Stephen D. Griffiths

Accuracy assessment of global barotropic ocean tide models

The accuracy of state-of-the-art global barotropic tide models is assessed using bottom pressure data, coastal tide gauges, satellite altimetry, various geodetic data on Antarctic ice shelves, and independent tracked satellite orbit perturbations. Tide models under review include empirical, purely hydrodynamic (“forward”), and assimilative dynamical, i.e., constrained by observations. Ten dominant tidal constituents in the diurnal, semidiurnal, and quarter-diurnal bands are considered. Since the last major model comparison project in 1997, models have improved markedly, especially in shallow-water regions and also in the deep ocean. The root-sum-square differences between tide observations and the best models for eight major constituents are approximately 0.9, 5.0, and 6.5 cm for pelagic, shelf, and coastal conditions, respectively. Large intermodel discrepancies occur in high latitudes, but testing in those regions is impeded by the paucity of high-quality in situ tide records. Long-wavelength components of models tested by analyzing satellite laser ranging measurements suggest that several models are comparably accurate for use in precise orbit determination, but analyses of GRACE intersatellite ranging data show that all models are still imperfect on basin and subbasin scales, especially near Antarctica. For the M2 constituent, errors in purely hydrodynamic models are now almost comparable to the 1980-era Schwiderski empirical solution, indicating marked advancement in dynamical modeling. Assessing model accuracy using tidal currents remains problematic owing to uncertainties in in situ current meter estimates and the inability to isolate the barotropic mode. Velocity tests against both acoustic tomography and current meters do confirm that assimilative models perform better than purely hydrodynamic models.

Modeling of Polar Ocean Tides at the Last Glacial Maximum: Amplification, Sensitivity, and Climatological Implications

Abstract Diurnal and semidiurnal ocean tides are calculated for both the present day and the Last Glacial Maximum. A numerical model with complete global coverage and enhanced resolution at high latitudes is used including the physics of self-attraction and loading and internal tide drag. Modeled present-day tidal amplitudes are overestimated at the standard resolution, but the error decreases as the resolution increases. It is argued that such results, which can be improved in the future using higher-resolution simulations, are preferable to those obtained by artificial enhancement of dissipative processes. For simulations at the Last Glacial Maximum a new version of the ICE-5G topographic reconstruction is used along with density stratification determined from coupled atmosphere–ocean climate simulations. The model predicts a significant amplification of tides around the Arctic and Antarctic coastlines, and these changes are interpreted in terms of Kelvin wave dynamics with the aid of an exact analytica...

Internal Tide Generation at the Continental Shelf Modeled Using a Modal Decomposition: Two-Dimensional Results

Abstract Stratified flow over topography is studied, with oceanic applications in mind. A model is developed for a fluid with arbitrary vertical stratification and a free surface, flowing over three-dimensional topography of arbitrary size and steepness, with background rotation, in the linear hydrostatic regime. The model uses an expansion of the flow fields in terms of a set of basis functions, which efficiently capture the vertical dependence of the flow. The horizontal structure may then be found by solving a set of coupled partial differential equations in two horizontal directions and time, subject to simple boundary conditions. In some cases, these equations may be solved analytically, but, in general, simple numerical procedures are required. Using this formulation, the internal tide generated by a time-periodic barotropic tidal flow over a continental shelf and slope is calculated in various idealized configurations. The topography and fluid motion are taken to be independent of one coordinate di...

Nonlinear Vertical Scale Selection in Equatorial Inertial Instability

Abstract The zonal flows of the earths equatorial stratosphere and mesosphere are prone to inertial instability when the horizontal shear Λ ≠ 0 at the equator. However, it is not clear why the vertical wavelength of the observed structures is much greater than that predicted by the simple linear theory, based on a vertical scale selection by molecular diffusion. Here, a nonlinear mechanism is described that can lead to upscaling from structures with short vertical wavelengths to long vertical wavelengths. The mechanism is dependent upon a secondary Kelvin–Helmholtz instability that develops as the inertial instability grows. Numerical simulations on an equatorial β plane, employing a simple parameterization of the vertical transport of horizontal momentum due to the secondary instability, are used to assess the likely degree of upscaling in a zonally symmetric system. For a fluid with buoyancy frequency N, it is shown that upscaling toward the buoyancy cutoff wavenumber 4Nβ/Λ2 can occur, even when diffus...

The limiting form of inertial instability in geophysical flows

The instability of a rotating, stratified flow with arbitrary horizontal cross-stream shear is studied, in the context of linear normal modes with along-stream wavenumber k and vertical wavenumber m. A class of solutions are developed which are highly localized in the horizontal cross-stream direction around a particular streamline. A Rayleigh-Schrodinger perturbation analysis is performed, yielding asymptotic series for the frequency and structure of these solutions in terms of k and m. The accuracy of the approximation improves as the vertical wavenumber increases, and typically also as the along-stream wavenumber decreases. This is shown to correspond to a near-inertial limit, in which the solutions are localized around the global minimum of fQ, where f is the Coriolis parameter and Q is the vertical component of the absolute vorticity. The limiting solutions are near-inertial waves or inertial instabilities, according to whether the minimum value of fQ is positive or negative. We focus on the latter case, and investigate how the growth rate and structure of the solutions changes with m and k. Moving away from the inertial limit, we show that the growth rate always decreases, as the inertial balance is broken by a stabilizing cross-stream pressure gradient. We argue that these solutions should be described as non-symmetric inertial instabilities, even though their spatial structure is quite different to that of the symmetric inertial instabilities obtained when k is equal to zero. We use the analytical results to predict the growth rates and phase speeds for the inertial instability of some simple shear flows. By comparing with results obtained numerically, it is shown that accurate predictions are obtained by using the first two or three terms of the perturbation expansion, even for relatively small values of the vertical wavenumber. Limiting expressions for the growth rate and phase speed are given explicitly for non-zero k, for both a hyperbolic-tangent velocity profile on an f-plane, and a uniform shear flow on an equatorial β-plane.

Kelvin wave propagation along straight boundaries in C-grid finite-difference models

Discrete solutions for the propagation of coastally-trapped Kelvin waves are studied, using a second-order finite-difference staggered grid formulation that is widely used in geophysical fluid dynamics (the Arakawa C-grid). The fundamental problem of linear, inviscid wave propagation along a straight coastline is examined, in a fluid of constant depth with uniform background rotation, using the shallow-water equations which model either barotropic (surface) or baroclinic (internal) Kelvin waves. When the coast is aligned with the grid, it is shown analytically that the Kelvin wave speed and horizontal structure are recovered to second-order in grid spacing h. When the coast is aligned at 45^o to the grid, with the coastline approximated as a staircase following the grid, it is shown analytically that the wave speed is only recovered to first-order in h, and that the horizontal structure of the wave is infected by a thin computational boundary layer at the coastline. It is shown numerically that such first-order convergence in h is attained for all other orientations of the grid and coastline, even when the two are almost aligned so that only occasional steps are present in the numerical coastline. Such first-order convergence, despite the second-order finite differences used in the ocean interior, could degrade the accuracy of numerical simulations of dynamical phenomena in which Kelvin waves play an important role. The degradation is shown to be particularly severe for a simple example of near-resonantly forced Kelvin waves in a channel, when the energy of the forced response can be incorrect by a factor of 2 or more, even with 25 grid points per wavelength.

The influence of modulational instability on energy exchange in coupled sine-Gordon equations

We consider a two-component system of coupled sine-Gordon equations, particular solutions of which represent a continuum generalization of periodic energy exchange in a system of coupled pendulums. Weakly nonlinear solutions describing periodic energy exchange between waves traveling in the two components are governed, depending on the length scale of the amplitude variation, either by two nonlocally coupled nonlinear Schrödinger equations, with different transport terms due to the group velocity, or by a model that is nondispersive to the leading order. Using both asymptotic analysis and numerical simulations, we show that the effects of dispersion significantly influence the structure of these solutions, causing modulational instability and the formation of localized structures but preserving the pattern of energy exchange between the components.

Shear flow instabilities in shallow-water magnetohydrodynamics

Within the framework of shallow-water magnetohydrodynamics, we investigate the linear instability of horizontal shear flows, influenced by an aligned magnetic field and stratification. Various classical instability results, such as H{\o}ilands growth rate bound and Howards semi-circle theorem, are extended to this shallow-water system for quite general profiles. Two specific piecewise-constant velocity profiles, the vortex sheet and the rectangular jet, are studied analytically and asymptotically; it is found that the magnetic field and stratification (as measured by the Froude number) are generally both stabilising, but weak instabilities can be found at arbitrarily large Froude number. Numerical solutions are computed for corresponding smooth velocity profiles, the hyperbolic-tangent shear layer and the Bickley jet, for a uniform background field. A generalisation of the long-wave asymptotic analysis of Drazin & Howard (1962) is employed in order to understand the instability characteristics for both profiles. For the shear layer, the mechanism underlying the primary instability is interpreted in terms of counter-propagating Rossby waves, thereby allowing an explication of the stabilising effects of the magnetic field and stratification.