The mechanisms by which wave energy may be absorbed in Alfvén resonant layers and cause heating of laboratory plasmas and coronal flux tubes are well known. Similar transfer of energy between waves and mean flux components also frequently takes place in large-scale dynamos. However, in the latter case the precise nature of this absorption mechanism has largely been overlooked in the interest of analysing the electromotive force (EMF) responsible for the magnetic induction in terms of mean-field quantities. In this study, a first-order solution valid in the vicinity of an Alfvén resonant layer is derived for a cylindrical plasma described by the coupled incompressible Navier–Stokes and induction equations. The solution is used to obtain an analytical expression for the EMF due to waves that arise from a magnetohydrodynamic (MHD) instability. This exposes the functional dependency of the EMF on the background state and the wave properties, and enables specific criteria for the existence of a non-zero wave EMF to be formulated. In addition, an expression for the group velocity and a detailed account of the energy absorption in the resonant layers are provided. The formulation is general and applicable to EMFs arising from waves due to either the magnetorotational instability (MRI) or the ‘kink’/Tayler instability. The concepts are illustrated in a one-dimensional model of a Keplerian disc.

In order to explore the effects of climate change on atmospheric convection and the water cycle, we develop and analyse an extension of the Rainy-Bénard model, which is itself a moist version of the Rayleigh-Bénard model of dry convection. Including moisture changes the character of the convection, with condensation providing a source of buoyancy via latent heating and the system exhibiting moist conditional instability. A range of idealised climate change scenarios are constructed by appropriate choice of both the radiative cooling rate and the surface temperature, and these scenarios are investigated over a wide range of surface relative humidity values. We impose moist-pseudoadiabatic conditions at the top boundary, which allow the temperature and specific humidity values to vary at the top boundary in response to convection. The model is analysed across the different climate change scenarios space by examining diagnostics of the model's basic state, and its stability, with Convective Available Potential Energy (CAPE) calculations and a linear stability analysis. We use the linear stability results to identify new parameters relevant for this moist convective system, and to understand how the linear instability responds to the climate parameters. In particular, we define the “Rainy number” as a scaled ratio of positive-area CAPE and diffusion parameters. An alternative radiative-based Rainy number also is shown to describe the parameter space, especially for problems relating to changes in flux conditions. The Rainy number acts like the traditional Rayleigh number for dry Rayleigh-Bénard convection, and provides a novel theoretical tool for understanding how the dynamics and scales of moist convection and hence precipitation will change under climate change. The linear analysis predicts an intensification of the hydrological cycle under climate change. The model set up and linear analysis provide a basis for future investigation into the non-linear dynamics of (idealised) moist convection.

Topographic effects on mixing in internal hydraulic jumps in stratified flow over complex geometries are investigated using idealised numerical simulations. The simulation geometry is motivated by flow conditions present in Hood Canal, Washington, with a channel expansion coincident with the sill. Simulation results show that mixing in the domain is typically increased by topographic variation due to vortices generated by an expansion or contraction coincident with the sill that forces the hydraulic jump. However, the associated velocity decrease in expansions counteracts this increase in mixing. Likewise, a contraction with an increased velocity causes more mixing due to the compounding effect of geometry change and velocity increase. Given the results described here, future numerical work investigating more realistic 3D topographies is recommended.

We study the nonlinear evolution of the magnetic buoyancy instability in rotating and non-rotating gas layers (with emphasis on the parameter range typical of spiral galaxies) using numerical solutions of non-ideal, isothermal MHD equations. The unstable magnetic field is either imposed through the boundary conditions or generated by an imposed α-effect. In the case of an imposed field, we solve for the deviations from the background state which satisfy periodic boundary conditions. We also include cosmic rays as a weightless fluid which exerts a dynamically significant pressure and somewhat amplifies magnetic buoyancy. This version of the instability is known as the Parker instability. Without rotation, systems with an imposed magnetic field evolve to a state with a very weak magnetic field, very different from the marginally stable eigenfunction, where the gas layer eventually becomes very thin as it is supported by the thermal and turbulent pressures alone. However, this does not happen when the magnetic field is maintained by the α-effect. Rotation fundamentally changes the development of the instability. A rotating system develops nonlinear oscillations, and the magnetic field direction changes even with an imposed magnetic field. We demonstrate that this is caused by the secondary α-effect at large altitudes as the gas flow produced by the instability becomes helical. The secondary α-effect has an anomalous sign with the α-coefficient being negative in the northern hemisphere, whereas the Coriolis force produces a positive α. The mean-field dynamo action outside the original gas layer can also lead to a switch in the magnetic field parity from quadrupolar (typical of the mean-field dynamo action in a thin layer) to dipolar. Altogether, the magnetic buoyancy instability and the mean-field dynamo action become separated as distinct physical effects in a nonlinear rotating system. We show that none of the assumptions used in analytic studies of the Parker instability is corroborated by numerical results.

In non-mirror-symmetric system, turbulent helicity 〈 u ′ ⋅ ω ′ 〉 enters the Reynolds-stress expression as the coupling coefficient of the mean absolute vorticity (anti-symmetric part of the mean velocity shear) ( u ′ : velocity fluctuation, ω ′ ( = ∇ × u ′ ) : vorticity fluctuation, 〈 ⋯ 〉 : ensemble average). This gives stark contrast to the turbulent energy, which plays an essential role in the eddy-viscosity representation of the Reynolds stress as the coupling coefficient of the mean velocity strain (symmetric part of the mean velocity shear). By considering the turbulent vortex-motive force 〈 u ′ × ω ′ 〉 in the mean vorticity equation, it is shown that a large-scale flow can be induced in the direction of the mean absolute vorticity (mean vorticity and system rotation) mediated by the inhomogeneous turbulent helicity. This inhomogeneous helicity effect is applied to the large-scale flow generation and sustainment in the stellar convective zone. The contribution of the inhomogeneous helicity effect to the angular-momentum transport in the stellar convection is discussed with the aid of some direct numerical simulations. Emphasis is also made on the turbulent helicity as a link between the large-scale flow structures, like differential azimuthal rotation and the meridional circulation, and the statistical properties of turbulence.

Two outstanding problems in the formulation of models for the evolution of dendritic or granular mushy zones in solidifying binary alloys are the self-selection of the micro-scale which controls the flow permeability, and the prescription of an appropriate boundary condition for the solid fraction at the mush liquid interface. We suggest that the microscale is selected to be that where diffusive removal of solute from the dendrite surfaces becomes rate-limiting, and the interface regains its stability. In freely growing mushes (but not necessarily in directional solidification), the microscale is then determined by the (large) value of the Lewis number. With the microscale selected in this way, we find that if the averaged model equations are derived by the method of homogenisation, the appropriate boundary condition for the solid fraction at the mush-liquid interface (that it equal zero) can be formally derived through consideration of solute conservation at the interface. A particular consequence of our study is the derivation of an explicit microscale model describing the evolution of the dendritic interface. In principle this model should be able to describe the formation of secondary dendrites, and we show that a uniform solution corresponding to a cellular-dendritic array is always unstable to the formation of secondary dendrites, although the linear response is ultimately damped. We use this assessment of the microscale to comment on possible pore spacings in dendritic solidification of the Earth's inner core.

We compare the efficiency and ease-of-use of the Sparse Identification of Nonlinear Dynamics (SINDy) algorithm and Sparse Physics-Informed Discovery of Empirical Relations (SPIDER) framework in recovering the relevant governing equations and boundary conditions from data generated by direct numerical simulations (DNS) of turbulent convective flows. In the former case, a weak-form implementation pySINDy is used. Time-dependent data for two- (2D) and three-dimensional (3D) DNS simulation of Rayleigh-Bénard convection and convective plane Couette flow is generated using the Dedalus PDE framework for spectrally solving differential equations. Using pySINDy we are able to recover the governing equations of 2D models of Rayleigh-Bénard convection at Rayleigh numbers, R, from laminar, through transitional to moderately turbulent flow conditions, albeit with increasing difficulty with larger Rayleigh number, especially in recovery of the diffusive terms (with coefficient magnitude proportional to 1 / R ). SPIDER requires a much smaller library of terms and we are able to recover more easily the governing equations for a wider range of R in 2D and 3D convection and plane flow models and go on to recover constraints (the incompressibility condition) and boundary conditions, demonstrating the benefits and capabilities of SPIDER to go beyond pySINDy for these fluid problems governed by second-order PDEs. At the highest values of R, discrepancies appear between the governing equations that are solved and those that are discovered by SPIDER. We find that this is likely associated with limited resolution of DNS, demonstrating the potential of machine-learning methods to validate numerical solvers and solutions for such flow problems. We also find that properties of the flow, specifically the correlation time and spatial scales, should inform the initial selection of spatiotemporal subdomain sizes for both pySINDy and SPIDER. Adopting this default position has the potential to reduce trial and error in selection of data parameters, saving considerable time and effort and allowing the end user of these or similar methods to focus on the importance of setting the power of the integrating polynomial in these weak-form methods and the tolerance of the optimiser technique selected.

The role of memory time fluctuations for instabilities in random media is considered. It is shown that fluctuations can result in infinitely fast growth of statistical moments. The effect is demonstrated in the framework of light propagation in the Universe, which contains curvature fluctuations while remaining homogeneous and isotropic on average.

In our earlier paper [Childress, S. and Gilbert, A.D., Eroding dipoles and vorticity growth for Euler flows in R 3 : the hairpin geometry as a model for finite-time blowup. Fluid Dyn. Res. 2018, 50, 011418.], we derived equations of motion for a vortex dipole in the shape of a hairpin, and subject to erosion of vorticity, that is the shedding of vorticity at the rear stagnation point of the dipole and consequent loss of circulation. We applied these to the question of Euler blow-up of vorticity in R 3 . In this paper, we shall calculate the axial flow within the vortex tubes of the hairpin and evaluate the resulting vorticity structure of their cores. The model is unusual in that it is not evolved from simple initial conditions. Rather, the hairpin structure is constructed at a time prior to possible blowup. It consists of a “nose”, where blow-up would occur, from which there extend two symmetric, quasi-two-dimensional “tails” of infinite length and infinitely large spatial scale. The quasi-self-similarity of the structure determines blow-up at the point of joining of the tails. During this growth, the dipole maintains a quasi-two-dimensional geometry. The analysis is believed to be the first study of blow-up incorporating both the deformation of the cores of the constituent vortex tubes and the axial flow within the tubes. The analysis raises problems which we will not be able to resolve fully here. Our results suggest that axial flow, coupled with erosion, may provide a mechanism preventing blow-up in finite time. The essential difficulty is that axial flow changes the local dipole structure and hence the dipole propagation speed. This expels the possibility of complete self-similarity. Possible ways to deal with this obstacle are discussed.

The mean-field induction equation ∂ t B ¯ − η Δ B ¯ = ∇ × F is considered in a conducting volume V , where B ¯ is the mean magnetic field, ∂ t is rate of change and η is magnetic diffusivity. Using the Green's function method and the second-order correlation approximation (SOCA), or the nearly axisymmetric methods of Braginskii [Self excitation of a magnetic field during the motion of a highly conducting fluid. Sov. Phys. JETP 1964, 20], then the electromotive force F is F = α ⋅ B ¯ . This work consists of two parts. Part I: The following antidynamo theorem (ADT) is derived: if there is no generation of azimuthal F from azimuthal B ¯ , that is 1 ϕ ⋅ α ⋅ 1 ϕ = α ϕϕ = 0 , where 1 ϕ is the unit vector in the ϕ direction, ( s , ϕ , z ) cylindrical polar coordinates, then an axisymmetric magnetic field ( B ¯ ( s , z ) ) will decay. Firstly, the magnetic field in meridional planes is shown to decay to zero. Then the azimuthal component of the magnetic field is shown to decay. As a weighted measure of the magnetic energy, ‖ b ‖ 2 = ∫ V b 2 d V , where b = B ¯ ( s , ϕ ) ⋅ 1 ϕ / s , is considered. The resulting ‖ b ‖ 2 magnetic energy analysis demonstrates that; for α = α ( s , z ) , and α ϕϕ = 0 , once the meridional field has decayed, diffusion decreases energy to more than account for the inductive contributions due to α , and, consistent with the α ϕϕ = 0 ADT, the field decays. Numerical results and field plots using the model α = s 1 z 1 ϕ , illustrate the interaction mechanisms responsible for the diffusive dominance as induction is increased. Using the SOCA and Green's function analysis, an explicit formulation for α ϕϕ is derived. Thus, physical mechanisms for the generation of α ϕϕ are established. Conditions are produced for which α ϕϕ = 0 , including a conductor filling all space with zero mean flow and co-axisymmetric perturbation flow and mean magnetic field. Part II: The Eulerian approach of Braginskii (1964), where the fields are analysed as perturbations from axisymmetry, is extended to compressible velocity fields for appropriate stellar and planetary dynamos. The hybrid Euler–Lagrange approach of Soward and Roberts [Eulerian and Lagrangian means in rotating, magnetohydrodynamic flows II. Braginsky's nearly axisymmetric dynamo. Geophys. Astrophys. Fluid Dyn. 2014, 108], following Soward [A kinematic theory of large magnetic Reynolds number dynamos. Phil. Trans. R. Soc. A 1972, 272], is continued to compressible flow and non-isochoric transformations in cylindrical polar coordinates, producing results that can be used for more general, higher-order approximations. These compressible extensions produce an α ϕϕ component and other effects for a reformulation of the problem into new effective mean, magnetic and velocity fields. Each of these approaches provides insight into mechanisms responsible for generating this critical α ϕϕ component. Common to all these approaches is the induction mechanism generated by the non-zero mean helicity of the meridional perturbation velocity field. Conclusions for non-magnetic stars are proposed and implications for hidden dynamos are drawn.