An official website of the United States government
Abstract The so-called traditional approximation, wherein the component of the Coriolis force proportional to the cosine of latitude is ignored, is frequently made in order to simplify the equations of atmospheric circulation. For velocity fields whose vertical component is comparable to their horizontal component (such as convective circulations), and in the tropics where the sine of latitude vanishes, the traditional approximation is not justified. We introduce a framework for studying the effect of diabatic heating on circulations in the presence of both traditional and nontraditional terms in the Coriolis force. The framework is intended to describe steady convective circulations on anfplane in the presence of radiation and momentum damping. We derive a single elliptic equation for the horizontal velocity potential, which is a generalization of the weak temperature gradient (WTG) approximation. The elliptic operator depends on latitude, radiative damping, and momentum damping coefficients. We show how all other dynamical fields can be diagnosed from this velocity potential; the horizontal velocity induced by the Coriolis force has a particularly simple expression in terms of the velocity potential. Limiting examples occur at the equator, where only the nontraditional terms are present, at the poles, where only the traditional terms appear, and in the absence of radiative damping where the WTG approximation is recovered. We discuss how the framework will be used to construct dynamical, nonlinear convective models, in order to diagnose their consequent upscale momentum and temperature fluxes.
more »
« less
This content will become publicly available on May 1, 2026
Convective Circulations and the Coriolis Force: A Mechanism for Upscale Momentum Fluxes in the Tropics
Abstract In the study of subgrid-scale tropical convection, the importance of retaining the frequently omitted “nontraditional” component of the Coriolis force is increasingly being recognized. A number of recent papers have developed linear theories examining the behavior of a diabatic heat-source-driven convective circulation in the presence of the full Coriolis force, and it was shown that the nontraditional Coriolis terms drive vertical shears on the large scales through upscale fluxes of momentum. In the present work, we generalize these results to the nonlinear regime, using a formal asymptotic theory based upon the fact that rotation is a second-order effect compared with advection by the vertical component of velocity at subgrid scales. Ultimately, we demonstrate that the same basic flow structures persist, with a particular emphasis on the counterrotating vortex pair induced by the nontraditional Coriolis terms which drive a westward tilt in convection. We compute the form of the upscale momentum flux convergence in the nonlinear regime, greatly extending the regimes of validity provided by the simple analytical expressions previously given in the linear case. This study constitutes an important step toward being able to accurately and consistently parameterize the large-scale vertical shear driven by nonlinear, subgrid convective processes under the influence of the nontraditional Coriolis force terms.
more »
« less
- Award ID(s):
- 2224293
- PAR ID:
- 10615449
- Publisher / Repository:
- American Meteorological Society
- Date Published:
- Journal Name:
- Journal of the Atmospheric Sciences
- Volume:
- 82
- Issue:
- 5
- ISSN:
- 0022-4928
- Page Range / eLocation ID:
- 849 to 867
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
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
-
Dynamos driven by rotating convection in the plane layer geometry are investigated numerically for a range of Ekman number ( $$E$$ ), magnetic Prandtl number ( $Pm$ ) and Rayleigh number ( $Ra$ ). The primary purpose of the investigation is to compare results of the simulations with previously developed asymptotic theory that is applicable in the limit of rapid rotation. We find that all of the simulations are in the quasi-geostrophic regime in which the Coriolis and pressure gradient forces are approximately balanced at leading order, whereas all other forces, including the Lorentz force, act as perturbations. Agreement between simulation output and asymptotic scalings for the energetics, flow speeds, magnetic field amplitude and length scales is found. The transition from large-scale dynamos to small-scale dynamos is well described by the magnetic Reynolds number based on the small convective length scale, $$\widetilde {Rm}$$ , with large-scale dynamos preferred when $$\widetilde {Rm} \lesssim O(1)$$ . The magnitude of the large-scale magnetic field is observed to saturate and become approximately constant with increasing Rayleigh number. Energy spectra show that all length scales present in the flow field and the small-scale magnetic field are consistent with a scaling of $$E^{1/3}$$ , even in the turbulent regime. For a fixed value of $$E$$ , we find that the viscous dissipation length scale is approximately constant over a broad range of $Ra$ ; the ohmic dissipation length scale is approximately constant within the large-scale dynamo regime, but transitions to a $$\widetilde {Rm}^{-1/2}$$ scaling in the small-scale dynamo regime.more » « less
-
Summary The large-scale dynamics of convection-driven dynamos in a spherical shell, as relevant to the geodynamo, is analyzed with numerical simulation data and asymptotic theory. An attempt is made to determine the asymptotic size (with the small parameter being the Ekman number, Ek) of the forces, and the associated velocity and magnetic fields. In agreement with previous work, the leading order mean force balance is shown to be thermal wind (Coriolis, pressure gradient, buoyancy) in the meridional plane and Coriolis-Lorentz in the zonal direction. The Lorentz force is observed to be weaker than the mean buoyancy force across a range of Ek and thermal forcing; the relative difference in these forces appears to be O(Ek1/6) within the parameter space investigated. We find that the thermal wind balance requires that the mean zonal velocity scales as O(Ek−1/3), whereas the meridional circulation is asymptotically smaller by a factor of O(Ek1/6). The mean temperature equation shows a balance between thermal diffusion and the divergence of the convective heat flux, indicating the presence of a mean temperature length scale of size O(Ek1/6). Neither the mean nor the fluctuating magnetic field show a strong dependence on the Ekman number, though the simulation data shows evidence of a mean magnetic field length scale of size O(Ek1/6). A consequence of the asymptotic ordering of the forces is that Taylor’s constraint is satisfied to accuracy O(Ek1/6), despite the absence of a leading-order magnetostrophic balance. Further consequences of the force balance are discussed with respect to the large-scale flows thought to be important for the geodynamo.more » « less
-
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.more » « less
