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Predicting
uncertainty in forecasts of weather and climate
(Also published as ECMWF Technical Memorandum No. 294)
By T.N. Palmer
Research Department, ECMWF
November 1999
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5 . Error growth in the linear and nonlinear phase
5.1 Singular vectors, eigenvectors and Lyapunov vectors
As discussed in Section 3, the initial phase-space directions which evolve into the major axes of the ellipsoid of forecast error pdf are determined by the dominant
singular vectors of the forward tangent propagator 
. Singular vectors of 
have been studied extensively in weather prediction models using simpler
metrics than 
(Lacarra and Talagrand,
1988; Molteni and Palmer,
1993; Buizza and Palmer,
1995). One simple choice is the energy metric 
. The 
singular vectors of 
are the dominant eigenvectors of 
where '*' now denotes the adjoint defined with respect to the inner product
where 
is the total energy of the perturbation 
. An explicit expression for 
is given in Buizza and Palmer
(1995). An example of a
singular vector is given in Fig.
5 , calculated using the forward and adjoint tangent propagators of
the ECMWF weather prediction model, optimised over a three-day period, with
starting conditions on 9 January 1993. The streamfunction (inverse Laplacian
of vorticity) associated with the singular vector is shown at three levels
in the atmosphere, at initial and optimisation time. The singular vector
has some qualitative resemblance to an idealised baroclinically unstable
eigenmode of the atmosphere (Charney,
1947; Eady, 1949); for
example the disturbance amplifies as it propagates through the region in
the west Atlantic where north-south surface temperature gradients are largest,
and the disturbance shows evidence of westward tilting phase with height,
consistent with a northward flux of heat (Gill,
1982). However, the singular vector also clearly illustrates non-modal characteristics. At initial time the disturbance is localised over the west Atlantic, at optimisation time the disturbance has propagated downstream to Europe. At initial time, maximum disturbance amplitude is located in the lower troposphere, whilst at final time amplitude is largest in the upper troposphere at the level of maximum winds. Finally, the horizontal scale of the initial disturbance is noticeably smaller at initial time than at optimisation time.
A simple way of understanding the non-modal vertical structure of the singular vector is to consider a steady zonally symmetric basic state flow
, which is slowly varying in the vertical. In such a flow, the wave action
of a small-amplitude disturbance with energy 
, zonal wavenumber 
and frequency 
will be conserved as it propagates vertically on the background flow (e.g.
Gill, 1982). Optimal
energy growth will therefore. tend to be associated with propagation away
from a region of small intrinsic frequency (near the so-called baroclinic
steering level usually located in the-lower troposphere), to a region of
large intrinsic frequency (such as would occur at the jet stream level in
the upper troposphere). The upscale evolution is consistent with an inverse energy cascade characteristic of two dimensional turbulence (see below). At first sight it might appear paradoxical that such a nonlinear property can be emulated by a linearised calculation. However, this non-modal characteristic of the linear perturbation is a consequence of the non-normality (i.e.
) of the tangent propagator (Farrell
and Ioannou, 1996). In turn non-normality derives its existence from
the advective nonlinearity in Eq.
(1). Hence, in some sense, the cascade properties associated with nonlinear
advection in Eq. (1) are
partially reflected in the non-normality of 
. 
The ability of singular vectors to describe upscale evolution can be illustrated in simpler `toy models'. Hansen (1998) has studied singular vectors of the two-scale dynamical system
|
(27) |
of Lorenz (1996). Here
are taken as large-scale variables, 
as small-scale variables. For each large-scale variable, there are 
small-scale variables. Hansen maximises
|
(28) |
where
is the forward tangent propagator of Eqs.
(27), and P projects the entire state vector onto the subspace of the
large-scale 
variables. For particular choices of the parameters 
, 
and 
, the dominant singular vector which maximises amplitude in the large-scale
variables at optimisation time, is comprised almost entirely of small-scale
components at initial time, emulating the upscale cascade exhibited in this
nonlinear model. It is also possible to compute singular vectors of the tangent propagator from intermediate coupled ocean-atmosphere models (e.g. Moore and Kleeman, 1996; Chen et al. 1997; Xue et al. 1997). As with the atmosphere-only singular vectors, the growth characteristics are extremely non-modal. The surface temperatures of the evolved singular vectors at optimisation time is characteristic of the EI Niño phenomenon itself, whilst at initial time the singular vector has little spatial correlation with the El Niño pattern. The singular values are strongly dependent on time of year, being largest from (boreal) spring to autumn, and weakest from (boreal) autumn to spring. As will be argued below, the seasonal cycle in these coupled ocean-atmosphere singular vectors may be vital in understanding how the climate responds to external forcing perturbations.
Let us consider the relationship between singular vector growth and eigenvector growth when Eq. (1) is linearised about a stationary solution. In this case, normalised eigenvectors
of 
with eigenvalues 
give rise to modal solutions 
.) (The propagator 
will also have eigenvectors 
but with eigenvalues 
.) Irrespective of normality, eigenvectors 
and eigenvalues 
of the adjoint Jacobian satisfy the biorthogonality condition
|
(29) |
where `cc' denotes the complex conjugate.
If an initial disturbance is written in terms of the eigenmodes
, i.e.
|
(30) |
then from Eq. (29)
|
(31) |
From Eq. (30), the fastest growing eigenmode
will ultimately contribute most to the growth of 
. In order to maximise the contribution of the first eigenmode, 
should be as large as possible. From Eq.
(31), this occurs when 
is parallel to 
(for a non self-adjoint operator, 
) Hence, in order to maximise the asymptotic amplitude of 
, the initial perturbation should not project onto the fastest-growing eigenmode,
but onto its adjoint. Fig. 6 illustrates schematically the crucial difference between eigenmode and singular vector growth. An idealised 2D system has two very non-orthogonal decaying eigenmodes
and 
; we take 
to have the larger real eigenvalue component. The adjoint eigenmodes 
and 
are also shown, consistent with the biorthogonality condition (29).
A normalised vector 
is shown parallel to 
; the sequence of vectors 
show the time evolution of 
. The projection of 
onto 
is much larger than that of a second vector 
which was initially normalised and aligned along 
. Another familiar measure of perturbation growth is given by the Lyapunov exponent. The system's Lyapunov exponents can be defined as
|
(32) |
where
is the eigenvalue of the 
-th eigenvector of the operator
|
(33) |
These eigenvectors, or Lyapunov vectors, correspond to (instantaneous) realisations of evolved singular vectors, for long optimisation times.
Although Lyapunov exponents are normally associated with mean growth over the attractor, local Lyapunov exponents can be defined as the growth rates
|
(34) |
Just as singular vector growth can exceed eigenmode growth for stationary flows, so it can also exceed local Lyapunov growth for transient flows (Trevisan and Legnani, 1995).
In fact in a multi-scale system, the concept of Lyapunov exponent growth is not a useful one. For example, in Eq. (27), the dominant Lyapunov exponent is determined (for
) by the growth of the small scale 
variables. However, if 
is sufficiently large, then their effect on the predictability of the 
variables will be small compared with the effect of initial uncertainty
in the large-scale 
variables themselves. In the atmosphere, Lyapunov exponent growth would
be associated with small-scale convective instability, rather than large-scale
more slowly growing baroclinic instability (which determines the structure
of extratropical weather systems). As such, the predictability of these
weather systems can be much longer than a dominant atmospheric Lyapunov
timescale. One way over overcoming this problem with the Lyapunov vector is through the so-called breeding-vector modification (Toth and Kalnay, 1993, 1997). A bred vector is obtained by integrating the model twice over a time ?t from initial conditions differing by a small random perturbation. The resulting difference field is rescaled to the initial perturbation amplitude, and the procedure repeated. The essential difference between a bred vector and a Lyapunov vector is that the time
is chosen to be sufficiently long that instabilities on scales much smaller
than weather, which would in practice determine the atmosphere's dominant
Lyapunov exponent, are damped by nonlinear saturation. 5.2 Error dynamics and scale cascades
A fundamental characteristic of error growth in climate and weather prediction is the upscale cascade of error. This characterises the essential element of the `butterfly effect' paradigm, as much as does amplitude growth (i.e. initial conditions are sensitive not only to small amplitude error, but also small-scale error). In this section, we briefly describe estimates of predictability arising from cascade processes, using from simple scaling arguments.
Let
denote a characteristic wavenumber in such a system, and 
denote the energy kinetic energy per unit wavenumber, at wavenumber 
. Assume that observations cannot resolve wavenumbers higher than 
. Hence the initial pdf will certainly be non-zero in the phasespace sub-manifold
associated with wavenumbers higher than kr. We present below a heuristic
argument for determining the time it takes this component of initial error
to infect the meteorological scales 
. Following Lorenz (1969)
and Lilly (1973), let
us assume that the time it takes before complete uncertainty at wavenumber
to strongly infect wavenumber 
, is proportional to the `eddy turn-over time' 
. The time 
taken for uncertainty to propagate from wavenumber 
to wavenumber 
is therefore given by
|
(35) |
In the case of a two-dimensional isotropic homogeneous turbulence in the inertial subrange between some large-scale (e.g. baroclinic) forcing scale and dissipation scale, then
, 
is independent of 
, and 
which diverges as 
. Hence, although uncertainty at small scales (within the inertial range)
can infect the larger meteorological scale, in theory long-range predictions
are possible providing the initial data has sufficient small-scale resolution.
This two dimensional paradigm is appropriate to weather systems where quasi-geostrophic
scaling is appropriate (Charney, 1971). By contrast, for homogeneous isotropic 3-dimensional flow
and 
, the familiar Kolmogorov inertial range (Frisch,
1995). In this case 
tends to a finite limit as 
, that is
|
(36) |
This remarkable result says that the predictability time for a large-scale system is on the order of an eddy turn-over time, a few days. In the limit of infinite Reynolds number, this is consistent with the hypothesised lack of existence of a unique smooth solution of the Navier-Stokes equations in the Euler limit (Frisch, 1995), since it implies that uncertainty on arbitrarily small scales can destroy predictability on large scales in finite time.
However, for the quasi-2D large-scale weather and climate systems discussed in this paper, this 3D paradigm does not appear to be directly relevant; for scales O(100 km) or larger, horizontal scales dominate the vertical scale (the aspect ratio is small). However, there is evidence for a
spectrum on scales of 
km (Nastrom and Gage,
1985; Gage and Nastrom,
1986); Cho et al., 1999).
This is likely to be associated with a 2D inverse energy cascade (Kraichnan,
1971) possibly forced by organised convective activity (Lilly,
1983), of the type discussed in section 4. This suggests that in the atmosphere
there are two sets of 2D inertial ranges, an 
enstrophy cascading range associated with forcing from weather systems at
large scale, and a smaller-scale 
inverse energy cascading range associated with forcing from mesoscale variability
on small scales (Lilly,
1983, 1989). As such, uncertainties on these mesoscales may influence the
predictability of large-scale weather systems within a few days, and reinforces
the need for a stochastic representation of such effects in global weather
and climate models (see Section
4).
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