## Main

The Jovian InfraRed Auroral Mapper (JIRAM) on board Juno, operating at 4.8 μm and capturing ammonia clouds1,2, revealed a set of weather phenomena present at Jovian high latitudes1,3. In contrast to mid-latitude cloud bands and zonal jets, the dominant features are numerous vortices of varying size (Fig. 1), including a cluster of large cyclonic vortices (eight in the north and five in the south) surrounding a single polar cyclone, located at about ±83°. These cyclones have a diameter of ~5,000 km, peak speeds of 100 m s−1 and a fast rotation period of ~10 h, and have endured since their discovery in 20171,3,4. Intermediate-scale cyclones and anticyclones, with a diameter of ~1,000 km and less, have a similar rotation period. The smallest observable vortices and filaments (<100 km) survive a couple of hours to several days only3 and are the signature of an intense cloud activity3,4 (Fig. 1). Lightning reported at northern latitudes in recent studies5,6,7, especially above the 2-bar level7, points to moist convection within ammonia clouds as a powerful source of energy in Jupiter’s polar regions. However, even though the dynamical link between small-scale moist convection and large-scale vortices has been investigated at high latitudes in recent modelling studies8,9,10,11, it has never been supported by observations.

Here, we investigate this link using high-resolution infrared images taken by JIRAM at the north pole of Jupiter on 2 February 2017, from which two complementary datasets are derived. First, we use wind velocities (Extended Data Fig. 1a) derived by tracking clouds in sequence of infrared images (see ref. 4 for the methodology) in a domain greater than 7,000 × 20,000 km (Fig. 1, black rectangle), where overlapping observations over short intervals are available. The wind field has a resolution of 10 km, which is slightly higher than previous measurements12. Second, the optical depth, defined as log(I0/I), with I the brightness and I0 = 0.55946 W m−2 the maximum brightness, is directly obtained from the infrared measurements3 with the same resolution as the wind field (Fig. 2a). Optical depth is related to variations in cloud thickness such that thick cloud produces high optical depth3,13.

The analysis of these two unique datasets using a theoretical framework (the surface quasi-geostrophy (SQG) framework14,15,16,17,18, described in Moist convection is a KE source at 100 km scales section) allows us to uncover a dynamical regime on Jupiter, which highlights the key role played by moist convection at Jovian high latitudes. Surprisingly, this regime is reminiscent of idealized studies of rapidly rotating Rayleigh–Bénard convection19,20,21,22,23,24 and laboratory experiments25 and even resembles parts of the Earth’s atmosphere26,27,28.

### Dynamical context

We analyse the properties of the wind measurements in terms of kinetic energy (KE), relative vorticity (ζ, that is, the local rate of spin of a vortex, Methods) and horizontal divergence (χ, Methods). The magnitude of ζ (Extended Data Fig. 1b) and χ (Extended Data Fig. 1c) reaches the local spin rate of the planet (f = 3.5 × 10−4 s−1, also called the Coriolis parameter), which is one fold higher than previous estimates29. First, we use a suite of spectral analyses (Methods) to investigate how KE, ζ and χ are distributed among length scales and what their spectral characteristics tell us about the underlying dynamical regime (Methods). The KE spectrum (Fig. 3a) exhibits two distinct scale ranges separated by a conspicuous spectral slope discontinuity: scales larger than 1,600 km are characterized by a steep slope in k−3 (with k the wavenumber), whereas scales smaller than 1,600 km are characterized by a shallower slope in k−4/3. This discontinuity is even more striking in ζ’s power spectrum (Fig. 3b, blue curve) since the slope changes sign at 1,600 km. Second, we retrieve the physical patterns captured by each scale range using a spectral filter (Methods). As expected, the ζ field associated with the k−3 scale range comprises the large vortices, including the persistent circumpolar and polar cyclones, as well as one large anticyclone (Fig. 2b). Since the χ spectral amplitude is a lot smaller than the ζ spectral amplitude in this scale range (Fig. 3b), large scales are likely governed by 2D turbulence30. The Rossby number, defined as Ro = ζ/f, reaches 0.5 and its distribution is positively skewed (Extended Data Fig. 2b), confirming the overwhelming dominance of the large cyclones in the k−3 scale range. In contrast, the k−4/3 scale range (length scales of ~100–1,600 km) includes numerous cyclones, anticyclones and elongated filaments embedded in between the vortices (Extended Data Fig. 1b). Unlike the k−3 scale range, χ and ζ variances are of the same order of magnitude, and are even equal for scales smaller than 300 km (Fig. 3b,c), suggesting that vortices in this scale range are forced by vertical motions and hence that their dynamics is three dimensional31. Finally, the Ro distribution is only slightly negatively skewed (Extended Data Fig. 2c), highlighting the absence of a cyclone–anticyclone preference in the k−4/3 scale range.

Overall, the analysis of the wind measurements highlights a turbulent regime that involves two classes of vortices separated by a conspicuous spectral slope discontinuity (Fig. 3a,b, blue curve) between large-scale vortices for which the surface dynamics is probably driven by horizontal motions and smaller vortices and filaments affected by vertical motions. However, vorticity and divergence diagnosed from the wind measurements contain small-scale noise (100 km, Extended Data Fig. 1b,c) as their computation involves both spatial and temporal derivatives. This noise prevents us from analysing the link between small-scale features and large-scale vortices solely from the wind measurements.

### Moist convection is a KE source at 100 km scales

We use the information provided by the optical depth anomaly, τ, in combination with the wind measurements. Positive values of τ correspond to thick clouds (that is, updraughts) and negative values to no or thin clouds (that is, downdraughts)3,13,32. Thus, τ can be interpreted as the signature of vertical motions and therefore as a vertical scale. Based on the assumption that cloud thickness is an indicator of tropopause height anomalies (as for the Earth’s tropopause15), tropopause height anomalies are estimated as h = H0τ, where H0 is the length scale of tropopause height anomalies. Then, τ can be interpreted in terms of APE, defined as N2h2, with N the Brunt–Väissälä frequency (Methods). APE and KE spectra have the same slope (k−4/3) for scales <1,600 km, which is reminiscent of the dynamical SQG framework14,15,16,17,18 (Methods). In the next paragraph, we use this framework and focus on the k−4/3 scale range to better understand the properties of the intermediate- and small-scale vortices.

In SQG, APE and KE spectra are identical, which leads to simple relationships between horizontal motions and h (ref. 18). As the SQG framework is relevant for the Earth’s tropopause14,15,17,18,27, we assume that it also applies to Jupiter. We adjust the APE spectral amplitude to match that of KE for scales <1,600 km by taking N constant and equal to 4 × 10−3 s−1 in Jupiter’s upper troposphere33 and choosing H0 = 9 km (Fig. 3a). Using SQG properties, relative vorticity can be diagnosed from τ (equation (10)). This relative vorticity field, ζτ, is displayed in Fig. 2c. As expected, ζ and ζτ spectra overlap in the k−4/3 scale range (Fig. 3b, orange and blue curves). To validate our assumption that SQG applies in the k−4/3 scale range, we now compare ζ and ζτ in physical space after application of a conservative band-pass filter that retains only length scales between 250 and 1,600 km (see Methods and Extended Data Fig. 3b,c for the filtered fields). Although ζ is noisier than ζτ (Methods), the correspondence between the two fields is striking, especially where ζ is well resolved. For instance, the streamer between the two anticyclones is characterized by a filament of negative relative vorticity surrounded by two filaments of positive relative vorticity in both ζ and ζτ (Extended Data Fig. 3b,c, black polygon). The relationship between ζ and ζτ is corroborated by the scatterplots in Extended Data Fig. 4, confirming the existence of a SQG regime and the fact that τ can be interpreted in terms of tropopause height anomalies, with H0 the typical depth scale. In addition, the good match between ζ and ζτ confirms the positive correlation between anticyclonic vorticity and positive cloud thickness anomaly (equation (10)). Finally, an estimation of the convective APE (CAPE)34,35 within NH3 clouds leads to an order of magnitude similar to our APE, confirming its convective origin (Methods).

Convection within clouds is known to be associated with updraughts, producing positive divergence and anticyclonic vorticity at the cloud top32,36. This is consistent with our dynamical analysis, which implies that thick clouds (that is, positive optical depth anomaly τ) are associated with anticyclonic vorticity and the fact that ζτ is negatively correlated with χ (Extended Data Fig. 5). Furthermore, χ has the same magnitude as the divergence estimated from CAPE within clouds (Methods) and as the ones reported for cloud convection in Jupiter35,37. Taken together, this indicates that thick clouds are associated with positive divergence and anticyclonic vorticity, consistent with a convective origin. To double check this result, we also investigate the direct relation between τ and χ (Extended Data Figs. 3a,d and 6), which confirms the positive correlation between cloud thickness and divergence, lending further support to moist convection at Jovian high latitudes. Moist convection is most significant at 100 km scales, as suggested by the maxima of the ζτ, ζ and χ spectra (Fig. 3b).

KE and APE spectra differ greatly for scales larger than 1,600 km (Fig. 3a). The APE spectrum (Fig. 3a, orange curve) has a shallower slope and a much smaller variance than the KE spectrum (Fig. 3a, blue curve). In addition, the divergence is weak in this range (Fig. 3b). Taken together, this implies that vortices in the k−3 scale range are not directly impacted by clouds (Dynamics in the k−3 scale range section), consistent with a character and generation mechanism independent of tropopause-level forcing2. A natural follow-up question is: Can moist convection, through vortices in the k−4/3 scale range, account for the emergence and persistence of the large cyclones present at Jovian high latitudes in the k−3 scale range? This is addressed in the next section.

### Upscale energy transfer

Here, we diagnose the KE transfer across scales to understand how small-scale moist convection impacts the large cyclones. The KE transfer, KEadv, is derived from the momentum equations and computed from the wind measurements (Methods). A negative (positive) value of KEadv(k) indicates a KE loss (gain) at the wavenumber k. Figure 4a reveals that scales smaller than 215 km, where the KE source due to convection is the most significant, lose KE. On the other hand, larger scales up to about 5,000 km gain KE. This indicates an upscale KE transfer in both the k−3 and k−4/3 scale ranges, from the smallest observable vortices (<100 km) up to the large cyclones (~5,000 km). Note that, while the zero-crossing value of 215 km (Fig. 4a) is weakly sensitive to the data processing and can vary between 100 and 400 km (Extended Data Fig. 7f), the upscale energy transfer up to the large cyclones’ scale is a robust feature of the flow. In addition, a downscale enstrophy (that is, the relative vorticity squared) transfer is observed at all scales (Fig. 4b), consistent with an upscale energy transfer in the classical theory of rotating turbulence38. Taken together, these results show that turbulence at Jovian high latitudes is forced by convection at 100 km scales and that the resulting KE cascades upscale to the large circumpolar and polar cyclones. This upscale KE transfer is consistent with previous analyses of Jovian mid-latitudes for scales larger than ~1,600 km (ref. 29) but has never been reported at smaller scales.

Our findings are remarkably consistent with recent idealized studies of rapidly rotating Rayleigh–Bénard convection19,20,21,21,22,23,24, as well as with laboratory experiments25 and numerical simulations of forced turbulence in a rotating frame39 with generic small-scale forcing. All these studies reveal the existence of KE and APE spectra similar to those in Fig. 3a as well as an upscale KE transfer from small-scale forcing up to the large 2D vortices. They further indicate that the upscale KE transfer is enhanced by a positive feedback loop whereby large-scale vortices organize intermediate and small-scale convective features. In other words, 2D vortices organize 3D vortices and filaments, themselves constraining the convection21. The noticeable spiral bands of relative vorticity and clouds around the polar vortex (Fig. 2a,c), as well as the aggregation of thick clouds within intermediate-scale anticyclones (Extended Data Fig. 4), is consistent with such a positive feedback loop. For instance, the streamer of negative vorticity and thick clouds (Fig. 2a,c and Extended Data Fig. 3a,b) suggests that small-scale moist convection is organized by intermediate-scale vortices and filaments, with the latter being advected and stretched by the large cyclones. However, verifying these results in Jupiter would require a longer time series of observations than currently available.

## Discussion

Our results show that moist convection at 100 km scales is associated with an upscale energy transfer strengthening the large cyclones at Jovian high latitudes. This energy route is expected to increase the heat transfer between deep and hot interior layers and colder upper layers, where heat gets converted into KE20,22,23,40, which is also consistent with deep convection and a deep extent of the large cyclones2,41,42. If this was the case, the presence of two convective regimes (a deep one responsible for the large cyclones and mostly forced from below, and a shallower one responsible for the smaller circulations and directly linked to clouds and winds) would point to an explicit stratification in the upper portion of Jupiter’s polar atmosphere. Other non-mutually exclusive mechanisms can also account for the genesis and persistence of the cluster of cyclones at Jupiter’s poles. Cyclones are expected in the polar regions because of the β effect, which forces cyclones (anticyclones) to drift polewards (equatorwards)43. An estimation of the β drift, or planetary Burger number8, agrees with a regime comprising multiple circumpolar cyclones9,10 (Methods). The stability of Jupiter’s cluster of cyclones may also be explained by more complex vorticity dynamics4,44,45.

This study contributes to our fundamental understanding of vortex dynamics and highlights a regime that has never been observed on another planet before. The closest dynamical analogues in the solar system may well be some parts of the Earth’s atmosphere26,28. Indeed, in close agreement with the Jovian wind measurements presented here, wind observations taken along a Lagrangian trajectory at the Earth’s tropopause display a conspicuous KE spectrum with a k−3 slope for large scales and k−5/3 for smaller scales26,46. The corresponding dynamical regime remains an open question27,46,47, mainly because of the lack of 2D observations. While a theoretical paper invokes the impact of internal waves47, recent numerical studies argue that moist convection is the leading mechanism28,46,48. The results of the present study highlight the power of combining high-resolution 2D cloud observations with wind measurements, which, if applied to the Earth’s atmosphere, should help us better understand the dynamics of our own planet.

## Methods

### Datasets

#### Infrared images

Infrared images are taken by JIRAM’s imaging channel in the M band, which operates at 4.8 μm wavelength. These are images of ammonia clouds shielding the thermal emission of the planet at different viewing angles2. During the perijove 4 orbit, JIRAM performed a series of overlapping observations of the poles of Jupiter in the hours bracketing perijove. In this paper, we consider only the north polar sequences because they provide better overlap among the images and shorter intervals between images than at the south pole (Extended Data Table 1). Observations were taken on 2 February 2017 and have a resolution of ~10 km per pixel, depending on foreshortening and spacecraft motion. The image processing, from the raw data to the polar orthographic projected maps, is thoroughly described in ref. 4. However, briefly, the first step in the processing is to determine the precise location of each pixel on the planet. This is done with Navigation and Ancillary Information Facility/Spacecraft Planet Instrument C-matrix Event (NAIF/SPICE) data from the spacecraft and precise geometric calibration of the JIRAM instrument49. The second step is to map the brightness patterns onto a 10-km gridded reference plane tangent to the planet at the pole. Mosaics are then assembled by combining 12 such infrared images separated by 30 s each. A total of four mosaics (that is, 48 infrared images, see Extended Data Table 1) are analysed, all yielding same results (Extended Data Figs. 8and 9). The projected infrared images are provided in the Supplementary Data under the filename ‘mapped brightness’. In Main, we show the results for one typical mosaic (n02, Fig. 2a), taken at mid-time between the two mosaics used to derive the wind field described below.

#### Wind measurements

Horizontal winds are computed by tracking cloud features between two overlapping infrared observations separated by 16 min using Tracker 3 software from JPL4 with a resolution of 10 km and an error of 2.1 m s−1. That dataset has filename ‘velocity vector’ in the Supplementary Data. To remove remaining small-scale noise, we apply a Butterworth filter of order 1 with a cutoff wavelength of 250 km. We analyse a total of two wind mosaics (derived from a total of 48 infrared images), yielding similar results (Extended Data Figs. 8 and 9). In Main, we show results for the area with broader coverage (Extended Data Fig. 1), using the mosaics n01 and n03. Note that using velocity vectors and infrared brightness maps projected onto a 15 km per pixel grid as done in ref. 4 yields similar results. An analysis of the error propagation indicates that the errors associated with the wind measurements are confined to scales less than ~50 km (Extended Data Fig. 7a–e) and therefore do not impact the conclusions of our study.

#### Optical depth

JIRAM infrared brightness I is primarily governed by cloud opacity50, with high I corresponding to thin clouds located at pressures of 1–5 bar (that is, within warm regions at depth), and low I corresponding to thick clouds that can reach the tropopause at a pressure of about 0.3 bar (that is, within colder upper regions)3,51,52. Thus, JIRAM observations are not taken along a constant-pressure surface and cannot be directly converted into a potential temperature from which one can infer horizontal motions at the tropopause, as done by ref. 15 for the Earth. Consequently, instead of directly using I, we use the optical depth, a proxy of cloud thickness3,13. The optical depth is computed as τa = log(I0/I), with I0 = 0.55946 W m−2 the maximum brightness measured in the infrared mosaic n02 displayed in Fig. 2a, following refs 3,13. In this study, we define the optical depth anomaly τ as $$\tau ={\tau }_{\mathrm{a}}-\overline{{\tau }_{\mathrm{a}}}$$, with an overbar denoting the domain average.

### Spectral methods

In this paper, all the computations are done in spectral space, which allow us to easily and accurately isolate different scale ranges53. Before taking the Fourier transform of any given variable, we apply two steps. First, we make the variable doubly periodic by applying a mirror symmetry in the x and y directions. Performing spectral analyses in a doubly periodic domain considerably improves the result’s accuracy compared with windowing methods, which damp both the large and small scales and can change the amplitude and slope of a power spectra density54. Second, the variable is detrended by subtracting the mean value.

#### Wavenumber spectrum

For a given doubly periodic and detrended variable ϕ, we first compute a discrete 2D fast Fourier transform $$\widehat{\phi }({k}_{x},{k}_{y})$$, with kx and ky the wavenumbers in the x and y direction, then compute a 1D spectral density $$| \widehat{\phi }(k){| }^{2}$$ from the 2D spectrum, with k the isotropic wavenumber defined as $$k=\sqrt{{k}_{x}^{2}+{k}_{y}^{2}}$$, following a standard procedure described in ref. 55.

#### Butterworth filter

Throughout the paper, we apply a low-pass Butterworth filter to the wind measurements to remove the remaining small-scale noise. After experimenting, we find that a filter of order 1 with a cutoff wavenumber of 250 km produces the best results in terms of the signal-to-noise ratio. We use the spectral characteristics of τ to infer the upper bound of the spectral variance at the smallest scales. To do so, $$\hat{\zeta }$$ does not exceed $$\hat{{\zeta }_{\tau }}$$ (defined below) for scales smaller than 100 km where most of the wind measurements’ noise associated with taking the temporal derivative between two infrared observations is located. Note that we applied a Butterworth filter of order 2 and cutoff wavenumber of 500 km to the wind field in Extended Data Fig. 1a for visualization purposes.

#### Band-pass filter

A band-pass filter is implemented in spectral space and retains a range of length scales by setting the wavenumbers outside of this range to zero. In Extended Data Figs. 36, we apply a band-pass filter that retains length scales between 250 and 1,600 km (that is, wavenumbers between 6 × 10−4 cpkm and 4 × 10−3 cpkm). A sensitivity analysis on these wavenumbers was performed and results remain unchanged when scales smaller than 250 km are included (not shown). In Fig. 2, we apply a low-pass (b) and high-pass (c) version of this filter with a cutoff at 6 × 10−4 cpkm (that is, at the spectral slope discontinuity shown in Fig. 3ab).

#### Relative vorticity and horizontal divergence

Relative vorticity $$\hat{\zeta }$$ and horizontal divergence $$\hat{\chi }$$ are computed in spectral space from horizontal wind measurements $$\hat{u}$$ and $$\hat{v}$$ as

$$\hat{\zeta }=-{\mathrm{i}}\, ({k}_{y}\, \hat{u}-{k}_{x}\, \hat{v})\ \ {{{\rm{and}}}}\ \ \hat{\chi }={\mathrm{i}}\, ({k}_{x}\, \hat{u}+{k}_{y}\, \hat{v}),$$

with i2 = −1. Note that the physical fields ζ and χ shown in Extended Data Fig. 1b,c are retrieved from the inverse 2D fast Fourier transform of $$\hat{\zeta }$$ and $$\hat{\chi }$$, respectively. In addition, the computation of ζ = vx − uy and the horizontal divergence χ = ux + vy, with subscripts denoting partial derivative, is done in physical space in ref. 4 and yields comparable results.

### Rossby number

Taking the divergence of the momentum equations and assuming that the divergence’s time derivative and friction terms are negligible at leading order leads to31

$$f\zeta -{\nabla }_{\mathrm{h}}\cdot \left({{{{\bf{u}}}}}_{\mathrm{h}}\cdot{\nabla }_{\mathrm{h}}\ {{{{\bf{u}}}}}_{\mathrm{h}}\right)-{{{\Delta }}}_{\mathrm{h}}P=0,$$
(1)

with h is the horizontal gradient operator, Δh is the horizontal Laplacian operator, uh is the horizontal velocity vector and P is the pressure. The Rossby number, defined as Ro $$\equiv \frac{U}{fL}$$, with U and L respectively a velocity and length scale31, quantifies in equation (1) the relative importance of the Coriolis term (that is, fζ) compared with the nonlinear term. A dimensional analysis of equation (1) indicates that there is a balance between the Coriolis and pressure terms for Ro  1, called the geostrophic balance. For Ro ~ 1, the balance also involves the nonlinear terms, called the gradient wind balance. In this study, we estimate the Rossby number as Ro = ζ/f. The vortices at Jovian high latitudes have Ro ~ 1, indicating that nonlinear terms and therefore nonlinear vortex interactions are significant.

### Available potential energy

Tropopause height anomalies, h, are related to the optical depth anomaly, τ, via h = H0τ, with H0 a scale depth of tropopause height anomalies. In the atmosphere, APE is defined as15,31$${\left(g\frac{{{\Theta }}}{{{{\Theta }}}_{0}}\right)}^{2}\frac{1}{{N}^{2}}$$, with g the gravitational constant, Θ the potential temperature taken on a constant-pressure surface, Θ0 the reference potential temperature and N2 the squared Brunt–Väissälä frequency defined as $$g\frac{\overline{{{{\Theta }}}_{z}}}{{{{\Theta }}}_{0}}$$. As the potential temperature is unknown (Optical depth section), we use a first-order Taylor series expansion $$g\frac{{{\Theta }}}{{{{\Theta }}}_{0}} \sim {N}^{2}h$$, yielding

$${{{\rm{APE}}}}={N}^{2}{h}^{2}\ .$$
(2)

τ is thus related to the APE via h = H0τ, with H0 = 9 km and N = 4 × 10−3 s−1 chosen such that KE and APE spectra are equal in the k−4/3 scale range. N has been chosen from ref. 33, yielding a value of H0 = 9 km. Using these values yields APE = 1,296 m2 s−2. Note that N might be slightly different at the pole, which in turn would change H0, without however affecting our computations.

We suggest interpreting APE in terms of CAPE34 in order to assign moist convection to condensation process within clouds35. CAPE = $$gZ\frac{{\mathrm{\delta}} T}{ < T > }$$, with δT the temperature difference between heated parcels and their background, Z the vertical length scale of the path followed by ascending parcels and <T> the background mean temperature34,35. We use the values provided by ref. 35 for NH3 clouds: <T> = 130 K at 0.5 bar, Z = 40 km (from 0.1 to 1 bar) and δT = 0.3 K. This leads to CAPE = 2,612 m2 s−2, a value close to our APE estimate. Note that the values for H2O clouds yield a CAPE magnitude ten times larger. As such, our CAPE estimation validates equation (2) and confirms that APE has a convective origin within NH3 clouds.

In addition, using the relation CAPE = $${w}_{\mathrm{max}}^{2}/2$$ and Z = 40 km (ref. 35) yields an upper bound for the divergence within clouds of wmax/Z ~ 7 × 10−4 s−1. This divergence value is consistent with χ estimated from the wind measurements as well as with modelling results35,37.

Note that, if we define a stream function ψ = fP, with P the pressure (as done in the next section), the hydrostatic approximation yields $$\frac{\partial \psi }{\partial z}=\frac{g}{f}\frac{{{\Theta }}}{{{{\Theta }}}_{0}}$$. APE can then be written as

$${{{\rm{APE}}}}=\frac{{f}^{2}}{{N}^{2}}{\left(\frac{\partial \psi }{\partial z}\right)}^{2}$$
(3)

### The quasi-geostrophic framework

The quasi-geostrophic (QG) framework is useful to study the dynamics of a flow field of Ro ~ O(1) despite relying on a small Ro approximation31. In particular, this framework has been successfully used to study thermal convection in the limit of rapid rotation21, reminiscent of the observations presented here. In the QG framework, potential vorticity (PV) is conserved along a Lagrangian trajectory and the PV anomaly is given by

$${\mathrm{PV}}={{\Delta }}\psi +\frac{{f}^{2}}{{N}^{2}}\frac{{\partial }^{2}\psi }{\partial {z}^{2}},$$
(4)

with ψ the stream function, Δ the horizontal Laplacian operator and f and N assumed constant. The relative vorticity ζ is given by Δψ. From PV conservation, ζ is related to the stretching term, $$\frac{{f}^{2}}{{N}^{2}}\frac{{\partial }^{2}\psi }{\partial {z}^{2}}$$, via the horizontal divergence as31,56

$$\frac{{\mathrm{d}}\zeta }{{\mathrm{d}}t}=-f\chi \ ,$$
(5)
$$\frac{{f}^{2}}{{N}^{2}}\frac{d}{dt}\frac{{\partial }^{2}\psi }{\partial {z}^{2}}=f\chi$$
(6)

with χ the horizontal divergence.

### Dynamics in the k−4/3 scale range

In the k−4/3 scale range, KE and APE spectra are equal, which is equivalent to PV = 0 (refs. 14,18). This configuration corresponds to a subset category of the QG framework, referred to as surface quasi-geostrophy (SQG)14,15,16,18 and initially used in ref. 15 to study the dynamics at the Earth’s tropopause. Using equations (2) and (3), the stream function at the tropopause (where z = 0) can be written as

$$\frac{\partial {\psi }_{\mathrm{T}}}{\partial z}=\frac{{N}^{2}}{f}h=\frac{{N}^{2}}{f}{H}_{0}\ \tau ,$$
(7)

with subscript ‘T’ denoting the tropopause. Integration of PV = 0 (equation (4), valid in the troposphere) using the boundary condition at the tropopause given by equation (7) and assuming that ψ vanishes at z = −  leads to the spectral solution

$${\widehat{\psi }}\left(k,z\right)={\widehat{\psi }}_{\mathrm{T}}\exp \left(k\frac{N}{f}z\right),$$
(8)
$$\frac{\partial \widehat{\psi }}{\partial z}=\frac{\partial {\widehat{\psi }}_{\mathrm{T}}}{\partial z}\exp (k\frac{N}{f}z).$$
(9)

From PV = 0 and equations (7), (8) and (9), the relative vorticity at the tropopause can now be linked to the optical depth anomaly τ as

$$\widehat{{\zeta }_{\tau }}+N\, {H}_{0}\, \widehat{\tau }\, k=0$$
(10)

The SQG framework also allows to infer the aspect ratio between horizontal and vertical scales. Indeed, equation (8) leads to an e-folding vertical scale given by $$k\frac{N}{f}H \sim 1$$, with H the depth scale. This indicates that the aspect ratio L/H of a structure is close to N/f ~ 10. In other words, the largest vortices in the k−4/3 scale range (~1,600 km) have a depth extension of about 160 km, further highlighting their shallow depth extension and therefore their 3D character.

The value of 1,600 km can be interpreted as an upper bound for the size of the baroclinic eddies, which positively depends on the convection’s strength19,21,28. Indeed, a stronger convection tends to increase the baroclinic spectrum’s amplitude (k−4/3 slope) but does not change the barotropic spectrum’s amplitude (k−3 slope) for the small scales21,28.

### Dynamics in the k−3 scale range

On the one hand, APE is associated with the stretching term in the PV expression and is therefore related to the depth dependence of the stream function, or what is called the baroclinic mode57. On the other hand, KE is associated with the relative vorticity in the PV expression57. The exchanges between both energies are driven by the divergence (equations (5) and (6)). The APE spectrum in the k−3 scale range has a much smaller variance than KE (Fig. 3a), and in addition, the divergence is small (Fig. 3b) in this range. These two characteristics indicate that the depth-dependent contribution to (or the baroclinic part of) the total stream function is small and therefore that the stream function is dominated by the depth-independent part (or the barotropic part)58. These arguments imply that vortices are 2D (depth independent) in this scale range.

### KE and enstrophy transfer

We diagnose the KE and enstrophy (ENS) transfer between wavenumbers in spectral space using the momentum equations at the tropopause (where vertical velocities are null). Multiplying these equations by the conjugate of the horizontal wind speed and without considering dissipation for simplicity’s sake leads to38,46,59

$$\frac{1}{2}\frac{\partial | \widehat{{{{{\bf{u}}}}}_{\mathrm{h}}}{| }^{2}}{\partial t}=-{{{\rm{Re}}}}\left({\widehat{{{{{\bf{u}}}}}_{\mathrm{h}}}}^{* }\cdot (\widehat{{{{{\bf{u}}}}}_{\mathrm{h}}\cdot {\nabla }_{\mathrm{h}}{{{{\bf{u}}}}}_{\mathrm{h}}})\right)-\frac{1}{{\rho }_{0}}{{{\rm{Re}}}}\left({\widehat{{{{{\bf{u}}}}}_{\mathrm{h}}}}^{* }\cdot {\nabla }_{\mathrm{h}}p\right)\ ,$$
(11)
$$\frac{1}{2}\frac{\partial | \widehat{\zeta }{| }^{2}}{\partial t}=-{{{\rm{Re}}}}\left({\widehat{\zeta }}^{* }\cdot (\widehat{{{{{\bf{u}}}}}_{\mathrm{h}}\cdot {\nabla }_{\mathrm{h}}\zeta })\right)-{{{\rm{Re}}}}\left(f{\widehat{\zeta }}^{* }\cdot \widehat{\chi }\right),$$
(12)

with asterisk indicates the complex conjugate and $${{{\rm{Re}}}}(\cdot)$$ the real part. Equations (11) and (12) are the equations for the time evolution for a given wavenumber k of the KE and ENS, respectively. The first terms on the right-hand side of equations (11) and (12) are nonlinear advection terms, whereas the seconds terms are sources and/or sinks.

$${{{{\rm{KE}}}}}_{{{{\rm{adv}}}}}(k)=-{{{\rm{Re}}}}\left({\widehat{{{{{\bf{u}}}}}_{\mathrm{h}}}}^{* }\cdot (\widehat{{{{{\bf{u}}}}}_{\mathrm{h}}\cdot {\nabla }_{\mathrm{h}}{{{{\bf{u}}}}}_{\mathrm{h}}})\right)\ ,$$
(13)
$${{{{\rm{ENS}}}}}_{{{{\rm{adv}}}}}(k)=-{{{\rm{Re}}}}\left({\widehat{\zeta }}^{* }\cdot (\widehat{{{{{\bf{u}}}}}_{\mathrm{h}}\cdot {\nabla }_{\mathrm{h}}\zeta })\right)\ \ .$$
(14)

More precisely, the terms in equations (13) and (14) represent the KE and ENS, respectively, gained (lost) by a wavenumber k from (to) other wavenumbers. Figure 4a, from which the upscale KE transfer is inferred, shows equation (13) in a variance-preserving form (that is, multiplied by k).

Figure 4b shows the enstrophy spectral flux estimated as

$${{{\Pi }}}_{{{{\rm{ENS}}}}}(k)=\int\nolimits_{k}^{{k}_{\mathrm{s}}}{{{{\rm{ENS}}}}}_{{{{\rm{adv}}}}}(k)\ {\mathrm{d}}k,$$
(15)

with k the isotropic wavenumber defined previously.

### Planetary Burger number

Here, we explore how our results compare with previous modelling studies8,9,10,11, which computed a planetary Burger number Bu = $${L}_{\mathrm{d}}^{2}/2{a}^{2}$$, with Ld an internal Rossby radius of deformation and a the planetary radius. Using an upper limit value of Ld = 1,600 km (that is, the size of the largest baroclinic eddies) yields Bu = 2.6 × 10−4, corresponding to a regime with multiple circumpolar cyclones9,10 and hence comparing favourably with refs 8,9,10,11.