The Numerical Solution of the Mellor-Yamada Level 2.5 Turbulent Kinetic Energy Equation in the Eta Model

Joseph P. Gerrity Jr. National Meteorological Center, Washington, D.C.

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Thomas L. Black National Meteorological Center, Washington, D.C.

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Russell E. Treadon General Sciences Corporation, Laurel, Maryland

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Abstract

A new method is presented for obtaining the numerical solution of the production-dissipation component of the turbulent kinetic energy equation that arises in the Mellor-Yamada level 2.5 turbulent closure model. The development of this new method was motivated by the occasional appearance of large temporal oscillations in the solution provided by a previously used method. Analysis of the equation revealed that the solution should tend toward a stationary asymptotic value, which is the equilibrium value of turbulent kinetic energy for the level 2, Mellor-Yamada model. Failure to identify the correct asymptotic value in the formalism underlying the numerical solution of the equation allows the solution to overshoot the equilibrium. Repeated overshooting gives rise to an oscillation in the solution from one time step to the next. The new method prevents this from happening.

Idealized cases are used to demonstrate the performance of the new method. It has been incorporated into the eta coordinate, numerical weather prediction model being used by the National Meteorological Center.

Although the new method corrects the particular deficiency of the previous method, the integration of the equation for turbulent kinetic energy remains subject to oscillatory solutions associated with rapid variations of the Richardson number. An example of this is provided.

Additionally, even with the new method, it is sometimes necessary to revert to the level 2 model when the numerical integration of the full system of equations yields a value of turbulent kinetic energy that falls below a value associated with a singularity of the level 2.5 model. In future work, modifications of the Mellor-Yamada turbulent closure system that avoid this limitation will be investigated.

Abstract

A new method is presented for obtaining the numerical solution of the production-dissipation component of the turbulent kinetic energy equation that arises in the Mellor-Yamada level 2.5 turbulent closure model. The development of this new method was motivated by the occasional appearance of large temporal oscillations in the solution provided by a previously used method. Analysis of the equation revealed that the solution should tend toward a stationary asymptotic value, which is the equilibrium value of turbulent kinetic energy for the level 2, Mellor-Yamada model. Failure to identify the correct asymptotic value in the formalism underlying the numerical solution of the equation allows the solution to overshoot the equilibrium. Repeated overshooting gives rise to an oscillation in the solution from one time step to the next. The new method prevents this from happening.

Idealized cases are used to demonstrate the performance of the new method. It has been incorporated into the eta coordinate, numerical weather prediction model being used by the National Meteorological Center.

Although the new method corrects the particular deficiency of the previous method, the integration of the equation for turbulent kinetic energy remains subject to oscillatory solutions associated with rapid variations of the Richardson number. An example of this is provided.

Additionally, even with the new method, it is sometimes necessary to revert to the level 2 model when the numerical integration of the full system of equations yields a value of turbulent kinetic energy that falls below a value associated with a singularity of the level 2.5 model. In future work, modifications of the Mellor-Yamada turbulent closure system that avoid this limitation will be investigated.

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