# ## 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).