Simulations of collision of ice particles

The objective of this paper is to develop a realistic model for ice–structure interaction. To this end, the experiments made by Bridges et al. [Bridges FG, Hatzes A, Liu DNC. Structure, stability and evolution of Saturn’s rings. Nature 1984;309:333–5] in order to measure the coefficient of restitution for ice particles are thoroughly analyzed. One particularly troublesome aspect of the aforementioned experiments is fracture of the ice particles during a collision. In the present effort, the collisional properties of the ice particles are investigated using a Finite Element approach. It is found that a major challenge in modeling collision of the ice balls is the prediction of the onset of fracture and crack propagation in them. In simulations of a block of ice collision to a structure, it is crucial that fracture is determined correctly, as it will influence the collisional properties of the ice particles. The results of the simulation, considering fracture criterion implemented into the Finite Element Model [Zamankhan P, Bordbar M-H. Complex flow dynamics in dense granular flows. Part I: experimentation. J Appl Mech (T-ASME) 2006;73:648–57; Zamankhan P, Huang J. Complex flow dynamics in dense granular flows. Part II: simulations. J Appl Mech (T-ASME) 2007;74:691–702] together with a material model for the ice, imply that most of the kinetic energy dissipation occurs as a result of fracturing at the contact surface of the ice particles. The results obtained in the present study suggest that constitutive models such as those proposed by Brilliantov et al. [Brilliantov NV, Spahn F, Hertzsch JM, Poschel T. Model for collisions in granular gases. Phys Rev E;1996;53:5382–92] for collisions of ice particles are highly questionable.     

Simulation of ice under mechanical and thermal load

The contact physics of the wheel-rail contact of a railway vehicle under presence of water and ice at low temperatures is still not completely understood. For the investigation of the particular process in the contact zone a simulation is required, which is able to calculate the normal and tangential contact, the temperature field and the fluid-structural interaction between wheel and rail at low temperatures under presence of snow and ice. For that purpose the behaviour of ice under wheel-rail contact conditions is an important part. In this paper the thermal dynamic model of TSHIJOV [1], [3] for an adiabatic ice probe is updated by the new IAPWS equations of state for water [5] and ice phase Ih [4]. In a first approximation an ice specimen is loaded by specific wheel-rail contact pressure distributions calculated by the half-space formulation to clarify if phase transitions of ice can exist. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A method of calculating ice loads when ice piles up on a fixed wall

A model for calculating ice loads on a shelf ice-resistant platform is developed which corresponds to a cyclic scenario of the interaction of ice with the wall of a platform observed in laboratory experiments. Using relations which follow from the laws of conservation of mass, momentum and energy written for ice which fills a ridge formed at the wall of the platform, an explicit formula is obtained for the ice load on the wall. The generalized forces characterizing the dependence of the energy dissipation accompanying ridging on the dimensions of the sail and keel of the ridge and on the shape of the wall are parametrized. It is shown that the calculated loads obtained using the assumption of hydrostatic equilibrium of the sail and the keel of the ridge correspond to the lower boundary of the experimentally measured loads. The calculated loads obtained using the assumption that the keel of a ridge is of constant size yield an upper estimate of the ice loads. In order to estimate the limit loads at which collapse of the ice into the water occurs, a problem on the formation of a flexural crack in an ice floe which is thrust onto the wall is considered. The average and maximum ice loads on the wall of a platform are calculated on a real scale.     

The choice of constitutive relations for an ice cover

The constitutive relations between the internal stresses and the deformation parameters of a sea ice cover, which are used in the AIDJEX elastoplastic model and Hibler's non-linearly viscous model, are investigated. It is shown that the structural instability of the ice cover with respect to plastic shear deformations is a consequence of the associated flow rule used in these models. The use of constitutive relations which violate the associated flow rule, but which are in good agreement with the physical properties of granular media, is suggested. An ice cover damage parameter and an empirical equation which describes the change in this parameter are introduced into the treatment. Energy relations are investigated.     

Parametrization of the homogeneous ice nucleation rate for the numerical simulation of multiphase flow

Homogeneous ice nucleation in supercooled water is studied with the classical nucleation theory (CNT). The nucleated ice particles are assumed to possess physical properties of metastable cubic ice. The comparison with literature data on homogeneous ice nucleation rates shows a good agreement between CNT predictions and experimental nucleation rates. The CNT predicted nucleation rate, critical cluster radius, and critical nucleation work are parametrized for use in numerical simulations of the freezing process of supercooled water droplets. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Discrete dipole approximation simulations on GPUs using OpenCL—Application on cloud ice particles

The global distribution and climatology of ice clouds are among the main uncertainties in climate modeling and prediction. In order to retrieve ice cloud properties from remote sensing measurements, the scattering properties of all cloud ice particle types must be known. The discrete dipole approximation (DDA) simulates scattering of radiation by arbitrarily shaped particles and is thus suitable for cloud ice crystals. The DDA models the particle as a collection of equal dipoles on a lattice, and is computationally much more expensive than approximations restricted to more regularly shaped particles. On a single computer the calculation for an ice particle of a specific size, for a given scattering plane at one specific wavelength can take several days. We have ported the core routines of the scattering suite “ADDA” to the open computing language (OpenCL), a framework for programming parallel devices like PC graphics cards (graphics processing units, GPUs) or multi-core CPUs. In a typical case we can achieve a speed-up on a GPU as compared to a CPU by a factor of 5 in double precision and a factor of 15 in single precision. Spreading the work load over multiple GPUs will allow calculating the scattering properties even of large cloud ice particles.     

估计海冰厚度函数的一种辨识方法

提出一种采用海冰和海水温度观测数据来估计海冰厚度的辨识方法, 避免了因使用厚度数据所带来的种种局限性. 首先建立一个拟线性海冰-海水热力学系统, 得到了系统解的存在唯一性; 然后以该系统中描述海冰厚度函数的参数为辨识量, 以系统输出的温度和实际观测温度的偏差为目标泛函, 建立了以目标泛函为最小的参数辨识模型; 最后构造了以半隐式差分格式、遗传算法和Hooke-Jeeves算法相结合的数值算法, 得到了海冰厚度函数, 并对辨识量做了敏感性分析. 结果表明: 这种方法是有效可行的.     

Resonance interactions of waves in an ice channel

The properties of low-amplitude surface waves propagating in an ice channel are investigated in the shallow-water approximation. The ice cover is modelled either by a rigid cap or by a thin elastic plate floating on a liquid surface. It is shown that an ice channel is a waveguide for surface waves. The dispersive properties of the natural oscillations of the liquid in the channel are investigated. The resonance velocities of the motion of the load on the channel surface, at which the amplitude of the forced oscillations of the liquid increases without limit in time, are determined. The decay instability of the natural oscillations of high harmonics with respect to waves of the first mode is demonstrated. The process is described by the standard equations for non-linear three-wave interaction. The investigations lead to the conclusion that critical modes of motion of a boat are realizable in an ice channel.     

Mechanism of ice formation in seeded convective storms

Water vapor transfer processes in clouds were examined as they apply to the formation of ice crystals on solver iodide particles acting as ice forming nuclei. Part of the water vapor released from freezing cloud droplets is available for recondensation on aerosol particles active as condensation nuclei at high supersaturations. Calculations have shown that the combination of a high liquid water content, an updraft, and a large concentration of submicron silver iodide particles is required for rapid formation of ice crystals in a seeded cloud. Ice crystal concentrations produced by the recondensation of water vapor on silver iodide particles are orders of magnitude larger than concentrations resulting from Brownian diffusion of silver iodide particles to cloud droplets. Ice formation by this mechanism provides a satisfactory explanation of seeding efficacy in severe storms.

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Zusammenfassung Die Transport-Mechanismen von Wasserdampf in Wolken wurden im Hinblick auf die Bildung von Eiskristallen an Silberjodidteilchen als Eiskerne untersucht. Ein Teil des Wasserdampfes der während des Gefrierens von Wolkentröpfchen abgegeben wird, kann an Aerosolteilchen, die als Kondensationskerne bei hohen Übersättigungen wirksam sind, wieder kondensieren.Rechnungen zeigen, daß ein Zusammenwirken von hohem Gehalt an Flüssigwasser, starken Aufwinden und einer hohen Konzentration von Submikron-Silberjodidteilchen nötig ist, um eine rasche Bildung von Eiskristallen in der geimpften Wolke hervorzurefen. Die Eiskristallkonzentration, die durch die Wiederkondensation von Wasserdampf an Silberjodidteilchen verursacht wird, ist um Größenordnungen höher, als die Konzentrationen, die durch Brownsche Diffusion von Silberjodidteilchen zu Wolkentröpfchen hervorgerufen werden. Die Eisbildung aufgrund dieses Mechanismus erklärt die Wirksamkeit der Silberjodidimpfung in schweren Gewittern zufriedenstellend.

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Nomenclature a, b constants - B ventilation factor - C heat capacity - c _p,c _pi,c _pn specific heat of air, ice, and water - D' diffusion coefficient of water vapor - D(s) Brownian diffusion coefficient - enn h(r) enhancement function - f _i fraction of ice in a freezing droplet - f (r) droplet number distribution - G mass of liquid water in a freezing droplet - g(r) ice crystal number distribution - h(s) aerosol particle number distribution - I proportionality factor indicating the dependence of particle capture rate on the droplet and on ice crystal size distribution - K thermal conductivity of air - K (r) particle concentration number density - L _f latent heat of freezing - LR lapse rate - m mass - N inactive AgI particle number density per unit volume of cloud water - N _a dr dt total number of active AgI particles received by a group of droplets - P(s) number density (aerosol) - q, Q heat - Q (r) number density (droplets) - r radius of cloud droplet or of ice crystal - r', l distance - Re(r) Reynolds number - r' (T) radius of AgI particles above which particles are active as ice forming nuclei - s radius of aerosol particle - S water vapor saturation - surface element - t time - T temperature of supercooling (environment) - T _r droplet surface temperature - v droplet terminal velocity - V(r, T) supersaturated volume produced by a single freezing droplet - V _a(t) total supersaturated volume per unit time - x, z distance - , , constants - thermoconductivity - environment water vapor density - _a density of air - _r water vapor density at the surface of a droplet - _w water density - boundary layer thickness - thermal diffusivity The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

The symmetry of the partition function of some square ice models

We consider the partition function Z(N; x ₁ , …, x_N, y ₁ , …, y_N) of the square ice model with domain wall boundary conditions. We give a simple proof that Z is symmetric with respect to all its variables when the global parameter a of the model is set to the special value a = e^iπ/3 . Our proof does not use any determinant interpretation of Z and can be adapted to other situations (e.g., to some symmetric ice models).     

The ice flow behavior in the neighborhood of the grounding line. Non-Newtonian case

In this paper we address one of the problems that has attracted much interest in the glaciological scientific community which is the grounding line dynamics. The grounding line is the line where transition between ice attached to the solid ground and ice floating over the sea takes place. We analyze a mathematical model describing the ice flow near the grounding line where the ice is considered a non-Newtonian fluid. This generalizes the results obtained in [M.A. Fontelos, A.I. Muñoz, A free boundary problem in glaciology: The motion of grounding lines, Interfaces Free Bound. (9) (2007) 67–93] for the Newtonian case and allows us to consider a more realistic rheological model. We prove the existence and uniqueness (in a class to be defined) of weak solutions with moving grounding lines and zero contact angle and also determine the shape and asymptotic properties of the free boundary. Finally some finite element numerical simulations will illustrate the local and global behavior of the problem solutions.     

The diffraction of plane gravitational waves by the edge of an ice cover

The diffraction of plane surface gravitational waves by the edge of an ice cover lying on the surface of an incompressible fluid of infinitely great depth is considered. The ice cover is simulated by a thin elastic plate. The wave reflection and transmission coefficients are determined when it interacts with the ice cover. A wave field is constructed in the fluid under the conditions that a periodic lumped force and a lumped moment act on the edge of the ice cover. It is shown that as the incident wavelength increases the reflection coefficient tends to zero and the transmission coefficient tends to unity.     

A coupled multivalued model for ice streams and its numerical simulation

^** Email: nati{at}dma.uvigo.es^*** Email: durany{at}dma.uvigo.es^**** Email: anaisabel.munoz{at}urjc.es^***** Email: emanuele.schiavi{at}urjc.es^****** Email: carlosv{at}udc.es This paper deals with the numerical solution of a non-linearmodel describing a free-boundary problem arising in modern glaciology.Considering a shallow, viscous ice sheet flow along a soft,deformable bed, a coupled non-linear system of differentialequations can be obtained. Particularly, an obstacle problemis then deduced and solved in the framework of its complementarityformulation. We present the numerical solution of the resultingmultivalued system modelling the ice sheet non-Newtonian dynamicsdriven by the underlying drainage system. Our numerical resultsshow the existence of fast ice streams when positive wave-likeinitial conditions are considered. The solutions are numericallycomputed with a decoupling iterative method and finite-elementtechnique. A duality algorithm and a projected Gauss–Seidelmethod are the alternatives used to cope with the resultingvariational inequality while the explicit treatment, Newtonmethod or a duality method are proposed to deal with the non-linearsource term. Finally, the numerical solutions are physicallyinterpreted and some comparisons among the numerical methodsare then discussed.     

Optimal control of ice formation in living cells during freezing

A mathematical model of ice formation in living cells during freezing is considered. Application of appropriate averaging techniques to partial differential equations describing the dynamics of water–ice phase transitions reduces spatially distributed relations to several ordinary differential equations with control parameters and uncertainties. Such equations together with an objective functional which expresses the difference between the amount of ice in the extracellular and intracellular liquids are treated as a differential game where the aim of the control is to maximize the objective functional and the aim of the disturbance is opposite. A stable finite-difference scheme for computing the value function is developed. Based on the computed value function, optimal controls are designed to produce cooling protocols ensuring simultaneous freezing inside and outside of living cells. Such a regime balances the pressures inside and outside of cells, which may prevent cells from injuring.     

The motion of a submerged sphere in a liquid under an ice sheet

The solution of the linear steady problem of the flow of an inviscid, incompressible and infinitely deep liquid around a sphere under an ice sheet, which is modelled by a thin elastic stressed plate of constant thickness is constructed. Special cases of this problem are the motion of a submerged sphere under broken ice, a membrane, and also under the free surface both in the presence and absence of capillary effects. The method of multipole expansions is used in the framework of the linear potential wave theory. The hydrodynamic loads (the wave drag and the buoyancy) acting on the body and also the distribution of the deflections of the ice sheet are calculated as a function of the body velocity, the ice thickness and the value of the compressing or stretching forces. It is shown that all the flow characteristics depend considerably on the ratio of the body velocity and the critical velocity of flexural-gravitational waves.     

Existence of weak solutions to a system of nonlinear partial differential equations modelling ice streams

This paper deals with the mathematical analysis of a nonlinear system of three differential equations of mixed type. It describes the generation of fast ice streams in ice sheets flowing along soft and deformable beds. The system involves a nonlinear parabolic PDE with a multivalued term in order to deal properly with a free boundary which is naturally associated to the problem of determining the basal water flux in a drainage system. The other two equations in the system are an ODE with a nonlocal (integral) term for the ice thickness, which accounts for mass conservation and a first order PDE describing the ice velocity of the system. We first consider an iterative decoupling procedure to the system equations to obtain the existence and uniqueness of solutions for the uncoupled problems. Then we prove the convergence of the iterative decoupling scheme to a bounded weak solution for the original system.     

Process of establishing a plane-wave system on ice cover over a dipole moving uniformly in an ideal fluid column

We consider a planar evolution problem for perturbations of the ice cover by a dipole starting its uniform rectilinear horizontal motion in a column of an initially stationary fluid. Using asymptotic Fourier analysis, we show that at supercritical velocities, waves of two types form on the water–ice interface. We describe the process of establishing these waves during the dipole motion. We assume that the fluid is ideal and incompressible and its motion is potential. The ice cover is modeled by the Kirchhoff–Love plate.     

Hopf bifurcation in a conceptual climate model with ice–albedo and precipitation–temperature feedbacks

In this paper we analyse a dynamical system based on the so-called KCG (Källén, Crafoord, Ghil) conceptual climate model. This model describes an evolution of the globally averaged temperature and the average extent of the ice sheets. In the nondimensional form the equations can be simplified facilitating the subsequent analysis. We consider the limiting case of a stationary snow line for which the phase plane can be completely analysed and the type of each critical point can be determined. One of them can exhibit the Hopf bifurcation and we find sufficient conditions for its existence. Those, in turn, have a straightforward physical meaning and indicate that the model predicts internal oscillations of the climate. Using the typical real-world values of appearing parameters we conclude that the obtained results are in the same ballpark as the conditions on our planet during the quaternary ice ages. Our analysis is a rigorous justification of a generalization of some previous results by KCG and other authors.     

An identification method to estimate surface shortwave fluxes on temperature and thickness observations of sea ice in CHINARE 2006

In this paper, a parameter identification method to determine surface shortwave fluxes using temperature and thickness measurements of sea ice in CHINARE 2006 is presented. Adopting a new standard for the calculation of the thermodynamic properties of seawater named TEOS‐10, the surface shortwave fluxes are calculated by the temperature and thickness observations that were measured at Nella Fjord around Zhongshan Station, Antarctica. New simulations for temperature profiles in a different measurement period are performed by three parameterization schemes including the present method, Zillman and Shine. All numerical results are compared with in situ measurements. Results show that better simulations of the surface shortwave radiations and temperature distributions are possible with the identification method than Zillman and Shine. Therefore this method is valid, and the obtained shortwave radiation function can be applied in sea ice modeling. Copyright © 2015 John Wiley & Sons, Ltd.