Balance of the Background-Error Variances in the Ensemble Assimilation System DART/CAM

N. Žagar University of Ljubljana and Center of Excellence SPACE-SI, Ljubljana, Slovenia

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J. Tribbia National Center for Atmospheric Research, Boulder, Colorado

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J. L. Anderson National Center for Atmospheric Research, Boulder, Colorado

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K. Raeder National Center for Atmospheric Research, Boulder, Colorado

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Abstract

This paper quantifies the linear mass–wind field balance and its temporal variability in the global data assimilation system Data Assimilation Research Testbed/Community Atmosphere Model (DART/CAM), which is based on the ensemble adjustment Kalman filter. The part of the model state that projects onto quasigeostrophic modes represents the balanced state. The unbalanced part corresponds to inertio-gravity (IG) motions. The 80-member ensemble is diagnosed by using the normal-mode function expansion. It was found that the balanced variance in the prior ensemble is on average about 90% of the total variance and about 80% of the wave variance. Balance depends on the scale and the largest zonal scales are best balanced. For zonal wavenumbers greater than k = 30 the balanced variance stays at about the 45% level. There is more variance in the westward- than in the eastward-propagating IG modes; the difference is about 2% of the total wave variance and it is associated with the covariance inflation. The applied inflation field has a major impact on the structure of the prior variance field and its reduction by the assimilation step. The shape of the inflation field mimics the global radiosonde observation network (k = 2), which is associated with the minimum variance reduction in k = 2. Temporal variability of the ensemble variance is significant and appears to be associated with changes in the energy of the flow. A perfect-model assimilation experiment supports the findings from the real-observation experiment.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: N. Žagar, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia. E-mail: nedjeljka.zagar@fmf.uni-lj.si

Abstract

This paper quantifies the linear mass–wind field balance and its temporal variability in the global data assimilation system Data Assimilation Research Testbed/Community Atmosphere Model (DART/CAM), which is based on the ensemble adjustment Kalman filter. The part of the model state that projects onto quasigeostrophic modes represents the balanced state. The unbalanced part corresponds to inertio-gravity (IG) motions. The 80-member ensemble is diagnosed by using the normal-mode function expansion. It was found that the balanced variance in the prior ensemble is on average about 90% of the total variance and about 80% of the wave variance. Balance depends on the scale and the largest zonal scales are best balanced. For zonal wavenumbers greater than k = 30 the balanced variance stays at about the 45% level. There is more variance in the westward- than in the eastward-propagating IG modes; the difference is about 2% of the total wave variance and it is associated with the covariance inflation. The applied inflation field has a major impact on the structure of the prior variance field and its reduction by the assimilation step. The shape of the inflation field mimics the global radiosonde observation network (k = 2), which is associated with the minimum variance reduction in k = 2. Temporal variability of the ensemble variance is significant and appears to be associated with changes in the energy of the flow. A perfect-model assimilation experiment supports the findings from the real-observation experiment.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: N. Žagar, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia. E-mail: nedjeljka.zagar@fmf.uni-lj.si

1. Introduction

In the ensemble Kalman filter (EnKF; Houtekamer and Mitchell 2005, and references therein) the background or forecast-error covariance is approximated by the sample covariance computed from an ensemble of prior forecasts. The size of the ensemble feasible for numerical weather prediction (NWP) purposes is O(102), which is tiny compared to the dimension of the state vector, O(107). Fortunately, most of the atmospheric variance at large scales occupies a model subspace with a much smaller dimension. Provided the ensemble samples that subspace, the EnKF includes information about the flow dependency of background-error covariances (e.g., Buehner 2005). Indeed, various applications of the EnKF have been demonstrated to provide competitive analyses with respect to variational methods, which are based on a stationary background-error covariance matrix (e.g., Kalnay et al. 2007).

As a consequence of the small ensemble size, implementations of the EnKF for NWP purposes require dealing with spurious long-range spatial correlations and insufficient ensemble spread. The former has been addressed by so-called localization (Gaspari and Cohn 1999; Houtekamer and Mitchell 2001; Hamill et al. 2001; Anderson 2007b). An unwanted side effect of localization is imbalance (e.g., Mitchell et al. 2002; Kepert 2009). An additional cause of too little ensemble spread is model error (e.g., Dee 1995; Houtekamer and Mitchell 2005; Houtekamer et al. 2009). Studies of model errors are usually based on the hypothesis that model error is due to some imperfect aspect of the forecast model (Houtekamer et al. 2009). In general, forecast models describe small-scale unbalanced motions less accurately than large scales. However, larger relative errors in small scales are compensated by their smaller energy so that global model errors are dominated by large scales. In the EnKF system Data Assimilation Research Testbed/Community Atmosphere Model (DART/CAM; Anderson et al. 2009), which is used in this study, model error is accounted for by covariance inflation, which increases the ensemble spread by multiplying the ensemble by spatially varying factors (Anderson 2007a).

Early analysis systems suffered from imbalances (noise) introduced by deficiencies of the assimilation methodology and the models. Detrimental impacts of imbalances on subsequent forecasts were handled by the initialization step; for example, nonlinear normal-mode initialization has been widely applied (e.g., Baer and Tribbia 1977; Wergen 1988). Present four-dimensional variational data assimilation (4DVAR) methods employed at major operational centers produce analyses in which the impact of initialization is nearly negligible (e.g., Gauthier and Thépaut 2001). At the same time, present-day NWP analyses have reached resolutions at which inertio-gravity (IG) waves over various scales (tropics, mesoscale, and middle atmosphere) are well represented (e.g., Shutts and Vosper 2011). Žagar et al. (2009a, 2010b) quantified the magnitude of large-scale IG energy in state-of-the-art analysis systems to be about 10% of the global wave energetics. Williams et al. (2008) reported a similar percentage based on laboratory observations of IG waves emitted from balanced flow.

IG waves are eastward- and westward-propagating solutions of the linearized primitive equations on the sphere (Kasahara 1976). Although they are represented predominantly by the velocity potential, some large-scale equatorial IG modes (e.g., the Kelvin wave) have a significant rotational component. The second kind of solutions is the westward-propagating waves represented primarily by the streamfunction [Rossby type (ROT)]. Because of their quasigeostrophic nature, ROT modes are often referred to as balanced motions in contrast to the IG waves which are referred to as unbalanced. The same terminology is applied in the present study, which quantifies balanced (ROT) and unbalanced (IG) motions in DART (Anderson et al. 2009), an EnKF assimilation system developed at the National Center for Atmospheric Research (NCAR).

The balance is studied by using a normal-mode function (NMF) expansion derived by Kasahara and Puri (1981). Quantification of the ensemble variance associated with the various modes is obtained by projecting model states onto a predefined set of three-dimensional orthogonal ROT and IG modes. Vertical and horizontal scales and mass and wind fields are analyzed together. The part of the model state that projects onto IG modes represents the unbalanced state. Three-dimensional orthogonality allows quantification of variance in each mode.

Our study of the ensemble balance is different from studies by Mitchell et al. (2002) and Houtekamer et al. (2009), which measured imbalance due to covariance localization by comparing digitally filtered and unfiltered 6-h forecasts states and by the ensemble mean square value of the second derivative of the surface pressure field, respectively. In contrast, IG motions of our concern span a range of scales and frequencies including slow large-scale tropical waves. Although some important components of tropical circulation (e.g., the Hadley cell) are not IG waves in any normal sense, but a balanced response to heating, that is, they project onto IG modes. Tropical IG modes (e.g., the Kelvin wave) represented by the forecast model used in this study have a significant balanced component; we refer to IG modes as unbalanced to make a distinction between them and Rossby modes as defined above.

The paper concentrates on the spatiotemporal characteristics of the ensemble variance fields. The questions addressed are the following:

  • How is the variance of analysis and short-range forecast ensembles divided between the balanced and unbalanced motions?

  • How large is the temporal variability of ensemble variances?

  • Are the observations more effective in reducing variance in certain scales and motion types?

  • How can the NMF representation assist in understanding the observational and modeling aspects that contribute to the ensemble spread?

The study focuses on weather forecasting application of the EnKF. Unlike operational NWP ensemble systems, our assimilation experiments start from a climatological ensemble and we assimilate only a subset of the observations used in NWP. Implications of the results are discussed in detail in order to highlight aspects of results relevant for NWP applications of the EnKF and balance issues in general. Furthermore, the setup of experiments also allows some discussion of model-error issues relevant for long-range forecasting.

The paper is divided into six sections. Basic properties of the DART ensemble assimilation system and the application of NMFs are presented in section 2. Sections 34 present results. Additional discussion is provided in section 5, and the main conclusions are stated in section 6.

2. The experimental environment and diagnostics

a. The DART/CAM system

The DART system (Anderson et al. 2009) was used to apply the ensemble adjustment Kalman filter to combine observations with an ensemble of short-range forecasts from the global atmospheric model to produce an ensemble of analyses. For details of the ensemble adjustment Kalman filter, the reader is referred to Anderson (2001) and references in Anderson et al. (2009). As in the latter paper, the ensemble of short-range forecasts is denoted as the “prior ensemble” while the ensemble of analyses is called the “posterior ensemble.”

The forecast model is the CAM, which is an atmospheric component of the Community Climate System Model (CCSM) of NCAR (Collins et al. 2006). The CAM model uses the hybrid σ–pressure vertical coordinate with the top model level located near 3.7 hPa. A 26-level version of the CAM model is used here, with 8 levels above 100 hPa and 3 levels below 900 hPa. Model dynamics is solved in spectral space with a T85 truncation. The model version used here (version 3.1) has an improved physics package with respect to the version documented in Hurrell et al. (2006). Time integration is Eulerian and the step is 10 min. At the surface, CAM is coupled to a land surface model and to interpolated, monthly mean, observed, sea surface temperature fields produced by the National Centers for Environmental Prediction (NCEP).

Like other implementations of the ensemble Kalman filter, the ensemble adjustment Kalman filter applies covariance localization to filter noisy background-error covariances associated with remote observations; the method proposed by Gaspari and Cohn (1999) is implemented as a three-dimensional ellipsoid. The ensemble spread is increased by covariance inflation (Anderson 2007a). In this study we use a time-constant, spatially varying covariance inflation applied to the posterior ensemble. The inflation fields were produced by assimilating the same observations into the same model using a temporally adaptive inflation scheme (Anderson 2009) until initial transients had been damped sufficiently. Then the fields of inflation values were stored for repeated use in the experiments described below. The inflated fields are obtained by the following formula: , where i represents the ensemble member, j is the state vector component, the overbar stands for the ensemble mean, and λ is a covariance inflation factor (Anderson 2007a). The variance about the mean of each element of the state vector xj is increased by λ while correlations between any pairs of elements remain unchanged. With these inflation factors, the posterior ensemble spread for an observation should be consistent with the expected difference between the posterior ensemble mean and the actual observed value. Figure 1 shows λ for zonal wind at a model level located in the upper troposphere where inflation over North America is largest. This figure reflects the global availability of the radiosondes and aircraft observations and it projects primarily onto zonal wavenumber 2, which is relevant for the subsequent discussion.

Fig. 1.
Fig. 1.

Zonal wind inflation factor at level 14 (~227 hPa). Isolines every 1.5, starting from 2. See text for details.

Citation: Monthly Weather Review 139, 7; 10.1175/2011MWR3477.1

1) The observational network

Every day, observations are assimilated at 0000, 0600, 1200, and 1800 UTC. In the experiments presented in the paper, observations used include radiosondes, aircraft measurements, and satellite cloud motion wind vectors. No satellite radiances, no moisture observations, and no surface measurements are assimilated. Thus, the observation network is relatively sparse compared to that used by operational systems, especially over the oceans, in the Southern Hemisphere and in the tropics. The number of observations actually assimilated over the Northern Hemisphere extratropics (20°–90°N) is about 12 000–13 000 for the radiosonde temperature and wind records (at 0000 and 1200 UTC), twice that large for temperature and wind observations made by aircraft, and 2000–6000 wind observations derived from cloud motions. For the tropics (20°S–20°N), the number of available wind observations from cloud motion vectors varies between 6000 and 13 000. The number of aircraft observations of temperature and wind assimilated varies between 800 and 2300, and up to 3000 measurements of wind and temperature made by radiosondes in the tropics were used in DART/CAM. A large spatiotemporal inhomogeneity in the observation coverage has an impact on the ensemble variance as discussed below. No observation above 100 hPa (close to model level 9) is assimilated. Above this level, vertical correlations extend the impact of observations for two more levels but above model level 7 (~72 hPa) all increments are zero.

The input observation files are those used for the NCEP–NCAR reanalysis project (Kistler et al. 2001). The observation errors from the NCEP–NCAR reanalysis project are used for all observation types. The error variance consists of the instrument error plus the representativeness error used in the NCEP–NCAR reanalysis system. No effort has been made to tune the representativeness error to better represent the DART/CAM system; this is not considered very important for the present application as the horizontal and vertical discretization of the two systems are rather similar (T85 vs T62 truncation and 26 vs 28 vertical levels in the DART/CAM and NCEP–NCAR systems, respectively). As the input observations come from the NCEP system, its observation quality control flags are also used, meaning that any observation that is rejected by NCEP–NCAR is not considered in DART. In addition, DART has its own quality control that compares observations with the prior ensemble and ignores observations that lie unusually far from the prior mean.

2) The assimilation experiments

Two experiments are carried out, the reference experiment and a perfect-model experiment and they will be referred to as the “reference” and “PM” experiments, respectively. In each case, the ensemble consists of 80 members. The initial ensemble is taken from a century-long, free run of CAM conducted for the Atmospheric Model Intercomparison Project (AMIP), which produced various realizations of CCSM due to different sea surface temperature fields. Various simulations valid on 1 July are taken to be valid on 1 July 2007 so that the first assimilation time step is performed for 0600 UTC 1 July. The initial ensemble is characterized by a large spread, especially in the zonally averaged state as will be illustrated in the next section. In the reference experiment the large initial spread reduces quickly and a stable variance behavior is obtained after 5 days. All statistical results presented in the paper are calculated for the 100 assimilation times starting at 0600 UTC 6 July.

In the PM experiment, the “truth” or “nature run” is simulated by the same model used for later assimilation experiments. In our case, a randomly chosen single AMIP realization valid at 0000 UTC 1 July 2007 is used as the initial condition for a 31-day-long CAM forecast. Simulated observations are then generated from the nature run by adding a random error from a Gaussian distribution of zero mean and with the standard deviation equal to the observation error used in the reference experiment.

Observations of the temperature and the two wind components are assimilated using the same model setup as in the reference experiment. The main difference in the assimilation step is the absence of the covariance inflation in the perfect-model experiment. The temporal and spatial locations of simulated observations correspond to the real observation network; the observation operator is bilinear for the horizontal interpolation and linear in pressure for the vertical interpolation. Since the conventional observations assimilated by DART can be considered largely unbiased and the simplicity of the observation operators does not introduce a bias, we expect the perfect-model experiment to illuminate the impact of model deficiencies on the background-error covariances.

Figure 2 compares root-mean-square errors (rmse) and systematic errors (bias) for the temperature and zonal winds using the North American radiosonde observations in the reference and PM experiments. For the reference experiment, the reduction in zonal wind rmse resulting from assimilation is between 0.5 and 1 m s−1 and between 0.2 and 1 K for the temperature field. The analysis error is between 1.5 (lower troposphere) and 3 m s−1 (upper troposphere), whereas the tropospheric temperature error varies between 0.8 and 1 K. Temperature bias in the short-range forecast ensemble is positive throughout the troposphere and in the range 0.4–0.8 K. Observations reduce this error the most in the lower troposphere. Systematic wind errors over North America are relatively small and they change sign in the midtroposphere. By design, the PM experiment contains no significant systematic errors.

Fig. 2.
Fig. 2.

Rmse and bias based on the radiosonde (a),(b) temperature and (c),(d) zonal wind observations over North America. Vertical profiles are shown for the background and analysis ensemble averages in (a),(c) reference and (b),(d) PM assimilation experiments.

Citation: Monthly Weather Review 139, 7; 10.1175/2011MWR3477.1

b. Diagnosis of ensemble variance by normal modes

The output DART/CAM fields are processed on a regular Gaussian N64 grid. The state vector of each field, therefore, used as input for the normal-mode diagnostics has 851 968 (=256 × 128 × 26) gridpoint variables. Each member of the posterior and prior ensemble is projected independently onto NMFs. For the formulation of the NMF expansion the reader is referred to Kasahara and Puri (1981) and references therein. A summary of their derivation is provided in the appendix of Žagar (2009).

Žagar et al. (2009a) and Žagar et al. (2009b) applied the NMF expansion to study the energy spectra of different analysis systems including the posterior ensemble mean of the present reference experiment. These studies showed that the levels of balanced (ROT) and IG energy in DART/CAM analyses are about the same as in the operational systems of the European Centre for Medium-Range Weather Forecasts (ECMWF) and NCEP. In Žagar et al. (2010) time-averaged analysis increments in DART/CAM were studied and compared to average increments in ECMWF, NCEP, and NCEP–NCAR systems. Temporally averaged analysis increments were interpreted as the analysis system bias. It was found that, while all systems have the largest biases in the tropics, DART/CAM is characterized by significant biases also in the extratropics, especially in the Southern Hemisphere. This result is of relevance for the present study.

The NMF expansion as presented in Žagar et al. (2009a) is applied here to each ensemble member to obtain time series of the complex expansion coefficients , which represent the wind and geopotential fields in terms of Hough functions and vertical structure functions (shown in Fig. 3 in Žagar et al. 2009a). The three indices k, n, and m represent the zonal wavenumber, meridional mode index, and the vertical mode index, respectively. The index p denotes the motion type; a value of 1 indicates eastward-propagating inertio-gravity motion (EIG), 2 stands for westward-propagating inertio-gravity waves (WIG), and 3 is the balanced (ROT) motions. A single mode is thus denoted by a four-component index ν = (k, n, m, p). The complex modal state vector has dimension 80 × 25 × 23 × 3, where the first three values correspond to truncation indices in the zonal (Nk), meridional (Nn), and vertical (Nm) directions, respectively. The factor 3 represents the three types of motion considered.

Each m is associated with a value of equivalent depth Heq(m), which connects the vertical and horizontal modes. Increasing values of m are associated with smaller depths and layers lower in the atmosphere. By taking Nm < Nz (the number of vertical levels) we do not represent input data at the lowest model levels well. The same applies to the high latitudes, especially the Arctic region where our projection method does not represent well the variance in the input data. See Žagar et al. (2009a) for details about the application of the NMF expansion.

The smoothing effect of the NMF expansion is illustrated in Fig. 3, which compares analysis increments for a randomly chosen ensemble member and date in physical space with fields obtained by inversion of the difference field, , separately for all modes, for balanced and for IG motions. This snapshot over Australia illustrates also the inaccuracy of the NMF expansion (i.e., filtering of smaller scales). Nevertheless, the comparison of Fig. 3a, which shows the difference between input analysis and background fields and Fig. 3b, which shows the result of inversion of , illustrates that the projection accounts for the majority of large-scale variance, which is our primary interest here. The other two panels (Figs. 3c,d) illustrate the main advantage of using normal-mode functions, that is, separation of increments into contributions from balanced and unbalanced motions. It can be seen that the increments are primarily balanced but the IG component is not negligible. In what follows, we quantify the differences in levels of ROT and IG variance. Figure 3d can be further split into contributions from the westward (WIG) and eastward-propagating (EIG) modes.

Fig. 3.
Fig. 3.

Analysis increments at 0000 UTC 26 Jul 2007 at model level 19 (~500 hPa) for ensemble member 1. (a) Increments in wind and geopotential P variables in physical space. (b) Wind and geopotential increments obtained after projection and its inverse. (c) Balanced increments. (d) Inertio-gravity part of increment field. The contour interval for the geopotential variable is 15 m, starting from ±15 m. Positive values are shaded and negative values are drawn by isolines. Scale of wind arrows in (d) applies also to (a)–(c).

Citation: Monthly Weather Review 139, 7; 10.1175/2011MWR3477.1

The nondimensional expansion coefficients χν(t) represent both wind and mass fields. The latter is represented by the modified geopotential height variable, defined as P = Φ + RTolnps, where all parameters have their usual meaning: Φ = gh is the geopotential, To is the globally averaged temperature on σ surfaces, and ps is the surface pressure field. Dimensional values (i.e., energy in a particular mode) are obtained through multiplication of χν by :
e1
The ensemble variance is obtained as follows. At each analysis time t the modal ensemble mean for each mode ν is computed first:
e2
The index i denotes ensemble members up to Ne = 80. Then the ensemble variance in each mode Vν(t) (in units of m2 s−2 or J kg−1) is obtained as
e3
The square root of this quantity represents the ensemble spread.
Total variance per motion type p is the sum:
eq1
It represents the variance in the wind and geopotential field summed over the whole model domain as captured by selected truncation limits of the normal-mode expansion.
In the results section we shall first discuss time-averaged variances for each motion type:
e4
The temporal variability of Vk,n,p ideally represents the flow-dependency of background-error variances in the DART/CAM system. In our reference experiment, variability of depends on the extent to which the flow variability was captured by the ensemble and on the assimilated observations as well as the assimilation methodology including the covariance localization and the covariance inflation. The majority of the results will present the distribution for the prior ensemble and its variability. We always sum up all vertical modes since giving a physical meaning to a particular vertical mode is difficult.

3. Variance and variance variability in the reference experiment

Figure 4 presents the total prior ensemble variance in ROT, EIG, and WIG modes as a function of time during 25 days of July 2007. Three main features of the variance are clearly seen in the figure. First, the variance is mainly contained in balanced modes all the time. Second, temporal variations appear significant. The total variance varies between about 1047 m2 s−2 on 10 July to 706 m2 s−2 on 20 July. Most of the variability is due to balanced mode variance that varies between 949 and 600 m2 s−2 on 10 and 26 July, respectively. Finally, there is more variance in WIG than in EIG modes and their relative difference increases with time. Dynamics of the IG variance is the same for EIG and WIG modes; their respective total variances slowly decrease until 21 July and increase afterward at the same time that their relative difference increases. The percentage of balanced variance has a minimum value of 85% on 26 July while the maximal percentage of 91% is reached on 10 July following the period of variance growth. Correspondingly, the IG variance percentage has an absolute minimum on 10 July (9%) and after 15 July the IG variance steadily increases until reaching a maximum on 26 July. The zigzag shape of the variance curves in Fig. 4a reflects the data availability; minima are at 0600 and 1800 UTC. In what follows, we discuss the structure of time-averaged variances and their temporal variability in association with the flow dynamics and data assimilation methodology.

Fig. 4.
Fig. 4.

Temporal evolution of the variance (in m2 s−2) in various modes in the prior ensemble in the reference experiment. (a) Total variance (all k) in balanced (R), eastward (E), and westward (W) inertio-gravity modes. (b) Ratios of R, E, and W variance and the total variance. In both (a) and (b) values for E and W variance are multiplied by 10.

Citation: Monthly Weather Review 139, 7; 10.1175/2011MWR3477.1

a. Time averages

If the curves in Fig. 4 are averaged in time, the balanced variance is 754 m2 s−2, which is 88.2% of the total average variance. The EIG and WIG motions contain on average 47 and 54 m2 s−2, that is, 5.5% and 6.3% of the total variance, respectively. These numbers apply to all scales. Among various scales the zonally averaged state (k = 0) dominates; on average it contains 48% of the total variance and this percentage varies between 41% (15 July) and 54% (30 July). The variance in k = 0 is practically all balanced (98%). When only waves are considered, the average difference between WIG and EIG mode variance is somewhat below 2% of the total wave variance with the WIG variance dominant.

1) The zonally averaged state

Because of the magnitude of variance in k = 0 we choose to discuss it first. Figure 5 shows the evolution of k = 0 variance as a function of the meridional mode starting on 1 July. It can be seen that the initial k = 0 variance was distributed rather homogeneously over all meridional scales. After the first 2 days the variance became nearly stationary with the maximum at n = 1–2; in other words, initial reduction of the ensemble spread occurred only for higher n whereas the spread at largest meridional scales was left practically untouched.

Fig. 5.
Fig. 5.

Variance evolution in the prior ensemble zonally averaged state as a function of meridional mode. Labels on the y axis are located at 1200 UTC.

Citation: Monthly Weather Review 139, 7; 10.1175/2011MWR3477.1

There are several reasons for this and they are illustrated in Fig. 6, which shows the zonal wind spread in physical space averaged for the period 6–30 July. Three panels in Fig. 6 illustrate three properties of our experiment that are responsible for the large spread in k = 0. The top panel, which displays model level 6 (~53 hPa), shows the stratospheric spread, which has remained unchanged from the initial AMIP ensemble as there are no analysis increments at this level. The middle panel shows the spread at level 10 (~118 hPa), which is around the top of the tropical troposphere; at this and the neighboring levels the spread is largest in the tropics. This uncertainty is again associated with the initial AMIP ensemble that contains large spread at these levels in the tropics; assimilated observations at these latitudes are insufficient to reduce the spread. Finally, the bottom panel in Fig. 6, which shows a midtropospheric level (level 18 located near 439 hPa), illustrates the permanent lack of observations in high latitudes of the Southern Hemisphere, which applies to all tropospheric levels. In both the stratosphere and troposphere, spread is largest in the Southern Hemisphere extratropics. The consequence for the variance distribution is that it is dominated by the asymmetric modes (even n), primarily mode n = 2 (figure not shown). In the rest of the paper, we keep these properties in mind and discuss most of the results without k = 0. Associated with later discussion of variance variability we also note that the k = 0 variance maxima occurred on 7 July and 30–31 July while the minimum occurred on 15–16 July. An unknown part of the spread, which remains in the largest meridional scales, is associated with the model bias as diagnosed in Žagar et al. (2010). The perfect-model assimilation experiment allows for an approximate estimation of the magnitude of the bias.

Fig. 6.
Fig. 6.

Time-averaged prior ensemble spread in the zonal wind at model level (top) 6 (~53 hPa), (middle) 10 (~118 hPa), and (bottom) 18 (~439 hPa) in physical space.

Citation: Monthly Weather Review 139, 7; 10.1175/2011MWR3477.1

Some other properties of the initial AMIP ensemble are worth mentioning. At the initial analysis step (0600 UTC 1 July) k = 0 variance accounted for about 31% of the total variance; this percentage increased to 40% at 1800 UTC 1 July and to 50% by 6 July. An explanation is that the initial ensemble contained too much wave variability in comparison to the real atmosphere. A balance shift also occurred during the first few steps. At 0600 UTC 1 July the balanced variance (all k) made up 96% of the total variance and after two more analysis times (1800 UTC 1 July) it was reduced to 84% and 77% for all k and k > 0, respectively. This suggests that the AMIP ensemble was too balanced compared to observations, especially given the fact that k = 0 variance, nearly all balanced, simultaneously decreased. Finally, we note that the initial AMIP ensemble contained more variance in the eastward than in the westward-propagating motions. This fact can be discussed in relation with the results from Žagar et al. (2009a), which showed that on average there is more energy in EIG than in WIG motions in the DART/CAM posterior mean analyses. For the wave motions, the difference is about 3% and it is due to the Kelvin wave in the tropics. The same applies also to the average prior ensemble energy. Furthermore, the same property was found in the operational analysis systems of ECMWF and NCEP (Žagar et al. 2009a). Thus, the prevalence of EIG over WIG energy is common to state-of-the-art general circulation models and analysis datasets. However, the DART/CAM ensemble variance on average behaves differently. After the first 2 days of the assimilation the WIG variance increased sufficiently to prevail over the EIG variance. Possible reasons for this will be investigated.

2) Wave variance

When only the wave variance (k ≠ 0) is considered, it is distributed among the three motion types as follows: balanced waves contain 79.5%, EIG 9.4%, and WIG 11.1% of the time-averaged wave variance. Figures 78 present the time-averaged variance divided among ROT, EIG, and WIG modes as a function of the zonal wavenumber and meridional mode. The two-dimensional distribution (k, n) is dominated by the largest zonal and meridional scales. The balanced variance distribution shows the dominance of meridional modes n = 2–3 and has an absolute maximum at (k, n) = (2, 3) (Fig. 7a). This is a symmetric mode, which means that the experiment characteristics responsible for the large variance in k = 0 do not dominate the distribution of the wave variance. Although there is no a priori reason for the ensemble variance distribution to agree with the energy distribution, we note that the average energy fields for the same period have maxima at (k, n) = (1, 1) and (k, n) = (1, 4) (not shown). For EIG modes, the similarity between the average ensemble energy and the ensemble variance is better. The EIG variance maximum is in the Kelvin wave (KW), at (k, n) = (1, 0), which contributes about 14% of the EIG variance. For comparison, the KW energy contributes about 23% of the EIG energy in the ensemble mean fields (Žagar et al. 2009b). The shapes of the (k, n) distributions appear similar for energy and variance fields. This is not the case for the WIG variance, which is largest at (k, n) = (1, 2) while the energy distribution is dominated by k = 0 with maxima at (k, n) = (0, 2–3) and (k, n) = (0, 5) (not shown). In all three motion types, the vertical mode m = 2 dominates. Figure 8 shows the prior variance distribution as a function of the vertical wavenumber that is defined here as the inverse of the equivalent depth . The largest amount of the balanced variance is contained in m = 1 and m = 2, which correspond to the equivalent depths of 10 and 6.2 km, respectively. A secondary maximum is seen for equivalent depth approximately 75 m. For both IG modes the variance maximum is for m = 2 and the WIG variance dominates over the EIG variance over all vertical scales. Shapes of corresponding vertical structure functions are presented in Fig. 3 in Žagar et al. (2009a).

Fig. 7.
Fig. 7.

(k, n) distribution of time-averaged prior ensemble variance (m2 s−2) for (a) balanced, (b) EIG, and (c) WIG modes. Note the difference in the zonal wavenumber range between (a), (b), and (c).

Citation: Monthly Weather Review 139, 7; 10.1175/2011MWR3477.1

Fig. 8.
Fig. 8.

Average variance of the prior ensemble divided among balanced, EIG and WIG modes as a function of the vertical wavenumber. Summation is performed over all n and all k > 0.

Citation: Monthly Weather Review 139, 7; 10.1175/2011MWR3477.1

Balance depends on the scale and the largest scales are most balanced as can be seen in Fig. 9. The percentage of the balanced variance steadily decreases up to k = 30. Beyond this wavenumber the balanced variance stays at about 45%. The rest of the variance is about equally divided between EIG and WIG. However, only 4% of the total wave variance is in scales k > 30. The lowest 10 wavenumbers contain 84% of the total wave variance. The most interesting feature concerning the IG variance is the difference between EIG and WIG for 4 < k < 35, where WIG variance dominates.

Fig. 9.
Fig. 9.

Ratio of the average variance in a particular k and motion type to the total variance in the same wavenumber. Summation is performed over all n and all m.

Citation: Monthly Weather Review 139, 7; 10.1175/2011MWR3477.1

b. Variance reduction

The cumulative impact of observations in DART is presented in Fig. 10. It is obtained by first summing up the variance over all m, n in each (k, p, t), then dividing the posterior variance by the prior variance and finally time averaging this quantity for each (k, p). Figure 10 presents this ratio subtracted from 1 as a function of the zonal wavenumber for each motion type. It shows that the variance decreases more in the balanced modes than in the IG modes across all scales. The largest reduction of the balanced variance happens at scales around k = 20–40 where there is relatively little variance. On the other hand, the reduction is smallest at k = 2 where there is most variance. There are two possible reasons. First, a large part of the variance in k = 2 belongs to the asymmetric meridional mode n = 2 (seen in Fig. 7a), which corresponds to the Southern Hemisphere extratropics. Because of the lack of observations the variance in this region remains stationary. Another reason contributing to this minimum is the shape of the inflation field that is associated with the observing network (k = 2). The extent to which the inflation defines the minimum variance reduction will be checked by comparison with the PM experiment. If instead of Fig. 10 we presented the time-averaged difference between the prior and posterior variance it would show the maximal absolute variance reduction at k = 0 and a sharp variance decay as a function of k according to a 1/k law.

Fig. 10.
Fig. 10.

Ratio of the average posterior to the prior variance in each k subtracted from 1.

Citation: Monthly Weather Review 139, 7; 10.1175/2011MWR3477.1

The amplitude of the variance reduction is defined by the observation density. The reduction at times 0000 and 1200 UTC is different from 0600 and 1800 UTC. Variance reduction shown in Fig. 11 corresponds to times 0000 and 1200 UTC. The shapes in (k, n) space appear the same at 0600 and 1800 UTC but the average reduction is about 10% smaller. The reduction minimum at (k, n) = (2, 2) is more apparent at 0000 and 1200 UTC than at 0600 and 1800 UTC, which supports the explanation that the relative scarcity of observations in the Southern Hemisphere is at least in part responsible for the minimal variance reduction at k = 2. In the IG modes the reduction is somewhat different for EIG and WIG modes. At scales k < 20 the reduction is not dependent on n and the homogeneity is better for WIG modes. Their variance is also a little more reduced than the EIG variance on all scales except at lowest k. As there is no clear dynamical reason for this it may be a property of the assimilation system as investigated next.

Fig. 11.
Fig. 11.

(k, n) distribution of time-average ratio of the posterior to the prior variance subtracted from 1 and multiplied by 100% at assimilation steps 0000 and 1200 UTC: (a) balanced, (b) EIG, and (c) WIG modes.

Citation: Monthly Weather Review 139, 7; 10.1175/2011MWR3477.1

c. Impact of the inflation

In an attempt to understand the difference between the EIG and WIG modes, we investigated scales affected by the inflation. Figure 12 compares average variances in inflated and uninflated posterior and prior ensembles for balanced and IG modes. It shows that inflation affects practically all scales. The most important impact of the inflation is that the prior variance maintains the scale distribution imposed by the inflated posteriors. Furthermore, the posterior variance is on average somewhat more unbalanced than the prior variance. In other words, the assimilation step introduces some imbalance compared to the prior variance. However, this occurs at scales where there is relatively little variance; reduction of the balanced variance is about 0.5% of the total wave variance. Although the spatially varying covariance inflation can cause a change in the spatial gradient of variance and a corresponding change in covariance for a pair of state variables, the correlations are left unchanged. Since DART/CAM analyses are not used for regular forecasting it is impossible to quantify the impact of increased imbalance on the forecast quality.

Fig. 12.
Fig. 12.

(a) Balanced, (b) EIG, and (c) WIG mode average variance ratio in the prior, posterior, and inflated posterior ensembles.

Citation: Monthly Weather Review 139, 7; 10.1175/2011MWR3477.1

On average, inflation increases the total posterior ensemble variance by about 26% of its value before the inflation. However, the inflation field “prefers” the zonally averaged state in such a way that the relative variance increase is larger in the k = 0 mode than in wave modes. The variance reduction by assimilation shows a trend that can clearly be noticed in Fig. 13. Figure 13 presents the WIG wave variance where the trend is most noticeable. The trend is associated with the inflation, which is illustrated by comparing the difference between the average prior and posterior variance and the difference between the posterior and inflated posterior variance. The two fields of differences contain the same signal. The trend from the inflated posteriors is carried forward by the prior ensemble.

Fig. 13.
Fig. 13.

Time evolution of differences between the prior and posterior WIG wave variance and inflated posterior and posterior WIG wave variance.

Citation: Monthly Weather Review 139, 7; 10.1175/2011MWR3477.1

d. Temporal evolution of the variance

In Fig. 8 we saw that variance is largest at the largest zonal scales, k = 1 and k = 2, and decreases as k increases. Figure 14 compares the variance evolution in k = 1 and k = 2. We choose to display the whole period of the experiment in order to illustrate the initial variance reduction in higher meridional modes that occurred first, followed by the reduction in modes n = 2–6, which were taking place until 5 July. Variance maxima occur on 11–12 July in k = 2 and on 15 July in k = 1 and they are associated with n = 3 and n = 2 for k = 2 and k = 1, respectively. Given the meridional mode structure in physical space, this implies that the variance maximum shifted to larger scales and higher latitudes. These two events of increased variance are reflected in the extended variance maximum in n = 2 between 10 and 15 July in the evolution of both total (all k) and wave variance in balanced modes. As mentioned earlier, k = 0 variance in the same period was decreasing to reach its minimum on 15–16 July. The maximum of k = 0 variance on 7 July coincides with the minima in k = 1 and n = 2 in Fig. 14. Ideally, we would like to understand this variance development in relation to the flow properties.

Fig. 14.
Fig. 14.

As in Fig. 5, but (a) k = 1, (b) k = 2, and (c) n = 2.

Citation: Monthly Weather Review 139, 7; 10.1175/2011MWR3477.1

After its initial reduction during the first 4 days the EIG variance remains homogeneously distributed in the lowest n. The WIG variance is similar except that it takes longer to reduce the initial WIG variance in the lowest n. After 20 July the WIG variance shows the tendency to increase as seen in Fig. 6. This property is a consequence of inflation which “favors” westward-propagating modes.

In general, the question of flow dependency is how much the time-averaged distribution shown in Fig. 7 varies in time and space. When the standard deviation of Fig. 7 is computed for 100 samples, the resulting distributions appear similar to the average variances except for the WIG modes. In this case maximal variability is in (k, n) = (2, 1), which again is likely associated with the inflation. One way to visualize average variability is shown in Fig. 15, which presents the ratio between the temporal standard deviation and the mean variance in percentages for the balanced variance. The dominant feature is a maximum centered at (k, n) = (2, 1) with a value of about 70%. On scales k < 10 the balanced variance on average varies up to 40% across all meridional scales. Variability in EIG and WIG modes in these scales is on average about 10% smaller. This can be expected since on these scales IG modes correspond to tropical motions that on average have smaller variability than the extratropical flow. For both EIG and WIG, maximal variability is in mode (k, n) = (2, 0) corresponding to the Kelvin wave and the lowest WIG mode. Except for the maximum around k = 2 and lowest n, variability appears more homogeneous meridionally than zonally for all motion types. Once again one wonders about the largest average variability in k = 2. This is the zonal wavenumber with most of the variance but also smaller balanced variance reduction (i.e., smallest impact of observations), which suggests that it is not real atmospheric variability. Furthermore, this feature is associated with all three motion types, supporting the idea that it is caused by inflation.

Fig. 15.
Fig. 15.

Ratio (%) between temporal deviation and time-averaged balanced prior variance.

Citation: Monthly Weather Review 139, 7; 10.1175/2011MWR3477.1

Variability on a day-to-day basis is much larger than the time-averaged variability. This is illustrated in Fig. 16 by 24-h variance differences normalized by the temporally averaged variance. Shown are results from four subsequent days (from a randomly chosen period) that correspond to 1200 UTC time, when the most observations were available. Variability with magnitudes over ±50% of the average variability appears on a daily basis in different regions of the modal space. As opposed to the balanced variance, changes in the IG variance are small and limited to the lowest n (tropics). In EIG modes, tendencies are usually limited to n = 0, the Kelvin mode, and smaller tendencies sometimes appear in n = 1–2; various zonal wavenumbers dominate on various days. For the WIG variance tendencies are limited to k = 3–10 and n = 0–4 (figures not shown).

Fig. 16.
Fig. 16.

24-h change of the balanced variance at 4 consecutive days between 11 and 14 Jul 2007, normalized by time-averaged variance for the whole study period and multiplied by 100.

Citation: Monthly Weather Review 139, 7; 10.1175/2011MWR3477.1

It is difficult to link variance changes with changes in the flow. First of all, we do not have a standard measure of the flow that could be used for comparison with the variance variability. Tendencies in the energy are very difficult to associate with the variance changes. However, there is a general similarity of energy and variance patterns on large scales, which is illustrated in Fig. 17 for k = 2. The main feature of average energy changes during the month is the reduction of wave energy that started on 10 July when wave energy made up 9% of the total energy. By 17 July the fraction of wave energy dropped below 6% and stayed around this level. In k = 2 an energy maximum occurred on 12 July, a day after the maximum in the variance field in the same wavenumber (see Fig. 14b). On the other hand, the total wave variance reaches its maximum on 15–16 July, simultaneously with total wave energy reaching the minimum (figures not shown). All in all, comparison of Figs. 17 and 14b suggests that the large-scale features of ensemble variance are not an artifact of our assimilation system.

Fig. 17.
Fig. 17.

As in Fig. 14b, but for energy evolution of the posterior mean.

Citation: Monthly Weather Review 139, 7; 10.1175/2011MWR3477.1

4. Perfect-model assimilation experiment

In the PM experiment the majority of systematic errors have been removed (Fig. 2). Given that observations applied are mainly conventional, the systematic errors in the reference experiment are likely primarily due to model errors (Žagar et al. 2010). The consequence is that a large part of the ensemble spread presented in Fig. 6 is removed in the PM experiment, both in the troposphere and the stratosphere. A figure equivalent to Fig. 6 (not shown) confirms that the spread in the Southern Hemisphere has been greatly reduced in the PM experiment and that there is relatively little spread left in the Northern Hemisphere. Best agreement between the reference and PM experiments in the upper troposphere is found in the tropics. Lower down in the troposphere the spread in the Southern Hemisphere is also reduced to less than half of that found in the reference experiment (not shown).

In Fig. 18 we present two relevant features of our PM experiment. First, Fig. 18a shows the prior ensemble spread and its mean rmse for a randomly chosen pressure level and variable, the meridional wind component at 925 hPa. The error is computed by comparison with the Northern Hemisphere radiosondes. The ensemble spread is on average tracking the rmse, which confirms the validity of the PM experiment. The second panel in Fig. 18 shows the evolution of the total variance in the PM experiment, which can qualitatively be compared with Fig. 4a. The feature clearly seen in this figure is a steady reduction of the variance, especially in the first part of the month. This loss of variance is the reason for very small differences between the analysis and background rmse in Fig. 2. It suggests that in our PM experiment that applies no inflation of covariances the error growth due to perturbations in the analysis does not suffice to counter the variance reduction due to observations. In spite of the variance loss, we believe that the main properties of this experiment, including its variance distribution in comparison with the reference experiment, can be discussed with confidence. One can notice that after 20 July the IG mode variance stays approximately constant while the balanced variance still shows a very small tendency to further decrease suggesting that in an extended experiment variance would stay approximately constant.

Fig. 18.
Fig. 18.

(a) Evolution of the prior ensemble spread and rmse with respect to the Northern Hemisphere radiosonde meridional wind at 925 hPa. (b) As in Fig. 4a, but for the PM experiment.

Citation: Monthly Weather Review 139, 7; 10.1175/2011MWR3477.1

a. Time-averaged variances

Approximately 89.8% of the variance corresponds to balanced motions, 5.0% to EIG, and 5.2% to WIG modes. These numbers are similar to those from the reference experiment except that levels of EIG and WIG variance are now about the same. For nonzero k, the variance fractions associated with ROT, EIG, and WIG are 81.7%, 9.1%, and 9.2%, respectively. The percentage of the balanced wave variance in the total wave variance is about 2% larger than in the reference experiment. In the reference experiment this difference belonged to the WIG variance. The relative percentage of the variance in k = 0 remains the same (i.e., 48%) and the k = 0 variance is nearly all balanced just like in the reference experiment.

The major difference between Figs. 18b and 4a is the lack of noticeable differences between the EIG and WIG modes. However, when the variance is summed up as a function of the zonal wavenumber the difference is seen in k = 1 where EIG modes contain more variance than WIG modes, which is a property of the energy spectra (Žagar et al. 2009a). The positive difference between EIG and WIG modes at k = 1 is compensated for by other k modes, where WIG is somewhat larger so that in total the variance levels are the same. The slope of the balanced variance spectra is −2 just like in the reference experiment (not shown).

With model errors removed, the (k, n) distribution for the balanced variance has changed. The ROT k = 0 variance is now concentrated at n = 1, a symmetric mode. The wave variance also has a maximum at a symmetric meridional mode, (k, n) = (1, 3), just like the average energy distribution (Fig. 19). Comparison of Figs. 19 and 7a illustrates how powerfully the model errors influence the covariance structures in the background-error covariance matrix. Variances for EIG and WIG modes appear more similar to the reference case, in particular in the EIG case where the largest variance is in the Kelvin mode.

Fig. 19.
Fig. 19.

As in Fig. 7a, but for the PM experiment.

Citation: Monthly Weather Review 139, 7; 10.1175/2011MWR3477.1

b. Variance reduction and time dependence

The average impact of observations on variance reduction is shown in Fig. 20. This figure illustrates the main differences between the two experiments. There is no minimum at k = 2, which supports the idea that this minimum is associated with model errors in the poorly observed Southern Hemisphere extratropics and with the applied inflation field. The variance is best reduced in scales k = 7–20. However, the reduction is small; it is less than half as large as in the reference experiment and the reduction at largest scales is negligible. The (k, n) distribution for the variance reduction shows that the maximum reduction is shifted toward smaller n (in comparison to the reference experiment) for the balanced modes while the reduction appears homogeneous for the IG modes. The maximum reduction for ROT variance at 0000 and 1200 UTC is up to 30% at scales (k, n) = (10, 7–8) (not shown).

Fig. 20.
Fig. 20.

As in Fig. 10, but for the PM experiment.

Citation: Monthly Weather Review 139, 7; 10.1175/2011MWR3477.1

Temporal deviations of average variances in the PM experiment look the same as the time-averaged variance distributions. Similar to the reference experiment, variability occurs over many scales but more at larger scales. The main difference between the two experiments is the lack of a variability maximum at k = 2 and on average larger variability in the PM experiment. For example, average variability in balanced modes is about 50% of average variance and it is nearly homogeneously distributed across all n for k < 8. The same (50%) average variability percentage applies to EIG and WIG modes, which is also different from the reference experiment where variability in these modes was smaller. To understand the reasons for this, one can look at the time evolution of various modes as was done for the reference experiment in Fig. 6. In the PM experiment there is no development of variance maxima in k = 1 and k = 2; instead, the variance in these modes keeps decreasing in agreement with Fig. 18. On the other hand the variance grows somewhat in the zonal mean state and n = 3–4 in the second part of the month. Forcing the ensemble to the randomly selected ensemble member that represents the truth makes the flow more uncertain in the zonally averaged state while removing the variance from the eddies. On a daily basis variability is even larger. However, we do not discuss these variations in the present experiment because the variance continually decreases.

5. Discussion

There is a discrepancy between the percentage of balanced motions in increment fields presented in Žagar et al. (2010) and the prior ensemble variance. In Žagar et al. (2010) we reported that about 50% of the energy of increment fields of the ensemble mean is balanced; here we show that the percentage of balanced prior variances (i.e., background-error variances) in waves is about 80%. The main reason for the discrepancy is the fact that increments are limited to the troposphere and the lowest levels in the stratosphere while our present study considers the whole model depth. Excluding stratospheric levels from our diagnosis would certainly make the two numbers more similar.

The DART/CAM assimilation is not based on NMFs so it is in principle difficult to understand in this way what is happening inside the DART system. DART includes covariance localization which we did not discuss here and which most likely causes differences in balanced variance between the prior and posterior ensembles, as demonstrated in other studies of the EnKF (e.g., Kepert 2009). The reduction of balance in the posterior ensemble occurs in both k = 0 and the waves. In Fig. 12 the relative change is more visible at scales beyond k = 4 but in our system most of the variance is in the largest zonal scales making changes there relatively more important. Inflation, as seen in Fig. 12 increases the relative percentage of balanced variance; however, it does so by increasing the relative percentage of balanced variance in k = 0 and decreasing it in wave motions. The prior ensemble maintains exact balance in the k = 0 state as in the inflated posterior ensemble. In summary, the percentage of the balanced variance is reduced by the assimilation step and increased by the inflation. Balance increase due to the latter is larger than the balance reduction by the former, but the prior ensemble balance is somewhat reduced with respect to the inflated posteriors, suggesting that forecasts are noisy. In each case the change in balance, as a percentage of total variance, is a small number and its importance is difficult to quantify without making extensive forecast experiments.

The ratio between total variance in the reference and PM experiments as well as the ratio between the wave variance in the two experiments is larger than a factor of 3 (cf. amplitudes on y axes in Figs. 4a and 18b). Since the time range of the PM experiment was characterized by a steady variance loss this ratio might be somewhat smaller but nevertheless large. If, instead of variance, we use the prior ensemble spread, the ratio between the reference and PM experiment is a factor of 2. The same ratio was reported by Houtekamer et al. (2009) in their study with the EnKF system of the Meteorological Service of Canada when not accounting for model errors. This tells us that in order to use DART/CAM for weather forecasting, model errors need to be simulated in some manner. The fact that without inflation of covariances the PM experiment steadily loses variance is suggestive of model error growth being a major component of forecast-error growth and of the main role of observations being to constrain the model error growth. Comparison of balanced variances in the reference and PM experiments (Figs. 7a and 19, respectively) is relevant for model-error covariance modeling. It suggests that model errors on large scales may be a dominant factor defining the background-error variance distribution. Consequently, a modeling approach that respects the structure of forecast errors is meaningful for large scales.

An interesting difference between the reference and PM experiments is in temporal variability, which is, in the latter case on average, equally large for all motion types. In the reference experiment, variability is larger in balanced modes. It remains for further investigation to analyze the extent to which variability is influenced by the inflation as compared to the flow properties. Overall our results show that more research is needed on the problem of inflation and the representation of model error in general.

We also note that our projection does not account for the whole ensemble variance in physical space. During most of the period there was a significant amount of variance in physical space in the Arctic that was not well projected onto the modal space. This was also true for the lowest model levels that are not well represented by the NMF expansion. In Žagar et al. (2009a) we discussed the accuracy of the projection. Since here we concentrated on large scales, for which NMF expansion applies well, we do not consider accuracy as a factor influencing our conclusions.

6. Conclusions

In this paper, a new approach to the diagnosis of balance in the EnKF system has been proposed, based on the normal modes of the linearized primitive equations on the sphere. The applied framework allows quantification of mass–wind field balance and its temporal variability. Unlike the conventional approach of studying flow dependency in physical space, the applied modal diagnosis allows a detailed insight into the temporal behavior of the ensemble variance in terms of various scales and motions types. The amount of observations assimilated into the 80-member DART/CAM ensemble does not suffice to constrain all aspects of large-scale circulation in July 2007. Nevertheless, results are relevant for the NWP application of the EnKF and the role of covariance inflation and model errors. The latter were diagnosed by comparing an experiment with real observations (reference experiment) and a perfect-model experiment without any model error representation. Even though in the perfect-model assimilation experiment variance steadily decreased in the first half of the period studied, the main properties of the perfect-model experiment are useful for understanding the results of the reference experiment. The conclusions are as follows:

  1. In the DART/CAM EnKF system, nearly 90% of the total variance is balanced. Almost half of the total variance is in the zonally averaged state, which is associated with the properties of our system. The remaining variance, from waves, is balanced at about the 80% level. The remaining 20% is divided between the westward- (WIG) and eastward-propagating (EIG) inertio-gravity modes. Balance is scale dependent and the largest scales are best balanced. Beyond zonal wavenumber k = 30 balanced variance stays at about the 45% level.

  2. The WIG variance dominates over EIG; the difference is about 2% of the total wave variance and it was shown to be associated with the inflation applied to the posterior ensemble. The prior variance maintains the scale distribution imposed by the inflated posteriors. In the perfect-model experiment, levels of EIG and WIG variance are about the same.

  3. Observations are more effective in reducing balanced variance than IG variance. Variance is most reduced at scales greater than k = 10. The minimum variance reduction occurs at k = 2, which is associated with inflation. The applied inflation field inflated the zonally averaged state variances more than the wave field variances. However, without longer forecasts it is impossible to quantify the effect of the inflation on the forecast quality.

  4. Temporal variability of the ensemble variance is significant, especially on a day-to-day basis. At large scales variability is associated with changes in the energy of the flow, especially interaction between the zonally averaged and wave motions. However, average temporal variability is influenced by the applied inflation. It remains for further studies to propose appropriate ways to quantify pure flow dependency in contrast to variability solely associated with the assimilation methodology.

Acknowledgments

The authors thank J. Whitaker and T. Hamill for pointing out the k = 2 structure of the conventional observing network, to A. Kasahara for his work on the Hough functions and for his interest in this study, and to T. Hoar and N. Collins for their work on the DART/CAM system. We also thank M. Reszka, Editor H. Mitchell, and two anonymous reviewers for their constructive comments. The Centre of Excellence for Space Sciences and Technologies SPACE-SI is an operation partly financed by the European Union, European Regional Development Fund and Republic of Slovenia, Ministry of Higher Education, Science and Technology.

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Save
  • Anderson, J. L., 2001: An ensemble adjustment Kalman filter for data assimilation. Mon. Wea. Rev., 129, 28842903.

  • Anderson, J. L., 2007a: An adaptive covariance inflation error correction algorithm for ensemble filters. Tellus, 59A, 210224.

  • Anderson, J. L., 2007b: Exploring the need for localization in ensemble data assimilation using an hierarchical ensemble filter. Physica D, 230, 99111.

    • Search Google Scholar
    • Export Citation
  • Anderson, J. L., 2009: Spatially and temporally varying adaptive covariance inflation for ensemble filters. Tellus, 61A, 7283.

  • Anderson, J. L., T. Hoar, K. Raeder, H. Liu, N. Collins, R. Torn, and A. Avellano, 2009: The Data Assimilation Research Testbed: A community facility. Bull. Amer. Meteor. Soc., 90, 12831296.

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  • Fig. 1.

    Zonal wind inflation factor at level 14 (~227 hPa). Isolines every 1.5, starting from 2. See text for details.

  • Fig. 2.

    Rmse and bias based on the radiosonde (a),(b) temperature and (c),(d) zonal wind observations over North America. Vertical profiles are shown for the background and analysis ensemble averages in (a),(c) reference and (b),(d) PM assimilation experiments.

  • Fig. 3.

    Analysis increments at 0000 UTC 26 Jul 2007 at model level 19 (~500 hPa) for ensemble member 1. (a) Increments in wind and geopotential P variables in physical space. (b) Wind and geopotential increments obtained after projection and its inverse. (c) Balanced increments. (d) Inertio-gravity part of increment field. The contour interval for the geopotential variable is 15 m, starting from ±15 m. Positive values are shaded and negative values are drawn by isolines. Scale of wind arrows in (d) applies also to (a)–(c).

  • Fig. 4.

    Temporal evolution of the variance (in m2 s−2) in various modes in the prior ensemble in the reference experiment. (a) Total variance (all k) in balanced (R), eastward (E), and westward (W) inertio-gravity modes. (b) Ratios of R, E, and W variance and the total variance. In both (a) and (b) values for E and W variance are multiplied by 10.

  • Fig. 5.

    Variance evolution in the prior ensemble zonally averaged state as a function of meridional mode. Labels on the y axis are located at 1200 UTC.

  • Fig. 6.

    Time-averaged prior ensemble spread in the zonal wind at model level (top) 6 (~53 hPa), (middle) 10 (~118 hPa), and (bottom) 18 (~439 hPa) in physical space.

  • Fig. 7.

    (k, n) distribution of time-averaged prior ensemble variance (m2 s−2) for (a) balanced, (b) EIG, and (c) WIG modes. Note the difference in the zonal wavenumber range between (a), (b), and (c).

  • Fig. 8.

    Average variance of the prior ensemble divided among balanced, EIG and WIG modes as a function of the vertical wavenumber. Summation is performed over all n and all k > 0.

  • Fig. 9.

    Ratio of the average variance in a particular k and motion type to the total variance in the same wavenumber. Summation is performed over all n and all m.

  • Fig. 10.

    Ratio of the average posterior to the prior variance in each k subtracted from 1.

  • Fig. 11.

    (k, n) distribution of time-average ratio of the posterior to the prior variance subtracted from 1 and multiplied by 100% at assimilation steps 0000 and 1200 UTC: (a) balanced, (b) EIG, and (c) WIG modes.

  • Fig. 12.

    (a) Balanced, (b) EIG, and (c) WIG mode average variance ratio in the prior, posterior, and inflated posterior ensembles.

  • Fig. 13.

    Time evolution of differences between the prior and posterior WIG wave variance and inflated posterior and posterior WIG wave variance.

  • Fig. 14.

    As in Fig. 5, but (a) k = 1, (b) k = 2, and (c) n = 2.

  • Fig. 15.

    Ratio (%) between temporal deviation and time-averaged balanced prior variance.

  • Fig. 16.

    24-h change of the balanced variance at 4 consecutive days between 11 and 14 Jul 2007, normalized by time-averaged variance for the whole study period and multiplied by 100.

  • Fig. 17.

    As in Fig. 14b, but for energy evolution of the posterior mean.

  • Fig. 18.

    (a) Evolution of the prior ensemble spread and rmse with respect to the Northern Hemisphere radiosonde meridional wind at 925 hPa. (b) As in Fig. 4a, but for the PM experiment.

  • Fig. 19.

    As in Fig. 7a, but for the PM experiment.

  • Fig. 20.

    As in Fig. 10, but for the PM experiment.

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