Method 1, EOF

We employ a method proposed by KM to distinguish the 2-D and 3-D regimes by using an Empirical Orthogonal Function (EOF), which is usually called as principal component analysis in the field of multivariate analysis. In the EOF expression used in the present study, the vertical velocities are given as the superposition of orthogonal modes in the vertical direction as,

,                           (1)

where and are the horizontal coordinates, is the vertical coordinate, the time, the vertical velocity, and the mode number, the vertical structure of the nth mode amplitudes, the spatial distribution of the mode. According to KM, an experiment for which the first EOF mode explains more than (less than) 60% of the total energy of the vertical velocities is classified into the 2-D (3-D) regime. EOF calculations are repeatedly conducted for each experiment, and the average of explained variances from 80 to 144 hours is adopted as the variance for the experiment. If we change the threshold from KM's 60% to another possible value, i.e., 50%, the conclusions of the present papers are essentially unchanged.