The spherical solution of the linear balance equation (LBE), Ñ × ¦ Ñ Y = Ñ 2 f , for the non-divergent stream function can be affected by the singularity at the equator. An non-terminating kinetic energy spectrum arises when the spectral LBE is truncated in a fully-determined system (Eliasen and Machenhauer, 1965; Merilees, 1968; Wiin-Nielsen, 1979; Daley, 1983). Baer (1977) solved an over-determined spectral LBE for the anti-symmetric components in a least square and obtained a kinetic energy spectrum which exhibits a minus-one power law in the inertial range. In this paper, we approximate the spectral LBE in an under-determined system which removed the tropical singularity.
The solution for an under-determined system consists of two parts: the regular solution is a function of the prescribed f , while the singular solution is a projection of the observed Y on the singular eigenmode derivable under the least square approximation. We will present the result of calculations showing that with the tropical singularity removed, the linearly-balanced kinetic energy spectrum exhibits a slope predicted by the theory of geostrophic turbulence (Charney, 1971).
The impact of the present numerical method on the spherical solution of the vorticity balance equation, Ñ × ( ¦ + z )Ñ c = B , for the irrotational velocity potential in a c problem (Sardeshmukh and Hoskins, 1988: Sardeshmukh, 1993) will be discussed.
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12th Conference on Atmospheric and Oceanic Fluid Dynamics