105
tors underpins this relationship. For example, under certain assumptions
(steady zonally symmetric basic state flow which is slowly varying in the
vertical) the wave-action E/(w - kuo) of a linear disturbance with energy
E, zonal wavenumber k and frequency w will be conserved as it propagates
vertically on the background flow Uo. Optimal energy growth will tend to
be associated with propagation from a region of small intrinsic frequency
( near the baroclinic steering level, to a region of large intrinsic frequency
(such as might occur near the jet level).
The horizontal-scale evolution of the singular vector is explored further
in Fig 6 which shows the energy distribution of the singular vector at
initial and final time, as a function of total wavenumber. Fig 6a shows
the spectral distribution of the disturbance shown in Fig 5 peaking near
the truncation limit at initial time (dashed line) and at about wavenumber
10 at optimisation time. This upscale energy transfer can occur because
the basic state (unlike those in many idealised calculations) is itself an
unrestricted solution to the equations of motion, and, in particular contains
scales comparable with those in the disturbance field. This allows triad
interactions between the disturbance field and the basic state.
Fig 6b,c shows the spectral distribution of two further singular vector
calculations made using the same trajectory. In these calculations, the
spectral projection operator (2.17) has been applied both at initial and
optimisation time. For both calculations, the operator at optimisation time
maximises energy between wavenumbers 0 and 10. The initial perturbation
is constrained to wavenumbers 0-10 in Fig 6b and to wavenumbers 11-20
in Fig 6c.
The results are quite dramatic. Constraining the perturbation to have
the same energy distribution in wavenumber space at initial and final time
(which an eigenmode solution, if it exists, must have), severely restricts
perturbation growth. On the other hand, constraining the perturbation
at initial and final time to have energy in non-overlapping wavenumber
intervals hardly restricts energy growth at all (see Hartmann et al, 1995,
for more details).
These calculations illustrate, in a linear context, the 'butterfly effect' in
its original sense (Lorenz, 1963b), ie that small-scale initial disturbances
can ultimately have an overwhelming influence on large-scale disturbances.
This is in addition to the commonly perceived meaning of the butterfly effect that small-amplitude initial disturbances will ultimately have an overwhelming effect on large-amplitude disturbances. (In fact, Lorenz refers to
tors underpins this relationship. For example, under certain assumptions
(steady zonally symmetric basic state flow which is slowly varying in the
vertical) the wave-action E/(w - kuo) of a linear disturbance with energy
E, zonal wavenumber k and frequency w will be conserved as it propagates
vertically on the background flow Uo. Optimal energy growth will tend to
be associated with propagation from a region of small intrinsic frequency
( near the baroclinic steering level, to a region of large intrinsic frequency
(such as might occur near the jet level).
The horizontal-scale evolution of the singular vector is explored further
in Fig 6 which shows the energy distribution of the singular vector at
initial and final time, as a function of total wavenumber. Fig 6a shows
the spectral distribution of the disturbance shown in Fig 5 peaking near
the truncation limit at initial time (dashed line) and at about wavenumber
10 at optimisation time. This upscale energy transfer can occur because
the basic state (unlike those in many idealised calculations) is itself an
unrestricted solution to the equations of motion, and, in particular contains
scales comparable with those in the disturbance field. This allows triad
interactions between the disturbance field and the basic state.
Fig 6b,c shows the spectral distribution of two further singular vector
calculations made using the same trajectory. In these calculations, the
spectral projection operator (2.17) has been applied both at initial and
optimisation time. For both calculations, the operator at optimisation time
maximises energy between wavenumbers 0 and 10. The initial perturbation
is constrained to wavenumbers 0-10 in Fig 6b and to wavenumbers 11-20
in Fig 6c.
The results are quite dramatic. Constraining the perturbation to have
the same energy distribution in wavenumber space at initial and final time
(which an eigenmode solution, if it exists, must have), severely restricts
perturbation growth. On the other hand, constraining the perturbation
at initial and final time to have energy in non-overlapping wavenumber
intervals hardly restricts energy growth at all (see Hartmann et al, 1995,
for more details).
These calculations illustrate, in a linear context, the 'butterfly effect' in
its original sense (Lorenz, 1963b), ie that small-scale initial disturbances
can ultimately have an overwhelming influence on large-scale disturbances.
This is in addition to the commonly perceived meaning of the butterfly effect that small-amplitude initial disturbances will ultimately have an overwhelming effect on large-amplitude disturbances. (In fact, Lorenz refers to