Donald N. Slinn

Large-eddy simulations of the wind-induced turbulent Ekman layer

At urbulent Ekman layer created by a steady wind near the water surface is investigated using the numerical method of large-eddy simulations. The classical case of a flow unaffected by density stratification and surface waves is revisited to understand the internal structure of the flow and implications of the traditional assumptions of constant effective viscosity and the ‘f -plane’ approximation. A series of numerical experiments reveals that the Ekman solution needs correcting even in this case. The examination of the effective viscosity hypothesis confirms its validity but shows that the viscosity varies strongly with depth. It increases in the subsurface layer of thickness about 1/4 the turbulent length scale and decreases below this level. A Bessel function solution is proposed that corresponds to the approximate effective viscosity profile and matches with the LES results. Strong flow dependence on the latitude and wind direction is detected and explained by the effects of redistribution of turbulent kinetic energy between the velocity components and modification of the vertical transfer of turbulent momentum. In this paper, we consider the classical problem of a turbulent flow generated near the ocean surface by a steady wind stress in the presence of Earth’s rotation. Interest in this flow goes back to Ekman’s landmark work published in 1905. (An interesting historical review of Fridtjof Nansen’s polar expedition and other events preceding Ekman’s paper is given by Walker (1991).) Ekman assumed a balance between the Coriolis force, viscous friction and the pressure gradient, adopted the approximation of constant vertical eddy viscosity Az ,a ndderived a solution now known as the ‘Ekman spiral’. In the case of a steady wind in the x-direction, the steady-state Ekman velocity profile in the open ocean is (for the northern hemisphere) u = V0 cos π + π D z

Alongshore currents over variable beach topography

The nonlinear dynamics of unstable alongshore currents in the nearshore surf zone over variable barred beach topography are studied using numerical experiments. These experiments extend the recent studies of Allen et al. [1996] and Slinn et al. [1998], which utilized alongshore uniform beach topographies by including sinusoidal alongshore variation to shore parallel sandbars. The model involves finite difference solutions to the nonlinear shallow water equations for forced, dissipative, initial value problems and employs periodic boundary conditions in the alongshore direction. Effects of dissipation are modeled by linear bottom friction. Forcing for the alongshore currents is provided by gradients in the radiation stress, which are specified using linear theory and the dissipation function for breaking waves formulated by Thornton and Guza [1983]. Distinct flows develop depending on the amplitude ∈ and wavelength λ of the topographic variability and the dimensionless parameter Q, the ratio of an advective to a frictional timescale. For Q greater than a critical value QC the flows are linearly stable. For ΔQ = QC - Q>0 the flow can be unstable. For small values of ΔQ the effect of increasing ∈; is to stabilize or regularize the flows and to cause the mean flow to approximately follow contours of constant depth. Equilibrated shear waves develop that propagate along the mean current path at phase speeds and wavelengths that are close to predictions for the most unstable mode from linear theory applied to alongshore-averaged conditions. At intermediate values of ΔQ, unsteady vortices form and exhibit nonlinear interactions as they propagate along the mean current path, occasionally merging, pairing, or being shed seaward of the sandbar. Eddies preferentially form in the mean current when approaching alongshore troughs of the sandbar and break free from the mean current when approaching alongshore crests of the sandbar. At the largest values of ΔQ examined the resulting flow fields resemble a turbulent shear flow and are less strongly influenced by the alongshore variability in topography. As the amplitude of the alongshore topographic variability increases, alongshore wavenumber-frequency spectra of the across-shore velocity show a corresponding increase in energy at both higher alongshore wavenumbers and over a broader frequency range with significant energy at wavenumbers of topographic variability and harmonics. Across-shore fluxes of mass and momentum generally increase with increasing topographic amplitude and increasing ΔQ. Time- and space-lagged correlations of the across-shore velocity show that correlation length scales decrease as topographic perturbation amplitudes increase. Terms from the vorticity equation show that the alongshore variation of the radiation stresses and the value of ΔQ are of importance to the flow behavior. Hybrid experiments separating effects of spatially variable forcing and the dynamic influence of topography on time-averaged currents show that the effects are generally comparable with the relative importance of each effect a function of ΔQ. The results show that topographic variability has a significant influence on nearshore circulation.

Along-slope current generation by obliquely incident internal waves

A series of numerical experiments is performed to investigate the breaking of obliquely incident internal waves propagating towards a bottom slope. The case of critical reflection is considered, where the angle between the wave group velocity vector and the horizontal matches the bottom slope angle. The flow evolution is found to be significantly different from the evolution observed previously in simulations of normally incident waves. The divergence of the Reynolds stress in the breaking zone causes a strong along-slope mean current, which changes the flow structure dramatically. The wave does not penetrate the current but breaks down at its upper surface as the result of a critical layer interaction. A continuously broadening mean along-slope current with an approximately constant velocity is produced. We propose a simple model of the process based on the momentum conservation law and the radiation stress concept. The model predictions are verified against the numerical results and are used to evaluate the possible strength of along-slope currents generated by this process in the ocean.

Turbulent convection driven by surface cooling in shallow water

We present the results of large-eddy simulations (LES) of turbulent thermal convection generated by surface cooling in a finite-depth stably stratified horizontal layer with an isothermal bottom surface. The flow is a simplified model of turbulent convection occurring in the warm shallow ocean during adverse weather events. Simulations are performed in a 6 x 6 x 1 aspect ratio computational domain using the pseudo-spectral Fourier method in the horizontal plane and finite-difference discretization on a high-resolution clustered grid in the vertical direction. A moderate value of the Reynolds number and two different values of the Richardson number corresponding to a weak initial stratification are considered. A version of the dynamic model is applied as a subgrid-scale (SGS) closure. Its performance is evaluated based on comparison with the results of direct numerical simulations (DNS) and simulations using the Smagorinsky model. Comprehensive study of the spatial structure and statistical properties of the developed turbulent state shows some similarity to Rayleigh-Benard convection and other types of turbulent thermal convection in horizontal layers, but also reveals distinctive features such as the dominance of a large-scale pattern of descending plumes and strong turbulent fluctuations near the surface.