Juan M. Restrepo

An asymptotic theory for the interaction of waves and currents in coastal waters

A multi-scale asymptotic theory is derived for the evolution and interaction of currents and surface gravity waves in water of finite depth, under conditions typical of coastal shelf waters outside the surf zone. The theory provides a practical and useful model with which wave–current coupling may be explored without the necessity of resolving features of the flow on space and time scales of the primary gravity-wave oscillations. The essential nature of the dynamical interaction is currents modulating the slowly evolving phase of the wave field and waves providing both phase-averaged forcing of long infra-gravity waves and wave-averaged vortex and Bernoulli-head forces and hydrostatic static set-up for the low-frequency current and sea-level evolution equations. Analogous relations are derived for wave-averaged material tracers and density stratification that include advection by horizontal Stokes drift and by a vertical Stokes pseudo-velocity that is the incompressible companion to the horizontal Stokes velocity. Illustrative solutions are analysed for the special case of depth-independent currents and tracers associated with an incident surface wave field and a vortex with O (1) Rossby number above continental shelf topography.

The Wave-Driven Ocean Circulation

Oceanic surface gravity waves have a mean Lagrangian motion, the Stokes drift. The dynamics of winddriven, basin-scale oceanic currents in the presence of Stokes drift are modified by the addition of so-called vortex forces and wave-induced material advection, as well by wave-averaged effects in the surface boundary conditions for the dynamic pressure, sea level, and vertical velocity. Some theoretical analyses previously have been made for the gravity wave influences on boundary-layer motions, including the Ekman currents. The present paper extends this theory to the basin-scale, depth-integrated circulation in a bounded domain. It is shown that the Sverdrup circulation relation, with the meridional transport proportional to the curl of the surface wind stress, applies to Lagrangian transport, while the associated Eulerian transport is shown to have a component opposite to the Stokes-drift transport. A wave-induced correction to the relation between sea level and surface dynamic pressure is also derived. Preliminary assessments are made of the relative importance of these influences using a global wind climatology and an empirical relationship between the wind and wave fields. Recommendations are made for further development and testing of this theory and for its inclusion in general circulation models.

Simulated two-dimensional red blood cell motion, deformation, and partitioning in microvessel bifurcations.

Movement, deformation, and partitioning of mammalian red blood cells (RBCs) in diverging microvessel bifurcations are simulated using a two-dimensional, flexible-particle model. A set of viscoelastic elements represents the RBC membrane and the cytoplasm. Motion of isolated cells is considered, neglecting cell-to-cell interactions. Center-of-mass trajectories deviate from background flow streamlines due to migration of flexible cells towards the mother vessel centerline upstream of the bifurcation and due to flow perturbations caused by cell obstruction in the neighborhood of the bifurcation. RBC partitioning in the bifurcation is predicted by determining the RBC fraction entering each branch, for a given partition of total flow and for a given upstream distribution of RBCs. Typically, RBCs preferentially enter the higher-flow branch, leading to unequal discharge hematocrits in the downstream branches. This effect is increased by migration toward the centerline but decreased by the effects of obstruction. It is stronger for flexible cells than for rigid circular particles of corresponding size, and decreases with increasing parent vessel diameter. For unequally sized daughter vessels, partitioning is asymmetric, with RBCs tending to enter the smaller vessel. Partitioning is not significantly affected by branching angles. Model predictions are consistent with previous experimental results.

Wave–Current Interaction: A Comparison of Radiation-Stress and Vortex-Force Representations

Abstract The vortex-force representation of the wave-averaged effects on currents is compared to the radiation-stress representation in a scaling regime appropriate to coastal and shelf waters. Three-dimensional and vertically integrated expressions for the conservative current equations are obtained in both representations. The vortex-force representation decomposes the main wave-averaged effects into two physically understandable concepts—a vortex force and a Bernoulli head. The vortex force is shown to be the dominant wave-averaged effect on currents. This effect can occur at higher order than the apparent leading order for the radiation-stress representation. Excluding nonconservative effects such as wave breaking, the lowest-order radiation or interaction stress can be completely characterized in terms of wave setup, forcing of long (infragravity) waves, and an Eulerian current whose divergence cancels that of the primary wave Stokes drift. The leading-order, wave-averaged dynamical effects incorpora...

Ensemble Filtering for Nonlinear Dynamics

Abstract A method for data assimilation currently being developed is the ensemble Kalman filter. This method evolves the statistics of the system by computing an empirical ensemble of sample realizations and incorporates measurements by a linear interpolation between observations and predictions. However, such an interpolation is only justified for linear dynamics and Gaussian statistics, and it is known to produce erroneous results for nonlinear dynamics with far-from-Gaussian statistics. For example, the ensemble Kalman filter method, when used in models with multimodal statistics, fails to track state transitions correctly. Here alternative ensemble methods for data assimilation into nonlinear dynamical systems, in particular, those with a large state space are studied. In these methods conditional probabilities at measurement times are calculated by applying Bayess rule. These results show that the new methods accurately track the transitions between likely states in a system with bimodal statistics,...

Stable and Unstable Solitary-Wave Solutions of the Generalized Regularized Long-Wave Equation

Summary. Investigated here are interesting aspects of the solitary-wave solutions of the generalized Regularized Long-Wave equation

Inner product computations using periodized Daubechies wavelets

u_t + u_x + \alpha \left( {u^p } \right)_x - \beta u_{xxt} = 0.

Shoreface-connected ridges under the action of waves and currents

For p>5, the equation has both stable and unstable solitary-wave solutions, according to the theory of Souganidis and Strauss. Using a high-order accurate numerical scheme for the approximation of solutions of the equation, the dynamics of suitably perturbed solitary waves are examined. Among other conclusions, we find that unstable solitary waves may evolve into several, stable solitary waves and that positive initial data need not feature solitary waves at all in its long-time asymptotics.