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Analytical Solution of Surface Layer Similarity Equations

Kun YangRiver Laboratory, Department of Civil Engineering, University of Tokyo, Tokyo, Japan

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Nobuyuki TamaiRiver Laboratory, Department of Civil Engineering, University of Tokyo, Tokyo, Japan

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Toshio KoikeRiver Laboratory, Department of Civil Engineering, University of Tokyo, Tokyo, Japan

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Abstract

Turbulent exchange between the surface and the atmosphere strongly depends on the stability of the surface layer. If surface radiometric temperature, rather than aerodynamic temperature, is used to parameterize the surface turbulent fluxes, the solution of the stability parameter is related to the thermal roughness length zT, which is generally not identical to the aerodynamic roughness length z0. This note derives the exact solution of the stability parameter equation for a stable surface layer and proposes an approximate analytical solution for an unstable surface layer. The solution can improve the computational efficiency of flux parameterization and is applicable in a wide range of z/z0 (50–104) and z0/zT (from less than 1 to greater than 104).

Corresponding author address: Kun Yang, River Lab., Dept. of Civil Engineering, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan. yangk@hydra.t.u-tokyo.ac.jp

Abstract

Turbulent exchange between the surface and the atmosphere strongly depends on the stability of the surface layer. If surface radiometric temperature, rather than aerodynamic temperature, is used to parameterize the surface turbulent fluxes, the solution of the stability parameter is related to the thermal roughness length zT, which is generally not identical to the aerodynamic roughness length z0. This note derives the exact solution of the stability parameter equation for a stable surface layer and proposes an approximate analytical solution for an unstable surface layer. The solution can improve the computational efficiency of flux parameterization and is applicable in a wide range of z/z0 (50–104) and z0/zT (from less than 1 to greater than 104).

Corresponding author address: Kun Yang, River Lab., Dept. of Civil Engineering, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan. yangk@hydra.t.u-tokyo.ac.jp

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