Numerical studies of the scattering of light from a two-dimensional randomly rough interface between two dielectric media

The scattering of polarized light incident from one dielectric medium on its two-dimensional randomly rough interface with a second dielectric medium is studied. A reduced Rayleigh equation for the scattering amplitudes is derived for the case where p- or s-polarized light is incident on this interface, with no assumptions being made regarding the dielectric functions of the media. Rigorous, purely numerical, nonperturbative solutions of this equation are obtained. They are used to calculate the reflectivity and reflectance of the interface, the mean differential reflection coefficient, and the full angular distribution of the intensity of the scattered light. These results are obtained for both the case where the medium of incidence is the optically less dense medium, and in the case where it is the optically more dense medium. Optical analogues of the Yoneda peaks observed in the scattering of x-rays from metal surfaces are present in the results obtained in the latter case. Brewster scattering angles for diffuse scattering are investigated, reminiscent of the Brewster angle for flat-interface reflection, but strongly dependent on the angle of incidence. When the contribution from the transmitted field is added to that from the scattered field it is found that the results of these calculations satisfy unitarity with an error smaller than $10^{-4}$.Comment: 25 pages, 14 figure

A comparison of Eulerian and Lagrangian methods for vertical particle transport in the water column

A common task in oceanography is to model the vertical movement of particles such as microplastics, nanoparticles, mineral particles, gas bubbles, oil droplets, fish eggs, plankton, or algae. In some cases, the distribution of the vertical rise or settling velocities of the particles in question can span a wide range, covering several orders of magnitude, often due to a broad particle size distribution or differences in density. This requires numerical methods that are able to adequately resolve a wide and possibly multi-modal velocity distribution. Lagrangian particle methods are commonly used for these applications. A strength of such methods is that each particle can have its own rise or settling speed, which makes it easy to achieve a good representation of a continuous distribution of speeds. An alternative approach is to use Eulerian methods, where the partial differential equations describing the transport problem are solved directly with numerical methods. In Eulerian methods, different rise or settling speeds must be represented as discrete classes, and in practice, only a limited number of classes can be included. Here, we consider three different examples of applications for a water column model: positively buoyant fish eggs, a mixture of positively and negatively buoyant microplastics, and positively buoyant oil droplets being entrained by waves. For each of the three cases, we formulate a model for the vertical transport based on the advection–diffusion equation with suitable boundary conditions and, in one case, a reaction term. We give a detailed description of an Eulerian and a Lagrangian implementation of these models, and we demonstrate that they give equivalent results for selected example cases. We also pay special attention to the convergence of the model results with an increasing number of classes in the Eulerian scheme and with the number of particles in the Lagrangian scheme. For the Lagrangian scheme, we see the convergence, as expected for a Monte Carlo method, while for the Eulerian implementation, we see a second-order () convergence with the number of classes

A comparison of Eulerian and Lagrangian methods for vertical particle transport in the water column

A common task in oceanography is to model the vertical movement of particles such as microplastics, nanoparticles, mineral particles, gas bubbles, oil droplets, fish eggs, plankton, or algae. In some cases, the distribution of vertical rise or settling velocities of the particles in question can span a wide range, covering several orders of magnitude, often due to a broad particle size distribution or differences in density. This requires numerical methods that are able to adequately resolve a wide and possibly multi-modal velocity distribution. Lagrangian particle methods are commonly used for these applications. strength of such methods is that each particle can have its own rise or settling speed, which makes it easy to achieve a good representation of a continuous distribution of speeds. An alternative approach is to use Eulerian methods, where the partial differential equations describing the transport problem are solved directly with numerical methods. In Eulerian methods, different rise or settling speeds must be represented as discrete classes, and in practice only a limited number of classes can be included. Here, we consider three different examples of applications for a water-column model: positively buoyant fish eggs, a mixture of positively and negatively buoyant microplastics, and positively buoyant oil droplets being entrained by waves. For each of the three cases we formulate a model for the vertical transport, based on the advection-diffusion equation with suitable boundary conditions and in one case a reaction term. We give a detailed description of an Eulerian and a Lagrangian implementation of these models, and we demonstrate that they give equivalent results for selected example cases. We also pay special attention to the convergence of the model results with increasing number of classes in the Eulerian scheme, and the number of particles in the Lagrangian scheme. For the Lagrangian scheme, we see the 1/√Np convergence as expected for a Monte Carlo method, while for the Eulerian implementation, we see a second order (1/N2k) convergence with the number of classes

Long-term fate of Ra-226 originating from offshore produced water discharges

Formation water in oil and gas fields contain elevated concentrations of several radionuclides compared with seawater. It is therefore of interest to track the fate of these radionuclides when they are released into the ocean along with the produced water stream. In this work we have simulated the fate of radionuclides in produced water discharge from all relevant installations on the Norwegian continental shelf for periods up to 20 years. We investigated separately the fate of the radionuclide Ra-226 in solid and in dissolved phase. Our findings show that when Ra-226 spreads as a solid it has the highest potential for reaching the sediments in the areas around the Oseberg and Ekofisk fields. The estimated contribution above background at these sites over a 30 year period was estimated to be from 2 Bq/kg to 10 Bq/kg, depending on the degree of burial, bioturbation and sediment mixing. Our simulations of dissolved Ra-226 showed transport along the Norwegian coast up past Svalbard. We found that the maximal contribution of dissolved Ra-226 to radioactivity in sediments over a 30 year period lies in the range 0.6 - 2 Bq/kg at several locations along the Norwegian coast.NRPApublishedVersio

Scattering of light from weaklyrough surfaces

A formalism is introduced for the non-perturbative, purely numerical, solution of the reduced Rayleigh equation for the scattering of light from two-dimensional penetrable rough surfaces. In the papers included in this thesis, we apply this formalism to study the scattering of p- or s-polarised light from two-dimensional dielectric or metallic randomly rough surfaces, or from two-dimensional randomly rough thin dielectric films on metallic substrates, by calculating the full angular distribution of the co- and cross-polarised intensity of the scattered light. We present calculations of the mean differential reflection coefficient for glass and silver surfaces characterised by (isotropic or anisotropic) Gaussian and cylindrical power spectra, and find a good match with experimental results, as well as results obtained from another numerical method. We also present a numerical calculation of the Mueller matrix for scattering from rough surfaces, based on the same method. We investigate the optical phenomena of enhanced backscattering, enhanced forward scattering and satellite peaks. Enhanced backscattering is a well known phenomenon, and is used as one among several indicators of correct results. The phenomenon of enhanced forward scattering has not previously been investigated in two-dimensional systems. We demonstrate its presence, and provide an explanation for why it is qualitatively different from the same phenomenon in one dimension. Regarding satellite peaks, there has been a dispute in the literature, where one group found they should be present in scattering from a thin dielectric film on a metallic substrate, while another group found they should not. We have demonstrated their presence, and shown how the one-dimensional phenomenon of satellite peaks become “satellite rings” in the two-dimensional case. The proposed method is found, within the validity of the Rayleigh hypothesis, to give reliable results. For a non-absorbing metal surface the conservation of energy was explicitly checked, and found to be satisfied to within 0.03%, or better, for the parameters assumed. This testifies to the accuracy of the approach and a satisfactory discretisation. We also perform a numerical investigation of the range of validity of the reduced Rayleigh equation for scattering from two-dimensionally rough silver and perfectly conducting surfaces. The advantage of using a numerical solution of the reduced Rayleigh equation, rather than a rigorous numerical method such as the surface integral method, lies in the required computational resources. The main limitation of these methods for considering two-dimensionally rough surfaces are their memory requirements. To calculate the scattering amplitude for a typical system studied in this thesis, by the reduced Rayleigh equation, requires 12 GB of memory. To solve a similarly sized system with a rigorous method requires one or two orders of magnitude more. The limitation of the reduced Rayleigh equation is that it can only be applied to weakly rough surfaces, due to the assumption of the Rayleigh hypothesis

Numerical integrators for Lagrangian oceanography

A common task in Lagrangian oceanography is to calculate a large number of drifter trajectories from a velocity field pre-calculated with an ocean model. Mathematically, this is simply numerical integration of an Ordinary Differential Equation (ODE), for which a wide range of different methods exist. However, the discrete nature of the modelled ocean currents requires interpolation of the velocity field in both space and time, and the choice of interpolation scheme has implications for the accuracy and efficiency of the different numerical ODE methods. We investigate trajectory calculation in modelled ocean currents with 800 m, 4 km and 20 km horizontal resolution, in combination with linear, cubic and quintic spline interpolation. We use fixed-step Runge-Kutta integrators of orders 1-4, as well as three variable-step Runge-Kutta methods (Bogacki-Shampine 3(2), Dormand-Prince 5(4) and 8(7)). Additionally, we design and test modified special-purpose variants of the three variable-step integrators, that are better able to handle discontinuous derivatives in an interpolated velocity field. Our results show that the optimal choice of ODE integrator depends on the resolution of the ocean model, the degree of interpolation, and the desired accuracy. For cubic interpolation, the commonly used Dormand-Prince 5(4) is rarely the most efficient choice. We find that in many cases, our special-purpose integrators can improve accuracy by many orders of magnitude over their standard counterparts, with no increase in computational effort. Equivalently, the special-purpose integrators can provide the same accuracy as standard methods, at a reduced computational cost. The best results are seen for coarser resolutions (4 km and 20 km), thus the special-purpose integrators are particularly advantageous for research using regional to global ocean models to compute large numbers of trajectories. Our results are also applicable to trajectory computations on data from atmospheric models

Convergence of Ensemble Simulations for Environmental Risk Assessment

Ensemble simulations of oil fate and transport for a hypothetical oil spill, sometimes referred to as "stochastic simulations", are frequently used for environmental risk assessment related to offshore operations. In this study we investigate the importance of the number of simulations and the effects of different sampling strategies on the results. We focus particularly on stranded oil, as this result is often assessed as a worst case criterion. We have run three ensembles for different discharge durations (3, 24 and 48 hours), all having the same simulation duration of 10 days. The ensembles were created by starting one simulation every hour throughout the four year period 2009 - 2019, resulting in a total number of 35064 simulations for each of the three durations. The complete set, as well as smaller subsets, were then used for further investigations. We find large variations in the probability distributions for amount of oil in the environmental compartments atmosphere, ashore, sea surface, water column and sediments and for the biodegraded fraction. When calculating the autocorrelations of the series of simulation results, we found the shorter releases to have shorter correlation times. Based on these findings, we discuss the relationships between release duration sampling rate and expected deviation caused by missing samples. Furthermore, we looked into sampling strategies, using both uniform and random sampling to create subsets from the complete set of simulations. We also briefly discuss the length of environmental data series to use for an environmental risk assessment, and briefly outline some future workpublishedVersio