A numerical investigation of an asymptotically reduced model for quasigeostrophic
Rayleigh-Bénard convection is conducted in which the depth-averaged flows are numerically suppressed by modifying the governing equations. At the largest accessible values
of the Rayleigh number Ra, the Reynolds number and Nusselt number show evidence
of approaching the diffusion-free scalings of Re ∼ RaE/Pr and Nu ∼ Pr−1/2Ra3/2E2,
respectively, where E is the Ekman number and Pr is the Prandtl number. For large
Ra, the presence of depth-invariant flows, such as large-scale vortices, yield heat and
momentum transport scalings that exceed those of the diffusion-free scaling laws. The
Taylor microscale does not vary significantly with increasing Ra, whereas the integral
length scale grows weakly. The computed length scales remain O(1) with respect to the
linearly unstable critical wave number; we therefore conclude that these scales remain
viscously controlled. We do not find a point-wise Coriolis-inertia-Archimedean (CIA)
force balance in the turbulent regime; interior dynamics are instead dominated by horizontal advection (inertia), vortex stretching (Coriolis) and the vertical pressure gradient. A
secondary, subdominant balance between the Archimedean buoyancy force and the viscous
force occurs in the interior and the ratio of the root mean square (rms) of these two forces
is found to approach unity with increasing Ra. This secondary balance is attributed to
the turbulent fluid interior acting as the dominant control on the heat transport. These
findings indicate that a pointwise CIA balance does not occur in the high Rayleigh number
regime of quasigeostrophic convection in the plane layer geometry. Instead, simulations
are characterized by what may be termed a nonlocal CIA balance in which the buoyancy
force is dominant within the thermal boundary layers and is spatially separated from the
interior Coriolis and inertial forces.
more »
« less
Connections between nonrotating, slowly rotating, and rapidly rotating turbulent convection transport scalings
In this study, we investigate and develop scaling laws as a function of external non-dimensional control parameters for heat and momentum transport for non-rotating, slowly rotating and rapidly rotating turbulent convection systems, with the end goal of forging connections and bridging the various gaps between these regimes. Two perspectives are considered, one where turbulent convection is viewed from the standpoint of an applied temperature drop across the domain and the other with a viewpoint in terms of an applied heat flux. While a straightforward transformation exist between the two perspectives indicating equivalence, it is found the former provides a clear set of connections that bridge between the three regimes. Our generic convection scalings, based upon an Inertial-Archimedean balance, produce the classic diffusion-free scalings for the non-rotating limit (NRL) and the slowly rotating limit (SRL). This is characterized by a free-falling fluid parcel on the global scale possessing a thermal anomaly on par with the temperature drop across the domain. In the rapidly rotating limit (RRL), the generic convection scalings are based on a Coriolis-Inertial-Archimedean (CIA) balance, along with a local fluctuating-mean advective temperature balance. This produces a scenario in which anisotropic fluid parcels attain a thermal wind velocity and where the thermal anomalies are greatly attenuated compared to the total temperature drop.
We find that turbulent scalings may be deduced simply by consideration of the generic non-dimensional transport parameters --- local Reynolds $Re_\ell = U \ell /\nu$; local P\'eclet $Pe_\ell = U \ell /\kappa$; and Nusselt number $Nu = U \vartheta/(\kappa \Delta T/H)$ --- through the selection of physically relevant estimates for length $\ell$, velocity $U$ and temperature scales $\vartheta$ in each regime. Emergent from the scaling analyses is a unified continuum based on a single external control parameter, the convective Rossby number\JMA{,} $\RoC = \sqrt{g \alpha \Delta T / 4 \Omega^2 H}$, that strikingly appears in each regime by consideration of the local, convection-scale Rossby number $\Rol=U/(2\Omega \ell)$. Thus we show that $\RoC$ scales with the local Rossby number $\Rol$ in both the slowly rotating and the rapidly rotating regimes, explaining the ubiquity of $\RoC$ in rotating convection studies. We show in non-, slowly, and rapidly rotating systems that the convective heat transport, parameterized via $Pe_\ell$, scales with the total heat transport parameterized via the Nusselt number $Nu$. Within the rapidly-rotating limit, momentum transport arguments generate a scaling for the system-scale Rossby number, $Ro_H$, that, recast in terms of the total heat flux through the system, is shown to be synonymous with the classical flux-based `CIA' scaling, $Ro_{CIA}$. These, in turn, are then shown to asymptote to $Ro_H \sim Ro_{CIA} \sim \RoC^2$, demonstrating that these momentum transport scalings are identical in the limit of rapidly rotating turbulent heat transfer.
more »
« less
- Award ID(s):
- 1853196
- NSF-PAR ID:
- 10211932
- Date Published:
- Journal Name:
- Physical review research
- Volume:
- 2
- ISSN:
- 2643-1564
- Page Range / eLocation ID:
- 043115
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
more » « less
SUMMARY We present investigations of rapidly rotating convection in a thick spherical shell geometry relevant to planetary cores, comparing results from quasi-geostrophic (QG), 3-D and hybrid QG-3D models. The 170 reported calculations span Ekman numbers, Ek, between 10−4 and 10−10, Rayleigh numbers, Ra, between 2 and 150 times supercritical and Prandtl numbers, Pr, between 10 and 10−2. The default boundary conditions are no-slip at both the ICB and the CMB for the velocity field, with fixed temperatures at the ICB and the CMB. Cases driven by both homogeneous and inhomogeneous CMB heat flux patterns are also explored, the latter including lateral variations, as measured by Q*, the peak-to-peak amplitude of the pattern divided by its mean, taking values up to 5. The QG model is based on the open-source pizza code. We extend this in a hybrid approach to include the temperature field on a 3-D grid. In general, we find convection is dominated by zonal jets at mid-depths in the shell, with thermal Rossby waves prominent close to the outer boundary when the driving is weaker. For the thick spherical shell geometry studied here the hybrid method is best suited for studying convection at modest forcing, $Ra \le 10 \, Ra_c$ when Pr = 1, and departs from the 3-D model results at higher Ra, displaying systematically lower heat transport characterized by lower Nusselt and Reynolds numbers. We find that the lack of equatorially-antisymmetric motions and z-correlations between temperature and velocity in the buoyancy force contributes to the weaker flows in the hybrid formulation. On the other hand, the QG models yield broadly similar results to the 3-D models, for the specific aspect ratio and range of Rayleigh numbers explored here. We cannot point to major disagreements between these two data sets at Pr ≥ 0.1, with the QG model effectively more strongly driven than the hybrid case due to its cylindrically averaged thermal boundary conditions. When Pr is decreased, the range of agreement between the hybrid and 3-D models expands, for example up to $Ra \le 15 \, Ra_c$ at Pr = 0.1, indicating the hybrid method may be better suited to study convection in the low Pr regime. We thus observe a transition between two regimes: (i) at Pr ≥ 0.1 the QG and 3-D models agree in the studied range of Ra/Rac while the hybrid model fails when $Ra\gt 15\, Ra_c$ and (ii) at Pr = 0.01 the QG and 3-D models disagree for $Ra\gt 10\, Ra_c$ while the hybrid and 3-D models agree fairly well up to $Ra \sim 20\, Ra_c$. Models that include laterally varying heat flux at the outer boundary reproduce regional convection patterns that compare well with those found in similarly forced 3-D models. Previously proposed scaling laws for rapidly rotating convection are tested; our simulations are overall well described by a triple balance between Coriolis, inertia and Archimedean forces with the length-scale of the convection following the diffusion-free Rhines-scaling. The magnitude of Pr affects the number and the size of the jets with larger structures obtained at lower Pr. Higher velocities and lower heat transport are seen on decreasing Pr with the scaling behaviour of the convective velocity displaying a strong dependence on Pr. This study is an intermediate step towards a hybrid model of core convection also including 3-D magnetic effects.
-
null (Ed.)Steady two-dimensional Rayleigh–Bénard convection between stress-free isothermal boundaries is studied via numerical computations. We explore properties of steady convective rolls with aspect ratios ${\rm \pi} /5\leqslant \varGamma \leqslant 4{\rm \pi}$ , where $\varGamma$ is the width-to-height ratio for a pair of counter-rotating rolls, over eight orders of magnitude in the Rayleigh number, $10^3\leqslant Ra\leqslant 10^{11}$ , and four orders of magnitude in the Prandtl number, $10^{-2}\leqslant Pr\leqslant 10^2$ . At large $Ra$ where steady rolls are dynamically unstable, the computed rolls display $Ra \rightarrow \infty$ asymptotic scaling. In this regime, the Nusselt number $Nu$ that measures heat transport scales as $Ra^{1/3}$ uniformly in $Pr$ . The prefactor of this scaling depends on $\varGamma$ and is largest at $\varGamma \approx 1.9$ . The Reynolds number $Re$ for large- $Ra$ rolls scales as $Pr^{-1} Ra^{2/3}$ with a prefactor that is largest at $\varGamma \approx 4.5$ . All of these large- $Ra$ features agree quantitatively with the semi-analytical asymptotic solutions constructed by Chini & Cox ( Phys. Fluids , vol. 21, 2009, 083603). Convergence of $Nu$ and $Re$ to their asymptotic scalings occurs more slowly when $Pr$ is larger and when $\varGamma$ is smaller.more » « less
-
The central open question about Rayleigh–Bénard convection – buoyancy-driven flow in a fluid layer heated from below and cooled from above – is how vertical heat flux depends on the imposed temperature gradient in the strongly nonlinear regime where the flows are typically turbulent. The quantitative challenge is to determine how the Nusselt number $Nu$ depends on the Rayleigh number $Ra$ in the $Ra\to \infty$ limit for fluids of fixed finite Prandtl number $Pr$ in fixed spatial domains. Laboratory experiments, numerical simulations and analysis of Rayleigh's mathematical model have yet to rule out either of the proposed ‘classical’ $Nu \sim Ra^{1/3}$ or ‘ultimate’ $Nu \sim Ra^{1/2}$ asymptotic scaling theories. Among the many solutions of the equations of motion at high $Ra$ are steady convection rolls that are dynamically unstable but share features of the turbulent attractor. We have computed these steady solutions for $Ra$ up to $10^{14}$ with $Pr=1$ and various horizontal periods. By choosing the horizontal period of these rolls at each $Ra$ to maximize $Nu$ , we find that steady convection rolls achieve classical asymptotic scaling. Moreover, they transport more heat than turbulent convection in experiments or simulations at comparable parameters. If heat transport in turbulent convection continues to be dominated by heat transport in steady rolls as $Ra\to \infty$ , it cannot achieve the ultimate scaling.more » « less
-
The connection between the heat transfer and characteristic flow velocities of planetary core-style convection remains poorly understood. To address this, we present novel laboratory models of rotating Rayleigh–Bénard convection in which heat and momentum transfer are simultaneously measured. Using water (Prandtl number, Pr≃6) and cylindrical containers of diameter-to-height aspect ratios of Γ≃3,1.5,0.75, the non-dimensional rotation period (Ekman number, E) is varied between 10−7≲E≲3×10−5 and the non-dimensional convective forcing (Rayleigh number, Ra) ranges from 107≲Ra≲1012. Our heat transfer data agree with those of previous studies and are largely controlled by boundary layer dynamics. We utilize laser Doppler velocimetry (LDV) to obtain experimental point measurements of bulk axial velocities, resulting in estimates of the non-dimensional momentum transfer (Reynolds number, Re) with values between 4×102≲Re≲5×104. Behavioral transitions in the velocity data do not exist where transitions in heat transfer behaviors occur, indicating that bulk dynamics are not controlled by the boundary layers of the system. Instead, the LDV data agree well with the diffusion-free Coriolis–Inertia–Archimedian (CIA) scaling over the range of Ra explored. Furthermore, the CIA scaling approximately co-scales with the Viscous–Archimedian–Coriolis (VAC) scaling over the parameter space studied. We explain this observation by demonstrating that the VAC and CIA relations will co-scale when the local Reynolds number in the fluid bulk is of order unity. We conclude that in our experiments and similar laboratory and numerical investigations with E≳10−7, Ra≲1012, Pr≃7, heat transfer is controlled by boundary layer physics while quasi-geostrophically turbulent dynamics relevant to core flows robustly exist in the fluid bulk.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- The DOI is not currently available.
