Boundary Layer Analysis for Navier-Slip Rayleigh–Bénard Convection: The Non-existence of an Ultimate State

  • G.-M. Gie
  • J. P. WhiteheadEmail author


We discuss the asymptotic behavior, at small viscosity and/or diffusivity, of the Rayleigh–Bénard convection problem governed by the Boussinesq equations. The velocity vector field and the temperature are supplemented respectively with the Navier friction boundary conditions and the fixed flux boundary condition in a 3D periodic channel domain. By explicitly constructing the boundary layer correctors, which approximate the difference between the viscous/diffusive solutions and the corresponding limit solution, we validate the asymptotic expansions, and prove the vanishing viscosity and diffusivity limit with the optimal rate of convergence. Correctors in this setting include higher order diffusive effects than considered previously and accurately account for the interplay between the viscous and thermal layers. The boundary layer correctors satisfy a linear evolution equation indicating that for these boundary conditions, there is no turbulence in the boundary layer. The impact of this fact on the existence of an ‘ultimate state’ of turbulent convection is discussed, particularly in light of recent upper bounds on the heat transport that indicate such a state may exist in this setting.


Boundary layers Navier boundary conditions Rayleigh–Bénard convection 



We thank the anonymous referee for some insightful comments and corrections which improved the presentation of this result. G-MG is partially supported by the Research—RI Grant, Office of the Executive Vice President for Research and Innovation, University of Louisville, and the Victor A. Olorunsola Endowed Research Award for Young Scholars, College of Arts and Sciences, University of Louisville. This collaboration arose following participation in a Mathematics Research Communities workshop sponsored by the American Mathematical Society, and further extended through a visit of G-MG sponsored by the Department of Mathematics at Brigham Young University.

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Conflict of interest

The authors declare that they have no conflict of interest.


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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of LouisvilleLouisvilleUSA
  2. 2.275 TMCBBrigham Young UniversityProvoUSA

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