no significant differences between the north and south polar vortices (J. Sharkey et al. 2021). However, the eccentricity of Saturn's orbit means that Titan's southern winter receives more insolation and is longer than the northern winter, so there may be asymmetry between Titan's northern and southern polar vortices. To build up a complete picture of Titan's winter polar vortex behavior, observations spanning the second half of Titan's year (L_s = 93°–293°), not covered by Cassini, would be required. Due to geometry, ground-based observatories are unable to observe Titan's winter polar region, highlighting the importance of a Titan polar orbiter to observe the polar vortices (C. A. Nixon et al. 2020; J. W. Barnes et al. 2021).

The meridional extent of the south polar vortex is observed to be larger at higher altitudes during 2012–2015 (L_s ≈ 30°–75°; Figure 13). This equatorward tilt in PV with increasing altitude has previously been observed in the PV structure of the polar vortices on Titan (J. Sharkey et al. 2021), Earth, and Mars. However, the PV feature tilts poleward as opposed to equatorward on Mars (D. M. Mitchell et al. 2015). The PV altitude tilt seen on Earth and Mars has been attributed to gravity waves (D. M. Mitchell et al. 2015). N. W. Kutsop et al. (2022) observe circumpolar haze bands in Titan's stratosphere at altitudes of ≳100 km (≲8 mbar) using Cassini/VIMS observations. At the 1 mbar level (Θ = 350 K; Figure 13, pink lines), we observe hints of secondary PV maxima at similar latitudes to where the haze bands are observed. In the northern hemisphere during 2012–2017 (L_s ≈ 30°–90°), we observe hints of local PV maxima broadly around 25°, 50°, and 75° north; however, again, these features should be interpreted with caution. N. W. Kutsop et al. (2022) suggest that haze bands form owing to haze particles being transported with the meridional circulation and collecting at mixing barriers. Our observed PV structure—that is, opposing PV gradients, which can act as barriers to mixing—suggest that such mixing barriers do indeed exist, confining the haze in bands. Alternatively, the enrichment of haze in these bands might affect the absorption of solar radiation, impacting the local temperature structure, circulation, and hence the PV structure. For example, temperature changes may generate atmospheric waves, affecting the PV structure, since PV is also a diagnostic of Rossby waves (B. J. Hoskins et al. 1985; E. R. Nash et al. 1996). There could be a positive feedback between the confinement of haze and the observed PV structure. Higher-resolution models and observations would be required to determine whether these observed PV features, including secondary PV annuli and the detached lower-stratosphere PV maximum, are real. Models may also require additional processes, for example, radiatively active tracer species, to fully understand the physical mechanisms responsible.

The Dragonfly mission is forecast to launch in 2028 and arrive at Titan in the mid-2030s. Before beginning operation at Titan's surface, Dragonfly must undergo a parachute descent through Titan's superrotating middle atmosphere without damage. Zonal winds have a significant impact on Dragonfly's landing ellipse (the area where Dragonfly is expected to land; R. D. Lorenz 2021). The east–west dimension of the ellipse largely depends on the uncertainty in the zonal wind field (R. D. Lorenz et al. 2021). Furthermore, due to the strength of Titan's zonal winds, the stratospheric tilt is a major component of the meridional wind field, so the north–south dimension of the landing ellipse is dependent on the stratospheric tilt axis. Improved knowledge of the tilt at the time and location of Dragonfly's descent will better inform the north–south dimension of the landing ellipse.

The north–south size of the ellipse is dependent on the uncertainty in the tilt offset. Consideration of the tilt offset could shift the predicted landing location of Dragonfly by around 2–3 km (R. D. Lorenz, in private communication), which should land near the Selk impact crater at around 7° north, 199° west (R. D. Lorenz et al. 2021). Dragonfly is planned to arrive at Titan in 2035 in late northern winter (L_s ≈ 310°). This coincides with the season observed early in the Cassini mission, which arrived at L_s = 293°. The stratosphere at Dragonfly's arrival is therefore expected to be similar to our 2005 panel in Figures 10 and 11. Assuming that the stratospheric tilt direction is fixed in the inertial frame, we can predict that the tilt axis will be within the uncertainty envelope ψ = 114°–125° (3σ error) west of the subsolar point at northern spring equinox. The magnitude of the tilt is more complicated. In the inertial-fixed frame, we determine the tilt to fluctuate. There is a suggestive seasonal trend in the tilt magnitude (Figure 8), so we loosely predict the tilt to have a similar magnitude at the same point in the next seasonal cycle. Therefore, it might be expected that the tilt during 2035, L_s ∼ 310°, will be similar to the tilt in 2005 (L_s ∼ 310°), which is β = 2$\mathop{.}\limits^{\unicode{x000b0}}$7 ± 1$\mathop{.}\limits^{\unicode{x000b0}}$1. However, with only half a Titan year of coverage, this seasonal trend is tentative, and for mission planning purposes a conservative estimate of the tilt would be β = 0$\mathop{.}\limits^{\unicode{x000b0}}$2–10$\mathop{.}\limits^{\unicode{x000b0}}$0.