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Project titleMulti-scale mathematical modeling of the Arctic ice evolution: the influence on climate change

AffiliationFederal State Autonomous Educational Institution of Higher Education "Ural Federal University named after the First President of Russia B.N. Yeltsin",

Research area 01 - MATHEMATICS, INFORMATICS, AND SYSTEM SCIENCES, 01-217 - Mathematical simulation of physical environments

KeywordsArctic ice, nonlinear dynamics, multi-scale modeling, phase field method, gradient-stable algorithms, melting, freezing, structural and phase transformations, dendrites

Annotation
The project is aimed at solving problems of structure formation for the analysis of dynamic changes in the climate of the Arctic zone due to the behavior of the ice cover. For this purpose, we shall develop theoretical framework for the multi-scale modeling of the dynamics of interface motion "ice - water", chemical content and phase composition of the ice structure, taking into account the radiative and convective heat and mass transfer. Presently, virtually no scientific data for climate prediction based on modeling processes of melting and freezing of ice. Therefore, the project seeks to develop a new formalism which will be the basis for the modeling of structure Arctic ice. Information about the structure and properties of the interface "ice-water", the resulting micro-level simulation is used to predict the formation and melting of the two-phase zone - the heterogeneous region of phase coexistence "ice - water" at the mesoscopic level of analysis. Properties and characteristics of two-phase zone are used to determine the parameters for modeling the global dynamics of the Arctic zone of the Earth. Analysis of the global formation and melting of ice in the Arctic zone includes the construction of bifurcation diagrams and phase portraits of the deterministic behavior of the seasonal changes in climate, taking into account deviations of possible stochastic parameters of the developing models. Solutions of the model equations will define the equilibrium region and dynamical climate changes leading to the evolution of the Arctic ice. Developed mathematical modeling techniques based on unconditionally gradient stable algorithms and their accelerated numerical solutions will allow us to obtain solutions of the dynamical and transport equations for global analysis and phase-field simulations. Models and modeling data will enable the creation of a software package for computer prediction of dynamic changes in ice cover and climate change Arctic zone of the Earth.

Expected results
The atomistic modeling on the basis of the molecular dynamics method will allow us to find the information about water’s structure and properties as well as the ice-water interface characteristics in conditions of thermodynamical equilibrium and phase transformations. The information obtained on the basis of this microscopic level of modeling will be directly used for the analytical and numerical calculations of the atomistic continuum modeling (based on the phase crystal field model) and of the mesoscopic modeling (based on the traditional phase-field model). The analytical model connected with the amplitude equations of the phase crystal field model is suitable for the description of structures and defects formation in nano-scales (1-100 nm) in the ice melting and freezing processes. This model will be used within the framework of the present project for the quick quantitative estimate and qualitative analysis of structural transformations. The developing non-isothermal phase field model will enable us to analyze the dendrite growth dynamics and the crystal melting in meso-scales. In addition, a progress in numerical solving of the second, fourth and sixth order equations with respect to the space variable and the second order equations with respect to the time used in the project, will be achieved. Namely, the unconditionally stable numerical scheme as well as high speed numerical algorithms will be detailed for such evolutionary and transport equations. They are required for the description of changes in structures and phase compositions of the Arctic ices taking into account variations in the ice meso- and microstructures. The results of mesoscopic modeling of the crystal structure will be used for the description of a mushy layer in Arctic ices as well as for the global modeling, which is a subject of the present project. A dynamic theory of sea ice freezing/melting with a phase transition layer filled with ice crystals will be formulated in the mesoscopic level. The temperature and salinity distributions, solid phase fraction in the mushy layer of sea ice, dynamic laws of the interface motion between the pure ice and the mushy layer as well as between the mushy layer and the ocean will be determined. This study will be carried out for different boundary conditions of the freezing/melting sea ice dynamics: temperature changes in the atmosphere, sea water salinity, oceanic flows under the sea ice etc. The latent heat flux appearing as a result of water freezing and conductive heat flux at the ice surface going to the atmosphere as ice extent increases will be found for the determination of the main parameters of the global dynamic model of the Arctic Region of the Earth. The theory will be developed for the description of the global dynamics of shelf ices. The estimates of main parameters of the global dynamical model of the Arctic region will be carried out on the basis of determined characteristics of the microscopic and mesoscopic models. An analysis of bifurcation diagrams and phase portraits of this deterministic model with special attention to possible stochastic parameters of seasonal changes will be done. The system equilibrium zones as well as its dynamical regimes describing the Arctic ice evolution will be found. The project results are important for the top-priority scientific problem on the ice formation and melting in the Arctic Region. The anticipated results of multi-scale analysis correspond to the world research level and represent the basis for the national software package development with the aim of numerical forecasting of dynamic changes in the ice cover and climate changes of the Arctic Region of the Earth. The solution of problems stated in the project will allow us to obtain the qualitative and quantitative estimates of the Arctic climate changes, to estimate possible variations in oceanic flows and in increased/decreased ocean level. This will allow us to determine the seasonal temperature and humidity changes in the Arctic zone with their possible influence on the Northern hemisphere climate. The national economic significance and practicability in determining the information about the evolution of the Arctic climate changes is clear. So, among others, the estimates of navigation information about the Northern Sea Route obtained on the basis of the present project research results will lead to essential additional economic gains (in addition to observational forecastings) for the national economy.

Annotation of the results obtained in 2018
1. A mathematical model of crystal growth for the n-th order of crystal symmetry is formulated and its analytical solutions are defined, where n = 2 - whiskers or prisms, n = 3 - triangles or pyramids, n = 4 - squares or cubes, n = 5 - mosaic crystals, i.e. quasicrystals, n = 6 - hexagons or snowflakes, n = 10 - typical quasicrystals. The solution of the model is in agreement with the results of solving the phase-field model for ice crystals having the symmetry n = 6. 2. A model of ice crystal growth under conditions of convective/turbulent heat and mass transfer at the ice surface was formulated, analytical solutions of the model were determined, and a criterion for stable crystal growth was found. The solutions found are in good agreement with experimental data on the kinetics of dendrite growth. 3. Equations are derived that determine the implicit dependencies of the tip velocity and the tip diameter of dendrites, depending on the total supercooling. Exact analytical solutions of these nonlinear equations are found in a parametric form. Asymptotic solutions are derived that describe crystal growth at small Peclet numbers. The theoretical predictions are confirmed by experimental data for ice dendrites growing in binary water-ethylene glycol solutions, as well as in pure water. 4. The theory of kinetic phase transition in the structure of a binary ordering crystal was developed. A phase-field model of the formation of the structure of a binary ordering crystal growing out of a supercooled liquid is formulated. The results are analyzed on the basis of solving the equations of the dynamics of the phase field and the relaxation of the long-range order parameter in the diffuse crystal-liquid interface and in the crystal volume. The critical temperatures of change in the velocity and the long-range parameter of a growing crystal during the kinetic transition are determined. A qualitative difference is found, and analogies are determined in the processes of non-equilibrium capture of an impurity and the formation of a disordered structure of rapidly growing crystals. The analysis is complete, because all stages of crystal formation have been obtained and analyzed: from a completely exhausted to its completely disordered structure. The results of the work will be published in 2019 in the article “Rapid solidification as non-ergodic process” (Physics Reports, Q1, impact factor 20.099, ISSN: 0370-1573) with acknowledgements to the Russian Science Foundation. 5. The diagrams of the structures of various modifications in equilibrium and metastable conditions are constructed. Analytical and numerical limiting transitions between micro- and mesoscopic levels of crystal modeling are determined and a phenomenological model of the interrelation of phase-structural properties and the formation of ice cover is created. A monograph with acknowledgements to the Russian Science Foundation is published: P.K. Galenko, V. Ankudinov and I. Starodumov, Phase-Field Crystals: Fast Interface Dynamics (de Gruyter, Berlin, 2018). 6. A multiparametric model of the movement of the Arctic ice edge is formulated based on the data on the main parameters of this model obtained at the meso- and micro-levels of multiscale modeling, as well as data from an existing experiment. The nonlinear dynamics of this model is investigated within the framework of deterministic equations. Stable equilibria and limit cycles were determined, phase portraits in the planes of the main parameters were constructed. A stochastic model is formulated, taking into account the presence of additive noise in the main parameters of the system in order to determine the influence of variations of these parameters on the dynamics of ice movement. Equilibria and limit cycles of the stochastic model are found; phase portraits of the system in the presence of noise of different intensities are plotted; dynamic trajectories are determined. The noise thresholds responsible for the removal of phase trajectories from attractors are defined. The possibility and conditions for chaotic oscillations of the ice cover are determined. The analysis of various stochastically-induced scenarios for the evolution of the global ice dynamics model is performed. The results will be published in 2019 in the article “Nonlinear Climate Dynamics: from Deterministic Behavior to Stochastic Excitability and Chaos” (Physics Reports, Q1, Impact Factor 20.099, ISSN: 0370-1573) with acknowledgements to the Russian Science Foundation. 4 computational programs are developed. These programs have been registered (the state registration of computer programs).

Publications

1. Alexandrov D.V., Galenko P.K. Thermo-solutal growth of an anisotropic dendrite with six-fold symmetry Journal of Physics Condensed Matter, Vol. 30, Iss. 10, Art. No. 105702. (year - 2018) https://doi.org/10.1088/1361-648X/aaab7b

2. Alexandrov D.V., Starodumov I.O., Pavlyuk E.V., Ivanov A.A. Образование дефектных и метастабильных структур при моделировании фазовых переходов методом кристаллического фазового поля Расплавы, номер 2, С. 247-256. (year - 2018)

3. Alexandrov D.V., Toropova L.V., Galenko P.K. Thermo-solutal growth of an anisotropic dendrite in the case of convective heat and mass transfer in a binary system AIP Conference Proceedings, Vol. 1978, Art. No. 470065. (year - 2018) https://doi.org/10.1063/1.5044135

4. Buchbinder G.L., Galenko P.K. Boundary conditions and heat resistance at the moving solid-liquid interface Physica A: Statistical Mechanics and its Applications, Vol. 489, P. 149-162. (year - 2018) https://doi.org/10.1016/j.physa.2017.08.001

5. Galenko P.K., Alexandrov D.V. From atomistic interfaces to dendritic patterns Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 376, Iss. 2113, Art. No. 20170210 (year - 2018) https://doi.org/10.1098/rsta.2017.0210

6. Galenko P.K., Alexandrov D.V., Toropova L.V., Rettenmayr M., Herlach D.M. Effect of forced convection on dendritic growth: theoretical modeling and analysis of recent experimental results Selected, peer reviewed papers from the SEVENTH INTERNATINAL CONFERENCE ON SOLIDIFICATION AND GRAVITY, Selected, peer reviewed papers from the SEVENTH INTERNATINAL CONFERENCE ON SOLIDIFICATION AND GRAVITY, Miskolc-Lillafüred, Hungary September 3-6, 2018, P. 259-264. (year - 2018)

7. Galenko P.K., Nizovtseva I.G., Reuther K., Rettenmayr M. Kinetic transition in the order-disorder transformation at a solid/liquid interface Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 376, Iss. 2113, Art. No. 20170207. (year - 2018) https://doi.org/10.1098/rsta.2017.0207

8. Galenko P.K., Nizovtseva I.G., Reuther K., Rettenmayr M. Кинетика формирования неупорядоченной структуры кристалла при высокоскоростном затвердевании Журнал экспериментальной и теоретической физики, том 154, вып. 1 (7), С. 124-133. (year - 2018) https://doi.org/10.1134/S0044451018070106

9. Makoveeva E.V., Alexandrov D.V. К теории нуклеации и роста кристаллов в метастабильной области фазового превращения при учете различных кинетических механизмов Расплавы, номер 2, С. 219-234. (year - 2018)

10. Nizovtseva I.G., Galenko P.K. Travelling-wave amplitudes as solutions of the phase-field crystal equation Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 376, Iss. 2113, Art. No. 20170202. (year - 2018) https://doi.org/10.1098/rsta.2017.0202

11. Starodumov I.O., Galenko P.K., Kropotin N.V., Alexandrov D.V. Об аппроксимации периодического решения уравнения кристаллического фазового поля при расчетах методом конечных элементов Программные системы: теория и приложения, Том 9, № 4(39), С. 265-278. (year - 2018) https://doi.org/10.25209/2079-3316-2018-9-4-265-278

12. Titova E.A. 2D dendrite shape in the large chemical Peclet number limit AIP Conference Proceedings, Vol. 2015, Art. No. 020102. (year - 2018) https://doi.org/10.1063/1.5055175

13. Titova E.A., Alexandrov D.V., Galenko P.K. Время релаксации к стационарному росту вторичных ветвей дендрита Расплавы, - (year - 2019)

14. Titova E.A., Alexandrov D.V., Galenko P.K. Исследование роста дендритного кристалла в форме эллиптического параболоида методом граничных интегральных уравнений Расплавы, номер 3, С. 312-319. (year - 2018)

15. Toropova L.V., Alexandrov D.V., Galenko P.K. К вопросу об устойчивом росте анизотропного дендрита при конвективном теплопереносе в жидкой фазе у поверхности дендрита Расплавы, номер 3, С. 320-329. (year - 2018)

16. Toropova L.V., Alexandrov D.V., Galenko P.K. How the convective heat transport at the solid/liquid phase interface influences the stable mode of dendritic growth AIP Conference Proceedings, Vol. 1997, Art. No. 020030. (year - 2018) https://doi.org/10.1063/1.5049024

17. Toropova L.V., Alexandrov D.V., Galenko P.K. Solvability criterion for stable mode of dendritic evolution in the case of convective heat and mass transfer in a binary alloy AIP Conference Proceedings, Vol. 1953, Art. No. 040005. (year - 2018) https://doi.org/10.1063/1.5032625

18. Alexandrov D.V., Galenko P.K., Toropova L.V. Thermo-solutal and kinetic modes of stable dendritic growth with different symmetries of crystalline anisotropy in the presence of convection Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 376, Iss. 2113, Art. No. 20170215 (year - 2018) https://doi.org/10.1098/rsta.2017.0215

19. Galenko P.K., Alexandrov D.V., Titova E.A. The boundary integral theory for slow and rapid curved solid/liquid interfaces propagating into binary systems Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 376, Iss. 2113, Art. No. 20170218. (year - 2018) https://doi.org/10.1098/rsta.2017.0218

20. Galenko P., Ankudinov V., Starodumov I. Phase-field crystals De Gruyter, Berlin/Boston, De Gruyter, Berlin/Boston (year - 2019)

21. - DendriteN-theta -, 2018616067 (year - )

22. - DendriteTurbulent -, 2018616310 (year - )

23. - DendriteShape -, 2018616311 (year - )

24. - Phase Field Crystal Simulator (PFC_Simulator) -, 2018617793 (year - )

25. - Ученые могут предсказывать изменения температуры с помощью математических моделей СМИ Уральского федерального университета, Новость номер 23804 (year - )

26. - Ученые могут предсказывать изменения температуры с помощью математических моделей СМИ РНФ, Новость номер 3141 (year - )

Annotation of the results obtained in 2016
For the primary analysis of growth kinetics of ice dendrites (in a pure and salt water), systematization and comparative analysis has been provided in examples of organic, metallic, semiconducting and alloying crystals. Namely, the growth kinetics of succinonitrile, Ni, Si-As, Mg-Al, Sn-Bi in comparison with accessible data for ice dendrites growing from undercooled water has been analyzed. Modeling of hexagonal crystalline structures melted or growing during crystallization is carried out using a phase field crystal model (PFC-model). The numeric solution has been made by parallelization of partial differential equations of the PFC-model with using unconditionally gradient stable algorithms of computational stability. A diagram of three dimensional crystalline structures (in co-ordinates “temperature – atomic density” is constructed in which region of hexagonal patterns is defined (in comparison with other possible structures). The diagram is plotted for structures selected by minimizing of free energy functional by atomic density and crystal lattice parameter at a given temperature. As a result, regions of stable structural phases are defined for: liquid, face centered crystal, body centered crystal, hexagonal closed packed crystal (ice), rods and bands. The dynamics of hexagonal pattern formation has been modeled which results in equilibrium crystals as well as possible metastable states of ice crystals. Amplitude dynamical equations for the hexagonal structure formation were derived using the PFC-model. Analytical traveling wave solutions for propagating amplitudes have been found for evaluation of front of hexagonal patter invading metastable (undercooled) liquid. The obtained analytical solutions have been considered as benchmarks for numerical solutions. As a result of numerical and analytical solutions of the PFC-model, kinetic data on growth and melting of ice crystals were obtained. These data are used in macro(meso)scopic model of two-phase mushy zone which has been used for qualitative estimations of macroscopic ice layer formation. Evolution of a phase transition layer is studied to analyze the mesoscopic structural and phase transitions. A generalized equation governing the dynamics of a curvilinear phase interface ice-water, which determines the condition of constitutional supercooling appearance, is derived. Taking into account nucleation and dendritic growth processes, we investigate the evolution of a supercooled phase transition layer. The processes of initiation and growth of ice crystals are described with allowance for their buoyancy effects. The kinetic equation for the density distribution function of crystals over sizes is solved for different kinetics of the solid phase growth. A thermodiffusion problem describing the anisotropic dendritic growth with convection is solved. A selection criterion, which determines a relation between the stable growth velocities of dendritic tips and their curvature radii is found. A theory of microscopic solvability with convection is detailed. The developed crystallization theory with a structural and phase transformation layer enables to obtain the solid fraction in a mushy layer as well as the dynamic laws of its boundaries (pure ice – mushy layer and mushy layer - ocean). The macroscopic modeling of the Earth’s ice cover evolution is carried out on the basis of the three-dimensional dynamical Saltzman model. The coefficients of macroscopic model are determined with allowance for results of microscopic and mesoscopic studies as well as of research works of other authors. The developed nonlinear model takes into consideration the changes in atmospheric carbon dioxide as well as a feedback between the climate system parameters. The bifurcation diagram and phase portraits of the deterministic system are plotted. Its attractors are found as well. A stochastic climate model is formulated. Its stochastic sensitivity is studied. The stochastic trajectories and dynamic dependencies of the main climate model parameters are found at different noise intensities. Analyzing the Lyapunov exponents we concluded that the climate system is chaotic at different nose intensities.

Publications

1. Alexandrov D.V. Mathematical modelling of nucleation and growth of crystals with buoyancy effects Philosophical magazine letters, 96 (4), 132-141 (year - 2016) https://doi.org/10.1080/09500839.2016.1177222

2. Alexandrov D.V. The large-time behaviour of coarsening of a particulate assemblage due to Ostwald ripening and coagulation Philosophical magazine letters, 96 (9), 355-360 (year - 2016) https://doi.org/10.1080/09500839.2016.1225996

3. Alexandrov D.V., Bashkirtseva I.A., Ryashko L.B. Comment on: Cyclic extrusion of a lava dome based on a stick-slip mechanism, by Costa et al. (2012) Earth and planetary science letters, - (year - 2017)

4. Alexandrov D.V., Bashkirtseva I.A., Ryashko L.B. Stochastic variability and noise-induced generation of chaos in a climate feedback system including the carbon dioxide dynamics EPL, 115 (4), 40009 (year - 2016) https://doi.org/10.1209/0295-5075/115/40009

5. Alexandrov D.V., Galenko P.K. Аналитическое решение задачи об обтекании параболического дендрита наклонным потоком вязкой жидкости в приближении Осеена Вестник Удмуртского университета. Математика. Механика. Компьютерные науки, 26 (3), 379-387 (year - 2016) https://doi.org/10.20537/vm160307

6. Alexandrov D.V., Galenko P.K. Boundary integral approach for propagating interfaces in a binary non-isothermal mixture Physica A, - (year - 2017)

7. D.V. Alexandrov, I.V. Alexandrova Solute Redistribution Around a Parabolic Dendrite in the case of Thermodiffusion (Soret Effect) and Temperature-Dependent Diffusivity AIP conference proceedings, - (year - 2017)

8. Galenko P.K., Danilov D.A., Reuther K., Alexandrov D.V., Rettenmayr M., Herlach D.M. Effect of convective flow on stable dendritic growth in rapid solidification of a binary alloy Journal of crystal growth, Vol. 457, pp. 349-355 (year - 2016) https://doi.org/10.1016/j.jcrysgro.2016.07.042

9. Gao J., Kao A., Bojarevics V., Pericleous K., Galenko P.K., Alexandrov D.V. Modeling of convection, temperature distribution and dendritic growth in glass- fluxed nickel melts Journal of crystal growth, - (year - 2016)

10. Gusakova O.V., Galenko P.K., Shepelevich V.G., Alexandrov D.V. Формирование микроструктуры быстрозатвердевших сплавов системы Sn-Bi Вестник Удмуртского университета. Математика. Механика. Компьютерные науки, 26 (3), 388-400 (year - 2016) https://doi.org/10.20537/vm160308

11. Nizovtseva I.G., Galenko P.K., Alexandrov D.V. The hyperbolic Allen–Cahn equation: exact solutions Journal of Physics A: Mathematical and Theoretical, 49 (43), 435201 (year - 2016) https://doi.org/10.1088/1751-8113/49/43/435201

12. Starodumov I., Kropotin N. Features in simulation of crystal growth using the hyperbolic PFC equation and the dependence of the numerical solution on the parameters of the computational grid AIP conference proceedings, 1759, 020136 (year - 2016) https://doi.org/10.1063/1.4959750

13. Starodumov I.O., Pavlyuk E.V., Abramov S.M., Klyuev L.V., Galenko P.K., Alexandrov D.V. Эффективность распараллеливания алгоритма решения уравнения PFC с использованием библиотеки PetIGA Вестник Удмуртского университета. Математика. Механика. Компьютерные науки, 26 (3), 445-450 (year - 2016) https://doi.org/10.20537/vm160312

14. Titova E.A., Alexandrov D.V., Galenko P.K. О времени нестационарности роста первичных дендритов Вестник Удмуртского университета. Математика. Механика. Компьютерные науки, 26 (3), 439-444 (year - 2016) https://doi.org/10.20537/vm160311

Annotation of the results obtained in 2017
In the work, mathematical modeling of the dynamic behavior of the ice cover is carried out, taking into account the variability of various physical parameters and processes affecting the climate (for example, the intensity of volcanic extrusions). The simulation takes into account the mutual influence of the poles of the planet on the basis of the model of coupled oscillators. The model coefficients included in the governing equations take into account the results of macroscopic modeling. Equilibria of the deterministic system are determined, phase portraits and dynamic trajectories are constructed. A stochastic model that takes into account fluctuations in the parameters of the system is formulated and studied. Its phase diagrams are constructed for various noise intensities. The decisive role of the presence of stochastic noise in the parameters of the climate model for its evolutionary behavior is demonstrated. As part of the work on the physical and mathematical PFC model (model of the phase field crystal), a software package has been developed that makes it possible to perform numerical calculations by this method using high-performance computational clusters. Verification of the results obtained with the software by comparing them with the theoretical analysis of the PFC theory is carried out. The analysis of the efficiency of the use of computer resources by the software is made and the requirements for computational resources for modeling large-scale tasks are determined. On the basis of a series of computer experiments, the mechanism for the formation of unstable and metastable structures, uncharacteristic for the full-scale experiment, was formulated, which allowed to justify further modification of the PFC method by including the member responsible for stochastic "noises" in the model. These noises will allow us to describe the natural background oscillations of the phase fields of temperature and atomic density, and more accurately describe the evolution of the microstructure of matter under metastability conditions with an extremely small region of stability. Using the nucleation theory of Skripov, an undercoolability for water and zirconium, as crystals having hexagonal close packed (HCP) crystal lattice, has been evaluated. Limits of metastability for these substances have been established. Using the phase field crystal model (PFC-model), amplitude equations for crystalline fronts invading undercooled liquid have been obtained. Formation of stable and metastable HCP-crystalline phase has been analyzed by the traveling wave solution of amplitude equations. Kinetic relationships “front velocity – undercooling” have been found during the growth of HCP-crystals in comparison with experimental data previously obtained on droplets crystallized in electromagnetic levitation facility (containerless method allowing us to reach deep undercoolings). The obtained results of theoretical modeling create a basis for robust predictions of formation of (meta)stable phases on the mesoscopic level of description with the possibility to suggest data for the macroscopic level of the ice cover analysis.

Publications

1. Alexandrov D.V. A transient distribution of particle assemblies at the concluding stage of phase transformations Journal of Materials Science, 52(12), 6987-6993 (year - 2017) https://doi.org/10.1007/s10853-017-0931-y

2. Alexandrov D.V. On the Theory of fragmentation process with initial particle volume Communications in Theoretical Physics, 68(2), 269-271 (year - 2017) https://doi.org/10.1088/0253-6102/68/2/269

3. Alexandrov D.V., Alexandrova I.V., Ivanov A.A. On the theory of self-similar phase transitions with a mushy layer IOP Conference Series: Materials Science and Engineering, 192(1), 012017 (year - 2017) https://doi.org/10.1088/1757-899X/192/1/012003

4. Alexandrov D.V., Alexandrova I.V., Ivanov A.A. On the theory of unsteady-state crystallization with a mushy layer IOP Conference Series: Materials Science and Engineering, 192(1), 012003 (year - 2017) https://doi.org/10.1088/1757-899X/192/1/012003

5. Alexandrov D.V., Bashkirtseva I.A., Ryashko L.B. Noise-induced variability of volcanic extrusions EPL, 116(4),40006 (year - 2016) https://doi.org/10.1209/0295-5075/116/40006

6. Alexandrov D.V., Galenko P.K. Selected mode of dendritic growth with n-fold symmetry in the presence of a forced flow EPL, 119(1), 16001 (year - 2017) https://doi.org/10.1209/0295-5075/119/16001

7. Alexandrov D.V., Galenko P.K. Dendritic growth with the six-fold symmetry: Theoretical predictions and experimental verification Journal of Physics and Chemistry of Solids, 108, 98-103 (year - 2017) https://doi.org/10.1016/j.jpcs.2017.04.016

8. Alexandrov D.V., Galenko P.K. Selected mode for rapidly growing needle-like dendrite controlled by heat and mass transport Acta Materialia, 137, 64-70 (year - 2017) https://doi.org/10.1016/j.actamat.2017.07.022

9. Galenko P.K., Reuther K., Kazak O.V., Alexandrov D.V., Rettenmayr M. Effect of convective transport on dendritic crystal growth from pure and alloy melts Applied Physics Letters, 111(3), 031602 (year - 2017) https://doi.org/10.1063/1.4985340

10. Nizovtseva I.G., Galenko P.K., Alexandrov D.V. Traveling wave solutions for the hyperbolic Cahn-Allen equation Chaos, Solitons and Fractals, 94, 75-79 (year - 2017) https://doi.org/10.1016/j.chaos.2016.11.010

11. Pavlyuk E., Starodumov I., Osipov S. Remote control system for high-perfomance computer simulation of crystal growth by the PFC method IOP Conference Series: Materials Science and Engineering, 192(1), 012016 (year - 2017) https://doi.org/10.1088/1757-899X/192/1/012016

12. Salhoumi A., Galenko P.K. Analysis of interface kinetics: Solutions of the Gibbs-Thomson-type equation and of the kinetic rate theory IOP Conference Series: Materials Science and Engineering, 192(1), 012014 (year - 2017) https://doi.org/10.1088/1757-899X/192/1/012014

13. Starodumov I., Derevianko A., Alexandrov D. Application of the Saint-Venant model and the modified Stefan model for modeling the formation of the ice cover at the thermal growth stage IOP Conference Series: Materials Science and Engineering, 192(1), 012033 (year - 2017) https://doi.org/10.1088/1757-899X/192/1/012033

14. Starodumov I., Galenko P., Alexandrov D., Kropotin N. Influence of computational domain size on the pattern formation of the phase field crystals IOP Conference Series: Materials Science and Engineering, 192(1), 012008 (year - 2017) https://doi.org/10.1088/1757-899X/192/1/012008

15. Titova E.A., Alexandrov D.V., Galenko P.K. Boundary integral approach for elliptical dendritic paraboloid as a form of growing crystals IOP Conference Series: Materials Science and Engineering, 192(1), 012025 (year - 2017) https://doi.org/10.1088/1757-899X/192/1/012025

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