95
L(tn' to) are related to the Lyapunov exponents by the equation
. 1
i· = hm[-ln(T·]
•
n"""*oo n
•
(2.15)
From this, some authors (eg Abarbanel et ai, 1991) define the local Lyapunov exponents li(tl, to) of the attractor between tl and to as
(2.16)
The corresponding singular vectors can therefore be referred to as local
Lyapunov vectors, although it should be noted that this terminology is not
universal (eg Toth and Kalnay, 1996).
2.5 Projection operators
Singular vectors are, in general, not modal. For the application in section
3.2, their shapes evolve not only in geographical space but also in their
spectral distribution of energy. As we shall see for extratropical weather
systems, this spectral evolution describes, in a linear context, the upscale
energy transfer associated with turbulent processes. In order to study this
upscale energy transfer more explicitly we introduce a spectral projection
operator P[nl,n2] where [nl, n2] denotes the total wavenumber interval nl ::;
n ::; n2. P[nl,n2] is defined as
P[nl,n2]Xn
P[nl,n2]xn
Xn if m[nl, n2]
o otherwise
(2.17)
Here Xn is the wavenumber n component of the spherical harmonic expansion of the (atmospheric) state vector. If we wish to find perturbations,
initially constrained to be in [n3, n4], with maximum energy in [nl, n2],
these are given by the singular vectors of P[nl,n2]LP[n3,n4]. A similar projection operator can be applied to study singular vectors whose energy is
optimised to a specific geographical area (for applications, see Buizza and
Palmer, 1995; Hartmann et ai, 1995).
2.6 Numerical solution
When systems with a large number of degrees of freedom (eg 0(10 4 ) or
more) are considered, the eigenvalue problem (2.6, 2.7) cannot be solved
L(tn' to) are related to the Lyapunov exponents by the equation
. 1
i· = hm[-ln(T·]
•
n"""*oo n
•
(2.15)
From this, some authors (eg Abarbanel et ai, 1991) define the local Lyapunov exponents li(tl, to) of the attractor between tl and to as
(2.16)
The corresponding singular vectors can therefore be referred to as local
Lyapunov vectors, although it should be noted that this terminology is not
universal (eg Toth and Kalnay, 1996).
2.5 Projection operators
Singular vectors are, in general, not modal. For the application in section
3.2, their shapes evolve not only in geographical space but also in their
spectral distribution of energy. As we shall see for extratropical weather
systems, this spectral evolution describes, in a linear context, the upscale
energy transfer associated with turbulent processes. In order to study this
upscale energy transfer more explicitly we introduce a spectral projection
operator P[nl,n2] where [nl, n2] denotes the total wavenumber interval nl ::;
n ::; n2. P[nl,n2] is defined as
P[nl,n2]Xn
P[nl,n2]xn
Xn if m[nl, n2]
o otherwise
(2.17)
Here Xn is the wavenumber n component of the spherical harmonic expansion of the (atmospheric) state vector. If we wish to find perturbations,
initially constrained to be in [n3, n4], with maximum energy in [nl, n2],
these are given by the singular vectors of P[nl,n2]LP[n3,n4]. A similar projection operator can be applied to study singular vectors whose energy is
optimised to a specific geographical area (for applications, see Buizza and
Palmer, 1995; Hartmann et ai, 1995).
2.6 Numerical solution
When systems with a large number of degrees of freedom (eg 0(10 4 ) or
more) are considered, the eigenvalue problem (2.6, 2.7) cannot be solved