On Saturn's six-sided polar jet stream

We derive the nonlinear governing equations for stratified circumpolar atmospheric jet flow in Saturn's upper troposphere. An exact solution is obtained in the material (Lagrangian) framework, by specifying its hypotrochoidal particle paths. The resulting flow pattern presents a striking resemblance to the hexagonal jet stream structure observed near Saturn's North Pole.


Introduction
One of the most unusual and easily recognisable features in the solar system, first discovered by NASA's Voyager mission in the early 1980s, is a six-sided narrow circumpolar jet stream on Saturn (see figure 1), lasting for the subsequent decades and showing no sign of abating.The hexagonal feature is relatively narrow (spanning the latitudes 74 • -78 • in Saturn's Northern hemisphere) and about 100 km deep, but its sides are about 14,500 km long.Its effects extend above the clouds of the upper troposphere into the stratosphere to altitudes of 0.5 mbar (Ingersoll 2020).The hexagon-shaped structure is almost static, barely moving relative to the planet's overall rotation.
Observations from the Voyager and Cassini spacecrafts, combined with ground-based observations and images from the Hubble Space Telescope, laboratory experiments and numerical simulations, allow a good understanding of many features of this startling atmospheric phenomenon (Ingersoll 2020).However, the issue of finding the exact cause of Saturn's startling six-sided jet stream is still unresolved, with Rossby wave theory (Sánchez-Lavega et al. 2014, Fletcher et al. 2018) and deep rotating convection (Yadav and Bloxham 2020) among the hypotheses, many other alternative explanations being already refuted (for example, the fact that Saturn's magnetic field plays a crucial role).In the present study we present an exact solution to the nonlinear governing equations for the stratified flow in Saturn's upper troposphere in the Lagrangian framework, by specifying the trajectories of the individual fluid parcels.These paths are hypotrochoidal curves that produce a flow pattern strikingly similar to that observed on Saturn.While the study of parcel paths is analytically intricate, the process permits a detailed study of the dynamical structures, revealing other features that replicate those on Saturn (for example, concerning the vorticity within the jet stream).Note that the Lagrangian approach proved to be very useful to locate the edge of the terrestrial polar vortex (Serra et al. 2017).
In section 2, we derive the governing equations for stratified circumpolar flow in Saturn's upper troposphere, showing that the leading-order dynamics is two-dimensional, nonlinear and inviscid.In section 3, we present an exact solution to the governing equations by specifying its hypotrochoidal trajectories that resemble a hexagon with rounded corners.A detailed study of the obtained flow pattern is pursued in section 4, highlighting its shape-invariance under translations in time and exploring the implications of a nonvanishing vertical component of the vorticity.A brief survey of the remarkable history of hypotrochoidal curves is also made available in section 4.

The nonlinear governing equations
We consider a rotating right-handed Cartesian coordinate system with the x -axis pointing from West to East, the y -axis from South to North and the z -axis upwards, using primes to denote physical/dimensional variables (they will be removed when we nondimensionalize).The Hexagon being embedded within a circumpolar strip spanning the latitudes 74 • − 78 • in the Northern hemisphere, we regard the Coriolis parameters as constant; here θ denotes the angle of latitude and Ω ≈ 1.62 × 10 −4 rad s −1 is the (constant) rate of rotation of Saturn around its polar axis.We denote by u , v , w , the corresponding fluid velocity components.If t stands for time, g ≈ 10.4 m s −2 is the (constant) gravitational acceleration near Saturn's tropopause, μ and ν are the (constant) horizontal and vertical eddy viscosity coefficients, respectively, ρ is the density, T is the temperature and P is the atmospheric pressure, and we denote by the material derivative, the governing equations are the Navier-Stokes equations (Vallis 2017) coupled with the equation of mass conservation the equation of state for an ideal gas, and the first law of thermodynamics Here R ≈ 4016.43 m 2 s −2 K −1 is the value of the gas constant for Saturn, c p is the specific heat, κ/c p is the thermal diffusivity, and Q is the heat-source term.It suffices to keep track of the velocity field (u , v , w ), of the pressure P and of the density ρ .The ideal gas law (2) then specifies the temperature T and the first law of thermodynamics (3) identifies the associated heat sources -for this viewpoint see also the discussions of terrestrial atmospheric flows in Constantin andJohnson (2021, 2022) and of stratospheric flows of the giant gas planets in Constantin and Germain (2022).
To nondimensionalize the governing equations we introduce the following representative physical scales (Dobrijevic et al. 2003, Catling 2015, Cosgrove and Forbes 2017): the vertical length scale H = 10 5 m, the horizontal length scale L = 1.56 × 10 7 m (the mean radius of the relevant latitudinal circumferences), U = 125 m s −1 as the horizontal speed scale, W = 10 −2 m s −1 as the vertical velocity scale, and ρ ≈ 1 g m −3 as the average density of Saturn's upper troposphere.We can thus introduce dimensionless variables t, x, y, z, u, v, w, ρ, P and T by with the normalisation factors L /U ≈ 1.5 × 10 5 s (about 43 h, corresponding to 4 days on Saturn), ρ U 2 ≈ 10 −1 mbar (with 1-10 mbar the pressure range in the upper troposphere), W H ≈ 10 3 m 2 s −1 (adequate for the eddy viscosity near the tropopause) and U 2 /R ≈ 4 • K (with 50-100 • K the temperature range in the upper troposphere).We obtain the nondimensional version of the governing equations: is the nondimensional material derivative and with θ 0 = 76 • .In the regime ε δ 1, by taking the limit δ → 0, we obtain that the leadingorder dynamics is described by the equations with the vertical velocity component w negligible at leading order O(1).For the purpose of flow visualisation, note that one can also derive the system ( 7)-( 11) by relying on the polar plane approximation (Cosgrove and Forbes 2017), whereby lines of latitude correspond to circles centred about the North Pole, with the Coriolis parameter varying quadratically with latitude: , where R ≈ 5.8232 × 10 7 m is the radius of Saturn.Given the scales involved in the nondimensionalization leading to ( 7)-( 11), for the relevant band of 4 • latitude width the quadratic correction to a constant Coriolis parameter f is negligible.
We show in the next section that for any specified density ρ = ρ(z), the system ( 7)-( 11) admits explicit solutions with hypotrochoidal particle paths that are markedly similar to the streamline pattern depicted in figure 1.Moreover, the corresponding pressure P and temperature T display the observed behaviour within Saturn's six-sided polar jet streamthey both decrease with increasing height z, and there is a positive meridional component of the temperature gradient.

Lagrangian description of the flow pattern
Given the density ρ = ρ(z) of the background state, in the form of a decreasing function of the height z, we claim that an explicit solution to the equations ( 7)-( 11) can be obtained by specifying, at time t, the particle positions in terms of the labelling variables (a, b), the height z, the westward wave speed c > 0, the phase α, and the parameters R > r > 0 that control the amplitude of the oscillations.The labelling variable a runs over the real numbers, with b ∈ (b 1 , b 2 ) for suitable b 1 < b 2 < 0 that capture the meridional width of the zonal strip to which the six-sided polar stream jet is confined (about 4 • of latitude), while z ∈ [z 0 , z 1 ], where z 0 and z 1 with z 0 < z 1 correspond to the bottom and top elevation of the six-sided polar jet, respectively.Note that c f (13) since, denoting by c its dimensional counterpart, cf. the data in Sayanagi et al. (2018).By imposing the constraint we ensure that the particle paths of the flow pattern (12a) are hypotrochoids with six bulges and no self-intersections (see figure 2 and section 4).We will show that to any density in the form of a decreasing function ρ = ρ(z) we can associate a pressure distribution of type so that the velocity field (u, v) determined by (12a) solves the system (7)-(10).From (15) we get ∂P ∂z = −gρ(z) < 0 so that formula (15) captures the observed pressure decrease with respect to the tropospheric height z.After deriving the explicit expression for the P 0 -term in (15), we can use (11) to also verify other observed features within Saturn's six-sided polar jet: a temperature decrease with height and a positive meridional component of the temperature gradient -see the data from the Cassini mission provided in Fletcher et al. (2018).In this context, let us point out that in Saturn's upper troposphere the North Pole is a hot spot -about 10 • K warmer than the mean temperature at 80 • N latitude, forcing a reversed meridional monotonicity of the temperature poleward of 80 • N.
To prove these claims, note that the Ansatz (15) validates ( 9), so that we only have to find a function P 0 (a − ct, b) so that ( 7)-( 8) and ( 10 Due to ( 14), the Jacobian determinant of the map relating at the instant t the particle positions to the labelling variables is strictly negative.If we denote by (x 0 , y 0 ) the initial data, the chain-rule identity Since ρ is only dependent on the z-variable, (10) holds.It remains to verify ( 7)-( 8) for a suitable pressure distribution of the form (15).
The velocity of a particle with labels (a, b, z) is obtained by taking the time derivative of its position vector, so that Note that the field data in Sayanagi et al. (2018) confirms that the zonal and meridional velocity components u and v are of the same order of magnitude within the hexagonal jet stream, as indicated by their nondimensional counterparts (18a).The horizontal acceleration of a particle with labels (a, b, z) is the total time-derivative of the horizontal velocity vector (u, v), and can be computed by taking the time derivative of the components (18a): Thus we can rewrite ( 7)-( 8) as We now invoke ( 15) and ( 16) to express (20) in the equivalent form Therefore the choice ith P * 0 a constant such that validates our solution.Indeed, if (22) holds, then the decrease of density with height in combination with ( 11) and ( 15) ensures since by ( 13) and ( 21) we have Furthermore, since (21) in combination with ( 13) and ( 14) yields ≤ cr 2 e 2b R r because the roots of the quadratic polynomial 11) and ( 15) we now get the claimed poleward decrease of temperature throughout the six-sided jet stream:

Discussion
We now present some of the main features of the obtained nonlinear flow pattern.
For fixed labels (a, b, z, α), the particle paths are time-parametrised hypotrochoids.The interest in such curves dates back to the astronomical studies of the ancient Greeks and over the last three centuries they attracted the attention of many researchers, including Bernoulli, Euler, Huygens, Newton (Brieskorn and Knörrer 1986;Simoson 2010).
Hypotrochoids are the curves traced out by a point P rigidly attached to a disk of radius r 1 > 0 that is rolling without slipping inside of a fixed circle of radius r 2 > 0, centred at the point O.If d > 0 is the distance from P to the centre D of the rolling disk, and if α ∈ [0, 2π) is the angle at O between the half-lines OD and DP (see figure 3), a parametric ).The phase α between the two circular motions does not alter the appearance of the six-sided closed curve -it merely rotates the hexagonal shape (Pook 2011).(Colour online) representation of the hypotrochoid is where τ is the central angle at O between the positive x-axis and the half-line OD, the origin being at O (Gray 2006).In the special case d = r 1 the curve parametrised by (23a) is called a hypocycloid (these being the only non-smooth hypothrochoids, as cusps occur), while for r 2 = 2r 1 we get an ellipse.The shape of a hypotrochoid depends on the parameters k = r 2 /r 1 and d.For example, a hypotrochoid is a closed curve if and only if k is a rational number, having k loops as τ ranges over the interval [0, 2π ] if k is an integer.Moreover, a hypotrochoid will not self-intersect if and only if k is an integer and r 1 ≥ d (Konkar 2022).
A peculiar aspect of the solution (12a) is its validity independent of the value of the wave speed c > 0. Saturn's six-sided flow pattern moves slowly westward relative to the nominal rotation rate for Saturn, and while precise speed predictions are very difficult, in Saturn's rotating reference frame the clockwise (westward) rotation of the hexagonal pattern with time is of the order of 10 −2 degrees of latitude per day (Fletcher et al. 2018).In this context, note the folowing shape-invariance property of the flow pattern (12a): performing a translation in time t → t + t 0 is equivalent to a rotation of the pattern by −ct 0 and a change of phase α → α − 6ct 0 since cos(ct 0 ) sin(ct 0 ) − sin(ct 0 ) cos(ct 0 ) Let us now discuss the vorticity of the flow (12a) relative to Saturn's surface.At leading order, since the vertical velocity component w vanishes, the relative vorticity vector is 25r 2 e 10b − R 2 e 2b < 0 , with the sign determined by ( 14).By the right-hand rule, a negative vertical vorticity component is indicative of a clockwise horizontal rotation.Note that small clouds are swept along with the six-sided jet stream -the hexagonal band is like a road, and these clouds are like cars moving along a racetrack (Ingersoll 2020).While the dynamics of these smaller features is not captured by the leading-order governing equations ( 7)-( 11), the above considerations are consistent with the observation of small vortices spinning clockwise within the background flow represented by the six-sided jet, as one can see in the high-resolution movie made from images obtained by Cassini's cameras, available at https://solarsystem.nasa.gov/missions/cassini/science/saturn/hexagon-in-motion/

Conclusions
The performed Lagrangian analysis shows that the nonlinear governing equations for stratified circumpolar flow in the upper troposphere of Saturn admit solutions with hypotrochoidal particle paths.We refrain from speculating on the atmospheric forcings that might generate such flows.Nonetheless, given that the flow pattern has a striking resemblance with the observed six-edge jet pattern on Saturn, it must be seen to represent a dynamically important type of large-scale atmospheric flow.

Figure 1 .
Figure 1.Natural-colour view of Saturn's north polar region down to about 72 • N, acquired on 27.XI.2012during NASA's Cassini mission, with Saturn's rings visible in the upper right corner (Image Credit: NASA/JPL-Caltech/SSI).The region surrounded by the six-sided jet stream comprises a massive hurricane centred on the pole, north of 88 • N, and numerous small vortices, with the biggest spanning about 3500 km.Some vortices spin clockwise while the six-sided jet stream and the hurricane spin counterclockwise.

Figure 2 .
Figure2.For fixed labels (a, b, z, α), the particle paths (12a), parametrised by time t in Saturn's rotating reference frame, trace clockwise an algebraic curve resembling a hexagon with rounded corners -a curtate hypotrochoid -centred at the North Pole, as in the photograph reproduced in figure1.Note that, due to (13), Saturn's hexagon rotates counterclockwise when viewed from the North Pole down.
) hold.Denoting ξ = a − ct, fact that D(b) is time-independent yields that the matrix determinant.Therefore the flow (12a) is area-preserving and thus ∂u ∂x + ∂v ∂y = 0 .(

Figure 3 .
Figure3.The hypotrochoids parametrised by (23a) are the curves traced by a point P rigidly attached to the disk centred at D and rolling without slipping inside the fixed circle centred at the origin O.The movement of P is composed of two uniform circular motions in opposite directions, with constant angular velocities: the counterclockwise rotation of the rolling disk and the clockwise motion of the disk's centre D along the circle centred at O and with radius (r 2 − r 1 ).The phase α between the two circular motions does not alter the appearance of the six-sided closed curve -it merely rotates the hexagonal shape(Pook 2011).(Colour online) 10b − R 2 e 2b 5re 5b sin(α − 5ξ) + Re b sin ξ −5re 5b cos(α − 5ξ) − Re b cos ξ 5re 5b cos(α − 5ξ) − Re b cos ξ 5re 5b sin(α − 5ξ) − Re b sin ξ , we can use the chain rule to compute