Is rotating turbulence behind the banding of Jupiter's atmosphere?

The self-organization of large-scale coherent flows is ubiquitous in nature and is an essential, cutting edge research topic in fundamental fluid dynamics, geophysics and planetary sciences. The strong zonal (i.e. east-west) winds, that form large-scale bands, the large vortices and the swirling clouds observed on the gas giants, i.e. Jupiter, Saturn, Uranus and Neptune are unmistakable markers of powerful, self-organized, atmospheric dynamics. Here we show, how and why these large-scale features develop in gas giant planets and we explain how to reproduce them in a lab experiment and in numerical models, using the most advanced tools of nowadays researches.

How to make a giant planet in the Lab?

With JUMP, we give a recipe for zonal jets that results from the culmination and combination of several of long-term strands of research. We identify three fundamental ingredients that leads to the formation of Jupiter like zonal jets and large scale vortices. These ingredients are: (I) First, a fluid layer highly energized to drive turbulent motions, for example, stirring in a bucket of water to inject highly energetic and random motions. (II) Second, we have to rotate the turbulent bucket at a constant speed to emulate planetary rotation. (III) Finally, the deformation of the free water surface (forming a paraboloidal shape) in the rotating bucket will ensure the growth of powerful zonal jets (symmetric around the rotation spin axis) and a host of waves and vortices. This recipe led to the first laboratory model of a gas giant and you can visit this lab here.

Rotate a turbulent fluid

The following figure represents ingredients (I) and (II) in an illustrative numerical simulation, the rotating bucket is represented by a 3D-turbulent cube and turbulent motions are randomly imposed in the simulated fluid. Rotation forces the flow to self-organize around the rotation spin axis and form a large-scale vortex. Rotation also leads to the in-variance of the velocity along the rotation spin axis, we usually say that rotation promotes flow bidimentionalization. In other words flow motions can freely evolve on the horizontal plan perpendicular to the rotation axis. We also note that the global flow is a stack of the four different cubes for which scales of motions have been separated for illustration purpose.

Deform to Mimic planetary sphericity

Next figure represents ingredient (III), surface deformation of a rotating flows impose a geometrical constrain that enforces a new geometry to the turbulent fluid. In planets, surface deformation is the spherical shape of the atmospheric layer, in the lab, it is easier to use the paraboloidal deformation of the water surface in bucket.

Surface deformation of a rotating flows adds a geometrical ingredient that enforces a new geometry to the turbulent fluid. In planets, surface deformation is the spherical shape of the atmospheric layer, in the lab, it is easier to use the paraboloidal deformation of the water surface in bucket.

> Click to visit our laboratory <

> Click to visit a 5m experiment <

JUMP's Data collection

Jets in the lab

In natural settings, zonal jets develop from rapidly rotating turbulence in the presence of strong boundary curvature. In particular, the Coriolis force must dominate the fluid’s inertia, which, in turn, must dominate the viscous dissipative effects. We gathered all these dynamical ingredients in a single laboratory device that emulate Jupiter like zonal jets. More about the experiment can find at this link.

We performed a series of laboratory experiments in a rotating tank at rotation rate Ω= 3.0 rad s−1 in a square tank of radius R = 29.7 cm. The mean water depth was 4 cm and the β-effect was stipulated by the parabolic curvature of the free surface. The small-scale turbulent forcing is applied, see link! We acquired velocity fields by analyzing images of passively advected styrene particles monitored by a video camera with a spatial resolution of 1023 × 1240 pixels at a frequency of 20 Hz. The particles were seeded at a fluid surface. We obtain velocity field on a cylindrical coordinate grid with a resolution of Nr = 60 radial points (radial resolution of about 0.5 cm) and Nθ = 90 points in the angular direction (angular resolution of 1°).

Jets from Cassini observations

In December 2000, the Cassini fly-by near Jupiter delivered high-resolution images of Jupiter’s clouds over the entire planet in a band between 50°N and 50°S. Three daily-averaged two-dimensional velocity snapshots extracted from these images. Recently, Galperin et al. (2015) calculated the horizontal wind field using an automated ‘‘cloud tracking’’ method based on Correlation Imaging Velocimetry (CIV) as detailed below.

The principle of the cloud tracking method is to take a pair of images of the same place but separated in time, and correlate the brightness of pixel patches as they move from the first image to the second. At wavelengths where the brightness is due to reflection from clouds, moving patches correspond to cloud movement from which the underlying wind velocity can be estimated. The algorithm works on the assumption that brightness features will retain their general appearance as they evolve over the time between images.

Jets from a Global Climate numerical Models

The emergence of high performance Global Climate Models (GCMs) for the Earth atmosphere can be used to model the atmospheric circulation of gas giants with appropriate physical parametrizations. To step in this direction, I used a high-resolution model named DYNAMICO, solving for 3D primitive equations of motion and spatially designed to reproduce Saturn atmosphere. I represent below a snapshot of the zonal velocity, on a latitudinal-longitudinal map, after 15 Saturn simulated years.

I ran a Saturn simulation covering 15 Saturn years using the Saturn DYNAMICO GCM. Wind fields are output every 20 Saturn days at 32 pressure levels onto 1/2° latitude-longitude grid maps. In the steady state, the zonal energy is an order of magnitude larger than the residual energy, and remains approximately constant over the 15 simulated years.

A statistical diagnostic to relate - experiments, observations and simulations..

How do axisymmetric jets emerge in gas giants? It is well accepted that in three-dimensional (3D) systems, turbulent kinetic energy is typically transferred from larger to smaller scales through a direct en­ergy cascade (the Kolmogorov cascade). However, this process can be reversed under certain conditions favoring coherent flow. In other words, non-axisymmetric turbulent flows (e.g. caused by flow bar­oclinicity, Salmon, 1980) can feed the large-scale jets. This is the case in both quasi-two-dimensional (2D) domains, where the flow is geomet­rically confined in depth to a shallow layer, or under rapid rotation in 3D domains, where the flow becomes invariant along the rotational spin axis and is dynamically confined to quasi-2D geostrophy. Two types of models have been proposed for jet formation in planetary atmospheres: shallow models where jet forcing is confined to the atmospheric weather layer (Cho and Polvani, 1996; Lian and Showman, 2008; Liu and Schneider, 2010; Schneider and Liu, 2009; Spiga et al., 2020; Young et al., 2019) and deep models where jet forcing extends into the convective molecular envelope of the planet (Cabanes et al., 2017; Christensen, 2002; Gastine et al., 2014; Heimpel et al., 2005). In both cases, flow bi-dimensionalization transfers kinetic energy from small to large scales via the so-called inverse energy cascade (Kraichnan, 1967). It is only recently that this theoretical framework has been generalized to 3D rotating turbulence by Sukoriansky and Galperin (2016). Using analytical tools, they showed the transition from a 3D direct energy cascade to rotation-dominated (quasi-2D) turbulence leading to an in­ verse energy cascade. This duality between 2D and 3D turbulence as well as geometrical and dynamical flow confinement, together with the associated transition from direct to inverse energetic cascade, is at the core of fundamental geophysical fluid dynamics.

Spherical geometry also carries an additional background of planetary vorticity denoted by the parameter β, The effect β acts on the flow to preferentially channels kinetic energy in the zonal (i.e. east-west) direction promoting large-scale zonal motions. The planetary vorticity β = (2Ω/R) cos θ directly results from the variation of the Coriolis force with latitude θ, the rotation rate Ω, and planetary radius R. This coupled action of the turbulent inverse cascade and of the β-effect in planetary atmospheric envelopes links large-scale features to smaller scale dynamics through the continuous process of scale-to-scale interactions. To sum-up, turbulent flows in planets undergo 2 successive anisotropies that channel energy from 3D isotropic turbulence to 2D horizontal anisotropy and finally to zonal anisotropy forming large scale zonal jets. The aim of JUMP is to draw a full statistical diagnostic that describe theoretically the energetic distribution at all scales in Jupiter like flows. The Figure below shows theoretical distribution of the total kinetic energy in planetary systems as a function of latitudinal wavenumbers n as coined by Sukoriansky et al. (2002) and detailed in Cabanes et al Icarus (2020).

Spectral projection of the velocity maps

Guidlines to access statistical analysis

Related publications:

[1] - S. Cabanes, A. Spiga and R. Young. “Global climate modeling of Saturn’s atmosphere. Part III: Global statistical picture of zonostrophic turbulence in a high-resolution 3D-turbulent simulations.” Icarus (2020). doi.org/10.1016/j.icarus.2020.113705 pdf

[2] - S. Espa, S. Cabanes, G. P. King, G. Di Nitto and B. Galperin. "Eddy–wave duality in a rotating flow". Physics of Fluids 32.7 (2020): 076604. https://doi.org/10.1063/5.0006206 pdf

[3] - D. Bardet, A. Spiga, S. Guerlet, S. Cabanes, E Millour and A. Boissinot. “Global climate modeling of Saturn's atmosphere. Part IV: Stratospheric equatorial oscillation.” Icarus (2020). https://doi.org/10.1016/j.icarus.2020.114042 pdf

[4] - S. Cabanes, S.Espa, B. Galperin, R. Young and P. Read. “Reveal the energetic power of turbulence in planetary atmospheres.” Submitted to Geophysical Research Letters, (In revision).

Data sets are in open source:

• You can find all data and velocity fields from JUMP's lab, here.

• You can find all data and velocity fields from Cassini, here.

• You can find all data and velocity fields from DYNAMICO, here.

Numerical codes to compute the statistical diagnostics are open source:

• You can find code for statistical analysis in spherical geometry on Github. --> https://github.com/scabanes/POST

• You can find code for statistical analysis in cylindrical geometry on Github. --> https://github.com/scabanes/JUMP

• You can find code for statistical analysis in cartesian geometry on Github. --> https://github.com/scabanes/JUMP

Find more presentations

• WITGAF 2019 : Waves, Instabilities and Turbulence in Geophysical and Astrophysical Flows is a conference where I gave a talk in July 2019. The full presentation is available in the PDF bellow.