Dali Kong

THERMAL-GRAVITATIONAL WIND EQUATION FOR THE WIND-INDUCED GRAVITATIONAL SIGNATURE OF GIANT GASEOUS PLANETS: MATHEMATICAL DERIVATION, NUMERICAL METHOD, AND ILLUSTRATIVE SOLUTIONS

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 Greens 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.

ON THE VARIATION OF ZONAL GRAVITY COEFFICIENTS OF A GIANT PLANET CAUSED BY ITS DEEP ZONAL FLOWS

Rapidly rotating giant planets are usually marked by the existence of strong zonal flows at the cloud level. If the zonal flow is sufficiently deep and strong, it can produce hydrostatic-related gravitational anomalies through distortion of the planet’s shape. This paper determines the zonal gravity coefficients, J2n ,n = 1,2,3 ,..., via an analytical method taking into account rotation-induced shape changes by assuming that a planet has an effective uniform density and that the zonal flows arise from deep convection and extend along cylinders parallel to the rotation axis. Two different but related hydrostatic models are considered. When a giant planet is in rigid-body rotation, the exact solution of the problem using oblate spheroidal coordinates is derived, allowing us to compute the value of its zonal gravity coefficients ¯ J2n ,n = 1,2,3 ,... , without making any approximation. When the deep zonal flow is sufficiently strong, we develop a general perturbation theory for estimating the variation of the zonal gravity coefficients, ΔJ2n = J2n − ¯ J2n ,n = 1,2,3 ,... , caused by the effect of the deep zonalflows for an arbitrarily rapidly rotating planet. Applying the general theory to Jupiter, we find that the deep zonal flow could contribute up to 0.3% of the J2 coefficient and 0.7% of J4. It is also found that the shape-driven harmonics at the 10th zonal gravity coefficient become dominant, i.e., ΔJ2n ¯ J2n for n5.

Swimming motion of rod-shaped magnetotactic bacteria: the effects of shape and growing magnetic moment.

We investigate the swimming motion of rod-shaped magnetotactic bacteria affiliated with the Nitrospirae phylum in a viscous liquid under the influence of an externally imposed, time-dependent magnetic field. By assuming that fluid motion driven by the translation and rotation of a swimming bacterium is of the Stokes type and that inertial effects of the motion are negligible, we derive a new system of the twelve coupled equations that govern both the motion and orientation of a swimming rod-shaped magnetotactic bacterium with a growing magnetic moment in the laboratory frame of reference. It is revealed that the initial pattern of swimming motion can be strongly affected by the rate of the growing magnetic moment. It is also revealed, through comparing mathematical solutions of the twelve coupled equations to the swimming motion observed in our laboratory experiments with rod-shaped magnetotactic bacteria, that the laboratory trajectories of the swimming motion can be approximately reproduced using an appropriate set of the parameters in our theoretical model.

A THREE-DIMENSIONAL NUMERICAL SOLUTION FOR THE SHAPE OF A ROTATIONALLY DISTORTED POLYTROPE OF INDEX UNITY

We present a new three-dimensional numerical method for calculating the non-spherical shape and internal structure of a model of a rapidly rotating gaseous body with a polytropic index of unity. The calculation is based on a finite-element method and accounts for the full effects of rotation. After validating the numerical approach against the asymptotic solution of Chandrasekhar that is valid only for a slowly rotating gaseous body, we apply it to models of Jupiter and a rapidly rotating, highly flattened star (α Eridani). In the case of Jupiter, the two-dimensional distributions of density and pressure are determined via a hybrid inverse approach by adjusting an a priori unknown coefficient in the equation of state until the model shape matches the observed shape of Jupiter. After obtaining the two-dimensional distribution of density, we then compute the zonal gravity coefficients and the total mass from the non-spherical model that takes full account of rotation-induced shape change. Our non-spherical model with a polytropic index of unity is able to produce the known mass of Jupiter with about 4% accuracy and the zonal gravitational coefficient J 2 of Jupiter with better than 2% accuracy, a reasonable result considering that there is only one parameter in the model. For α Eridani, we calculate its rotationally distorted shape and internal structure based on the observationally deduced rotation rate and size of the star by using a similar hybrid inverse approach. Our model of the star closely approximates the observed flattening.

Equatorial Zonal Jets and Jupiter’s Gravity

The depth of penetration of Jupiters zonal winds into the planets interior is unknown. A possible way to determine the depth is to measure the effects of the winds on the planets high-order zonal gravitational coefficients, a task to be undertaken by the Juno spacecraft. It is shown here that the equatorial winds alone largely determine these coefficients which are nearly independent of the depth of the non-equatorial winds.

ON THE GRAVITATIONAL FIELDS OF MACLAURIN SPHEROID MODELS OF ROTATING FLUID PLANETS

Hubbard recently derived an important iterative equation for calculating the gravitational coefficients of a Maclaurin spheroid that does not require an expansion in a small distortion parameter. We show that this iterative equation, which is based on an incomplete solution of the Poisson equation, diverges when the distortion parameter is not sufficiently small. We derive a new iterative equation that is based on a complete solution of the Poisson equation and, hence, always converges when calculating the gravitational coefficients of a Maclaurin spheroid.

A FULLY SELF-CONSISTENT MULTI-LAYERED MODEL OF JUPITER

We construct a three-dimensional, fully self-consistent, multi-layered, non-spheroidal model of Jupiter consisting of an inner core, a metallic electrically conducting dynamo region, and an outer molecular electrically insulating envelope. We assume that the Jovian zonal winds are on cylinders parallel to the rotation axis but, due to the effect of magnetic braking, are confined within the outer molecular envelope. We also assume that the location of the molecular-metallic interface is characterized by its equatorial radius , where R e is the equatorial radius of Jupiter at the 1 bar pressure level and H is treated as a parameter of the model. We solve the relevant mathematical problem via a perturbation approach. The leading-order problem determines the density, size, and shape of the inner core, the irregular shape of the 1 bar pressure level, and the internal structure of Jupiter that accounts for the full effect of rotational distortion, but without the influence of the zonal winds; the next-order problem determines the variation of the gravitational field solely caused by the effect of the zonal winds on the rotationally distorted non-spheroidal Jupiter. The leading-order solution produces the known mass, the known equatorial and polar radii, and the known zonal gravitational coefficient J 2 of Jupiter within their error bars; it also yields the coefficients J 4 and J 6 within about 5% accuracy, the core equatorial radius and the core density corresponding to 3.73 Earth masses; the next-order solution yields the wind-induced variation of the zonal gravitational coefficients of Jupiter.

The sidewall-localized mode in a resonant precessing cylinder

We investigate, via direct numerical simulation using a finite-element method, the precessionally driven flow of a homogeneous fluid confined in a fluid-filled circular cylinder that rotates rapidly about its symmetry axis and precesses about a different axis that is fixed in space. Our numerical simulation, after validating with the asymptotic analytical solution for a weakly precessing cylinder and with the constructed exact solution for the strongly nonlinear problem, focuses on the strongly precessing flow at asymptotically small Ekman numbers. An unusual form of the resonant precessing flow is found when the precessing rate is sufficiently large and the corresponding nonlinearity is sufficiently strong. The nonlinear precessing flow is marked by a sidewall-localized non-axisymmetric traveling wave and a wall-localized axisymmetric shear together with an overwhelmingly dominant interior rigid-body rotation whose direction and magnitude substantially reduce the angular momentum of the rotating fluid sy...

On the transition from the laminar to disordered flow in a precessing spherical-like cylinder

We investigate, through both asymptotic analysis and direct numerical simulation, precessionally driven flow of a homogeneous fluid confined in a fluid-filled circular cylinder that rotates rapidly about its symmetry axis and precesses about a different axis that is fixed in space. A particular emphasis is placed on a spherical-like cylinder whose diameter is nearly the same as its length. At this special aspect ratio, the strongest direct resonance occurs between the spatially simplest inertial mode and the precessional Poincaré forcing. An asymptotic analytical solution in closed form describing weakly precessing flow is derived in the mantle frame of reference for asymptotically small Ekman numbers. We also construct a nonlinear three-dimensional finite element model – which is validated against both the asymptotic solution and a constructed exact solution – for elucidating the nonlinear transition leading to disordered flow in the precessing spherical-like cylinder. Properties of both weakly and strongly precessing flows are investigated with the aid of a complete inertial-mode decomposition of the fully nonlinear solution. Despite a large effort being made, the well-known triadic resonance is not found in the precessing spherical-like cylinder. The energy contained in the precessionally forced inertial mode is primarily transferred, through nonlinear effects in the viscous boundary layers, to the geostrophic flow that becomes predominant when the precessional Poincaré force is sufficiently large. It is found that the nonlinear flow evolutes gradually and progressively from the laminar to disordered as the precessional force increases.

On fluid flows in precessing narrow annular channels: asymptotic analysis and numerical simulation

We consider a viscous, incompressible fluid confined in a narrow annular channel rotating rapidly about its axis of symmetry with angular velocity Omega that itself precesses slowly about an axis fixed in an inertial frame. The precessional problem is characterized by three parameters: the Ekman number E, the Poincare number E and the aspect ratio of the channel Gamma. Dependent upon the size of Gamma, precessionally driven flows can be either resonant or non-resonant with the Poincare forcing. By assuming that it is the viscous effect, rather than the nonlinear effect, that plays an essential role at exact resonance, two asymptotic expressions for epsilon << 1 and E << 1 describing the single and double inertial-mode resonance are derived under the nonslip boundary condition. An asymptotic expression describing non-resonant precessing flows is also derived. Further studies based on numerical integrations, including two-dimensional linear analysis and direct three-dimensional nonlinear simulation, show a satisfactory quantitative agreement between the three asymptotic expressions and the fuller numerics for small and moderate Reynolds numbers at an asymptotically small E. The transition from two-dimensional precessing flow to three-dimensional small-scale turbulence for large Reynolds numbers is also investigated.