## 1. Introduction

To account for uncertainties in both the initial conditions and forecast model and to combat the associated forecast error and flow-dependent predictability, ensemble methods have become popular for producing numerical weather forecasts (Molteni and Palmer, 1993; Toth and Kalnay, 1993) and for performing data assimilation (Evensen, 1994). The idea behind ensemble-based methods is that the nonlinear chaotic dynamics of the underlying forecast model and the associated sensitivity to initial conditions cause an ensemble of trajectories to explore sufficiently large parts of the phase space in order to deduce meaningful statistical properties of the dynamics. However, all currently operational ensemble systems are underdispersive, meaning that the true atmospheric state is on average outside the statistically expected range of the forecast or analysis (e.g. Buizza *et al.*, 2005; Hamill and Whitaker, 2011). This detrimental underdispersiveness can be linked to two separate sorts of model error: a *dynamical* model error due to a misrepresentation of the physics of the forecast model and a *numerical* model error due to the choice of the numerical method used to simulate those forecast models.

It has been argued (Palmer, 2001) that dynamical model error is largely due to a misrepresentation of unresolved subgrid-scale processes. The very active field of stochastic parametrization is aimed at reintroducing statistical fluctuations of the unresolved degrees of freedom and a direct coupling between the resolved and unresolved degrees of freedom into the equations of motion (Palmer and Williams, 2010, provide a recent review on current trends). Indeed, using a low-dimensional toy model, Mitchell and Gottwald (2012) showed that stochastic parametrizations can lead to superior skill performance in ensemble data assimilation schemes in situations where the dynamics involve rapid large-amplitude excursions such as in regime switches or in intermittent dynamics.

Numerical model error is often introduced deliberately to control numerical instabilities arising in the simulation of geophysical fluids, such as in numerical weather prediction or climate dynamics. In such simulations, numerical codes tend to produce large errors linked to numerical instabilities at the grid resolution level, which limits the reliability of the forecast over the forecast window. In a data assimilation context, this numerical ‘noise’ can lead to an unrealistic overestimation of forecast-error covariances at small scales. The standard approach to dealing with these numerical instabilities is to add various forms of artificial dissipation to the model (Durran, 1999). However this has several well-documented drawbacks. In the context of ensemble data assimilation, artificial viscosity diminishes the spread of the forecast ensemble, causing detrimental underestimation of the error covariances (Burgers *et al.*, 1998; Houtekamer and Mitchell, 1998; Anderson and Anderson, 1999; Houtekamer *et al.*, 2005; Charron *et al.*, 2010). Most notably, artificial dissipation implies unrealistic and excessive drainage of energy out of the system (Shutts, 2005), effecting processes ranging from frontogenesis to large-scale energy budgets. For example, Palmer (2001) reports on the unphysical dissipation of kinetic turbulent energy caused by mountain gravity-wave drag parametrizations. Skamarock and Klemp (1992) introduce divergence damping to control unwanted gravity wave activity in hydrostatic and non-hydrostatic elastic equations to stabilize the numerical scheme, leading to numerical energy dissipation (also Durran, 1999). Blumen (1990) controls the collapse of near-surface frontal width by artificially introducing momentum diffusion in a semi-geostrophic model of frontogenesis, with the effect of producing unrealistic energy dissipation rates. In semi-Lagrangian advection schemes, Côté and Staniforth (1988) and Ritchie (1988) find unrealistic energy dissipation rates that are caused by interpolation. Besides affecting the small-scale dynamics, numerical dissipation also has a major effect on the large scales and their energy spectra (Palmer, 2001; Shutts, 2005). In ideal fluids, which often serve as a good approximation of the large-scale dynamics (Salmon, 1998), numerical dissipation destroys dynamically important conservation laws which may be undesirable in long time simulations such as in climate modelling (Thuburn, 2008). Using statistical mechanics, Dubinkina and Frank (2007), Dubinkina and Frank (2010) show how the choice of the conservation laws respected by a numerical scheme affects the statistics of the large-scale fields. There exists a plethora of methods to remedy the unphysical energy loss due to artificial diffusion and reinject energy back into the numerical model, for example via energy backscattering (Frederiksen and Davies, 1997; Shutts, 2005) or systematic stochastic model reductions (Majda *et al.*, 1999).

Here we will look at these issues in the context of ensemble data assimilation. Rather than controlling numerical instabilities of the dynamic core via appropriate discretization schemes of the numerical forecast model, we will modifiy the actual ensemble data assimilation procedure, avoiding excessive numerical dissipation. Underdamping in dynamic cores causes the ensemble to acquire unrealistically large spread. This overdispersion is exacerbated in sparse observational networks where, for the unobserved variables, unrealistically large forecast covariances are poorly controlled by observational data (Liu *et al.*, 2008; Whitaker *et al.*, 2009). The questions we address in this article are: (i) Can one use numerical forecasting models which are not artificially damped, yet still control the resulting overestimation of forecast covariances within the data assimilation procedure? (ii) Is it possible to use the increased spread induced by numerical instabilities in a controlled fashion to improve the skill of ensemble data assimilation schemes?

In a recent article, Gottwald *et al.* (2011) proposed a variance-limiting ensemble Kalman filter (VLKF) which controls overestimation in sparse observational networks by including climatological information of the unobserved variables. In a perfect model data assimilation scheme, unrealistic overestimation of the error covariances occurs in sparse observational networks as a finite size effect; it was shown that, for short observational intervals of up to 10 h, the VLKF produces superior skill when compared to the standard ensemble transform Kalman filter (ETKF) (Bishop *et al.*, 2001; Tippett *et al.*, 2003; Wang *et al.*, 2004). Here we will apply the VLKF in the case of an imperfect, underdamped forecast model, and for large observation intervals. Unlike in the perfect model case where for large observation intervals the forecast-error covariances approach the climatic variance, when forecasting with underdamped forecast models the error covariances will be larger than the climatic variance. This will be controlled by the VLKF, invoking a weak constraint onto the mean and the variance of the unobserved variables, leading to an improvement in the analysis skill. We will study this effect using the Lorenz-96 model (Lorenz, 1996), where we assume that the truth is evolving to the standard set of parameters. To investigate whether underdamped models can be used as forecast models, we then use forecast models with smaller values of the linear damping term of the Lorenz-96 system to generate ensembles with forecast spreads exceeding the true climatology.

Our main result is that VLKF produces significantly better skill than ETKF, increasing with the strength of the model error, the sparsity of the observational network and the observational noise covariance. We will establish that a variance-limiting weak constraint may be used as an adaptive alternative to artificial viscosity damping, counteracting model error ‘on the fly’ when underdamping of the numerical forecast model causes an unrealistic overestimation of the forecast ensemble. However, we will see that there is a trade-off since for large model error the skill of VLKF becomes comparable to the skill of an analysis involving no forecast but using an analysis comprised of the observations and the climatological mean for the observed and unobserved variables, respectively. It will be shown that this is important for large observation times when the analysis is not tracking anymore due either to model error or to the chaotic nature of the underlying dynamics, and one can only improve the analysis, assuring a reliable reproduction of statistical properties of the dynamics.

In the next section we briefly describe the VLKF. In section 3, we introduce the Lorenz-96 model and describe the underdamped forecast model used. In section 4, we present results showing under what conditions the VLKF produces better skill than the usual ETKF. We conclude with a discussion and outlook in section 5.