This paper describes the ROMS model, especially the tools based on the tangent linear (Eq. 6 in Errico 1997) and adjoint (Eq. 9 in Errico 1997) versions of the model and the new types of analysis that can be performed.
“For the discrete dynamical systems described by numerical models, the tangent linear model provides the Jacobian of the dynamical operators that are tangent linear to a solution trajectory of the nonlinear system. The adjoint of this Jacobian operator provides information about the sensitivity of the system to variations in the model state vector, boundary conditions or model parameters. “
Use in oceanography: see Junge and Haine (2001), Galanti and Tziperman (2003) and Galanti et al. (2003).
“Tangent linear and adjoint models can also be used to explore the stability of dynamical systems, and to understand the dynamics of perturbation growth. The traditional approach to stability analysis involves computing the eigenvalues and eigenvectors of the tangent linear model (Pedlosky, 1979, his chapter 7). These eigenvectors represent dominant patterns of variability that might be expected if the associated eigenvalues have a positive real part. Such eigenvector (normal mode) analyses have been used in meteorology to understand various aspects of the large scale circulation (e.g. Charney, 1947; Eady, 1949; Simmons and Hoskins, 1976; Frederiksen, 1982). In general, the eigenvectors of observed and modelled circulation fields do not form an orthonormal basis, in which case the eigenvectors of the associated adjoint model are also of interest since they represent the optimal excitations for the corresponding eigenvectors of the tangent linear model (Branstator, 1985). In general these two sets of eigenvectors have very different structures. Thus the adjoint eigenvectors indicate how a particular eigenvector of the tangent linear system can be forced or excited (Frederiksen, 1997).”
See Moore and Farrell (1993; Gulf Stream), Moore and Mariano (1999; Gulf Stream) and Moore et al. (2002; wind-driven ocean circulation) for an application of the singular vectors (that give the fastest growing of all possible perturbations): “while the AFTEs are optimal for exciting the corresponding FTE, they are not the fastest growing of all possible perturbations. This latter class of perturbations are the singular vectors (SVs) of the tangent linear propagator” (p. 242).
About non-normality of systems (p. 241):
“[T]he FTEs and their optimal excitations, the AFTEs, have very different spatial structures. This is an important feature of nonnormal (and non-self-adjoint) systems. [...] It is the inhomogeneities in the basic state circulation that render the system non-normal, and in fact, except under very special circumstances, non-autonomous systems will always be non-normal (Farrell and Ioannou, 1999). Following Farrell and Ioannou (1999) the degree of non-normality of the tangent linear system can be quantified by considering the dot product of the FTEs and AFTEs.”