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  • View in gallery
    Fig. 1.

    A yz cross section of the external forcing (m s−1 day−1) exerted on the u-wind component. The forcing is symmetric in the x direction. Contour intervals are 0.2 m s−1 day−1

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    Fig. 2.

    (top) An xy cross section (z = 30) of the u-wind component (m s−1) truth run initial condition. The nine circles indicate the location and range of the Doppler radars. (bottom) A yz cross section (x = 30) of the u-wind component (m s−1) truth run initial condition. Contour intervals are 10 m s−1

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    Fig. 3.

    An xz cross section (y = 15) of the truth run initial condition. Dark patches indicate regions where w ≥ 5 cm s−1, i.e., where radars are able to measure radial velocities if their range allows it

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    Fig. 4.

    (top left) The u-wind ensemble mean error of the analyses (averaged over the two ensembles and over space) for experiments A8 (dotted–dashed thick line) and A2 (dotted thick line). The solid thick line is the mean error of the forecast without assimilating data (experiment CONTROL). (top right) As in top-left panel, but for the υ-wind component. (bottom left) As in top-left panel, but for the w-wind component. (bottom right) As in top-left panel, but for temperature. Thin curves are the ensemble standard deviations

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    Fig. 5.

    (top) An xz cross section of the w wind of the first member of the first ensemble (y = 15) after the 10th assimilation cycle for experiment A8 (cm s−1). (bottom) As in top panel, but for the truth run. Contour intervals are 20 cm s−1

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    Fig. 6.

    (top left) The u-wind ensemble mean error in experiment A8 after the first assimilation cycle, i.e., at t = 30 min (in m s−1, averaged over the two ensembles and over the vertical) as a function of x and y. Darker areas indicate lower values. Contour intervals are 0.4 m s−1. (top right) As in top-left panel, but for the υ-wind component. Contour intervals are 0.4 m s−1. (bottom left) As in top-left panel, but for the w-wind component (cm s−1). Contour intervals are 1 cm s−1. (bottom right) As in top-left panel, but for temperature (K). Contour intervals are 0.15 K

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    Fig. 7.

    (top left) The u-wind ensemble standard deviation in experiment A8 after the first assimilation cycle, i.e., at t = 30 min (m s−1, averaged over the two ensembles and over the vertical) as a function of x and y. Darker areas indicate lower values. Contour intervals are 0.4 m s−1. (top right) As in top-left panel, but for the υ-wind component. Contour intervals are 0.4 m s−1. (bottom left) As in top-left panel, but for the w-wind component (cm s−1). Contour intervals are 0.5 cm s−1. (bottom right) As in top-left panel, but for temperature (K). Contour intervals are 0.05 K

  • View in gallery
    Fig. 8.

    As in Fig. 6, but after the 10th assimilation cycle. Contour intervals are (top) 0.4 m s−1, (bottom left) 4 cm s−1, and (bottom right) 0.2 K

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    Fig. 9.

    As in Fig. 7, but after the 10th assimilation cycle. Contour intervals are (top) 0.4 m s−1, (bottom left) 4 cm s−1, and (bottom right) 0.2 K

  • View in gallery
    Fig. 10.

    (top left) The u wind at point (x = 15, y = 15, z = 15) as a function of time for the first member of the first ensemble (m s−1). The solid line represents the evolution of u starting at the initial time and ending at the first analysis time; the dotted–dashed line represents the evolution of u without external gravity mode filtering of the w and T increments (experiment A8), starting after the first analysis and ending before the second analysis; the dashed line represents the evolution of u with external gravity mode filtering of the w and T increments, starting after the first analysis and ending before the second analysis. (top right) As in top-left panel, but for the υ-wind component. (bottom left) As in top-left panel, but for the w-wind component (cm s−1). (bottom right) As in top-left panel, but for temperature (K)

  • View in gallery
    Fig. 11.

    Horizontal wind power spectrum averaged over the vertical of experiments CONTROL (thick solid line) and sd_CONTROL (thin solid line) at time equals 48 h. Reference spectra with slope −5/3 (dashed) and −3 (dotted–dashed) are also shown; n is the total horizontal wavenumber multiplied by 1000/(2π) km

  • View in gallery
    Fig. 12.

    As in Fig. 4, but for the case with no large-scale forcing, a reduced dissipation, and an assimilation period of 2 days (experiments sd_CONTROL, sd_A8, and sd_A2). Only ensemble mean errors are shown. Standard deviations (not shown) are very close to mean errors for all variables

  • View in gallery
    Fig. 13.

    As in Fig. 4, except that the number of assimilated observations per cycle has been reduced by a factor of about 4. The solid lines are for experiment CONTROL; the dotted–dashed lines are for experiment lo_A8; the dashed lines are for experiment flo_A8; the double-dotted–dashed lines are for experiment nflo_A8. Only mean errors are shown. The ensemble std devs (not shown) are very close to the ensemble mean errors for all variables. Note that the curves for u, υ, and T for experiments lo_A8, flo_A8, and nflo_A8 are almost overlying

  • View in gallery
    Fig. 14.

    As in Fig. 4, except that the number of assimilated observations per cycle has been reduced by a factor of about 4 and the initial ensembles are obtained from 2N long integrations. The solid lines are for experiment ic_CONTROL; the dashed lines are for experiment iclo_A8; the double-dotted–dashed lines are for experiment ficlo_A8. Only mean errors are shown. The ensemble std devs (not shown) are very close to the ensemble mean errors for all variables. Note that the curves for u, υ, and T for experiments iclo_A8, and ficlo_A8 are almost overlying

  • View in gallery
    Fig. 15.

    As in Fig. 14, except that the solid lines are for experiment hiclo_A8 and the dashed lines are for experiment iclo_A8_2. The thin lines represent the ensemble std devs

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    Fig. 16.

    (top left) The solid curve is the forecasted u-wind ensemble std dev (m s−1; averaged over the two ensembles and over space) as a function of time when the initial conditions are the analyses at t = 52 h from experiment A8; the dotted–dashed curve is as in Fig. 4 for 52 h < t < 100 h. (top right) As in top-left panel, but for the υ-wind component. (bottom left) As in top-left panel, but for the w-wind component (cm s−1). (bottom right) As in top-left panel, but for temperature (K)

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Assimilation with an Ensemble Kalman Filter of Synthetic Radial Wind Data in Anisotropic Turbulence: Perfect Model Experiments

Martin CharronDivision de la Recherche en Météorologie, Service Météorologique du Canada, Dorval, Québec, Canada

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P. L. HoutekamerDivision de la Recherche en Météorologie, Service Météorologique du Canada, Dorval, Québec, Canada

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Peter BartelloDepartment of Atmospheric and Oceanic Sciences, and Department of Mathematics and Statistics, McGill University, Montréal, Québec, Canada

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Abstract

The ensemble Kalman filter (EnKF) developed at the Meteorological Research Branch of Canada is used in the context of synthetic radial wind data assimilation at the mesoscale. A dry Boussinesq model with periodic boundary conditions is employed to provide a control run, as well as two ensembles of first guesses. Synthetic data, which are interpolated from the control run, are assimilated and simulate Doppler radar wind measurements.

Nine “radars” with a range of 120 km are placed evenly on the horizontal 1000 km × 1000 km domain. These radars measure the radial wind with assumed Gaussian error statistics at each grid point within their range provided that there is sufficient upward motion (a proxy for precipitation). These data of radial winds are assimilated every 30 min and the assimilation period extends over 4 days.

Results show that the EnKF technique with 2 × 50 members performed well in terms of reducing the analysis error for horizontal winds and temperature (even though temperature is not an observed variable) over a period of 4 days. However the analyzed vertical velocity shows an initial degradation. During the first 2 days of the assimilation period, the analysis error of the vertical velocity is greater when assimilating radar observations than when scoring forecasts initialized at t = 0 without assimilating any data. The type of assimilated data as well as the localization of the impact of the observations is thought to be the cause of this degradation of the analyzed vertical velocity. External gravity modes are present in the increments when localization is performed. This degradation can be eliminated by filtering the external gravity modes of the analysis increments.

A similar set of experiments is realized in which the model dissipation coefficient is reduced by a factor of 10. This shows the level of sensitivity of the results to the kinetic energy power spectrum, and that the quality of the analyzed vertical wind is worse when dissipation is small.

Corresponding author address: Dr. Martin Charron, Division de la Recherche en Météorologie, 2121 route Transcanadienne, Dorval, Québec H9P 1J3, Canada. Email: Martin.Charron@ec.gc.ca

Abstract

The ensemble Kalman filter (EnKF) developed at the Meteorological Research Branch of Canada is used in the context of synthetic radial wind data assimilation at the mesoscale. A dry Boussinesq model with periodic boundary conditions is employed to provide a control run, as well as two ensembles of first guesses. Synthetic data, which are interpolated from the control run, are assimilated and simulate Doppler radar wind measurements.

Nine “radars” with a range of 120 km are placed evenly on the horizontal 1000 km × 1000 km domain. These radars measure the radial wind with assumed Gaussian error statistics at each grid point within their range provided that there is sufficient upward motion (a proxy for precipitation). These data of radial winds are assimilated every 30 min and the assimilation period extends over 4 days.

Results show that the EnKF technique with 2 × 50 members performed well in terms of reducing the analysis error for horizontal winds and temperature (even though temperature is not an observed variable) over a period of 4 days. However the analyzed vertical velocity shows an initial degradation. During the first 2 days of the assimilation period, the analysis error of the vertical velocity is greater when assimilating radar observations than when scoring forecasts initialized at t = 0 without assimilating any data. The type of assimilated data as well as the localization of the impact of the observations is thought to be the cause of this degradation of the analyzed vertical velocity. External gravity modes are present in the increments when localization is performed. This degradation can be eliminated by filtering the external gravity modes of the analysis increments.

A similar set of experiments is realized in which the model dissipation coefficient is reduced by a factor of 10. This shows the level of sensitivity of the results to the kinetic energy power spectrum, and that the quality of the analyzed vertical wind is worse when dissipation is small.

Corresponding author address: Dr. Martin Charron, Division de la Recherche en Météorologie, 2121 route Transcanadienne, Dorval, Québec H9P 1J3, Canada. Email: Martin.Charron@ec.gc.ca

1. Introduction

It is appropriate to use a probabilistic framework (e.g., Epstein 1969) to describe the chaotic flow of the atmosphere and ocean (Lorenz 1993). The notion of probability in this case is not inherent to the governing dynamical equations (which are deterministic), but is related to the sensitivity of the solutions to slightly different initial conditions, to unknown external forcings, and to deficiencies of the representing models. For a given forecast, the initial probability density function (PDF) of the fluid state depends on the quality of the assimilating system, and is thus technology dependent. Solving the equation governing the evolution of the PDF (the Fokker–Planck equation or the Liouville equation; see Ehrendorfer 1994a, b) is not feasible for systems with large dimensional phase space (Hamill 2001). Leith (1974) proposed to use a small sample of realizations for the description and estimation of the PDF. By using different realizations for the initial conditions, for the model error, and for the external forcings, we obtain differences in the resulting ensemble of forecasts. In the particular case of meteorological forecasting at the global scale, it has been shown that despite the relatively small number of realizations commonly used [O(100)] (e.g., Buizza 1997; Toth and Kalnay 1997), such an approach can provide useful probabilistic information on the atmospheric state up to lead times of about a week or two.

The ensemble Kalman filter (EnKF) (Evensen 1994) can be used to obtain an initial PDF of a dynamical system. One assimilates an ensemble of perturbed observations (Burgers et al. 1998; Houtekamer and Mitchell 1998) into an ensemble of background fields to obtain an ensemble of equally likely analyses. The EnKF technique has been applied in several contexts: Evensen (1994) developed and used it to assimilate ocean data with a quasigeostrophic model; in a complementary study, Evensen and van Leeuwen (1996) used the technique to assimilate Geostat data; Evensen (1997) studied the EnKF behavior using the chaotic Lorenz system (Lorenz 1963); Houtekamer and Mitchell (1998) investigated the feasibility of assimilating atmospheric data at the global scale with an EnKF by using a three-level quasigeostrophic model at resolution T21; Reichle et al. (2002) used the EnKF technique to assimilate hydrologic data.

Snyder and Zhang (2003) investigated the assimilation of simulated Doppler radar data at the convective scale using the EnKF technique. They assimilated data every 5 min (and during 2 h) and obtained convergence of the analyses to a reference run after about seven assimilation cycles. Their experiment also shows significant correlations between observed and unobserved variables.

In the present paper, the focus is on employing the EnKF to assimilate simulated radial wind data at longer time and length scales than in Snyder and Zhang (2003), but shorter than the synoptic time and length scales. In real situations, Doppler radar data can constitute an important source of observations at scales commonly used for mesoscale weather forecasting. The feasibility of assimilating these data with an EnKF in the meso-β range (20–200 km, i.e., the length scales resolved by the experiments described in this paper) is investigated. The specific interest of using the EnKF technique at the mesoscale range lies in its ability to deal with nongeostrophic and intermittent dynamics that can be ubiquitous at these scales. Another interesting feature of the meso-β dynamics is that its turbulence is not fully isotropic, as opposed to convective-scale turbulence, which is isotropic. A background error covariance matrix that evolves with the dynamics is likely more appropriate than a prespecified (often geostrophic) background error covariance matrix. However, it remains to be investigated if the EnKF behaves well if one has relatively short time scales and corresponding non-Gaussian background error statistics.

The approach taken is similar to Houtekamer and Mitchell (1998) and Snyder and Zhang (2003) in the sense that data are simulated (or synthesized) from a truth run that is considered perfect. A dry three-dimensional Boussinesq model comprising only dynamical and small-scale dissipative processes is employed to produce the truth run and the background fields. To simulate conditions somewhat representative of current Doppler radar observation networks, a set of nine radars is placed on a 1000 km × 1000 km domain.

The paper is organized as follows. Section 2 presents a quick review of the ensemble Kalman filter formulation and the particular settings used in the experiments. Section 3 presents the numerical model from which data are synthesized, explains the methodology used to generate the initial ensemble, and describes the performed experiments. Section 4 shows results of 4 days of assimilation cycles as well as sensitivity tests, and general conclusions are drawn in section 5.

2. Formulation of the ensemble Kalman filter

A short overview of the assimilation algorithm used in this work is presented here. More details can be found in Houtekamer and Mitchell (1998, 2001). The best linear unbiased estimator (BLUE) constitutes the basis on which the EnKF relies. Given some observation vector y and background (or first guess) field Xb, one is interested in generating an analysis Xa that is the best estimate of the true state Xt. Minimizing the variance of the analysis error, one obtains the well-known BLUE equations, which are the basis of modern data assimilation:
i1520-0493-134-2-618-e21a
i1520-0493-134-2-618-e21b
The gain matrix 𝗞 plays the role of a weighting function between the observations and the background field; 𝗛 is the forward operator that projects the background field into the observation space; and 𝗣 and 𝗥 are the background and observation error covariance matrices, respectively. The essence of the Kalman filter technique is to employ a matrix 𝗣 that is flow dependent. In the EnKF, 𝗣𝗛T as well as 𝗛𝗣𝗛T are sampled from an ensemble of realizations of the background, and an analysis is performed on each member of the ensemble. As shown in Burgers et al. (1998) and Houtekamer and Mitchell (1998), a random error with statistics given by 𝗥 must be added to the observations when performing the analysis of each ensemble member. Moreover, inbreeding occurs when performing the ensemble of analyses with a gain matrix that is not obtained from an independent ensemble of realizations (Houtekamer and Mitchell 1998, 1999; van Leeuwen 1999), resulting in an underestimation of the analysis error. This led to the formulation of what is sometimes referred to as the double EnKF (Houtekamer and Mitchell 1998) in which two independent ensembles are used in parallel, the gain matrix calculated from one ensemble being used in the analysis scheme of the other ensemble, and vice versa. This method provides better estimates of the analysis error.

A potentially noisy analysis results from the limited number of ensemble members used in the evaluation of the gain matrix 𝗞. To prevent observations and background field information far from a region of interest from influencing the analysis at that region, a localization procedure is applied on the background error covariance matrix. A Hadamard product of a localization covariance matrix ρ (Gaspari and Cohn 1999) with 𝗣𝗛T and 𝗛𝗣𝗛T is performed at each analysis cycle (𝗣𝗛T and 𝗛𝗣𝗛T being replaced by their Hadamard product with ρ in the analysis algorithm). The Hadamard product of ρ with any valid covariance matrix 𝗕 (denoted 𝗖 = ρo𝗕, where 𝗖ij = ρij𝗕ij) is a valid covariance matrix. The localization is performed in the three-dimensional physical space [this differs from Houtekamer and Mitchell (2001) where the localization is performed in the horizontal only].

The EnKF analysis scheme used in this paper is described by the following equations:
i1520-0493-134-2-618-e22a
i1520-0493-134-2-618-e22b
i1520-0493-134-2-618-e22c
i1520-0493-134-2-618-e22d
i1520-0493-134-2-618-e22e
The superscript (α, i) stands for the ith member of ensemble α; β equals one (two) when α is two (one), and N is the number of ensemble members (N = 50 throughout the paper). Details on the numerical solver can be found in Houtekamer and Mitchell (1998, 2001).

3. Model description and experiment settings

At the meso-β scale (20–200 km) and in the context of the present study, acoustic and Rossby waves are neglected in the development and evolution of the relevant dynamics. This dynamics is mainly nongeostrophic and can give rise to intermittent phenomena. A nonhydrostatic Boussinesq model with constant Coriolis parameter has been used to generate the truth run from which simulated observations are generated. The same model is used to generate the background fields needed by the EnKF analysis scheme (hence the problem of model error in data assimilation with the EnKF is not addressed here). The model equations are
i1520-0493-134-2-618-e31a
i1520-0493-134-2-618-e31b
i1520-0493-134-2-618-e31c
where u = (u, υ, w) is the three-dimensional wind vector, T is the perturbed (potential) temperature, ab is the square of the buoyancy frequency (a is gravity divided by a reference potential temperature, and b is a vertical reference potential temperature gradient), î = (1, 0, 0) is the unit vector along the x direction, = (0, 0, 1) is the vertical unit vector, f is the constant Coriolis parameter, Φ is pressure divided by density, F = F(y, z) is a large-scale constant external forcing depicted in Fig. 1, and
i1520-0493-134-2-618-eq1
are the horizontal and vertical viscosity coefficients, respectively. The numerical model solves these equations with the spectral method and Eulerian time stepping (see, e.g., Bartello et al. 1996). The horizontal domain covers an area of 1000 km × 1000 km, and the vertical domain extent is 10 km. The number of grid points is 60 in the three physical directions, but the effective resolution is 25 km in the horizontal (0.25 km in the vertical) due to the choice of spectral truncation at wavenumber 20. The time step is 4.2 s. This very small time step is necessary to account for the fastest waves with frequencies close to the buoyancy frequency. The boundary conditions are periodic in all directions to permit error dynamics at the boundaries. The Coriolis parameter is set to 10−4 s−1 and the buoyancy frequency ab is 10−2 s−1 (a and b are chosen to be equal, but this choice is arbitrary since only their product is dynamically relevant).

The initial conditions for the truth run and the ensemble members starting the first cycle of the assimilation process (the initial ensemble) are obtained in such a way that strong, unrealistic oscillations are avoided. An initialization with simple random noise as in Snyder and Zhang (2003) would not produce realistic ensembles because, as is described later in section 4, the dynamics at large scales is in approximate hydrostatic balance. Disrupting this approximate balance with perturbations that are strongly out of balance will generate realizations contaminated with gravity waves.

The result of a long previous model run is randomly perturbed by normal modes calculated with respect to a resting basic state (u0 = 0 and T0 = constant). The polarization and dispersion relations of the nondissipative, unforced, linearized Boussinesq equations are used to find the normal modes in Fourier space:
i1520-0493-134-2-618-e32a
i1520-0493-134-2-618-e32b
i1520-0493-134-2-618-e32c
i1520-0493-134-2-618-e32d
i1520-0493-134-2-618-e32e
where k = (k, l, m) is the wavenumber, ω is the frequency, and hatted variables are Fourier components. Fourier fields Φ̂(k) = C(k)e(k)/m—with real random Gaussian amplitudes C(k) and uniformly distributed random phases δ(k)—scaling as m−1 are generated for wavelengths greater than five grid points for each ensemble member and for the truth run. All spectral elements with m = 0 or k = l = 0 are set to zero. The sign of ω is also chosen randomly. The reality condition (û, υ̂, ŵ, )(k) = (û, υ̂, ŵ, )*(−k) is imposed on the wind and temperature perturbations obtained from Eqs. (3.2). They are then transformed to physical space and added to a previous long model run, forming the initial ensemble of background fields and the truth run initial conditions. Note that for this ensemble initialization scheme [also considering the specific form of Φ̂(k)], it can be shown that the correlation between horizontal winds and temperature of the initial ensembles tends to zero when the number of ensemble members tends to infinity. The mean amplitude of the pressure perturbations is tuned such that the ensemble standard deviation for the horizontal wind components averaged over the domain is close to 2 m s−1. The polarization relations and the chosen vertical spectral shape of Φ̂(k) imply that the ensemble standard deviation for the vertical wind and temperature averaged over the domain are 4.4 cm s−1 and 1.2 K, respectively. This way of generating the initial conditions allows one to start the numerical integrations without producing fast oscillations.

The simulated observational network consists of nine Doppler radars located in a 3 × 3 regular pattern on the horizontal domain at the first model level (see Fig. 2). Each radar has a range of 120 km and measures the radial velocity at the model grid points. The radars provide a measurement at a given point within their range if and only if the vertical velocity at this point is rising and equal to or greater than 5 cm s−1 (this is a proxy for precipitation). The observations are obtained at each analysis time by calculating, from the truth run, the radial velocities with respect to the radar locations and by adding Gaussian noise with standard deviation of 1 m s−1 and zero mean (unbiased observations) in order to incorporate observational errors.

Figure 2 shows cross sections of the u-wind initial condition of the truth run, as well as the location and range of the Doppler radars. The u-wind initial condition has the form of two jets with opposite directions. The presence of the second opposing jet is a consequence of the choice of periodic boundary conditions. The space average Rossby (obtained from the modulus of the vertical vorticity divided by the Coriolis parameter) and Froude (modulus of the total horizontal vorticity divided by the buoyancy frequency) numbers are initially 1.0 and 0.45, respectively. The Rossby (Froude) number remains between 0.8 and 1.1 (0.4 and 0.5) during the integrations. These values are typical of dynamical conditions in the meso-β range. The u- and υ-wind initial conditions of the ensemble background fields are similar to the truth run initial condition with random differences of the order of 22 m s−1. Figure 3 shows an xy cross section at initial time with patches where measurements could potentially be made, that is, where the ascending velocity of the truth run is equal to or greater than 5 cm s−1. Note that the data coverage is relatively widespread, and that it is likely that all radars will provide data at each assimilation period. Although this can be seen as a limitation to the experimental setting in regard to real radar observations, we show in section 4 that sparser data do not alter the main results that are presented in this study.

Three main experiments are performed: a control experiment in which no data are assimilated, referred to as CONTROL; an experiment in which data are assimilated and the localization of 𝗣 is relatively weak—that is, the correlations are zero for distances greater than eight grid points (133 km in the horizontal direction and 1.33 km in the vertical direction)—referred to as A8; an experiment in which data are assimilated and the localization of 𝗣 is relatively strong—that is, the correlations are zero for distances greater than two grid points (33 km in the horizontal direction and 0.33 km in the vertical direction)—referred to as A2. Table 1 describes all the experiments, including the sensitivity tests (see section 4) performed in this paper. The external forcing F(y, z) and diffusion coefficients υh and υz were the same for experiments CONTROL, A8, and A2. The aspect ratio of the domain (1/100), and consequently of the dynamics, explains why the imposed decorrelation distance (kilometers) has been chosen to be 100 times shorter in the vertical direction compared to the horizontal direction. Tests without any vertical localization were performed with unsatisfying results, showing the necessity of using vertical localization with 50 ensemble members (see section 4).

Data are assimilated every 30 min, and 200 cycles are performed. The total assimilation period is 100 h. At t = 0, the initial ensemble members obtained from Eqs. (3.2) are integrated for a period of 30 min. The first assimilation procedure is performed at t = 30 min. Data are synthesized and valid at the assimilation time, meaning that the problem of data validity at periods different from the analysis time is not addressed here.

4. Results

a. Ensemble mean analysis error and standard deviation

The minimal condition measuring if an assimilation system produces an improved analysis is that the domain average analysis error must be smaller when data are assimilated than when no data are assimilated. Figure 4 shows the time evolution of the domain average difference modulus d between the truth run and the analysis ensemble mean, where
i1520-0493-134-2-618-e41
for winds and temperature for the three experiments CONTROL, A8, and A2. Here, d(t) is also referred to as the ensemble mean error, and J = 60 is the number of grid points in each spatial direction. Note that the mean over the 2N ensemble members has smaller ensemble mean error, but this error is not estimated by any ensemble standard deviation that can be computed by the double EnKF and has therefore not been considered here. The dashed–dotted (A8) and dotted (A2) curves both lie below the solid curves (CONTROL) for horizontal winds and temperature, indicating that both experiments A8 and A2 showed improvements over the CONTROL experiment for these variables. The analysis for the variables u, υ, and T of experiment A8 shows a better quality than for experiment A2, indicating that, under the present dynamical conditions, the covariances for distances greater than two grid points contribute significantly to the improvement of the analysis. The analyzed vertical velocity in experiment A8 is however considerably degraded with respect to the CONTROL experiment during the first 2 days of the assimilation period.

Although the model equations are nonhydrostatic, the geometry of the domain (aspect ratio of 1/100) ensures that the flow is very close to being in hydrostatic balance, at all scales but the smallest horizontal ones. A scale analysis of Eqs. (3.1) would show that nonhydrostaticity is confined to deep modes with high horizontal wavenumbers. A relatively strong initial adjustment perceivable in the vertical velocity occurs in experiment A8, producing oscillations with frequencies close to the buoyancy frequency in the background vertical velocity fields. This adjustment is related to a slight hydrostatic imbalance of the increments that generates strong vertical motion. It appears that during the first 10 h, the nonhydrostatic modes of the analyzed vertical velocity are of low quality in experiment A8, producing the initial degradation of the analyzed velocity. Figure 5 compares a vertical (xz) cross section of the analyzed vertical velocity for the first member of the first ensemble with the truth run at t = 5 h. Deep structures with high horizontal wavenumbers are visible in the analysis, indicating that nonhydrostatic modes are responsible for a large part of the error in the analyzed vertical velocity.

Experiment A2 does not show any sign of initial degradation of the analyzed vertical velocity. After 10 h (20 assimilation cycles), the initial adjustment in experiment A8 is attenuated and the analysis quality of the vertical velocity is similar in experiments A2 and A8 after 4 days of assimilation. As will be seen later on, the analysis quality of the vertical velocity in experiment A8 depends strongly on the shape of the horizontal wind power spectrum.

One remarkable feature is the quality of the temperature analysis. Although temperature is not an observed variable, its analysis is surprisingly good in both experiments A8 and A2. We will argue below that this is mainly caused by the improved horizontal wind fields advecting the temperature.

In an acceptable ensemble Kalman filter, the ensemble standard deviation must be of the same order as the analysis error of the ensemble mean. Figure 4 also shows the analysis ensemble standard deviation σ averaged over the two analysis ensemble standard deviations as a function of time for u, υ, w, and T. The equation for σ is
i1520-0493-134-2-618-e42
Note that both d(t) [Eq. (4.1)] and σ(t) [Eq. (4.2)] are first calculated on each separate ensemble, and then the results for the two ensembles are averaged. The subscript “b” (for background) replaces the subscript “a” when calculating quantities related to the guess fields. Examining Fig. 4, it is seen that the standard deviations in experiments A8 and A2 are very similar to their respective ensemble mean errors. Figures 4 does not show the values for guess field ensemble standard deviations and ensemble mean errors since they follow very closely the curves shown on this figure. This is due to the relatively short time between each assimilation procedure compared with the doubling time of small errors (see Table 2, described below, and section 4h).

An examination of the ensemble mean error and ensemble standard deviation as a function of space at different times for experiment A8 shows how the radar data impact locally on the analyses. Figures 6 and 7 depict the vertical average of the ensemble mean error and ensemble standard deviation, respectively, after the first assimilation cycle. These calculations are performed first on each separate ensemble, then the two results are averaged. As expected, the ensemble mean error and ensemble standard deviation are smaller at the radar locations for the two components of the horizontal velocity (the radars measure radial velocities formed mainly with horizontal winds) than at places located outside the range of the radars. Patches of smaller analysis error and standard deviation are elongated in the x (y) direction for the u (υ) wind component, according to the radar line of sight. Figure 7 also reveals that the standard deviation of analyses of the vertical velocity (before the model is restarted for the first time from the result of the EnKF analyses) is smaller at the radar locations than at places located outside the range of the radars. The ensemble standard deviation of the temperature analyses does not seem to be reduced at the radar locations (see lower-right panel of Fig. 7). This is expected since it was noted earlier that the correlation between the horizontal winds and temperature of the initial ensembles is close to zero. The ensemble mean error is inherently more noisy than the ensemble standard deviation since the former is calculated from the difference between the ensemble mean and the unique truth field [similar to Eq. (4.1), except that J3 is replaced by J and the sum is over the vertical only] whereas the latter is calculated from the difference between the ensemble mean and an ensemble of perturbation fields [similar to Eq. (4.2)]. This makes it difficult to base conclusions on the lower panels of Fig. 6.

After 5 h of assimilation, the temperature ensemble mean error and standard deviation are significantly reduced slightly downstream of the radar locations, as is visible from Figs. 8 and 9 around Y = 10 and Y = 50, indicating that the more accurate horizontal wind fields have a lot to do in the reduction of temperature ensemble mean error and standard deviation. Figure 9 also shows the increase of vertical velocity standard deviation near the radar locations, pointing out the adjustment problem discussed previously. Table 2 shows the domain average impact of the assimilation procedure on winds and temperature at the first and tenth assimilations. The guess field and analysis mean error and standard deviation obtained from Eqs. (4.1) and (4.2) are shown for experiment A8. It is seen that the impact of the assimilation procedure per se on temperature is rather modest, and that the improved temperature analysis seen on Fig. 8 is very likely due to the quality of the analyzed horizontal winds advecting the temperature, as stated earlier. Table 2 also shows the modest impact of the assimilation procedure on improving the background vertical winds.

Table 2 also shows that the background temperature standard deviation is slightly smaller than the analysis temperature standard deviation during the first assimilation cycle for experiment A8. In principle, this should not happen when the number of members tends to infinity. In practice, due to sampling errors, both the single and double EnKF with perturbed observations can produce a standard deviation of the analysis greater than the background standard deviation. For instance, nonzero correlation between the perturbed observations and the background field and/or sampling error in the estimation of observational errors can produce larger analysis variance than background variance. Finally, we may note that when the true correlation between temperature and horizontal wind fields is zero by design, wind observations cannot lead to a reduction of the temperature error, as confirmed here by the very slight increase of temperature standard deviation at the first analysis.

Despite the low quality of the vertical velocity analysis during the first 2 days of assimilation, there is no obvious feedback on the quality of the analysis of the other dynamical variables.

b. Impact of external gravity modes

External gravity modes are particular solutions of Eqs. (3.1) provided that the dynamics is unforced, linear, and nondissipative. Moreover, these modes are independent of the altitude z. From these conditions, it follows that u = υ = 0 and that ω2 = ab for external gravity modes.

Two factors suggest that the presence of external gravity modes is responsible for the initial degradation of vertical wind analyses in experiment A8: 1) the vertical velocity oscillates with frequencies very close to the buoyancy frequency, and 2) as is shown in Fig. 5, the vertical velocity of the ensemble members has deep structures. Moreover, since radars measure mainly horizontal winds in the experiments described in this paper, the assimilated data can hardly describe external gravity modes for which u = υ = 0 (in the linear limit). This implies that once the increments of vertical velocity are contaminated with external gravity modes, the subsequent assimilation of more radar data can hardly help to eliminate this contamination.

We believe that the localization procedure in the assimilation can have a strong negative impact on the vertical velocity in experiment A8, although the ensemble initialization described by Eqs. (3.2) could potentially contribute to this degradation. The three experiments (CONTROL, A8, and A2) employ the same initial ensemble, but only A8 is affected by a degradation of the vertical velocity as data are assimilated. Experiments CONTROL and A2 do not show unrealistically strong gravity modes. This could be due to the fact that the first initialization of the ensemble is performed on scales larger than five grid points only. The localization procedure in A8 allows nonzero correlations up to a distance of eight grid points, allowing the initial ensemble to generate an impact, if any. On the other hand, experiment A2 does not allow any nonzero correlations for distances greater than two grid points, thus reducing the initial ensemble interaction with the assimilation procedure (remember that the initial ensemble members are identical on scales shorter than five grid points). In section 4f, the question of the initial ensembles is studied further.

This problem could have been reduced by filtering the external gravity modes from the vertical wind and temperature increments. It can be done by imposing w and T increments with null spectral elements for m = 0. This procedure has been applied on the first analyses of experiment A8 with satisfying results, as is seen from Fig. 10. This figure depicts the time evolution of the four dynamical fields at a randomly selected point of an ensemble member in experiment A8, from the initial time (t = 0) to the end of the second ensemble of 30-min forecasts (t = 60 min). It is noted that the horizontal wind components (without external gravity mode filtering of the w and T increments) slightly oscillate with frequency ab after the first restart. This has to be due to nonlinearities. It is seen that external gravity mode filtering of the analysis increments of w and T reduces the contaminating oscillation of the four dynamical fields. The same result applies equally to all the other ensemble members (not shown). A series of assimilation cycles with external gravity mode filtering has been shown to improve greatly the vertical velocity analyses (see section 4e).

c. Sensitivity to the kinetic energy power spectrum

The large-scale forcing and dissipation coefficients of experiments CONTROL, A8, and A2 were tuned to keep the total energy approximately constant during the 100 h of integration, while generating no small-scale accumulation in the high-wavenumber range of the horizontal wind power spectrum. Sensitivity experiments were conducted to test the robustness of the previous results to the shape of the wind power spectrum, in particular to the increase of small-scale motion.

In this set of new experiments, the large-scale forcing of the truth run and the background fields is turned off and all the dissipation coefficients are decreased by a factor of 10. Similar experiments to CONTROL, A8, and A2 are performed for a period of 50 h (100 assimilation cycles). These experiments are referred to as sd_CONTROL, sd_A8, and sd_A2, respectively (“sd” stands for “small dissipation”). In this set of new experiments, there is a loss of 10% of the initial total energy after 50 h of integration.

Figure 11 depicts the horizontal wind power spectra of the two truth runs (moderate dissipation with large-scale forcing versus weak dissipation and no large-scale forcing) at time equals 48 h, showing the increase of small-scale motion in the new set of experiments. Figure 11 also shows a reference −3 slope, which is characteristic of two-dimensional turbulence, and a −5/3 slope, which is characteristic of three-dimensional turbulence. Note that the slope of experiment sd_CONTROL in the high-horizontal-wavenumber range is less steep than the reference −5/3 slope, indicating that the presence of small-scale motion is overestimated in experiments sd_CONTROL, sd_A8, and sd_A2 as compared to fully developed three-dimensional turbulence.

It turns out that the quality of the analysis is fairly sensitive to the shape of the high-wavenumber range of the power spectrum. Figure 12 shows that the analysis error of the vertical velocity in experiment sd_A8 is degraded and that there is no obvious sign of recovery within 2 days. The quality of the vertical velocity analysis in experiment sd_A2 shows no sign of degradation when compared to sd_CONTROL. For the first 15 h or so, the analysis in experiment sd_A8 is better than in experiment sd_A2 for the horizontal winds and temperature, but the bad vertical velocity analysis in experiment sd_A8 feeds back on the horizontal wind and temperature fields, degrading the analyses of these fields when compared to experiment sd_A2. The presence of energetic and unpredictable small-scale motion shortens the correlation lengths calculated from the ensembles and renders the use of a flow-dependent background error covariance matrix almost irrelevant.

d. Sensitivity to the number of assimilated observations

Results from experiments A8 and A2 were obtained by assimilating around 22 000 radial wind observations at each assimilation time. To simulate measurement conditions with the dry Boussinesq model resembling those of Doppler radars, a proxy for precipitation was set by allowing measurements only when the vertical velocity is equal or greater than 5 cm s−1 (wwc = 5 cm s−1).

Tests have been performed in which the number of radial wind observations has been reduced by a factor of around 4, that is, around 5500 observations at each assimilation time, by choosing wc = 12.5 cm s−1 instead of 5 cm s−1. A sensitivity experiment called lo_A8 (“lo” stands for “less observations”) employs the same settings as in experiment A8, except for the value of wc, and has been performed for 100 assimilation cycles. Figure 13 shows the ensemble mean errors (averaged over the domain) as a function of time for experiment lo_A8. It turns out that the ensemble mean error (as well as the ensemble standard deviation; not shown) for the variables u, υ, and T in experiment lo_A8 continue to be reduced by the assimilation, but the reduction is at a slower rate than in experiment A8. The ensemble mean error (and standard deviation; not shown) of w in lo_A8 continue to be degraded by the assimilation procedure, and it appears that the recovery time scale is much longer than 2 days.

e. Impact of filtering the external gravity modes of the increments

The filtering procedure described in section 4b has been applied in a sensitivity experiment employing the same settings as experiment lo_A8. This experiment using increment filtering is called flo_A8. It is seen in Fig. 13 that this filtering procedure is successful in reducing the level of noise in the vertical velocity increments and that the mean error of the analyzed vertical velocity is reduced by the assimilation procedure using filtered increments. Figure 13 also shows that the quality of the horizontal wind and temperature analyses is not affected by this filtering.

f. Sensitivity to the choice of initial ensembles

To verify if the degradation of the analyzed vertical velocity is caused by the specific choice of initial ensembles [essentially, by Eqs. (3.2)], an alternative method of generating the initial ensembles has been tested. It consists of integrating 2 × N + 1 (in this case, 101) members from t_ = −2 days to t0 = 0 and taking the initial ensembles starting the assimilation procedure at t0 to be the 2 × N first realizations at t = t0. The 101st realization at t0 is the initial condition of the truth. At t_, the 101 initial conditions of the 101 realizations are achieved by slightly perturbing the amplitudes of u and υ from an analytic initial condition. Other than that, the same settings as in experiment lo_A8 are used. This experiment is referred to as iclo_A8. An experiment, called iclo_CONTROL and in which no data are assimilated, was also performed for comparison.

Figure 14 shows that the mean error of the analyzed vertical velocity also increases initially when this alternative method is employed to generate the initial ensembles. Thus, the ensemble initialization described by Eqs. (3.2) is likely not responsible for the degradation of the analyzed vertical velocity in experiments A8 and lo_A8. The hydrostatic imbalance present in the analysis increments, most probably caused by localization, seems to be at the source of the problem.

Moreover, another test was performed to validate this result. Twenty-five cycles of assimilation were done starting from the 76th cycle of experiment flo_A8 (after 38 h of assimilation), but without filtering the analysis increments (experiment nflo_A8). It turns out that despite being relatively far from t0 (far from the initial conditions), the vertical velocity analysis keeps being degraded by the assimilation procedure (see Fig. 13).

Note that Fig. 14 shows that the filtering procedure described above is still efficient in increasing the quality of the analyzed vertical velocity when using the alternative initial ensembles (experiment ficlo_A8).

g. Sensitivity to the vertical localization

Two more experiments were done to quantify the impact of vertical localization alone on experiment iclo_A8. In one of these two experiments, the horizontal localization was kept at eight grid points, but the vertical localization was removed (experiment hiclo_A8). In the other one, the horizontal localization was also kept at eight grid points, but the vertical localization was set to two grid points (experiment iclo_A8_2).

From Fig. 15, it is seen that experiment hiclo_A8 is underdispersive and that the ensemble mean error of the four dynamical variables is not reduced (or even increases) after one day of assimilation. This shows the necessity of using vertical localization in the present experiments with 50 ensemble members. Moreover, the underdispersive nature of experiment hiclo_A8 can be an indication that the rank of the background covariance matrix is too low without vertical localization.

Figure 15 also shows that for strong vertical localization (experiment iclo_A8_2), the vertical velocity analysis continues to be degraded by the assimilation procedure (as opposed to experiment A2 versus experiment A8), but the degradation is less acute than when the vertical localization is done over eight grid points (iclo_A8, comparing with Fig. 14). On the other hand, the horizontal wind components and the temperature analyses are degraded compared with the experiment performed using three-dimensional localization at eight grid points (again comparing with Fig. 14).

h. Small-error doubling time scale of the Boussinesq model

To characterize the error growth of the Boussinesq model in a configuration identical to experiment A8, an ensemble of 2-day forecasts has been performed for which the initial conditions of an ensemble of integrations were given by the analyses of experiment A8 at t = 52 h. The ensemble standard deviation at t = 52 h is relatively small, allowing one to calculate the doubling time of small errors. The error doubling time of global numerical prediction models (with horizontal resolution of about 50–80 km) is of the order of 2 days for the 500-hPa geopotential (e.g., Lorenz 1982, 1990; Simmons et al. 1995; Charron 2002). This small-error doubling time is due to imperfect initial conditions, as well as model deficiencies. Following arguments given in Lorenz (1982, 1990), a rough estimate of the small-error doubling time solely due to imperfect initial conditions can be obtained from differences between i-day and (i + l)-day forecasts valid at the same time. In Charron (2002), this time scale is of the order of 3 days for the 500-hPa geopotential for the Global Environmental Multiscale (GEM) operational model at the Canadian Meteorological Centre.

In the context of the present work, the small-error growth is solely due to imperfect initial conditions since the truth run comes from the same model as the background fields. It is interesting to see how the small-error doubling time of the Boussinesq model in the present configuration compares with the small-error doubling time of numerical weather prediction models. Figure 16 depicts the u, υ, w, and T ensemble standard deviations of experiment A8 between t = 52 h and t = 100 h, and of an ensemble 2-day forecast with initial conditions given by the ensemble of analyses at t = 52 h. One can see that the ensemble standard deviation doubling time due to imperfect initial conditions is about 24 h for υ, and 36 h for u and T, that is, 2–3 times faster than the estimated doubling time due to imperfect initial conditions of the 500-hPa geopotential of typical global numerical weather prediction models. Note that in a numerical prediction model, errors in u and υ are also expected to grow faster than errors in the geopotential due to the presence of finer scales. However, the ensemble standard deviation of the forecasted vertical velocity decreases with time, indicating that the relative quality of the vertical velocity field after day 2 in experiment A8 is not due to a good vertical velocity analysis per se, but rather to the feedback on w of the other dynamical variables, and possibly to the cascade of external gravity modes toward the dissipative scale.

These results of estimates of small-error doubling time scales of the Boussinesq model are an a posteriori indication that the model has realistic internal variability when compared to the situation found in the real atmosphere.

5. Summary and conclusions

Simulated radial wind data have been assimilated in the context of a perfect nonhydrostatic Boussinesq model with an EnKF. The small aspect ratio of the chosen domain imposes that the flow of the truth is very close to being in hydrostatic balance at large (more energetic) scales, and nonhydrostatic modes (less energetic) are confined to deep motion with high horizontal wavenumbers. It turns out that because of the limitations of the analysis procedure, essentially the lack of hydrostatic balance of the analysis increments, and because of the type of observations that has been employed (the Doppler radars do not “see” external gravity modes), a part of the analysis increments generates strong vertical adjustments when the covariance matrix 𝗣 is localized. This tends to initially degrade the quality of the vertical wind analysis because of external gravity mode contamination. It has been shown that filtering the external gravity modes from the analysis increments of w and T improves the assimilation procedure and that contaminating fast gravity waves can be almost completely eliminated. Note that external gravity modes do not exist when the lower and upper boundaries are rigid and when the flow is incompressible. At the model resolution employed in this study, it is expected that the problem of increment contamination by these modes would be reduced when using rigid upper and lower boundaries.

Horizontal winds are successfully analyzed with 𝗣 not too strongly localized and without any direct filtering of their analysis increments. The high quality of the temperature analysis, although no temperature data are used during the assimilation process, is most likely due to the more accurate horizontal wind fields advecting the temperature. These results are dependent on the horizontal wind power spectral shape. Too much small-scale motion (experiment sd_A8) prevents the recovery of the initially bad vertical velocity analysis, and this in turn feeds back on the horizontal winds and temperature.

These results have been shown to hold when employing an alternative method of initializing the ensembles at t = 0. In particular, it has been shown that when starting the assimilation procedure with 2N members that are the result of 2N relatively long integrations [not using Eqs. (3.2) to create the initial ensembles], the vertical velocity analyses continue to be degraded. Again, this degradation can be removed by filtering the external gravity modes from the analysis increments of w and T.

The impact of using different localization length scales of the 𝗣 matrix have also been quantified. It turns out that moderate localization (eight grid points) in the three physical directions provided the best results for the experimental settings used in this paper, as long as external gravity waves are filtered out from the analysis increments. The usefulness of employing moderate vertical localization to improve the horizontal wind and temperature analyses has been demonstrated.

The inclusion of moist processes to the present experiments would allow a more realistic treatment of the synthetic observations and might modify the variability of the model.

In an operational context, it is likely that the model used to simulate the meso-β scale would be hydrostatic, making the issue of the adjustment caused by the increments less acute. Such a model would also contain more physical processes that would alter the predictability of the flow through, for example, self-organization. The question of knowing the impact of the physical processes on the behavior of the EnKF at the meso-β scale has not been investigated here.

The results of this study indicate that the EnKF can be an efficient assimilation technique at the meso-β scale. Low rank issues associated with a small number of ensemble members compared with the large degrees of freedom of the atmosphere can be controlled with three-dimensional localization of the impact of the observations. The importance of employing a flow-dependent error covariance matrix 𝗣 as opposed to a flow-independent one in the assimilation procedure at the meso-β scale has not been specifically quantified here, but will be the topic of a subsequent study.

Acknowledgments

The authors are thankful to S. Laroche, A. Zadra, R. de Elia, L. Fillion, and two anonymous reviewers for suggestions that led to improvements of the manuscript. Part of this work was funded by the Canadian Foundation for Climate and Atmospheric Sciences (CFCAS).

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Fig. 1.
Fig. 1.

A yz cross section of the external forcing (m s−1 day−1) exerted on the u-wind component. The forcing is symmetric in the x direction. Contour intervals are 0.2 m s−1 day−1

Citation: Monthly Weather Review 134, 2; 10.1175/MWR3081.1

Fig. 2.
Fig. 2.

(top) An xy cross section (z = 30) of the u-wind component (m s−1) truth run initial condition. The nine circles indicate the location and range of the Doppler radars. (bottom) A yz cross section (x = 30) of the u-wind component (m s−1) truth run initial condition. Contour intervals are 10 m s−1

Citation: Monthly Weather Review 134, 2; 10.1175/MWR3081.1

Fig. 3.
Fig. 3.

An xz cross section (y = 15) of the truth run initial condition. Dark patches indicate regions where w ≥ 5 cm s−1, i.e., where radars are able to measure radial velocities if their range allows it

Citation: Monthly Weather Review 134, 2; 10.1175/MWR3081.1

Fig. 4.
Fig. 4.

(top left) The u-wind ensemble mean error of the analyses (averaged over the two ensembles and over space) for experiments A8 (dotted–dashed thick line) and A2 (dotted thick line). The solid thick line is the mean error of the forecast without assimilating data (experiment CONTROL). (top right) As in top-left panel, but for the υ-wind component. (bottom left) As in top-left panel, but for the w-wind component. (bottom right) As in top-left panel, but for temperature. Thin curves are the ensemble standard deviations

Citation: Monthly Weather Review 134, 2; 10.1175/MWR3081.1

Fig. 5.
Fig. 5.

(top) An xz cross section of the w wind of the first member of the first ensemble (y = 15) after the 10th assimilation cycle for experiment A8 (cm s−1). (bottom) As in top panel, but for the truth run. Contour intervals are 20 cm s−1

Citation: Monthly Weather Review 134, 2; 10.1175/MWR3081.1

Fig. 6.
Fig. 6.

(top left) The u-wind ensemble mean error in experiment A8 after the first assimilation cycle, i.e., at t = 30 min (in m s−1, averaged over the two ensembles and over the vertical) as a function of x and y. Darker areas indicate lower values. Contour intervals are 0.4 m s−1. (top right) As in top-left panel, but for the υ-wind component. Contour intervals are 0.4 m s−1. (bottom left) As in top-left panel, but for the w-wind component (cm s−1). Contour intervals are 1 cm s−1. (bottom right) As in top-left panel, but for temperature (K). Contour intervals are 0.15 K

Citation: Monthly Weather Review 134, 2; 10.1175/MWR3081.1

Fig. 7.
Fig. 7.

(top left) The u-wind ensemble standard deviation in experiment A8 after the first assimilation cycle, i.e., at t = 30 min (m s−1, averaged over the two ensembles and over the vertical) as a function of x and y. Darker areas indicate lower values. Contour intervals are 0.4 m s−1. (top right) As in top-left panel, but for the υ-wind component. Contour intervals are 0.4 m s−1. (bottom left) As in top-left panel, but for the w-wind component (cm s−1). Contour intervals are 0.5 cm s−1. (bottom right) As in top-left panel, but for temperature (K). Contour intervals are 0.05 K

Citation: Monthly Weather Review 134, 2; 10.1175/MWR3081.1

Fig. 8.
Fig. 8.

As in Fig. 6, but after the 10th assimilation cycle. Contour intervals are (top) 0.4 m s−1, (bottom left) 4 cm s−1, and (bottom right) 0.2 K

Citation: Monthly Weather Review 134, 2; 10.1175/MWR3081.1

Fig. 9.
Fig. 9.

As in Fig. 7, but after the 10th assimilation cycle. Contour intervals are (top) 0.4 m s−1, (bottom left) 4 cm s−1, and (bottom right) 0.2 K

Citation: Monthly Weather Review 134, 2; 10.1175/MWR3081.1

Fig. 10.
Fig. 10.

(top left) The u wind at point (x = 15, y = 15, z = 15) as a function of time for the first member of the first ensemble (m s−1). The solid line represents the evolution of u starting at the initial time and ending at the first analysis time; the dotted–dashed line represents the evolution of u without external gravity mode filtering of the w and T increments (experiment A8), starting after the first analysis and ending before the second analysis; the dashed line represents the evolution of u with external gravity mode filtering of the w and T increments, starting after the first analysis and ending before the second analysis. (top right) As in top-left panel, but for the υ-wind component. (bottom left) As in top-left panel, but for the w-wind component (cm s−1). (bottom right) As in top-left panel, but for temperature (K)

Citation: Monthly Weather Review 134, 2; 10.1175/MWR3081.1

Fig. 11.
Fig. 11.

Horizontal wind power spectrum averaged over the vertical of experiments CONTROL (thick solid line) and sd_CONTROL (thin solid line) at time equals 48 h. Reference spectra with slope −5/3 (dashed) and −3 (dotted–dashed) are also shown; n is the total horizontal wavenumber multiplied by 1000/(2π) km

Citation: Monthly Weather Review 134, 2; 10.1175/MWR3081.1

Fig. 12.
Fig. 12.

As in Fig. 4, but for the case with no large-scale forcing, a reduced dissipation, and an assimilation period of 2 days (experiments sd_CONTROL, sd_A8, and sd_A2). Only ensemble mean errors are shown. Standard deviations (not shown) are very close to mean errors for all variables

Citation: Monthly Weather Review 134, 2; 10.1175/MWR3081.1

Fig. 13.
Fig. 13.

As in Fig. 4, except that the number of assimilated observations per cycle has been reduced by a factor of about 4. The solid lines are for experiment CONTROL; the dotted–dashed lines are for experiment lo_A8; the dashed lines are for experiment flo_A8; the double-dotted–dashed lines are for experiment nflo_A8. Only mean errors are shown. The ensemble std devs (not shown) are very close to the ensemble mean errors for all variables. Note that the curves for u, υ, and T for experiments lo_A8, flo_A8, and nflo_A8 are almost overlying

Citation: Monthly Weather Review 134, 2; 10.1175/MWR3081.1

Fig. 14.
Fig. 14.

As in Fig. 4, except that the number of assimilated observations per cycle has been reduced by a factor of about 4 and the initial ensembles are obtained from 2N long integrations. The solid lines are for experiment ic_CONTROL; the dashed lines are for experiment iclo_A8; the double-dotted–dashed lines are for experiment ficlo_A8. Only mean errors are shown. The ensemble std devs (not shown) are very close to the ensemble mean errors for all variables. Note that the curves for u, υ, and T for experiments iclo_A8, and ficlo_A8 are almost overlying

Citation: Monthly Weather Review 134, 2; 10.1175/MWR3081.1

Fig. 15.
Fig. 15.

As in Fig. 14, except that the solid lines are for experiment hiclo_A8 and the dashed lines are for experiment iclo_A8_2. The thin lines represent the ensemble std devs

Citation: Monthly Weather Review 134, 2; 10.1175/MWR3081.1

Fig. 16.
Fig. 16.

(top left) The solid curve is the forecasted u-wind ensemble std dev (m s−1; averaged over the two ensembles and over space) as a function of time when the initial conditions are the analyses at t = 52 h from experiment A8; the dotted–dashed curve is as in Fig. 4 for 52 h < t < 100 h. (top right) As in top-left panel, but for the υ-wind component. (bottom left) As in top-left panel, but for the w-wind component (cm s−1). (bottom right) As in top-left panel, but for temperature (K)

Citation: Monthly Weather Review 134, 2; 10.1175/MWR3081.1

Table 1.

List of experiments and their main characteristics

Table 1.
Table 2.

Domain average mean error and ensemble std dev for analyses and guess fields in experiment A8 at the first and tenth assimilations. Guess field values are in parentheses

Table 2.
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