92
values are complex, there is an arbitrary phase factor that has to be defined
to make this correspondence precise.)
In order to maximise the contribution of the first eigenmode at optimisation time, C1 in (2.11) should be as large as possible. If x(to) equals {I
then from (2.12), C1 = 1 which could be highly sub-optimal. In fact, ifx(ta)
projects onto '1/1, then C! is maximised and is given by the projectibility
factor 1/ ( co sal ) (Zhang, 1988).
Hence, for indefinitely long optimisation time, the dominant singular
vector, at initial time, is determined by the first adjoint eigenmode, whilst
the dominant singular vector at optimisation time is determined by the first
eigenmode itself. The singular value will depend on both the e-folding time
of the dominant eigenmode and its projectibility. For finite optimisation
time, the dominant singular vectors will no longer project onto individual
eigenmode solutions (and their adjoints), and the amplitude of finite-time
instabilities need not be bounded by properties of the dominant eigenmodes
alone.
Fig 2 illustrates schematically the crucial difference between eigenmode
and singular vector growth, and the relationship between singular vectors
and adjoint modes. An idealised 2-D system has two very non-orthogonal
decaying eigenmodes {I and {2· We take {I to have the larger real eigenvalue component. The adjoint eigenmodes '1/1 and '1/2 are shown with '1/1,'1/2
orthogonal to {2,{1 respectively (according to the biorthogonality condition
2.10). A normalised vector Va is shown parallel to '1/1. Its time evolution
can be estimated by mapping the tip and tail of Va along the {I and {2
directions (shown as dashed lines) using the modal decay rates. The sequence of vectors Vn , n = 1,2, ... giving the time evolution of Va increases
in amplitude up to some finite n = N and is aligned almost entirely with
{I for large n. The projection of Vn onto {I for large n is much larger
than that associated with the evolution of a second normalised vector J-ln
which is initially aligned along {I. The sequence V n , n = 1,2, ... describes
singular vector growth over a long time interval. The transient growth of
the singular vectors in such systems was first noted by Orr (1907), and a
review of this process in plane parallel shear flow (and its relationship to
wave overreflection) is discussed in Lindzen (1988).
values are complex, there is an arbitrary phase factor that has to be defined
to make this correspondence precise.)
In order to maximise the contribution of the first eigenmode at optimisation time, C1 in (2.11) should be as large as possible. If x(to) equals {I
then from (2.12), C1 = 1 which could be highly sub-optimal. In fact, ifx(ta)
projects onto '1/1, then C! is maximised and is given by the projectibility
factor 1/ ( co sal ) (Zhang, 1988).
Hence, for indefinitely long optimisation time, the dominant singular
vector, at initial time, is determined by the first adjoint eigenmode, whilst
the dominant singular vector at optimisation time is determined by the first
eigenmode itself. The singular value will depend on both the e-folding time
of the dominant eigenmode and its projectibility. For finite optimisation
time, the dominant singular vectors will no longer project onto individual
eigenmode solutions (and their adjoints), and the amplitude of finite-time
instabilities need not be bounded by properties of the dominant eigenmodes
alone.
Fig 2 illustrates schematically the crucial difference between eigenmode
and singular vector growth, and the relationship between singular vectors
and adjoint modes. An idealised 2-D system has two very non-orthogonal
decaying eigenmodes {I and {2· We take {I to have the larger real eigenvalue component. The adjoint eigenmodes '1/1 and '1/2 are shown with '1/1,'1/2
orthogonal to {2,{1 respectively (according to the biorthogonality condition
2.10). A normalised vector Va is shown parallel to '1/1. Its time evolution
can be estimated by mapping the tip and tail of Va along the {I and {2
directions (shown as dashed lines) using the modal decay rates. The sequence of vectors Vn , n = 1,2, ... giving the time evolution of Va increases
in amplitude up to some finite n = N and is aligned almost entirely with
{I for large n. The projection of Vn onto {I for large n is much larger
than that associated with the evolution of a second normalised vector J-ln
which is initially aligned along {I. The sequence V n , n = 1,2, ... describes
singular vector growth over a long time interval. The transient growth of
the singular vectors in such systems was first noted by Orr (1907), and a
review of this process in plane parallel shear flow (and its relationship to
wave overreflection) is discussed in Lindzen (1988).