Onset of convection in a rapidly rotating compressible fluid spherical shell

Drew, Steve; Jones, C. A.; Zhang, K.

doi:10.1080/03091929508228957

Cited by 14 publications

(33 citation statements)

References 10 publications

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“…Gilman and Glatzmaier (1981) and Glatzmaier and Gilman (1981a) used the anelastic equations to investigate the onset of convection and nonlinear convection in a rotating spherical shell geometry. Drew et al (1995) extended the Glatzmaier and Gilman (1981a) linear stability analysis to reach larger Taylor numbers and a broader range of Prandtl numbers; surprisingly, they observed negative critical Rayleigh numbers for sufficiently small Prandtl numbers in both the spherical shell and plane layer geometry. Whereas many studies have performed one-to-one comparisons of the anelastic and compressible equations for investigating stably stratified gravity wave dynamics (e.g., Davies et al 2003, Klein et al 2010, Brown et al 2012), there appears to be few comparisons in the literature between the two equation sets for the unstably stratified case of thermal convection.…”

Section: Introductionmentioning

confidence: 93%

Onset of rotating and non-rotating convection in compressible and anelastic ideal gases

Calkins

Julien

Marti

2014

Geophysical & Astrophysical Fluid Dynamics

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A linear stability analysis for compressible convection in a plane layer geometry both with and without the influence of rotation is presented. For the rotating cases we employ the tilted f -plane geometry that allows for varying angles between the rotation and gravity vectors. The stability criteria for compressible and anelastic ideal gases is compared. As expected, the critical parameters for the compressible equations approach those of the anelastic equations as the background stratification approaches the adiabatic (anelastic) limit. For the rotating cases, we observe asymptotic scaling behavior in the critical parameters in both compressible and anelastic fluids as the Taylor number becomes large. In contrast to the incompressible limit, finite tilt angles between the gravity and rotation vectors result in propagating compressible Rossby waves as the most unstable eigenmode and the critical parameters are established for a range of stratification levels and Taylor numbers; all wave orientations are found to propagate in prograde and equatorward directions for non-isothermal background states. We also compare the linear stability of the thermodynamically rigorous anelastic equations with an anelastic model that replaces thermal diffusion with an entropy diffusion-like term in the energy equation; it is shown that the linear stability of the entropy diffusion model yields qualitatively similar results for the critical parameters in comparison to the full anelastic set. We show that a thermodynamically rigorous alternative to the entropy diffusion model is the isothermal adiabatic background state in which temperature and entropy become equivalent thermodynamic quantities and viscous heating becomes subdominant in the energy equation; the stability characteristics of this model are also presented.

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Section: Introductionmentioning

confidence: 93%

Onset of rotating and non-rotating convection in compressible and anelastic ideal gases

Calkins

Julien

Marti

2014

Geophysical & Astrophysical Fluid Dynamics

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“…One of their important discoveries was that the depth-dependence of the thermal conductivity strongly influences the localization of the region where the convective instability first sets in and in the case when thermal diffusivity is assumed constant, anelastic convection develops small structures close to the outer boundary. The onset of compressible convection in spherical geometry with constant kinematic or dynamic viscosity was also investigated by Drew et al (1995) and an asymptotic linear theory under the anelastic approximation was developed for rapidly rotating spherical shells by . A very detailed derivation and comprehensive discussion of the compressible convection equations in the geophysical context was presented by Braginsky and Roberts (1995).…”

Section: Introductionmentioning

confidence: 99%

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

Mizerski

Tobias

2011

Geophysical & Astrophysical Fluid Dynamics

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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 ( À1/12 ( jj ( 1, where is the Taylor number and 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 1999Fluids , 11, 1469Fluids -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 is larger than two.

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“…We thus find that the range of stratification for which the AE can approximate the NSE becomes increasingly small as the Prandtl number is reduced. As first noted by Drew et al [24], the Taylor number is required to be sufficiently large to observe the spurious behaviour of the AE. For Pr = 0.1 and N ρ = 5, the AE yield critical parameters that closely approximate the NSE critical parameters up to Ta ≈ 10 8 .…”

Section: Geostrophy)mentioning

confidence: 99%

“…With the exception of the anelastic study of [24], there have been no investigations of rapidly rotating compressible convection in low Prandtl number gases, despite the astrophysical relevance of this regime. In this work, we investigate this parameter regime with both the NSE and the AE, and show that the fundamental instability consists of compressional quasi-geostrophic (low frequency) oscillations whose existence intrinsically depends upon the presence of the time derivative of the density perturbation present in the mass conservation equation.…”

Section: Introductionmentioning

confidence: 99%

The breakdown of the anelastic approximation in rotating compressible convection: implications for astrophysical systems

2015

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The linear theory for rotating compressible convection in a plane layer geometry is presented for the astrophysically relevant case of low Prandtl number gases. When the rotation rate of the system is large, the flow remains geostrophically balanced for all stratification levels investigated and the classical (i.e. incompressible) asymptotic scaling laws for the critical parameters are recovered. For sufficiently small Prandtl numbers, increasing stratification tends to further destabilize the fluid layer, decrease the critical wavenumber and increase the oscillation frequency of the convective instability. In combination, these effects increase the relative magnitude of the time derivative of the density perturbation contained in the conservation of mass equation to non-negligible levels; the resulting convective instabilities occur in the form of compressional quasi-geostrophic oscillations. We find that the anelastic equations, which neglect this term, cannot capture these instabilities and possess spuriously growing eigenmodes in the rapidly rotating, low Prandtl number regime. It is shown that the Mach number for rapidly rotating compressible convection is intrinsically small for all background states, regardless of the departure from adiabaticity.

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Onset of convection in a rapidly rotating compressible fluid spherical shell

Cited by 14 publications

References 10 publications

Onset of rotating and non-rotating convection in compressible and anelastic ideal gases

Onset of rotating and non-rotating convection in compressible and anelastic ideal gases

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

The breakdown of the anelastic approximation in rotating compressible convection: implications for astrophysical systems

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