Stratified Background Research Articles

A rigorous bound on the vertical transport of heat in Rayleigh-Bénard convection at infinite Prandtl number with mixed thermal boundary conditions

A rigorous upper bound on the Nusselt number is derived for infinite Prandtl number Rayleigh-Bénard convection for a fluid constrained between no-slip, mixed thermal vertical boundaries. The result suggests that the thermal boundary condition does not affect the qualitative nature of the heat transport. The bound is obtained with the use of a nonlinear, stably stratified background temperature profile in the bulk, notwithstanding the lack of boundary control of the temperature due to the Robin boundary conditions.

Calculation of a point source radiation in a flat or non-flat stratified background: an alternative to Sommerfeld integrals

In this paper, we present an alternative method to Sommerfeld integrals for the calculation of a point source radiation in a multilayered background, based on the method of auxiliary sources, also called fictitious sources method. The method lies upon the decomposition of reflected and transmitted fields on a basis of secondary sources, which amplitudes are determined by applying boundary conditions at each multilayer interface, solving an over-determined system of equations. We present a generalization of the classical FSM to open stratified domains, with arbitrary number of layers and arbitrary position of the point source in the multilayer. Two and three dimensional formalisms are proposed, and compared to exact Sommerfeld formulation. Computation time reduction factors as high as 100 are obtained for 2D problems. Moreover, this method enables the study of non-flat interfaces, which is not possible with Sommerfeld approach. This method can be used for example in order to determine radiation efficiency of emitting devices, or it can be integrated in discrete source type electromagnetic methods for calculation of scattering by a complex target.

Transport and mixing of density in a continuously stratified shear layer

Scalar transport and mixing by active turbulence in a high Reynolds number inhomogeneous stratified shear layer are investigated using three-dimensional Direct Numerical Simulation. Two density profiles are considered: (i) two layers of homogenous fluid with different density, namely the two-layer case, and (ii) a continuously stratified background ambient, namely the Jd case. The evolution of the mixing layer includes shear instability, formation of Kelvin–Helmholtz rollers, transition to turbulence, fully developed active turbulence, and, finally, decay toward a laminar state. In the Jd case, internal gravity waves carrying momentum and energy are observed to propagate away from the shear layer. Although different during the initial evolution, the eddy diffusivity and mixing efficiency when plotted as a function of buoyancy, Reynolds number takes similar values between the two cases later in time during the stage when turbulence decays. During this stage, the mixing efficiency computed based on the buoy...

Resonant Wave Interactions in the Presence of a Diurnally Varying Heat Source

Abstract Resonant interactions among equatorial waves in the presence of a diurnally varying heat source are studied in the context of the diabatic version of the equatorial β-plane primitive equations for a motionless, hydrostatic, horizontally homogeneous and stably stratified background atmosphere. The heat source is assumed to be periodic in time and of small amplitude [i.e., O(ɛ)] and is prescribed to roughly represent the typical heating associated with deep convection in the tropical atmosphere. In this context, using the asymptotic method of multiple time scales, the free linear Rossby, Kelvin, mixed Rossby–gravity, and inertio-gravity waves, as well as their vertical structures, are obtained as leading-order solutions. These waves are shown to interact resonantly in a triad configuration at the O(ɛ) approximation, and the dynamics of these interactions have been studied in the presence of the forcing. It is shown that for the planetary-scale wave resonant triads composed of two first baroclinic equatorially trapped waves and one barotropic Rossby mode, the spectrum of the thermal forcing is such that only one of the triad components is resonant with the heat source. As a result, to illustrate the role of the diurnal forcing in these interactions in a simplified fashion, two kinds of triads have been analyzed. The first one refers to triads composed of a k = 0 first baroclinic geostrophic mode, which is resonant with the stationary component of the diurnal heat source, and two dispersive modes, namely, a mixed Rossby–gravity wave and a barotropic Rossby mode. The other class corresponds to triads composed of two first baroclinic inertio-gravity waves in which the highest-frequency wave resonates with a transient harmonic of the forcing. The integration of the asymptotic reduced equations for these selected resonant triads shows that the stationary component of the diurnal heat source acts as an “accelerator” for the energy exchanges between the two dispersive waves through the excitation of the catalyst geostrophic mode. On the other hand, since in the second class of triads the mode that resonates with the forcing is the most energetically active member because of the energy constraints imposed by the triad dynamics, the results show that the convective forcing in this case is responsible for a longer time scale modulation in the resonant interactions, generating a period doubling in the energy exchanges. The results suggest that the diurnal variation of tropical convection might play an important role in generating low-frequency fluctuations in the atmospheric circulation through resonant nonlinear interactions.

Geodetic and Structural Research in La Palma, Canary Islands, Spain: 1992–2007 Results

We review the results of the geodetic and structural studies carried out on La Palma Island using geodetic and geophysical data during the period 1992-2007. InSAR and GPS observation techniques were applied to study the existence of deformation on the island and gravity observations were carried out for structural studies. Gravity data were inverted using a nonlinear three-dimensional gravity inversion approach to obtain the geometry of the anomalous bodies constructed in a random growth process with respect to an exponentially stratified background. The main structural feature is a large central body (under the Caldera de Taburiente) with high density, which was interpreted as the Pliocene-age uplifted seamount and a relatively dense intrusive plutonic complex/magma body. The Cumbre Vieja series is characterized by elongated minima distributed according to the rift structure. InSAR results show a clear subsidence located on the Teneguia volcano, where the last eruption took place in 1971. A thermal source is the most probable origin for this deformation. A GPS network composed of 26 stations covering the total island surface was set up. Vertical displacements determined comparing the GPS coordinates obtained in 2007 with coordinates determined in 1994 are consistent with the InSAR results obtained in the southern part of the island. This is not the case for the northern part. From the comparison of 2006 and 2007 coordinates it is clear that more time is needed to obtain significant displacements, but observed trends are also consistent with InSAR results. All the observed significant displacements are in stations located outside of the large high-density central body.

Seismic Halos around Active Regions: A Magnetohydrodynamic Theory

Comprehending the manner in which magnetic fields affect propagating waves is a first step toward constructing accurate helioseismic models of active region subsurface structure and dynamics. Here we present a numerical method for computing the linear interaction of waves with magnetic fields embedded in a solar-like stratified background. The ideal magnetohydrodynamic (MHD) equations are solved in a three-dimensional box that straddles the solar surface, extending from 35 Mm below the photosphere to 1.2 Mm into the atmosphere. One of the challenges in performing these simulations involves generating a magnetohydrostatic (MHS) state wherein the stratification assumes horizontal inhomogeneity in addition to the strong vertical stratification associated with the near-surface layers. Keeping in mind that the aim of this effort is to understand and characterize linear MHD interactions, we discuss a means of computing statically consistent background states. Power maps computed from simulations of waves interacting with thick flux tubes that have peak photospheric field strengths of 600 and 3000 G are presented. Strong modal power reduction in the umbral regions of the flux tube enveloped by a halo of increased wave power is seen in the simulations with the thick flux tubes. These enhancements are also seen in Doppler velocity power maps of active regions observed in the Sun, which leads us to propose that the halo has MHD underpinnings.

Stability Properties of Magnetic Tower Jets

Stability properties of jets propagating in a gravitationally stratified background have been examined by performing three-dimensional magnetohydrodynamic simulations. The current-carrying, Poynting flux-dominated magnetic tower jet, which possesses a highly wound helical magnetic field, is subject to the current-driven instability (CDI). We find that, under general physical conditions including small perturbations in the initial background profiles, the propagating magnetic tower jets develop the nonaxisymmetric, m = 1 kink mode of the CDI. The kink mode grows on the local Alfven crossing timescale. In addition, two types of kink modes appear in the system. At the central region where external thermal pressure confinement is strong, only the internal kink mode is excited and will grow. A large distance away from the central region where the external thermal pressure becomes low, the external kink mode is observed. As a result, the exterior of magnetic tower jets will be deformed into a large-scale wiggled structure. We also discuss extensively the different physical processes that contribute to the overall stability properties of the magnetic tower jets. Specifically, when the jet propagates in an initially unperturbed background, we find that it can survive the kink mode beyond the point predicted by the well-known Kruskal-Shafranov (K-S) criterion. The stabilization in this case comes mainly from the dynamical relaxation of magnetic twists during the propagation of magnetic towers; the magnetic pitch is reduced, and the corresponding K-S critical wavelength becomes longer as the tower jet proceeds. Furthermore, we show that the pressure-driven and Kelvin-Helmholtz instabilities do not occur in the magnetic tower jets. This strongly suggests that the CDI is the primary reason for the wiggling structures in jets.

Anomalous response of stratified fluid to forcing

The time-independent disturbances induced by nonuniformly distributed surface shear stress in a binary fluid (salt water) are analyzed in the linear approximation. It is shown that correct treatment of two scalar fields (stratified background temperature and salinity distributions) may lead to counterintuitive qualitative predictions even if the thermal and salt diffusivities are equal. In particular, when stable salinity stratification is imposed on stable temperature stratification, both the amplitude and depth of the disturbance may substantially increase rather than decrease (contrary to intuitive expectations). A previously unknown mechanism of convective instability is revealed for a stable density-stratified binary fluid.

Turbulent diffusion and transport from a CO2 lake in the deep ocean

If liquid CO2 is stored as a dense “lake” on the deep ocean floor, it is expected to dissolve in seawater. Ocean currents and turbulence can increase the net rate of CO2 release by several orders of magnitude compared to molecular diffusion. However, density stratification in the seawater created by dissolved CO2 will tend to reduce vertical mixing. A two‐dimensional numerical study with a high‐resolution advection‐diffusion model, coupled with a general turbulence model, reveals significant modifications of the boundary layer structure above a generic CO2 lake taken to be 500 m in length and placed in a 10 km domain that is subject to specified far‐field currents in the range of 0.05–0.20 m s−1. Dissolution rates of order 0.07 μmol cm−2 s−1, reaching approximately 0.5 μmol cm−2 s−1 during a benthic storm, are predicted. The friction velocity is reduced above the CO2 lake, and CO2 concentrations corresponding to excess water densities of up to 0.5 kg m−3 occur in the lower 10 m of the water column. The persistency of this low‐pH, CO2‐enriched water layer in weakly stratified and neutral background conditions in the model suggests that gravity current dynamics are important over considerable distances and should be considered in future larger‐scale models and impact studies.

Estimating statistical parameters of an elastic random medium from traveltime fluctuations of refracted waves

Traveltime fluctuations of diving-type refracted waves are studied in the framework of geometrical optics in order to estimate the statistical parameters of an elastic random medium. A stratified background medium is considered in which the velocity increases linearly with depth. Smooth and strongly anisomeric (statistically anisotropic) inhomogeneities are embedded in this medium. The covariance and the variance of traveltime fluctuations are derived and subsequently used to estimate the standard deviation of the medium fluctuations and the inhomogeneity scale lengths in horizontal and vertical directions. The theoretical estimation procedure is verified by performing numerical calculations and it is observed that, under the considered conditions, the traveltime variance decreases at large offsets. This new phenomenon has not been observed before either in acoustics and optics, or in radio wave propagation.

Reference Forms of the Complete Spectral Electric and Magnetic Dyadic Green's Functions (DGF's) for Planar Stratified Media

The classical problem of determining the complete spectral electric and magnetic Dyadic Green's Functions (DFG's) for 3D current sources imbedded within planar stratified background media is performed in a modern concise manner and notation. The expressions are then rearranged into "reference forms". This annotation is justified by examining the correspondence with the familiar free space expressions and clarifying some difficulties encountered in the formulation and numerical solution of Surface and Volume Integral Equations (S/V-IE's).

A Green's tensor approach to the modeling of nanostructure replication and characterization

We use the Green's tensor technique to study the optical processes taking place in configurations typically used for the replication and characterization of nanostructures. For the replication process we investigate light‐coupling masks for optical contact lithography and for the characterization process the mode scattered by a defect or a short grating in a planar waveguide. Both configurations consist of structures embedded in a stratified background composed of a stack of material layers with different permittivities. We perform calculations for two‐dimensional and three‐dimensional structures and compare their optical behavior. Our results show that the additional material interfaces in three‐dimensional systems can lead to significantly different field distributions and must be taken into account for a complete understanding of the electromagnetic properties of the systems.

Approximate solution for Euler equations ofstratified water via numerical solution of coupled KdV system

We consider Euler equations with stratified background state that is valid for internal water waves. The solution of the initial‐boundary problem for Boussinesq approximation in the waveguide mode is presented in terms of the stream function. The orthogonal eigenfunctions describe a vertical shape of the internal wave modes and satisfy a Sturm‐Liouville problem. The horizontal profile is defined by a coupled KdV system which is numerically solved via a finite‐difference scheme for which we prove the convergence and stability. Together with the solution of the Sturm‐Liouville problem, the stream functions give the internal waves profile.

Scattering experiments with a diving cylinder

We present numerical experiments of light scattering by a circular dielectric cylinder embedded in a stratified background, using the Green's tensor technique. The stratified background consists of two or three dielectric layers, the latter forming an anti-reflection system. We show movies of the scattered field as a function of different parameters: polarization, angle of incidence, and relative position of the cylinder with respect to the background interfaces.

Green's tensor technique for scattering in two-dimensional stratified media.

We present an accurate and self-consistent technique for computing the electromagnetic field in scattering structures formed by bodies embedded in a stratified background and extending infinitely in one direction (two-dimensional geometry). With this fully vectorial approach based on the Green's tensor associated with the background, only the embedded scatterers must be discretized, the entire stratified background being accounted for by the Green's tensor. We first derive the formulas for the computation of this dyadic and discuss in detail its physical substance. The utilization of this technique for the solution of scattering problems in complex structures is then illustrated with examples from photonic integrated circuits (waveguide grating couplers with varying periodicity).

Light propagation and scattering in stratified media: a Green’s tensor approach

We present a new technique for computing the electromagnetic field that propagates and is scattered in three-dimensional structures formed by bodies embedded in a stratified background. This fully vectorial technique is based on the Green’s tensor associated with the stratified background. Its advantage lies in the fact that only the scatterers must be discretized, the stratified background being accounted for in the Green’s tensor. Further, the boundary conditions at the different material interfaces as well as at the edges of the computation window are perfectly and automatically fulfilled. Several examples illustrate the utilization of the technique for the modeling of photonic circuits (integrated optical waveguides), the study of the optics of metal (surface plasmons), and the development of new optical lithography techniques.

A fully vectorial technique for scattering and propagation in three-dimensional stratified photonic structures

We present a three-dimensional (3D) technique for computing light scattering and propagation in complex structures formed by scatterers embedded in a stratified background. This approach relies on the Green's tensor associated with the background and requires only the discretization of the scatterers, the entire stratified background being accounted for in the Green's tensor. Further, the boundary conditions at the edges of the computation window and at the different material interfaces in the stratified background are automatically fulfilled. Different examples illustrate the application of the technique to the modeling of photonic integrated circuits: waveguides with protrusions (single element ‘grating’) and notches. Subtle effects, like polarization crosstalks in an integrated optics device are also investigated.

Three-wave coupling in a stratified MHD plasma

Abstract. A general coupling coefficient for three wave interactions in an ideal MHD plasma is presented. Using a special stratified background atmosphere, an explicit symmetric form of the coupling coefficient is derived.

Buoyancy‐Driven Rotating Boundary Currentsa

Abstract: The structure of boundary currents formed from intermediately dense water introduced into a rotating, stably stratified, two‐layer environment is investigated in a series of laboratory experiments, performed for Froude numbers ranging from 0.01 to 1. The thickness and streamwise velocity profiles in quasi‐steady currents are measured and found to agree with analytic solutions and numerical models. A sloping boundary in a stratified background results in robust stability at all experimental Froude numbers. Such stability is in sharp contrast to the rapid breakup into eddies of such a current flowing against a vertical boundary, or of a current flowing along a slope in a uniform background. The possibility of wave radiation supported by a variety of wave mechanisms found in the ambient fluid might account for the observed stability.

Toward a Unified Theory of Gravity Wave Stability

Abstract This paper reveals relationships among linear instabilities of internal gravity waves often supposed to be independent. Using a Floquet analysis of a monochromatic wave propagating in a uniformly stratified background without shear, which accounts for finite wave amplitude, spatial and temporal periodicity, tilted phase planes, and 3D disturbances, it is demonstrated that the dominant instabilities of overturning waves are the large-wave-amplitude manifestations of resonant and slantwise instabilities of small amplitude waves, and that they possess no threshold amplitudes. An energy budget analysis examines the relation of parametric instabilities at large wave amplitude to vertical dynamic and static instabilities; however, the instability characteristics for propagating waves are very different from those inferred by analogy to Kelvin–Helmholtz instability and Benard convection in simpler backgrounds. At small amplitudes, resonant instabilities rely on horizontal or slantwise gradients of wave ...