Diagnosing Summertime Mesoscale Vertical Motion - AMS Journals

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MONTHLY WEATHER REVIEW

VOLUME 135

Diagnosing Summertime Mesoscale Vertical Motion: Implications for Atmospheric Data Assimilation CHRISTIAN PAGÉ Department of Earth and Atmospheric Sciences, University of Quebec in Montreal, Montreal, Quebec, Canada

LUC FILLION Data Assimilation and Satellite Meteorology Division, Environment Canada, Dorval, Quebec, Canada

PETER ZWACK Department of Earth and Atmospheric Sciences, University of Quebec in Montreal, Montreal, Quebec, Canada (Manuscript received 27 December 2005, in final form 1 June 2006) ABSTRACT Balance omega equations have recently been used to try to improve the characterization of balance in variational data assimilation schemes for numerical weather prediction (NWP). Results from Fisher and Fillion et al. indicate that a quasigeostrophic omega equation can be used adequately in the definition of the control variable to represent synoptic-scale balanced vertical motion. For high-resolution limited-area data assimilation and forecasting (1–10-km horizontal resolution), such a diagnostic equation for vertical motion needs to be revisited. Using a state-of-the-art NWP forecast model at 2.5-km horizontal resolution, these issues are examined. Starting from a complete diagnostic partial differential equation for omega, the rhs forcing terms were computed from model-generated fields. These include the streamfunction, temperature, and physical time tendencies of temperature in gridpoint space. To accurately compute one term of secondorder importance (i.e., the ageostrophic vorticity tendency forcing term), a special procedure was used. With this procedure it is shown that Charney’s balance equation brings significant information in order to deduce the geostrophic time tendency term. Under these conditions, results show that for phenomena of length scales of 15–100 km over convective regions, a diagnostic equation can capture the major part of the model-generated vertical motion. The limitations of the digital filter initialization approach when used as in Fillion et al. with a cutoff period reduced to 1 h are also illustrated. The potential usefulness of this study for mesoscale atmospheric data assimilation is briefly discussed.

1. Introduction Current operational variational data assimilation systems for numerical weather prediction (NWP) all rely on the use of approximate “dynamical balance” concepts, which are considered essential so as to enforce a delicate balance between mass and winds in order to avoid the excitation of fast gravity wave noise in the early stage of the forecast. Before the advent of operational variational data assimilation approaches,

Corresponding author address: Christian Pagé, Département des Sciences de la Terre et de l’Atmosphère, Université du Québec à Montréal, P.O. Box 8888, Stn “Downtown” Montreal, QC H3C 3P8, Canada. E-mail: [email protected] DOI: 10.1175/MWR3371.1 © 2007 American Meteorological Society

MWR3371

the notion of dynamical balance was represented by the nonlinear normal mode initialization (NNMI) method based on Machenhauer’s (1977) or Baer and Tribbia’s (1977) balance schemes (see Temperton 1988). In the prevariational analysis period (i.e., when optimal interpolation analysis schemes were used), NNMI schemes were applied to fields of initial conditions (e.g., winds, temperature, moisture, and surface pressure). The resulting analysis fields were then referred to as initialized initial conditions (i.e., balanced initial conditions). In early operational variational analysis context, however, balance concepts were introduced as simple relationships applied to analysis increments rather than to the full predictive variables. With respect to these balance concepts, Parrish and Derber (1992) first introduced the notion of balanced and unbalanced analysis

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variable components before correlation modeling assumptions were introduced. Similar simple balance concepts were adopted and used in the Canadian threedimensional variational data assimilation (3DVAR) analysis system (see Gauthier et al. 1999). Later, Derber and Bouttier (1999) exploited inherent balance relationships assumed to exist in their ensemble of lagged forecasts. A regression approach was developed to capture the linear relationship contained in the forecast error samples between mass and winds, for instance. Sophisticated linear regression relationships were introduced in the then operational European Centre for Medium-Range Weather Forecasts (ECMWF) 3DVAR analysis system with significant success. As part of this new approach, it was shown that linear regressions were capable of representing expected geostrophic relationships among mass and rotational wind components from an ensemble of lagged forecasts. Recently, in an attempt to improve ECMWF background error covariances, Fisher (2003, hereafter referred to as F03) introduced the quasigeostrophic (QG) set of balanced equations explicitly in the definition of the balanced part of the analysis variables. As shown by F03, quite similar analysis increments resulted from this new analysis procedure as compared with the previous operational use of linear regressions mentioned above. A noticeable distinction, however, was obtained for the divergent part of the wind increments and, consequently, vertical motion increments. As one would expect from balanced considerations at synoptic scales treated by ECMWF analysis, the QG omega equation enforced coherent extratropical divergent flow structures depending on the basic state considered (see Fig. 17 from F03). Very recently, additional major NWP centers joined the group of centers operating with benefits a fourdimensional variational data assimilation (4DVAR) system after initial implementations performed at ECMWF and Meteo-France; for example, the Japan Meteorological Agency now runs 4DVAR for their operational mesoscale and global data assimilation systems, the Met Office in the United Kingdom has run 4DVAR since 5 October 2004, and the Canadian Meteorological Centre (CMC) in Dorval has run a global operational 4DVAR system since 15 March 2005. As a natural extension for operational regional data assimilation, Environment Canada has recently developed a limited-area 4DVAR analysis system (hereafter referred to as LAM4D-Var; Fillion et al. 2005). The primary goal of this development project is to replace the current operational variational analysis system, which suffers from being totally dependent on the global

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variational analysis system and especially upon its low horizontal resolution [triangular truncation at wavenumber 108 (T108)]. For this system, Fillion et al. (2005, hereafter referred to as F05) introduced an extension to F03’s ideas together with the introduction of diabatic forcing of balanced vertical motion. These ideas are currently being examined in what is called the continental version of LAM4D-Var with inner loop resolution set at 35 km. Because LAM4D-Var contains only a small fraction of the Tropics, as opposed to a global implementation as in F03, implementing the QG omega equation (tangent linear version) suffers less from limitations of the QG approximation (Daley 1991, sections 7.9 and 10.7, e.g.). What is more of an unknown at this point, however, is the degree of extension of the original ideas of F05 in the context of truly mesoscale versions of the 3D- and 4DVAR systems. This directly concerns the local version of the LAM4D-Var project at the CMC where the forecast model is run at 2.5 km and the inner loop of the 3DVAR analysis system still needs to be optimized. This local version covers, for instance, the southern part of the Quebec region and is intended to be used for forecasts up to 24 h. Figure 1 shows both the continental and local analysis/forecast domains just described. With regard to the above discussion on balance operators introduced in the analysis variables, it is not yet known if a similar diagnostic omega equation can be used to accurately represent the balanced part of the background error correlations at such fine scales. In view of these primary goals, it becomes important to examine the possibility of establishing whether some form of omega diagnostic equation exists at the mesoscale, for instance. Without going into sophisticated scaling arguments typical of previous theoretical studies on balanced mesoscale flows, we pose the diagnostic issue rather directly in terms of examining summertime mesoscale systems as they emerge from mature runs with a state-of-the-art mesoscale forecast model. Starting from a complete nonlinear omega equation, dominant contributions are isolated and used to derive an accurate procedure to diagnose the divergent part of the wind. It can already be expected that the presence of moist physics, for instance, will play a key role at these scales. In addition, we expect that the relative importance of the terms in the complete omega equation is quite different for the synoptic scale (phenomena of length scales of 150–1000 km) compared with the meso-␤ scale (15–100 km). The diagnosed mesoscale vertical motion is compared with a real summer convective case using a multinested mesoscale model (reso-

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FIG. 1. Target grid for regional 48-h forecast at 10 km using the GEM-LAM.

lutions of 1–15 km). It is important to mention that the diagnosed mesoscale vertical motion from the omega equation represents the effect of the diabatic heating on the vertical motion at scales larger than the individual convective elements whose vertical motion is driven by local buoyancy. Using a specially designed set of experiments, the importance of information derived from divergent winds in the analysis of a forecast in convective situations is assessed. The numerical model and simulations will be presented in section 2 along with the nonlinear balance omega (NLB) equation. The case study is presented in section 3a. The accuracy of the NLB omega diagnostics, calculated with the model’s convection-induced temperature tendencies, along with comparisons against digitally filtered and unfiltered omega vertical motion

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and the experiments on divergent winds will be discussed in section 4.

2. Methodology a. Omega diagnostic approach Our objective is to characterize the balance of terms that may exist under summertime mesoscale convection in view of possible diagnostic relationships that could be used for advanced atmospheric data assimilation at such spatial and temporal resolutions. It can be shown (see appendix) that an omega equation can be derived by combining the equations of motion, energy, ideal gas, and continuity, using only the hydrostatic approximation. The resulting equation takes the form

冊

R R ⭸ ⭸␻ ⭸␷ ⭸␻ ⭸u R 2 ⭸2␻ ⭸2␨ ⭸T ⵜ S␻ ⫹ f 共 f ⫹ ␨ 兲 2 ⫺ f␻ 2 ⫺ f ⫺ ⫽ ⫺ ⵜ2共⫺V · ⵱T 兲 ⫺ ⵜ2 p ⭸p ⭸x ⭸p ⭸y ⭸p p p ⭸t ⭸p ⭸p ⫺f

冉冊

冉冊

⭸ ⭸ ⭸ ⭸␨ag 1 ⭸ ⭸␾ ⫺ ␤ . 关⫺V · ⵱共 f ⫹ ␨兲兴 ⫺ f 共k · ⵱ ⫻ F兲 ⫹ f ⭸p ⭸p ⭸p ⭸t f ⭸y ⭸t

All the symbols have their usual meteorological meaning (e.g., Holton 1992, 476–479) except for F, which represents frictional and orographic forcings, and S the

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phys

共1兲

static stability. The subscript ag stands for ageostrophic. Such an equation was used in the past for diagnostic studies, for instance by Pauley and Nieman (1992) and

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Räisänen (1995) at synoptic scales. Because nothing precludes it, we use (1) for mesoscale conditions. To compute each term in (1), whenever we refer to a diagnosed forcing or diagnosed vertical motion from (1) at a given time, the following forecast output fields are assumed available at the given time on 20 equally spaced pressure vertical levels ranging from 50 to 1000 hPa: horizontal winds, temperature, surface pressure, diabatic temperature tendencies, and surface height. These output fields are assumed available every 5 min. In theory, if one has available all the fields required to compute and solve (1), a good correspondence between this solution and the forecast model computed omega field at the same time would be observed. The sources of the differences expected would be the different numerical spatial and temporal discretization of (1) as compared with the model itself, which uses a different approach to solve the original primitive equations. There is also the fact that the forecast fields are output on pressure coordinates (which is not the model’s vertical coordinate used for the discretization) and the vertical resolution in pressure coordinates is lower than what the model is effectively using. Additionally, we note that (1) uses the hydrostatic assumption whereas our model forecast simulations (as described in detail in section 3) are nonhydrostatic. Given these recognized numerical discrepancies, there are also additional specific problems one encounters when trying to recover the model forecast omega field using (1). The following discussion clarifies this point. Three time derivatives are involved in (1), ⭸T ⭸t

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冉冊

⭸ ⭸␨ag , f , ⭸p ⭸t phys

冉冊

1 ⭸ ⭸␾ and ␤ . f ⭸y ⭸t

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The term ⳵T/⳵t| phys is set equal to the numerical model diabatic temperature tendencies. Because of the smallscale variability of deep convection, we can expect that the rhs diabatic forcing in (1) represents significant small-scale forcing. This type of forcing generates small-scale noise in the solution of the predictive system of equations (e.g., Browning and Kreiss 1994, hereafter referred to as BK94). We shall nevertheless attempt to use this forcing as it appears on the rhs of (1) and give indications on the presence of such noise in our solution for ␻. The terms f

冉冊

⭸ ⭸␨ag ⭸p ⭸t

and

冉冊

1 ⭸ ⭸␾ ␤ , f ⭸y ⭸t

when approached in a straightforward manner, lead to difficulties that degrade the accuracy of the omega diagnostic. We describe herein the precise nature of this problem. By definition, the ageostrophic vorticity tendency is given by ⭸␨ag ⭸␨ ⭸␨g ⫽ ⫺ , ⭸t ⭸t ⭸t

where ␨ag, ␨, and ␨g are the ageostrophic, relative, and geostrophic vorticities, respectively. Given the available forecast fields, the first term on the rhs of (2) is easily computed. The problem appears in the second rhs term of (2). First, (1) is solved to get ␻0b using the approximations

and

⭸␨ag ⫽0 ⭸t

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冉冊

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⭸ ⭸␾ ⫽0 ⭸y ⭸t that is,

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⭸ ⭸␻b0 ⭸␷ ⭸␻b0 ⭸u R R R 2 0 ⭸2␻b0 ⭸2␨ ⭸T 0 ⫺ f ␻ ⫺ f ⵜ S␻b ⫹ f 共 f ⫹ ␨兲 ⫺ ⫽ ⫺ ⵜ2共⫺V · ⵱T 兲 ⫺ ⵜ2 b 2 2 p ⭸p ⭸x ⭸p ⭸y ⭸p p p ⭸t ⭸p ⭸p ⫺f

共2兲

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phys

⭸ ⭸ 关⫺V · ⵱共 f ⫹ ␨兲兴 ⫺ f 共k · ⵱ ⫻ F兲. ⭸p ⭸p

The subscript b in ␻0b stands for balance (the precise meaning of the word balance will become clear below) and the superscript 0 identifies iteration 0. Using the geostrophic wind relationship 1 Vg ⫽ k ⫻ ⵱␾, f

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and applying the operator (k · ⵱⫻) to (6) gives 1 1 ⭸␾ , ␨g ⫽ ⵜh2 ␾ ⫺ 2 ␤ f ⭸y f

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where ⵜ2h is a horizontal Laplacian operator and the Coriolis parameter is assumed to be only a function of the y coordinate. Applying ⳵/⳵t to (7) gives

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冉冊

冉冊

1 ⭸ ⭸␾ ⭸␨g 1 2 ⭸␾ ⫺ 2␤ . ⫽ ⵜh ⭸t f ⭸t ⭸y ⭸t f

ⵜ2␾ ⫽ f␨ ⫺

共8兲

For horizontal scales considered here, we stress that our calculations of the term ⳵␾/⳵t convinced us that it cannot be evaluated directly using geopotential output fields as mentioned previously; that is, the computed omega field is strongly contaminated by spatial noise components. Because of the presence of fast (advective) time-scale noise in the solution of the predictive equations (e.g., see BK94), a special procedure is required for estimating this forcing term accurately and in order that it represents the major slow component of the solution. The following iterative scheme is used. Similar to BK94 [their (2.10)], and Browning and Kreiss (1997, hereafter referred to as BK97, system 5.3), a diagnostic equation can be used to derive geopotential

ⵜ2

冉冊| ⭸␾ ⭸t

⫽ f ⵜ2

b

冉冊冋冉冊 ⭸␺ ⭸t

⭸2

⫹2

⭸x

2

⭸␺ ⭸t

⭸2␺ ⭸y

2

⫹

冉冊

in (1). The boundary condition used to invert the ⵜ2 in (10) is ␾f ⫺1. On the other hand, a Dirichlet boundary condition is used, ⳵␾/⳵t ⫽ 0 to invert the ⵜ2 in (11). The first term on the rhs of (2) can be computed for iteration 0 using the complete vorticity tendency equation. Setting ␻ ⫽ ␻0b; ␨b ⬃ ␨ and we get ⭸␨ ⭸t

| ⫽ ⫺V · ⵱ f ⫹ ␨ ⫹ f ⫹ ␨ 共

b

⫺

冉

兲

共

冊

兲

⭸␻b1 ⭸p

⫺ ␻b1

冋

ⵜ2␾b ⫽ f ⵜ2␺ ⫹ 2

⭸␨ ⭸p

⭸␻b1 ⭸␷ ⭸␻b1 ⭸u ⫺ ⫹ 共k · ⵱ ⫻ F兲 ⭸x ⭸p ⭸y ⭸p 共12兲

and thus ⳵␨ag/⳵t|b in (2) can be estimated. Once ⳵␨ag/⳵t |b is known, it is used in (1) to obtain a second estimate, noted ␻1b. Only one iteration was needed to produce good accuracy. As will be shown in section 4, using only the time derivative of Charney’s equation in (2) proves to be sufficient to significantly improve ␻ diagnostics for mature convective systems. A refinement of the procedure would be to keep terms involving divergence in (9) [or (2.10) in BK94] as an estimate for the current

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⭸2␺ ⭸2␺

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⭸2␺ ⫺ ⭸x⭸y ⭸x2 ⭸y2

2

⫹

⭸f ⭸␺ ⭸f ⭸␺ ⫹ . ⭸y ⭸y ⭸x ⭸x 共10兲

For the iteration 0, we use (10) to derive balanced geopotential time tendencies in (8); that is,

⭸x ⭸y

1 ⭸ ⭸␾ ␤ f ⭸y ⭸t

⭸f dD u ⫹ 2J ⫺ ⫺ D2, ⭸y dt

where J is the Jacobian of the horizontal winds and D is horizontal divergence (the latter being related to omega via the continuity equation as shown in the appendix). At iteration 0, we neglect the divergent part of the wind, which amounts to Charney’s (1955) nonlinear balance equation

⭸2␺ ⭸2

The term ⳵␺/⳵t is evaluated using the numerical model streamfunction tendencies computed using Helmholtz decomposition of the horizontal wind tendencies. The ⵜ2 operator can be inverted in (11) to evaluate ⳵␾/⳵t|b, and thus ⳵␨g/⳵t|b in (8) and a balanced estimate of

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2

2

冉冊 ⭸␺ ⭸t

⭸2 ⫺2 ⭸x⭸y

冉冊册 ⭸␺ ⭸t

⭸2␺ ⭸ ⫹␤ ⭸x⭸y ⭸y

冉冊

⭸␺ . ⭸t

共11兲

iteration, that is, ␻kb. This refinement is not considered in the present study, however, because of the sufficient accuracy obtained. Because of the preceding discussion and use of (9), we will refer to (1) as the “nonlinear balance” omega equation. In the proposed iterative scheme, (1) can exhibit problems when the ellipticity condition is not met, that is, Sⱕ0

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or

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f 2 ⭸V 2 RSf . 共␨ ⫹ f 兲 ⱕ p 4 ⭸p

共14兲

To ensure (1) is of elliptic type, the stability S and/or relative vorticity ␨ are slightly modified [following Räisänen (1995, p. 2450)] when one or both conditions are met. Ellipticity classification is discussed in Mikhaïlov (1980). In the case discussed here, a mean (per output time step) of 9% of the points in the domain exhibited this problem and needed these modifications. Of these points, some were located in the regions studied in the following sections. Over these regions, the static stability was left untouched by the procedure but the vorticity field was changed slightly. We note, however, that the terms on the lhs of the complete nonlinear omega equation [(1)] responsible for nonellipticity and involving vorticity are (as will be shown in the results) negligible as compared to the dominant terms. Going back to the mesoscale data assimilation prob-

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lem, the full equation [(1)] may appear too complicated to use in a minimization context like the variational analysis approach, especially if it must be solved at each simulation in the mesoscale extension of F05’s scheme. However, as will be demonstrated in section 4c, a simpler form can be used over strong convective regions.

b. Numerical filtering aspects For comparisons and calculations, the output of the Limited-Area Model (LAM) simulation, which is internally filtered by using cubic splines, is first vertically interpolated to 20 vertical pressure levels (50–1000 hPa with a uniform interval of 50 hPa), then temporally interpolated between the output time steps (every 5 min), and finally horizontally filtered using a Shuman numerical filter that removes wavelengths equal to or less than 5 grid points. This filtered model output is then used in the NLB omega equation, which is solved iteratively and whose results are then filtered again using the same Shuman filter. This latter filtering was necessary because the calculations in (1) are quite sensitive to numerical noise generated by the high-order spatial derivatives. It is to be noted that a careful examination of the filtering effects revealed that the most important filtering processes are the time interpolation and the five-gridpoint Shuman filter on NLB diagnostics. Other filters have very limited impact (results not shown). The digital filter initialization is of widespread use in variational analysis schemes for synoptic scales at operational centers referred to in section 1. Because of the clear separation of time scales between gravity wave noise and signal at synoptic scales, the use of Huang and Lynch’s (1993) digital filter approach is justified and very efficient. At the mesoscale, however, it is clear that an overlap of time scales exists and, except for significantly reducing the cutoff period, its application needs to be revisited. For these reasons, our NLB vertical motions will also be compared to digitally filtered vertical motions (3-h time span for both 3-h and 1-h cutoff periods), as well as against the unfiltered model’s vertical motion.

3. Numerical experiments Case study The accuracy of the NLB solution of (1) as compared with the model’s vertical motion will be examined (i) for ensembles of rapidly developing storm cells and (ii) for ensembles that have lasted for a longer period of time. The case studied (7 July 2004) is a region of very weak pressure and temperature gradients within a very

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weak surface low pressure system over the middle Mississippi Valley and adjacent states, where deep convection occurred (Fig. 2). This weak low pressure region was connected to a weak trough from a low pressure center over the Great Lakes. There was also a weak temperature gradient and higher humidity in the region where convection was initiated later in the simulation. Horizontal winds at the surface were light, and on the order of 10–20 m s⫺1 in the rest of the troposphere with some maxima on the order of 25 m s⫺1 in the upper troposphere. Several mesoscale “fronts” (significant humidity, pressure, and temperature gradients over small distances) developed during the integration and one of these mesoscale fronts was associated with very strong convection. Numerical simulations are performed using the Global Environmental Multiscale (GEM) model in its regional configurations (REG; Côté et al. 1998) and limited area (Yeh et al. 2002). The GEM-LAM is run for 6 h, using as initial conditions a 12-h forecast of the GEM-REG valid at 1200 UTC 7 July 2004. Because GEM-LAM is of limited area, it needs boundary conditions throughout the simulation; hence, the GEMREG forecasts from 12 h to 18 h are used for boundary conditions (the GEM-REG is thus called the pilot model). The LAM grid has 564 ⫻ 494 horizontal grid points with 58 eta vertical levels, with a resolution of 2.5 km and an integration time step of 1 min. Convection is explicit and the model is nonhydrostatic. The LAM horizontal boundary conditions are updated every hour. A 3-h cutoff time digital filter is used for initialization of the LAM to ensure that gravity waves resulting from the initial conditions do not contaminate the forecast, because we are not interested here in the precision of the forecast but rather in creating a model state that is not contaminated by artificial gravity waves resulting from the initial conditions. The model forecast fields are saved every 5 min. To assess the impact of the information from divergent winds on a forecast, as well as to validate the diagnosed balanced divergent winds, special experiments were designed. These experiments are summarized in Table 1, where a LAM2 domain is nested within the LAM domain described above. The LAM2 domain is centered in the LAM domain and 3D grid points in both domains are collocated. The dimensions are 534 ⫻ 464 grid points, which are smaller than the LAM pilot grid for nesting purposes. No time digital filters normally used in the early stage of the forecast to filter gravity wave noise were used in these LAM2 simulations. The initial conditions used for these LAM2 simulations were those of the LAM 5-h forecast, valid at 1700 UTC, except for the divergent winds. Hence, ther-

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FIG. 2. A portion of the GEM-REG pilot model domain is shown, along with mean sea level pressure (solid) and geopotential height at 500 hPa (dashed). This is a 17-h forecast valid at 1700 UTC 7 Jul 2004. The GEM-LAM domain is labeled (A), along with regions of interest (B and C).

modynamic fields (temperature and humidity) as well as pressure and rotational winds are obtained from the LAM 5-h forecast. The LAM2 simulations are initiated at 1700 UTC, and the LAM simulation is used as the pilot model. The nonlinear balance diagnostics of the divergent winds use explicit convection temperature–time tendencies from the pilot LAM simulation (perfect case), combined with the pressure, rotational wind, temperature, and moisture of the LAM2 initial conditions.

4. Results and discussion In an attempt to validate our diagnostics of balanced divergent wind and vertical motion, a special set of numerical experiments was designed.

a. Importance of initial divergent winds As a simple illustration of the impact of divergent wind information on convection evolution, three numerical simulations have been performed. One of the simulations is using the nonlinear balance diagnosed

divergent part of the winds in the analysis, while the other is using the pilot model (reference), and another one has zero divergent winds (see Table 1). The vertical motions as output from these simulations are then compared. We note that in the context of a data assimilation cycle, there is always a background divergent wind prior to the analysis. The zeroing of the initial divergence field thus represents an extreme case. However, because the central focus of the mesoscale forecast is convection evolution, we use the initial zero divergence simulation case as a clear demonstration of the importance of avoiding divergence spinup with regard to convection lifetime. The LAM2S2 simulation (balanced divergent winds), TABLE 1. Configurations for special divergent wind experiments. Expt

Divergent winds

LAM2S1 LAM2S2 LAM2S3

From pilot LAM (reference) From nonlinear balance diagnostics of pilot LAM Set to zero

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FIG. 3. The 45-min forecast valid at 1745 UTC 7 Jul 2004: GEM-LAM model LAM2S1 (solid) and LAM2S2 (dashed) vertical motions at 750 hPa over a subdomain of the model around the Mississippi Valley (region B highlighted in Fig. 2). Contours from ⫺50 to 50 Pa s⫺1 are shown (interval 5).

when compared with the LAM2S1 reference simulation, shows that the regions of significant upward vertical motion are slightly displaced spatially and that this displacement increases slowly with time. The 3D vertical motion structure is quite similar (see Figs. 3 and 4), even at the smallest scales, which is confirmed by a

mean correlation of 0.70 after 45 min of simulation (0.83 after 30 min). On the other hand, the LAM2S3 simulation (zero divergent winds) shows that convection is generated correctly in the first minutes of integration, but after 20 min the 3D structures and the regions affected by con-

FIG. 4. Same as in Fig. 3, but for a zoom over the eastern part of the domain at the vertical center.

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FIG. 5. Same as in Fig. 3, but for LAM2S1 (solid) and LAM2S3 (dashed).

vection are very different from the reference simulation (see Figs. 5 and 6). Convection is also generated in locations different from the reference simulation. It is only at the largest scales that the coverage and intensity of convection is spatially similar, but only for the first 30 min of simulation. This is confirmed with the correlation against the reference run, which is only 0.13 after 45 min (0.20 after 30 min). These results emphasize clearly the important role of

divergent winds for the evolution of convection at these scales. Even when all the other pieces of information (temperature, moisture, rotational winds, and pressure) are correctly taken into account, a lack of proper treatment of divergent winds can be detrimental. At synoptic scale, the ratio of the spinup time of the numerical model to the lifetime of meteorological systems that involve convection is quite small. On the contrary, at the scales examined here, this ratio is much larger be-

FIG. 6. Same as in Fig. 5, but for a zoom over the eastern part of the domain at the vertical center.

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FIG. 7. The 5-h forecast valid at 1700 UTC 7 Jul 2004: GEM-LAM (solid) and NLB diagnosed (dashed) vertical motions at 750 hPa over a subdomain of the model around the Mississipi Valley (region B highlighted in Fig. 2). Contours from ⫺50 to 50 Pa s⫺1 are shown (interval 5).

cause the lifetime of ensembles of convective cells is on the order of 45 min. This has drastic impacts on atmospheric data assimilation at the mesoscale. Often of secondary importance in the design of analysis schemes at synoptic scale, it now becomes of primary importance at the mesoscale.

b. Accuracy of NLB omega diagnostics At 1700 UTC, after 5 h of integration, intense convection occurs throughout much of the LAM domain. Figure 7 shows the model’s 750-hPa omega vertical motion superimposed on the NLB diagnosed omega vertical motion in a region of active convection (region B highlighted in Fig. 2). For the majority of convective ensembles (especially the older ones), the agreement between the model’s and the nonlinear balanced omega vertical motion is very good. This is also confirmed by vertical cross sections and profiles (Figs. 8 and 9). This means that the model vertical motion is nearly balanced even at these scales and that the model’s temperature tendencies due to convection (directly from the model’s physics) can be used in the omega equation [(1)]. For the period between 1520 UTC and 1800 UTC (between 3 h 20 min and 6 h of integration), the mean correlation between the nonlinear and the model omega is 0.92, while the mean ratio of the root-mean-squares is 1.07. Other experiments have also been performed using the GEM-REG at 15-km resolution, which uses a

parameterized convective scheme, and the results show a similar accuracy of the diagnosed balanced omega vertical motion (not shown). We stress here that this result has important implications for the introduction of a balanced coupling between diabatic effects and vertical motion in a data assimilation context. It stresses the fact that the use of model-generated time tendencies of temperature, for instance, can actually be used in a numerical procedure such as used here for inferring balanced vertical motion. This supports F05’s methodology to the extent that these physical time tendencies can actually be incorporated in the analysis procedure.

c. Dominant terms of the NLB omega equation With the exception of the forcing due to friction, the relative magnitudes of the lhs and rhs terms of the NLB omega equation [(1)] were evaluated at 1700 UTC (5-h forecast). These magnitudes are shown in Table 2 for grid points at which the NLB vertical motion has an absolute value of at least 5 Pa s⫺1. Clearly from (1), only the first lhs term and the diabatic rhs terms are dominant in regions of strong vertical motion: R ⭸T R 2 ⵜ S␻ ⫽ ⫺ ⵜ2 p p ⭸t

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共15兲

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FIG. 8. Same as in Fig. 7, but for a vertical cross section following the arrow in Fig. 7.

where Dirichlet homogeneous boundary conditions are used. However, the Laplacian of temperature advection as well as the ageostrophic tendency terms on the rhs become significant in a few convective ensembles, accounting for as much as 20% of the total vertical motion. It follows that the inclusion of the balanced ageostrophic vorticity tendency term in the omega equation can improve the balanced vertical motion by as much as 20%. Physically, however, a simple balance of terms such as in (15) is interesting to apply simply in a practical data assimilation system. If boundary conditions were biperiodic, for instance, the Laplacian in (15) could be inverted on both sides of the equation to get the physical result that whenever heating (cooling) occurs via moist diabatic forcing, an upward (downward) vertical motion is associated with it in a mature deep convective situation. The statement that the NLB vorticity tendency [(11)] is a good approximation of the model vorticity tendency is confirmed in Figs. 10 and 11. This is representative of the overall results. It is also confirmed by a mean correlation of 0.67 and a mean root-mean-square fraction of 0.84 over the forecast hours from 3 h 20 min to 6 h of the simulation and between the vertical levels from 300 to 800 hPa. It is then possible to calculate a balanced height tendency using only the rotational part of the wind, which is less affected by unbalanced motions, using (2)–(11). Thus, using this methodology, it is possible to recover the balanced part of the model’s height tendency, and, through the NLB equation, it improves the NLB vertical motion representation. This is

especially true at the center of very active ensembles of storm cells where the ageostrophic vorticity tendency term is more important.

d. Comparison with digital filter vertical motion As mentioned in the previous section, because of its important use in current synoptic-scale 3D–4DVAR data assimilation schemes at operational centers, we examine the use of the digital filter technique in our context. In this section, a comparison will be made between the diagnosed balanced vertical motion, filtered spatially using a 5-gridpoint horizontal Shuman filter at 1.5 h into the simulation, and vertical motion calculated using a digital filter with two different cutoffs (1 h and 3 h) as well as with the unfiltered instantaneous model vertical motion, which as previously mentioned, was spatially filtered internally using cubic splines. It is to be noted that the effect of using the model’s internal cubic spline filter, as well as using the Shuman filter, on model output prior to diagnostic calculations is not very significant overall (not shown). They primarily affect convective cloud top (in the vertical) where the vertical gradients can be very strong in some places. A comparison, shown for region C highlighted in Fig. 2, between the balanced and 3-h digitally filtered vertical motions (Figs. 12 and 13) shows that the latter is much weaker than the former and the spatial correlation is not very good (Table 3). The unfiltered instantaneous vertical motion is, on the other hand,

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FIG. 9. Same as in Fig. 7, but for a vertical profile at X indicated in Fig. 7. Horizontal axis is in Pa s⫺1.

better spatially and vertically correlated (Figs. 14 and 15) with the balanced one. However, the amplitudes are much larger, which is possibly the result of gravity wave oscillations and balance adjustments. Temporal animations (not shown) also reveal what are probably gravity wave signatures (especially in the cell complex in the left part of the domain of Fig. 14). The use of 3-h cutoffs, although justifiable for coarser grids simulations, is as expected, not appropriate here. The 1-h cutoff digitally filtered vertical motion has a behavior between these two extremes, and it is better correlated (Figs. 16 and 17) with the balanced vertical motion than the 3-h cutoff case. However, the amplitudes are too weak. Furthermore, the 1-h cutoff digital filter cannot distinguish between the diabatically forced local circulation and some freely propagating gravity waves that have a comparable time scale, and thus it filters out incorrectly the diabatically balanced

part of the information, which is thus lost (see Figs. 14 and 16). It seems to imply that further care is needed in the application of digital filter initialization (DFI) in mesoscale models. Table 3 shows the mean correlations and absolute mean values of these vertical motions and TABLE 2. Relative magnitude of the NLB omega equation terms using the LAM simulation. Nonlinear balance omega equation (R/p)ⵜ S␻ ⫹f ( f ⫹ ␨)(⳵2␻ /⳵p2) ⫺f␻ (⳵2␨/⳵p2) ⫺f (⳵/⳵p)[(⳵␻/⳵x)(⳵␷/⳵p) ⫺ (⳵␻ /⳵y)(⳵u/⳵p)] ⫽⫺(R/p)ⵜ2(⫺V · ⵱T ) ⫺(R/p)ⵜ2(⳵T/⳵t)| phys ⫺f (⳵/⳵p)关⫺ V · ⵱( f ⫹ ␨)] ⫹f (⳵/⳵p)(⳵␨ag/⳵t) 2

10⫺11 10⫺16 10⫺16 10⫺15 10⫺12 10⫺11 10⫺15 10⫺12 at max

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FIG. 10. Same as in Fig. 7, but for vorticity tendency. Contour interval is 5 ⫻ 10⫺7 s⫺2.

summarizes those results. It is clear from these results that the use of the NLB omega equation produces more accurate results than the digital filter as used here.

5. Conclusions Balance issues in advanced atmospheric data assimilation schemes are a challenge at the mesoscale (model resolutions of 1–10 km). The current practice is that either a digital filter approach is used or these issues are

simply not dealt with. It is of primary importance to address diabatic balance issues because balance provides the link between the model’s variable and modelgenerated physical time tendencies, a critical aspect for maintaining storm cells after the analysis time. A set of experiments using either the balance derived in this study or zero divergent winds showed that even with all the exact thermodynamic and dynamic fields as initial conditions, the accuracy of the divergent part of the wind is crucial. Nonlinear balance diagnostics pro-


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FIG. 12. The 1.5-h forecast valid at 1330 UTC 7 Jul 2004: GEM-LAM digitally filtered 3-h cutoff (solid) and NLB diagnosed (dashed) vertical motions at 750 hPa over a model subdomain (shown for region C highlighted in Fig. 2). Contours from ⫺50 to 50 Pa s⫺1 are shown (interval 5).

duced with the method presented here were able to retrieve, from the other model’s variables and the convective scheme, much of the needed divergent wind information with respect to the forecast of the evolution of convection. The work presented here demonstrates that for sustained summer convective circulations, an accurate di-

agnostic equation for vertical motion ␻ can be used for meso- ␤ scales (phenomena of 15–100-km length scales). It also shows that for such cases, the numerical model’s physical tendencies, through its convective temperature tendencies, can be used directly in a nonlinear balance omega equation to calculate balanced omega vertical motion, and therefore, balanced


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TABLE 3. Comparing vertical motions with NLB vertical motion digitally filtered using 1- and 3-h cutoff periods (3-h time span) as well as not digitally filtered.

Digital filter 3 h Digital filter 1 h No digital filter

Correlation

Ratio of mean RMS

0.70 0.84 0.87

0.56 0.97 1.28

horizontal divergent winds. This vertical motion is in dynamical balance with the model’s convective diabatic temperature tendencies. Using a complete form of the omega equation [see (1)], the results reveal that for convective situations, only the first term on the lefthand side of the nonlinear omega equation is significant. On the right-hand side, apart from the Laplacian of diabatic temperature tendencies (which is the dominant term), the Laplacian of temperature advection and the ageostrophic vorticity tendency terms can be important (up to 20% of the total for the latter) over some regions. Overall, however, they are much smaller. The resulting omega equation, where all the rhs terms of (1) (the beta term can be neglected) are included but only the lhs term ⵜ2S␻ is retained, takes the form ⵜ2S␻ ⫽ ⫺ⵜ2共⫺V · ⵱T 兲 ⫺ ⵜ2 ⫺

|

fp ⭸ ⭸T ⫺ 关⫺V · ⵱共 f ⫹ ␨兲兴 ⭸t phys R ⭸p

冉冊

fp ⭸ fp ⭸ ⭸␨ag . 共k · ⵱ ⫻ F兲 ⫹ R ⭸p R ⭸p ⭸t

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It is important to note that the ageostrophic vorticity tendency ⳵␨ag/⳵t depends on ⵜ2⳵␾/⳵t [see (8)]. Using the numerical model’s geopotential produces spatial noise when evaluating horizontal derivatives over localized convection with scales comparable to the grid length. To overcome this problem, a diagnostic equation for balanced geopotential as in Browning and Kreiss (1994, 1997; system 5.3) is used to evaluate the slow component of the solution, which in our approximation is reduced to Charney’s (1955) nonlinear balance equation for iteration 0. Thus, it also means that ⳵␨ag/⳵t depends on an accurate ⳵␺/⳵t. In our study, we were given the vorticity field at regular time steps so this term could be computed and it accurately represented the adjusted mature convective cells studied here. This piece of information would obviously be missing in a realistic operational data assimilation context so that approximations would have to be used. It may also be useful to exploit the simplified (typically 20% less accurate) diagnostic equation [(16)] as a first implementation step. This needs to be explored as an extension to this study. Also, our results tend to indicate that the balanced part of the computed omega solution projects mostly onto deep vertical scales. However, the use of Charney’s nonlinear balance equation as a means of computing one important rhs forcing term of the diabatic omega equation is not restricted to large vertical scales. It also involves shallow internal mode structures that may be important (see particularly Figs. 8–11 and related discussion in section 4c). This result can be seen as a complement to the discussion given in Errico (1991,

FIG. 14. Same as in Fig. 12, except that the model vertical motion is unfiltered (black).

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p. 96) on mesoscale balance issues. We keep for a forthcoming study the analysis of the results presented here in terms of modal structure, to better complement Errico’s (1991) study. A comparison was made between balanced vertical motion with the unfiltered model and digitally filtered vertical motion (1-h and 3-h cutoff periods using a 3-h time span and 1-min time steps; i.e., 180 temporal points used in the filter, making the filter’s response

sharp). Results indicate that a 3-h cutoff period is much too long for simulations using a horizontal resolution of 2.5 km. Although the amplitudes are still damped, a 1-h cutoff for the digital filter is a compromise because it retains most of the 3D structure. However, the amplitudes are too weak. Furthermore, the 1-h cutoff digital filter cannot distinguish between the diabatically forced local circulation and some freely propagating gravity waves that have comparable time scales, and thus it

FIG. 16. Same as in Fig. 12, except that the model vertical motion is using a 1-h cutoff digital filter (black).

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incorrectly filters out the diabatically balanced part of the information that is thus lost (see Figs. 14 and 16). Table 3 shows the mean correlations and absolute mean values of these vertical motions and summarizes those results. It is clear from these results that the use of the NLB omega equation produces more accurate results than the digital filter as used here. These results show that even if convective temperature tendencies from the model’s physics are very local and involve sharp gradients, they can be used to calculate a balanced vertical motion and therefore horizontal divergent winds, which are in dynamical balance with the model’s physics. Spatial and temporal filtering are needed to achieve these results. This result was found to be representative of both explicit and parameterized convective schemes, at model resolutions ranging from 2.5 to 15 km. Although the results presented here have significant accuracy to be envisaged as a useful tool for balance constraint in mesoscale data assimilation schemes, increasing the vertical resolution as compared with the one used here (i.e., 20 vertical levels) can potentially bring higher accuracy of diagnosed vertical motion. The case study presented here is almost purely convective. Hence this work should be extended to more (similar) cases, as well as to baroclinic cases and others that would involve significant upper-atmosphere dynamics, such as jets. We leave these aspects for a future study. Acknowledgments. We dedicate this study on atmospheric vertical motion to Prof. Peter Zwack, our

teacher and very enthusiastic friend who deeply loved to transfer his knowledge to others, who passed away recently. The subject treated here was one of his favorites and was largely stimulated by him. This work is part of the Ph.D. thesis of the first author. Thanks are owed to Amin Erfani, Sylvie Gravel, and Michel Desgagné for GEM model support, as well as to Yves Chartier for help on several technical issues. This work was funded partially by the Canadian Foundation for Climate and Atmospheric Sciences (CFCAS).

APPENDIX Complete Omega Equation Derivation Starting with the equation of motion in isobaric coordinates and in vectorial form, dV ⫹ f k ⫻ V ⫹ ⵱p␾ ⫹ F ⫽ 0, dt

共A1兲

applying the operator k · ⵱p ⫻ to (A1), one can obtain the vorticity equation. Arranging in a familiar form and omitting the p subscript to the ⵱ operator gives ⫺共 f ⫹ ␨兲⵱ · V ⫺ ␻

冉

冊

⭸␨ ⭸␨ ⭸V ⫹k· ⫻ ⵱␻ ⫽ ⭸p ⭸p ⭸t

⫹ V · ⵱共f ⫹ ␨兲 ⫺ k · ⵱ ⫻ F.

共A2兲

Taking the vertical derivative ⳵/⳵p of the vorticity equation [(A2)], and expanding the tilting–twisting term gives

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冊冉冊

⭸ ⭸␻ ⭸␨ ␻⭸2␨ ⭸ ⭸␻ ⭸␷ ⭸␻ ⭸u ⭸ ⭸␨ ⭸ ⭸ ⫺ ⫺ ⫺ ⫽ ⫹ 关共 f ⫹ ␨兲⵱ · V兴 ⫺ 关V · ⵱共 f ⫹ ␨兲兴 ⫺ 共k · ⵱ ⫻ F兲. 2 ⭸p ⭸p ⭸p ⭸p ⭸x ⭸p ⭸y ⭸p ⭸p ⭸t ⭸p ⭸p ⭸p 共A3兲

Using the continuity equation ⵱ · V ⫽ ⳵␻/⳵p, the definition of the geostrophic vorticity, ␨ ⫽ ␨g ⫹ ␨ag, and 1 1 ⭸␾ ␨g ⫽ ⵜ2␾ ⫺ 2 ␤ f ⭸y f gives

冋

册

冉

冊

冉

冊

⭸␻ ⭸␨ ⭸ ⭸␻ ⭸␷ ⭸␻ ⭸u 1⭸ ⭸ ⭸ ⭸␻ ⭸2␨ ⭸␾ 1 ⭸␾ ⫺␻ 2⫺ ⫺ ⫺ ␤ ⫺ ⫽ ⵜ2 ⫹ 共 f ⫹ ␨兲关V · ⵱共 f ⫹ ␨兲兴 ⭸p ⭸p ⭸p ⭸p ⭸p ⭸x ⭸p ⭸y ⭸p f ⭸t ⭸p f ⭸y ⭸p ⭸p ⫺

冉冊

⭸ ⭸ ⭸␨ag . 共k · ⵱ ⫻ F兲 ⫹ ⭸p ⭸p ⭸t

共A4兲

Expanding the first two lhs terms gives 共 f ⫹ ␨兲

冉

⭸2␻

冊

冉

冊

冉冊

⭸ ⭸␻ ⭸␷ ⭸␻ ⭸u 1⭸ ⭸ ⭸ ⭸ ⭸␨ag ⭸2␨ ⭸␾ 1 ⭸␾ ⫺ ⫺ ␤ ⫺ ␻ ⫺ ⫽ ⵜ2 ⫹ 关V · ⵱共 f ⫹ ␨兲兴 ⫺ 共k · ⵱ ⫻ F兲 ⫹ . 2 2 ⭸p ⭸x ⭸p ⭸y ⭸p f ⭸t ⭸p f ⭸y ⭸p ⭸p ⭸p ⭸t ⭸p ⭸p 共A5兲

Using the hydrostatic approximation ⳵␾/⳵p ⫽ 1/␳ and ␳ ⫽ p/RT and the thermodynamic equation ⭸T ⭸T ⫽ S␻ ⫺ V · ⵱T ⫹ ⭸t ⭸t

冏

,

phys

the equation becomes

冉

冊

⭸ ⭸␻ ⭸␷ ⭸␻ ⭸u R R ⭸2␻ ⭸2␨ ⭸T R 2 ⵜ S␻ ⫹ 共f ⫹ ␨兲 2 ⫺ ␻ 2 ⫺ ⫺ ⫽ ⵜ2共⫺V · ⵱T 兲 ⫺ ⵜ2 pf ⭸p ⭸x ⭸p ⭸y ⭸p pf pf ⭸t ⭸p ⭸p ⫺

冏

⫹

phys

⭸ 关V · ⵱共 f ⫹ ␨兲兴 ⭸p

冉冊冉

冊

⭸ ⭸ ⭸␨ag ⭸ 1 ⭸␾ ⫺ ␤ . 共k · ⵱ ⫻ F兲 ⫹ ⭸p ⭸p ⭸t ⭸t f 2 ⭸y

共A6兲

Finally, multiplying by f, modifying the sign of the vorticity advection term, and rearranging the terms gives the final equation

冉

冊

⭸ ⭸␻ ⭸␷ ⭸␻ ⭸u R R ⭸2␻ ⭸2␨ ⭸T R 2 ⵜ S␻ ⫹ f 共 f ⫹ ␨兲 2 ⫺ f␻ 2 ⫺ f ⫺ ⫽ ⫺ ⵜ2共⫺V · ⵱T 兲 ⫺ ⵜ2 p ⭸p ⭸x ⭸p ⭸y ⭸p p p ⭸t ⭸p ⭸p ⫺f

冉冊

冉冊

⭸ ⭸ ⭸ ⭸␨ag 1 ⭸ ⭸␾ ⫺ ␤ . 关⫺V · ⵱共 f ⫹ ␨兲兴 ⫺ f 共k · ⵱ ⫻ F兲 ⫹ f ⭸p ⭸p ⭸p ⭸t f ⭸y ⭸t

Thus, this complete omega equation can be derived using the equations of motion, energy, ideal gas, and continuity by using only the hydrostatic approximation.

冏

phys

共A7兲

REFERENCES Baer, F., and J. J. Tribbia, 1977: On complete filtering of gravity modes through nonlinear initialization. Mon. Wea. Rev., 105, 1536–1539.

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Browning, G. L., and H.-O. Kreiss, 1994: The impact of rough forcing on systems with multiple time scales. J. Atmos. Sci., 51, 369–383. ——, and ——, 1997: The role of gravity waves in slowly varying in time mesoscale motions. J. Atmos. Sci., 54, 1166–1184. Charney, J., 1955: The use of the primitive equations of motion in numerical prediction. Tellus, 7, 22–26. Côté, J., S. Gravel, A. Méthot, A. Patoine, M. Roch, and A. Staniforth, 1998: The operational CMC–MRB Global Environmental Multiscale (GEM) model. Part I: Design considerations and formulation. Mon. Wea. Rev., 126, 1373–1395. Daley, R., 1991: Atmospheric Data Analysis. Cambridge University Press, 471 pp. Derber, J. D., and F. Bouttier, 1999: A reformulation for the background error covariance in the ECMWF global data assimilation system. Tellus, 51, 195–222. Errico, R. M., 1991: Theory and application of nonlinear normal mode initialization. NCAR Tech. Note NCAR/TN-3441IA, National Center for Atmospheric Research, 137 pp. [Available from NCAR, P.O. Box 3000, Boulder, CO 80307.] Fillion, L., M. Tanguay, N. Ek, C. Pagé, and S. Pellerin, 2005: Balanced coupling between vertical motion and diabatic heating for variational data assimilation. Int. Symp. on Nowcasting and Very Short Range Forecasting, Toulouse, France, World Meteorological Organization. [Available online at http://www.meteo.fr/cic/wsn05/DVD/resumes/longs/3.1089.pdf.] Fisher, M., 2003: Background error covariance modeling. Proc. ECMWF Seminar on Recent Developments in Data Assimilation for Atmosphere and Ocean, Reading, United Kingdom, 45–63.

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Gauthier, P., C. Charette, L. Fillion, P. Koclas, and S. Laroche, 1999: Implementation of a 3D variational data assimilation system at the Canadian Meteorological Centre. Part I: The global analysis. Atmos.–Ocean, 37, 103–156. Holton, J. R., 1992: An Introduction to Dynamic Meteorology. 3d ed. Academic Press, 511 pp. Huang, X.-Y., and P. Lynch, 1993: Diabatic digital-filtering initialization: Application to the HIRLAM model. Mon. Wea. Rev., 121, 589–603. Machenhauer, B., 1977: On the dynamics of gravity oscillations in a shallow water model with application to normal mode initialization. Contrib. Atmos. Phys., 50, 253–271. Mikhaïlov, V., 1980: Equations aux Dérivées Partielles. Editions Mir, 390 pp. Parrish, D. F., and J. D. Derber, 1992: The National Meteorological Center spectral statistical interpolation analysis system. Mon. Wea. Rev., 120, 1747–1763. Pauley, P. M., and S. J. Nieman, 1992: A comparison of quasigeostrophic and nonquasigeostrophic vertical motions for a model-simulated rapidly intensifying marine extratropical cyclone. Mon. Wea. Rev., 120, 1108–1134. Räisänen, J., 1995: Factors affecting synoptic-scale vertical motions: A statistical study using a generalized omega equation. Mon. Wea. Rev., 123, 2447–2460. Temperton, C., 1988: Implicit normal mode initialization. Mon. Wea. Rev., 116, 1013–1031. Yeh, K.-S., J. Côté, S. Gravel, A. Méthot, A. Patoine, M. Roch, and A. Staniforth, 2002: The CMC–MRB Global Environmental Multiscale (GEM) model. Part III: Nonhydrostatic formulation. Mon. Wea. Rev., 130, 339–356.