Xinhao Liao

On inertial waves in a rotating fluid sphere

Several new results are obtained for the classical problem of inertial waves in a rotating fluid sphere which was formulated by Poincare more than a century ago. Explicit general analytical expressions for solutions of the problem are found in a rotating sphere for the first time. It is also discovered that there exists a special class of three-dimensional inertial waves that are nearly geostrophic and always travel slowly in the prograde direction. On the basis of the explicit general expression we are able to show that the internal viscous dissipation of all the inertial waves vanishes identically for a rotating fluid sphere. The result contrasts with the finite values obtained for the internal viscous dissipation for all other cases in which inertial waves have been studied.

On inertial waves and oscillations in a rapidly rotating spheroid

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 Poincare (1885) and Bryan (1889). Upon realizing the limitation of Bryans 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 less than or equal to N less than or equal to 2, an explicit solution can be derived by using Kudlicks procedure. When 3 less than or equal to N less than or equal to 4, the procedure cannot be practically used because the analytical expression for the N distinct roots becomes too complicated. When N > 4, Kudlicks 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 less than or equal to N less than or equal to 2 with the azimuthal wavenumbers 1 less than or equal to m less than or equal to 2. We have found the first explicit general analytical solution of this classical problem valid for 0 less than or equal to N < infinity and O less than or equal to m < infinity. 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

A Three-dimensional Spherical Nonlinear Interface Dynamo

A fully three-dimensional, nonlinear, time-dependent spherical interface dynamo is investigated using a finite-element method based on the three-dimensional tetrahedralization of the spherical system. The spherical interface dynamo model consists of four zones: an electrically conducting and uniformly rotating core, a thin differentially rotating tachocline, a uniformly rotating turbulent convection envelope, and a nearly insulating exterior. The four regions are coupled magnetically through matching conditions at the interfaces. Without the effect of a tachocline, the conventional nonlinear α2 dynamo is always stationary, axisymmetric, and equatorially antisymmetric even though numerical simulations are always fully three-dimensional and time dependent. When there is no tachocline, the azimuthal field is confined to the convection zone while the poloidal magnetic field penetrates into the radiative core. The effects of an interface dynamo with a tachocline having a purely axisymmetric toroidal velocity field are as follows: (1) the action of the steady tachocline always gives rise to an oscillatory dynamo with a period of about 2 magnetic diffusion units, or about 20 yr if the magnetic diffusivity in the convection zone is 108 m2 s-1; (2) the interface dynamo solution is always axisymmetric, selects dipolar symmetry, and propagates equatorward (for the assumed form of α) although the simulation is fully three-dimensional; (3) the generated magnetic field mainly concentrates in the vicinity of the interface between the tachocline and the convection zone; and (4) the strength of the toroidal magnetic field is dramatically amplified by the effect of the tachocline. Extensions of Cowlings theorem and the toroidal flow theorem to multilayer spherical shell regions with radially discontinuous magnetic diffusivities are presented.

A new asymptotic method for the analysis of convection in a rapidly rotating sphere

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 much less than 1 and Pr/E much greater than 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)partial derivative/partial derivativephi = O (E-1/3), partial derivative/partial derivativez = O(1), R-c = O(E-1/3), 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 much less than 1 and Pr/E much greater than 1, indicating the radial asymptotic scaling partial derivative/partial derivatives = O (E-1/3), was recently developed by Jones et al. (J. Fluid Mech. vol. 405, 2000, p. 157). In the different double limit E much less than 1 and Pr/E much less than 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 partial derivative/partial derivatives similar to (1/s)partial derivative/partial derivativephi = O(1), partial derivative/partial derivativez = O(1) and R-c = O(E) for stress-free spheres.

Asymptotic solutions of convection in rapidly rotating non-slip spheres

Asymptotic solutions describing the onset of convection in rotating, self-gravitating Boussinesq fluid spheres with no-slip boundary conditions, valid for asymptotically small Ekman numbers and for all values of the Prandtl number, are derived. Central to the asymptotic analysis is the assumption that the leading-order convection can be represented, dependent on the size of the Prandtl number, by either a single quasi-geostrophic-inertial-wave mode or by a combination of several quasi-geostrophicinertial-wave modes, and is controlled or influenced by the effect of the oscillatory Ekman boundary layer. Comparisons between the asymptotic solutions and the corresponding fully numerical simulations show a satisfactory quantitative agreement.

On fluid motion in librating ellipsoids with moderate equatorial eccentricity

The motion of a homogeneous fluid of viscosity nu confined in a librating ellipsoidal cavity with semi-axes a and moderate equatorial eccentricity E is investigated. The ellipsoid rotates with an angular velocity Omega(0)(1 + d delta sin (omega) over capt), where Omega(0) is the mean rate of rotation, (omega) over cap is the libration frequency and Omega(0)delta represents the amplitude of longitudinal libration. When delta << E(2) and E(1/2) << E(2) << 1, where E is the Ekman number defined as E = nu/(a(2)Omega(0)), an explicit analytical solution describing fluid motion in librating ellipsoids is derived for any size of the libration frequency (omega) over cap. Three-dimensional numerical simulations of the same problem are also performed, revealing the generation of mean zonal flow in librating ellipsoidal cavities and showing a satisfactory agreement between the asymptotic and numerical analyses.

Nonaxisymmetric Instabilities of a Toroidal Magnetic Field in a Rotating Sphere

It has been suggested that nonaxisymmetric instabilities of an axisymmetric toroidal magnetic field, independent of the existence of convective turbulence, provide an important dynamo effect for stars when the convective turbulence is largely suppressed by the magnetic field. We investigate analytically such three-dimensional magnetic instabilities arising from a purely toroidal magnetic field whose strength is proportional to distance from the rotation axis of a star. A three-dimensional perturbation approach is used in our stability analysis. The leading-order problem neglects dissipative effects and describes two- or three-dimensional hydromagnetic waves in spherical geometry. For the first time we are able to obtain general explicit solutions for the problem. Interesting properties of the hydromagnetic waves are revealed by the explicit solutions. We investigate the effect of ohmic dissipation on the stability of the toroidal magnetic field by first deriving the magnetic boundary layer solution and then matching it onto the interior leading-order solution. Furthermore, for the first time we are able to derive an explicit analytic expression for the growth rate of the magnetic instabilities. We find that the most unstable mode of the instabilities is always three-dimensional and characterized by small radial and latitudinal scales. Implications of these magnetic instabilities for the solar dynamo are discussed.

Pacific warm pool excitation, earth rotation and El Niño southern oscillations

The interannual changes in the Earths rotation rate, and hence in the length of day (LOD), are thought to be caused by the variation of the atmospheric angular momentum (AAM). However, there is still a considerable portion of the LOD variations that remain unexplained. Through analyzing the non-atmospheric LOD excitation contributed by the Western Pacific Warm Pool (WPWP) during the period of 1970-2000, the positive effects of the WPWP on the interannual LOD variation are found, although the scale of the warm pool is much smaller than that of the solid Earth. These effects are specifically intensified by the El Nino events, since more components of the LOD-AAM were accounted for by the warm pool excitation in the strong El Nino years. Changes in the Earths rotation rate has attracted significant attention, not only because it is an important geodetic issue but also because it has significant value as a global measure of variations within the hydrosphere, atmosphere, cryosphere and solid Earth, and hence the global changes.

On the completeness of inertial wave modes in rotating annular channels

An important unanswered mathematical question in the theory of rotating fluids has been the completeness of the inviscid eigenfunctions which are usually referred to as inertial waves or inertial modes. We provide for the first time a mathematical proof for the completeness of the inertial modes in a rotating annular channel by establishing the completeness relation, or Parseval’s equality, for any piecewise continuous, differentiable velocity of an incompressible fluid.

On the initial-value problem in a rotating circular cylinder

The initial-value problem in rapidly rotating circular cylinders is revisited. Four different but related analyses are carried out: (i) we derive a modified asymptotic expression for the viscous decay factors valid for the inertial modes of a broad range of frequencies that are required for an asymptotic solution of the initial value problem at an arbitrarily small but fixed Ekman number; (ii) we perform a fully numerical analysis to estimate the viscous decay factors, showing satisfactory quantitative agreement between the modified asymptotic expression and the fuller numerics; (iii) we derive a modified time-dependent asymptotic solution of the initial value problem valid for an arbitrarily small but fixed Ekman number and (iv) we perform fully numerical simulations for the initial value problem at a small Ekman number, showing satisfactory quantitative agreement between the modified time-dependent solution and the numerical simulations.