## 1. Introduction

The use of tangent linear (TL) and, in particular, adjoint models has been very useful in several applications in numerical weather prediction (NWP) (e.g. Errico, 1997, 2003; Errico and Ehrendorfer, 2007, give an overview). For example, at the European Centre for Medium-range Weather Forecasts (ECMWF) these linear models play a crucial role in the computation of initial condition perturbations used in the ensemble prediction system (Leutbecher and Palmer, 2008) and in their four-dimensional data assimilation system (4D-Var; Courtier *et al.*, 1994). One of the major limitations to the application of linear models is that the results are useful only when the linear approximation is valid (Errico, 1997). By this we mean that the difference between two runs of the nonlinear model can be described by the associated linearized version of the nonlinear model. To achieve this, great effort is taken to develop linearized models which capture as many features as possible of the full nonlinear model (Janisková *et al.*, 1999). Despite these efforts, the use of TL and adjoint models is restricted to ‘short’ time spans. The time span for which the TL model can be considered accurate will be referred to as the TL regime.

The duration of the TL regime depends on many factors. Typically the difference between two nonlinear forecasts is compared with the linear forecast by a scalar index, and it is said that the TL assumption is violated when the index has reached a threshold value. So the measure which is employed to compare forecast fields is already important in the definition of the TL regime. But also the size of the initial condition perturbation, the orientation of the perturbation, the background trajectory around which the TL model is linearized and the physical processes taken into account in the TL model all play a role. Another issue which influences the usefulness of linear models is whether we are considering forecast problems, where error growth is determined by the singular value spectrum of the propagator, or estimation problems that are typically characterized by the reciprocal of the singular value spectrum. In general, the spectrum of reciprocal of the singular values attains higher values (Reynolds and Palmer, 1998) and therefore the usefulness of the TL model in estimation problems is shorter. This effect becomes even more pronounced by the fact that the typical size of perturbations used in backward integrations is larger than in forward mode.

In this forward mode, the TL assumption is generally believed to be valid for 2–3 days at the synoptic scale. However, Gilmour *et al.* (2001) argue that 1 day is perhaps a better estimate. On the cloud-resolving scale, the TL assumption probably holds for much shorter time periods on the order of 1.5 h (Hohenegger and Schär, 2007). As models reach higher resolutions, the validity of the TL assumption is therefore a major concern. We will show that for bilinear systems the usefulness of the TL model can be greatly extended by modifying the linearization trajectory and therefore one of the major limitations on using TL models can be eliminated.

In section 2, the definition of bilinear differential equations is given. In section 3, we show that for bilinear systems there is an optimal linearization trajectory such that, if the TL model is linearized around this trajectory, the perturbation growth in the TL model is equal to the nonlinear perturbation growth. Knowing that such a trajectory exists, we show in section 4 that there is an iterated map based purely on TL integrations that converges to this linearization trajectory. In section 5, we show how the iterative method can be used in forecast sensitivity experiments using the inverse of the TL model. In section 6, the experimental results using a quasi-geostophic (QG) model (Marshall and Molteni, 1993, described in Appendix A) and the Lorenz 96 model (Lorenz, 1996, described in Appendix B) are given. In the discussion in section 7, the prospects for using the method in realistic NWP models and a method to regularize the error growth in the TL model are discussed. The conclusions are given in section 8.

To keep the notation simple, we use the convention that lower-case variables are perturbations (also referred to as increments) to upper-case variable, e.g. is a perturbation to the state vector .