## New Research In

### Physical Sciences

### Social Sciences

#### Featured Portals

#### Articles by Topic

### Biological Sciences

#### Featured Portals

#### Articles by Topic

- Agricultural Sciences
- Anthropology
- Applied Biological Sciences
- Biochemistry
- Biophysics and Computational Biology
- Cell Biology
- Developmental Biology
- Ecology
- Environmental Sciences
- Evolution
- Genetics
- Immunology and Inflammation
- Medical Sciences
- Microbiology
- Neuroscience
- Pharmacology
- Physiology
- Plant Biology
- Population Biology
- Psychological and Cognitive Sciences
- Sustainability Science
- Systems Biology

# Origin of Jupiter’s cloud-level zonal winds remains a puzzle even after Juno

Contributed by Gerald Schubert, June 18, 2018 (sent for review April 6, 2018; reviewed by Andrew Jackson and Johannes Wicht)

## Significance

How the Jovian cloud-level zonal winds are generated and maintained has been a major scientific puzzle for decades. There are two main contenders to explain the origin of the winds: (*i*) They are maintained and generated by deep thermal convection and extend deep into Jupiter’s interior and (*ii*) they are associated with horizontal temperature differences between belts and zones and confined to a very thin stably stratified weather layer below which there exists an unknown convective circulation. We show that the Juno gravitational measurements alone cannot discriminate between these two different scenarios. The origin of the winds is still a mystery.

## Abstract

How far Jupiter’s cloud-level zonal winds penetrate into its interior, a question related to the origin of the winds, has long been a major puzzle about Jupiter. There exist two different views: the shallow scenario in which the cloud-level winds are confined within the thin weather layer at cloud top and the deep scenario in which the cloud-level winds manifest thermal convection in the deep interior. We interpret, using two different models corresponding to the two scenarios, the high-precision measurements of Jupiter’s equatorially antisymmetric gravitational field by the Juno spacecraft. We demonstrate, based on the thermal-gravitational wind equation, that both the shallow and deep cloud-level winds models are capable of explaining the measured odd gravitational coefficients within the measured uncertainties, reflecting the nonunique nature of the gravity inverse problem. We conclude that the high-precision Juno gravity measurements cannot provide an answer to the long-standing question about the origin of Jupiter’s cloud-level zonal winds.

Even though alternating cloud-level zonal winds on Jupiter have been accurately measured for several decades (1⇓–3), their depth of penetration into its interior, a question closely linked with the origin of the winds, is still uncertain. Since an internal zonal flow with sufficiently large amplitude can generate an externally measurable gravitational signature by inducing substantial density anomalies (4⇓–6), it was hoped that the high-precision measurements of Jupiter’s gravitational field by the Juno spacecraft (7⇓–9) would provide an opportunity to answer this long-standing question (4, 10⇓–12).

There are two profoundly different views on the origin of the cloud-level zonal winds of Jupiter, suggesting two different models for the interpretation of the gravitational coefficients provided by the Juno spacecraft (9). One view is that the cloud-level zonal winds are shallow and confined within a thin, stably stratified weather layer about 70 km thick at cloud top in which the winds are associated with horizontal temperature differences between belts and zones (13, 14) and there exists a thick convection layer (15⇓–17) beneath the thin weather layer. In this scenario, the fast cloud-level winds do not penetrate into the deep interior and the thin weather layer contains less than

The equatorially symmetric and antisymmetric components of Jupiter’s gravitational field contain different information about its interior. The equatorially symmetric gravitational field, represented by even gravitational coefficients

Following the two profoundly different views discussed above, we construct two different models, shallow and deep, for interpreting the four nonzero odd gravitational coefficients. The shallow cloud-level winds model assumes that the cloud-level winds are confined in the thin weather layer and, hence, its contribution to the measured gravitational signal is negligible. We then determine an unknown zonal flow in the underlying convection region constrained by the measured equatorially antisymmetric gravitational field without making a priori assumptions about the nature and structure of the flow. The deep cloud-level winds model, according to the deep convection scenario, assumes that the cloud-level winds structure extends into Jupiter’s interior and, hence, is responsible for the measured gravitational signal. This assumption allows us to construct a parameterized zonal flow whose cloud-level and internal structure/amplitude is constrained by the cloud-level profile.

The problem of determining fluid flow in the interior of a planet from its externally measured physical fields, such as its magnetic and gravitational fields, is characteristically nonunique. A well-known example is the determination of the fluid flow at the top of the Earth’s core from the measured external geomagnetic field (27⇓–29). The core flow inferred from the geomagnetic field is necessarily nonunique, although the ambiguity can be reduced by placing various restrictions, such as the frozen-flux hypothesis and the tangentially geostrophic approximation, on the permitted flow (28, 29). In this study, we demonstrate that fluid flow inferred from the externally measured gravitational field of Jupiter is also necessarily nonunique: Both the shallow and deep zonal winds models are able to fully interpret the odd gravitational coefficients

## Governing Equations

We assume that the equatorially antisymmetric zonal flow of Jupiter is characterized by small Rossby number, viscous forces are much smaller than the Coriolis forces, and its compressible fluid is described by the polytropic equation of state with index unity (4, 6, 30, 31). We also assume that Jupiter is uniformly rotating about the symmetry z axis with the angular velocity **3**, are generally of the same order of magnitude.

A further assumption is that the electric current J is sufficiently weak such that the Lorentz force **3** then leads to the thermal-gravitational wind equation (32) describing a mathematical relationship between the equatorially antisymmetric zonal flow **4** should be highlighted. First, the two terms on the left side of Eq. **4** are generally comparable in size and, hence, the integral term cannot be neglected. Second, the zonal flow **4** must satisfy the required solvability condition: Its solution exists if and only if its inhomogeneous term (the term on its right side) satisfies the solvability condition (11, 33). As in many physical problems governed by inhomogeneous differential or integral equations, it is the solvability condition that helps to select mathematically acceptable and physically relevant solutions.

It should be pointed out that the results of Kaspi (10) and Kaspi et al. (12) are based on the thermal wind equation,**4** and, hence, represents a diagnostic relation. A “solution” **5** always exists for any given **4** and **5**, which will be discussed further.

## Models and Methods

Our shallow model assumes that the cloud-level zonal winds are confined in the thin weather layer and, because it contains too little mass, cannot produce the measured gravitational signal. Since the nature of the zonal flow in the underlying convection layer that produces the gravitational signal is unknown (17), we cannot make a priori assumptions about the amplitude and structure of the flow. We thus expand, following the theory of spherical inertial eigenfunctions in rotating spheres—whose general explicit analytical expressions are available (34) and which are mathematically complete (35)—an equatorially antisymmetric zonal flow **7** has k real distinct positive eigenvalues within **6** can be used, because of the mathematical completeness of spherical inertial eigenfunctions, to represent an arbitrary equatorially antisymmetric zonal flow that is continuous and differentiable. We truncate the expansion Eq. **6** at **6** satisfies the solvability condition required for the inhomogeneous integral Eq. **4**.

With the flow **6**, we then solve the integral Eq. **4** numerically, using an extended spectral method together with a special set of **4**, between an equatorially antisymmetric zonal flow **4** marked by a 2D kernel integral in the form of Green’s function. The zonal flow inferred from the four odd gravitational coefficients in this way is necessarily nonunique. But the ambiguity is largely removed by a further restriction that the Lorentz force

Our second model follows the deep convection scenario by constructing a highly constrained zonal flow. The structure and amplitude of the flow at cloud level **4**, between the constructed flow given by Eq. **8** and the four odd gravitational coefficients (9) to determine the optimized values of H, h, and

It should be noted that the zonal flow **6** or Eq. **8**, is continuous and differentiable everywhere and satisfies the required solvability condition. Consequently, solutions of Eq. **4** exist and are numerically convergent. By contrast, the constructed zonal flow used by Kaspi et al. (12) (their equations **12**–**14**) is discontinuous across the equatorial plane and violates the required solvability condition. When their constructed flow is used in solving Eq. **4**, solutions of Eq. **4** would be numerically divergent, reflecting a key mathematical property of the inhomogeneous integral Eq. **4**. The nonphysical effect of the equatorial discontinuity on solutions of the thermal wind Eq. **5** is discussed in detail in Kong et al. (36).

## Results

The physical and mathematical principles of the problem are well understood (4, 11). If an equatorially antisymmetric zonal flow **4**. After obtaining **4** with a given **4** (connecting the antisymmetric flow

We first discuss the results of the shallow cloud-level winds model, which are computationally much more challenging. Through an iterative procedure, via the thermal-gravitational wind Eq. **4**, between the expansion coefficients and the measured odd gravitational coefficients, we are able to derive an antisymmetric zonal flow that not only produces the measured odd coefficients *A*, decreases rapidly from the outer surface and, then, extends slowly with a small amplitude to a depth of about

We now discuss the results of the deep cloud-level winds model, which, because it is highly constrained, is computationally much less challenging. For given values of H, h, and **4**, between the constructed flow given by Eq. **8** and the four odd gravitational coefficients (9) to determine the optimized values of H, h, and **8** with *B*, showing that

Finally, we mention two significant points. First, while both our shallow and deep cloud-level winds models are able to produce the odd coefficients to within the stated uncertainties (9), we cannot rule out the existence of other models that are also able to fit the odd coefficients to within the uncertainties. Second, the reported uncertainties (9) are already an inflation by a factor of 3 of the formal uncertainties, indicating that ample freedom has already been given for a model to fit the data.

## Conclusions and Remarks

A great mystery of our solar system is how deeply Jupiter’s cloud-level zonal winds penetrate into its interior, a question closely linked with the origin of the winds. We have constructed two profoundly different models, the shallow cloud-level winds model or the

Our conclusion is based on solutions of the thermal-gravitational wind Eq. **4** in which the two terms on its left side are generally comparable in size. For example, using the flow shown in Fig. 1 which satisfies the required solvability condition, Eq. **4** gives **4** [i.e., the thermal wind Eq. **5** used by Kaspi et al. (12)] yields **4** even though its solution is computationally much more difficult and demanding. Some special profile **4** is small compared with the first term and, thus, negligible. In this special case, however, the integral term can be neglected only a posteriori. There are no physical or mathematical reasons to justify the a priori neglect of the integral term in Eq. **4**. We are also unable to obtain any numerically convergent solution of Eq. **4** when the constructed flow used by Kaspi et al. (12) is adopted, a consequence of the violation of the solvability condition by their constructed flow.

Since the rotational distortion of Jupiter, because of its equatorial symmetry, does not contribute to the odd coefficients, the four odd coefficients measured by the Juno spacecraft directly reflect, although nonuniquely, the structure and amplitude of the equatorially antisymmetric zonal flow taking place in the interior of Jupiter. Since the equatorially antisymmetric zonal flow is usually produced by the instabilities of an equatorially symmetric zonal flow (25, 26, 39), the structure and amplitude of the equatorially symmetric zonal flow in the Jovian interior are closely coupled with the equatorially antisymmetric flow.

What do we learn about the Jovian interior from our two different models that are constrained by the equatorially antisymmetric gravitational field measured by the Juno spacecraft? Our shallow cloud-level winds model paints the following picture. In the outermost stably stratified weather layer about 70 km thick, there are fast zonal winds of

## Acknowledgments

We thank Dr. S. Thomson for helpful discussions about the

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: schubert{at}ucla.edu or K.Zhang{at}exeter.ac.uk. ↵

^{2}Retired.

Author contributions: D.K., K.Z., G.S., and J.D.A. designed research; D.K., K.Z., G.S., and J.D.A. performed research; G.S. analyzed data; K.Z. and G.S. wrote the paper; and D.K. ran the numerical model.

Reviewers: A.J., Institut fur Geophysik; and J.W., Max-Planck Institute for Solar System Research.

The authors declare no conflict of interest.

Published under the PNAS license.

## References

- ↵
- Limaye SS

- ↵
- ↵
- Porco CC, et al.

- ↵
- ↵
- ↵
- Kong D,
- Liao X,
- Zhang K,
- Schubert G

- ↵
- Bolton SJ, et al.

- ↵
- Folkner WM, et al.

- ↵
- ↵
- ↵
- Kong D,
- Zhang K,
- Schubert G

- ↵
- ↵
- ↵
- Showman AP

- ↵
- ↵
- Warneford E,
- Dellar PJ

- ↵
- ↵
- Adriani A, et al.

- ↵
- ↵
- Iacono R,
- Struglia MV,
- Ronchi C

- ↵
- Scott RK,
- Polvani LM

- ↵
- O’Neill ME,
- Emanuel KA,
- Flierl GR

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- Gubbins D

- ↵
- ↵
- ↵
- ↵
- ↵
- Kong D,
- Zhang K,
- Schubert G,
- Anderson J

- ↵
- Zhang K,
- Earnshaw P,
- Liao X,
- Busse FH

- ↵
- Ivers DJ,
- Jackson A,
- Winch D

- ↵
- ↵
- Duarte L,
- Gastine T,
- Wicht J

- ↵
- Jones CA

- ↵
- Zhang K,
- Schubert G

- ↵

## Citation Manager Formats

## Sign up for Article Alerts

## Article Classifications

- Physical Sciences
- Applied Mathematics