3D Primitive Equations Research Articles

EARLY EXPERIENCES WITH THE 360TF IBM BLUE GENE/L PLATFORM

The High Order Method Modeling Environment is a scalable, spectral-element-based prototype for the Community Atmospheric Model component of the Community Climate System Model. The 3D moist primitive equations are solved on the cubed sphere with a hybrid pressure η vertical coordinate using an Emanuel convective parametrization for moist processes. Semi-implicit time integration, based on a preconditioned conjugate gradient solver, circumvents the time step restrictions associated with gravity waves. Benchmarks for two standard tests problems at 10 km horizontal resolution have been run on Blue Gene/L. Results obtained on a 32-rack Blue Gene/L system (65,536 processors, 183.5-teraflop peak) show sustained performance of 8.0 teraflops on 32,768 processors for the moist Held–Suarez test problem in coprocessor mode and 11.3 teraflops on 32,768 processors for the aquaplanet test problem, running in virtual node mode.

Uniform gradient bounds for the primitive equations of the ocean

In this paper, we consider the 3D primitive equations of the ocean in the case of the Dirichlet boundary conditions on the side and bottom boundaries. We provide an explicit upper bound for the $H^{1}$ norm of the solution. We prove that, after a finite time, this norm is less than a constant which depends only on the viscosity $\nu$, the force $f$, and the domain $\Omega$. This improves our previous result from [7] where we established the global existence of strong solutions with an argument which does not give such explicit rates.

The 3D Primitive Equations in the absence of viscosity: Boundary conditions and well-posedness in the linearized case

In this article we consider the 3D Primitive Equations (PEs) of the ocean, without viscosity and linearized around a stratified flow. As recalled in the Introduction, the PEs without viscosity ought to be supplemented with boundary conditions of a totally new type which must be nonlocal. In this article a set of boundary conditions is proposed for which we show that the linearized PEs are well-posed. The proposed boundary conditions are based on a suitable spectral decomposition of the unknown functions. Noteworthy is the rich structure of the Primitive Equations without viscosity. Our study is based on a modal decomposition in the vertical direction; in this decomposition, the first mode is essentially a (linearized) Euler flow, then a few modes correspond to a stationary problem partly elliptic and partly hyperbolic; finally all the other modes correspond to a stationary problem fully hyperbolic.

Real-time forecasting of ecosystem dynamics during the CROZEX experiment and the roles of light, iron, silicate, and circulation

Real-time coupled physical and biological forecasting was conducted prior to and during the CROZet natural bloom and EXport experiment (CROZEX) programme in the Southern Ocean (SO) between November 2004 and January 2005. The programme was aimed at investigating, through both measurements and modelling, the origin and fate of an intense and long-lasting phytoplankton bloom that is unusual for the otherwise high-nutrient low-chlorophyll conditions that characterise the SO. It was hypothesised that dissolved iron supplied to the system from shallow sediments accumulates downstream of the Crozet Islands during winter and creates conditions that favour phytoplankton production. Regular 14-day forecasts and reanalysis of the biogeochemical fields were issued on a weekly basis using a 3D primitive equation model coupled with an 11-component ecosystem model, along with assimilation of remotely sensed and in situ data. Real-time modelling assisted not only in optimisation of the cruise strategy, but also provided an ongoing synthesis of observations and test of the working hypotheses proposed to explain bloom dynamics in the study area.The model results indicated that a complex interplay between light, iron, and silicate limitation on primary production and grazing control determined the spatial extent and dynamics of the phytoplankton bloom. The model suggested that natural fertilisation of the area by dissolved iron was insufficient on its own to initiate a phytoplankton bloom. The persistence of deep mixing in the southern part of the iron-enriched waters was delaying, or in some areas preventing, the development of the bloom.

Forcing a three‐dimensional, hydrostatic, primitive‐equation model for application in the surf zone: 1. Formulation

An Eulerian analysis for wave forcing of three‐dimensional (3D) wave‐averaged mean circulation in the surf zone is presented. The objective is to develop a dynamically consistent formulation for applications in a 3D primitive equation model. The analysis is carried out for the case of shallow‐water linear waves interacting with wave‐averaged depth‐independent horizontal currents that vary on larger space scales and timescales. Variations in wave properties are governed by a wave action equation that includes wave‐current interactions and dissipation representative of wave breaking. Wave forcing of the mean currents consists of a surface stress and a body force. The surface stress is proportional to the wave energy dissipation. The body force includes one term that is related to gradients of part of the radiation stress tensor and a second term that is related to the vortex force and is proportional to a product of the mean wave momentum and the vertical component of the mean vorticity vector. In addition, there is a nonzero normal velocity at the mean surface that arises from the divergence of the mean Eulerian wave mass flux. This velocity results in an additional momentum flux forcing of the mean flow. Applications of this formulation to the DUCK94 field experiment are presented in part 2.

Hydrodynamics in the Gulf of Aigues-Mortes, NW Mediterranean Sea: In situ and modelling data

The Gulf of Aigues-Mortes (NW Mediterranean Sea) is a midshelf zone whose scale is an intermediate between the nearshore scale (0–10 m depth) and the coastal scale (including the whole continental shelf). Its hydrodynamics is investigated for the first time. ADCP, CTD and thermosalinograph data were collected during three short cruises (HYGAM; March 6–7, 20–21, April 5–6, 2005). They were scheduled approximately every 15 days to sample the gulf circulation under different weather conditions. Moreover, the cruise data were used to validate the Symphonie model, a 3D primitive equations circulation model. The circulation features displayed by in situ data were well reproduced by Symphonie. A downscaling modelling approach was implemented, the largest scale being obtained by the replay of the MFSTEP regional model of the North-Western Mediterranean Sea. The analysis, closely coupling in situ measurements and model results, provides information that would not have been obtained using data separately. The great variability of the oceanic circulation at this scale is well evidenced. Winds are the main forces, not only locally but also at a larger scale. North winds generate coherent structures interacting with the general Mediterranean circulation. The explanation of the induced currents is then not straightforward, some of them being, for example, northward. South and southeast winds reinforce the sloping surface, the latter allowing the geostrophic equilibrium with longshore currents. This study, focused on the end of winter 2004–2005, also enhances the heat fluxes from offshore to nearshore, as well as the initiation of shelf dense waters. The knowledge of the midshelf oceanic circulation is then applied to run a numerical experiment which describes both the dispersion of a fictive contaminant from a point S (at the centre of the GAM), and evaluates the risk of contamination of the beaches along the GAM shores.

The global attractor for the solutions to the 3D viscous primitive equations

Existence of the global attractor is proved for the <em> strong </em> solutions to the 3D viscous Primitive Equations (PEs) modeling large scale ocean and atmosphere dynamics. This result is obtained under the natural assumption that the external heat source $Q$ is square integrable. Furthermore, it is shown in [20] that the fractal and Hausdroff dimensions of the global attractor for 3D viscous PEs are both finite.

Phytoplankton size classes competitions at sub-mesoscale in a frontal oceanic region

The effects of mesoscale and sub-mesoscale dynamics on the competition between two different phytoplankton size classes are investigated with a 3D primitive equations model. The model reproduces realistic simulations of mesoscale turbulence generated by a westward current in the southern hemisphere at statistical equilibrium in a summer situation. Effects of two different grazing pressures on phytoplankton competitions are compared and the role of eddy variability is quantified comparing high and low resolution simulations. High resolution simulations reveal a filamentary distribution of biomass and nutrients induced by the combination of vertical advection and horizontal stirring. This fine scale variability is observed not only on the horizontal but also on the vertical into the subsurface chlorophyll maximum. One of the key results is that such a dynamics induces a spatial segregation of the phytoplankton in the southern part of the frontal region that is mainly filamentary. This spatial segregation consists in biomass maxima for large phytoplankton in rich nutrients filaments and maxima for small phytoplankton outside these filaments. This anti-correlation is particularly strong when grazing pressure is low and is confirmed by statistical analysis. In the central frontal region, dominated by mesoscale dynamics, the two phytoplankton classes are strongly correlated together and biomass maxima are located close to downwelling regions that are poor in nutrients. It is shown that the effect of grazing is significantly amplified by the fine scale dynamics and that the combination of these two mechanisms is responsible of a switch of the ecosystem dominance in the surface layers. In addition, the effect of frontal dynamics on the detritus export is very sensitive to grazing pressure: increasing grazing induces a significant decrease of the export in the presence of frontal dynamics whereas it induces an increase of the export without small-scale variability.

Evaluating hypotheses for the initiation and development of Alexandrium fundyense blooms in the western Gulf of Maine using a coupled physical–biological model

A coupled physical/biological model and observations are used to investigate the factors governing the initiation and development of an Alexandrium fundyense bloom in the western Gulf of Maine (WGOM) during the spring of 1993 (March 19–June 6). The physical circulation is simulated using a 3D primitive equation model forced by climatological sea-surface elevation and observed winds, irradiance, and river outflow. This is coupled with a biological model constructed from laboratory and field data that estimates the germination and growth rates of A. fundyense as a function of environmental conditions. Four biological model structures of increasing complexity are considered, with each structure representing a hypothesis for factors controlling bloom initiation and development. The model/data fit is optimized over the uncertainty in the parameters to which the model is most sensitive. The significance of changes in the model/data fit between model structures is quantified using a maximum likelihood ratio test. The baseline biological model, which parameterizes growth as only a function of temperature, salinity, and light, severely over-estimates observed A. fundyense abundance in the late spring. It is thus rejected with greater than 99% confidence in favor of biological models that include a mortality term or a dependence of growth on dissolved inorganic nitrogen (DIN). The overall best-fit simulation uses both nitrogen dependence and mortality. However, simulations using one or the other of these factors could not be differentiated from the best-fit case with greater than 90% confidence. The best-fit model captures the general timing and magnitude of the observed bloom and some of its secondary features. However, considerable misfits may exist in the point-to-point comparison, and some regional misfits remain. Diagnosis of the cell budget suggests that germination from a large cyst bed offshore of Casco Bay provides the majority of cells comprising spring A. fundyense populations within the WGOM. The size of the modeled bloom is largely set by the size of this cyst-driven source. Transport of cells from the eastern Gulf of Maine becomes increasingly important later in the spring. Net growth of the modeled A. fundyense populations is first limited by low water temperatures and then by the combined influence of nitrogen limitation and mortality. This results in low domain-averaged net growth rates (o0.05 day � 1 ) throughout much of the simulation. However, rates are sometimes elevated locally and therefore add notable spatial structure to the bloom. Primary uncertainties within the biological model include spatial and temporal variability in mortality, and the influence of sediment dynamics and inter-annual variability in the cyst abundance on the

FoSSI: the family of simplified solver interfaces for the rapid development of parallel numerical atmosphere and ocean models

The portable software FoSSI is introduced that—in combination with additional free solver software packages—allows for an efficient and scalable parallel solution of large sparse linear equations systems arising in finite element model codes. FoSSI is intended to support rapid model code development, completely hiding the complexity of the underlying solver packages. In particular, the model developer need not be an expert in parallelization and is yet free to switch between different solver packages by simple modifications of the interface call. FoSSI offers an efficient and easy, yet flexible interface to several parallel solvers, most of them available on the web, such as PETSC, AZTEC, MUMPS, PILUT and HYPRE. FoSSI makes use of the concept of handles for vectors, matrices, preconditioners and solvers, that is frequently used in solver libraries. Hence, FoSSI allows for a flexible treatment of several linear equations systems and associated preconditioners at the same time, even in parallel on separate MPI-communicators. The second special feature in FoSSI is the task specifier, being a combination of keywords, each configuring a certain phase in the solver setup. This enables the user to control a solver over one unique subroutine. Furthermore, FoSSI has rather similar features for all solvers, making a fast solver intercomparison or exchange an easy task. FoSSI is a community software, proven in an adaptive 2D-atmosphere model and a 3D-primitive equation ocean model, both formulated in finite elements. The present paper discusses perspectives of an OpenMP-implementation of parallel iterative solvers based on domain decomposition methods. This approach to OpenMP solvers is rather attractive, as the code for domain-local operations of factorization, preconditioning and matrix–vector product can be readily taken from a sequential implementation that is also suitable to be used in an MPI-variant. Code development in this direction is in an advanced state under the name ScOPES: the Scalable Open Parallel sparse linear Equations Solver.

Control of the Tropical Tropopause and Vertical Transport across It

Abstract A 3D primitive equation model is used to investigate how the tropical tropopause is influenced by cumulus convection in the troposphere and mean upwelling in the stratosphere. The model simulates the residual mean circulation explicitly, whereas it represents the influence of convection on large-scale structure through statistical properties of cloud. Within this global framework, a change of extratropical planetary waves induces a large change of downwelling over the winter hemisphere, compensated at lower latitudes by a change of upwelling. This exerts a major influence on thermal structure in the extratropics, but only a minor one in the Tropics. On seasonal time scales, however, the influence in the Tropics is significant. During northern winter, extratropical planetary waves are sharply amplified. The accompanying intensification of tropical upwelling, while small compared to the intensification of extratropical downwelling, accounts for about half of the observed seasonal change of the trop...

A Scalable Spectral Element Eulerian Atmospheric Model (SEE-AM) for NWP: Dynamical Core Tests

A new dynamical core for numerical weather prediction (NWP) based on the spectral element method is presented. This paper represents a departure from previously published work on solving the atmospheric primitive equations in that the horizontal operators are all written, discretized, and solved in 3D Cartesian space. The advantages of using Cartesian space are that the pole singularity that plagues the equations in spherical coordinates disappears; any grid can be used, including latitude–longitude, icosahedral, hexahedral, and adaptive unstructured grids; and the conversion to a semi-Lagrangian formulation is easily achieved. The main advantage of using the spectral element method is that the horizontal operators can be approximated by local high-order elements while scaling efficiently on distributed-memory computers. In order to validate the 3D global atmospheric spectral element model, results are presented for seven test cases: three barotropic tests that confirm the exponential accuracy of the horizontal operators and four baroclinic test cases that validate the full 3D primitive hydrostatic equations. These four baroclinic test cases are the Rossby–Haurwitz wavenumber 4, the Held–Suarez test, and the Jablonowski–Williamson balanced initial state and baroclinic instability tests. Comparisons with four operational NWP and climate models demonstrate that the spectral element model is at least as accurate as spectral transform models while scaling linearly on distributed-memory computers.

Space-time structure and dynamics of the forecast error in a coastal circulation model of the Gulf of Lions

The probability density function (pdf) of forecast errors due to several possible error sources is investigated in a coastal ocean model driven by the atmosphere and a larger-scale ocean solution using an Ensemble (Monte Carlo) technique. An original method to generate dynamically adjusted perturbation of the slope current is proposed. The model is a high-resolution 3D primitive equation model resolving topographic interactions, river runoff and wind forcing. The Monte Carlo approach deals with model and observation errors in a natural way. It is particularly well-adapted to coastal non-linear studies. Indeed higher-order moments are implicitly retained in the covariance equation. Statistical assumptions are made on the uncertainties related to the various forcings (wind stress, open boundary conditions, etc.), to the initial state and to other model parameters, and randomly perturbed forecasts are carried out in accordance with the a priori error pdf. The evolution of these errors is then traced in space and time and the a posteriori error pdf can be explored. Third- and fourth-order moments of the pdf are computed to evaluate the normal or Gaussian behaviour of the distribution. The calculation of Central Empirical Orthogonal Functions (Ceofs) of the forecast Ensemble covariances eventually leads to a physical description of the model forecast error subspace in model state space. The time evolution of the projection of the Reference forecast onto the first Ceofs clearly shows the existence of specific model regimes associated to particular forcing conditions. The Ceofs basis is also an interesting candidate to define the Reduced Control Subspace for assimilation and in particular to explore transitions in model state space. We applied the above methodology to study the penetration of the Liguro-Provençal Catalan Current over the shelf of the Gulf of Lions in north-western Mediterranean together with the discharge of the Rhône river. This region is indeed well-known for its intense topographic and atmospheric forcings.

An Unstructured Grid, Finite-Volume, Three-Dimensional, Primitive Equations Ocean Model: Application to Coastal Ocean and Estuaries

An unstructured grid, finite-volume, three-dimensional (3D) primitive equation ocean model has been developed for the study of coastal oceanic and estuarine circulation. The model consists of momentum, continuity, temperature, salinity, and density equations and is closed physically and mathematically using the Mellor and Yamada level-2.5 turbulent closure submodel. The irregular bottom slope is represented using a s-coordinate transformation, and the horizontal grids comprise unstructured triangular cells. The finite-volume method (FVM) used in this model combines the advantages of a finite-element method (FEM) for geometric flexibility and a finite-difference method (FDM) for simple discrete computation. Currents, temperature, and salinity in the model are computed in the integral form of the equations, which provides a better representation of the conservative laws for mass, momentum, and heat in the coastal region with complex geometry. The model was applied to the Bohai Sea, a semienclosed coastal ocean, and the Satilla River, a Georgia estuary characterized by numerous tidal creeks and inlets. Compared with the results obtained from the finite-difference model (ECOM-si), the new model produces a better simulation of tidal elevations and residual currents, especially around islands and tidal creeks. Given the same initial distribution of temperature in the Bohai Sea, the FVCOM and ECOM-si models show similar distributions of temperature and stratified tidal rectified flow in the interior region away from the coast and islands, but FVCOM appears to provide a better simulation of temperature and currents around the islands, barriers, and inlets with complex topography.

A synopticity test of a sampling pattern in the Alboran Sea

Hydrographic surveys are usually considered as synoptic, as if the corresponding data set had been sampled at the same time. However, this assumption might lead to strong biases in subsequent analyses. To obtain synoptic data at a given moment, we integrate a Lagrangian motion equation towards a ‘modal sampling time’, to relocate the sampling stations both in time and space, by combining objective analysis and geostrophic velocities, assumed to be either stationary or unstationary over the period of integration. The relocation algorithms are applied to a data set covering the entire Alboran Sea. The relocation of data points leads to a significant change on vertical velocities computed with the OMEGA equation. For validation purpose, the algorithm is also applied to a synthetic data set obtained from a 3D Primitive Equation (PE) model output. Raw and pseudosynoptic analysed temperature–salinity (T/S) fields and derived quasigeostrophic (QG) vertical velocities are compared. It is shown that the relocation method might correct significantly the errors arising from the synopticity hypothesis and that the sensitivity to the level of no-motion and the analysis parameters is less important than the nonsynopticity.

Space and time distributions of phosphate in the Mediterranean Sea

Statistical modelling was applied to a large number of historical nutrient data to assess the significance of human perturbations in the Mediterranean Sea. All available phosphate data were grouped into subsets representative of averaged values of the measured vertical profiles in the surface and deep water layers. In contrast to earlier predictions, the statistical analysis of the phosphate concentrations in a deep water layer does not indicate any particular trend in time for the last 30 years. These data sets were then used as an input to an inverse model and a 3D primitive equation model (PEM). The former redistributes the measured concentrations by means of a variational principle and reconstructs average horizontal space distributions of the phosphate data fields as gridded solutions over the whole area. The spatial and temporal distributions thus obtained are visualised graphically and compared with other existing data, providing the first overall view of phosphate in the whole Mediterranean Sea and revealing an increasing oligotrophy towards the eastern basin. The primitive equation model is then used to assess the variability and upwelling fluxes.

Non-synoptic versus pseudo-synoptic data sets: an assimilation experiment

Several first-order correction methods are implemented to compute pseudo-synoptic data sets from non-synoptic raw data sets. These include a geostrophic relocation method, a linear and a quadratic interpolation method, and a method using spatio-temporal correlation functions. The relocation method involves analyses and geostrophic velocity computations to allow the relocation of stations in time and space to a particular analysis time. Interpolation methods involve several almost identical and consecutive surveys interpolated in time. Temporal weighting methods are based upon a spatio-temporal function modifying the weight on data with respect to the time at which they have been sampled. These techniques are tested on the OMEGA data set and are validated by simple nudging assimilation into a 3D primitive equation model (PE). It is shown that, under certain hypothesis, these methods are able to correct the lack of synopticity in hydrographic data sets, and improve the diagnosis of vertical velocities computed from the Omega equation. These methods are of particular interest for the scientific community. They might be used together with diagnostic models. They might provide suitable pseudo-synoptic fields required by 3D PE models as initial conditions, boundary conditions or assimilation data sets. They may also be useful in the design of mesoscale samplings.

A Modeling Study of the Episodic Cross-Frontal Water Transport over the Inner Shelf of the South Atlantic Bight

Abstract Physical processes that control the episodic cross-frontal transport over the inner shelf of the South Atlantic Bight were examined using a 3D primitive turbulent closure equation model. The model results showed that significant cross-frontal water exchange occurred under upwelling-favorable wind conditions through the detachment of isolated low-salinity lenses at the outer edge of the front. The formation of isolated lenses was a 3D feature associated with complex nonlinear processes. It depended on 1) the amount of multiple river discharges, 2) direction and magnitude of the wind, and 3) tidal mixing. As the total amount of river discharges increased, stronger winds were required to form the isolated lenses. When the river discharges were larger, tidal mixing tended to delay the occurrence of isolated lenses through 1) increasing the cross-shelf gradients of density and surface elevation and thus enhancing the along-shelf current and reducing the along-shelf variation of salinity, and 2) streng...

Fast Singular Oscillating Limits and Global Regularity for the 3D Primitive Equations of Geophysics

Fast singular oscillating limits of the three-dimensional primitive equations of geophysical fluid flows are analyzed. We prove existence on infinite time intervals of regular solutions to the 3D primitive Navier-Stokes equations for strong stratification (large stratification parameter N ). This uniform existence is proven for periodic or stress-free boundary conditions for all domain aspect ratios, including the case of three wave resonances which yield nonlinear dimensional limit equations for N → +∞; smoothness assumptions are the same as for local existence theorems, that is initial data in Hα , α ≥ 3/4. The global existence is proven using techniques of the Littlewood-Paley dyadic decomposition. Infinite time regularity for solutions of the 3D primitive Navier-Stokes equations is obtained by bootstrapping from global regularity of the limit resonant equations and convergence theorems.

Seasonal variation of temperature and salinity fields and water masses in the Bien Dong (South China) sea

Seasonal and bimonthly analyses of the CTD and STD data in the Bien Dong Sea have been performed using a variational inverse method (VIM). The results, materialized as gridded data sets (horizontal resolution at 1 4 of a degree at 26 depth levels from 5m to 4500m), show the basic features of the hydrology of the Bien Dong Sea and its trend of seasonal variabilities affecting the properties of the water masses. The seasonally reversing monsoon wind and the variation of net air-sea fluxes play an important role in determining the transformation and spreading of the water masses. The present work is conceived as a basic support to more advanced studies, including diagnostic calculations and assimilation of hydrological data in 3D primitive equation models for the Bien Dong Sea.