102
product, the PDF of analysis error is relatively isotropic). This assumption
has been verified for local European forecasts by Barkmeijer et al (1993).
There is one other important difference between the breeding and singular vector methodology. The singular vectors are computed over the period
in the immediate future of the initial analysis, whilst the breeding method
refers to the period leading up to the initial analysis. This is why singular
vector perturbations define, amongst all the a priori equally-likely initial
directions, those which give rise to the largest possible forecast error.
Nevertheless, perturbations using both breeding vectors and singular
vectors have proven useful for medium-range ensemble prediction. It is
possible that differences between the two techniques becomes less important as one evolves towards the 'strongly nonlinear' PDF regime (see Fig
1).
3 Examples of singular vectors
3.1 A baroclinic singular vector
Fig 5 shows the dominant singular vector calculated using the ECMWF
primitive equation model (Simmons et al, 1989; Courtier et al, 1991), for a
3-day trajectory portion made from initial conditions on 9 January 1993, at
three levels in the atmosphere (200 hPa, 700 hPa and 850 hPa) at initial and
optimisation time. The figure illustrates some features which bear qualitative resemblance to an idealised baroclinic eigenmode (Charney, 1947;
Eady, 1949): the disturbance clearly amplifies as it propagates through
the region of maximum baroclinity (where north-south temperature gradients are strongest), and the disturbance shows evidence of westward tilting
phase with height, consistent with a northward flux of heat.
On the other hand, the figure also clearly illustrates the non-modal nature of the disturbance. At initial time the disturbance is localised near
the north Atlantic jet entrance region; at optimisation time the disturbance
has propagated downstream to Europe. At initial time, maximum disturbance amplitude is located in the lower troposphere, whilst at optimisation
time maximum amplitude is located in the upper troposphere at the level
of maximum winds (cl Farrell, 1989). Finally, the horizontal scale of the
initial disturbance is noticeably smaller at initial time than at optimisation
time.
As discussed in section 2, singular-vector structure can be related to the
product, the PDF of analysis error is relatively isotropic). This assumption
has been verified for local European forecasts by Barkmeijer et al (1993).
There is one other important difference between the breeding and singular vector methodology. The singular vectors are computed over the period
in the immediate future of the initial analysis, whilst the breeding method
refers to the period leading up to the initial analysis. This is why singular
vector perturbations define, amongst all the a priori equally-likely initial
directions, those which give rise to the largest possible forecast error.
Nevertheless, perturbations using both breeding vectors and singular
vectors have proven useful for medium-range ensemble prediction. It is
possible that differences between the two techniques becomes less important as one evolves towards the 'strongly nonlinear' PDF regime (see Fig
1).
3 Examples of singular vectors
3.1 A baroclinic singular vector
Fig 5 shows the dominant singular vector calculated using the ECMWF
primitive equation model (Simmons et al, 1989; Courtier et al, 1991), for a
3-day trajectory portion made from initial conditions on 9 January 1993, at
three levels in the atmosphere (200 hPa, 700 hPa and 850 hPa) at initial and
optimisation time. The figure illustrates some features which bear qualitative resemblance to an idealised baroclinic eigenmode (Charney, 1947;
Eady, 1949): the disturbance clearly amplifies as it propagates through
the region of maximum baroclinity (where north-south temperature gradients are strongest), and the disturbance shows evidence of westward tilting
phase with height, consistent with a northward flux of heat.
On the other hand, the figure also clearly illustrates the non-modal nature of the disturbance. At initial time the disturbance is localised near
the north Atlantic jet entrance region; at optimisation time the disturbance
has propagated downstream to Europe. At initial time, maximum disturbance amplitude is located in the lower troposphere, whilst at optimisation
time maximum amplitude is located in the upper troposphere at the level
of maximum winds (cl Farrell, 1989). Finally, the horizontal scale of the
initial disturbance is noticeably smaller at initial time than at optimisation
time.
As discussed in section 2, singular-vector structure can be related to the