Orthogonal signed-distance coordinates and vector calculus near evolving curves and surfaces

We provide an elementary derivation of an orthogonal coordinate system for boundary layers around evolving smooth surfaces and curves based on the signed-distance function. We go beyond previous works on the signed-distance function and collate useful vector calculus identities for these coordinates. These results and provided code enable consistent accounting of geometric effects in the derivation of boundary layer asymptotics for a wide range of physical systems.Comment: 24 pages, 3 figures, Mathematica code available at https://github.com/ericwhester/signed-distance-cod

Double-diffusive instabilities of a shear-generated magnetic layer

Previous theoretical work has speculated about the existence of double-diffusive magnetic buoyancy instabilities of a dynamically evolving horizontal magnetic layer generated by the interaction of forced vertically sheared velocity and a background vertical magnetic field. Here we confirm numerically that if the ratio of the magnetic to thermal diffusivities is sufficiently low then such instabilities can indeed exist, even for high Richardson number shear flows. Magnetic buoyancy may therefore occur via this mechanism for parameters that are likely to be relevant to the solar tachocline, where regular magnetic buoyancy instabilities are unlikely.Comment: Submitted to ApJ

The Evolution of a Double Diffusive Magnetic Buoyancy Instability

Recently, Silvers, Vasil, Brummell, & Proctor (2009), using numerical simulations, confirmed the existence of a double diffusive magnetic buoyancy instability of a layer of horizontal magnetic field produced by the interaction of a shear velocity field with a weak vertical field. Here, we demonstrate the longer term nonlinear evolution of such an instability in the simulations. We find that a quasi two-dimensional interchange instability rides (or "surfs") on the growing shear-induced background downstream field gradients. The region of activity expands since three-dimensional perturbations remain unstable in the wake of this upward-moving activity front, and so the three-dimensional nature becomes more noticeable with time.Comment: 9 pages; 3 figures; accepted to appear in IAU symposium 27

On the stability of isothermal shocks in black hole accretion disks

Most black holes possess accretion disks. Models of such disks inform observations and constrain the properties of the black holes and their surrounding medium. Here, we study isothermal shocks in a thin black hole accretion flow. Modelling infinitesimal molecular viscosity allows the use of multiple-scales matched asymptotic methods. We thus derive the first explicit calculations of isothermal shock stability. We find that the inner shock is always unstable, and the outer shock is always stable. The growth/decay rates of perturbations depend only on an effective potential and the incoming--outgoing flow difference at the shock location. We give a prescription of accretion regimes in terms of angular momentum and black hole radius. Accounting for angular momentum dissipation implies unstable outer shocks in much of parameter space, even for realistic viscous Reynolds numbers of the order $\approx 10^{20}$.Comment: 26 page

A study of the double pendulum using polynomial optimization

In dynamical systems governed by differential equations, a guarantee that trajectories emanating from a given set of initial conditions do not enter another given set can be obtained by constructing a barrier function that satisfies certain inequalities on phase space. Often these inequalities amount to nonnegativity of polynomials and can be enforced using sum-of-squares conditions, in which case barrier functions can be constructed computationally using convex optimization over polynomials. To study how well such computations can characterize sets of initial conditions in a chaotic system, we use the undamped double pendulum as an example and ask which stationary initial positions do not lead to flipping of the pendulum within a chosen time window. Computations give semialgebraic sets that are close inner approximations to the fractal set of all such initial positions