91
2.3 Correspondence with 'normal' mode instability
Singular vector analysis is in some sense a generalisation of classical normal
mode instability analysis. This can be made explicit by linearising about
a stationary solution of (2.1), so that normalised eigenvectors !;.i of Ml
with eigenvalues /-li give rise to modal solutions !;.ieJti(t-tO) of (2.2). The
integral operator L(t, to) can be written as e(t-tO)MI, with eigenvectors!;.i
and eigenvalues e(t-tO)Jti.
For application to atmosphere-ocean dynamics, the linear evolution operators associated with realistic basic state flows are never normal (ie
L *L =f. LL *) because of vertical and horizontal shear (eg Farrell and Ioannou, 1996). Now, it is common meteorological parlance to call any modal
eigenvectors as 'normal' modes. However, for any meteorological basic
state these eigenvectors are not eigenvectors of a normal operator (and
hence not normal). In future we refer to such eigenvectors as 'eigenmodes'
(hence the quotation marks in the title of this sub-section). However, irrespective of normality, eigenvectors "'i and eigenvalues ()i of the adjoint
operator L * satisfy the biorthogonality condition
(2.10)
where 'cc' denotes complex conjugate. This condition ensures that the
eigenvalues of an eigenvector/adjoint eigenvector pair that are not orthogonal, must form a complex conjugate pair. The magnitude of the inner
product ("'i; !;.i) for such eigenvector pairs equals the cosine of the angle,
ai, they subtend in phase space.
If an initial disturbance comprises a linear combination of the eigenmodes !;.i so that
x(t) = L Ci!;.ieJti(t-t O )
i
then from the biorthogonality condition (2.10)
(2.11)
(2.12)
From (2.11), the fastest growing eigenmode will ultimately dominate the
linear combination. Hence for sufficiently long optimisation times, the
dominant singular vector at optimisation time will correspond to the most
unstable eigenmode. (Since the singular values are real, whilst the eigen-
2.3 Correspondence with 'normal' mode instability
Singular vector analysis is in some sense a generalisation of classical normal
mode instability analysis. This can be made explicit by linearising about
a stationary solution of (2.1), so that normalised eigenvectors !;.i of Ml
with eigenvalues /-li give rise to modal solutions !;.ieJti(t-tO) of (2.2). The
integral operator L(t, to) can be written as e(t-tO)MI, with eigenvectors!;.i
and eigenvalues e(t-tO)Jti.
For application to atmosphere-ocean dynamics, the linear evolution operators associated with realistic basic state flows are never normal (ie
L *L =f. LL *) because of vertical and horizontal shear (eg Farrell and Ioannou, 1996). Now, it is common meteorological parlance to call any modal
eigenvectors as 'normal' modes. However, for any meteorological basic
state these eigenvectors are not eigenvectors of a normal operator (and
hence not normal). In future we refer to such eigenvectors as 'eigenmodes'
(hence the quotation marks in the title of this sub-section). However, irrespective of normality, eigenvectors "'i and eigenvalues ()i of the adjoint
operator L * satisfy the biorthogonality condition
(2.10)
where 'cc' denotes complex conjugate. This condition ensures that the
eigenvalues of an eigenvector/adjoint eigenvector pair that are not orthogonal, must form a complex conjugate pair. The magnitude of the inner
product ("'i; !;.i) for such eigenvector pairs equals the cosine of the angle,
ai, they subtend in phase space.
If an initial disturbance comprises a linear combination of the eigenmodes !;.i so that
x(t) = L Ci!;.ieJti(t-t O )
i
then from the biorthogonality condition (2.10)
(2.11)
(2.12)
From (2.11), the fastest growing eigenmode will ultimately dominate the
linear combination. Hence for sufficiently long optimisation times, the
dominant singular vector at optimisation time will correspond to the most
unstable eigenmode. (Since the singular values are real, whilst the eigen-