David W. Hughes

Staircase formation and layering is studied in simplified, potential vorticity (PV) conserving models of plasmas and geophysical fluids, by investigating turbulent self-organization and nonlinear saturation with different mechanisms of free energy production—forcing or linear instability—and with standard or modified zonal flow (ZF) responses. To this end, staircase formation in both the standard and modified Charney–Hasegawa–Mima (CHM) equations with stochastic forcing, along with two different simple instability-driven models—one from a plasma and one from a geophysical context—are studied and compared. In these studies, it is observed that β-plane turbulence that does not distinguish between zonal and non-zonal perturbations (i.e. standard zonal response) gradually forms large-scale, elliptic zonal structures that merge progressively, regardless of whether it is driven by forcing (though it should be weak and slow enough to allow wave couplings) or by baroclinic instability, using, for example, a two-layer model. Conversely, the plasma system, with its modified zonal response, can rapidly form straight, stationary jets of well-defined size, again regardless of the way it is driven: by stochastic forcing or by the dissipative drift-wave instability. Furthermore, the instability-driven plasma system exhibits a subcritical zonal bifurcation, which resembles a phase transition, between a ZF dominated state and an eddy dominated state. In both states, saturation is possible without large-scale friction. This article is part of the theme issue ‘Anti-diffusive transport and layering: from sub-cellular to astrophysical scales’.

The formation of layered profiles (or ‘staircases’) in temperature, salinity and density, resulting from double-diffusive convection (DDC), is a widely observed oceanographic phenomenon. In the past two decades, new developments, both in numerical simulations and theory, have significantly advanced our under-standing of this fascinating process. Nonetheless, obtaining a complete description of layering—precisely when and why it occurs—still presents a sizeable theoretical challenge to fluid dynamicists. Here, we review the current state of the field, summarizing briefly the observations and simulations, before concentrating on simplified theoretical modelling approaches. Layering is described as an essentially one-dimensional ‘anti-diffusive’ process acting on a quasi-homogeneous background state (which may result from a classical double-diffusive instability). A key ingredient of the models discussed is the dependence of the mixing length on an emergent, or dynamical, scale. An instability of the basic turbulent state leads initially to regular sinusoidal perturbations of the buoyancy gradient, which, over time, can sharpen, leading to the formation of a staircase comprised of well-mixed layers separated by sharp interfaces. The long-time evolution is marked by a gradual coarsening of the staircase through layer mergers. The reduced models perform well in replicating the qualitative behaviour exhibited by direct numerical simulations. This article is part of the theme issue ‘Anti-diffusive transport and layering: from sub-cellular to astrophysical scales’.

Anti-diffusion refers to a remarkable class of processes, in which particles, fluid elements, etc., tend to migrate up-gradient, contrary to expected diffusive relaxation. A particularly dramatic and fascinating anti-diffusive phenomenon is layering or staircase formation, in which turbulent stirring leads to the formation of mixed regions, interspersed by layers of steepened gradients. Anti-diffusive dynamics is of interest across a wide range of physical systems—e.g. in geophysical and astrophysical fluid dynamics, in magnetically confined plasmas and in active matter. To bring together researchers from a variety of different backgrounds, the Isaac Newton Institute hosted a six-month programme in 2024 entitled Anti-Diffusive Dynamics: from Sub-Cellular to Astrophysical Scales. This special issue is a product of that programme, containing papers aimed at a broad readership covering observations, experiments, and theoretical and computational developments in a range of topics in fluid dynamics, plasma physics and active matter. This article is part of the theme issue ‘Anti-diffusive transport and layering: from sub-cellular to astrophysical scales’.

Double-diffusive convection, in which the density of a fluid is dependent on two fields that diffuse at different rates (such as temperature and salinity), has been widely studied in areas as diverse as the oceans and stellar atmospheres. Under the assumption of classical Fickian diffusion for both heat and salt, the evolution of temperature and salinity is governed by parabolic advection–diffusion equations. In reality, there are small additional terms in these equations that render them hyperbolic (the Maxwell–Cattaneo (M–C) effect). Although these corrections are nominally small, they represent a singular perturbation, and hence can lead to significant effects when the underlying differences of salinity and temperature are large. In an earlier paper (Hughes, Proctor & Eltayeb, J. Fluid Mech. , vol. 927, 2021, p. A13), we investigated the linear stability of a double-diffusive fluid layer subject to the M–C effect in either the temperature or the salinity equation (but not both). Here we consider the general, and much more complicated, case in which the M–C effect influences both temperature and salinity. We find that, as in the earlier paper, oscillatory instability is indeed facilitated (and in fact made possible when the salinity gradient is destabilising, where the classical problem has no oscillatory instability) when the salinity gradients are sufficiently large. The scalings that emerge from the earlier paper, however, are not necessarily representative of those in the general case, thus justifying the present study. In addition, we have found a remarkable singular situation when the ratio of the M–C effects is equal to the ratio between the heat and salinity diffusivities, near which the critical wavenumber is sharply reduced. In addition to determining the stability boundaries we have also investigated the growth rates of unstable modes and shown that these are on a par with those of classical double-diffusive convection.

The shallow-water equations are widely used to model interactions between horizontal shear flows and (rotating) gravity waves in thin planetary atmospheres. Their extension to allow for interactions with magnetic fields – the equations of shallow-water magnetohydrodynamics (SWMHD) – is often used to model waves and instabilities in thin stratified layers in stellar and planetary atmospheres, in the perfectly conducting limit. Here we consider how magnetic diffusion should be added to the equations of SWMHD. This is crucial for an accurate balance between advection and diffusion in the induction equation, and hence for modelling instabilities and turbulence. For the straightforward choice of Laplacian diffusion, we explain how fundamental mathematical and physical inconsistencies arise in the equations of SWMHD, and show that unphysical dynamo action can result. We then derive a physically consistent magnetic diffusion term by performing an asymptotic analysis of the three-dimensional equations of magnetohydrodynamics in the thin-layer limit, giving the resulting diffusion term explicitly in both planar and spherical coordinates. We show how this magnetic diffusion term, which allows for a horizontally varying diffusivity, is consistent with the standard shallow-water solenoidal constraint, and leads to negative semidefinite Ohmic dissipation. We also establish a basic type of antidynamo theorem.

Understanding the generation of large-scale magnetic fields and flows in magnetohydro-dynamical (MHD) turbulence remains one of the most challenging problems in astrophysical fluid dynamics. Although much work has been done on the kinematic generation of large-scale magnetic fields by turbulence, relatively little attention has been paid to the much more difficult problem in which fields and flows interact on an equal footing. The aim is to find conditions for long-wavelength instabilities of stationary MHD states. Here, we first revisit the formal exposition of the long-wavelength linear instability theory, showing how long-wavelength perturbations are governed by four mean field tensors; we then show how these tensors may be calculated explicitly under the ‘short-sudden’ approximation for the turbulence. For MHD states with relatively little disorder, the linear theory works well: average quantities can be readily calculated, and stability to long-wavelength perturbations determined. However, for disordered basic states, linear perturbations can grow without bound and the purely linear theory, as formulated, cannot be applied. We then address the question of whether there is a linear response for sufficiently weak mean fields and flows in a dynamical (nonlinear) evolution, where perturbations are guaranteed to be bounded. As a preliminary study, we first address the nature of the response in a series of one-dimensional maps. For the full MHD problem, we show that in certain circumstances, there is a clear linear response; however, in others, mean quantities – and hence the nature of the response – can be difficult to calculate.

Instabilities driven by some combination of rotation, velocity shear, and magnetic field in a stratified fluid under gravity play an important role in many astrophysical settings. Of particular note are the centrifugal instability, the magnetorotational instability, and magnetic buoyancy instability. Here, we consider a Cartesian model of an equatorial region incorporating all the physical ingredients necessary to study their competition. We investigate the linear instability to interchange (“axisymmetric”) modes of an inviscid, perfectly conducting, isothermal gas, including the effects of rotation, velocity shear, and poloidal and toroidal magnetic fields. The stability problem can be reduced to a second-order boundary value problem, with the growth rate as the eigenvalue. We can make analytic progress through consideration of the physically relevant regime in which the transverse horizontal wavenumber k ≫ 1. Via a perturbation analysis, with 1/k as the small parameter, we can derive the growth rate and the spatial dependence of the eigenfunctions: the unstable modes are strongly localized in the vertical direction, being either wall modes (localized near a boundary of the domain) or body modes (localized in the interior). We describe the conditions under which the joint action of the separate instability mechanisms leads to enhancement or suppression of the instability. Our analytical results are supplemented by numerical solutions of the stability problem. The most unstable mode found analytically is typically in excellent agreement with that found numerically through consideration of a wide range of wavenumbers. Finally, we discuss how our results relate to the solar tachocline.

Understanding the dynamics of staircases in salt fingering convection presents a long-standing theoretical challenge to fluid dynamicists. Although there has been significant progress, particularly through numerical simulations, there are a number of conflicting theoretical explanations as to the driving mechanism underlying staircase formation. The Phillips effect proposes that layering in stirred stratified flow is due to an antidiffusive process, and it has been suggested that this mechanism may also be responsible for salt fingering staircases. However, the details of this process, as well as mathematical models to predict the evolution and merger dynamics of staircases, have yet to be established. We generalise the theory of the Phillips effect to a three-component system (e.g. temperature, salinity, energy) and demonstrate a regularised nonlinear model of layering based on mixing length parametrisations. The model predicts both the inception of layering and its long-term evolution through mergers, while generalising, and remaining consistent with, previous results for double-diffusive layering based on flux ratios. Our model of salt fingering is formulated using spatial averaging processes, and closed by a mixing length parametrised in terms of the kinetic energy and the ratio of the temperature and salt gradients. The model predicts a layering instability for a bounded range of parameter values in the salt fingering regime. Nonlinear solutions show that an initially unstable linear buoyancy gradient develops into layers, which proceed to merge through a process of stronger interfaces growing at the expense of weaker ones. Our results indicate that these mergers are responsible for the characteristic increase of buoyancy flux through thermohaline staircases.