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themselves solutions of the equations of motion. With regard to the second
approximation, the error involved in using a stationary basic state should
at least be smallest for those states corresponding to observed cluster centroids, since, by construction, these are, in some ensemble sense, closest to
stationary.
The approximation involved in using the barotropic model is actually
not as bad as might be imagined at first sight. According to results in
Molteni and Palmer (1993), 8-day singular vectors of a realistic timevarying baroclinic basic state are (at final time) more accurately represented by a time-averaged barotropic basic state than by a time-averaged
baroclinic basic state. The reason for this is that the instability of a timeaveraged baroclinic flow will tend to overly dominated by the baroclinic instabilities which relate directly to the meridional thermal contrasts. Moreover, since the Rossby-wave structures of such time-averaged flows are
under-represented, the upscale cascade process discussed in section 3.1 is
weak. As a result, singular vectors from time-averaged baroclinic flows
tend, at final time, to be dominated by smaller scales than would occur
with a time-varying flow. By contrast singular vectors from time-averaged
barotropic basic states (which also evolve through upscale energy evolution) are dominated at final time by larger, more realistic scales.
It is of interest to ask, however, how one might go about calculating regime instabilities without using such approximations. Let us represent the regime centroid by the a given normalised large- scale pattern
E(x, y, z). One possibility is to compute the pseudo-inverse of E(x, y, z)
using the dominant singular vectors, for a variety of finite-time trajectories. A less computationally demanding, but largely equivalent calculation
can be achieved with a single integration of the adjoint model. Suppose
we want to find a perturbation, with unit norm at initial time, and maximum projection onto the given pattern E at final time. In symbols, we
want a perturbation e at t = to which at t = to + !::.t = tl maximises
< Le, E > I < e, e >. This perturbation will be given at t = to by
e = L* E. We can refer to e as the sensitivity pattern (cf Marchuk, 1974;
Cacuci, 1981; Rabier et al, 1995) for E, and the growth IILell/llel1 as the
instability index of E. Such work is currently in progress using a three-level
quasi-geostrophic model whose climatology, as shown above, has realistic
regime structures (Susanna Corti, personal communication). These calculations could be of particular interest in studying the sensitivity of observed
climate change patterns (see section 6).
themselves solutions of the equations of motion. With regard to the second
approximation, the error involved in using a stationary basic state should
at least be smallest for those states corresponding to observed cluster centroids, since, by construction, these are, in some ensemble sense, closest to
stationary.
The approximation involved in using the barotropic model is actually
not as bad as might be imagined at first sight. According to results in
Molteni and Palmer (1993), 8-day singular vectors of a realistic timevarying baroclinic basic state are (at final time) more accurately represented by a time-averaged barotropic basic state than by a time-averaged
baroclinic basic state. The reason for this is that the instability of a timeaveraged baroclinic flow will tend to overly dominated by the baroclinic instabilities which relate directly to the meridional thermal contrasts. Moreover, since the Rossby-wave structures of such time-averaged flows are
under-represented, the upscale cascade process discussed in section 3.1 is
weak. As a result, singular vectors from time-averaged baroclinic flows
tend, at final time, to be dominated by smaller scales than would occur
with a time-varying flow. By contrast singular vectors from time-averaged
barotropic basic states (which also evolve through upscale energy evolution) are dominated at final time by larger, more realistic scales.
It is of interest to ask, however, how one might go about calculating regime instabilities without using such approximations. Let us represent the regime centroid by the a given normalised large- scale pattern
E(x, y, z). One possibility is to compute the pseudo-inverse of E(x, y, z)
using the dominant singular vectors, for a variety of finite-time trajectories. A less computationally demanding, but largely equivalent calculation
can be achieved with a single integration of the adjoint model. Suppose
we want to find a perturbation, with unit norm at initial time, and maximum projection onto the given pattern E at final time. In symbols, we
want a perturbation e at t = to which at t = to + !::.t = tl maximises
< Le, E > I < e, e >. This perturbation will be given at t = to by
e = L* E. We can refer to e as the sensitivity pattern (cf Marchuk, 1974;
Cacuci, 1981; Rabier et al, 1995) for E, and the growth IILell/llel1 as the
instability index of E. Such work is currently in progress using a three-level
quasi-geostrophic model whose climatology, as shown above, has realistic
regime structures (Susanna Corti, personal communication). These calculations could be of particular interest in studying the sensitivity of observed
climate change patterns (see section 6).