Moist convection drives an upscale energy transfer 1 at Jovian high latitudes 2

13 Jupiter’s atmosphere is one of the most turbulent places in the solar system. While lightning and thunderstorm observations 14 point to moist convection as a small-scale energy source for Jupiter’s large-scale vortices and zonal jets, it has never been 15 demonstrated due to the coarse resolution of pre-Juno measurements. Since 2017, the Juno spacecraft discovered that Jovian 16 high-latitudes host a cluster of large cyclones (diameter of ∼ 5,000 km each) associated with intermediate ( ∼ 1,600–500 17 km) and smaller-scale vortices and ﬁlaments ( ∼ 100 km). Here, we analyze Juno-infrared images with an unprecedented 18 high-resolution of 10 km. We unveil a new dynamical regime associated with a signiﬁcant energy source of convective origin 19 that peaks at 100 km–scales and in which energy gets subsequently transferred upscale to the large circumpolar and polar 20 cyclones. While this energy route has never been observed on another planet, it is surprisingly consistent with idealized 21 studies


1
A natural follow up question that begs to be answered is: can moist convection, through vortices in the k −4/3 scale range, 111 account for the emergence and persistence of the large cyclones present at Jovian high latitudes in the k −3 scale range? This is 112 addressed in the next section.

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Upscale energy transfer 114 Here, we diagnose the KE transfer across scales to understand how small-scale moist convection impacts the large cyclones. 115 The KE transfer, KE adv , is derived from the momentum equations and computed from the wind measurements (see Methods). 116 A negative (positive) value of KE adv (k) indicates a KE loss (gain) at the wavenumber k.   However, verifying these results in Jupiter would require a longer time series of observations than currently available.

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Our results show that moist convection at 100 km-scales is associated with an upscale energy transfer strengthening the large 141 cyclones at Jovian high latitudes. This energy route is expected to increase the heat transfer between deep and hot interior layers 142 and colder upper layers, where heat gets converted into KE 8-10, 23 , which is also consistent with deep convection and a deep 143 origin of the large cyclones 24, 25 . If this was the case, the presence of two convective regimes (a deep one responsible for the 144 large cyclones and mostly forced from below, and a shallower one responsible for the smaller circulations and directly linked to  The cluster of vortices may also owe its stable structure to optimal shielding -an anticyclonic ring around each cyclone 3, 27 .

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This study contributes to our fundamental understanding of vortex dynamics and highlights a regime that has not been reported 151 on another planet before. The closest dynamical analogs in the solar system may well be some parts of the Earth's atmosphere. e2019JE006098 (2020).                    Wavenumber spectrum 271 For a given doubly-periodic and detrended variable φ , we first compute a discrete 2D fast Fourier transform φ (k x , k y ), with k x 272 and k y the wavenumbers in the x and y direction, and then we compute a 1D spectral density | φ (k)| 2 from the 2D spectrum, 273 with k is the isotropic wave number defined as k = k 2 x + k 2 y , following a standard procedure described in ref 37 .

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Butterworth filter 275 Throughout the paper, we apply a low-pass Butterworth filter to the wind measurements to remove the remaining small-scale 276 noise. After experimenting, we find that a filter of order 1 with a cutoff wavenumber of 250 km produces the best results in 277 terms of the signal to noise ratio. We use the spectral characteristics of τ to infer the upper bound of the spectral variance at 278 the smallest scales. To do so,ζ does not exceedζ τ (defined below) for scales smaller than 100 km where most of the wind 279 measurements' noise associated with taking the temporal derivative between two infrared observations is located. Note that  negligible at leading order, leads to 13 : taken on a constant pressure surface, Θ 0 the reference potential temperature and N 2 the squared Brunt-Väissälä frequency 307 defined as g Θ z Θ 0 . As the potential temperature is unknown (see Methods section "optical depth"), we use a first order Taylor 308 series expansion g Θ Θ 0 ∼ N 2 h, yielding: τ is thus related to APE via h = H 0 τ, with H 0 chosen such that KE and APE spectra are equal in the k −4/3 scale range. If 310 we define a streamfunction ψ = f P, with P the pressure (as done in the next section), the hydrostatic approximation yields 311 ∂ ψ ∂ z = g f Θ Θ 0 . APE can then be written as: The quasi-geostrophic framework 313 The quasi-geostrophic (QG) framework is useful to study the dynamics of a flow field of Ro ∼ O(1) despite relying on a small 314 Ro approximation 13 . In particular, this framework has been successfully used to study thermal convection in the limit of rapid 315 rotation 22 , reminiscent of the observations presented here. In the QG framework, potential vorticity (PV) is conserved along a 316 Lagrangian trajectory and is given by: with ψ the streamfunction, ∆ the horizontal Laplacian operator and f and N assumed constant. The relative vorticity ζ is given 318 by ∆ψ. From PV conservation, ζ is related to the stretching term, f 2 N 2 ∂ 2 ψ ∂ z 2 , via the horizontal divergence as 13, 38 : with χ the horizontal divergence. can be written as: with subscript T denoting the tropopause. Integration of PV= 0 (eq. 4, valid in the troposphere) using the boundary condition 326 at the tropopause given by equation (7) and assuming that ψ vanishes at z = −∞, leads to the spectral solution: From PV= 0 and equations (7), (8) and (9), the relative vorticity at the tropopause can now be linked to the cloud thickness as: The SQG framework also allows to infer the aspect ratio between horizontal and vertical scales. Indeed, (8) leads to an e-folding As mentioned before, APE is associated with the stretching term in the PV expression and is therefore related to the depth 336 dependence of the streamfunction, or what is called the baroclinic mode 13, 38 . APE spectrum in the k −3 scale range has a 337 much smaller variance than KE and, in addition, has a much shallower spectral slope (Fig. 3a). These two characteristics 338 indicate that the depth dependent contribution (or the baroclinic part) to the total streamfunction is small and therefore that the 339 streamfunction is dominated by the depth-independent part (or the barotropic part) 40 . These arguments imply that vortices are 340 2D (depth-independent) in this scale range, consistent with the weak χ-variance (Fig. 3b). This can also be understood with 341 equation (6), in which a small χ-variance corresponds to a small stretching. 343 We diagnose the KE and enstrophy (ENS) transfer between wavenumbers in spectral space using the momentum equations at 344 the tropopause (where vertical velocities are null). Multiplying these equations by the conjugate of the horizontal wind speed 345 and without considering dissipation for simplicity's sake, leads to 20, 29, 41 :

KE and enstrophy transfer
with . * the complex conjugate and Re(.) the real part. Equations (11) and (12) are the equations for the time evolution for a 347 given wavenumber k of the KE and ENS, respectively. The first terms on the right hand side of equations (11) and (12) are 348 nonlinear advection terms, whereas the seconds terms are sources and/or sinks.   Fig. 1) after application of a low-pass filter that retains lengthscales greater than 1,600 km (i.e., wavenumbers k < 6.10 −4 cpkm) and c) small-scale relative vorticity ζ τ derived from τ after application of a high-pass filter that retains lengthscales smaller than 1,600 km (i.e., wavenumbers k > 6.10 −4 cpkm, see Methods). As ζ τ is directly related to cloud thickness, it is the signature of cloud convection. Large-scale vortices in b) gain their energy from small-scale vortices in c) via an upscale energy transfer (see main text).

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Extended Data Figure 2. Relative vorticity and horizontal divergence Maps of a) relative vorticity ζ and b) horizontal divergence χ corresponding to the spectra shown in Figure 2 b (blue and green curves, respectively). These fields were derived from wind measurements to which a Butterworth filter of order 1 and cutoff lengthscale of 250 km was applied (see Extended Data Fig. 1). The seams between the mosaic' strips are particularly visible in b).

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Extended Data Figure 5. Scatter plots between ζ τ and ζ Scatter plots between ζ τ and ζ for the fields in Extended Data Fig.  4 per subdomain : in a) the polar vortex (blue rectangle in Extended Data Fig. 4), b) the lower left filament (orange rectangle in Extended Data Fig. 4), c) the streamer subdomain (black polygon in Extended Data Fig. 4), d) the entire domain. Each point represents the average over each grid interval on the abscissa (that has a total of 200 grid intervals), and thin vertical lines show std dev around the averages. Straight lines indicate the least-square regression line between the points. The slope and r 2 of the regression line is shown in each panel.

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Extended Data Figure 6. Scatter plots between ζ τ and χ Same as Extended Data Fig. 5 but for χ instead of ζ .

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Extended Data Figure 7. Enstrophy spectral flux Enstrophy spectral flux derived from wind measurements (see Methods) showing an ubiquitous direct cascade from large to small scales, consistent with an upscale KE transfer in classical theory of rotating turbulence 20 .