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Examples: Determine the fastest growing perturbations of a coupled ocean
atmosphere model

In climate research, numerical simulation models are used to estimate the
future state of the climate system. These models solve the system's dynamical
equations, using the initial state of the system as a starting point. Any
estimate of this initial state, however, is subject to errors. Amplification
of these errors is one of the factors that limits the predictability of the
future state of the system.

In this context, for a given estimate of the initial state, it is interesting
to determine the 'patterns' of potential errors that amplify fastest because
they can do the maximum damage to a prediction of the model's future state.
Knowing these patterns (and the rates at which they amplify), the estimate
of the initial state (and thus the predictability of the system) can be
systematically improved by collecting additional observations that optimally
constrain these patterns.

Representing the linearisation of the model (tangent linear model) by
an operator *M *that transforms the initial state into the predicted
state for a future time, one can show that the perturbations that grow fastest
are the singular vectors corresponding to the largest singular values of
the matrix *M*M *(*M* *denotes the adjoint of *M,
*defined
for or a reasonable inner product). For state of the art climate models
these matrices have huge dimensions. Fortunately, to determine only the
singular vectors belonging to the leading singular values, there are algorithms
that avoid a computation of the matrix. Instead, these algorithms work by
computing the product of *M*M* and selected
vectors a number of times. These products can, however, be evaluated efficiently provided
the tangent linear and adjoint of the model are available. In this case
one runs the tangent linear model first and subsequently uses its output
as input to the adjoint model.

With Ralf's advice and the use of the Tangent linear and Adjoint Model Compiler (TAMC), Christian Eckert generated the tangent linear model and the adjoint model
for a climate model (Hybrid Coupled Model) used for simulation of the
El
Niño/ Southern Oscillation (ENSO) phenomenon. Christian then applied these models to study the limits of predictability
of ENSO (see Eckert 1998).