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.

The problem of fluid motions in the form of inertial waves or inertial oscillations in an incompressible viscous fluid contained in a rotating spheroidal cavity was first formulated and studied by Poincaré (1885) and Bryan (1889). Upon realizing the limitation of Bryan's implicit solution using complicated modified spheroidal coordinates, Kudlick (1966) proposed a procedure that may be used to compute an explicit solution in spheroidal coordinates. However, the procedure requires an analytical expression for the $N$ real and distinct roots of a polynomial of degree $N$, where $N$ is a key parameter in the problem. When $0\,{\le}\,N\,{\le}\,2$, an explicit solution can be derived by using Kudlick's procedure. When $3\,{\le}\,N\,{\le}\,4$, the procedure cannot be practically used because the analytical expression for the $N$ distinct roots becomes too complicated. When $N\,{>}\,4$, Kudlick's procedure cannot be used because of the non-existence of an analytical expression for the $N$ distinct roots. For the inertial wave problem, Kudlick thus restricted his analysis to several modes for $1\,{\le}\,N\,{\le}\,2$ with the azimuthal wavenumbers $1\,{\le}\,m\,{\le}\,2$. We have found the first explicit general analytical solution of this classical problem valid for $0\,{\le}\,N\,{<}\,\infty$ and $0\,{\le}\,m\,{<}\,\infty$. The explicit general solution in spheroidal polar coordinates represents a possibly complete set of the inertial modes in an oblate spheroid of arbitrary eccentricity. The problem is solved by a perturbation analysis. In the first approximation, the effect of viscosity on inertial waves or oscillations is neglected and the corresponding inviscid solution, the pressure and the three velocity components in explicit spheroidal coordinates, is obtained. In the next approximation, the effect of viscous dissipation on the inviscid solution is examined. We have derived the first explicit general solution for the viscous spheroidal boundary layer valid for all inertial modes. The boundary-layer flux provides the solvability condition that is required to solve the higher-order interior problem, leading to an explicit general expression for the viscous correction of all inertial modes in a rapidly rotating, general spheroidal cavity. On the basis of the general explicit solution, some unusual and intriguing properties of the spheroidal inertial waves or oscillation are discovered. In particular, we are able to show that $\int_V ( {\bm u} \,{\bm \cdot}\, \nabla^2 {\bm u} )\,{\rm d}V \equiv 0$, where ${\bm u}$ is the velocity of any three-dimensional inviscid inertial waves or oscillations in an oblate spheroid of arbitrary eccentricity and $\int_V$ denotes three-dimensional integration over the volume of the spheroidal cavity.

Convective instabilities in a fluid-filled circular cylinder heated from below and rotating about its vertical axis are investigated both analytically and numerically under experimental boundary conditions. It is found that there exist two different forms of convective instabilities: convection-driven inertial waves for small and moderate Prandtl numbers and wall-localized travelling waves for large Prandtl numbers. Asymptotic solutions for both forms of convection are derived and numerical simulations for the same problem are also performed, showing a satisfactory quantitative agreement between the asymptotic and numerical analyses.

The standard thermal wind equation (TWE) relating the vertical shear of a flow to the horizontal density gradient in an atmosphere has been used to calculate the external gravitational signature produced by zonal winds in the interiors of giant gaseous planets. We show, however, that in this application the TWE needs to be generalized to account for an associated gravitational perturbation. We refer to the generalized equation as the thermal-gravitational wind equation (TGWE). The generalized equation represents a two-dimensional kernel integral equation with the Green's function in its integrand and is hence much more difficult to solve than the standard TWE. We develop an extended spectral method for solving the TGWE in spherical geometry. We then apply the method to a generic gaseous Jupiter-like object with idealized zonal winds. We demonstrate that solutions of the TGWE are substantially different from those of the standard TWE. We conclude that the TGWE must be used to estimate the gravitational signature of zonal winds in giant gaseous planets.

Thermal convection in rapidly rotating, self-gravitating Boussinesq fluid spheres is characterized by three parameters: the Rayleigh number $R$, the Prandtl number Pr and the Ekman number $E$. Two different asymptotic limits were considered in the previous studies of the linear problem. In the double limit $E \,{\ll}\, 1$ and $\hbox{\it Pr}/E \,{\gg}\, 1$, the local asymptotic theory showed that the convective motion is strongly non-axisymmetric, columnar, highly localized and described by the asymptotic scalings, $({1}/{s}) {\zpt}/{\zpt \phi} \,{=} {O}\big(E^{-1/3}\big),\; {\zpt}/{\zpt z} \,{=}\, {O}(1), \; R_c\,{=}\,{O}\big(E^{-1/3}\big)$, where $R_c$ denotes the critical Rayleigh number and $(s,\phi,z)$ are cylindrical polar coordinates with the axis of rotation at $s\,{=}\,0$. A global asymptotic theory with novel features for the limit $E \,{\ll}\, 1$ and $\hbox{\it Pr}/E \,{\gg}\, 1$, indicating the radial asymptotic scaling ${\zpt}/{\zpt s} \,{=}\, {O}\big(E^{-1/3}\big)$, was recently developed by Jones et al. (J. Fluid Mech. vol. 405, 2000, p. 157). In the different double limit $E \,{\ll}\, 1$ and $ \hbox{\it Pr}/E \,{\ll}\, 1 $, an asymptotic theory for the onset of convection building upon the theory of inertial waves was developed by Zhang (J. Fluid Mech. vol. 268, 1994 p. 211). It was shown that the convective motion at the leading-order approximation is represented by a single inertial-wave mode with a quadratic polynomial of $s$ and $z$, obeying the asymptotic dependence ${\zpt}/{\zpt s} \,{\sim}\, ({1}/{ s}) {\zpt}/{\zpt \phi} \,{=}\, {O}(1), {\zpt}/{\zpt z} \,{=}\, {O}(1)$ and $ R_c\,{=}\,{O}(E)$ for stress-free spheres. There exist no simple asymptotic scalings for $ E \,{\ll}\, 1 $ appropriate to all values of $\hbox{\it Pr}/E$. For an arbitrary small but non-zero $E$, the highly localized convection spreads out spatially with decreasing $\hbox{\it Pr}$, suggesting that the scaling laws such as ${\zpt}/{\zpt s} \,{=}\, {O}\big(E^{-1/3}\big)$ are no longer valid when $\hbox{\it Pr}/E$ is not sufficiently large. This paper represents an attempt to develop a new asymptotic method for the analysis of convection in rapidly rotating spheres valid for asymptotically small $E$ and for $0 \,{\le}\,\hbox{\it Pr}/E \,{<}\, \infty$. The new method is based on the following three hypotheses. The first is that the leading-order velocity of convection for $0 \,{\le}\,\hbox{\it Pr}/E \,{<}\, \infty$ at $E \,{\ll}\, 1$ is represented by either a single quasi-geostrophic-inertial-wave mode or by a combination of several quasi-geostrophic-inertial-wave modes convectively excited and sustained. Secondly, we assume that the convective motion for $0 \,{\le}\,\hbox{\it Pr}/E \,{<}\, \infty$ at $E \,{\ll}\, 1$ always has columnar structure, i.e. ${\zpt}/{\zpt z} \,{\sim}\, {O}(1)$, but without the general asymptotic scalings in the radial and azimuthal direction. Thirdly, we assume that there always exists a boundary flow that is non-zero only in the Ekman boundary layer on the bounding spherical surface and plays an important role even in the case of stress-free boundaries. Comparison between the result of the new method and the corresponding fully numerical simulation demonstrates a satisfactory quantitative agreement for all values of $0 \,{\le}\,\hbox{\it Pr}/E \,{\le}\,{O}(10^6)$ when $ {O} (10^{-5}) \le E \,{\le}\,{O}(10^{-6})$. The new method is asymptotic in the sense that it is valid only for an asymptotically small $E \,{\ll}\, 1$. In addition to the linear problem of thermal convection in rapidly rotating spheres, the corresponding weakly nonlinear problem is also solved to obtain an analytical expression for the convection-driven differential rotation generated by the nonlinear interaction of quasi-geostrophic-inertial-wave modes through the Reynolds stresses. The new method not only reveals the underlying nature of thermal convection in rapidly rotating spheres but also unites the two previously disjointed subjects in rotating fluids: the inertial-wave problem and the convective instability problem.

Mars’ magnetic field has been measured at large scale by orbiting spacecraft and at very small scale via Martian meteorites. Here we report on a ground magnetic survey on metre to kilometre scales. The Zhurong rover made vector measurements at 16 sites along a 1,089 m track in the Utopia Basin on Mars. It recorded an extremely weak magnetic field, with an order of the average intensity less than that inferred from orbit, in contrast to the large magnetic field in Elysium Planitia measured by InSight. A spacecraft measurement samples an area with radius comparable to its altitude, while a ground measurement samples an area with radius comparable to the depth of the magnetized body. The weak magnetic field measured by Zhurong indicates no magnetization anomalies for a depth of many kilometres around and below the rover’s traverse. We suggest two possible explanations for the weak magnetic field: the entire Utopia Basin may have remained unmagnetized since its formation about 4 billion years ago or that the 5-km-radius ghost crater where Zhurong landed may have been been demagnetized by impact.