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A Compact Model for the Stability Dependency of TKE Production–Destruction–Conversion Terms Valid for the Whole Range of Richardson Numbers

Ivan Bašták ĎuránONPP/CHMI, Prague, Czech Republic

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Jean-François GeleynONPP/CHMI, Prague, Czech Republic

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Filip VáňaONPP/CHMI, Prague, Czech Republic

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Abstract

Recently, observational and numerical evidence has accumulated against the concept of a critical Richardson number Ricr beyond which too-stable stratification would extinguish turbulence. It also appeared that the characteristics of the “weak turbulent regime” where the Prandtl number σt increases proportionally to the Richardson number Ri can be explained via the conservation of total turbulent energy in a strongly anisotropic flow. Having a “No Ri(cr)” situation together with due consideration of the anisotropy thus leads to the correct asymptotic behavior at high stabilities in several recent proposals [revisit of the Mellor–Yamada basic system, non-Reynolds-type quasi-normal scale elimination (QNSE) theory, and energy and flux budget (EFB) theory leading to a fully self-consistent hierarchy of increasingly prognostic schemes]. The present work derives a simple unique analytical framework for these various alternatives, simplifying, in two complementary but surprisingly converging ways, the revisited Mellor–Yamada formulation and emulating with high accuracy the relevant solutions within QNSE and EFB. The simplification or emulation steps differ from one case to the next, but the obtained common framework is very compact, valid for Ri going from −∞ to +∞, depending only on four free parameters and on three “functional dependencies.” Each functional dependency corresponds either to a constant value or to a regular function of the flux Richardson number Rif depending on the complexity of the considered hypotheses. Four realizations of this codification are representative of all related possibilities, the analytical scheme thus possessing high transversal validity. Extension toward higher-order solutions and/or moist turbulence can be envisaged in such a unified framework.

Denotes Open Access content.

Current affiliation: CNRM-GAME, Météo-France/CNRS UMR 3589, Toulouse, France.

Current affiliation: ECMWF, Reading, United Kingdom.

Corresponding author address: Ivan Bašták Ďurán, Czech Hydrometeorological Institute, Na Šabatce 2050/17, 143 06 Prague, Czech Republic. E-mail: ivan.bastak@chmi.cz

Abstract

Recently, observational and numerical evidence has accumulated against the concept of a critical Richardson number Ricr beyond which too-stable stratification would extinguish turbulence. It also appeared that the characteristics of the “weak turbulent regime” where the Prandtl number σt increases proportionally to the Richardson number Ri can be explained via the conservation of total turbulent energy in a strongly anisotropic flow. Having a “No Ri(cr)” situation together with due consideration of the anisotropy thus leads to the correct asymptotic behavior at high stabilities in several recent proposals [revisit of the Mellor–Yamada basic system, non-Reynolds-type quasi-normal scale elimination (QNSE) theory, and energy and flux budget (EFB) theory leading to a fully self-consistent hierarchy of increasingly prognostic schemes]. The present work derives a simple unique analytical framework for these various alternatives, simplifying, in two complementary but surprisingly converging ways, the revisited Mellor–Yamada formulation and emulating with high accuracy the relevant solutions within QNSE and EFB. The simplification or emulation steps differ from one case to the next, but the obtained common framework is very compact, valid for Ri going from −∞ to +∞, depending only on four free parameters and on three “functional dependencies.” Each functional dependency corresponds either to a constant value or to a regular function of the flux Richardson number Rif depending on the complexity of the considered hypotheses. Four realizations of this codification are representative of all related possibilities, the analytical scheme thus possessing high transversal validity. Extension toward higher-order solutions and/or moist turbulence can be envisaged in such a unified framework.

Denotes Open Access content.

Current affiliation: CNRM-GAME, Météo-France/CNRS UMR 3589, Toulouse, France.

Current affiliation: ECMWF, Reading, United Kingdom.

Corresponding author address: Ivan Bašták Ďurán, Czech Hydrometeorological Institute, Na Šabatce 2050/17, 143 06 Prague, Czech Republic. E-mail: ivan.bastak@chmi.cz
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