Plane Layer Of Fluid Research Articles

Correction: Tolmachev et al. Algorithmic Aspects of Simulation of Magnetic Field Generation by Thermal Convection in a Plane Layer of Fluid. Mathematics 2023, 11, 808

There was an error in the original publication [...]

Algorithmic Aspects of Simulation of Magnetic Field Generation by Thermal Convection in a Plane Layer of Fluid

We present an algorithm for numerical solution of the equations of magnetohydrodynamics describing the convective dynamo in a plane horizontal layer rotating about an arbitrary axis under geophysically sound boundary conditions. While in many respects we pursue the general approach that was followed by other authors, our main focus is on the accuracy of simulations, especially in the small scales. We employ the Galerkin method. We use products of linear combinations (each involving two to five terms) of Chebyshev polynomials in the vertical Cartesian space variable and Fourier harmonics in the horizontal variables for space discretisation of the toroidal and poloidal potentials of the flow (satisfying the no-slip conditions on the horizontal boundaries) and magnetic field (for which the boundary conditions mimick the presence of a dielectric over the fluid layer and an electrically conducting bottom boundary), and of the deviation of temperature from the steady-state linear profile. For the chosen coefficients in the linear combinations, the products satisfy the respective boundary conditions and constitute non-orthogonal bases in the weighted Lebesgue space. Determining coefficients in the expansion of a given function in such a basis (for instance, for computing the time derivatives of these coefficients) requires solving linear systems of equations for band matrices. Several algorithms for determining the coefficients, which are exploiting algebraically precise relations, have been developed, and their efficiency and accuracy have been numerically investigated for exponentially decaying solutions (encountered when simulating convective regimes which are spatially resolved sufficiently well). For the boundary conditions satisfied by the toroidal component of the flow, our algorithm outperforms the shuttle method, but the latter proves superior when solving the problem for the conditions characterising the poloidal component. While the conditions for the magnetic field on the horizontal boundaries are quite specific, our algorithms for the no-slip boundary conditions are general-purpose and can be applied for treating other boundary-value problems in which the zero value must be admitted on the boundary.

Onset of oscillatory magnetoconvection under rapid rotation and spatially varying magnetic field

The onset of periodic time-varying magnetoconvection in a regime relevant to the Earth's outer core is investigated in this study. A rapidly rotating plane fluid layer subject to an axially varying horizontal magnetic field is considered under the dynamical regimes of stronger magnetic diffusion compared to thermal and viscous diffusion rates. Dynamically specific convective instabilities, both inside and outside the tangent cylinder regions of the Earth's core, have been investigated by using appropriate patterns of the imposed mean magnetic field. The hallmark of convection onset, with such axially varying mean field, is a viscous oscillatory mode weakly modified by the magnetic field. This modified viscous oscillatory (mVO) mode is observed to exist over a wide range of the strength of the imposed field, making it a dynamically appropriate flow structure with characteristics of outer core convection. An optimal Prandtl number Pr* is found through numerical experiments where the mVO mode can be the most unstable to infinitesimal perturbations. It is further shown that this optimal state admits oscillations for earthlike regimes where thermal diffusion is much less than magnetic diffusion. Also, the formation of columnar convection rolls from isolated vortices is demonstrated as a result of combinations of the classical viscous oscillatory and mVO modes in the rapidly rotating limit (Ekman number E→0). Overall, the qualitative characteristics of magnetoconvection modes for the various imposed patterns are found to be similar despite representing distinct regions in the Earth's outer core.

Influence of boundary conditions on rapidly rotating convection and its dynamo action in a plane fluid layer

We investigate the influence of thermal, mechanical, and magnetic boundary conditions (BCs) on convective dynamos in a rapidly rotating plane fluid layer using direct numerical simulations. While the velocity BCs largely control whether large-scale flows and fields are generated, the magnetic BCs affect the magnetic field topology. The role of the thermal BCs is of note: For no-slip boundaries, the Nusselt number increases significantly when a fixed heat flux is imposed instead of a given temperature difference. We explain this effect, which applies to both dynamos and nonmagnetic, rotating convection, by an interplay of Ekman pumping and the internal structure of the thermal boundary layer.

TRIPLY RESONANT DOUBLE DIFFUSIVE CONVECTION IN A FLUID LAYER

We study the problem of double-diffusive convection in a horizontal plane fluid layer when there is a heat sink/source which is linear in the vertical coordinate which is in the opposite direction to gravity. The thresholds for linear instability are found and compared to those derived by a global nonlinear energy stability analysis. A region is discovered where a very sharp increase in Rayleigh number is observed. In addition to a linearized instability analysis, two global (unconditional) nonlinear stability thresholds are derived.

Stability of advective flow in an inclined plane fluid layer bounded by solid planes with a longitudinal temperature gradient 2. Stable stratification

Stability of advective flow in an inclined plane fluid layer bounded by solid planes with a longitudinal temperature gradient 2. Stable stratification

Stability of advective flow in an inclined plane fluid layer bounded by solid planes with a longitudinal temperature gradient. 1. Unstable stratification

The stability of steady convective flow in an inclined plane fluid layer bounded by ideally heat conducting solid planes is studied in the presence of a homogeneous longitudinal temperature gradient under unstable stratification conditions where the layer is inclined so that the temperature is higher in the lower part than in the upper part. It is shown that the inclination leads to the transition from critical perturbations to long-wavelength helical perturbations. Flow stability maps are given for the entire range of Prandtl numbers and inclination angles corresponding to unstable stratification.

Stability of Advective Flow in an Inclined Plane Fluid Layer Bounded by Solid Planes with a Longitudinal Temperature Gradient. 1. Unstable Stratification

Stability of Advective Flow in an Inclined Plane Fluid Layer Bounded by Solid Planes with a Longitudinal Temperature Gradient. 1. Unstable Stratification

Approximate two-dimensional equations of vortex flow in a plane layer with solid boundaries

Approximate two-dimensional equations governing turbulent vortex flows in plane fluid layers are considered. The equations were derived by the author in his earlier studies using the shallow water approximation and neglecting circulatory flows in the layer cross-sections. It is shown that, due to the centrifugal effect in the vortex flow, return flows in the layer cross-sections have only a slight influence on the fluid flow in the plane layer and can be neglected.

A precessionally-driven dynamo in a plane layer

A simple model of precessionally-forced flow has been proposed and studied by Mason and Kerswell (Mason, R.M. and Kerswell, R.R., Chaotic dynamics in a strained rotating flow: a precessing plane fluid layer. Geophys. Astrophys. Fluid Dynam., 491, 2002, 71–106). The fluid is viscous and incompressible and occupies a plane layer that spins about its normal that itself precesses about an axis fixed in space. The present article confirms the linear stability results of Mason and Kerswell and investigates in greater detail the complicated bifurcation that arises at the onset of two-dimensional instability. Three-dimensional modes are studied and it is shown that, if the fluid is sufficiently electrically conducting, the motions can, for sufficiently strong precessional forcing, maintain a magnetic field by dynamo action. The Lorentz force is found to reduce the vigor of the motions but to modify them in a way that assists magnetic field generation.

Chaotic dynamics in a strained rotating flow: a precessing plane fluid layer

The nonlinear dynamics exhibited by a planar layer of precessing fluid is examined as a canonical example of a strained rotating flow. The simple basic flow, Ubasic = −YXˆ+(X−2εZ)Yˆ in a frame rotating at εXˆ, consists of sheared circular streamlines (where ε measures the shearing) which are linearly unstable through the pairwise resonance of two inertial waves in a fashion similar to elliptical flow. Direct numerical simulation shows that the weakly nonlinear regime is quickly disrupted by further instabilities which lead to a multitude of co-existing solution branches, some of which represent chaotic flows. All these solutions remain within O(ε) (in an energy norm) of Ubasic so that energy is not apparently withdrawn from the fluid's underlying rotation. Further increases in the precession rate cause the flow to branch-switch randomly between these now quasi-stable states so that a new form of ‘slow’ dynamics emerges. The implication of this and the fact that these instabilities can nevertheless be classed as ‘strong’ is discussed from the perspective of the closely related problem of the precessing Earth and laboratory models thereof.

From Penetrative Convection to Teleconvection

In the deep interiors of rapidly rotating stars and planets the Coriolis force plays an essential role in determining the depth of penetrative convection from a convectively unstable region into a neighboring stably stratified region. The linear Coriolis force, not the nonlinear inertial force, dominates the dynamics. Accordingly, this paper investigates linear penetrative convection in a rapidly rotating, convectively unstable plane or spherical fluid layer bounded above by a less unstable or a stable corotating plane or spherical fluid layer. For the plane-layer geometry, exact solutions are obtained for two stress-free bounding walls. Asymptotic relations for the onset of penetrative convection in a rapidly rotating plane-layer system are obtained for an asymptotically small Ekman number. It is shown that the interface between the stable and unstable layer forms an effective wall that prevents convective flows from penetrating deeply into the stable fluid layer. This is the ordinary situation of penetrative convection. In spherical-layer geometry, the problem is fundamentally different. Several new and novel forms of convection are discovered. In one phenomenon that we term teleconvection, convection is thermally driven in the deep unstable interior, but the resulting convective motions concentrate primarily in the stable region near the outer spherical surface far away from the location of the thermal forcing. In another case we find a multilayer roll structure in rotating spherical two-layer convection as a result of a stable spherical outer layer. Our findings suggest that observed motions in the atmospheres of planets or stars could be driven by remote energy sources in their deep interiors.

Types of discrete symmetries of convection in a plane fluid layer

The Rayleigh–Benard convection in a plane fluidlayer is actively studied both theoretically and experi-mentally. Diverse types of structures observed undernatural and laboratory conditions [1, 2] should be clas-sified, and the conditions of their existence should bedetermined. In this paper, in order to determine permis-sible types of convection structures in a fluid layerheated uniformly from below, the method of calculat-ing discrete symmetries is first applied. It unites themethods of continuous-group theory, the formalism ofdifferential forms, and the method of imbedding inhigher dimension space [3, 4].We consider the steady motion of a single fluidwhose density depends only on temperature as ρ = ρ

On the validity of the geostrophic approximation for calculating the changes in the angular momentum of the core

The changes in the length of day at decade periods are attributable to exchanges of angular momentum between the core and the mantle. It is commonly assumed that the changes in angular momentum in the fluid are carried by geostrophic flows which are characterized by the rotation of rigid cylindrical shells aligned with the Earth's rotation axis. In this work, we explore the validity of this assumption using the simple geometry of a plane fluid layer between two solid boundaries. Coupling at the boundaries is done by electromagnetic interactions. We quantify the conditions under which the motion of the fluid that results from this coupling is accurately described by geostrophic flows. We find that for the region outside the so-called tangent cylinder, the geostrophic approximation is valid for all periods of oscillation longer than a few days. For the region inside the tangent cylinder, the geostrophic nature of the flow is broken at annual and shorter periods by the propagation of MC waves.

Thermovibrational convective instability of mechanical equilibrium of an inclined fluid layer

The convective instability of mechanical equilibrium of an inclined plane layer of fluid developing under the action of a static gravity field and high-frequency vibration is studied. Configurations corresponding to four directions of the equilibrium temperature gradient — vertical, longitudinal, horizontal, and transverse — are considered for an arbitrary orientation of the vibration axis. The stability limits and the characteristics of the critical perturbations are determined.

Linear Penetrative Spherical Rotating Convection

The onset of penetrative convection in a rotating, gravitationally unstable spherical fluid layer bounded above or below by a stable spherical corotating layer is investigated. Rapid rotation and spherical geometry produce new phenomena in the context of penetrative convection that are fundamentally different from what has been observed in the well-studied plane-layer model. In a slowly rotating or nonrotating spherical system, the character of penetration is qualitatively similar to that in plane fluid layers. In a rapidly rotating spherical system with a spherical layer of stable fluid bounded above by an unstable spherical layer, the stable fluid prevents penetration of convection across the interface between the stable and unstable spherical layers. In the reciprocal situation, however, when a spherical layer of stable fluid is bounded below by an unstable spherical layer, convective motions penetrate from the unstable layer all the way through the outermost stable layer with nearly the same amplitude as in the underlying unstable layer. The phenomena can be explained as a direct consequence of the combined effects of rapid rotation and spherical layer geometry.

The Rayleigh–Taylor instability of viscous fluid layers

The effects of viscosity and surface tension on the nonlinear evolution of Rayleigh–Taylor instability of plane fluid layers are investigated. Full two-dimensional incompressible Navier–Stokes equations and exact boundary equations are solved simultaneously for a precise prediction of this phenomenon. An accurate flux line segment model (FLAIR) for fluid surface advection is employed for the interface reconstruction. The instability is characterized by three stages of development, which are defined based on the competition of the bubble and spike growth. This competition is responsible for the development of different spike and bubble morphologies and is decided based on geometrical factors, mainly the amplitude and wavelength of the initial perturbation, and on the fluid properties, mainly viscosity and surface tension. It is addressed and explained why the spike sometimes grows faster than the bubble, and vice versa. The cutoff and the most unstable wave numbers are identified numerically based on the Weber number. The effect of Weber and Reynolds numbers on the growth rate of instability and the role of viscosity in dragging the development of instability are also investigated.

Convective instability in the presence of volume energy release

The problem of the convective instability of a plane fluid layer bounded by rigid walls with heating in a narrow layer running parallel to the walls inside the volume in question is solved. Instability criteria depending on the location of the heated layer and the Rayleigh numbers of the upper and lower layers are found. The results are compared with those for a plane layer with uniform energy release inside the volume.

Mechanical quasi-equilibrium and thermovibrational convective instability in an inclined fluid layer

The linear stability of mechanical quasi-equilibrium of a long inclined plane fluid layer, in the presence of a constant temperature gradient, subject to a static gravity field and high frequency vibration is investigated theoretically. The layer is oriented in an arbitrary respect to the vertical. The boundaries of the layer are assumed to be rigid and highly conducting. Each of two vectors-the temperature gradient and the axis of vibration-can have one of the four orientations: vertical ( v), longitudinal ( l), horizontal ( h), and transversal ( t). Thus a total of sixteen situations are studied. The consideration is based on the equations system describing mean (averaged) fields in the frame of an averaging method. The possibility and necessary conditions of mechanical quasi-equilibrium existance are studied. The spectral amplitude problem for small two-dimensional normal disturbances is formulated. In the case of long-wave instability, the spectral problem is solved asymptotically using the wave number as a small parameter for expansion. In the case of an arbitrary value of wave number, the spectral problem is solved numerically. The boundaries of stability and critical disturbance characteristics are determined for all the sixteen cases mentioned before.

Towards a poloidal magnetic field theorem

Abstract The paper presents evidence that a kinematic dynamo with a purely poloidal magnetic field does not exist. In particular, it is shown that the poloidal scalar S, if it satisfies a certain regularity condition, decays monotonically with respect to the norm maxz (max x, y S—min x, y S) in the case of a plane fluid layer and with respect to the norm maxr[r(max|r|=r S—min|r|=r S)] in the case of a spherical fluid volume. The result is valid for nonsteady, radially varying conductivity, possibly moving boundaries and a nonsteady, compressible flow field constrained in such a way that no toroidal magnetic field is generated. Similar decay results apply to the nonradial derivatives of S. Since the above norm is equivalent to the uniform max-norm, we have also (not necessarily monotonic) decay of S in the max-norm. For steady or time-periodic flow fields these results imply exponential decay of S and its derivatives to zero. The steady case, finally, is ruled out by application of standard maximum princip...