Dmitri V. Alexandrov

Sea Ice Dynamics Induced by External Stochastic Fluctuations

The influence of stochastic fluctuations in the atmosphere and in the ocean caused by different occasional phenomena (noises) on dynamic processes of sea ice growth with a mushy layer is studied. It is shown that atmospheric temperature variances substantially increase the sea ice thickness, whereas dispersion variations of turbulent flows in the ocean to a great extent decrease the ice content produced by false bottom evolution.

Stochastically driven transitions between climate attractors

The classical non-linear climatic model previously developed by Saltzman with co-authors and Nicolis is analysed in both the deterministic and stochastic cases in a wider domain of system parameters. A detailed analysis of the deterministic model shows a co-existence of a stable cycle and equilibrium phase points of the climate system localisation. A fine structure of attraction basins existing around stable equilibria is studied. The model under consideration possesses the noise-induced transitions between possible system attractors (limit cycle and two equilibria) in the case of stochastic dynamics caused by temperature fluctuations. A new phenomenon of stochastic generation of large amplitude oscillations around two equilibrium points in the absence of a limit cycle is revealed. The co-existence of large-, small- and mixed-mode stochastic transitions between the climate system attractors is found.

The boundary integral theory for slow and rapid curved solid/liquid interfaces propagating into binary systems

The boundary integral method for propagating solid/liquid interfaces is detailed with allowance for the thermo-solutal Stefan-type models. Two types of mass transfer mechanisms corresponding to the local equilibrium (parabolic-type equation) and local non-equilibrium (hyperbolic-type equation) solidification conditions are considered. A unified integro-differential equation for the curved interface is derived. This equation contains the steady-state conditions of solidification as a special case. The boundary integral analysis demonstrates how to derive the quasi-stationary Ivantsov and Horvay–Cahn solutions that, respectively, define the paraboloidal and elliptical crystal shapes. In the limit of highest Péclet numbers, these quasi-stationary solutions describe the shape of the area around the dendritic tip in the form of a smooth sphere in the isotropic case and a deformed sphere along the directions of anisotropy strength in the anisotropic case. A thermo-solutal selection criterion of the quasi-stationary growth mode of dendrites which includes arbitrary Péclet numbers is obtained. To demonstrate the selection of patterns, computational modelling of the quasi-stationary growth of crystals in a binary mixture is carried out. The modelling makes it possible to obtain selected structures in the form of dendritic, fractal or planar crystals. This article is part of the theme issue ‘From atomistic interfaces to dendritic patterns’.

Selection criterion for the growing dendritic tip at the inner core boundary

A free dendrite growth at the Earth’s inner core boundary is analyzed using a sharp interface model. A new selection criterion of the dendrite tip’s stable growth into non-isothermal binary melt with convection and pressure effects for the 2D and 3D axisymmetric models is derived. The criterion obtained combines known analytic results for the dendrite growth under forced convection in a pure system and dendrite growth in a stagnant binary system. The generalized selection criterion represents a condition connecting the main physical parameters of the Earth’s inner core.

A complete analytical solution of the Fokker–Planck and balance equations for nucleation and growth of crystals

This article is concerned with a new analytical description of nucleation and growth of crystals in a metastable mushy layer (supercooled liquid or supersaturated solution) at the intermediate stage of phase transition. The model under consideration consisting of the non-stationary integro-differential system of governing equations for the distribution function and metastability level is analytically solved by means of the saddle-point technique for the Laplace-type integral in the case of arbitrary nucleation kinetics and time-dependent heat or mass sources in the balance equation. We demonstrate that the time-dependent distribution function approaches the stationary profile in course of time. This article is part of the theme issue ‘From atomistic interfaces to dendritic patterns’.

Thermo-solutal and kinetic modes of stable dendritic growth with different symmetries of crystalline anisotropy in the presence of convection

Motivated by important applications in materials science and geophysics, we consider the steady-state growth of anisotropic needle-like dendrites in undercooled binary mixtures with a forced convective flow. We analyse the stable mode of dendritic evolution in the case of small anisotropies of growth kinetics and surface energy for arbitrary Péclet numbers and n-fold symmetry of dendritic crystals. On the basis of solvability and stability theories, we formulate a selection criterion giving a stable combination between dendrite tip diameter and tip velocity. A set of nonlinear equations consisting of the solvability criterion and undercooling balance is solved analytically for the tip velocity V and tip diameter ρ of dendrites with n-fold symmetry in the absence of convective flow. The case of convective heat and mass transfer mechanisms in a binary mixture occurring as a result of intensive flows in the liquid phase is detailed. A selection criterion that describes such solidification conditions is derived. The theory under consideration comprises previously considered theoretical approaches and results as limiting cases. This article is part of the theme issue ‘From atomistic interfaces to dendritic patterns’. This article is part of the theme issue ‘From atomistic interfaces to dendritic patterns’.

Selected mode of dendritic growth with n-fold symmetry in the presence of a forced flow

The effect of n-fold crystal symmetry is investigated for a two-dimensional stable dendritic growth in the presence of a forced convective flow. We consider dendritic growth in a one-component undercooled liquid. The theory is developed for the parabolic solid-liquid surface of dendrite growing at arbitrary growth Peclet numbers keeping in mind small anisotropies of surface energy and growth kinetics. The selection criterion determining the stable growth velocity of the dendritic tip and its stable tip diameter is found on the basis of solvability analysis. The obtained criterion includes previously developed theories of thermally and kinetically controlled dendritic growth with convection for the case of four-fold crystal symmetry. The obtained nonlinear system of equations (representing the selection criterion and undercooling balance) for the determination of dendrite tip velocity and dendrite tip diameter is analytically solved in a parametric form. These exact solutions clearly demonstrate a transition between thermally and kinetically controlled growth regimes. In addition, we show that the dendrites with larger crystal symmetry grow faster than those with smaller symmetry.

The hyperbolic Allen–Cahn equation: exact solutions

Using the first integral method, a general set of analytical solutions is obtained for the hyperbolic Allen–Cahn equation. The solutions are presented by (i) the class of continual solutions described by -profiles for traveling waves of the order parameter, and (ii) the class of singular solutions which exhibit unbounded discontinuity in the profile of the order parameter at the origin of the coordinate system. It is shown that the solutions include the previous analytical results for the parabolic Allen–Cahn equation as a limited class of -functions, in which the inertial effects are omitted.

Stochastic variability and noise-induced generation of chaos in a climate feedback system including the carbon dioxide dynamics

In this work, a non-linear dynamics of a simple three-dimensional climate model in the presence of stochastic forcing is studied. We demonstrate that a dynamic scenario of mixed-mode stochastic oscillations of the climate system near its equilibrium can be formed. As this takes place, a growth of noise intensity increases the frequency of interspike intervals responsible for the abrupt climate changes. In addition, a certain enhancement of stochastic forcing abruptly increases the atmospheric carbon dioxide and decreases the Earths ice mass. A transition from order to chaos occurring at a critical noise is shown.

Nonlinear dynamics of mushy layers induced by external stochastic fluctuations

The time-dependent process of directional crystallization in the presence of a mushy layer is considered with allowance for arbitrary fluctuations in the atmospheric temperature and friction velocity. A nonlinear set of mushy layer equations and boundary conditions is solved analytically when the heat and mass fluxes at the boundary between the mushy layer and liquid phase are induced by turbulent motion in the liquid and, as a result, have the corresponding convective form. Namely, the ‘solid phase–mushy layer’ and ‘mushy layer–liquid phase’ phase transition boundaries as well as the solid fraction, temperature and concentration (salinity) distributions are found. If the atmospheric temperature and friction velocity are constant, the analytical solution takes a parametric form. In the more common case when they represent arbitrary functions of time, the analytical solution is given by means of the standard Cauchy problem. The deterministic and stochastic behaviour of the phase transition process is analysed on the basis of the obtained analytical solutions. In the case of stochastic fluctuations in the atmospheric temperature and friction velocity, the phase transition interfaces (mushy layer boundaries) move faster than in the deterministic case. A cumulative effect of these noise contributions is revealed as well. In other words, when the atmospheric temperature and friction velocity fluctuate simultaneously due to the influence of different external processes and phenomena, the phase transition boundaries move even faster. This article is part of the theme issue ‘From atomistic interfaces to dendritic patterns’. This article is part of the theme issue ‘From atomistic interfaces to dendritic patterns’.