Papers by Sergiy Vasylkevych
Quarterly Journal of the Royal Meteorological Society
Physical Review B
We report the existence of a new regime for domain wall motion in uniaxial and near-uniaxial ferr... more We report the existence of a new regime for domain wall motion in uniaxial and near-uniaxial ferromagnetic nanowires, characterised by applied magnetic fields sufficiently strong that one of the domains becomes unstable. There appears a new stable solution of the Landau-Lifshitz-Gilbert equation, describing a nonplanar domain wall moving with constant velocity and precessing with constant frequency. Even in the presence of thermal noise, the new solution can propagate for distances on the order of 500 times the field-free domain wall width before fluctuations in the unstable domain become appreciable.
Physical Review Letters
The large curvature effects on micromagnetic energy of a thin ferromagnetic film with nonlocal di... more The large curvature effects on micromagnetic energy of a thin ferromagnetic film with nonlocal dipolar energy are considered. We predict that the dipolar interaction and surface curvature can produce perpendicular anisotropy which can be controlled by engineering a special type of periodic surface shape structure. Similar effects can be achieved by a significant surface roughness in the film. We show that in general the anisotropy can point in an arbitrary direction depending on the surface curvature. We provide simple examples of these periodic surface structures to demonstrate how to engineer particular anisotropies in the film.
Journal of Physics A: Mathematical and Theoretical
A ribbon is a surface swept out by a line segment turning as it moves along a central curve. For ... more A ribbon is a surface swept out by a line segment turning as it moves along a central curve. For narrow magnetic ribbons, for which the length of the line segment is much less than the length of the curve, the anisotropy induced by the magnetostatic interaction is biaxial, with hard axis normal to the ribbon and easy axis along the central curve. The micromagnetic energy of a narrow ribbon reduces to that of a one-dimensional ferromagnetic wire, but with curvature, torsion and local anisotropy modified by the rate of turning. These general results are applied to two examples, namely a helicoid ribbon, for which the central curve is a straight line, and a Möbius ribbon, for which the central curve is a circle about which the line segment executes a 180 • twist. In both examples, for large positive tangential anisotropy, the ground state magnetization lies tangent to the central curve. As the tangential anisotropy is decreased, the ground state magnetization undergoes a transition, acquiring an in-surface component perpendicular to the central curve. For the helicoid ribbon, the transition occurs at vanishing anisotropy, below which the ground state is uniformly perpendicular to the central curve. The transition for the Möbius ribbon is more subtle; it occurs at a positive critical value of the anisotropy, below which the ground state is nonuniform. For the helicoid ribbon, the dispersion law for spin wave excitations about the tangential state is found to exhibit an asymmetry determined by the geometric and magnetic chiralities.
We previously introduced a general variational framework for the derivation of balance models for... more We previously introduced a general variational framework for the derivation of balance models for nearly geostrophic flow which is based on changing coordinates such that consistent asymptotics in the trans-formed Lagrangian leads to a degenerate structure. In this paper, we study first order models for shallow water with spatially varying Corio-lis parameter. These models can be formulated in terms of an advected potential vorticity with a nonlinear vorticity inversion relation. The associated solvability conditions are easily satisfied and, in a certain special case, no more restrictive than on an f -plane.
The construction of modified equations is an important step in the backward error analysis of sym... more The construction of modified equations is an important step in the backward error analysis of symplectic integrator for Hamiltonian systems. In the context of partial differential equations, the standard construction leads to modified equations with increasingly high frequencies which increase the regularity requirements on the analysis. In this paper, we consider the next order modified equations for the implicit midpoint rule applied to the semilinear wave equation to give a proof-of-concept of a new construction which works directly with the variational principle. We show that a carefully chosen change of coordinates yields a modified system which inherits its analytical properties from the original wave equation. Our method systematically exploits additional degrees of freedom by modifying the symplectic structure and the Hamiltonian together.
Geophysical & Astrophysical Fluid Dynamics, 2013
ABSTRACT In this paper, we derive and study approximate balance models for nearly geostrophic sha... more ABSTRACT In this paper, we derive and study approximate balance models for nearly geostrophic shallow water flow where the Coriolis parameter is permitted to vary across the domain as long as it remains nondegenerate. This situation includes, for example, the β-plane approximation to the shallow water equations at mid-latitudes. Our approach is based on changing configuration space coordinates in the underlying variational principle in such a way that consistent asymptotics in the transformed Lagrangian leads to a degenerate Lagrangian structure. In this paper, we restrict our attention to first-order models. We show that the resulting models can be formulated in terms of an advected potential vorticity with a nonlinear vorticity inversion relation. We study the associated solvability conditions and identify a subfamily of models for which these conditions are satisfied without additional restrictions on the data. Finally, we provide the link between our framework and the theory of constrained Hamiltonian systems.
Dynamics of Partial Differential Equations, 2005
This paper provides a precise sense in which the time t map for the Euler equations of an ideal f... more This paper provides a precise sense in which the time t map for the Euler equations of an ideal fluid in a region in R n (or a smooth compact n-manifold with boundary) is a Poisson map relative to the Lie-Poisson bracket associated with the group of volume preserving diffeomorphism group. This is interesting and nontrivial because in Eulerian representation, the time t maps need not be C 1 from the Sobolev class H s to itself (where s > (n/2) + 1). The idea of how this difficulty is overcome is to exploit the fact that one does have smoothness in the Lagrangian representation and then carefully perform a Lie-Poisson reduction procedure. Contents 1 Introduction 2 Solutions of the Euler Equation 3 Motivation: The Poisson Reduction Theorem 4 Poisson Structures on Weak Riemannian Manifolds 5 Geometric Properties of the Flow of the Euler Equations 12 6 Conclusions 22
Discrete and Continuous Dynamical Systems, 2011
This paper presents a first rigorous study of the so-called largescale semigeostrophic equations ... more This paper presents a first rigorous study of the so-called largescale semigeostrophic equations which were first introduced by R. Salmon in 1985 and later generalized by the first author. We show that these models are Hamiltonian on the group of H s diffeomorphisms for s > 2. Notably, in the Hamiltonian setting an apparent topological restriction on the Coriolis parameter disappears. We then derive the corresponding Hamiltonian formulation in Eulerian variables via Poisson reduction and give a simple argument for the existence of H s solutions locally in time.
Archive for Rational Mechanics and Analysis, 2013
We prove the existence and uniqueness of global classical solutions to the generalized large-scal... more We prove the existence and uniqueness of global classical solutions to the generalized large-scale semigeostrophic equations with periodic boundary conditions. This family of Hamiltonian balance models for rapidly rotating shallow water includes the L 1 model derived by R. Salmon in 1985 and its 2006 generalization by the second author. The results are, under the physical restriction that the initial potential vorticity is positive, as strong as those available for the Euler equations of ideal fluid flow in two dimensions. Moreover, we identify a special case in which the velocity field is two derivatives smoother in Sobolev space as compared to the general case. Our results are based on careful estimates which show that, although the potential vorticity inversion is nonlinear, bounds on the potential vorticity inversion operator remain linear in derivatives of the potential vorticity. This permits the adaptation of an argument based on elliptic L p theory, proposed by Yudovich in 1963 for proving the existence and uniqueness of weak solutions for the two dimensional Euler equations, to our particular nonlinear situation.
Journal of Physics A: Mathematical and Theoretical
We derive a family of balance models for rotating stratified flow in the primitive equation setti... more We derive a family of balance models for rotating stratified flow in the primitive equation setting. By construction, the models possess conservation laws for energy and potential vorticity and are formally of the same order of accuracy as Hoskins' semigeostrophic equations. Our construction is based on choosing a new coordinate frame for the primitive equation variational principle in such a way that the consistently truncated Lagrangian degenerates. We show that the balance relations so obtained are elliptic when the fluid is stably stratified and certain smallness assumptions are satisfied. Moreover, the potential temperature can be recovered from the potential vorticity via inversion of a non-standard Monge-Ampère problem which is subject to the same ellipticity condition. While the present work is entirely formal, we conjecture, based on a careful rewriting of the equations of motion and a straightforward derivative count, that the Cauchy problem for the balance models is well posed subject to conditions on the initial data. Our family of models includes, in particular, the stratified analog of the L 1 balance model of R. Salmon.
Journal of Physics A: Mathematical and Theoretical
We derive a family of balance models for rotating stratified flow in the primitive equation setti... more We derive a family of balance models for rotating stratified flow in the primitive equation setting. By construction, the models possess conservation laws for energy and potential vorticity and are formally of the same order of accuracy as Hoskins' semigeostrophic equations. Our construction is based on choosing a new coordinate frame for the primitive equation variational principle in such a way that the consistently truncated Lagrangian degenerates. We show that the balance relations so obtained are elliptic when the fluid is stably stratified and certain smallness assumptions are satisfied. Moreover, the potential temperature can be recovered from the potential vorticity via inversion of a non-standard Monge-Ampère problem which is subject to the same ellipticity condition. While the present work is entirely formal, we conjecture, based on a careful rewriting of the equations of motion and a straightforward derivative count, that the Cauchy problem for the balance models is well posed subject to conditions on the initial data. Our family of models includes, in particular, the stratified analog of the L 1 balance model of R. Salmon.
In this article we derive the equations for a rotating stratified fluid governed by inviscid Eule... more In this article we derive the equations for a rotating stratified fluid governed by inviscid Euler–Boussinesq and primitive equations that account for the effects of the perturbations upon the mean. Our method is based on the concept of the geometric generalized Lagrangian mean recently introduced by Gilbert and Vanneste, combined with generalized Taylor and horizontal isotropy of fluctuations as turbulent closure hypotheses. The models we obtain arise as Euler–Poincaré equations and inherit from their parent systems conservation laws for energy and potential vorticity. They are structurally and geometrically similar to Euler–Boussinesq-α and primitive equations-α models, however feature a different regularizing second order operator.
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Papers by Sergiy Vasylkevych