V. B. Zalesny

Numerical simulation of large-scale ocean circulation based on the multicomponent splitting method

Abstract The problem of mathematical modelling of the baroclinic dynamics of the World Ocean and its large areas is considered. The mathematical model has been developed at the Institute of Numerical Mathematics RAS (INMRAS) and is based on the equations of general circulation represented in the generalized σ-system of coordinates with a free surface in the hydrostatics and Boussinesq approximation. The computational solution technique for this problem is based on multicomponent splitting and has a flexible, hierarchically scalable modular structure. The main ‘cycle’ is the decomposition of a complicated system of equations into a series of large and energetically balanced subsystems called modules, according to physical processes. The complete numerical model is determined by the set of split subsystems and approximate the original model with some order of time accuracy. Along with the solution of direct prognostic problems, the method of multicomponent splitting can be efficiently used for solution of four-dimensional problems of observation data variational assimilation. The computing methodology is illustrated by several problems of large-scale ocean circulation, including the problem with four-dimensional variational initialization of hydrophysical fields.

Numerical model of the circulation of the Black Sea and the Sea of Azov

The problem of mathematical modelling of the dynamics of the Black Sea and the Sea of Azov is considered with the use of the INMOM model developed at the Institute of Numerical Mathematics (INM) of the Russian Academy of Sciences. The model is based on the equations of general circulation written in a spherical σ -system of coordinates with a free surface boundary in the hydrostatics and Boussinesq approximations. The equations of sea dynamics are written in a symmetrized form. The numerical algorithm is based on the method of multicomponent splitting and has a flexible modular structure. Splitting with respect to physical processes and spatial coordinates is used. The problem is split into a series of energy-balanced subsystems called modules. Each particular module can be split further into modules of a simpler structure. The numerical experiment consists in the calculation of hydrophysical fields of the Black Sea and the Sea of Azov with the spatial resolution of ∼ 4 km, and 40 σ -levels non-uniformly distributed in depth are used in the vertical direction. Atmospheric forcing is calculated according to the EraInterim data, the calculation period is 3 years, from 2006 to 2008. The results of numerical simulation demonstrate good concordance with the observation data and also with the calculation of the Black Sea dynamics by the model of the Marine Hydrophysical Institute of the National Academy of Sciences of Ukraine. It is proposed to use the model presented here in the development of a monitoring system and for real-time forecast of water circulation in the Black Sea and the Sea of Azov. The problem of operative forecast [16] becomes nowadays one of the central problems of the mathematical modelling of seas and oceans. The base of real-time forecasts is a water circulation model for a given basin. At present, the system of modelling and real-time forecast for the dynamics of the Black Sea has been developed at the Marine Hydrophysical Institute of the National Academy of Sciences of Ukraine (MHI NASU) [16]. A finite difference model with explicit time integration schemes is used for this purpose (see [8–10]). The model adequately represents the structure of hydrophysical fields and is supplied with a block of observation data assimilation (see [10, 16]). The observation data assimilation is based on the application of the ∗Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow 119333, Russia †Marine Hydrophysical Institute of the National Academy of Sciences of Ukraine, Sevastopol 99011, Ukraine The work was supported by the Program of the Presidium of the Russian Academy of Sciences No. 21.1 ‘The Black Sea as a simulation ocean model’, by the Federal Target Program ‘Scientific and Pedagogical Staff for Innovative Russia’, by grants of the Russian Foundation for Basic Research and by the Grant Council of the President of the Russian Federation. 96 V. B. Zalesny et al. Kalman filter. The system gives satisfactory forecast accuracy in an open sea with the largest errors near the frontal zones. The further improvement of the forecast accuracy requires the use of more efficient numerical calculation methods, modern schemes of observation data assimilation, and the ability to perform calculations on remote computers. The products of the Institute of Numerical Mathematics of the Russian Academy of Sciences (INM RAS) in the field of sea circulation modelling based on the methods of splitting and adjoint equations [18] seem to be very promising for the prognostic system of the Black Sea. The application of such models allows one to improve the stability of calculations, increase the spatial resolution of the model, develop an efficient method of four-dimensional variational data assimilation. As the result, this should lead to a qualitative improvement of operative forecasts. The use of modern numerical methods, in particular, non-structured grids [7] may also provide us with the ability for an environment evolution prognosis in the coastal zone of the Black Sea within the framework of an integrated model. At present, simulation methods for the sea and ocean dynamics and algorithms of variational assimilation of observation data based on methods of splitting and adjoint equations are developed at the INM RAS (see [3, 18, 19, 27, 30]). The method of splitting has the two basic functions: first, it allows one to solve the problem in time efficiently; second, to construct a flexible, hierarchically developed information computing system (see [3, 20, 30]). The essence of the method is the representation of a complicated problem as a set of simpler modules. The model can be simplified, or physically enriched by rejecting or adding some modules. The method of adjoint equations is the base of analysis of complex systems (see [3, 19]). It essentially decreases the dimension of the space of solutions in the problem of the optimal choice of the initial parameters of the model aimed at bringing the model solution closer to the observation data. However, the following essential difficulty arises in the use of the method: it is necessary to construct and implement an adjoint model whose equations have a more complicated form comparing to the direct prognostic system. If the method of solution of the prognostic problem, or the form of equations, or their spatial approximation are changed, we have to change the adjoint analogue of the model too. This is rather laborious from the technological viewpoint. In our approach combining two methods, i.e., splitting and adjoint equations techniques, we are able to solve the variational assimilation problem more efficiently. We have to construct an adjoint analogue to a particular split module of the direct model. The direct model is composed from direct split problems. The adjoint one is composed from the corresponding modules of adjoint subproblems. Our approach simplifies the construction of the adjoint model and gives us the ability to calculate an algebraically precise gradient of the minimized cost function. In this paper we present the direct prognostic circulation model of the Black Sea and the Sea of Azov. Further we will construct the model of four-dimensional assimilation of observation data on its base. Numerical model of the circulation 97 1. Circulation model of the Black Sea and the Sea of Azov The model of hydrodynamics of the Black Sea and the Sea of Azov is the extension of the basic general circulation model of the World Ocean developed at the INM RAS (see [11, 30]). Let us present here a brief description of the problem of the dynamics of the Black Sea and the Sea of Azov and the methods of its solution. The dimensionless variable σ ∈ [0,1] is used in the model as the vertical coordinate: σ = z−ζ (x,y, t) H(x,y)−ζ (x,y, t) (1.1) where z is the vertical depth coordinate measured from the unperturbed sea surface towards the center of the Earth, H(x,y) is the depth of the sea, ζ (x,y, t) is the sea level deviation from its unperturbed state, (x,y) are the horizontal coordinates, in our case these are the longitude and latitude, respectively. It is worth noting that in the general case, INMOM can utilize any orthogonal coordinates, including those with a grid refinement in given subdomains. t is the time. In the notation of the system of primitive sea dynamics equations in the σ -system of coordinates, it is useful to introduce the function of geopotential surfaces expressed according to (1.1) as Z = (H−ζ )σ +ζ , Zσ ≡ ∂Z ∂σ . (1.2) Then the system of sea hydrothermodynamic equations takes the following form in this system of coordinates: Dtu−Zσ (l+ξ) v =− Zσ ρ0rx [ ∂ ∂x ( p− g

Numerical model of the hydrodynamics of the Black Sea and the Sea of Azov with variational initialization of temperature and salinity

A numerical primitive-equation model of the hydrodynamics of the Black Sea and the Sea of Azov in σ-coordinates is proposed. The model has a resolution of ∼4 × 4 km in horizontal coordinates with 40-σ levels in the vertical and includes the four-dimensional variational initialization of temperature and salinity fields. A numerical initialization algorithm combines splitting methods and adjoint equations. Flow, temperature, sea level, and salinity fields driven by atmospheric forcing are calculated for the year 2008. The calculations are made in a variational initialization — prediction regime. Temperature and salinity fields are initialized at the end of each month. The optimality system includes forward and adjoint transport-diffusion equations for heat and salt that are linearized on the assimilation interval. Results of three numerical experiments with different sets of assimilated data in comparison with the prediction obtained from the forward model are discussed.

Data-computing technologies: A new stage in the development of operational oceanography

An analysis is given of the methods of operational oceanography based on measurements derived from satellite data, observations acquired by drifters and passing vessels, and modern simulations of marine and oceanic circulations. In addition, a historical review is conducted of the previous and current research in this field carried out in the Soviet Union, Ukraine, and Russia. A discussion is given of the principles underlying the design of an effective data-computing system (DCS) for solving the problems of operational oceanography and the implementation of the prototype system for the Black Sea within the joint research project of the Russian Academy of Sciences (RAS) and the National Academy of Sciences of Ukraine (NASU) “The Black Sea as an Ocean Simulation Model.” The effectiveness of applying the multicomponent splitting method in the construction of sea circulation models and specialized DCSs with integrated algorithms of variational assimilation of observational data is estimated. The concept of using the Black Sea as a testing site for innovations is developed. The underlying idea of the concept is the similarity of the Black Sea dynamics with processes in the oceans. The numerical Black Sea circulation models used in the project are described, their development areas are discussed, and the requirements to a Black Sea observing system are defined.

Modeling of the World Ocean circulation with the four-dimensional assimilation of temperature and salinity fields

The problem of modeling the World Ocean circulation with the four-dimensional assimilation of temperature and salinity fields is considered. A mathematical model of the ocean general circulation and a numerical algorithm for its solution are formulated. The model equations are written in a σ coordinate system on the sphere with the North Pole shifted to the point of the continent (60° E, 60.5° N). The model has a flexible numerical structure and consists of two parts: the forward prognostic model and its adjoint analog. The numerical algorithm for solving the forward and adjoint problems is based on the method of multicomponent splitting. This method includes splitting with respect to physical processes and geometric coordinates. Three series of numerical experiments are performed: (1) a test solution to the problem of the four-dimensional variational assimilation, (2) modeling of the World Ocean circulation with the variational assimilation of climatic temperature and salinity fields, and (3) modeling of the World Ocean circulation with the variational assimilation of climatic temperature and salinity fields and the data of Argo buoys. The results of calculations demonstrate the expediency of using the model of World Ocean circulation with the procedure of assimilating observational data for a description of the general structure of thermohaline fields.

The 3D physical-biological model study in the Egyptian Mediterranean coastal sea

The ecosystem of the Egyptian Mediterranean coastal area between longitudes 29°45′ E and 33°45′ E was seasonally investigated. The obtained data sets for a variety of environmental variables were previously interpolated over the studied area, and used in the coupled physical-biological model Fin-Est. Here, the model has been implemented to solve the central problems in case of modeling local areas and the handling of open boundaries and initial conditions, which almost have significant influence on the results, of both the hydrodynamic and the ecosystem models calculation. In the present study, a multi-step calculations procedure was used in applying the 3 D coupled model at first for the entire Egyptian Mediterranean coastal area. Then, it runs for local areas obtaining the initial conditions and the open boundary ones from larger area calculations. In particular, the zoom-in approach was used for the detailed study of the Eastern Harbour of Alexandria city-Egypt (model grid step = 50 m.) selected from the larger Alexandria Sea (model grid step = 1.25 Km), which in turn was chosen from the Egyptian Mediterranean coastal area (model grid step = 12.5 km). The sensitivity of the model simulations to both physical and biological main factors of the studied areas is tested. Both climatic conditions and land sources effects are considered among the model forcing factors. The resultant simulations are compared with the actual measured values of the parameters. The model sensitivity test results are discussed in the context of the models capabilities and limitations, and with reference to the available knowledge of the ecosystems of the study areas.

Numerical simulation of the North Atlantic – Arctic Ocean – Bering Sea circulation in the 20th century

Abstract A model of the joint circulation of the North Atlantic, the Arctic Ocean, and the Bering Sea is presented with the resolution of 0.25° in latitude and longitude. The numerical technique of solving this problem and the organization of numerical experiments are described. Numerical calculations have been performed using this model for the period of 1958–2006. The results are compared with observation data and with the results of simulation by other models. Model estimates of the evolution of the Atlantic water incoming into the Arctic basin through the Fram Strait and the Barents Sea are presented. A positive trend of Atlantic water incoming into the Arctic basin through the Fram Strait is revealed. The evolution of the fresh water layer thickness in the Beaufort Gyre is considered. Three periods of increased thickness correlated with increased anticyclonic vorticity are pointed out: 1960s, 1980s, and the period from 1999 until now. The evolution of the anticyclonic vorticity is ahead of the changes in the fresh water layer thickness by 1.75 years. Long-term positive trends of fresh water layer thickness and the anticyclonic field vorticity in the Beaufort Gyre have been observed from the middle of 1970s. This period is characterized by a negative model trend of the ice area in the Arctic, which corresponds to observation data.

Modeling sea dynamics and turbulent zones on high spatial resolution nested grids

The problem of simulating sea dynamics in areas comprising near-shore zones and zones of high turbulence is considered. A mathematical model and the numerical algorithm of its solving are formulated. The model is based on the equations for nonhydrostatic dynamics and includes (k-ε) and (k-ω) parameterization of turbulent processes. The equations of the model are written in a σ-coordinate system. The numerical algorithm for solving the problem is based on the use of implicit schemes owing to the splitting with respect to the physical processes and space coordinates. The model calculations were performed for four nested sea basins with different spatial resolution: the Baltic Sea (3.7-km space resolution), the Gulf of Finland (1.85-km resolution), the Tallinn-Helsinki area (560-m resolution), and Tallinn Bay (93-m resolution). The results of the experiment show that the model well simulates the processes of enhanced turbulent activity in the near-shore zones that affect the local features of the sea characteristics.

The Baltic Sea circulation modelling and assessment of marine pollution

Abstract The problem of mathematical modelling of the large-scale circulation of the Baltic Sea is considered. Marine hydrodynamics equations are written in the spherical coordinate system with a displaced point of the North Pole. The geographical North Pole is shifted to the vicinity of St. Petersburg to increase the spatial resolution of the Gulf of Finland. The free surface, sigma-coordinate primitive equation model under the Boussinesq, continuity, and hydrostatic assumptions is solved numerically. The problem of estimation of the pollution of some ‘protected’ marine sub-area by a passive tracer by means of the introducing an adjoint equation for the sensitivity function is formulated. The sensitivity function specifies the contribution of each basin point to the total pollution of the ‘protected area’.

Numerical modelling of sea currents and tidalwaves

Abstract We consider a mathematical model of sea currents and tidal waves based on the marine dynamics primitive equations. The equations are written in the orthogonal coordinate system on sphere with arbitrary position of the poles. It makes it possible to increase horizontal resolution due to placement of a pole into vicinity of the considered sub-area. Two problems are solved: (1) joint computation of wind-generated, baroclinic and tidal currents in the Black and Azov Seas; (2) simulating mesoscale variability of coastal currents in the Black Sea. The second problem is solved with increased horizontal resolution in the coastal zone of Gelendzhik.