26 research outputs found
Anelastic Versus Fully Compressible Turbulent Rayleigh-B\'enard Convection
Numerical simulations of turbulent Rayleigh-B\'enard convection in an ideal
gas, using either the anelastic approximation or the fully compressible
equations, are compared. Theoretically, the anelastic approximation is expected
to hold in weakly superadiabatic systems with , where denotes the superadiabatic temperature drop over the
convective layer and the bottom temperature. Using direct numerical
simulations, a systematic comparison of anelastic and fully compressible
convection is carried out. With decreasing superadiabaticity , the
fully compressible results are found to converge linearly to the anelastic
solution with larger density contrasts generally improving the match. We
conclude that in many solar and planetary applications, where the
superadiabaticity is expected to be vanishingly small, results obtained with
the anelastic approximation are in fact more accurate than fully compressible
computations, which typically fail to reach small for numerical
reasons. On the other hand, if the astrophysical system studied contains
regions, such as the solar photosphere, fully compressible
simulations have the advantage of capturing the full physics. Interestingly,
even in weakly superadiabatic regions, like the bulk of the solar convection
zone, the errors introduced by using artificially large values for
for efficiency reasons remain moderate. If quantitative errors of the order of
are acceptable in such low regions, our work suggests that
fully compressible simulations can indeed be computationally more efficient
than their anelastic counterparts.Comment: 24 pages, 9 figure
The sensitivity of rapidly rotating Rayleigh--B\'enard convection to Ekman pumping
The dependence of the heat transfer, as measured by the nondimensional
Nusselt number , on Ekman pumping for rapidly rotating Rayleigh-B\'enard
convection in an infinite plane layer is examined for fluids with Prandtl
number . A joint effort utilizing simulations from the Composite
Non-hydrostatic Quasi-Geostrophic model (CNH-QGM) and direct numerical
simulations (DNS) of the incompressible fluid equations has mapped a wide range
of the Rayleigh number - Ekman number parameter space within the
geostrophic regime of rotating convection. Corroboration of the -
relation at from both methods along with higher covered by
DNS and lower by the asymptotic model allows for this range of the heat
transfer results. For stress-free boundaries, the relation has the dissipation-free scaling of for all
. This is directly related to a geostrophic turbulent interior
that throttles the heat transport supplied to the thermal boundary layers. For
no-slip boundaries, the existence of ageostrophic viscous boundary layers and
their associated Ekman pumping yields a more complex 2D surface in
parameter space. For results suggest that the surface can be
expressed as indicating the
dissipation-free scaling law is enhanced by Ekman pumping by the multiplicative
prefactor where . It follows for
that the geostrophic turbulent interior remains the flux bottleneck
in rapidly rotating Rayleigh-B\'enard convection. For , where DNS
and asymptotic simulations agree quantitatively, it is found that the effects
of Ekman pumping are sufficiently strong to influence the heat transport with
diminished exponent and .Comment: 9 pages, 14 figure
The effects of Ekman pumping on quasi-geostrophic Rayleigh-Benard convection
Numerical simulations of 3D, rapidly rotating Rayleigh-Benard convection are
performed using an asymptotic quasi-geostrophic model that incorporates the
effects of no-slip boundaries through (i) parameterized Ekman pumping boundary
conditions, and (ii) a thermal wind boundary layer that regularizes the
enhanced thermal fluctuations induced by pumping. The fidelity of the model,
obtained by an asymptotic reduction of the Navier-Stokes equations that
implicitly enforces a pointwise geostrophic balance, is explored for the first
time by comparisons of simulations against the findings of direct numerical
simulations and laboratory experiments. Results from these methods have
established Ekman pumping as the mechanism responsible for significantly
enhancing the vertical heat transport. This asymptotic model demonstrates
excellent agreement over a range of thermal forcing for Pr ~1 when compared
with results from experiments and DNS at maximal values of their attainable
rotation rates, as measured by the Ekman number (E ~ 10^{-7}); good qualitative
agreement is achieved for Pr > 1. Similar to studies with stress-free
boundaries, four spatially distinct flow morphologies exists. Despite the
presence of frictional drag at the upper and/or lower boundaries, a strong
non-local inverse cascade of barotropic (i.e., depth-independent) kinetic
energy persists in the final regime of geostrophic turbulence and is dominant
at large scales. For mixed no-slip/stress-free and no-slip/no-slip boundaries,
Ekman friction is found to attenuate the efficiency of the upscale energy
transport and, unlike the case of stress-free boundaries, rapidly saturates the
barotropic kinetic energy. For no-slip/no-slip boundaries, Ekman friction is
strong enough to prevent the development of a coherent dipole vortex
condensate. Instead vortex pairs are found to be intermittent, varying in both
time and strength.Comment: 20 pages, 10 figure
Kinetic Energy Transport in Rayleigh--B\'enard Convection
The kinetic energy balance in Rayleigh--B\'{e}nard convection is investigated
for the Prandtl number range and for fixed Rayleigh number
. The kinetic energy balance is divided into a dissipation, a
production and a flux term. We discuss profiles of all terms and find that the
different contributions to the energy balance can be spatially separated into
regions where kinetic energy is produced and where kinetic energy is
dissipated. Analysing the Prandtl number dependence of the kinetic energy
balance, we show that the height-dependence of the mean viscous dissipation is
closely related to the flux of kinetic energy. We show that the flux of kinetic
energy can be divided into four additive contributions, each representing a
different elementary physical process (advection, buoyancy, normal viscous
stresses and viscous shear stresses). The behaviour of these individual flux
contributions is found to be surprisingly rich and exhibits a pronounced
Prandtl number dependence. Different flux contributions dominate the kinetic
energy transport at different depth, such that a comprehensive discussion
requires a decomposition of the domain into a considerable number of
sub-layers. On a less detailed level, our results reveal that advective kinetic
energy fluxes play a key role in balancing the near-wall dissipation at low
Prandtl number, whereas normal viscous stresses are particularly important at
high Prandtl number. Finally, our work reveals that classical velocity boundary
layers are deeply connected to the kinetic energy transport, but fail to
correctly represent regions of enhanced viscous dissipation
A nonlinear model for rotationally constrained convection with Ekman pumping
It is a well established result of linear theory that the influence of
differing mechanical boundary conditions, i.e., stress-free or no-slip, on the
primary instability in rotating convection becomes asymptotically small in the
limit of rapid rotation. This is accounted for by the diminishing impact of the
viscous stresses exerted within Ekman boundary layers and the associated
vertical momentum transport by Ekman pumping. By contrast, in the nonlinear
regime recent experiments and supporting simulations are now providing evidence
that the efficiency of heat transport remains strongly influenced by Ekman
pumping in the rapidly rotating limit. In this paper, a reduced model is
developed for the case of low Rossby number convection in a plane layer
geometry with no-slip upper and lower boundaries held at fixed temperatures. A
complete description of the dynamics requires the existence of three distinct
regions within the fluid layer: a geostrophically balanced interior where fluid
motions are predominately aligned with the axis of rotation, Ekman boundary
layers immediately adjacent to the bounding plates, and thermal wind layers
driven by Ekman pumping in between. The reduced model uses a classical Ekman
pumping parameterization to alleviate the need for spatially resolving the
Ekman boundary layers. Results are presented for both linear stability theory
and a special class of nonlinear solutions described by a single horizontal
spatial wavenumber. It is shown that Ekman pumping allows for significant
enhancement in the heat transport relative to that observed in simulations with
stress-free boundaries. Without the intermediate thermal wind layer the
nonlinear feedback from Ekman pumping would be able to generate a heat
transport that diverges to infinity. This layer arrests this blowup resulting
in finite heat transport at a significantly enhanced value.Comment: 38 pages, 14 figure