A comparison of Lyapunov and optimal vectors in a low-resolution GCM
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We compare the first local Lyapunov vector (LLV) and the leading optimal vectors in a T10/18 level truncated version of the National Centers for Environmental Prediction global spectral model. The leading LLV is a vector toward which all other perturbations turn and hence it is characterized by the fastest possible growth over infinitely long time periods, while the optimal vectors are perturbations that maximize growth for a finite time period, with respect to a chosen norm. Linear tangent 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 with the Lanczos algorithm, using the total energy norm. For optimization periods shorter than the transition period (about 3 days), the horizontal structure of the leading initial optimal vectors differs substantially 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, changes in their horizontal structure are very slow. As a consequence, their amplification factor drops and stays well below that of the LLV for an extended period after the optimization period ends. This may have an 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 these vectors, the fastest growing initial perturbation has a horizontal structure similar to that of the leading LLV, and its major difference from the LLV, in the vertical structure, tends to disappear by the end of the optimization period.Initially, the optimal vectors are highly unbalanced and the rapid changes in their vertical structure are associated with geostrophic adjustment. The kinetic energy of the initial 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 field of the optimal vectors, with respect to their corresponding temperature field, is rapidly shifted 180. And, due to drastic changes that also take place in the vertical temperature distribution, the maximum baroclinic shear shifts from the lower tropo-sphere to just below the jet level. Just after initial time, when the geostrophic adjustment dominates, the leading optimal vectors exhibit a growth rate significantly higher than that of the LLV. By the end of the period of optimization, however, the growth rate associated with the leading optimal vectors drops to or below the level of the Lyapunov exponent. The transient super-Lyapunov growth associated with the 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 norm 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 are "repelled". By contrast, the leading LLV is the vector toward which all random perturbations, including the optimal vectors, are attracted, which gives the LLV a unique role in linear perturbation development.
