The effect of stratification and compressibility on anelastic convection in a rotating plane layer

We study the effect of stratification and compressibility on the threshold of convection and the heat transfer by developed convection in the nonlinear regime in the presence of strong background rotation. We consider fluids both with constant thermal conductivity and constant thermal diffusivity. The fluid is confined between two horizontal planes with both boundaries being impermeable and stress-free. An asymptotic analysis is performed in the limits of weak compressibility of the medium and rapid rotation (τ^?1/12???|θ|???1, where τ is the Taylor number and θ is the dimensionless temperature jump across the fluid layer). We find that the properties of compressible convection differ significantly in the two cases considered. Analytically, the correction to the characteristic Rayleigh number resulting from small compressibility of the medium is positive in the case of constant thermal conductivity of the fluid and negative for constant thermal diffusivity. These results are compared with numerical solutions for arbitrary stratification. Furthermore, by generalizing the nonlinear theory of Julien and Knobloch [Fully nonlinear three-dimensional convection in a rapidly rotating layer. Phys. Fluids 1999, 11, 1469–1483] to include the effects of compressibility, we study the Nusselt number in both cases. In the weakly nonlinear regime we report an increase of efficiency of the heat transfer with the compressibility for fluids with constant thermal diffusivity, whereas if the conductivity is constant, the heat transfer by a compressible medium is more efficient than in the Boussinesq case only if the specific heat ratio γ is larger than two.     

Validity of sound-proof approaches in rapidly-rotating compressible convection: marginal stability versus turbulence

The validity of the anelastic approximation has recently been questioned in the regime of rapidly-rotating compressible convection in low Prandtl number fluids (Calkins, Julien and Marti, Proc. R. Soc. A, 2015, vol. 471, 20140689). Given the broad usage and the high computational efficiency of sound-proof approaches in this astrophysically relevant regime, this paper clarifies the conditions for a safe application. The potential of the alternative pseudo-incompressible approximation is investigated, which in contrast to the anelastic approximation is shown to never break down for predicting the point of marginal stability. Its accuracy, however, decreases close to the parameters corresponding to the failure of the anelastic approach, which is shown to occur when the sound-crossing time of the domain exceeds a rotation time scale, i.e. for rotational Mach numbers greater than one. Concerning the supercritical case, which is naturally characterised by smaller rotational Mach numbers, we find that the anelastic approximation does not show unphysical behaviour. Growth rates computed with the linearised anelastic equations converge toward the corresponding fully compressible values as the Rayleigh number increases. Likewise, our fully nonlinear turbulent simulations, produced with our fully compressible and anelastic models and carried out in a highly supercritical, rotating, compressible, low Prandtl number regime show good agreement. However, this nonlinear test example is for only a moderately low convective Rossby number of 0.14.     

Large eddy simulations of two-dimensional turbulent convection in a density-stratified fluid

Large eddy simulations (LES) of two-dimensional turbulent convection within the anelastic approximation are presented for Rayleigh number Ra?=?10⁹, Prandtl number Pr?=?1 with free-slip boundary conditions. Various subgrid-scale (SGS) models are investigated such as a similarity model, a dynamic similarity model, a dynamic eddy-viscosity model, a hyperdiffusion model and a hybrid model (dynamic similarity hyperdiffusion model). To study the effects of density stratification on the models, we have also carried out simulations for a Boussinesq flow. The SGS models are compared to direct numerical simulation (DNS) data on the basis of kinetic energy and entropy variance spectra, mean entropy profiles, r.m.s. entropy profiles and r.m.s. kinetic energy density profiles. The results show that for the Boussinesq flow, all the SGS models agree fairly well with the high resolution DNS data. However, for the strongly density-stratified flow, only the hyperdiffusion and the hybrid model show good performance.     

The Boussinesq and anelastic liquid approximations for convection in the Earth’s core

Convection in the Earth’s core is usually studied in the Boussinesq approximation in which the compressibility of the liquid is ignored. The density of the Earth’s core varies from ICB to CMB by approximately 20%. The question of whether we need to take this variation into account in core convection and dynamo models is examined. We show that it is in the thermodynamic equations that differences between compressible and Boussinesq models become most apparent. The heat flux conducted down the adiabat is much smaller near the inner core boundary than it is near the core-mantle boundary. In consequence, the heat flux carried by convection is much larger nearer the inner core boundary than it is near the core-mantle boundary. This effect will have an important influence on dynamo models. Boussinesq models also assume implicitly that the rate of working of the gravitational and buoyancy forces, as well as the Ohmic and viscous dissipation, are small compared to the heat flux through the core. These terms are not negligible in the Earth’s core heat budget, and neglecting them makes it difficult to get a thermodynamically consistent picture of core convection. We show that the usual anelastic equations simplify considerably if the anelastic liquid approximation, valid if αT?1, where α is the coefficient of expansion and T a typical core temperature, is used. The resulting set of equations are not significantly more difficult to solve numerically than the usual Boussinesq equations. The relationship of our anelastic liquid equations to the Boussinesq equations is also examined.     

A nonlinear dynamo driven by rapidly rotating convection

In Kim et al. (Kim, E., Hughes, D.W. and Soward, A.M., “An investigation into high conductivity dynamo action driven by rotating convection”, Geophys. Astrophys. Fluid Dynam. 91, 303–332 ().) we investigated kinematic dynamo action driven by rapidly rotating convection in a cylindrical annulus. Here we extend this work to consider self-consistent nonlinear dynamo action in which the back-reaction of the Lorentz force on the flow is taken into account. In particular, we investigate, as a function of magnetic Prandtl number, the evolution of an initially weak magnetic field in two different types of convective flow – one chaotic and the other integrable. On saturation, the latter shows a systematic dependence on the magnetic Prandtl number whereas the former appears not to. In addition, we show how, in keeping with the findings of Cattaneo et al. (Cattaneo, F., Hughes, D.W. and Kim, E., “Suppression of chaos in a simplified nonlinear dynamo model”, Phys. Rev. Lett. 76, 2057–2060 ().), saturation of the growth of the magnetic field is brought about, for the originally chaotic flow, by a strong suppression of chaos.     

Large-scale convective dynamos in a stratified rotating plane layer

We study the effect of stratification on large-scale dynamo action in convecting fluids in the presence of background rotation. The fluid is confined between two horizontal planes and both boundaries are impermeable, stress-free and perfectly conducting. An asymptotic analysis is performed in the limit of rapid rotation (τ???1 where τ is the Taylor number). We analyse asymptotic magnetic dynamo solutions in rapidly rotating systems generalising the results of Soward [A convection-driven dynamo I. The weak field case. Philos. Trans. R. Soc. Lond. A 1974, 275, 611–651] to include the effects of compressibility. We find that in general the presence of stratification delays the efficiency of large-scale dynamo action in this regime, leading to a reduction of the onset of dynamo action and in the nonlinear regime a diminution of the large-scale magnetic energy for flows with the same kinetic energy.     

Numerical simulations of mantle convection: Time-dependent,three-dimensional,compressible, spherical shell

Abstract

We describe nonlinear time-dependent numerical simulations of whole mantle convection for a Newtonian, infinite Prandtl number, anelastic fluid in a three-dimensional spherical shell for conditions that approximate the Earth's mantle. Each dependent variable is expanded in a series of 4,096 spherical harmonics to resolve its horizontal structure and in 61 Chebyshev polynomials to resolve its radial structure. A semiimplicit time-integration scheme is used with a spectral transform method. In grid space there are 61 unequally-spaced Chebyshev radial levels, 96 Legendre colatitudinal levels, and 192 Fourier longitudinal levels. For this preliminary study we consider four scenarios, all having the same radially-dependent reference state and no internal heating. They differ by their radially-dependent linear viscous and thermal diffusivities and by the specified temperatures on their isothermal, impermeable, stress-free boundaries. We have found that the structure of convection changes dramatically as the Rayleigh number increases from 10⁵ to 10⁶ to 10⁷. The differences also depend on how the Rayleigh number is increased. That is, increasing the superadiabatic temperature drop, δT, across the mantle produces a greater effect than decreasing the diffusivities. The simulation with a Rayleigh number of 10⁷ is approximately 10,000 times critical, close to estimates of that for the Earth's mantle. However, although the velocity structure for this highest Rayleigh number scenario may be adequately resolved, its thermodynamic structure requires greater horizontal resolution. The velocity and thermodynamic structures of the scenarios at Rayleigh numbers of 10⁵ and 10⁶ appear to be adequately resolved. The 10⁵ Rayleigh number solution has a small number of broad regions of warm upflow embedded in a network of narrow cold downflow regions; whereas, the higher Rayleigh number solutions (with large δT) have a large number of small hot upflow plumes embedded in a broad weak background of downflow. In addition, as would be expected, these higher Rayleigh number solutions have thinner thermal boundary layers and larger convective velocities, temperatures perturbations, and heat fluxes. These differences emphasize the importance of developing even more realistic models at realistic Rayleigh numbers if one wishes to investigate by numerical simulation the type of convection that occurs in the Earth's mantle.     

The structure of separated flow past a circular cylinder in a rotating frame

Abstract

This paper examines the detailed E ^1/4-layer structure of separated flow past a circular cylinder in a low-Rossby-number rotating fluid as the Ekman number E tends to zero. This structure is based on an initial proposal by Page (1987) but with some modifications in response to further evidence, outlined both in this paper and elsewhere, on the behaviour of E ^1/4-layer flows in this context. Numerical calculations for flow in an E ^1/4 shear layer along the separated free streamline are described and the mass flux from this layer is then used to calculate the higher-order flow within the separation bubble. The flow structure is found to have two forms, depending on the value of the O(1) parameter λ, and these are compared with results from published “Navier-Stokes” type calculations for the flow at small but finite values of E.     

Large wavenumber convection in the rotating annulus

Abstract

The annulus model considers convection between concentric cylinders with sloping endwalls. It is used as a simplified model of convection in a rapidly rotating sphere. Large azimuthal wavenumbers are preferred in this problem, and this has been exploited to develop an asymptotic approach to nonlinear convection in the annulus. The problem is further reduced because the Taylor-Proudman constraint simplifies the dependence in the direction of the rotation vector, so that a nonlinear system dependent only on the radial variable and time results. As Rayleigh number is increased a sequence of bifurcations is found, from steady solutions to periodic solutions and 2-tori, typically ending in chaotic behaviour. Both the magnetic (MHD convection) and non-magnetic problem has been considered, and in the non-magnetic case our bifurcation sequence can be compared with those found by previous two-dimensional numerical simulations.     

Baroclinic nonlinear exchanges of energy and potential enstrophy in the quasi-geostrophic two-layer model

Abstract

Merilees and Warn's (1975) nonlinear interaction analysis of two-dimensional nondivergent flow is extended to examine the quasi-geostrophic two-layer model. Two sets of triads exist in this model (Salmon, 1978). The purely barotropic triads are the same as the triads examined by Merilees and Warn. Baroclinic-barotropic triads are found to exchange more energy or potential enstrophy with smaller or larger scales depending on the scale of motion as compared with the internal Rossby deformation radius and the relative wavenumber position of baroclinic and barotropic components.     

Convection in rotating annular channels heated from below. Part 1. Linear stability and weakly nonlinear mean flows

Convection in a Boussinesq fluid in an annular channel rotating about a vertical axis with lateral rigid sidewalls, stress-free top and bottom, uniformly heated from below is investigated. The sidewalls are assumed to be either perfectly insulating or conducting. Three different types of convection are identified when the channel is rotating sufficiently fast: (i) global oscillatory convection preferred for small Prandtl numbers in channels with intermediate or large aspect ratios (width to height ratio), (ii) wall-localized oscillatory convection representing the most unstable mode for moderate or large Prandtl numbers in channels with intermediate or large aspect ratios and (iii) global stationary convection preferred in channels with sufficiently small aspect ratios regardless of the size of the Prandtl number. The corresponding weakly nonlinear problem describing differential rotation and meridional circulation is also examined, showing that geostrophic, multiple-peaked (two prograde and two retrograde) differential rotation can be maintained by the Reynolds stresses in wall-localized convective eddies in a rapidly rotating channel.     

Two-and three-dimensional linear convection in a rotating annulus

Abstract

An exceptional case to the model-independent theory of Knobloch (1995) is presented, by investigating a rotating cylindrical annulus of height H and side wall radii r _o and r _i, with non-slip, perfectly thermally conducting side walls and thermally insulating stress-free ends. Radial heating permits the possibility of either two- or three-dimensional convective solutions being the preferred mode. An analytical solution is obtained for the two-dimensional case and a numerical solution for the three-dimensional solution, which is also applied to the two-dimensional solution. It is shown that both two- and three-dimensional solutions can be realized depending on the aspect ratio, γ = H/d, where d = r _o-r _i is the thickness of the annulus, the radii ratio λ = r _i/r _o and the rotation rate of the model. For γ = O(1) and λ = 0.4, the preferred convective solution is three-dimensional when the Taylor number, T < 10² and two-dimensional for T > 10². For small aspect ratios, γ ? 1, the preferred mode is two-dimensional for all rotation rates.     

On the onset of convection in rotating spherical shells

Abstract

The linear problem of the onset of convection in rotating spherical shells is analysed numerically in dependence on the Prandtl number. The radius ratio η=r _i/r _o of the inner and outer radii is generally assumed to be 0.4. But other values of η are also considered. The goal of the analysis has been the clarification of the transition between modes drifting in the retrograde azimuthal direction in the low Taylor number regime and modes traveling in the prograde direction at high Taylor numbers. It is shown that for a given value m of the azimuthal wavenumber a single mode describes the onset of convection of fluids of moderate or high Prandtl number. At low Prandtl numbers, however, three different modes for a given m may describe the onset of convection in dependence on the Taylor number. The characteristic properties of the modes are described and the singularities leading to the separation with decreasing Prandtl number are elucidated. Related results for the problem of finite amplitude convection are also reported.     

Effects of boundary conditions on the onset of convection with tilted magnetic field and rotation vectors

The problem of the onset of thermal convection is considered, firstly when a uniform tilted magnetic field is present, and secondly in a frame rotating about an oblique axis. If up–down symmetry is broken we expect to find only bifurcations that lead to travelling waves. Numerical studies show, however, that in a Boussinesq fluid the spectrum of eigenvalues can be symmetrical about the real axis, even when the boundary conditions are asymmetrical. Here we show analytically that this symmetry property indeed holds for a wide range of boundary conditions and hence that both steady solutions and standing waves are allowed.     

Nonlinear internal waves in ideal rotating basins

Abstract

Starting from Euler's equations of motion a nonlinear model for internal waves in fluids is developed by an appropriate scaling and a vertical integration over two layers of different but constant density. The model allows the barotropic and the first baroclinic mode to be calculated. In addition to the nonlinear advective terms dispersion and Coriolis force due to the Earth's rotation are taken into account. The model equations are solved numerically by an implicit finite difference scheme. In this paper we discuss the results for ideal basins: the effects of nonlinear terms, dispersion and Coriolis force, the mechanism of wind forcing, the evolution of Kelvin waves and the corresponding transport of particles and, finally, wave propagation over variable topography. First applications to Lake Constance are shown, but a detailed analysis is deferred to a second paper [Bauer et al. (1994)].     

Thermal convection in differentially rotating systems

Abstract

The onset of convection in a cylindrical fluid annulus is analyzed in the case when the cylindrical walls are rotating differentially, a temperature gradient in the radial direction is applied, and the centrifugal force dominates over gravity. The small gap approximation is used and no-slip conditions on the cylindrical walls are assumed. It is found that over a considerable range of the parameter space either convection rolls aligned with the axis of rotation or rolls in the perpendicular (azimuthal) direction are preferred. It is shown that by a suitable redefinition of parameters, results for finite amplitude Taylor vortices and for convection rolls in the presence of shear can be applied to the present problem. Weakly nonlinear results for transverse rolls in a Couette flow indicate the possibility of subcritical bifurcation for Prandtl numbers P less than 0.82. Heat and momentum transports are derived as functions of P and the problem of interaction between transverse and longitudinal rolls is considered. The relevance of the analysis for problems of convection in planetary and stellar atmospheres is briefly discussed.     

旋转流体中的二维惯性重力波-涡相互作用

本文利用f-平面浅水模型讨论了二维情况下快速惯性重力波与平衡涡旋之间的相互作用问题.当快速的惯性重力波传入涡旋区域后,波－涡之间发生了相互作用,涡旋可被加深,涡旋区域的风场等可以出现非对称分布,当惯性重力波全部传出涡旋区域并远离之后,涡旋恢复到其初始状态,但此时惯性重力波可以出现非对称结构,这与一维情况不同.在波－涡发生相互作用过程中,其物理量变化的最大区域分布基本与波动传播方向一致,在波动进入或移出涡旋时,相对应存在两个最大物理量变化区域,此外,相互作用导致的物理量的最大变化与波动结构、强度、传播方向以及涡旋特性有关.非线性条件下的波－涡相互作用要强于线性过程.     

Penetrative convection in rotating fluids: A model for the base of stellar convection zones

Abstract

To model penetrative convection at the base of a stellar convection zone we consider two plane parallel, co-rotating Boussinesq layers coupled at their fluid interface. The system is such that the upper layer is unstable to convection while the lower is stable. Following the method of Kondo and Unno (1982, 1983) we calculate critical Rayleigh numbers R_c for a wide class of parameters. Here, R_c is typically much less than in the case of a single layer, although the scaling R_c～T^2/3 as T → ∞ still holds, where T is the usual Taylor number. With parameters relevant to the Sun the helicity profile is discontinuous at the interface, and dominated by a large peak in a thin boundary layer beneath the convecting region. In reality the distribution is continuous, but the sharp transition associated with a rapid decline in the effective viscosity in the overshoot region is approximated by a discontinuity here. This source of helicity and its relation to an alpha effect in a mean-field dynamo is especially relevant since it is a generally held view that the overshoot region is the location of magnetic field generation in the Sun.     

Soret and Dufour effects on thermohaline convection in rotating fluids

Using linear and weakly nonlinear stability theory, the effects of Soret and Dufour parameters are investigated on thermohaline convection in a horizontal layer of rotating fluid, specifically the ocean. Thermohaline circulation is important in mixing processes and contributes to heat and mass transports and hence the earth’s climate. A general conception is that due to the smallness of the Soret and Dufour parameters their effect is negligible. However, it is shown here that the Soret parameter, salinity and rotation stabilise the system, whereas temperature destabilises it and the Dufour parameter has minimal effect on stationary convection. For oscillatory convection, the analysis is difficult as it shows that the Rayleigh number depends on six parameters, the Soret and Dufour parameters, the salinity Rayleigh number, the Lewis number, the Prandtl number, and the Taylor number. We demonstrate the interplay between these parameters and their effects on oscillatory convection in a graphical manner. Furthermore, we find that the Soret parameter enhances oscillatory convection whereas the Dufour parameter, salinity Rayleigh number, the Lewis number, and rotation delay instability. We believe that these results have not been elucidated in this way before for large-scale fluids. Furthermore, we investigate weakly nonlinear stability and the effect of cross diffusive terms on heat and mass transports. We show the existence of new solution bifurcations not previously identified in literature.