An extension of Mellor-Yamada type model to Terra Incognita scale

Conventional parameterizations of turbulence in numerical weather prediction models need to be somehow modified if a horizontal resolution is increased to be the “Terra Incognita” scale (Wyngaard, 2004), in which a numerical model starts to resolve some of the larger eddies which play significant roles in turbulent transports. This study presents an extension of a Mellor-Yamada-type (MY) model to the “Terra Incognita” scale.

The MY models (e.g. Mellor and Yamada 1982; Nakanishi and Niino 2009) of level 3, 2.5 or 2 are one of the most prevailed second-order turbulence closure schemes in numerical weather prediction models. One may easily implement the extension presented in this study into currently working numerical models that employ MY models for their turbulence closure.

The extension changes nothing if turbulence is unresolved. However, the extended MY model seamlessly adjusts its length scales and closure constants to diagnose reasonable sub-grid scale turbulent fluxes in the Terra Incognita scale and even inertial sub-range which has been resolved by large eddy simulations (LES). As in previous studies (e.g., Honnert et al. 2012), LES results of idealized convective mixed layers are used to design the extended model. The LES model has uniform grid spacing of 25 m. Uniform heat flux is given at the surface to develop convective mixed layers and geostrophic winds of various strength are also given. Horizontal top-hat filters of various size, dx^*, are imposed on the LES results to obtain sub-filter scale covariance and fluxes which are to be parameterized in a numerical model having a horizontal resolution dx^*. Hereafter all results are presented in a universal non-dimensional form under the free convection scaling (Deardorff 1970).

The dissipation length scale is found to decrease with dx=dx^*/h where h is the depth of the convective mixed layer. The manner of decrease is consistent with the fact that sub-grid scale turbulent kinetic energy (TKE) dissipation does not significantly depend on dx even in the “Terra Incognita” scale. If an empirical formula of subgrid-scale TKE varying with dx (c.f. partition function in Honnert 2012 et al.) is introduced, the dissipation length for dx in the Terra Incognita scale can be determined.

In MY Level 3 or less models, a local equilibrium between production and pressure correlation is assumed to diagnose turbulent fluxes, where length scales are also required to obtain the pressure correlation. Using pressure correlation and turbulent fluxes explicitly calculated from the filtered LES data, we can estimate the length scales to be used to predict vertical and horizontal turbulent flux, respectively. The length scale used for the vertical turbulent flux is found to vary in proportion to the dissipation length. On the other hand, the length scale for the horizontal turbulent flux decreases with decreasing dx, but its vertical profile is different from that of the dissipation length scale. The effects of shear and buoyancy on the pressure correlation (rapid part) turn out to be insignificant for smaller dx. As a result, the proposed parameterization approaches asymptotically a conventional sub-grid scale model of LES (Smagorinsky, 1963).

The proposed extension of a MY model has been implemented in a numerical model that has a horizontal resolution of 500 m or 1 km, which in a typical “Terra Incognita” scale. The model results for simulating a growth of a convective mixed layer demonstrate that the resolved-scale convection and subgrid-scale turbulent fluxes reasonably agree with those obtained by the filtered LES data.