## 1. Introduction

In numerical weather prediction and atmospheric modelling in general, the question of predictability has received considerable attention since the work of Thompson (1957) and Lorenz (1963, 1969). The investigation of the potential for growth of small errors superimposed on certain atmospheric flows is the prime subject of atmospheric predictability studies (Zhang *et**al.*, 2003; Tan *et al.*, 2004), as well as studies of atmospheric instability (Molteni and Palmer, 1993). If a system is particularly unstable, small errors become amplified in model forecasts and severely limit the accuracy of the prediction of the flow. Mu *et al.* (2004) considered the term ‘predictability’ as the analysis of factors and mechanisms that yield the uncertainties of forecast results, and exploration of methods and approaches to reduce these uncertainties.

Predictability studies can be performed by the impact method or optimal perturbation method. In the first method, the value of an input parameter is changed by a small amount, and a new model solution is evaluated for this new parameter value and then compared to the control solution (Thompson, 1957; Zhang *et al.*, 2002, 2003). This method is applied successfully in studies for which the number of input parameters is small.

The linear singular vector (LSV) is one of the classic optimal perturbation methods whose evolutions are well described by the tangent linear version of the nonlinear model (Lorenz, 1965; Molteni and Palmer, 1993; Ehrendorfer and Errico, 1995; Ehrendorfer *et al.*, 1999; Frederiksen, 2000). This method is based on the assumption of the validity of the tangent linear model. However, Zhang *et al.* (2002) found that error growth rates at small scales depend on the difference amplitude, suggesting that nonlinearity is important. Errico *et al.* (2002) declared that the evolution of the perturbations with structures consistent in size and shape with initial condition uncertainty can be substantially nonlinear, even after only a single day's growth. Adjoint technique fails to reproduce fully the evolution of forecast differences from the convective scale to larger scale that is present in their experiments (Zhang *et al.*, 2003). The maximum possible error growth over a finite time interval in which fully nonlinear error evolution is considered may be assessed by solving a nonlinear optimization problem (Ehrendorfer *et al.*, 1999).

Mu and Duan (2003) proposed a new concept called conditional nonlinear optimal perturbation (CNOP), which can be obtained numerically by solving a nonlinear optimization problem. Mu and Zhang (2006) pointed out that, when the initial perturbation is small enough and the evolution time is not too long, CNOPs are very similar to LSVs. On the other hand, when the tangent linear model fails to approximate the original nonlinear model because of the effects of nonlinearity, there exists a remarkable difference between CNOPs and LSVs. Jiang *et al.* (2008) revealed that CNOPs, similar to LSVs, have the property of norm dependence. Afterwards, Rivière *et al.* (2008) explored the behaviour of CNOPs in a baroclinic unstable flow, and pointed out that CNOPs essentially differ from LSVs in the presence of a positive zonal-mean shear at initial time and in a broader meridional extension. Furthermore, Rivière *et al.* (2009) described a new nonlinear technique to compute the sensitivity of synoptic perturbation growth to environmental moisture, which considered the impact of moisture on the growth of CNOPs. Till now, the CNOP method has been widely used in many atmospheric and oceanic research fields (Duan *et al.*, 2004; Mu and Jiang, 2008; Mu *et al.*, 2009).

As a midlatitude synoptic system, cut-off low-pressure is frequently observed on the south side of the west-wind belt in the Northern Hemisphere. Hsieh (1949) pointed out that the conditions for the formation of such a cold vortex over North America were found to be a rapid intrusion of cold air into the upper west-wind belt. Chen and Chou (1994) conducted an investigation of 60 cold vortices over the western North Pacific during the warm seasons of 1982–1987. One of the interesting findings of that study was the existence of a northwest-type and/or a south-type jet streak for the majority of cold lows. The northwest-type jet mainly formed in the northwest sector of the vortex and then propagated downstream and dissipated in its southwest sector, whereas the south-type jet usually formed in the southwest and southeast of the vortex and then propagated downstream and dissipated in its northeast or northwest sector. Zhao and Sun (2007) investigated cut-off low-pressure systems during the period June to August 1998 leading up to the record flood in Northeast Asia and pointed out that baroclinicity played an important role in the initiation of cut-off low-pressure systems. Hu *et al.* (2010) investigated the seasonal climatology of cut-off low-pressure systems and associated precipitation patterns over northeast China during 1979–2005. It seems that most work on cut-off low-pressure systems focused on the climatological and synoptic characteristics of these systems (Zheng, 1992; Sun *et**al.*, 2002; Wu *et al.*, 2010). However, the forecasting of the formation of cold vortices and their associated precipitation is still an important and difficult issue for operational forecasting researchers. On the basis of these considerations, it is necessary to explore the impact of initial errors on the evolution of cut-off low-pressure systems over northeast China so as to better understand the forecast error sources and the related initial error evolution mechanisms.

In this article, an optimization problem related to the maximum uncertainties caused by initial errors during the formation of a cold vortex is formulated and a CNOP method is adopted to explore the above problem. The organization of this article is as follows. In section 2, the method of conditional nonlinear optimal perturbation is introduced. The dry total energy and moisture energy norms are illustrated. Section 3 contains a brief description of the model used and the cold vortices studied, one 2006 case and one 2007 case. Linear singular vector and conditional nonlinear optimal perturbation are compared in section 4. Furthermore, the properties of CNOP are illustrated. Section 5 explores the linkage of CNOP signals with the background field characteristics. The article is concluded with a summary and their possible implications on mesoscale predictability and targeting observations.