Search for: All records

Abstract Starting from the fully compressible fluid equations in a plane-parallel atmosphere, we demonstrate that linear internal gravity waves are naturally pseudo-incompressible in the limit that the wave frequencyωis much less than that of surface gravity waves, i.e., $\omega \ll \sqrt{{\mathit{gk}}_h}$, wheregis the gravitational acceleration andk_his the horizontal wavenumber. We accomplish this by performing a formal expansion of the wave functions and the local dispersion relation in terms of a dimensionless frequency $\epsilon =\omega /\sqrt{{\mathit{gk}}_h}$. Further, we show that, in this same low-frequency limit, several forms of the anelastic approximation, including the Lantz–Braginsky–Roberts formulation, poorly reproduce the correct behavior of internal gravity waves. The pseudo-incompressible approximation is achieved by assuming that Eulerian fluctuations of the pressure are small in the continuity equation—whereas, in the anelastic approximation, Eulerian density fluctuations are ignored. In an adiabatic stratification, such as occurs in a convection zone, the two approximations become identical. However, in a stable stratification, the differences between the two approximations are stark and only the pseudo-incompressible approximation remains valid. 

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. 

Gyroscopic alignment gives rise to highly spatially anisotropic columnar structures that in combination with complex domain boundaries pose challenges for efficient numerical discretizations and computations. We define gyroscopic polynomials to be three-dimensional polynomials expressed in a coordinate system that conforms to rotational alignment. We remap the original domain with radius-dependent boundaries onto a right cylindrical or annular domain to create the computational domain in this coordinate system. We find the volume element expressed in gyroscopic coordinates leads naturally to a hierarchy of orthonormal bases. We build the bases out of Jacobi polynomials in the vertical and generalized Jacobi polynomials in the radial. Because these coordinates explicitly conform to flow structures found in rapidly rotating systems the bases represent fields with a relatively small number of modes. We develop the operator structure for one-dimensional semi-classical orthogonal polynomials as a building block for differential operators in the full three-dimensional cylindrical and annular domains. The differentiation operators of generalized Jacobi polynomials generate a sparse linear system for discretization of differential operators acting on the gyroscopic bases. This enables efficient simulation of systems with strong gyroscopic alignment. 

The competition between turbulent convection and global rotation in planetary and stellar interiors governs the transport of heat and tracers, as well as magnetic field generation. These objects operate in dynamical regimes ranging from weakly rotating convection to the “geostrophic turbulence” regime of rapidly rotating convection. However, the latter regime has remained elusive in the laboratory, despite a worldwide effort to design ever-taller rotating convection cells over the last decade. Building on a recent experimental approach where convection is driven radiatively, we report heat transport measurements in quantitative agreement with this scaling regime, the experimental scaling law being validated against direct numerical simulations (DNS) of the idealized setup. The scaling exponent from both experiments and DNS agrees well with the geostrophic turbulence prediction. The prefactor of the scaling law is greater than the one diagnosed in previous idealized numerical studies, pointing to an unexpected sensitivity of the heat transport efficiency to the precise distribution of heat sources and sinks, which greatly varies from planets to stars. 

The observational absence of giant convection cells near the Sun’s outer surface is a long-standing conundrum for solar modelers. We herein propose an explanation. Rotation strongly influences the internal dynamics, leading to suppressed convective velocities, enhanced thermal-transport efficiency, and (most significantly) relatively smaller dominant length scales. We specifically predict a characteristic convection length scale of roughly 30-Mm throughout much of the convection zone, implying weak flow amplitudes at 100- to 200-Mm giant cells scales, representative of the total envelope depth. Our reasoning is such that Coriolis forces primarily balance pressure gradients (geostrophy). Background vortex stretching balances baroclinic torques. Both together balance nonlinear advection. Turbulent fluxes convey the excess part of the solar luminosity that radiative diffusion cannot. We show that these four relations determine estimates for the dominant length scales and dynamical amplitudes strictly in terms of known physical quantities. We predict that the dynamical Rossby number for convection is less than unity below the near-surface shear layer, indicating rotational constraint.