On the spatial and temporal discretization of vertical diffusion in the turbulent planetary boundary layer

Numerical models behavior can be very sensitive to changes in time-step and spatial resolution [1]. In particular such dependency can be observed for turbulent planetary boundary layer schemes when running standard single column experiments. In this talk we propose to investigate if part of this sensitivity could be attributed to inadequate spatial and temporal discretizations of the nonlinear vertical diffusion term. The usual second-order centered discretizations are generally not suitable for problems characterized by sharp boundary layers. Various finite-difference and finite-volume alternatives with similar stencil are investigated. The very large vertical parabolic Courant numbers usually found in numerical simulations make it hard also to formulate a temporal scheme that is robust to changes in model parameters. For large parabolic Courant numbers and implicit Euler scheme, the amount of diffusion we would expect from physical principles (i.e. as diagnosed by the PBL scheme) is very different from the amount of diffusion actually ”seen” by the model because of numerical errors [2]. Alternatives proposed by [3] could help to mitigate this issue even if they have been derived mainly with nonlinear stability constraints in mind. We will conclude by discussing how a finite-volume approach for the vertical diffusion term could allow to jointly ensure the proper regularity of the numerical solutions while satisfying the underlying physical principles (e.g. the Monin-Obukhov theory). Numerical results using standard single column experiments are shown to illustrate our findings for 0-equation and 1-equation turbulent schemes.

[1] Gross, M. et al.: Physics–Dynamics Coupling in Weather, Climate, and Earth System Models: Challenges and Recent Progress. Mon. Wea. Rev., 146, 3505–3544 (2018)

[2] F. Lemarié, L. Debreu, G. Madec, J. Demange, J.-M. Molines, and M. Honnorat: Stability constraints for oceanic numerical models: implications for the formulation of time and space discretizations. Ocean Model. (2015)

[3] Wood N., M. Diamantakis and A. Staniforth: A monotonically-damping second-order- accurate unconditionally-stable numerical scheme for diffusion. Q. J. Roy. Met. Soc. (2007)