## 1. Introduction

During the last several years, considerable efforts have been devoted to understanding and to solving variational data assimilation problems. In meteorology, the potential usefulness of variational methods was first introduced by Sasaki (1958, 1970). This approach consists of minimizing a given scalar function by measuring the distance between a model state and the observations. In this respect, it requires the computation of the gradient of the scalar function with respect to the variables of the initial state.

The usage of the adjoint method described by Lewis and Derber (1985), Le Dimet and Talagrand (1986), and Talagrand and Courtier (1987a, b) enables the computation of the gradient of any objective function with respect to any initial model parameter. Their pioneering work provided for various numerical weather prediction centers a solid basis to design and further develop an efficient data assimilation system.

As stated by Errico (1997), the principal application of the adjoint model (ADM) is sensitivity analysis, and all its other applications may be considered as derived from it. Errico provides a complete discussion on the definition of adjoint sensitivity and its interpretation. The sensitivity of one aspect of the forecast with respect to small perturbations of initial conditions was the subject of the study of Errico and Vukicevic (1992) on real data cases.

Preliminary results on the sensitivity of *realistic* forecast errors to initial conditions were obtained by Rabier et al. (1993) where the objective function was the square norm of the difference between the operational 48-h forecast and the verifying analysis. In their experiments, a low-resolution (triangular truncation 63 and 31 vertical levels), near-adiabatic global model with surface drag and horizontal and vertical diffusion as in Buizza (1994) was used. The adjoint integrations were performed in the vicinity of a trajectory derived from a model employing the same simple physics. To improve these preliminary results, Rabier et al. (1996) computed trajectories with the fall operational physical package in the forward nonlinear model. The adjoint integration was performed to investigate the sensitivity of the 48-h forecast errors to the initial conditions of the forecast. The objective function was taken as the square total energy norm of the forecast errors, computed over the extratropical Northern Hemisphere. In their sensitivity experiments with a limited-area model, Gustafsson et al. (1998) employed the same objective function and norm. In addition, they studied the sensitivity of the forecast error with respect to the lateral boundary conditions. For a regional weather prediction model, lateral boundary condition error is another important error source distinct from the errors in the initial conditions. The specification of lateral boundary conditions is a weak point of limited-area prediction models, most of them using a boundary relaxation scheme (Davies 1983). Also, sensitivity studies with respect to both initial and lateral boundary conditions were performed by Errico et al. (1993) or more recently by Lewis et al. (2001). In addition, using a limited-area model, Zou et al. (1998) showed that the 5-day forecast of an explosive extratropical cyclone is not only sensitive to the errors in the initial conditions but also to the model physics and the intrinsic instability of the atmosphere. However, all these studies mostly have addressed synoptic-scale features predicted by global or large-sized limited-area models.

Sensitivity analysis using the adjoint of the high-resolution model Aire Limitée Adaptation Dynamique Développement International (ALADIN) has been initiated by Horányi and Joly (1996). Using an *adiabatic* ADM, they studied the sensitivity of two idealized frontal waves. They showed that the steady fronts are associated with very strong amplifications of the fastest-growing eigenvectors.

This paper provides an experimental framework designed to assess the performance and the evolutions of a high-resolution ADM for the mesoscale range. The main goal is to use and assess the behavior of the diabatic adjoint ALADIN model at the same horizontal and vertical resolution than the nonlinear forecast model. Our experiments are performed on real cases. The objective function is taken as the square total “dry” or “moist” energy norm of the forecast errors. The adjoint integration is performed along a diabatic basic state computed with the entire operational physical package of the nonlinear model.

The physical processes and their parameterization are described in section 2. The distinctions between the linearized and the nonlinear schemes are briefly pointed out. In the ADM, two physical parameterization schemes are employed by turn. Primarily, simple physics only describing the surface drag and vertical diffusion (Buizza 1994) are used. This simple physics, initially developed for a lower-resolution model, is found to trigger numerical instability in the high-resolution model. In section 3, the approach used to overcome this difficulty is described. There, the high-resolution (Δ*x* ∼ 10 km, 31 vertical levels) adjoint model is used to compute the gradients of an 18-h forecast error cost function with respect to the initial conditions. The investigations are performed using as a case study a rapidly evolving cyclone that occurred on 27 December 1999, the so-called Second French Christmas storm. This is a difficult forecast case especially for a limited-area model because it combines a complex upstream large-scale flow partially outside of the model domain with an active frontal zone in the lower troposphere. Thus, in terms of sensitivity and forecast errors, one expects both a signal in the dynamically sensitive regions of the initial conditions and in the lateral boundary data.

In section 4, the experiments are designed in order to determine whether the mesoscale sensitivity can be used for improving high-resolution precipitation forecasts. There, the potential benefit of using more complex simplified physical parameterizations (Janisková et al. 1999) in the gradient computations is assessed. As a case study, a misforecasted rainfall event that occurred in France on 3 May 2001 is chosen. We focus on the sensitivity of a mesoscale convective precipitation system, embedded in a larger subsynoptic frontal band. Thus, both the utility of the simplified physics and the impact of the ADM solutions on the forecast of precipitation are investigated in this situation. Finally, some conclusions and perspectives on the potential benefit of the adjoint technique at the 10-km scale, with a four-dimensional variational data assimilation (4DVAR) type of application in mind, are drawn.

## 2. Physical parameterizations in the ALADIN adjoint model

The purpose of this section is to briefly describe the parameterization of the physical processes, which is addressed in the last section of this study. The main differences between the schemes developed for the operational nonlinear forecast model and for the ADM are stressed. An important detail here is that the physical parameterization schemes employed in the ALADIN limited-area model are the same as those used in the global model Action de Recherche Petite Echelle Grande Echelle (ARPEGE), which is the operational global model at Météo-France (Courtier et al. 1991). Furthermore, all numerical details listed in this section refer to anterior work with the ARPEGE model, especially the original developments presented in Janisková et al. (1999). Hereafter, the simplified parameterization schemes are referred to as the *improved simplified physics.*

### a. Vertical turbulent diffusion

The complete parameterization scheme of the atmospheric turbulence simulates the effect of the turbulent vertical mixing, namely, the vertical redistribution of momentum, mass, and heat. The aim is to compute physical tendencies of horizontal wind components (*u* and *υ*), dry static energy (*s*), and specific humidity of water vapor (*q*). Hereafter, *ψ* denotes any of these conservative variables.

A common feature for both complete and linear schemes is the separate treatment of the turbulent transfer between the surface and the lowest model level and the turbulent transport between the model levels.

The comparison between the linearized schemes proposed by Buizza (1994) and Janisková et al. (1999) is performed with the aim of assessing their performances in the frame of the high-resolution model.

#### 1) Simple vertical diffusion scheme

The scheme is developed by Buizza (1994), and it is referred to as a simple scheme because it uses constant exchange coefficients. The surface drag and the upper-air fluxes of the horizontal wind components and dry static energy are computed separately. Note that this scheme does not treat the specific humidity of water vapor.

*ρ*is the reference density of the air, and

_{o}*κ*is the von Karman constant whereas

*ψ*stands for the horizontal wind components at the height

_{N}*z*of the lowest model level

_{N}*N.*The wind velocity scaling factor

*u*

_{*}=

*f*(

*u*

^{l}

_{*},

*u*

^{s}

_{*}) and the roughness length

*z*

_{o}=

*f*(

*z*

^{l}

_{*},

*z*

^{s}

_{*}) are functions of the land–sea mask. The superscripts

*l*and

*s*refer to the parameter values over land and sea, respectively.

*ψ*readsHere

*l*(

*z*) stands for the mixing length defined as in Blackadar (1976). In our experiments the function

*f*(

*z*/

*h*

_{o}) =

*e*

^{−(z/ho)}is chosen. The

*f*functions are described in Buizza (1994).

Note that hereafter, the *simple physics* or *simple initial physics* will refer to the simple vertical diffusion scheme with the following default values: *u*^{l}_{*} = 0.5 m s^{−1}, *u*^{s}_{*} = 0.2 m s^{−1}, and *h _{o}* = 1000 m.

#### 2) Simplified vertical diffusion

This scheme developed by Janisková et al. (1999) is the linearization of the operational version of the vertical diffusion parameterization. Thus, the exchange coefficients for momentum and heat are computed as functions of mixing length, local wind gradient, and Richardson number (Ri).

The approach used for simplifying the computation of the surface fluxes is to take the soil variables (e.g., surface temperature and surface humidity) from a model basic state previously computed with the complete nonlinear physical package. This simplification allows one to separately handle the inversion problem in the turbulence scheme for each soil variable. The perturbations of the soil variables are not computed when integrating the linearized models.

When developing linear physical parameterizations, spurious noise can appear in the model due to the presence of discontinuities of some functions or derivatives. For example, spurious noise appeared in the tangent linear model because of the exchange coefficient depending on Ri. It was found that large linearization errors of the diffusion coefficients for momentum and heat occur when the atmosphere changes from stable to unstable or vice-versa. A first solution adopted to overcome the occurrence of the artificial noise was to modify the Richardson number dependency when −2 ≤ Ri ≤ 2.

In this range for Ri, the amplitude of the diffusion coefficients is reduced by a factor of 10 around the point of singularity. Another solution, similar to that in Mahfouf (1999), was to set the diffusion coefficients to zero independently of the value of Ri. This zeroing creates an unconditionally stable numerical scheme.

### b. Gravity wave drag

The *complete* scheme used in ALADIN is a modified version of one described in Boer et al. (1984) and simulates the physical mechanism through which the orography perturbs the mean atmospheric flow by the generation of gravity waves. Thus, a fraction of the energy of the main stream is transferred to the gravity waves whereas a significant remaining part will dissipate. The global result of this phenomenon is the damping of the flow by friction. Also, the scheme simulates the effect on the flow produced by the subgrid orography, which is not taken into account by the friction scheme.

The *simplified* scheme for simulating the gravity wave drag only contains the real linear part of the orographic blocking effect on momentum (Lott and Miller 1997) and the orographic anisotropy effect (Geleyn et al. 1995). The nonlinear processes, including resonant damping, the enhancement of the surface stress, and the trapping effect of the waves below an unstable layer, are not treated.

### c. Large-scale precipitation

*p*becomes saturated. For a given layer

*j*, the total precipitation flux at the bottom,

*F*

^{b}

_{j}, is a function of the precipitation flux at the top of the layer

*F*

^{t}

_{j}, the specific wet-bulb humidity

*q*

_{wj}, the local moisture

*q*, the time step of the physics Δ

_{j}*t*, and can be written aswhere

*g*stands for the acceleration due to gravity. Since a distinction between liquid and solid precipitation is performed inside the model, it turns out that the computation of the total precipitation fluxes combines several threshold processes.

In the *simplified* scheme, the computation of total precipitation fluxes was initially performed under the following assumptions: 1) the supersaturation is immediately removed from the layer, and 2) precipitations will totally evaporate if falling into an unsaturated layer. These assumptions were found to be insufficient since the linearized scheme then produces spurious noise. Thus, in order to overcome the numerical instability in the ADM, Janisková et al. (1999) proposed two regularization functions, both of them applied to each model level: 1) a function derived from Eq. (3) that eliminates the second assumption, and 2) a smoothing function applied for a nonzero precipitation flux, which aims to lessen the perturbation size in the vicinity of the threshold point.

### d. Deep convection

The linearized convective scheme is developed such as to avoid as much as possible threshold problems. As presented in Janisková et al. (1999), the simplified parameterization scheme simulating deep convection is of mass-flux type (Geleyn et al. 1982; Bougeault 1985; Motte and Ruiz 1987). The goal is to estimate the convection that may develop with respect to the large-scale forcing and to infer its effect on the environment (wind, precipitation, etc).

The necessity of having a smoother vertical profile of the mass fluxes than in the complete nonlinear scheme has led to the following differences: 1) the detrainment rate *D* is computed as a function of the convective mass flux *M _{c}*; 2) inside the cloud, the updraft is computed using an entrainment rate

*E*, consistent with both

*D*and the local gradient of the convective mass flux,

*F*= −∂

*M*/∂

_{c}*p*; and 3) the vertical turbulent fluxes of dry static energy and moisture are treated separately for convection and for diffusion. To avoid nonlinear instability, the condition

*F*= max[0, −(∂

*M*

_{c}/∂

*p*)] is verified together with

*F*Δ

*t*< 1, Δ

*t*being the physics time step. Another simplification consists in omitting the evaporation in the subcloud layer and setting all upper-air convective fluxes to zero in the case of zero precipitation at the surface.

### e. Radiation and cloudiness

The complete parameterization scheme simulating the radiative processes is described in Geleyn and Hollingsworth (1979), Ritter and Geleyn (1992), and Geleyn et al. (1995). Based on the solution of the *δ*-two-stream version of the radiative transfer equation, the scheme is found to be too complex for a tangent linear and adjoint coding. Consequently, for developing the simplified radiative parameterization scheme, Janisková et al. (1999) follow the approach described by Motte and Ruiz (1987). Thus, to avoid the problem of nonlinearities and discontinuities and to simulate as accurately as possible the radiative transfer process, several assumptions are made: 1) the local effects of gases are constant over durations up to 24 h, 2) the equivalent transmission of two layers is given by the product of the transmissions of these layers, 3) the clouds are blackbodies, and 4) there is no scattering.

The cloudiness parameterization scheme (Piriou 1996) from the nonlinear model is used in the computation of cloudy solar and thermal radiation fluxes. In the case of large-scale cloudiness, the cloud water is a function of relative humidity whereas the convective cloudiness is computed from convective precipitation fluxes. The perturbation of cloudiness is not computed in the linearized models.

## 3. Features of the high-resolution ADM solutions

The main objective of this section is to assess the behavior of the diabatic adjoint model at the same resolution (around 10 km) as the nonlinear model. The interest is to evaluate the ADM dynamics as well as the potential benefit of employing the linearized physical parameterization schemes in the gradient computation. The sensitivity experiments are performed following the classical steps described in Gustafsson et al. (1998).

*J*used for the adjoint sensitivity integration is a scalar total energy weighted forecast error norm, defined on a verification domain Σ, and which readsHere

**x**(

*t*) =

*M*(

**x**

_{0}) is the model forecast valid at time

*t*, started at time

*t*

_{0}

*< t*from the initial conditions

**x**

_{0}=

**x**(

*t*

_{0}), while

**x**

*(*

^{a}*t*) denotes the verifying analysis valid at time

*t*. Note that in ALADIN, the state vector

**x**is composed of the horizontal wind components

*u*and

*υ*, the temperature

*T*, the specific humidity

*q*, and the surface pressure

*p*, respectively.

_{s}*δ*

**x**is a vector representing a perturbation of the atmospheric flow. The weights given to its components are a function of the reference temperature

*T*, the specific heat of dry air at constant pressure

_{r}*c*, and the gas constant for dry air

_{p}*R*;

_{d}*L*is the latent heat of condensation and

_{c}*p*(

*η*) is the pressure at

*η*levels,

*η*being the vertical coordinate (

*η*= 1 at the surface and

*η*= 0 at the top of the atmosphere). The norm is either called the moist total energy when it is written in the form described by Eq. (5), or, as in Rabier et al. (1996), the dry total energy if the q term is missing.

The gradient of the forecast error cost function with respect to the initial condition, **∇***J*, is used to perturb the initial model state, **x*** _{o}*. Except for the perturbation of the specific humidity, all the model dynamical fields are corrected by subtraction of a fraction

*α*

**∇**

*J.*The correction of specific humidity has the form

*δ*

**q**

_{o}= −

*α*(

*σ*

_{q}/

_{q}) ·

**∇**

_{q}

*J*. Here,

*σ*is the background error standard deviation of specific humidity as tuned for the 3DVAR assimilation system of the same limited-area model, and

_{q}_{q}represents the vertically averaged background error standard deviation over the whole model atmosphere and acts as a weighting factor (see Berre 2000; Široka et al. 2003). The perturbation in the humidity fields is only considered when the sensitivities are computed with a moist norm. The scaling factor

*α*is 0.1 for all the model state fields.

The model used in this study is the operational ALADIN-France model. ALADIN is a high-resolution bi-Fourier hydrostatic primitive equation model (Horányi et al. 1996). The initial and boundary conditions are provided by the global model ARPEGE (Courtier et al. 1991). Any sensitivity forecast is performed starting from the perturbed initial model state fields. The nonlinear model is run using the semi-Lagrangian (SL) time-integration scheme with a 450-s time step. We use the operational physical parameterization package, that is, horizontal and vertical diffusion (Louis et al. 1981; Geleyn et al. 1995), gravity wave drag (Lott and Miller 1997; Geleyn et al. 1995), large-scale precipitation (Geleyn et al. 1995), shallow and deep convection (Bougeault 1985; Geleyn et al. 1982), shortwave and longwave radiation (Ritter and Geleyn 1992), a diagnostic cloud scheme (Piriou 1996), and a land surface scheme (Noilhan and Mahfouf 1996). A digital filter of Dolph–Chebyshev type (Lynch et al. 1997) is applied at the start of the nonlinear integration. In the gradient computations, the trajectory and the adjoint models are run using a Eulerian time-integration scheme with a 60-s time step (unless otherwise stated), since the adjoint of the SL is not yet developed for the ALADIN model.

Our studies are performed on real meteorological events. Thus, before discussing the results obtained with the high-resolution ADM, a brief description of the synoptic situations for each case study is presented.

### a. Description of the synoptic situations

The first case study discussed in this paper is a rapid evolving cyclogenesis called the Second French Christmas storm (27 December 1999). It is chosen because the 18-h ALADIN forecast failed in predicting both the location and the central pressure of the storm. Another reason for selecting this case is its strong gradients and wind intensities over orographic terrain. The damages caused by the devastating winds associated with the storm were very serious. As reported by Météo-France, while the cyclone moved with a speed of about 100 km h^{−1}, the associated wind gusts were larger than 140 km h^{−1} over France and 170 km h^{−1} over the Atlantic coast. This severe, extreme meteorological event offers a good opportunity to assess the behavior of the adjoint version for the mesoscale range.

The examined phenomenon starts at 0000 UTC 27 December 1999 and covers an 18-h period. In Fig. 1, fields of mean sea level pressure for the (a) verifying analysis, valid at 1800 UTC 27 December 1999, and (b) the 18-h operational forecast are plotted for comparison. Neither the location of the center of the low (located about 300 km too far to the west) nor its depth (overestimated by 6 hPa) are well predicted. Globally, the forecast errors are significant, reaching a magnitude of 14 hPa over the continent, while over the Atlantic Ocean the pressure field is underestimated by about 8 hPa.

The second case study covers the period 0000–1200 UTC 3 May 2001. It was selected because the ALADIN model overestimated the accumulated total precipitation field. At 0000 UTC, the upper-level flow is characterized by an isolated quasi-stationary pool of cold air situated over the Iberian Peninsula. At the surface, the low has an associated cyclone with a relatively high pressure core of about 1012 hPa (not shown). This cyclonic flow conveys moist air over the Mediterranean Sea to southeastern France and far into the central and northern part of the country.

Within the next 6 h, while the cold pool remains almost over the same geographical area, another low is acting over the British Isles. At the end of the period of interest (0600–1200 UTC), the two synoptic systems are interacting, giving birth to a line of convergence over the Benelux and the northern part of France. Between 0600 and 1200 UTC, the unstable air existing in the lower layers allows the development of thunderstorms especially over the Paris basin. The operational model failed to predict both the area where the storms occurred and the quantity of rainfall. This is illustrated in Fig. 2, where both the 6-h numerical forecast and the amount of the ground-measured precipitation are superimposed.

### b. Experimental setup

In the sensitivity experiments carried out in this study, both nonlinear and adjoint versions of ALADIN have 31 vertical levels and the same elliptical truncation: E95 for the 27 December 1999 case for a domain with 288 × 288 grid points, which corresponds to a mesh size of 9.92 km, and E99 for the 3 May 2001 case on a horizontal domain with 300 × 300 grid points and a grid size of 9.5 km. The geographical area is the same for both domains and is illustrated in Fig. 1. The experiments are summarized in Tables 1 and 2.

For the 3 May 2001 case study, as depicted in Fig. 3, the verification time is selected at 6-h range while a 12-h sensitivity integration is carried out with the goal of improving the last 6-h forecast of accumulated total precipitation. Thus, by using mesoscale sensitivities of the 6-h forecast error to the initial conditions, several experiments are initiated in order to improve the precipitation forecast within the range of 0600–1200 UTC. The 6-h time window used for computing the gradients is enforced by the huge computer memory required for the experiments. However, within the specified time interval, the ADM is able to point to the dynamically unstable areas as we will show in the sequel of this paper. In addition, we will show that the changes introduced through the gradient fields are crucial for some meteorological parameters and can lead to a significantly different high-resolution forecast at longer range, outside the sensitivity time window. For this case, we address both the properties of the ADM solutions (as in the 27 December 1999 case) and the sensitivity of the forecast error with respect to the initial conditions.

### c. Simple physics in the sensitivity computations

In this section, the experiments are designed for validating the adjoint of the ALADIN model. An additional goal is to assess the behavior of the simple physics when employed for high-resolution sensitivity computations. In this respect, we first discuss the results obtained with the adiabatic version of the ADM. Then we investigate the sensitivity fields from the diabatic ADM. The experiments are performed using as a test bed the Second French Christmas storm. The cost function for the adjoint sensitivity integration is computed in the verification domain defined by the latitudes 43°–50°N and longitudes 5°W and 4°E.

#### 1) Adiabatic gradients

In the experiment labeled Gr1 (see Table 1), the sensitivities of the 18-h forecast error to the initial conditions at 0000 UTC 27 December 1999 are computed. The examination of the gradient structure within the whole model atmosphere reveals large values of the gradient confined to the lowest model levels near the surface, while at higher model levels, these structures have vanished (not shown). This is a common and strong feature of the adiabatic sensitivities and it appears to be independent of the model resolution. Indeed, using a low-resolution global model, 19 vertical levels, and a horizontal triangular truncation T21, Buizza (1994) noticed similar structures. In addition, he showed that these structures are unrealistic and are not representative of an error field evolution but are due to a lack of description of the physical processes in the planetary boundary layer. Indeed, by using the simple vertical diffusion and surface drag scheme in the ADM, these unrealistic patterns vanish.

#### 2) Diabatic gradients

Fields of sensitivity of the 18-h forecast error cost function with respect to temperature, **∇*** _{T}J*, at model level 23 (∼1600 m), diabatically computed along a basic state obtained with the operational physical parameterization package are shown in Fig. 4 (corresponding to experiment Gr2 in Table 1). The gradient structure reveals a strong numerical instability triggered in the ADM employing the simple linearized physics. Noisy and spurious wavy gradient patterns develop especially over the Alps. These structures are not only limited to the lowest model levels, but are slightly spread vertically throughout the atmosphere. Thus, the noise has a different nature than for the adiabatic gradients.

To find out which numerical instability triggers the spurious gradients over the Alps, a number of tests have been performed. Primarily, the connection with the length of the integration time step is investigated. By default, in all the tests performed at high resolution, a 40-s time step is chosen in this case in both nonlinear (trajectory) and adjoint Eulerian models. Thus, in order to investigate if the instability is triggered by an unsuitable time step, additional sensitivity experiments are carried out over 3 h of integration with time steps of 30, 20, and 10 s. The results show that there are always model levels with very large and unrealistic gradient values independently of the time step. Moreover, these spurious gradients develop below model level 20 (about 2800 m). This particularity points toward a link with a poor description in the ADM of the physical processes in the PBL.

Second, the physical parameterizations package is switched off from the nonlinear model. Thus, the sensitivity is calculated along an adiabatic basic state, but with the simple physics still activated in the ADM (experiment labeled Gr3 in Table 1). The results indicate (not shown) that the trajectory is partially responsible for the instability. The magnitude of the gradient has significantly decreased although a spotted gradient structure over the Alps still exists. This simple experiment reveals that the trajectory plays a very important role in the sensitivity computation. At the same time, the need to adapt the simple linearized physics from a coarser- to a higher-resolution model becomes an important issue.

#### 3) Simple physics for a high-resolution model

To adapt the simple physics scheme [see section 2a(1)] to the high-resolution model, several 18-h forecasts with the nonlinear model using the Eulerian advection scheme are compared: a control experiment with the operational version of the ALADIN vertical diffusion and surface drag scheme; an experiment in which the simple physics as tuned for low resolution are employed; and additional experiments in which the values of the wind velocity scaling factor, *u _{*}*, and the reference height,

*h*, are modified. The values of the mentioned parameters are changed in order to test their impact in the solution of the nonlinear model.

_{o}The examination of the results reveals that the nonlinear model with the simple physics scheme produces a realistic forecast (not shown). However, the winds are stronger than in the control forecast, although globally the flow patterns are rather similar. Thus, *u*_{*} and *h _{o}* are changed in order to decrease the wind magnitude. After several tests, the following values for the high-resolution model are adopted:

*h*= 1500 m,

_{o}*u*

^{l}

_{*}= 1.0 m s

^{−1}, and

*u*

^{s}

_{*}= 0.4 m s

^{−1}. Note that hereafter, “simple modified physics” will refer to the simple physics scheme together with the adopted modified values for

*h*and

_{o}*u*

_{*}. The original values, calibrated for a low-resolution global model, were

*h*= 1000 m,

_{o}*u*

^{l}

_{*}= 0.5 m s

^{−1}, and

*u*

^{s}

_{*}= 0.2 m s

^{−1}.

### d. Improved simplified physics in the gradient computations

Concerning the gradient computations in this paragraph, there are two main features distinct from the experiments previously described. On the one hand, in the ADM, the complex simplified physical parameterizations are employed and on the other hand, the time window for the sensitivity computation is reduced to 6-h maximum. This is the limit enforced by the huge computer resources required for the experiments.

Prior to an intensive investigation of the behavior of the improved simplified physics package in the ALADIN adjoint model, a comparison with the simple physics is initiated. The tests are carried out on the Second French Christmas storm because of the already known problems linked to the flow over orography. Thus, a qualitative validation of the dissipative schemes (vertical diffusion and orographic drag) is made possible. In this respect, four experiments have been performed in order to compute the gradients of the 3-h forecast error cost function with respect to the initial conditions. In the first experiment, the ADM is adiabatic while in the second, the simple initial physics is employed. As reported in Table 1, the other two experiments are carried out with the simple modified physics, (label Gr4), and with the dissipative schemes of the improved simplified physics (label Gr5). The basic state is computed with the complete set of the physical parameterizations used in the nonlinear model.

A special attention is devoted to the gradients developed over the Alps, particularly their amplitude. Results from the above-enumerated experiments are presented in Fig. 5 where the vertical profiles of maximum values of **∇*** _{T}J* over the Alps are plotted. As expected, the largest sensitivity values are found on the lowest model levels when computed with the adiabatic ADM. As discussed in section 3c, below level 20 (∼2800 m), the diabatic sensitivities calculated with simple initial physics are also large when compared with the other two diabatic sets of gradients. When gradients are computed with simple modified physics or improved simplified physics (gravity wave drag and vertical diffusion schemes), the gap between the curves significantly decreases. The latter comparison indicates that both the simple modified physics and the dissipative schemes of the improved simplified physics are properly adjusted for the model resolution. The numerical stability of the ADM, when the dissipative schemes are employed, allows us to now to further concentrate on the behavior of the moist schemes at high resolution.

In the following section, we will concentrate on the experiments on the second selected case of 3 May 2001. The cost function *J* for the adjoint integration is defined on the same domain as in the previous experiment.

Sensitivities of the 6-h forecast error to the initial conditions at 0000 UTC 3 May 2001 are shown in Fig. 6. This experiment is labeled Gr6 in Table 1. The figure illustrates fields of **∇*** _{T}J* at model level 20. The gradient pattern reveals a “radiating” noisy spot developed over the Atlantic Ocean. In its core, the maximum value reaches about 160 × 10

^{16}J K

^{−1}. On the same level (not shown), unrealistic values are found for the gradients of the cost function with respect to the zonal wind, −34 × 10

^{16}J (s m)

^{−1}, and to the meridional wind, −42 × 10

^{16}J (s m)

^{−1}, respectively. It turns out that the spurious pattern with almost concentric alternating positive and negative rings is the result of a numerical instability triggered during the adjoint integration by the large-scale precipitation scheme. Further investigations show that the instability is created by strong discontinuities in the vertical precipitation fluxes, as obtained in the adjoint of the stratiform precipitation scheme. Such numerical problems are also known from large-scale applications in the ARPEGE global model (M. Janisková and F. Bouyssel 2001, personal communications). Indeed, by using the function derived from Eq. (3) (see Janisková et al. 1999), which removes the evaporation processes below clouds, the noisy structure over the ocean partially vanishes.

In addition, a wavy and still noisy pattern becomes visible over France with a maximum value reaching 1022 J K^{−1}. This spurious signal can be controlled by modifying the parameters of the shape and shift of the smoothing function defined in Janisková et al. [1999, see their Eq. (45)]. Several values for *b*, the shape, and *c*, the shift, parameters have been tested. In Fig. 7, the original flux for large-scale precipitation and two modified fluxes are presented. The finally adopted values to cure, at least partially, the noise problem are *b* = 10^{4} and *c* = 10^{−3} whereas the default values are *b* = 7 × 10^{4} and *c* = 7 × 10^{−5} (Janisková et al. 1999). This new setting forces the intense precipitation fluxes to diminish and reduces the perturbation close to the threshold point where large perturbations are a source of noise. The maximum amplitude of the fields of ∂*J*/∂*T* obtained with the modified precipitation fluxes substantially decreases to 44 J K^{−1} and the wavy pattern disappears. However, with the new values, the nonlinear precipitation flux is also reduced substantially.

The results of the adjoint integrations presented in section 3 emphasize the importance to adapt the linearized physics to high resolution. In the absence of any reformulation of the large-scale precipitation scheme, our results show the necessity of diminishing the nonlinear precipitation flux. Furthermore, the experiments show the role of the trajectory in triggering numerical instability during the gradient computations, especially when the improved simplified physical parameterization schemes are used.

## 4. Toward the improvement of precipitation forecasts

The a posteriori improvement of numerical precipitation forecasts using mesoscale sensitivities is the topic of this section. The approach is to “correct” the initial model state with a fraction of the high-resolution sensitivity computed by employing the improved simplified physics, including the moist schemes. In this context, the potential usefulness of a moist forecast error norm in the gradient computations is shown. This section furthermore concentrates on the 3 May 2001 situation, introduced in section 3a and already used in the discussions of section 3d.

### a. Gradient computations

In this section, we deal with the computation of the 6-h gradients of the forecast error cost function using the complete package of simplified physics. In this respect, additional schemes, namely the convection, radiation, and cloudiness are employed in the ADM. Two experiments, T4 and Q2 as reported in Table 2, will be discussed here. The cost functions in T4 and Q2 are the dry and the moist forecast error norm, respectively. The results from these experiments illustrate the benefit of using the improved simplified physics for high-resolution sensitivity computations. Also, the setup of T4 is used for the contradictory experiments described in section 4c.

At the mesoscale range, the moist processes play a leading part. The evolution of the humidity field is the complex result of a wide range of scale interactions. Hence, it is likely that small errors in the initial humidity field may amplify more quickly for a high-resolution model describing more scales than for a coarser-grid numerical model. In addition, at high resolution, a better description of the model orography may lead to a faster release of latent heat by condensation due to a stronger local forcing.

An ADM including a description of the moist processes, together with a forecast error norm with an explicit term for humidity, can provide additional key aspects of initial condition errors. Indeed, this is illustrated in Fig. 8, in which fields of the 6-h gradient with respect to the initial conditions at 0000 UTC 3 May 2001 computed using a dry cost function, in panel (a), are plotted for comparison with those in panel (b) derived from a moist *J*. Both panels show fields of **∇*** _{T}J* at model level 20. In Fig. 8b, a distinctive alternating negative and positive dipole is visible in the northern part of France, which is not noticed in Fig. 8a when the forecast error norm is dry. A closer examination of Fig. 8b reveals that using a moist forecast error norm causes the development and the expansion of the negative area of sensitivity toward south and west, with an increase by about 40% of the maximum amplitude of the gradient both in the main core located in the eastern part of France and in the secondary positive gradient core (central-west of France), when compared to Fig. 8a.

Although it is difficult to claim that the increase of absolute values of the sensitivity is significant, one can notice that the moist forecast error norm provides additional information about the region in which to seek the errors in the initial conditions.

### b. Sensitivity forecasts

In this section we examine the 12-h sensitivity forecasts started with the perturbed initial state obtained by subtracting a perturbation *α***∇***J* from the original data. Note that **∇***J* represents the gradient of the 6-h forecast error to the initial conditions. In Fig. 9a, the precipitation field is obtained from a nonlinear model integration started with an initial state “corrected” by a fraction *α* = 0.1 of the gradient labeled T4 in Table 2. The cost function is dry and an ADM with the full set of improved simplified physics is utilized. In Fig. 9b, a moist forecast error cost function is used (label Q2).

Note that Figs. 9a and 9b reveal a narrow area with an intense precipitation core [∼20 mm (6 h)^{−1}] developed inside the precipitating system and located in the northeastern part of France (see also Fig. 2 showing the observations). Hereafter we will refer to this convective core as system C. The forecast of the maximum precipitation core is misplaced by about 150 km to the east. In contrast with the control run, which produced about 30 mm (6 h)^{−1} in the main part of the system (hereafter system A) over western France, the amount of precipitation in the sensitivity forecasts decreases by about 35%, to 20 mm (6 h)^{−1} (Fig. 9a), and by 50%, to 15 mm (6 h)^{−1} (Fig. 9b), respectively. The rain gauges, however, did not register more than 7 mm (6 h)^{−1}. Just about 100 km to the south of the main system, over the Vendée region, there is a secondary maximum of precipitation (system B) that is equally misforecast by both the control and the sensitivity integrations, which overestimate the rainfall. Thus, the sensitivity forecasts succeed in decreasing the precipitation in the main system and in producing a new local maximum closer to an actually observed event. Only the secondary maximum over the Vendée is too strong in the sensitivity integrations. So far, Fig. 9b reveals that the best sensitivity forecast is obtained when “correcting” the initial fields of specific humidity as well.

Globally, in Figs. 9a and 9b, one can see that the shape of the precipitation field is kept the same as regards the three main components of the rainfall system (the main core over western France and the secondary maxima over the Vendée and over northeastern France).

The results obtained in this particular case are encouraging despite the fact that the convective system was not triggered exactly in the right region in any of the experiments. The positive aspects of the sensitivity experiments are the simple fact that a convective activity can be initiated, which is totally absent from the operational forecast (see Fig. 2), and which in turn causes a decrease of the rainfall amount predicted in system A.

### c. Contradictory experiments

Contradictory experiments are designed to strengthen the results, especially by assessing or reassessing the initial assumptions. Since, for this particular case, an encouraging improvement of the precipitation forecast is observed, additional tests are performed in order to address the following questions: 1) Are indeed the errors in the initial conditions responsible for the forecast failure? 2) Is the verifying analysis necessary as a true state of the atmosphere for computing the gradients of the forecast error cost function? 3) To what extent do the results sustain the traditional approach of a perturbation method (modified forecast, 4DVAR)?

Two types of experiments have been designed. First, the initial conditions are modified by *adding* instead of subtracting the perturbation. The gradients used for creating the perturbation are those computed in experiment labeled T4 in Table 2. The effect of adding the perturbation is that it should tend to maximize instead of minimize the forecast error cost function. Hence, a direction in the phase space along which the initial condition errors evolve into large forecast errors is likely to be generated. Indeed, the comparison of the 6-h accumulated total precipitation field for the corresponding sensitivity forecast (Fig. 10) with the control reveals that the amount of rain in the main system (A) increases by 10% whereas to the south of it, the core of the secondary maximum (B) increases by 50%, to 15 mm (6 h)^{−1}. As in the control forecast, the system C does not develop. Thus, adding the perturbation instead of subtracting it leads to a forecast that is even slightly worse than the control as concerns precipitations.

Second, using the setup of the experiment T4 in Table 2, the real verifying analysis is substituted by turn with an ALADIN forecast valid at the verification time and with a future ARPEGE analysis valid at a totally different date. In Table 2, these experiments are labeled C1 and C2, respectively. In Figs. 8c and 8d, the gradients of *J* with respect to the temperature at model level 20 (from C1 and C2) are plotted for comparison with those in Fig. 8a. Although the initial conditions in the ADM are very different, the 6-h sensitivity patterns show similar features. The positive fields are characterized by cores with elliptical or quasi-circular shapes and values larger than 30 J K^{−1} concentrated rather in the same region whereas the negative areas are spread over a wide area. The highest negative values do not exceed 10 J K^{−1}. Thus, despite the fact that the experiments C1 and C2 produce these wide and fairly strong negative gradients in the south of France, the overall structures in C1 and C2 compare well with those of T4, especially as concerns the succession of the strong positive cores. These similarities suggest that the ADM solution is fully constructed by the adjoint model dynamics. Since the ADM is simply a linear projection operator, the experiments also suggest that any initial condition perturbation will be rapidly projected onto a very specific unstable subspace.

A negative temperature perturbation in the areas of positive sensitivity causes a decrease of the temperature, and the converse is true for a positive temperature perturbation. The changes in the initial temperature field due to a negative perturbation suggest a global cooling in the center of the integration domain. This cooling in turn produces a more stable air mass in the “corrected” initial conditions than in the original fields.

The 6-h total accumulated precipitation fields for the sensitivity runs using the gradient fields of C1 and C2 are shown in Figs. 9c and 9d, respectively. As regards the systems A, B, and C, the precipitation patterns are similar with those plotted in Figs. 9a and 9b while the cumulated quantities are globally increased.

To physically understand why the sensitivity forecasts are improved, additional investigations are performed on the first 6 h of the model prediction. Fields of convective available potential energy (CAPE), a measure of the atmospheric instability, are analyzed. For a better comparison, CAPE differences between the reference and perturbed forecasts at initial, 0000 UTC, and at verification time, 0600 UTC, are plotted. The differences are computed as CAPE(PERT) − CAPE(REF). As illustrated in Fig. 11a, at the initial time, when the perturbation is subtracted, CAPE decreases in the region where the convective system is to develop.

At first sight, it seems contradictory to decrease the CAPE on the one hand, while strengthening the onset of a strong convective nucleus on the other hand. Our explanation is that the global decrease of CAPE, illustrated in Fig. 11b, causes less activity in the main part of the precipitating system (system A), which in turn allows for a local progressive evolution of the wind and humidity fields toward convection in the region where system C should appear. In other words, the reference run seems to have too strong CAPE associated with the large mesoscale system and at early stages, which only feeds the main system A. If this CAPE is diminished, the dynamical evolution of the flow is then able to promote a new system C, separate from the main part (A). This interpretation is consistent with two other findings:

- The ability of the dynamical evolution in the ADM to point to the unstable subspace that contains the information about the presence of conditional convective instability (though the adjoint processes only can capture a linear evolution). This ability is backed by the already shown projection of gradients into the sensitive area in the ADM (see Fig. 8).
- When, instead of subtracting, the perturbation is added to the initial conditions, then CAPE is increased (Fig. 12a). System A then benefits from the bigger conditional instability and produces even more precipitation. The 6-h evolved CAPE is decreased in the surroundings where system C would develop (Fig. 12b).

Unfortunately, there is no accurate observational data to further sustain our interpretation, especially concerning the vertical stability of the atmosphere over central-east France.

## 5. Summary and discussion

In this article, we primarily investigate the behavior of the ALADIN adjoint model at 10-km horizontal resolution. The interest is to evaluate the ADM solutions as well as the potential benefit of employing the linearized physical parameterization schemes in the gradient computation. Second, in order to identify the initial condition errors that may lead to forecast failure, sensitivity studies at high resolution on real meteorological events are carried out. The misfit between the forecast and the verifying analysis is measured by the total energy norm.

The general results derived from the adjoint computations suggest several conclusions, which can be summarized as follows:

- Independently of the model horizontal resolution, unrealistic patterns of adiabatic gradients without meteorological significance are always confined to the very first model levels near the surface.
- To be used in a high-resolution ADM, the linearized physical parameterization schemes, developed for a coarser-grid model, have to be improved. In the absence of any retuning or reformulation, numerical instability can be triggered in the ADM. While the simple vertical diffusion scheme (Buizza 1994) requires some additional tuning, the vertical diffusion and orographic drag parameterization of Janisková et al. (1999) is efficient in our 10-km experiments without any extra adaptation.
- The large-scale precipitation scheme is a strong source of instabilities in precipitation areas, and this potential problem increases with resolution. Thus, the ADM solution will be contaminated by gradient structures without any meteorological relevance. This problem is overcome at a rather high cost, namely, a substantial reduction of the nonlinear precipitation flux. Yet, we cannot firmly state that by using this scheme for another particular local forcing, the numerical instability would not outbreak again. If thinking of global or limited-area applications with (incremental) resolutions from 50 to 10 km, then more work would probably have to be devoted for improving the physical parameterization schemes.
- The basic state around which the gradients are calculated plays a very important role, especially when the ADM is diabatic. The high-resolution ADM solutions depend on localized changes in the trajectory computation. More precisely, we show the potential difficulties over complex orography in the presence of a very strong nonlinear flow regime. With increasing resolution, stronger downslope wind gusts and vertical shear are produced in the trajectory, which in turn increase the amplitude of the response in the adjoint of the vertical diffusion scheme. The potential numerical problems were controlled in our case by realistic and robust dissipative physics.

The first case study discussed in this paper is the so-called Second French Christmas storm, which represents a difficult forecast case for a limited-area model. It is characterized by a very complex upstream large-scale flow partially outside of the domain boundary. Tentatively performed sensitivity forecasts failed most likely due to the fact that the gradients do not provide information within the whole model atmosphere. Indeed, during the adjoint integration, part of the gradient in the upper atmosphere leaves the western border of the integration area. In fact, rather than the sensitivity approach for itself, the Christmas storm case is of interest as a test bed to evaluate the numerical problems that may appear in the adjoint physics. The situation is extreme with respect to flow over complex orography, and as discussed before, it allows a thorough tuning of the dissipative processes, which was a useful step prior to the evaluation of the moist processes.

For the particular meteorological event that occurred on 3 May 2001, the results from the sensitivity experiments raise fundamental questions. Despite the fact that none of the predictions performed by subtracting the perturbation succeed in fitting the real evolution of the atmosphere, the forecasts are globally improved. Thus, the onset of the convective system in the northeastern part of France and the decrease of the precipitation amount in the main part of the system are positive impacts of the mesoscale gradients in “correcting” the model initial state. The extent to which the subsequent forecast actually is *deterministic* is, however, questionable. The question arises after contradictory experiments in which the verifying analysis is substituted with one totally out-of-date analysis or with a simple forecast instead of an analysis at verification time. Actually, it is found that a perturbation of sufficiently large amplitude, adequately introduced in the initial conditions, can lead to an improved forecast. On the one hand, this fact is a good indication that the dynamics and the physical parameterization of the ALADIN model are robust and sufficient to resolve the relevant processes of the atmosphere if the initial data are not contaminated by large errors. On the other hand, our findings show that even in the absence of a dense and good-quality observation network, the ADM still can be useful. Indeed, we surmise that the ADM can project a poor-quality signal coming from a few observations or poor-quality observations onto a dynamically sensitive subspace, thus providing a meteorologically consistent increment for correction.

The study of the 3 May 2001 convective system, embedded in a larger-scale frontal evolution, shows perhaps both the potential and the limitations of the linear adjoint tools. The ADM solution and the sensitivity studies have highlighted a strong dilemma for the variational techniques at high resolution: while convection, where vertical stability is crucial, still can be targeted by the linear models, the nonlinear nature of its dynamics creates nontrivial responses in the sensitivity experiments. If we now transpose this thinking into 4DVAR applications, for instance, then we may fear that the linear propagation of increments leads to unrealistic responses, far from the probably bounded nonlinear evolution in the full model. Also, in a multi-incremental system, the switch from one trajectory solution to one very different in a subsequent outer loop (where the full trajectory is recomputed from the new provisional solution) may disable the convergence of the outer loops. Reversely, a few observations at key positions may project on strong dynamically unstable directions and thus significantly alter the analysis solution. Whether a high-resolution 4DVAR system mostly benefits or not from these behaviors is out of the range of our study, and is left as an open debate.

From the number of experiments carried out, it appears that the 3 May 2001 case essentially possesses two forecast solutions: the operational one characterized by strong initial CAPE and a very active main system, and the alternative one with lower initial CAPE and an active second convective nucleus. These results suggest that a combination of adjoint techniques with an ensemble approach could have been an alternative solution to the problems investigated in this article. Indeed, for the 3 May 2001 situation, one important finding is that all forecasts basically can be classified in a two-member ensemble. Since any high-resolution observational network hardly can give a very accurate image of the atmosphere, this convective system might have been better sampled and understood in the frame of an ensemble approach.

Eventually, our study raises the following topics for future work:

- The development of an efficient dynamical core (semi-Langrangian).
- More case studies to assess the robustness of the ADM and the improved simplified physics. Possibly, a panel of situations could reveal systematic shortcomings of the simplified schemes (which would be very detrimental for 4DVAR type of applications).
- Future studies can further confirm the potential positive role played by the adjoint dynamics to project “increments” onto the relevant unstable subspace. We have shown one case where the sensitive area for convection was actually found by the ADM, but we cannot firmly state that this behavior is general or frequent.

## Acknowledgments

This paper is based on part of the Ph.D. thesis of the first author who was on leave from the NMA.

We are grateful to Marta Janiskov’a and François Bouyssel for fruitful discussions and relevant suggestions. We thank Météo-France for use of computing facilities. This work was supported by the ALATNET Grant HPRNCT-1999-00057. ALATNET is a TMR/IHP Programme of the European Community but the information provided here is the sole responsibility of the ALATNET team and does not reflect the Community’s opinion. The Community is not responsible for any use that might be made of data appearing here. Also, we are in debt to the ALADIN NWP international project (with, presently, 15 partner countries). We are grateful to all colleagues from the partner institutes who have contributed to maintain and develop the numerical tools used in this study. The authors wish to thank the anonymous reviewers for their careful reading and helpful comments that led to a significant improvement of the manuscript.

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Description of the experiments performed for computing high-resolution sensitivities. Labels Gr1 to Gr5 for 27 Dec 1999 case. Label Gr6 for 3 May 2001 case.

Description of the experiments performed for improving the 6-h precipitation forecast. Case 3 May 2001. The scaling factor *α* = 0.1 for all the experiments.