Characteristics of quadrant and octant advection schemes in wave models

We point out one problem of the grid advection schemes when used in wave models in coastal areas. The deficiency of the schemes is investigated by means of the ‘third' generation WAM wave model, in which the wave energy is advected by a first order upwind scheme. Two similar, alternative modifications of this scheme are analyzed, the second of which is shown to solve most of the problems encountered with advection along the co-ordinate axes.     

Improved representation of topographic effects by a vertical adaptive grid in Vector-Ocean-Model (VOM). Part II: Simulations in unstructured adaptive grids

A static adaptive grid approximates the topography and defines the vertical resolution in Vector-Ocean-Model (VOM). The adaptation to topography creates unstructured grids, which are organised in a one-dimensional vector by column-wise storage of only wet cells. The model’s name reflects this data structure. The intention of VOM is better resolving flow and stratification near topographic boundaries in Z-coordinates. This is the second part of a publication that describes the generation of adaptive grids (part I), and simulations with VOM in unstructured grids (this part). Adaptive grids generated for a synthetic topography in part I include shelf, continental slope, and ocean. Three of those grids are here utilised in upwelling simulations. Under the same forcing increased vertical resolution at seabed and slopes yields a significant increase in flow energy as compared to coarser grids. Results allow explaining the surface intensification of a continental slope jet by vertical displacements of water masses in the seabed Ekman layer. Results in unstructured grids are almost identical to reference simulations in equidistant grids where the respective smallest grid size of unstructured grids was used. Negative effects of grids on predicted flow and stratification are absent also over particularly rough topography, as demonstrated by using vertical velocity as most sensitive indicator. In a further simulation an overflow governed by the advection of water mass properties is presented to demonstrate the conservation properties of the model. After 5 months of simulation the predicted domain average temperature deviated by 10⁻⁸ from the initial temperature field. Compared to equidistant grids the advection/diffusion scheme looses about one order of magnitude in accuracy when used in an unstructured grid. The results of VOM, being defined in Z-coordinates, are void of coordinate transformation errors. In an arbitrary topography unforced zero-flow remains quiescent in a stratification that only varies in the vertical. VOM due to its depth-independent vertical resolution appears particularly suitable for simulations of ocean-shelf exchange.     

Energetics of lateral eddy diffusion/advection：Part Ⅲ. Energetics of horizontal and isopycnal diffusion/advection

Gravitational Potential Energy （GPE） change due to horizontal/isopycnal eddy diffusion and advection is examined. Horizontal/isopycnal eddy diffusion is conceptually separated into two steps： stirring and sub scale diffusion. GPE changes associated with these two steps are analyzed. In addition, GPE changes due to stirring and subscale diffusion associated with horizontal/isopycnal advection in the Eulerian coordinates are analyzed. These formulae are applied to the SODA data for the world oceans. Our analysis indicates that horizontal/isopycnal advection in Eulerian coordinates can introduce large artificial diffusion in the model. It is shown that GPE source/sink in isopycnal coordinates is closely linked to physical property distribution, such as temperature, salinity and velocity. In comparison with z-coordinates, GPE source/sink due to stir ring/cabbeling associated with isopycnal diffusion/advection is much smaller. Although isopycnal coordi nates may be a better choice in terms of handling lateral diffusion, advection terms in the traditional Eule rian coordinates can produce artificial source of GPE due to cabbeling associated with advection. Reducing such numerical errors remains a grand challenge.     

The effects of horizontal advection on the spring bloom of phytoplankton in the central Southern Huanghai Sea

An algal bloom is defined as a relatively rapid increase in the biomass of phytoplankton in an aquatic system. During 30 March to 24 April 2007, a cruise was conducted in the central Southern Huanghai Sea to investigate the spring bloom processes. The spatial and temporal variations of phytoplankton are discussed based on the in-situ observations and simultaneous remote sensing data. The explosive algal blooming varied quickly in temporal and spatial scales, due to the highly patchy distribution. Data obtained at the 2 anchor stations (BM1 and BM2) were analyzed in the present study. Horizontal advection is speculated to be responsible for the abrupt decrease in the concentration of chlorophyll-a at stations BM1 and BM2. At station BM2, the intermediate high chlorophyll-a concentration, coinciding with the low temperature, was found to be advected from the inshore colder water mass located to the east of the site.     

Comparative quantification of physically and numerically induced mixing in ocean models

A diagnostic method for calculating physical and numerical mixing of tracers in ocean models is presented. The physical mixing is defined as the turbulent mean tracer variance decay rate. The numerical mixing due to discretisation errors of tracer advection schemes is shown to be the decay rate between the advected square of the tracer variance and the square of the advected tracer and can be easily implemented into any ocean model. The applicability of the method is demonstrated for four test cases: (i) a one-dimensional linear advection equation with periodic boundary conditions, (ii) a two-dimensional flat-bottom lock exchange test case without mixing, (iii) a two-dimensional marginal sea overflow study with mixing and entrainment and (iv) the DOME test case with a dense bottom current propagating down a broad linear slope. The method has a number of advantages over previously introduced estimates for numerical mixing.     

Winter Oceanographic Conditions in the Southwestern Part of the Okhotsk Sea and Their Relation to Sea Ice

In the southwestern part of the Okhotsk Sea, oceanographic and sea-ice observations on board the icebreaker Soya were carried out in February 1997. A mixed layer of uniform temperature nearly at the freezing point extending down to a depth of about 300 m was observed. This is much deeper than has previously been reported. It is suggested that this deep mixed layer originated from the north (off East Sakhalin), being advected along the shelf slope via the East Sakhalin Current, accompanied with the thick first-year ice (average thickness 0.6 m). This vertically uniform winter water, through mixing with the surrounding water, makes the surface water more saline (losing a characteristic of East Sakhalin Current Water) and the water in the 100–300 m depth zone less saline, colder, and richer in oxygen (a characteristic of the intermediate Okhotsk Sea water). The oceanographic structure and a heat budget analysis suggest that new ice zone, which often appears at ice edges, can be formed through preconditioning of thick ice advection and subsequent cooling by the latent heat release due to its melting. This revised version was published online in August 2006 with corrections to the Cover Date.     

Second-order moment advection scheme applied to Arctic Ocean simulation

We apply the second-order moment (SOM) advection scheme of (Prather, M.J. 1986. Numerical advection by conservation of second-order moments. J. Geophys. Res. 91, 6671–6681.) to the simulation of the large-scale circulation of the Arctic Ocean with a coupled ocean–sea-ice model. Compared to three other advection schemes commonly employed in ocean simulations (centred differences, flux corrected transport, and multidimensional positive definite advection transport), the SOM method helps preserve the vertical structure of Arctic water masses. The depth, thickness and hydrographic properties of the Arctic Surface Water and the Arctic Atlantic Layer are better represented with SOM than with any of the other three advection algorithms. We also present a convenient method for calculating the implicit numerical diffusivity of upstream based schemes, such as the SOM method, and discuss three approaches for improving the monotonicity properties of the SOM algorithm.     

Dispersion and stability analyses of the linearized two-dimensional shallow water equations in Cartesian coordinates

In the present study, a Fourier analysis is used to develop expressions for phase and group speeds for both continuous and discretized, linearized two-dimensional shallow water equations, in Cartesian coordinates. The phase and group speeds of the discrete equations, discretized using a three-point scheme of second order, five-point scheme of fourth order and a three-point compact scheme of fourth order in an Arakawa C grid, are calculated and compared with the corresponding values obtained for the continuous system. The three-point second-order scheme is found to be non-dispersive with grid resolutions greater than 30 grids per wavelength, while both the fourth-order schemes are non-dispersive with grid resolutions greater than six grids per wavelength. A von Neumann stability analysis of the two- and three-time-level temporal schemes showed that both schemes are stable. A wave deformation analysis of the two-time-level Crank–Nicolson scheme for one-dimensional and two-dimensional systems of shallow water equations shows that the scheme is non- dispersive, independent of the Courant number and grid resolution used. The phase error or the dispersion of the scheme decreases with a decrease in the time step or an increase in grid resolution.     

Distribution of zooplankton biomass at three seamounts in the NE Atlantic

During different seasons of the years 2003–2005 in the NE Atlantic, zooplankton were sampled with a MOCNESS (multiple opening/closing net and environmental sensing system, mesh size 333 μm) above the slopes and summits of Seine, Sedlo and Ampère seamounts and at remote reference sites outside the influence of the seamounts (far field). Wet weights of different zooplankton size classes (<0.5, 0.5–2, >2 cm) were measured. Night and day hauls were analysed in order to detect diel vertical migrations of the zooplankton, as well as a possible trapping effect due to the shallow topography.Biomass concentrations, independent of daytime, season and summit height, were reduced above the summits at all three seamounts compared to the slope and far-field sites. No trapping effect or retention of biomass was apparent above the seamounts. The vertical distribution patterns of the size class <0.5 cm did not differ between night and day hauls at most sites, but indications of diel vertical migrations were found in the larger size fractions. With the exception of gelatinous organisms, zooplankton >0.5 cm were nearly absent above the summits of Seine and Ampère seamounts, but considerable numbers were found above the slopes and at the far-field sites. Possible explanations for the observed distribution patterns of zooplankton biomass and size classes are discussed, including retention and lateral advection due to the hydrography at the seamounts, as well as predation by resident seamount fish.     

Numerical simulation of a mesoscale convective system over the east coast of South Africa

Weather stations over northern KwaZulu-Natal, South Africa, recorded over 100 mm of rainfall during the night of 11–12 February 2005. This heavy rainfall was associated with a mesoscale convective system (MCS) that was initiated from small convective storms beginning early in the afternoon on 11 February 2005. An analysis of Medium Range Forecast (MRF) model output suggests that a combination of synoptic features contributed to the evolution of the system. It is likely that the high-lying topography of the eastern escarpment and high diurnal surface heating provided the trigger for the event. Although MCSs are not uncommon in this region, very little work has been done on South African cases.
The MCS is investigated with a non-hydrostatic numerical model (mesoscale model version 3; MM5) to help determine which processes were important in its initiation and development, as well as what factors contributed to the associated heavy rainfall. The model results suggest that the eastern escarpment played a key role in triggering the convective event, as well as influencing the low-level winds that advected moisture into the region. It was also apparent that the Agulhas Current played an important role in supplying moisture to fuel the extreme rainfall. The development of the MCS and the heavy nocturnal rainfall was due to a combination of the continuous moisture supply into the region, a conditionally unstable atmosphere, and uplift due to low-level convergence and the local topography.     

Measurements and modelling of the advection of suspended sediment in the swash zone by solitary waves

Novel laboratory experiments and numerical modelling have been performed to study the advection scales of suspended sediment in the swash zone. An experiment was designed specifically to measure only the sediment picked up seaward of the swash zone and during bore collapse. The advection scales and settling of this sediment were measured during the uprush along a rigid sediment-free beach face by a sediment trap located at varying cross-shore positions. Measurements were made using a number of repeated solitary broken waves or bores. Approximately 25% of the pre-suspended sediment picked up by the bores reaches the mid-swash zone (50% of the horizontal run-up distance), indicating the importance of the sediment advection in the lower swash zone. The pre-suspended sediment is sourced from a region seaward of the shoreline (still water line) which has a width of about 20% of the run-up distance. An Eulerian–Lagrangian numerical model is used to model the advection scales of the suspended sediment. The model resolves the hydrodynamics by solving the non-linear shallow water equations in an Eulerian framework and then solves the advection–diffusion equation for turbulence and suspended sediment in a Lagrangian framework. The model provides good estimates of the measured mass and distribution of sediment advected up the beach face. The results suggest that the correct modelling of turbulence generation prior to and during bore collapse and the advection of the turbulent kinetic energy into the lower swash is important in resolving the contribution of pre-suspended sediment to the net sediment transport in the swash zone.     

Comparison of forms of the viscous shallow-water equations in the boundary-fitted coordinates

In order to determine robustness of the models based on the viscous shallow-water equations in the boundary-fitted coordinates the solution characteristics of the boundary-value problem are compared for different forms of equations in contravariant, covariant and Cartesian presentations of the velocity. Distinctions between these forms associated with presentation of advection and eddy viscosity in their approximation on a curvilinear grid are established. A boundary-value problem for the tidal dynamics computation in the Strait of Messina (Mediterranean Sea) is formulated with use of different forms of equations in the boundary-fitted coordinates. An estimation and comparison of the modelling results are performed.     

Improved representation of topographic effects by a vertical adaptive grid in vector-ocean-model (VOM). Part I: Generation of adaptive grids

This is the first part of a publication that describes the generation of adaptive grids (this part), and simulations with vector-ocean-model (VOM) in unstructured grids resulting from the adaptation (part II). A static vertical adaptive grid in z-coordinates allows improving the approximation of topography and vertical resolution at slopes. Adaptive grids use elements from a set of grid sizes by multiplying a basic smallest cell size with powers of two, as in cell division. Grids with locally isotropic vertical resolution at surface, seabed, and slopes can be generated whereby resolution decreases towards the ocean interior. The adaptation to topography yields unstructured grids that are organised in a one-dimensional vector by column-wise storage of cells, discarding land cells. The vector storage suggested the model’s name. Grids are generated by an iterative procedure that relies on rules, i.e. criteria and directives to control the grid structure in favour of a good representation of physics and smooth numerical operations. The directives govern vertical resolution at sea surface and seabed, and at slopes. For the latter vertical resolution is extended in the horizontal. In the ocean interior horizontal distances between changes in grid size can be controlled for the sake of smooth numerics. The use of a z-grid that avoids transformation errors, the depth-independence of vertical resolution, and the lateral extension of vertical resolution at slopes towards the ocean interior are the most significant differences of adaptive grids in comparison to vertical coordinate transformations. Unstructured grids do not rely on a smoothing of topography and can be used within any of the horizontal Arakawa-grids. For the same topography directives allow creating various grids as demonstrated for a shelf-ocean topography. The number of cells per column in two unstructured grids generated for the North Atlantic may locally well exceed typical layer numbers in conventional model matrices. But the domain average is similar to layer numbers of today’s ocean models. Thus, with the same investment of cells per domain a higher resolution in slope regions can be achieved by unstructured grids as compared to conventional z-grids.     

Implementation of high-order particle-tracking schemes in a water column model

Stochastic differential equations (SDEs) offer an attractively simple solution to complex transport-controlled problems, and have a wide range of physical, chemical, and biological applications, which are dominated by stochastic processes, such as diffusion. As for deterministic ordinary differential equations (ODEs), various numerical scheme exist for solving SDEs. In this paper various particle-tracking schemes are presented and tested for accuracy and efficiency (time vs. accuracy). To test the schemes, the particle tracking algorithms are implemented into a community wide used 1D water column model. Modelling individual particles allows a straightforward physical interpretation of the involved processes. Further, this approach is strictly mass conserving and does not suffer from the numerical diffusion that plagues grid-based methods. Moreover, the Lagrangian framework allows to assign properties to the individual particles, that can vary spatially and temporally. The movement of the particles is described by a stochastic differential equation, which is consistent with the advection–diffusion equation. Here, the concentration profile is represented by a set of independent moving particles, which are advected according to the velocity field, while the diffusive displacements of the particles are sampled from a random distribution, which is related to the eddy diffusivity field.The paper will show that especially the 2nd order schemes are accurate and highly efficient. At the same level of accuracy, the 2nd order scheme can be significantly faster than the 1st order counterpart. This gain in efficiency can be spent on a higher resolution for more accurate solutions at a lower cost.     

A new adaptive Cartesian-grid CIP method for computation of violent free-surface flows

A new adaptive Cartesian-grid for the CIP (constrained interpolation profile) method is proposed and applied to two-dimensional numerical simulations of violent free-surface flows. The CCUP (CIP combined and unified procedure) method is employed and combined with this adaptive Cartesian-grid for robust and efficient computation. This adaptive grid is capable of tracking regions where flows vary violently, and a much finer grid is then concentrated automatically on these regions to adapt to the violent changing of the flow. Unlike the abacus-like Soroban grid which is an adaptive meshless grid with complicated algorithms and inefficiency of evaluation of frequently computed spatial derivatives, the present approach not only simplifies computational algorithm but also enhances efficiency of frequently-computed spatial derivatives. It is also different from most of the remeshing schemes that no additional CPU-time for the value-mapping from the old grid to the new grid is taken in this adaptive grid system provided that the advection velocity is interpolated, since the value-mapping process is accomplished simultaneously within the advection process. To validate the accuracy and efficiency of this newly-proposed CFD model, several two-dimensional benchmark problems are performed, and the results are compared with experimental measurements and other published numerical results. Numerical simulations show that the proposed numerical model is robust, accurate, and efficient for strongly nonlinear free-surface flows.     

A BFG model for calculation of tidal current and diffusion of pollutants in nearshore areas

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Energetics of lateral eddy diffusion/advection：Part Ⅳ. Energetics of diffusion/advection in sigma coordinates and other coordinates

Gravitational potential energy （GPE） source and sink due to stirring and cabbeling associated with sigma dif fusion/ advection is analyzed. It is shown that GPE source and sink is too big, and they are not closely linked to physical property distribution, such as temperature, salinity and velocity. Although the most frequently quoted advantage of sigma coordinate models are their capability of dealing with topography; the exces sive amount of GPE source and sink due to stirring and cabbeling associated with sigma diffusion/advec tion diagnosed from our analysis raises a very serious question whether the way lateral diffusion/advection simulated in the sigma coordinates model is physically acceptable. GPE source and sink in three coordinates is dramatically different in their magnitude and patterns. Overall, in terms of simulating lateral eddy diffu sion and advection isopycnal coordinates is the best choice and sigma coordinates is the worst. The physical reason of the excessive GPE source and sink in sigma coordinates is further explored in details. However, even in the isopycnal coordinates, simulation based on the Eulerian coordinates can be contaminated by the numerical errors associated with the advection terms.     

Nonhydrostatic tidal dynamics in the area of a seamount

The nonhydrostatic boundary problem for an arbitrary three-dimensional domain with a seamount is considered. The problem is integrated into curvilinear boundary-fitted coordinates on a nonuniform grid. In order to identify nonhydrostatic effects the grid is condensed on the slopes of the seamount preserving a coarse resolution in the rest of the domain, where the problem is solved in the hydrostatic approximation. Calculation results for the nonhydrostatic tidal dynamics and hydrology of the Strait of Messina in the area of a seamount are presented.     

Horizontal Pressure Gradient Errors of the Monterey Bay Sigma Coordinate Ocean Model with Various Grids

A coastal ocean -coordinate model of Monterey Bay (MOB) with realistic bottom topography and coastlines is developed using the Princeton Ocean Model (POM) and grid generation technique (GGT) to study the horizontal pressure gradient errors associated with the MOB steep topography. The submarine canyon in MOB features some of the steepest topography encountered anywhere in the world oceans. The MOB grids are designed using the EAGEAL View and GENIE⁺⁺ grid generation systems. A grid package developed by Ly and Luong (1993) is used in this study to couple grids to the model. The MOB model is tested with both orthogonal and curvilinear nearly-orthogonal (CNO) grids. The CNO grid has horizontal resolution which varies from 300 m to 2 km, while the resolution of the orthogonal grid is uniform with x = 1.25 km and y = 1.38 km. These grids cover a domain of 180 × 160 km with the same number of grid points of 131 × 131. Vertical resolutions of 25, 35 and 45 vertical sigma levels are tested. The error in the MOB are evaluated in terms of mean kinetic energy and velocity against various grids, vertical, horizontal resolution and distributions, and bottom topography smoothing. Simulations with various grids show that GGT can be used as another tool in reducing -coordinate errors in coastal ocean modeling besides increasing resolution and smoothing bottom topography. Topographical smoothing not only reduces topographic slope, but changes realistic topography. A CNO grid with a high grid density packed along steep slopes and Monterey Submarine Canyon reduces the errors by 40% compared to a rectangular grid with the same number of grid points. The CNO grid is more efficient than the rectangular grid, since it has most of its grids over water. The simulations show that the presented MOB -coordinate model can be used with a confidence regarding horizontal pressure gradient error.