A COMPARISON OF LYAPUNOV AND OPTIMAL VECTORS IN A LOW-RESOLUTION GCM

We compare the first local Lyapunov vector (LLV) and the leading optim al vectors in a T10/18 level truncated version of the National Centers for Environmental Prediction global spectral model. The leading LLV i s a vector toward which all other perturbations turn and hence it is c haracterized by the fastest possible growth over infinitely long time periods, while the optimal vectors are perturbations that maximize gro wth for a finite time period, with respect to a chosen norm. Linear ta ngent model breeding experiments without convection at T10 resolution show that arbitrary random perturbations converge within a transition period of 3 to 4 days to a single LLV. We computed optimal vectors wit h the Lanczos algorithm, using the total energy norm. For optimization periods shorter than the transition period (about 3 days), the horizo ntal structure of the leading initial optimal vectors differs substant ially From that of the leading LLV, which provides maximum sustainable growth, There are also profound differences between the two types of vectors in their vertical structure. While the 24-hour optimal vectors rapidly become similar to the LLV in their vertical structure, change s in their horizontal structure are: very slow. As a consequence, thei r amplification factor drops and stays well below that of the LLV for an extended period after the optimization period ends, This may have a n adverse effect when optimal vectors with short optimization periods are used as initial perturbations for medium-range ensemble forecasts. The optimal vectors computed for 3 days or longer are different. In t hese vectors, the fastest growing initial perturbation has a horizonta l structure similar to that of the leading LLV, and its major differen ce from the LLV, in the vertical structure, tends to disappear by the end of the optimization period. Initially, the optimal vectors are hig hly unbalanced and the rapid changes in their vertical structure are a ssociated with geostrophic adjustment. The kinetic energy of the initi al optimal vectors peaks in the lower troposphere, whereas in the LLV the maximum is around the jet level. During the integration the phase of the streamfunction held of the optimal vectors, with respect to the ir corresponding temperature field, is rapidly shifted 180 degrees. An d, due to drastic changes that also rake place in the vertical tempera ture distribution, the maximum baroclinic shear shifts from the lower troposphere to just below the jet level. Just after initial rime, when the geostrophic adjustment dominates, the leading optimal vectors exh ibit a growth rate significantly higher than that of the LLV. By the e nd of the period of optimization, however, the growth rate associated with the leading optimal vectors drops to or below the level of the Ly apunov exponent. The transient super-lyapunov growth associated with t he leading optimal vectors is due to a one-time-only rapid rotation of the optimal vectors toward the leading LLVs. The nature of this rapid rotation depends on the length of the optimization period and the nor m chosen, We speculate that the initial optimal vectors computed with commonly used norms may not be realizable perturbations in a dynamical system (that is not forced by random noise) since the leading optimal vectors are directions in the phase space from which perturbations ar e ''repelled''. By contrast, the leading LLV is the vector toward whic h all random perturbations, including the optimal vectors, are attract ed, which gives the LLV a unique role in linear perturbation developme nt.