This review of the dynamics of the flow of glaciers and ice sheets focusses on the mathematical models which have been developed to explain a number of observations in the behaviour of these large ice masses. It takes a personal view, starting with an account of the historical development of the subject, followed by a tour of some of the observational puzzles and theoretical solutions which I have found most intriguing. Most of the discourse involves basal processes: glacier sliding, subglacial hydrology, bedform generation. I describe in some detail wave ogives, surging glaciers, subglacial floods and subglacial bedforms. I variously explore issues in sliding theory: sub-temperate sliding, slip or till deformation; the rheology of till; the status of the instability theory of drumlin formation. And in an effort to broaden the field of view, I include discussion of Dansgaard-Oeschger and Heinrich events, grounding line stability, and the formation of eskers and, particularly, flutes.

Two-dimensional ideal incompressible magnetohydrodynamic (MHD) linear waves at the surface of a rotating sphere are studied as a model to imitate the outermost layer of the Earth's core or the solar tachocline. This thin conducting layer is permeated by a toroidal magnetic field the magnitude of which depends only on the latitude. The Malkus background field, which is proportional to the sine of the colatitude, provides two well-known groups of branches; on one branch, retrograde Alfvén waves gradually become fast magnetic Rossby (MR) waves as the field amplitude decreases, and on the other, prograde Alfvén waves undergo a gradual transition into slow MR waves. In the case of non-Malkus fields, we demonstrate that the associated eigenvalue problems can yield a continuous spectrum instead of Alfvén and slow MR discrete modes. The critical latitudes attributed to the Alfvén resonance eliminate these discrete eigenvalues and produce an infinite number of singular eigenmodes. The theory of slowly varying wave trains in an inhomogeneous magnetic field shows that a wave packet related to this continuous spectrum propagates toward a critical latitude corresponding to the wave and is eventually absorbed there. The expected behaviour whereby the retrograde propagating packets pertaining to the continuous spectrum approach the latitudes from the equatorial side and the prograde ones approach from the polar side is consistent with the profiles of their eigenfunctions derived using our numerical calculations. Further in-depth discussions of the Alfvén continuum would develop the theory of the “wave–mean field interaction” in the MHD system and the understanding of the dynamics in such thin layers.

Zonal flows driven by rotation and convection are found commonly within astrophysical and geophysical bodies. Multiple jets are most famously observed on the surface of Jupiter, with prograde and retrograde zonal flows making up part of its banded structure of zones and belts. Non-magnetic studies of convection in a rotating annulus model have previously been shown to produce multiple jet structures, similar to those observed in atmospheres of giant planets such as Jupiter. These giant planets have strong magnetic fields which impact on convection in the electrically conducting regions of their atmospheres. Therefore the effect of a magnetic field on the multiple jet solutions of the annulus model is of astrophysical interest. In this work we impose a uniform azimuthal magnetic field to the annulus model and vary its strength and other control parameters to determine the parameter space where multiple jets exist. Zonal flows and multiple jet solutions are found at weak magnetic field strength and, indeed, the magnetic field can promote the production of these features. Simulations with a weak magnetic field or no magnetic field are also found to match well with the Rhines scaling theory. For cases with multiple jets, a strong inertial force must be present, combined with either a weak Lorentz force, or a Lorentz force entering the main balance with increased contributions from the viscous force. At strong magnetic field strengths two regimes are found. For weak magnetic diffusion zonal flows can be retained, albeit without the multiple jet structure. For larger magnetic diffusion non-axisymmetric solutions without clear bands are found. In these regimes, the Lorentz force appears in the primary balance of forces and zonal flows are only retained at strong driving when inertia is able to also enter the balance at leading order.

Time-dependent meridional circulation and differential rotation in radiative zones are central open issues in stellar evolution theory. We streamline this challenging problem using the “downward control principle” of atmospheric science, under a geostrophic f-plane approximation. We recover the known stellar physics result that the steady-state meridional circulation decays on the length scale proportional to N / f × Pr , assuming molecular viscosity is the dominant drag mechanism. Prior to steady-state, the meridional circulation and the zonal wind (= differential rotation) spread together via radiative diffusion, under thermal wind balance. The corresponding (fourth-order) hyperdiffusion process is reasonably well approximated by regular (second-order) diffusion on scales of order a pressure scale-height. We derive an inhomogeneous diffusion (equiv. advection-diffusion) equation for the zonal flow which admits closed-form time-dependent solutions in a finite depth domain, allowing for rapid prototyping of differential rotation profiles. In the weak drag limit, we find that the time to rotational steady-state can be longer than the Eddington-Sweet time and be instead determined by the longer drag time. Unless strong enough drag operates, the internal rotation of main-sequence stars may thus never reach a steady-state. Streamlined meridional circulation solutions provide leading-order internal rotation profiles for studying the role of fluid/MHD instabilities (or waves) in redistributing angular momentum in the radiative zones of stars. Despite clear geometrical limitations and simplifying assumptions, one might expect our thin-layer geostrophic approach to offer qualitatively useful results to understand deep meridional circulation in stars.

The governing equations for a viscous, incompressible fluid, in the thin-shell approximation, written in a coordinate frame which ensures validity at the North Pole, is simplified by invoking the f-plane approximation. Together with suitable stress conditions at the surface, describing the interaction of wind over ice and ice over water, a solution is developed for the Transpolar Drift current. This involves prescribing the near-surface geostrophic ocean flow and also, for ease of calculation, a simple version of the Ekman flow. Then, via the stress conditions at the surface, the near-surface wind consistent with these flows is obtained. An example, which exhibits the reduction in ice thickness as the flow moves over the North Pole towards the Fram Strait, and also describes the relative directions and speeds of the wind, ice and water, is presented in graphical form; this solution recovers what is observed. The considerable freedom available in this approach, allowing choices to be made for the geostrophic flow and the wind (both variable), and the ice profile at some initial position, is explained, opening the way to further, extensive investigations.

The time evolution of the mother body of a planar, uniform vortex that moves in an incompressible, inviscid fluid is investigated. The vortex is isolated, so that its motion is just due to self-induced velocities. Its mother body is defined as the part internal to the vortex of the singular set of the Schwarz function of its boundary. In the present analysis, it is an arc of curve (branch cut), starting and ending in the two internal branch points of this function, across any point of which the Schwarz function experiences a finite jump. By looking at the mother body from the outside of the vortex, it behaves as a vortex sheet having density of circulation given by the jump of the Schwarz function. Its name (mother body) is taken from Geophysics, and it is here used due to its property of generating, outside the vortex and on its boundary, the same velocity as the vortex itself. The shape of the branch cut and the jump of the Schwarz function across any point of it change in time, by following the motion of the vortex boundary. As it happens for a physical vortex sheet, the mother body is not a material line, so that it does not move according to the velocities induced by the vortex. In the present paper, the cut shape, the above jumps, as well as the cut velocities are deduced from the time evolution equation of the Schwarz function. Numerical experiments, carried out by building the branch cut and calculating the limit values of the Schwarz function on its sides during the vortex motion, confirm the analytical calculations. Some global quantities (circulation, first and second order moments) are here rewritten as integrals on the cut, and their conservation during the vortex motion is analytically and numerically verified. Indeed, the numerical simulations show that they behave in the same way as their classical contour dynamics forms, written in terms of integrals on the vortex boundary. This proves that the shape of the cut, as well as the limit values of the Schwarz function on its sides, are correctly calculated during the motion.

We present three results on stability of rolls in Boussinesq convection in a plane horizontal layer with rigid boundaries that is rotating about an inclined axis with the angular velocity Ω = ( Ω 1 , Ω 2 , Ω 3 ) . (i) We call the full problem the set of equations governing the temporal behaviour of the flow and temperature for an arbitrary Ω , and by the reduced problem the set of equations for the angular velocity ( Ω 1 , 0 , Ω 3 ) . Here x,y are horizontal Cartesian coordinates in the layer and z is the vertical one. We prove that a y-independent solution to one of the two problems is also a solution to the second one. (ii) We calculate the critical Rayleigh number for the monotonic onset of convection. The instability mode in the form of rolls (a flow independent of a horizontal direction) is assumed. Let β be the angle between the horizontal projection of Ω and the rolls axes. We show that β = 0 for the least stable mode. When the axis of rotation is horizontal, this is proven analytically, and for Ω 3 ≠ 0 , the result is obtained numerically. Taking i into account, we conclude that the critical Rayleigh number for the onset of convection is independent of Ω 1 and Ω 2 and the emerging flow are rolls with axis aligned with the horizontal component of the rotation vector. (iii) We study the behaviour of convective flows by integrating numerically the three-dimensional equations of convection for Ω = ( 0 , Ω 2 , Ω 3 ) and a range of the Rayleigh numbers, other parameters of the problem being fixed. We assume square horizontal periodicity cells, whose sides are equal to the period of the most unstable mode. The computations indicate that, in general, in the nonlinear regime convective rolls become more stable as Ω 2 increases. Namely, on increasing Ω 2 , the interval of the Rayleigh numbers for which convective rolls are stable increases.

We investigate numerically the evolution of a baroclinic vortex in a two-level surface quasi-geostrophic model. The vortex is composed of two circular patches of uniform buoyancy, one located at each level. We vary the vortex radii, the magnitude of buoyancy, and the vertical distance between the two levels. We also study different radial profiles of buoyancy for each vortex. This article considers two main situations: firstly, initially columnar vortices with like-signed buoyancies. These vortices are contra-rotating, are linearly unstable and may break. Secondly, we consider initially tilted vortices with opposite-signed buoyancies, which may align vertically. Numerical experiments show that (1) identical contra-rotating vortices break into hetons when initially perturbed by low azimuthal modes; (2) unstable, vertically asymmetric, contra-rotating vortices can stabilise nonlinearly more often than vertically symmetric ones, and can form quasi-steady baroclinic tripoles; (3) co-rotating vortices can align when the two levels are close to each other vertically, and when the vortices are initially horizontally distant from each other by less than three radii; (4) for initially more distant vortices, two such vortices rotate around the plane center; and(5) in all cases, the vortex boundaries are disturbed by Rossby waves. These results compare favorably to earlier results with internal quasi-geostrophic vortices. Further modelling efforts may extend the present study to fully three-dimensional ocean dynamics.

ABSTRACT The fastest growing modes in anisotropic rotating magnetoconvection (RMC) processes are presented. After reminding the state of the art, we present our new approach that applies to isotropic as well as anisotropic diffusivities' conditions. We describe three cases: T (only temperature perturbation is time dependent), Q (temperature and velocity field perturbations are time dependent, but magnetic field perturbation is time independent) and G (temperature, magnetic and velocity field perturbations are time dependent). Isotropies as well as anisotropies are further distinguished by values of molecular and turbulent diffusivities. We show that T case does not describe properly convection in the Earth's outer core conditions, because it implies too huge Ekman numbers for the transition between the RMC modes of weak and strong magnetic field types. In G case, the convection is usually much more facilitated than in the T case: instabilities may arise with much smaller values of Ekman number and, in general, all types of convections occur with values of the physical parameters of the Earth consistent with the most reliable estimations. We demonstrate (and indicate) that Q and G cases can be well suited for the magnetic field of the Earth (and for other planetary magnetic fields), but only G case may correspond to turbulent stay of the Earth's core. We prove that, analogously like in the marginal modes, the value of anisotropic parameter α (the ratio between horizontal and vertical diffusivities) crucially influences the convection. The cases of α < 1 ( 1 $ ]]> α > 1 ) strongly decrease (increase) the Ekman numbers at which the RMC modes of weak and strong magnetic field types change between each other. Finally, we show and stress that not all types of anisotropies in the fastest growing modes can be equally strong. More specifically, we show that a fixed Rayleigh number puts a constraint on the maximum value of α, but do not put any lower positive limit on the minimum value of α. This special constraint is given by the necessary positiveness of the growth rate of the fastest growing modes. Our RMC approach allows to easily deal with very huge wave numbers and Rayleigh numbers as well as with very small Ekman numbers, what is usually not possible in the standard geodynamo simulations.

We numerically investigate thermochemically-driven convection in a spherical shell at a high Ekman number for a range of Rayleigh numbers, Prandtl numbers and buoyancy ratios. Rotating convection displays a nearly axisymmetric large-scale structure and appears mostly unaffected by the nature (thermal or chemical) of the driving buoyancy source. We observe both precession directions, prograde and retrograde, of convection structures and identify a snap-through transition between the prograde and retrograde flows in the parameter space spanned by the Rayleigh and Prandtl numbers. Equatorially wall-attached and spiralling columnar convection structures, which typically emerge in rapidly rotating shells, are not observed.