Singular vectors: the effect of spatial scale on linear growth of disturbances

The scale dependence of rapidly growing perturbations is investigated by studying the dominant singular vectors of T21 and T42 versions of the ECMWF model, which show the most linear energy growth in a 3-day period. A spectral filter is applied to the optimization process to determine which spatial scales are most effective in promoting energy growth. When the initial perturbation is confined to the top half of the total spherical harmonic wavenumber spectrum (high wavenumber end), the growth rates and final structures of the disturbances are changed very little from the case in which all wavenumbers are included. These results indicate that synoptic waves that become fully developed in a period of three days can arise from initial perturbations that are entirely contained at subsynoptic scales. Rapid growth is associated with initial perturbations that consist of smaller spatial scales concentrated near the effective steering level. The linear evolution of these initial perturbations in a highly complex basic flow leads to disturbances of synoptic scale that extend through most of the depth of the troposphere. Growth rates are approximately doubled when the model resolution is increased from T21 to T42, which is consistent with greater growth being associated with smaller spatial scales. When the initial perturbation is confined to the lower half of the total wavenumber spectrum, which describes the larger horizontal scales, the growth rates are significantly reduced and the initial and final structures are very different from the case in which all wavenumbers are included. These low wavenumber perturbations tend to be more barotropic in structure and in growth characteristics. As expected from their linear growth rates, when the low-wavenumber perturbations are inserted in the T63 forecast model, they grow more slowly and result in less forecast dispersion than the high wavenumber perturbations.