Oceanography, mathematical problems in

2020 Mathematics Subject Classification: Primary: 86A05 [MSN][ZBL]

mathematical problems in oceanology

Mathematical problems in the fields of marine physics, chemistry, geology, and biology. In marine physics, the problems mainly concern geophysical hydrodynamics (defined as the hydrodynamics of natural currents of rotating baroclinic stratified liquids). The Earth's rotation, essentially affecting large-scale currents (on global and synoptic scales), and its stratification, i.e. the change in density of the medium in the direction of the force of gravity (vertical), create a specific anisotropy of the individual hydrodynamic fields in the sea or of their statistical characteristics, which must be considered, for example, when selecting base functions to describe these fields by the Galerkin method through objective analysis (interpolation, extrapolation, smoothing) of empirical data on these fields and when choosing statistical models for vertical-heterogeneous random fields of turbulence and internal waves (cf. also Turbulence, mathematical problems in).

An analytic description of the sea's eigen oscillations using linearized equations of hydrodynamics is complicated by the irregular shape of its boundaries, the seabed and the shore, which makes it impossible to use solutions with separated variables. Therefore, in the theory of tides, in which the sea can have a resonant response to tidal forcing, there are analytic calculations only for model-seas of regular shape (e.g. bounded by segments of meridians and parallels). In real geometry, successive (growing) eigen frequencies must be defined as the extrema of quadratic integral functionals, related to energy, on extremals which can be selected by the Galerkin method; this approach has not yet been fully realized. In the theory of tides, both the linear and non-linear response of the sea to tidal forcing is noted, and can be described by representing the height of the tide as a functional power series with respect to the tidal forcing; the functional coefficients of this series describe the sea's properties as a resonant system.

There is a specific difference between the solutions of the equations of hydrodynamics for several classes of waves: acoustic, surface (capillary and gravitational), internal gravitational, inertial (including barotropic and baroclinic Rossby–Blinova waves, formed as a result of the change with latitude of the vertical projection of the angular speed of the Earth's rotation), and, finally, hydromagnetic waves arising from movements of an electroconductive fluid (salt sea-water) in the geomagnetic field. Separate classes of wave-solutions (and of dynamic equations for them) are constructed using asymptotic methods of non-linear mechanics, related to the van der Pol method, in a multiple time scale analysis. An example of this is the quasi-geostrophysic series, filtering fast waves from solutions of equations of hydrodynamics and separating classes of Rossby–Blinova waves.

Waves in the sea, as a rule, are non-linear. For long non-linear waves, both surface and internal, the Korteweg–de Vries equation holds good and its periodic (cnoidal) solutions and solitons can be used. For short waves, no general methods for finding solitons and periodic solutions have yet been constructed, and only individual examples exist (capillary Slezkin–Crapper waves, gravitational Gerstner and Stokes waves, barotropic and baroclinic Rossby solitons). Statistical theories of non-linear wave fields are also insufficiently developed, particularly those relating to the description of surface and internal gravitational waves (internal waves generating turbulent spots and blurring the spots in the layers of the vertical microstructure) and Rossby waves (with evolution of quasi-two-dimensional turbulence in a non-linear wave field).

One of the most important problems of marine hydrodynamics is the mathematical modelling of the sea's circulation (in the most general formulation, in its interaction with the atmosphere through the so-called upper mixed layer of the sea and the boundary layer of the atmosphere), while as a consequence of the broad spectrum of the scales of spatial heterogeneity (from millimetres to $ 10 ^ {4} $ kilometres) the system to be modelled here has a huge number of degrees of freedom (for millimetric elementary volumes of the order of $ 10 ^ {28} $), which invariably need to be taken together, for example by the method of parametrization of small-scale processes.

When approximating continuous hydrodynamic equations by difference equations, questions arise relating to the order of the approximation and the convergence and stability of the difference scheme. In contemporary so-called eddy-resolving models of the sea's circulation, spatial grids with horizontal steps of tens of kilometres are used. Schemes using spectral (including Galerkin) representations of spatial hydrodynamic fields can be competitive.

The basic problems of the mathematical processing of measurement data in marine hydrodynamics are divided into problems of sounding (functions of depth, their decomposition into modes and spectra), towing (horizontal and space-time spectra with Doppler effects) and polygon measurements (time mutual spectra, objective analysis, synchronized spatial pictures and four-dimensional analysis of wave fields).

In marine acoustics, typical problems of wave distribution in a stratified environment are examined; for an analytic description of the vertical structure of wave fields in a number of cases, the WKB approximation is used (see WKB method). In marine optics, multiple light scattering is a specific process for the description of which asymptotic analytical solutions and numerical solutions of the radiation transfer equation (cf. also Radiative transfer theory), obtained by Monte-Carlo methods, are used. In marine chemistry, the major mathematical problem is the calculation of convective diffusion of non-conservative mixtures with specific sources and sinks.

In marine geology problems have arisen in connection with the development of plate tectonics (tectonics of lithospheric plates), concerning kinematic calculations of movements of hard plates on the surface of a sphere and their genetic explanation using a mathematical modelling of processes of density convection on the Earth's surface (arising from the transfer of heavy substances from the mantel to the core). One of the important special problems of geology is biostratigraphy, i.e. recognition of the age of strata of sedimentary rocks by means of their content of micro-palaeontological assemblies using self-teaching programs (whereas in most of the work this problem is solved approximately, without the use of a computer).

Very substantial calculating problems in the estimation of statistical characteristics of signals, noise, filtration, and form recognition arise in registering and processing data of marine multi-channel continuous deep seismo-profiling and vibro-radioscopy of the seabed, while in a number of cases both in the formulation of measurement grids (spatial distributions of emitters and receivers of signals) and in their registration, holographic methods using Fourier transforms are promising.

In marine biology the great importance of the problem of controlling the sea's biological productivity is reflected in the mathematical modelling of the structure and functioning of ecosystems and, in particular, involves population dynamics. An example of this is the problem of evolution with time $ t $ of the vertical distributions $ q _ {i} ( z, t) $ of the components $ q _ {i} $ of an ecosystem (including the concentration of a number of forms of phyto- and zoo-plankton, oxygen, carbonate, phosophoric and nitric salts, the temperature and salinity of water, and the illumination by photosynthetically active radiation), described by equations in the form

$$ \dot{q} _ {i} = A _ {i \alpha } q _ \alpha + B _ {i \alpha \beta } q _ \alpha q _ \beta + \frac \partial {\partial z } K( z) \frac{\partial q _ {i} }{\partial z } , $$

where $ A _ {i \alpha } $, $ B _ {i \alpha \beta } $ are biological and biophysical parameters. Of interest here are numerical solutions of the Cauchy problem with specific initial data and conclusions of the qualitative theory of differential equations on the behaviour of solutions as a whole and on their dependence on the parameters contained in the equations.

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A general introduction to hydrodynamical problems in oceanography is given in [a1]–[a4]. The modelling and numerical analysis of waves in the ocean is presented in [a5]–[a7].

In the study of physical and biological processes in the ocean, some topics recently got special attention from scientists working in different fields. First, the interaction between ocean and atmosphere appears to play an important role in climatological phenomena. A well-known example is "El Niño" , a name that is used for the occurrence of excessive rainfall in certain years caused by a sea-surface temperature anomaly, see [a8]. Secondly, the storage (or buffering) of carbonic acids by the oceans is also part of this interaction. In the balance of $ \mathop{\rm CO} _ {2} $ in the air this has to be taken into consideration. Moreover, marine organisms may deposit a part of the carbonicacids in sediments at the ocean floor.

Sediments transport (sand and mud) near coasts and in estuaries are studied in relation with human activities at these locations [a9]–[a11]. The study of pollution at the outflow of rivers is analyzed in the same context. The series of which [a8] is a volume treats many aspects of the ocean: its physics, chemistry, biology, and geology.

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