1. Introduction

Large-scale motions in the atmosphere and oceans, affected by Earth's rotation as well as density stratification associated with Earth's gravity, are almost two-dimensional flows in the horizontal plane. In geophysical and planetary fluid dynamics, the framework of 2-dimensional flows has been successful in reconstructing features of actual phenomena to some extent.

The major characteristic of turbulence within a 2-dimensional flow is that, in a decaying turbulence (i.e., energy is only supplied initially and no external forcing is applied except for dissipation), coherent vortices are formed spontaneously from random initial fields (McWilliams, 1984).

However, his result is applicable only for non-rotational conditions, and the temporal evolution differs significantly when the effect of planetary rotation in taken into account. Rhines (1975) carried out experiments on 2-dimensional turbulence on a beta plane (a model that incorporates latitudinal dependence of planetary rotation), and found that a zonal band structure appeared. Furthermore, Williams (1978) conducted numerical experiments of forced turbulence in a rotating sphere (in which energy is continuously supplied to a specific wavenumber range), a model which targeted planetary atmospheres. While the model was strongly bound in terms of its meridional symmetry and longitudinal eightfold symmetry, it clearly revealed that band structures also appeared in a spherical model.

Numerical experiments covering the entire surface of a sphere at high resolution was first performed by Yoden and Yamada (1993) on decaying turbulence, and in their study it was discovered that easterly polar vortex is formed spontaneously when the planetary rotation is sufficiently rapid. Later, Nozawa and Yoden (1997) carried out high-resolution experiments on forced turbulence and confirmed the formation of the zonal band structure in 2-dimensional forced turbulence on a rotating sphere.

The studies introduced so far have all dealt with 2-dimensional non-divergent systems. However, more recently, Cho and Polvani (1996) have performed an experiment on decaying turbulence in a shallow-water equation system that incorporated the divergence effect of the system, and their study indicated the zonal band structures are also formed in decaying turbulence in shallow-water systems. While zonal band structures did appear in the decaying turbulence experiment in a non-divergent system in Yoden and Yamada (1993), we believe that improvements could be made on the resolution of the numerical calculations and assignment of initial values, which were not quite satisfactory at the time, with the computational environment of the present day. Thus, it is desirable to carry out a new set of experiments in order to eliminate any doubt as to whether or not zonal band structures appear spontaneously even in decaying turbulence.

With the history briefly described above being taken into consideration, the present study attempted a numerical experiment on 2-dimensional decaying turbulence on a rotating sphere at ultra-high resolution, and a detailed investigation was performed to determine the conditions at which characteristic patterns such as zonal band structures and easterly polar vortex appear. It should be noted that in the experiment in the present study were carried out to resolution up to a total wavenumber of 682 for spherical harmonics using the numerical codes by Ishioka (1999). To our knowledge, this is the highest resolution in the world to date for 2-dimensional spherical calculations.

The composition of the present paper is as follows. Section 2 describes the model equations and the method of numerical calculation. Section 3 presents the results of time evolution of decaying turbulence and analyses associated with them. Section 4 gives discussion and conclusions.

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