## 1. Introduction

Given a model in the form of an ordinary differential equation and measurements at discrete instances in time, data assimilation attempts to find the best possible approximation to the true dynamics of the physical system under consideration (Evensen, 2006). Data assimilation by sequential filtering techniques achieves such an approximation by discontinuous-in-time adjustments of the model dynamics due to available measurements. While optimality of the induced continuous–discrete filtering process can be shown for linear systems, discontinuities can lead to unphysical readjustment processes under the model dynamics for nonlinear systems and under imperfect knowledge of the error probability distributions: see, for example, Houtekamer and Mitchell (2005) and Kepert (2009) in the context of operational models as well as Neef *et al.* (2006) and Oke *et al.* (2007) for a study of this phenomenon under simple model problems. These observations have led to the consideration of data assimilation systems that seek to incorporate data in a more ‘continuous’ manner. In this article we focus on a novel continuous data assimilation procedure based on ensemble Kalman filters. In contrast to the widely used incremental analysis updates (IAU) of Bloom *et al.* (1996), which first perform a complete analysis step then distribute the increments evenly over a given time window, our approach performs the analysis step itself over a fixed window. Our novel filter technique is based on the continuous formulation of the Kalman analysis step in terms of ensemble members (Bergemann and Reich, 2010) and mollification of the Dirac delta function by smooth approximations (Friedrichs, 1944). The proposed mollified ensemble Kalman (MEnK) filter is described in section 2. The MEnK filter may be viewed as a ‘sophisticated’ form of nudging (Hoke and Anthes, 1976; Macpherson, 1991) with the nudging coefficients being obtained from a Kalman analysis perspective instead of heuristic tuning. However, for certain multiscale systems, nudging with prescribed nudging coefficients might still be advantageous (Ballabrera-Poy *et al.*, 2009). We also point to the closely related work by Lei and Stauffer (2009), which proposes a nudging-type implementation of the ensemble Kalman filter with perturbed observations (Burgers *et al.*, 1998).

To demonstrate the properties of the new MEnK filter, we propose a slow–fast extension of the popular Lorenz-96 model (Lorenz, 1996) in section 4. In contrast to other multiscale extensions of the Lorenz-96 model, the fast dynamics is entirely conservative in our model and encodes a dynamic balance relation similar to geostrophic balance in primitive equation models. Our model is designed to show the generation of unbalanced fast oscillations through standard sequential ensemble Kalman filter implementations under imperfect knowledge of the error probability distributions. Imperfect knowledge of the error probability distribution can arise for various reasons. In this article, we address in particular the aspect of small ensemble size and covariance localization (Houtekamer and Mitchell, 2001; Hamill *et al.*, 2001). It is also demonstrated that both IAU and the newly proposed MEnK filter maintain balance under assimilation with small ensembles and covariance localization. However, IAU develops an instability in our model system over longer assimilation cycles, which requires a relatively large amount of artificial damping in the fast dynamics. We also note that the MEnK filter is cheaper to implement than IAU, since no complete assimilation cycle and repeated model integration need to be performed. It appears that the MEnK filter could provide a useful alternative to the widely employed combination of data assimilation and subsequent ensemble re-initialization. See, for example, Houtekamer and Mitchell (2005).