Rigorous bounds on the heat transport of rotating convection with Ekman pumping

Abstract

We establish rigorous upper bounds on the time-averaged heat transport for a model of rotating Rayleigh-Benard convection between no-slip boundaries at infinite Prandtl number and with Ekman pumping. The analysis is based on the asymptotically reduced equations derived for rotationally constrained dynamics with no-slip boundaries, and hence includes a lower order correction that accounts for the Ekman layer and corresponding Ekman pumping into the bulk. Using the auxiliary functional method we find that, to leading order, the temporally averaged heat transport is bounded above as a function of the Rayleigh and Ekman numbers Ra and Ek according to $Nu\le 0.3704R{a}^{2}E{k}^2$. Dependent on the relative values of the thermal forcing represented by $Ra$ and the effects of rotation represented by $Ek$, this bound is both an improvement on earlier rigorous upper bounds, and provides a partial explanation of recent numerical and experimental results that were consistent yet surprising relative to the previously derived upper bound of $Nu\lesssim R{a}^3{k}^4$.

References in corpus (16)

• Large scale dynamics in turbulent Rayleigh-Benard convection
• Approaching the Asymptotic Regime of Rapidly Rotating Convection: Boundary Layers vs Interior Dynamics
• Heat transport in the geostrophic regime of rotating Rayleigh-B{é}nard convection
• The effects of Ekman pumping on quasi-geostrophic Rayleigh-Benard convection
• A nonlinear model for rotationally constrained convection with Ekman pumping
• Zonal flow scaling in rapidly-rotating compressible convection
• Optimal bounds and extremal trajectories for time averages in nonlinear dynamical systems
• Bounds on mean energy in the Kuramoto-Sivashinsky equation computed using semidefinite programming
• The sensitivity of rapidly rotating Rayleigh--Bénard convection to Ekman pumping
• Relationship between the methods of bounding time averages
• Bounds for Rayleigh-Bénard convection between free-slip boundaries with an imposed heat flux
• Optimal bounds with semidefinite programming: an application to stress driven shear flows
• Bounds on Heat Transport in Rapidly Rotating Rayleigh-Bénard Convection
• Asymptotic Behavior of Heat Transport for a Class of Exact Solutions in Rotating Rayleigh-Bénard Convection
• Sharp transitions in rotating turbulent convection: Lagrangian acceleration statistics reveal a second critical Rossby number
• Optimization with affine homogeneous quadratic integral inequality constraints

Cited by in corpus (3)

• The background method: Theory and computations
• Internally heated convection with rotation: bounds on heat transport
• Asymptotic approximations for convection onset with Ekman pumping at low wavenumbers