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Recent studies have demonstrated the possibility of constructing magnetostrophic dynamo models, which describe the slowly evolving background state of Earth's magnetic field when inertia and viscosity are negligible. Here we explore the properties of steady, stable magnetostrophic states as a leading order approximation to the slow dynamics within Earth's core. For the case of an axisymmetric magnetostrophic system driven by a prescribed α-effect, we confirmed the existence of four known steady states: ± B d , ± B q , where B d is purely dipolar and B q is purely quadrupolar. Importantly, here we show that in all but the most weakly driven cases, an initial magnetic field that is not purely dipolar or quadrapolar never converges to these states. Despite this instability, we also show that there are a plethora of instantaneous solutions that are quasi-steady, but nevertheless unstable. If the dynamics in Earth's core are reasonably modelled by a strongly driven α-effect, this work suggests that the background state can never be steady. We discuss the difficulties in comparing our magnetostrophic models with geomagnetic timeseries. [ABSTRACT FROM AUTHOR]

Mounting evidence from both seismology and numerical experiments on core composition suggests the existence of a layer of stably stratified fluid at the top of Earth's outer core. In such a layer, a magnetostrophic force balance and suppressed radial motion lead to stringent constraints on the magnetic field, named Malkus constraints , which are a much more restrictive extension of the well known Taylor constraints. Here, we explore the consequences of such constraints for the structure of the core's internal magnetic field. We provide a new simple derivation of these Malkus constraints, and show solutions exist which can be matched to any external potential field with arbitrary depth of stratified layer. From considerations of these magnetostatic Malkus constraints alone, it is therefore not possible to uniquely infer the depth of the stratified layer from external geomagnetic observations. We examine two models of the geomagnetic field defined within a spherical core, which obey the Taylor constraints in an inner convective region and the Malkus constraints in an outer stratified layer. When matched to a single-epoch geomagnetic potential field model, both models show that the toroidal magnetic field within the outer layer is about 100 times stronger compared to that in the inner region, taking a maximum value of 8 mT at a depth of 70 km. The dynamic regime of such a layer, modulated by suppressed radial motion but also a locally enhanced magnetic field, may therefore be quite distinct from that of any interior dynamo. [ABSTRACT FROM AUTHOR]

We develop an algorithm for computing the solution of a large system of linear ordinary differential equations (ODEs) with polynomial inhomogeneity. This is equivalent to computing the action of a certain matrix function on the vector representing the initial condition. The matrix function is a linear combination of the matrix exponential and other functions related to the exponential (the so-called ϕ-functions). Such computations are the major computational burden in the implementation of exponential integrators, which can solve general ODEs. Our approach is to compute the action of the matrix function by constructing a Krylov subspace using Arnoldi or Lanczos iteration and projecting the function on this subspace. This is combined with time-stepping to prevent the Krylov subspace from growing too large. The algorithm is fully adaptive: it varies both the size of the time steps and the dimension of the Krylov subspace to reach the required accuracy. We implement this algorithm in the matlab function phipm and we give instructions on how to obtain and use this function. Various numerical experiments show that the phipm function is often significantly more efficient than the state-of-the-art. [ABSTRACT FROM AUTHOR]

We present a numerical method for computing the pure-point spectrum associated with the linear stability of multidimensional traveling fronts to parabolic nonlinear systems. Our method is based on the Evans function shooting approach. Transverse to the direction of propagation we project the spectral equations onto a finite Fourier basis. This generates a large, linear, one-dimensional system of equations for the longitudinal Fourier coefficients. We construct the stable and unstable solution subspaces associated with the longitudinal far-field zero boundary conditions, retaining only the information required for matching, by integrating the Riccati equations associated with the underlying Grassmannian manifolds. The Evans function is then the matching condition measuring the linear dependence of the stable and unstable subspaces and thus determines eigenvalues. As a model application, we study the stability of two-dimensional wrinkled front solutions to a cubic autocatalysis model system. We compare our shooting approach with the continuous orthogonalization method of Humpherys and Zumbrun. We then also compare these with standard projection methods that directly project the spectral problem onto a finite multidimensional basis satisfying the boundary conditions. [ABSTRACT FROM AUTHOR]

The Alekseev–Gröbner lemma is combined with the theory of modified equations to obtain an a priori estimate for the global error of numerical integrators. This estimate is correct up to a remainder term of order h2p, where h denotes the step size and p the order of the method. It is applied to nonlinear oscillators whose behaviour is described by the Emden–Fowler equation y″+tνyn=0. The result shows explicitly that later terms sometimes blow up faster than the leading term of order hp, necessitating the whole computation. This is supported by numerical experiments. [ABSTRACT FROM PUBLISHER]