Breakdown of hydrostatic balance at convective ... - Wiley Online Library

Quarterly Journal of the Royal Meteorological Society

Q. J. R. Meteorol. Soc. 138: 1709–1720, October 2012 A

Breakdown of hydrostatic balance at convective scales in the forecast errors in the Met Office Unified Model S. Vetra-Carvalho,a *† M. Dixon,b‡ S. Migliorini,c† N. K. Nicholsa† and S. P. Ballard b‡ a

Department of Mathematics, University of Reading, UK b MetOffice@Reading, University of Reading, UK c Department of Meteorology, University of Reading, UK *Correspondence to: S. Vetra-Carvalho, Department of Mathematics, School of Mathematical and Physical Sciences, University of Reading, PO Box 220, Reading, RG6 6AX, UK. E-mail: [email protected] † The work of these authors was supported in part by a NERC CASE award with the Met Office and in part by the NERC National Centre for Earth Observation (NCEO). ‡ The contribution of these authors was written in the course of their employment at the Met Office, UK, and is published with the permission of the Controller of HMSO and the Queen’s Printer for Scotland.

For data assimilation in numerical weather prediction, the initial forecast-error covariance matrix Pf is required. For variational assimilation it is particularly important to prescribe an accurate initial matrix Pf , since Pf is either static (in the 3D-Var case) or constant at the beginning of each assimilation window (in the 4D-Var case). At large scales the atmospheric flow is well approximated by hydrostatic balance and this balance is strongly enforced in the initial matrix Pf used in operational variational assimilation systems such as that of the Met Office. However, at convective scales this balance does not necessarily hold any more. Here we examine the extent to which hydrostatic balance is valid in the vertical forecast-error covariances for highresolution models in order to determine whether there is a need to relax this balance constraint in convective-scale data assimilation. We use the Met Office Global and Regional Ensemble Prediction System (MOGREPS) and a 1.5 km resolution version of the Unified Model for a case study characterized by the presence of convective activity. An ensemble of high-resolution forecasts valid up to three hours after the onset of convection is produced. We show that at 1.5 km resolution hydrostatic balance does not hold for forecast errors in regions of convection. This indicates that in the presence of convection hydrostatic balance should not be enforced in the covariance matrix used for variational data assimilation at this scale. The results show the need to investigate covariance models that may be better suited for convective-scale data assimilation. Finally, we give a measure of the balance present in the forecast perturbations as a function of the horizontal scale (from 3–90 km) using a set of c 2012 Royal Meteorological Society and British Crown diagnostics. Copyright Copyright, the Met Office Key Words: high resolution numerical weather prediction; background covariances; ensemble data assimilation; MOGREPS Received 29 May 2010; Revised 24 November 2011; Accepted 14 February 2012; Published online in Wiley Online Library 12 April 2012 Citation: Vetra-Carvalho S, Dixon M, Migliorini S, Nichols NK, Ballard SP. 2012. Breakdown of hydrostatic balance at convective scales in the forecast errors in the Met Office Unified Model. Q. J. R. Meteorol. Soc. 138: 1709–1720. DOI:10.1002/qj.1925

c 2012 Royal Meteorological Society and Copyright British Crown Copyright, the Met Office

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1.

S. Vetra-Carvalho et al.

Introduction

Due to continuing increases in computer power, it has become possible for meteorological centres to run high-resolution models. These models are expected to produce more realistic forecasts because of their better representation of small-scale forcing from orography and land use, as well as their explicit representation of convection. Modelling at high resolution should thus lead to more accurate forecasting of high-impact weather events such as flooding, with potential social and economic benefits. Another important advantage of high-resolution models is that these may be used to assimilate high-resolution observations, e.g. from radar. To do so, it is important that the assumptions made for assimilation are still valid at these scales. For example, data-assimilation systems used by operational meteorological centres such as the Met Office assume that the errors in the state variables of the model are hydrostatically balanced. While this assumption is reasonable for large-scale models, it does not necessarily hold for perturbations at convective scales. It is then of interest, and an aim of this article, to determine whether (and to what extent) forecast errors are hydrostatically balanced for a 1.5 km horizontal resolution version of the Met Office Unified Model (UM). To this end, a set of high-resolution forecasts is generated for a case study on 27 July 2008 using the Met Office ensemble prediction system and the 1.5 km Unified Model over convective and non-convective regions of the southern UK for a period of up to three hours after the onset of convection. The validity of hydrostatic balance in the forecast errors is examined for four selected vertical columns in the domain of the high-resolution 1.5 km model. In addition, the degree to which hydrostatic balance holds in the forecast perturbations is investigated as a function of the horizontal scale, by coarsening the 1.5 km data to produce 3–90 km resolution datasets. The ensemble prediction system at convective scales used for this work is briefly described in section 2. The derivation of the hydrostatic balance relations for perturbations of the relevant fields is presented in section 3. In section 4 the details of a case study comparing hydrostatically balanced and full perturbations are presented. In sections 5 and 6 the results of this study are analysed and discussed in terms of their vertical correlation structures, amplitudes and mass-weighted standard deviations of perturbations. 2.

Ensemble prediction system at convective scales

Over the past few years, the Met Office has developed an ensemble prediction system, the Met Office Global and Regional Ensemble Prediction System (MOGREPS), which employs an ensemble transform Kalman filter (ETKF) to produce a set of initial conditions for generating predictions using the UM (Bowler et al., 2008). In this article, the 24 km horizontal resolution initial conditions from the operational North Atlantic European (NAE) version of MOGREPS are interpolated to a 1.5 km resolution grid and used to integrate forward a high-resolution version of the UM for three hours. In this section descriptions of the ETKF algorithm, the convective-scale version of the UM and the interpolation strategy used to downscale the initial condition ensemble are provided. c 2012 Royal Meteorological Society and Copyright British Crown Copyright, the Met Office

2.1.

The Ensemble Transform Kalman Filter

The ensemble Kalman filter (EnKF) (Evensen, 1994) is a useful approximation to the Kalman filter (KF) (Kalman, 1960), as it represents the errors in the state of a given system – composed, in the case of operational numerical weather prediction (NWP) models, of a number of variables of the order of 107−8 – with a basis spanning only a much lower dimensional subspace of the state space. As in the case of the Kalman filter, the EnKF assumes all errors to be Gaussian. In the case of forecast errors of nonlinear models, such as those used for NWP, this assumption may often not be valid. Nevertheless, there are indications that the EnKF behaves well even when the errors are not Gaussian (Bowler et al., 2008). In the EnKF, the ensemble matrix X is defined as X = (x1 , x2 , . . . , xN ) ,

(1)

where xi ∈ Rk is a state vector representing the ith ensemble member, k is the number of components in the state vector and N is the number of ensemble members (equal to 24 in the version of MOGREPS used in this study). The ensemble mean is represented by x¯ and the ensemble perturbation matrix is given by X = (x1 − x¯ , x2 − x¯ , . . . , xN − x¯ ).

(2)

Superscripts ‘f’ and ‘a’ are used as in Xf , Xa to mean forecast and analysis, respectively. Thus, from the ensemble forecasts the background-error covariance can be estimated by Pf =

X f (X f )T . N −1

The ensemble mean analysis is updated by x¯ a = x¯ f + K y − H(¯xf ) ,

(3)

(4)

where y are the observations and H is the observation operator. The Kalman gain K is given by −1 K = Pf HT HPf HT + R ,

(5)

where R is the observation-error covariance matrix and H is a linearized version of H. The ETKF is very closely related to the EnKF, allowing for a rapid calculation of the analysis perturbations (Bowler et al., 2008). The analysis-ensemble perturbations in the ETKF are given by X = X Tm , a

f

(6)

where T is a transform matrix and m is an inflation factor (discussed below) for forecast cycle m. The transform matrix is computed according to Wang et al. (2004) as T = C ( + I)−1/2 CT ,

(7)

where C, are matrices composed of the eigenvectors and eigenvalues respectively of the matrix −1/2 f T −1/2 f R Z Z R E= . N −1 Q. J. R. Meteorol. Soc. 138: 1709–1720 (2012)

(8)

Hydrostatic Balance Breakdown at Convective Scales

The columns of Zf are given by zfi = H(xif ) − H(¯xf ), which are the ensemble perturbations in observation space for each ensemble member i and are found using the nonlinear observation operator. The variance in an ensemble generated by an EnKF is often smaller than required, due to an insufficient ensemble size (Bowler et al., 2008). To overcome this problem a variable inflation factor is used, which seeks to ensure that the perturbation spread matches the root-mean-square error (RMSE) of the mean forecast and is given by 1/2 [tr(dm dTm ) − tr(R)]tr(Sm−1 ) m = m−1 , tr(Sm )

(9)

where dm = y − H(Xf ) is the ’average’ innovation vector and Sm = HPf HT is the spread of the forecast ensemble in observational space for forecast cycle m. (Here H(Xf ) denotes the matrix with columns given by H(xif ) and H(Xf ) denotes the vector average of these columns.) The values of m used here are vertically and horizontally uniform and typical values for m are in the range 2–8 (see figures 8 and 10 of Migliorini et al., 2011). 2.2. The Met Office Unified Model The 1.5 km version of the UM applied in this work uses non-hydrostatic deep-atmosphere equations with a hybrid height/terrain-following vertical coordinate (Davies et al., 2006). The model has staggered grids in the horizontal and the vertical. The Arakawa C-grid is used for horizontal staggering, where the zonal velocity component u is east–west staggered and the temperature and the meridional velocity component v are north–south staggered. The Charney–Phillips grid is used for vertical staggering, where potential temperature is on the same levels as the vertical velocity. The high-resolution model has a grid length of 1.5 km with 360 grid points in latitude and 288 in longitude covering southern England and Wales. The model has a grid with 70 vertical levels, where only about the 50 lowermost levels are affected by orography. The 24 km NAE model has 360 grid points in latitude, 215 in longitude and 38 vertical levels. The current data-assimilation (DA) system for the 1.5 km model is similar to that used with the operational UK 4 km Met Office model, which is discussed in detail in Dixon et al. (2009). In summary, the DA combines a threedimensional variational (3D-Var) scheme, used to assimilate the conventional observations producing the large-scale analysis, and nudging systems, used to update the highresolution moisture and surface precipitation data. The system uses a cloud-nudging (CN) procedure to nudge humidity increments, whereas surface precipitation rates are assimilated via latent-heat nudging (LHN). The increments produced by 3D-Var and nudging procedures are used to correct the model trajectory at each time step during the DA window. The main differences between the 1.5 km and 4 km model DA configurations are as follows. The former uses hourly assimilation cycles, rather than the three-hourly ones used in the operational 4 km system, starting from fields interpolated to the southern England and Wales 1.5 km grid from the operational UK 4 km forecast. The 1.5 km DA system also uses more frequent cloud (hourly) and precipitation (every 15 min) observations. c 2012 Royal Meteorological Society and Copyright British Crown Copyright, the Met Office

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2.3. Generation of high-resolution perturbations To obtain the initial-condition ensemble at 1.5 km resolution, the following steps are performed (see Figure A1 in Appendix A). First, an ensemble of atmospheric fields is formed by adding an ensemble of operational MOGREPS NAE analysis perturbations at 24 km resolution valid at a given time (1800 UTC), denoted as X 24 km , to the operational 4D-Var atmospheric analysis at 1800 UTC over the NAE domain, denoted as x¯ 24 km , resampled at a resolution of 24 km from its original resolution of 12 km. This 24 km resolution ensemble, denoted as X24 km , and the 4D-Var control, x¯ 24 km , are interpolated to a 1.5 km resolution grid to obtain X1.5/24 km and x¯ 1.5/24 km , respectively. The ensemble of perturbations at 1.5 km resolution, denoted as X 1.5/24 km , is then obtained by subtracting the reconfigured 4D-Var analysis x¯ 1.5/24 km from each column of the ensemble matrix X1.5/24 km . Finally, to obtain the ensemble of initial conditions at 1.5 km resolution at 1800 UTC, denoted as X1.5 km , the 1.5 km ensemble perturbations X 1.5/24 km are combined with a 3D-Var high-resolution analysis–including nudging of precipitation and cloud observations–valid at 1800 UTC from the 1.5 km model and data-assimilation system over the southern UK (Dixon et al., 2009). The 1800 UTC analysis was generated as part of a 1.5 km assimilation experiment with hourly DA cycles that were started at 0200 UTC on 27 July 2008 and initialized with a 4 km model forecast valid at the same time and started at 0000 UTC. To initiate the ensemble forecast at 1.5 km resolution, a 1.5 km version of the UM is integrated forward in time, starting with the 1800 UTC 3D-Var 1.5 km analysis and adding in the perturbations over 20 min using the incremental analysis update procedure (Bloom et al., 1996). Forecasts are produced at 1900 UTC, 2000 UTC and 2100 UTC on the same day. All ensemble members use the same boundary conditions taken from the operational 4 km UK forecasts. The full ensemble matrix has dimensions of X ∈ R(360×288×70)×24 . However, in this article we consider only the vertical part of the state by extracting vertical components from each ensemble member at a given location. For a given location and a given state variable, the ensemble matrix is then given by X ∈ R70×24 .

3. Hydrostatic balance in the perturbations In this section we derive the approximation of hydrostatic balance for perturbations by determining, for each of the ensemble members, the hydrostatically balanced potential temperature perturbations θH (where subscript ‘H’ stands for hydrostatically balanced), which are representative of dynamically important pressure-gradient perturbations. In section 4 we use θH to measure how close the perturbations are to hydrostatic balance. Available fields at a given location at a given vertical level are Exner pressure , potential temperature θ and specific humidity q. The ensemble-mean values for each of these ¯ θ¯ and q¯ , and , θ and q denote fields are denoted by , the perturbation from the mean of an individual ensemble member at the given location and height. Hydrostatic balance in the shallow-atmosphere approximation (hereafter denoted as simply hydrostatic balance) is Q. J. R. Meteorol. Soc. 138: 1709–1720 (2012)

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given by (e.g. see Wallace, 2006) dp gp =− , dz Rd Tv

(10)

where p(z) is pressure, Tv (z) is virtual temperature, z is vertical height and Rd = 287.06 J K−1 kg−1 is the gas constant for dry air. We can rewrite Eq. (10) in terms of the available variables, i.e. and θ , using the definition of the Exner pressure,

=

p p0

=

Tv , θv

Rd /cp (11) (12)

where cp = 1005 J kg−1 K−1 is the specific heat at constant pressure, p0 = 1000 hPa is a reference pressure at the ground level z0 and θv is the virtual potential temperature, given by (Wallace, 2006)

θv = θ 1 + ( −1 − 1)q ,

(13)

Hence, using the ensemble perturbations and mean, i.e. q , q¯ , and θ¯ , we can calculate the perturbation values of the potential temperature θH that are in hydrostatic balance with these variables at each location and height. The perturbation ensemble of each variable is defined as follows. • potential temperature ensemble

= θ 1 , θ 2 , ..., θ 24 ,

(20)

• specific humidity ensemble

, Q = q1 , q2 , ..., q24

(21)

• Exner pressure ensemble

= 1 , 2 , ..., 24 ,

(22)

where .i , i = 1, ..., 24, denote the ensemble members, i.e. vectors containing the fields at all vertical grid points at a given location. Then the hydrostatically balanced potential temperature perturbation, θH , is computed for each element of each ensemble member, i = 1, ..., 24, leading to the ensemble vertical-error covariance matrix Pe =

H TH , N −1

(23) with being the ratio of the mass of liquid water to dry air in the atmosphere and q being the specific humidity. Using θv instead of θ allows us to take the moisture of the atmosphere where H is the hydrostatically balanced potential into account. Hence, we may write temperature perturbation ensemble. The correlation matrix is defined as Ce = −1/2 Pe −1/2 , where e is a diagonal e e matrix with ensemble variances on the diagonal (i.e. p = p0 cp /Rd , (14) diag( e ) = diag(Pe )). (15) Tv = θv . 4. Case study Substituting Eqs (14)–(15) into Eq. (10) and using Eq. (13), we find that the hydrostatic balance may be expressed in In this section we give a description of the case study and the methods used to obtain the results presented in sections terms of the available variables and θ as 5 and 6.

−1 −1 d g The model was initialized with a set of ensemble −1 =− 1 + ( − 1)q θ . (16) atmospheric states valid at 1800 UTC on 27 July 2008 and dz cp determined as explained in section 2 and Appendix A. This equation may be linearized, giving a hydrostatic Ensemble forecasts were produced each hour for the following three hours, at 1900 UTC, 2000 UTC and equation for the mean variables, 2100 UTC on the same day. This case was selected as ¯ convection had already occurred before 1800 UTC and at 1 d g , (17) the time of initialization, 1800 UTC, the system was fully =− dz cp [1 + ( −1 − 1)¯q]θ¯ convective with convection moving in the domain over the and a first-order approximation to the hydrostatic equation next three hours. The degree of hydrostatic balance present in the forecast for the perturbations, perturbations was computed for various vertical columns in convective and non-convective cases, where the presence of ( −1 − 1)q g d rain in the ensemble mean forecast at a given location was = + dz cp (1 + ( −1 − 1)¯q)2 θ¯ used to indicate the presence of convection. Furthermore, possible deviations from hydrostatic balance for forecast θ , (18) perturbations over high terrain were also investigated. In −1 2 ¯ (1 + ( − 1)¯q)θ order to highlight effectively the resulting differences in the hydrostatically balanced perturbations, we limit our ¯ + , θ = θ¯ + θ and q = q¯ + q . where = discussion here to cases corresponding to columns in which Equation (18) may be rearranged to give rain rates were the highest in the mean (see Appendix B for ensemble mean precipitation rates). The columns for which ( −1 − 1)q θ¯ the ensemble mean precipitation was the highest for each + θH = − 1 + ( −1 − 1)¯q hour§ are labelled Conv1, Conv2 and Conv3, respectively. cp d 2 θ¯ [1 + ( −1 − 1)¯q]. (19) § For example, the column Conv1 has the highest rain rate at 1900 UTC. g dz c 2012 Royal Meteorological Society and Copyright British Crown Copyright, the Met Office

Q. J. R. Meteorol. Soc. 138: 1709–1720 (2012)


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and Conv1 at 1.5 km resolution are shown in Figure 2 for 1900 UTC. As discussed in section 3, the balanced variable θH is computed using Eq. (19) for each ensemble member and each vertical level. From Figure 2 we can see that at 1.5 km resolution in the case of no convection (Figure 2(a) and (b)) the ensemble correlation matrices of the perturbations and H are indistinguishable, meaning that hydrostatic balance holds very well in the perturbations when convection is not present. However, in the presence of convection (Figure 2(c) and (d)), vertical levels where the ensemble correlation matrices of perturbations and H are different can be clearly identified. In particular, above the boundary layer in the lower part of the free atmosphere (between vertical levels 20 and 35), where the convection is strongest¶ , the balanced profiles θH (see Figure 2(d)) are positively correlated over a smaller vertical region, while they are mainly negatively correlated with the region Figure 1. 1.5 km resolution domain and the chosen vertical columns for below level 20. In contrast, the values of full perturbations θ in the lower part of the free atmosphere are less testing. (negatively) correlated with the boundary layer, but more correlated within levels 20–35, implying that there is more These are compared with the column, labelled Non-Conv, convective mixing between the levels where convection is for which the ensemble mean precipitation was zero over the present. entire three hours. The location of these columns is shown This qualitative comparison between full and balanced in Figure 1. correlations of potential temperature indicates that the To find the degree to which hydrostatic balance holds hydrostatic balance is no longer valid in the perturbations in the perturbations as a function of horizontal scale, the where convection is present. original 1.5 km resolution data were aggregated into boxes with side ranging from 3–90 km resolution around the nonconvective point Non-Conv and the convective column 5.2. Amplitude of unbalanced perturbations relative to full Conv1. perturbation for high resolution The following quantities were calculated from both the 1.5 km and the coarsened resolution data: We can diagnose the amplitude of unbalanced perturbations • correlation matrices for and H (see sections 5.1 and 6.1); • amplitude of unbalanced perturbations relative to full perturbations computed for each vertical level at 1900 UTC, 2000 UTC and 2100 UTC for high resolution (see sections 5.2 and 6.2); • mass-weighted standard deviation of unbalanced perturbations, computed for each of the four columns over the three-hour forecast window (see sections 5.3 and 6.3).

relative to the full perturbations at a given height and for each hour. This ratio we define as follows: R(z, t) =

σU (z, t) , σ (z, t)

(24)

where z is the vertical level, σ is the grid-point standard deviation of full perturbations (i.e. the square root of the diagonal elements of Pe = T /(N − 1)) and σU is the grid-point standard deviation of the unbalanced perturbations. The unbalanced perturbations are defined by

The dependence of hydrostatic balance can be ascertained in a subjective way by inspection of the correlation matrices and in a quantitative way by means of the amplitude measure U = − H , (25) and the mass-weighted standard deviation. Results discussed in sections 5 and 6 are plotted against model levels; the corresponding height and pressure levels i.e. the difference between the full and balanced perturbations. are shown in Appendix C. The full ensemble perturbations are close to hydrostatic balance if R ≈ 0 and the larger the ratio R, the less balanced 5. Results for high resolution the full perturbations. To evaluate the degree of hydrostatic balance at 1.5 km This section discusses the comparison of the full resolution as a function of time, we plot the ratio R in perturbations obtained from the ensemble of 1.5 km percent (i.e. R × 100) in Figure 3 for each vertical level and forecasts and the hydrostatically balanced perturbations for each hour in our case study. The figure shows how the computed by applying Eq. (19) to the data, as described in amplitudes move with convection in space over the three previous sections. hours at 1.5 km resolution. From Figure 3(a) we see that 5.1. High-resolution correlation matrices for and H The full and balanced ensemble correlation matrices of the perturbations and H for the columns Non-Conv c 2012 Royal Meteorological Society and Copyright British Crown Copyright, the Met Office

¶ From Figures 3(a) and 4(a) we can see that where convection is present in Conv1 at 1900 UTC there are marked differences between balanced and unbalanced variables between levels 10 and 45, with the largest between levels 20 and 35.

Q. J. R. Meteorol. Soc. 138: 1709–1720 (2012)

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(a) 70

1

(a) 70 60

60 0.5

40 0 30 20

0 30 −0.5

10

1

10

20

30 40 50 Vertical levels

60

1

−1

70

(c) 70

1

1

10

20


60

70

(d) 70

−1

1

60

60 0.5

40 0 30 20

−0.5

0.5

50 Vertical levels

50 Vertical levels

40

20

−0.5

10

40 0 30 20

−0.5

10

10 1

0.5

50 Vertical levels

Vertical levels

50

1

1

1

10

20


60

−1

70

1

1

10

20


60

70

−1

Figure 2. Auto-correlations for and the corresponding H at 1900 UTC for non-convective column Non-Conv and convective column Conv1 at 1.5 km data resolution. (a) , Non-Conv; (b) H , Non-Conv; (c) , Conv1; (d) H , Conv1. Table 1. Mass-weighted standard deviation of unbalanced perturbations (×10−2 K) for selected vertical columns at 1.5 km resolution. Column index

1900 UTC

2000 UTC

Non-Conv Conv1 Conv2 Conv3

0.6174878 15.577047 2.533369 1.8783554

0.6606426 3.5001994 7.6961603 6.1030643

perturbations (see Eq. (25)) is larger than the magnitude of the full perturbations. In particular, from Figures 3 and 4 we see that R attains a maximum of 135% in the column 2100 UTC Conv3 at 2100 UTC. We can see the actual values of R in 1.1721997 Figure 4. 2.7601854 1.6352061 23.089188

Conv1 has the largest amplitude ratio at 1900 UTC, while at 2000 UTC and 2100 UTC Conv1 is very much in balance. In Figure 3(b) we see that Conv2 is in balance at 1900 UTC as convection has not reached this region yet, and it is in balance at 2100 UTC as convection has left the region; the largest amplitudes are at 2000 UTC, i.e. when convection is strongest at this location. Similar results hold for column Conv3 (see Figure 3(c)). On the other hand, the amplitudes of the unbalanced perturbations are negligible in the NonConv column for all three hours, meaning that θ ≈ θH for all t and for all z, as observed in the correlations (see Figure 2). We remark that R is not constrained to be in the range [0, 1]; it becomes larger than unity when the magnitude of the difference between the hydrostatically balanced perturbations (diagnosed from and q) and the full c 2012 Royal Meteorological Society and Copyright British Crown Copyright, the Met Office

5.3. Mass-weighted standard deviation of unbalanced perturbations for high resolution Another measure of the hydrostatic balance in the perturbations is the mass-weighted average standard deviation of unbalanced perturbations, given by w (t) =

69

w(z, t)σU (z, t),

(26)

z=0

where w(z) is a partial column mass density between two adjacent vertical levels divided by the total column mass density, given by w(z, t) =

p(z + 1, t) − p(z) . p(top, t) − p(surf, t)

(27)

Here p(z, t) is the pressure at level z, time t, p(surf, t) is the surface pressure and p(top, t) is pressure at the Q. J. R. Meteorol. Soc. 138: 1709–1720 (2012)


60 60

vertical levels

50 50 40 40 30 30 20 20 10 10 0

0 1900

2000

60 60 50 50 40 40 30 30 20 20 10 10 0

2100

0 1900

time, UTC

vertical levels

50 50 40 40 30 30 20 20 10 10

2000

2100

2100

time, UTC

140 130 120 110 100 90 80 70 60 50 40 30 20 10 0

(d) 70 60 60 50 50 vertical levels

140 130 120 110 100 90 80 70 60 50 40 30 20 10 0

60 60

0 1900

2000 time, UTC

(c) 70

0

140 130 120 110 100 90 80 70 60 50 40 30 20 10 0

(b) 70

vertical levels

140 130 120 110 100 90 80 70 60 50 40 30 20 10 0

(a) 70

1715

40 40 30 30 20 20 10 10 0

0 1900

2000

2100

time, UTC

Figure 3. Percentage amplitude of unbalanced perturbations relative to full perturbations as a function of time and height at 1.5 km resolution. (a) Conv1; (b) Conv2; (c) Conv3; (d) Non-Conv.

top of the atmosphere at time t. The mass-weighted standard deviation provides a measure of how far on average the full perturbations are from the hydrostatic perturbations. From Table 1 we see that for the non-convective column the mass-weighted amplitude of unbalanced perturbations grows slightly over the three-hour period. However, for the convective columns, hydrostatic balance holds much better in each column when convection is either not present or weak. For example, for column Conv2 w is relatively small at 1900 UTC since convection is very weak at this time in the region; however, at 2000 UTC, when convection in this column is strongest in the ensemble mean, the amplitude of the unbalanced perturbations is three times larger. Finally, at 2100 UTC the convection has moved out of the column and w for Conv2 has decreased again. Similar results hold for columns Conv1 and Conv3. These results indicate that the average absolute amplitude of unbalanced perturbations (measured by the massweighted standard deviation) increases in convective areas at 1.5 km resolution. c 2012 Royal Meteorological Society and Copyright British Crown Copyright, the Met Office

6. Results for coarse resolutions In this section we examine the differences in the full and balanced perturbations as the resolution of the data is decreased. 6.1. Coarse-resolution correlation matrices for and H By coarsening the grid we expect the perturbations to become more hydrostatically balanced. This is visually confirmed in Figure 5, where correlation matrices of and H for convective columns at 4.5 km and 12 km resolutions are shown. Even though at 4.5 km resolution (see Figure 5(a) and (b)) the perturbations are not in hydrostatic balance in the lower part of the free troposphere (levels 20–40), they appear to be more in balance in the boundary layer (below level 20) since negative correlations between levels 1–20 and 20–30 become more similar in the top left and right panels. However, again the full ensemble perturbations are more correlated than the balanced perturbations within the lower part of the free atmosphere. The comparison is more consistent for 12 km resolution, where not only correlations at the boundary layer but also correlations Q. J. R. Meteorol. Soc. 138: 1709–1720 (2012)

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(b) 1.5

(c) 1.5

1.0

1.0

1.0

σU/σ

σU/σ

σU/σ

(a) 1.5

0.5

0.5

0.5

0.0

0.0 0

10

20

30

40

50

60

70

0.0 0

10

20

z levels

30

40

50

60

70

0

10

z levels

20

30

40

50

60

70

z levels

Figure 4. Values of amplitudes of unbalanced perturbations relative to full perturbations at 1.5 km for each vertical level at (a) 1900 UTC, (b) 2000 UTC and (c) 2100 UTC.

in the lower part of the free atmosphere are very similar for balanced and full perturbations (see Figure 5(c) and (d)). The cross-correlations between the boundary layer and the lower part of the free atmosphere for both resolutions are also more consistent between the balanced and full perturbations.

resolution. This is reflected in the amplitude ratios where, for example in Figure 6(b), column Conv1 at 9 km resolution has higher amplitude ratio than at 3 km at some vertical levels. 6.3. Mass-weighted standard deviation of unbalanced perturbations for coarse resolutions

6.2. Amplitude of unbalanced perturbations relative to full perturbations for coarse resolutions Computing the mass-weighted standard deviation of unbalanced perturbations as in section 5.3, we find from Again performing a statistical diagnostic of the perturba- Figure 7 that at 1900 UTC the column Conv1 has the tions, we display in Figure 6 the amplitude of unbalanced strongest imbalance at the original resolution of 1.5 km, perturbations relative to the full perturbations R (see section due to the convection present in the column at this time. 5) as a function of height for the convective column Conv1 However, as the resolution is decreased (for column Conv1 from 3 km up to 22.5 km coarsened data at 1900 UCT, at 1900 UTC) the column becomes more balanced. We see 2000 UTC and 2100 UTC. This ratio shows the degree to that near 22 km resolution the mass-weighted amplitude which hydrostatic balance (in convective regions) increases of the unbalanced perturbations in Conv1 is equivalent as a function of horizontal scale and demonstrates that at for all three hours, 1900 UTC, 2000 UTC and 2100 UTC, resolutions coarser than about 20 km hydrostatic balance which confirms that 22 km is a threshold for hydrostatic holds very well. Similar results hold for the other convec- balance. At 2000 UTC and 2100 UTC the rain rate in the tive columns examined in this study. It follows that the column Conv1 has reduced significantly and the column is resolution at which hydrostatic balance holds in the perin near-hydrostatic balance, even at 1.5 km resolution, and turbations over the whole domain, i.e. for convective and only a very slight decrease in the mass-weighted amplitude non-convective regions, is coarser than 20 km. At 45 km of unbalanced perturbations is obtained by coarsening and 90 km resolutions (not shown) the perturbations are consistent with hydrostatic balance, as are perturbations for the grid at these times. The column Non-Conv is very the non-convective columns for all coarsened resolutions close to hydrostatic balance for all three hours and all (not shown) and in the stratosphere in all columns at all resolutions. resolutions. The convective coarsened column data was obtained by coarsening around the position Conv1, i.e. the column with the highest rain rate in the ensemble mean at 1900 UTC. As the convection moves out from the Conv1 column over the next two hours (see Figure 6(b) and (c)), the standard-deviation ratio R becomes closer to zero, thus showing that the unbalanced perturbations are close to zero at 2000 UTC and 2100 UTC, i.e. ≈ H at these hours. However, since with coarser horizontal resolution a larger area is encompassed, a column at a coarser resolution may have more convection on average than at a less coarse c 2012 Royal Meteorological Society and Copyright British Crown Copyright, the Met Office

7.

Summary and conclusions

To investigate how well hydrostatic balance holds for forecast errors at convective scales, we used an ensemble with 24 members obtained by running the 1.5 km resolution version of the UM, initialized according to the procedure described in section 2.3 and Appendix A. The ensemble was initialized at 1800 UTC on 27 July 2008 when convection was fully developed and data for analysis were available at 1900 UTC, 2000 UTC and 2100 UTC on the same day. Q. J. R. Meteorol. Soc. 138: 1709–1720 (2012)


(a) 70

1

(b) 70

60 0.5

40 0 30 20

0.5

50 Vertical levels

Vertical levels

1

60

50

40 0 30 20

−0.5

−0.5

10

10 1

1

10

20

30

40

50

60

1

−1

70

1

10

20

(c) 70

1

40

50

60

70

(d) 70

60

−1

1

60 0.5

40 0 30 20

−0.5

0.5

50 Vertical levels

50

40 0 30 20

−0.5

10

10 1

30

Vertical levels

Vertical levels

Vertical levels

1717

1

10

20


60

1

−1

70

1

10

20


60

70

−1

Figure 5. Auto-correlations for and the corresponding H at 1900 UTC for convective column Conv1 at 4.5 km and 12 km resolution. (a) , Conv1 4.5 km; (b) H , Conv1 4.5 km; (c) , Conv1 12 km; (d) H , Conv1 12 km.

1.0

1.0

1.0

0.8

0.8

0.8

0.6

σU/σ

(c) 1.2

σU/σ

(b) 1.2

σU/σ

(a) 1.2

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0.0

0.0 0

10

20

30 40 z levels

50

60

70

0.0 0

10

20

30 40 z levels

50

60

70

0

10

20

30 40 z levels

50

60

70

Figure 6. Values of amplitudes of unbalanced perturbations relative to full perturbations for resolutions of 3–22.5 km of convective Conv1 column only for each vertical level at (a) 1900 UTC, (b) 2000 UTC and (c) 2100 UTC.


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0.18 Conv1 1900UTC

0.16

Conv1 2000UTC Conv1 2100UTC

0.14

Non-Conv 1900UTC

0.12

Non-Conv 2000UTC Non-Conv 2100UTC

Σw

0.1 0.08 0.06 0.04 0.02 0 1.5 km 3 km 4.5 km 6 km

9 km

12 km 18 km 22 km 45 km 90 km

Resolution

Figure 7. Mass-weighted standard deviation of unbalanced perturbations in K over all vertical levels with coarsened resolutions around Conv1 and Non-Conv columns.

balance does not hold in the perturbations in the regions of convection, but does hold in those regions where convection is not present. From the relative amplitudes of unbalanced perturbations in the vertical it was also established quantitatively that at vertical levels 10–35 the perturbations are very far from being balanced. Not only the extent to which the perturbations are not in hydrostatic balance but also the fact that perturbations are in balance at all resolutions in all columns in the stratosphere (above vertical level 55) was demonstrated. Using the standard-deviation ratio for coarsened data, we also showed that 20 km horizontal resolution is the limit at which hydrostatic balance becomes valid over the entire domain. Our results indicate that hydrostatic balance should be relaxed around convective levels in the correlation matrices at 1.5 km resolution. In a recent study, Montmerle and Berre (2010) have developed a new 3D-Var scheme to assimilate satellite and radar data at mesoscales. They have shown that using heterogeneous background-error covariances with small correlation length-scales in precipitating areas is likely to improve the assimilation of precipitation data. To assess the impact of covariance models in the UM that are better suited for convective-scale data assimilation, a series of data-assimilation experiments is needed. This would require the Met Office’s control-variable transform in the variational data assimilation at convective scale to be redesigned.

In this article we focused on the vertical analysis of forecast errors. Four vertical columns from the whole domain were selected for testing purposes for each forecast hour: a column with no precipitation and three columns with the highest rain rates in the ensemble mean, for each forecast hour 1900 UTC, 2000 UTC and 2100 UTC, respectively. Data around two columns, one non-convective and one convective, were aggregated from 3–90 km resolution. For each of these columns the hydrostatically balanced potential temperature perturbations θH were calculated using the approximated hydrostatic equation for perturbations, expressed in terms of the available fields: potential temperature θ , Exner pressure and specific humidity q. By constructing the correlation matrices for these columns, it was shown that at 1.5 km resolution hydrostatic Appendix A. Flow of the high-resolution system

Initial X a24 km (t0 )

Initial 4D-Var xa24 km (t0 )

a

a a a Create ensemble states X24 (t ) = x (t ), ..., x (t ) 24 km 0 24 km 0 + X 24 km (t0 ) km 0 a a Reconfigure X24 (t ) − > X km 0 1.5/24 km (t0 ) Reconfigure

xa24 km (t0 )

− > xa1.5/24 km (t0 )

a a X a1.5/24 km (t0 ) = X1.5/24 (t ) − x (t ), ..., xa1.5/24 km (t0 ) 1.5/24 km 0 km 0 Available 3D-Var xa1.5 km (t0 )

i = 1, 2, 3

f Forward integration: UM(xa1.5 km (t0 ), X a1.5/24 km (t0 )) − > X1.5 km (ti )

Figure A1. Set-up and flow of the reconfigured ensemble system. Here t0 =1800 UTC, t1 =1900 UTC, t2 =2000 UTC and t3 =2100 UTC, all on 27 July 2008. The UM forward integration step at 1.5 km includes the addition of perturbations over 20 min via incremental analysis update (Bloom et al., 1996) from 1800 UTC onwards. All ensemble members use the same operational 4 km boundary conditions.


Q. J. R. Meteorol. Soc. 138: 1709–1720 (2012)


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Appendix B. Mean precipitation rate

(a)

(c)

(b)

Figure B1. Mean precipitation rates (mm h−1 ) at 1900 UTC, 2000 UTC and 2100 UTC. Black dots are coordinates of the selected vertical columns.

Appendix C. Pressure, height and model levels Table C1. Vertical height and pressure corresponding to each model. Here pressure is a hydrostatically balanced pressure exponentially decreasing with height, with surface pressure p0 = 1013.25 hPa and constant scale height H = 7400 m. Model level

Height (m)

Pressure (hPa)

Model level

Height (m)

Pressure (hPa)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

2.50 13.33 33.33 60.00 93.33 133.33 180.00 233.33 293.33 360.00 433.33 513.33 600.00 693.33 793.33 900.00 1013.33 1133.33 1260.33 1393.33 1533.33 1680.00 1833.33 1993.33 2160.00 2333.33 2513.33 2700.00 2893.33 3093.33 3300.00 3513.33 3733.33 3960.00 4193.33

1012.91 1011.43 1008.70 1005.07 1000.56 995.20 988.90 981.84 973.91 965.14 955.66 945.39 934.34 922.67 910.28 897.22 883.98 869.76 854.61 839.73 823.99 807.46 791.26 774.33 756.75 739.56 721.79 703.49 685.66 667.37 648.70 630.55 612.08 593.35 575.19

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69

4433.33 4680.00 4933.33 5193.33 5460.00 5733.33 6013.33 6300.08 6593.82 6895.16 7205.09 7525.14 7857.42 8204.76 8570.75 8959.93 9377.85 9831.21 10327.99 10877.55 11490.83 12180.45 12960.85 13848.50 14862.01 16022.34 17352.94 18879.97 20632.44 22646.46 24945.39 27580.09 30589.10 34018.87 37922.50

556.84 538.34 520.45 502.45 484.48 467.12 449.78 432.49 415.87 398.81 382.45 366.26 350.29 334.56 318.24 301.90 285.25 268.42 251.90 232.28 214.19 194.86 174.89 156.97 135.29 116.60 96.50 78.80 62.62 47.79 35.02 24.32 16.21 10.24 6.05


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References Bloom SC, Takacs LL, da Silva AM, Ledvina D. 1996. Data assimilation using incremental analysis updates. Mon. Weather Rev. 124: 1256–1271. Bowler NE, Arribas A, Mylne KR, Robertson KB, Beare SE. 2008. The MOGREPS short-range ensemble prediction system. Q. J. R. Meteorol. Soc. 134: 703–722. Davies T, Cullen MJP, Malcolm AJ, Mawson MH, Staniforth A, White AA, Wood N. 2006. A new dynamical core for the Met Office’s global and regional modelling of the atmosphere. Q. J. R. Meteorol. Soc. 131: 1759–1782. Dixon M, Zhihong L, Lean H, Roberts N, Ballard SP. 2009. Impact of data assimilation on forecasting convection over the United Kingdom using a high-resolution version of the Met Office Unified Model. Mon. Weather Rev. 137: 1562–1584. Evensen G. 1994. Sequential data assimilation with a nonlinear quasi-


geostrophic model using Monte-Carlo methods to forecast error statistics. J. Geophys. Res. 99: 10143–10162. Kalman RE. 1960. A new approach to linear filtering and prediction problems. Trans. AMSE – J. Basic Engineering 82D: 35–45. Migliorini S, Dixon M, Bannister R, Ballard SP. 2011. Ensemble prediction for nowcasting with a convection-permitting model – I: description of the system and the impact of radar-derived surface precipitation rates. Tellus A 63: 468–496. Montmerle T, Berre L. 2010. Diagnosis and formulation of heterogeneous backgroud-error covariances at the mesoscale. Q. J. R. Meteorol. Soc. 136: 1408–1420. Wallace JM. 2006. Atmospheric Science: An Introductory Survey, 2nd edn. Elsevier: Amsterdam. Wang XG, Bishop CH, Julier SJ. 2004. Which is better, an ensemble of positive–negative pairs or a centered spherical simplex ensemble? Mon. Weather Rev. 132: 1590–1605.

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