Explicit Adaptive Symplectic (Easy) Integrators: A Scaling Invariant Generalisation of the Levi-Civitaand KS Regularisations

We present a generalisation of the Levi-Civita and Kustaanheimo-Stiefel regularisation. This allows the use of more general time rescalings. In particular, it is possible to find a regularisation which removes the singularity of the equations and preserves scaling invariance. In addition, these equations can, in certain cases, be integrated with explicit symplectic Runge-Kutta-Nyström methods. The combination of both techniques gives an explicit adaptive symplectic (EASY) integrator. We apply those methods to some perturbations of the Kepler problem and illustrate, by means of some numerical examples, when scaling invariant regularisations are more efficient that the LC/KS regularisation.     

Future Benefits of Time-Varying Gravity Missions to Ocean Circulation Studies

A summary is offered of the potential benefits of future measurements of temporal variations in gravity for the understanding of ocean dynamics. Two types of process, and corresponding amplitudes are discussed: ocean basin scale pressure changes, with a corresponding amplitude of order 1 cm of water, or 1 mm of geoid height, and changes in along-slope pressure gradient, at cross-slope length scales corresponding to topographic slopes, with a corresponding amplitude of order 1 mm of water, or a maximum of about 0.01 mm of geoid. The former is feasible with current technology and would provide unprecedented information about abyssal ocean dynamics associated with heat transport and climate. The latter would be a considerable challenge to any foreseeable technology, but would provide an exceptionally clear, quantitative window on the dynamics of abyssal ocean currents, and strong constraints on ocean models. Both options would be limited by the aliassing effect of rapid mass movements in the earth system, and it is recommended that any future mission take this error source explicitly into account at the design stage. For basin-scale oceanography this might involve a higher orbit than GRACE or GOCE, and the advantages of exact-repeat orbits and multiple missions should be considered.     

Another look at theR-modes

Possible oscillation modes for a rotating star are listed. The only assumption made is that oscillations are adiabatic and that rotation is uniform. It is found that two modes not present for non-rotating stars are possible. Oscillation frequencies of these modes are rather different from those given in the literature for so-calledr-modes.     

The Nyquist frequency for irregularly spaced time-series: a calculation formula

Eyer & Bartholdi showed that the Nyquist frequency of irregularly sampled time-series can be very high. In this paper, a calculation formula for the Nyquist frequency is presented. In practice there is an upper limit of 0.5/Δ on this frequency, where Δ is the best accuracy with which time is recorded.