The response of a floating ice sheet to an accelerating line load

The two-dimensional response of a thin, floating sheet of ice to a line load that accelerates from rest at $t = 0$ to a uniform velocity V for $t \geq T$ is determined through an integral-transform solution of the linearized equations of motion. If $T = 0$ – i.e. if the load is impulsively started with velocity V – the solution exhibits singularities at $V = c_0$, the shallow-water-gravity-wave speed, and $V = c_{\min}$, the minimum speed for transverse motion of the ice, but these singularities are avoided by the acceleration of the load through the critical speeds

Instability of a plane conducting free surface submitted to an alternating magnetic field

This paper considers the stability of a horizontal liquid-metal free surface in the presence of a horizontal alternating magnetic field. A weak formulation is used to derive a generalized Mathieu–Hill equation for the evolution of surface perturbations. Previous studies which rely on time-averaging the electromagnetic force over one field cycle have predicted a generally weak instability, but we find much larger growth rates near the resonances, where the surface wave frequency is an integral multiple of the field frequency. The method can be extended to include viscous and ohmic damping; the former has little effect, while the latter damps all waves except those whose frequency is close to the field frequency. Growth rates can be closely approximated by simple algebraic formulae, as can the critical magnetic field strength for the onset of instability

The starfish experiment: a Lagrangian approach

The present paper analyses the free surface deformation of a liquid metal drop under the influence of an alternating magnetic field. The analysis is restricted to the first axi- symmetric mode oscillation. In the low frequency case, the electromagnetic forces are of gradient type and purely oscillatory. Without any viscous damping, it is then possible to build a Lagrangian function, which involves the kinetic energy, the gravitational energy, the surface energy and the electromagnetic energy. The time evolution of the pool height is easily obtained from the Lagrange equation. It is shown that the pool height behaves like a non-linear forced oscillator

Nonlinear development of the kink instability in coronal flux tubes

Solar prominences and flares are believed to be caused by rapid release of magnetic energy stored in coronal magnetic fields. Recent studies of the linear phase of ideal MHD instabilities has shown that energy release is slow and weak, so it is therefore important to study the nonlinear phase to see if this provides a mechanism for significant energy release. In this paper we describe a Lagrangian numerical scheme for simulating nonlinear evolution of ideal MHD equilibria and apply it to an unstable finite Gold-Hoyle flux tube, line-tied to perfectly conducting endplates. The ensuing kink instability develops considerably faster than linear theory would predict, and eventually (over typically 100 Alfvén time scales) a new kinked equilibrium is attained, in which current sheets appear to be present. Little magnetic energy is lost in the ideal MHD phase, but resistive instabilities in the current sheets could lead to much more explosive energy release. Numerical studies of nonlinear interactions indicate that growth of the unstable kink mode is suppressed by the presence of other modes, which offers a possible explanation of the observed longevity of coronal loops

Sheared coronal arcades: An evaluation of recent studies

We show that the family of magnetic force-free equilibria obtained by Low using the generating function method is really a sequence of Gold-Hoyle flux tubes. This sequence is stable under a wide range of solar conditions since each member, specified by the shear parameter μ, is anchored to the photosphere along an axial slice. We go on to demonstrate that recent magnetic relaxation simulations by Klimchuk and Sturrock are fundamentally incapable of representing the unconnected helical field lines inherent in the high-shear (μ > 1) Low solutions. Nonetheless, we believe that the numerical simulations are more likely to describe the equilibria of highly sheared arcades since they involve no change in topology with increasing shear. This view is reinforced by magnetic energy calculations which confirm that the Gold-Hoyle solutions are more energetic for μ > 1 than the numerical equilibria of Klimchuk and Sturrock

Completeness, conservation and error in SPH for fluids

Smoothed particle hydrodynamics (SPH) is becoming increasingly common in the numerical simulation of complex fluid flows and an understanding of the errors is necessary. Recent advances have established techniques for ensuring completeness conditions (low-order polynomials are interpolated exactly) are enforced when estimating property gradients, but the consequences on errors have not been investigated. Here, we present an expression for the error in an SPH estimate, accounting for completeness, an expression that applies to SPH generally. We revisit the derivation of the SPH equations for fluids, paying particular attention to the conservation principles. We find that a common method for enforcing completeness violates a property required of the kernel gradients, namely that gradients with respect the two position variables be equal and opposite. In such models this means conservation principles are not enforced and we present results that show this. As an aside we show the summation interpolant for density is a solution of, and may be used in the place of, the discretized, symmetrized continuity equation. Finally, we examine two examples of discretization errors, namely numerical boundary layers and the existence of crystallized states. Copyright ©2007 John Wiley & Sons, Ltd