Dynamical kernel of the Aladin NH spectral limited-area model: Revised formulation and sensitivity experiments

Transcription

1 Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. 136: , January 2010 Part A Dynamical kernel of the Aladin NH spectral limited-area model: Revised formulation and sensitivity experiments P. Bénard, a *J.Vivoda, b J. Mašek, b P. Smolíková, c K. Yessad, a Ch. Smith, d R. Brožková c and J.-F. Geleyn a,c a Météo-France, Toulouse, France b Slovak Hydro-Meteorological Institute, Bratislava, Slovakia c Czech Hydro-Meteorological Institute, Prague, Czech Republic d Met Office, Exeter, UK *Correspondence to: P. Bénard, CNRM/GMAP, 42 Avenue G. Coriolis, F Toulouse Cedex, France. pierre.benard@meteo.fr The contribution of C. Smith was written in the course of his employment at the Met Office, UK and is published with the permission of the controller of HMSO and the Queen s Printer for Scottland. Drawing from the results of theoretical studies about the behaviour of constantcoefficients semi-implicit schemes, the dynamical kernel of the Aladin NH spectral limited-area numerical weather prediction (NWP) model has been modified in order to allow for a stable and efficient integration of the fully elastic Euler equations. The resulting dynamical kernel offers the possibility to use semi-lagrangian transport schemes together with two-time-level discretizations at kilometric scales for NWP purposes. The main characteristics of the adiabatic part of the model formulation and the space and time discretization are described in this article. In order to illustrate the dependence of the results on adjustable parameters of the dynamical kernel, some real-case dynamical-adaptation forecasts performed with a basic physical parameterization package are presented. The results obtained with this model in real-case experiments fully confirm the conclusions drawn in previous numerical analysis studies. The good quality of the results is found to be compatible with a routine exploitation in a NWP framework. The Aladin NH dynamical kernel has been used in the operational NWP AROME model since December 2008 at the kilometric scale, with an appropriate physical parameterization package and data assimilation system. Copyright c 2010 Royal Meteorological Society and Crown Copyright. Key Words: Euler equations; semi-implicit scheme; semi-lagrangian scheme; stability Received 13 May 2009; Revised 27 August 2009; Accepted 27 August 2009; Published online in Wiley InterScience 19 January 2010 Citation: Bénard P, Vivoda J, Mašek J, Smolíková P, Yessad K, Smith Ch, Brožková R, Geleyn J-F Dynamical kernel of the Aladin NH spectral limited-area model: Revised formulation and sensitivity experiments. Q. J. R. Meteorol. Soc. 136: DOI: /qj Introduction The use of the efficient, while rather simple, semi-implicit (SI) and semi-lagrangian (SL) class of discretizations to solve the compressible Euler equation (EE) system for meteorological models was first advocated by Tanguay et al. (1990). These SISL schemes, by relaxing most computational stability constraints on advection and fast adjustment processes, are very attractive for numerical weather prediction (NWP) and climate simulation purposes, and a lot of effort has been dedicated in many weather services to applying this technique for the integration of EE Copyright c 2010 Royal Meteorological Society and Crown Copyright.

2 156 P. Bénard et al. systems (Davies et al., 2004; Caya and Laprise, 1999, and others). However, the EE system has faster adjustment terms than its predecessor in NWP applications, the hydrostatic primitive equation (HPE) system, and the use of the SISL technique for the integration of EE systems has often led to specific problems that were diversely overcome in different forecast models. A first attempt to integrate the fully elastic EE system for the atmosphere in a spectral SI mass-based coordinate model with the perspective of a possible future NWP purpose was presented in Bubnová et al. (1995, BHBG95 hereafter) as an experimental non-hydrostatic (NH) version of Aladin, referred to as Aladin NH. The Aladin model (ALADIN International Team, 1997) is a spectral limited-area mesoscale NWP model built in cooperation between the national meteorological services of sixteen European and Northern African countries, through a Consortium structure. This model is also the limited-area version of the global IFS/ARPEGE system jointly developed at the European Centre for Medium- Range Weather Forecasts (ECMWF) and Météo-France and used operationally in these two meteorological centres. The current applications of Aladin, exploited daily in the various National Meteorological Services of the ALADIN consortium at resolutions of about 5 10 km, are based on the HPE version of the model. The mass-based coordinate η used in Aladin NH is a hybrid terrain-following coordinate classically derived from the hydrostatic-pressure coordinate π, as proposed by Laprise (1992). The dynamical core of the experimental tool presented in BHBG95 was built on an Eulerian transport scheme and a three-time-level (3-TL) leap-frog time discretization. The fast dynamical processes were controlled by a semi-implicit scheme in which the linearly partitioned implicit system was designed with constant coefficients in the sense discussed in section 1 of Bénard et al. (2004, BLVS04 hereafter) and Bénard (2004a, B04a hereafter). The terminology constant coefficients refers to an approach in which the linearized operator associated with the implicit problem to be solved has constant and horizontally uniform coefficients, leading to simple and efficient techniques for the inversion problem. As mentioned in section 6.b of BHBG95, it was necessary to modify the semi-implicit scheme in order to achieve a more complete implicit treatment of the whole three-dimensional divergence field (noted D 3 therein), for which nonlinear explicitly treated residuals had appeared to be unstable. This modification consisted of a partial iterative treatment of the unstable source terms involving D 3,as symbolically summarized by (12) (13) in Bénard (2004b). With this latter modification, the 3-TL SI Eulerian model was found to perform well in idealized and real-data NWP cases. However, new sources of instability were experienced when the extension of this dynamical kernel to more efficient schemes was attempted. The transformation from a 3-TL Eulerian scheme to a 3-TL SL scheme resulted in a moderate but systematic instability. Although the SL model behaved similarly to the Eulerian one with identical values of time step, integrations usually failed after a few hours when large time steps were used, with Courant numbers greater than 1. For two time-level (2-TL) SL schemes, the situation was even worse: the model was found to be dramatically unstable regardless of the time-step value. The scheme generally blew upafterlessthan10timesteps. Theoretical studies were undertaken to understand better the sources of these problems and to propose possible remedies: the instabilities were successfully reproduced in simple idealized frameworks lending themselves to analytic investigations, and were identified as being of a similar nature to those presented in Simmons et al. (1978), that is instabilities linked to a sometimes deleterious impact of the explicit treatment of nonlinear residuals. From these theoretical studies, it appeared that for the chosen class of semi-implicit schemes (with constant coefficients), the choice of the prognostic variables and of the type of reference linear operator may have a crucial impact on the stability: the prognostic variables for which the differences between D 3 and its linearized counterpart are minimized were found to result in the most stable schemes (Bénard, 2003, hereafter B03; BLVS04; Bénard et al., 2005, hereafter BMS05). Additionally, it was shown in B04a that a further dramatic increase of stability was possible when more degrees of freedom were allowed in the constantcoefficients linear-system partitioning of the semi-implicit scheme (through an additional cold reference temperature Te ). Finally, as advocated in earlier articles (Côté et al., 1998; Cullen, 2001), implicit schemes with a larger part of the source terms treated implicitly (through an iterative process) have also been shown to be potentially beneficial to the stability. Drawing from the results of the theoretical studies reported in B03, B04a, BLVS04 and BMS05, alternative formulations were proposed for the dynamical kernel (see Appendix for notation): new pressure prognostic variable ˆq = ln(p/π) in replacement for the original variable ˆP = (p π)/π used in BHBG95; new vertical momentum prognostic variables d and dl in replacement for the original variable ˆd; new linear system partitioning, using an additional elastic reference temperature Te for vertical elastic wave terms; new iterative centred implicit (ICI) scheme in replacement for the scheme used in BHBG95. When implemented in Aladin NH, these new features led to a dramatically enhanced stability of the dynamical kernel. The model successfully passed a large variety of idealized test cases, such as vertical-plane 2D simulations of orographic flows, including trapped lee waves (Keller, 1994), bubble convection cases at decametric scales (Robert, 1993) and 3D flows with and without Earth rotation over idealized and real orography, at various scales. During these tests, the model constantly showed its ability to perform stable integrations with typical CFL numbers up to 10 at kilometric resolutions, thus indicating a reasonable appropriateness for further NWP use. The dynamical kernel of Aladin NH was then validated for routine real cases at Météo-France, and was chosen as the dynamical kernel of the so-called AROME mesoscale NWP application. The AROME NWP facility is built by coupling the dynamical kernel Aladin NH to a specific physics parameterization package targeted to kilometric resolutions, and to a data assimilation system adapted from the Aladin 3D variational system (Fischer et al., 2005). AROME became operational in December 2008 with a 2.5 km horizontal resolution, over a limited-area domain covering the French territory.

3 Dynamical Kernel of the Aladin NH Model 157 The NH and HPE versions of the Aladin dynamical kernel are implemented in a single software package (activated through a logical switch), which allows clean comparisons of the responses. The Aladin HPE dynamical kernel and its non-hydrostatic counterpart share the same general characteristics: both have a horizontal spectral representation of fields based on a double Fourier decomposition assuming an extension zone for periodizing fields, a mass-based hybrid terrain-following coordinate η (Simmons and Burridge, 1981; Laprise, 1992), and use a conformal projection of the shallow-atmosphere equations on a sphere. Moreover, both versions aim toward numerically efficient schemes such as 2-TL semi-lagrangian schemes. In both Aladin and Aladin NH models, two types of vertical discretization have been implemented as separate options: finite differences (Simmons and Burridge, 1981) and finite elements (Untch and Hortal, 2004). The finiteelements option is used in the operational HPE versions of Aladin, whereas for the EE versions of the model this discretization is still under research. Only results with vertical finite-difference discretizations are reported in this article; 3-TL schemes, becoming obsolete, are not examined here. The results obtained with the revised dynamical kernel of Aladin NH are reported in this article. The model formulation with the proposed prognostic variables and the new space and time discretizations is presented in section 2. The properties of the dynamical kernel with various options are discussed for real cases in section 3. Finally, some comments and concluding remarks are given in section Model formulation All the changes and options listed in the Introduction were actually implemented in the dynamical kernel of the Aladin NH model, under logical switches, in order to investigate the validity of the conclusions drawn in more idealized frameworks. As discussed in Bénard (2004b), the choice of the spectral method strongly suggests the use of constant coefficients (in time and space) for the partitioning of the linear implicit problem linked to the SI scheme, and therefore all options examined here for Aladin NH pertain to this class of schemes. In particular, it must be outlined that the orography is not represented in the linear model used for the implicit formulation. Since the primary goal of this article is to describe the dynamical kernel of the Aladin NH model and its behaviour, little emphasis is put on the moist formulation for fine-scale applications with detailed microphysics, a topic that is left for a future article. However, in order to present our results for real-case experiments, we chose to describe the moist dynamical kernel coupled with the physics package of the currently operational hydrostatic ARPEGE/Aladin models (ARPEGE is the global coupling model for Aladin, both models sharing most of their direct forecast code). Similarly, for clarity and conciseness, little emphasis is put in this article on the geometrical aspects of the rotating spherical atmosphere. Although options are available in Aladin to solve the meteorological equations in various conformal projected plane geometries, the model description presented here is restricted to the case of a pure non-rotating Euclidian geometry with a Cartesian framework, and therefore does not constitute an exact counterpart of all the possible model formulations from this point of view. For instance, as for most midlatitude applications, the real-case experiments discussed hereafter are made with a Lambert projection of the equation on the plane. The aspects linked to the Earth s sphericity and rotation are not believed to have a specific impact on non-hydrostatic motion and therefore are not mentioned in the model description. It should finally be noted that the formulation of the EE system described here has also been jointly implemented (with ECMWF) in the global model IFS/ARPEGE for stretched and unstretched spherical geometries (Wedi and Smolarkiewicz, 2009), but this also is not further developed here, mainly because global NWP modelling does not currently allow resolutions fine enough for NH effects to become clearly beneficial. A version of the NH dynamical kernel for a deep atmosphere has been implemented for both the global and limited-area versions of the dynamical kernel following Wood and Staniforth (2003), but for simplicity it is not used or documented here Model equations/variables In BHBG95, the EE system was solved using the set of non-hydrostatic prognostic variables ˆP and ˆd given by (11) and (12) therein. As seen in BLVS04, this choice was shown to result in significant instability in the presence of nonlinear explicitly treated thermal residuals. Numerical analyses suggested a change of non-hydrostatic variables (derived from pressure and vertical velocity respectively) in order to restore stability. For the flatterrain case, examined in BLVS04, the following nonhydrostatic variables (derived from pressure and vertical velocity respectively) were then proposed to circumvent the problem: ˆq = ln(p/π), (1) d = p w g mr a T η, (2) where the notation follows BLVS04 (see Appendix also). Here R a is the perfect gas constant of dry air, not to be confused with R, the perfect gas constant of the air/water-vapour mixture. For the case with orography, examined in BMS05, a new variable dl (defined below) was proposed in replacement of d, to remove instabilities linked to nonlinear orographic terms in the equations. It was believed in BMS05 that dl is potentially better than d for NWP use, when orography is present. However, numerical stability analysis in idealized contexts only provides rough indications about the likely behaviour of a model in real-case conditions; therefore it is important to investigate in detail the impact of solutions suggested by numerical analyses, in order to justify convincingly the final choices for NWP. Consequently, formulations and results with both d and dl variables are presented and compared in detail here.

4 158 P. Bénard et al. Using ˆq and d as prognostic variables, the EE system becomes dv dt + RT p p + 1 p φ = V, (3) m η ( ) dd dt + p 1 (p π) g2 mr a T η m η p V g mr a T η. w p W d(.v D 3 ) = g mr a T η, (4) dt dt + dˆq dt + π s t + C p RT C v D 3 = Q C v, (5) C v D 3 + π π = 0 Q C v T, (6). (mv)dη = 0, (7) where V,W and Q are the physical contributions for V, w and T respectively, and m = ( π/ η), (8) p = π exp(ˆq), (9) mrt φ = φ s + dη, (10) p D 3 = m η = π = g w = + 1 ( ) p V m RT φ. + η B 0.mV dη ( V. π η η 0 η 0.V ) R d, (11) ( Ra.mV dη, (12) ).mv dη, (13) mr a T g w s d dη η p ( mra T d p η ) dη. (14) The geopotential φ in (10) is obtained through an upward integration of atmospheric depths from the surface geopotential φ s. π s is the surface hydrostatic pressure (see Appendix for other notations). The quantities m, η and π in (12) (13) are the diagnosed mass-based verticalvelocity and parcel-mass-coordinate derivative respectively. Equation (14) results from the vertical integration of (2) from the surface to the current level, followed by a horizontal derivation. All horizontal derivatives are evaluated spectrally. The domain of integration is defined vertically by π [π Top = 0, π s ]andη [η Top = 0, 1], and the system is physically subjected to a material elastic boundary condition p Top = 0 at the top of the domain and to a material rigid boundary condition at the surface, denoted by the subscript s. Fromthenativew equation (see (28) below), the surface boundary condition can be written, in mass coordinates, as a Neumann condition for ˆq (through p): [ g m (p π) η ] s =ẇ s, (15) where the material rigid condition also links kinematic fields through gw s = V s. φ s. (16) Thislatter condition comesfromgw = dφ/dt at the surface, where ( φ/ t) vanishes due to the rigid boundary. The linear system used to solve implicitly the evolution of fast (gravity and elastic) modes is directly derived from the above complete system, through a linearization around a stationary reference state. Classically, the chosen reference state is resting, hydrostatically balanced, dry, isothermal, with no orography. The value of the surface hydrostatic pressure for this reference state is denoted π s. Following B04a, two different reference temperature values are introduced: T e for the term involved in the vertical propagation of elastic waves and T for all other terms. Save for the introduction of T e in (18) below, the derivation of the linear system follows the same principle as for example in Simmons and Burridge (1981). The linear system for the perturbed variables (D, d, T, ˆq, π s )maythusbewritten D t d t = T t ˆq t = π s t = R a G 2 T + R a T G 2 ˆq R a T 2 ˆq R at 2 π s, (17) = R at π s g2 R a Te L ˆq, (18) C va (D + d ), (19) S D C pa C va (D + d ), (20) = π s N D, (21) where G, S, N and L are spatial linear operators defined in the Appendix, similar to those introduced in Simmons and Burridge (1981). The elimination of all variables but d in the above system leads to the structure equation, the spatially discretized counterpart of which must be inverted in the numerical model to solve the implicit evolution of the perturbation: [ 1 4 ) ] ( c 2 t t 2 + L rh 2 + N2 r 2 d = 0, (22) where r = (T e /T ), H = (R a T /g), N 2 = g2 /(C pa T )and c 2 = (C pa/c va )R a T. As mentioned above, the model has also been implemented with the alternative prognostic variable dl defined by where dl = d + X = p mrt φ. p mrt φ. ( ) V = d + X, (23) η ( ) V. (24) η

5 Dynamical Kernel of the Aladin NH Model 159 Compared with (3) (14), the modifications brought to the EE system by the use of the variable dl only involve and ddl dt + g2 p ( 1 η m ) (p π) η mr a T p V g mr a T η. w (dl X)(.V D 3) p W Ẋ = g (25) mr a T η D 3 =.V + R a (dl X) + X, (26) R mr a T g w = g w s + (dl X)dη η p ( ) mra T (dl X) dη. (27) p η As pointed out in BLVS04, for example, it is important to note that no matter what choice is made for non-hydrostatic variables examined here ( ˆP, ˆq, ˆd, d and dl), the solution of the implicit problem is left formally unchanged, that is the structure and the coefficients of the implicit linear system are the same in all cases. For instance, the system (17) (21) is formally unchanged when the variable dl is used in place of d. As a consequence, the modifications associated with any change of these variables are restricted to terms pertaining to the explicit model computed in the physical space of the collocation grid, thus leaving spectral computations unchanged. The computation of the term Ẋ in (25) requires a special treatment which is explained in section (2.4). In the above systems, the prognostic equations for other optional prognostic variables (water-vapour content, etc.) have been omitted, since these variables have no dynamical sources Transform of prognostic variables The use of d or dl as a prognostic variable instead of the vertical velocity w leads to practical difficulties for the computation of the explicitly treated nonlinear part of the system: as seen from (4) and (25), the source terms for the vertical momentum equation are cumbersome compared with the original w equation; moreover, a dynamically consistent discretization is made more difficult to achieve, which may lead, if care is not taken, to the appearance of erroneous responses (Klemp et al., 2003). Finally, when using dl as a prognostic variable, the knowledge of X in (27) requires an additional spectral transform for X.Beside these points, it might be suspected that using vertically derived variables such as d or dl instead of a native variable such as w could have a detrimental impact on the model performance. In order to avoid these problems and uncertainties, a formulation using w as a prognostic variable for the explicit system was also implemented. However, d or dl are still used in the implicit system, for sake of stability. This option with grid point w is referred to as the GP-w option hereafter. A transformation from w to d or dl and vice versa is therefore performed at the beginning and the end of the explicit computations. These transformations are based on the definition of d and dl, that is (2) and (23). Starting from grid-point space computations, an explicit guess for w is first computed, using the following equation: dw dt = g m ( (p π) η ) + W, (28) instead of (4) or (25). Then, the transformation is applied to obtain an explicit guess for d or dl. The explicit part of the SI linear system is computed exactly as in the original version, for the d or dl variable, and is then added (unmodified with respect to the original form) in order to obtain the total forcing term of the linear implicit system. Back to spectral space, the inverse transformation is applied to recover w at the new time level Spatial discretization The basic principles for the vertical discretization are the same as in BHBG95: the finite-difference technique is used, for a domain divided into L layers [labelled (1,..., L) from top to bottom], and bounded by L + 1 interfaces [labelled ( 0,..., L)]. The modifications of the spatial discretization of the Aladin NH model with respect to the initial formulation presented in BHBG95 are restricted to the nonlinear part of the model, and are listed below. The staggering of the prognostic variables is the same as in BHBG95. However, when the equation of w is used in the explicit part, the gridpoint prognostic variable w is defined at interfaces levels ( 0,..., L). The definition of the hydrostatic pressure and directly related quantities are unchanged with respect to BHBG95: π l = A l + B l π s, (29) δπ l = π l π l 1, (30) π l = π l 1 π l, (31) δ l = δπ l /π l, (32) α l = 1 π l 1 /π l, (33) where A l and B l are the values of the functions that define the coordinate, taken at the interface labelled l Pressure gradient force The formulation of the pressure gradient force as in (53) of BHBG95 leads to a discrepancy in the shape of the nonlinear term (given by (54) in BHBG95) with respect to its linear counterpart (given by the third right-hand side (RHS) term of (26) therein). This results in instabilities for large time steps. To avoid this problem, the nonlinear pressure gradient force term (i.e. (54) in BHBG95) is discretized as ( RT p ) ( ) π = R l T l + R l T l ˆq l, (34) p l π l where the hydrostatic part keeps its original form, derived from the specification of conservation of the global atmospheric angular momentum in the absence of sources (Simmons and Burridge, 1981): ( ) π π l = 1 δπ l [ αl B l + (δ l α l )B l 1 ] π s. (35)

6 160 P. Bénard et al. A consequence of (34) is that the formal conservation of angular momentum by the discrete scheme in the absence of sources is ensured only for the hydrostatic part of the flow: non-hydrostatic perturbations of the pressure force may now act as a source of angular momentum. However, it is assumed that this potential source of spurious drift is acceptable, because its magnitude is smaller than the hydrostatic part of the source (the typical magnitude of ˆq is found to be about at the 10 km resolution used here, and about 10 4 in real-case integrations made with 2.5 km and 0.5 km resolutions). Additionally, a lack of formal conservation for the hydrostatic part itself does not seem to create problems worth reporting in other NWP models: as an illustration, the relaxation of the formal conservation of angular momentum in the vertical finite-element scheme of the hydrostatic IFS model was found not to be detrimental to the quality of the forecasts (Untch and Hortal, 2004). Another element supporting this is that, to our knowledge, this physical constraint is not even taken into consideration in any other NWP NH model, for any part of the flow. The impact of such a feature in long-term global NH integrations is beyond the scope of this article and is left for future studies Top and bottom boundary conditions for the horizontal wind An artificial vertical averaging of the horizontal wind near top and bottom boundaries was used in BHBG95 to prevent some local instability there. In the revised version of the dynamical core, such averaging was found to be no longer needed, and was consequently removed. The top and bottom conditions for the wind are natural free slip conditions in the outermost half-layers: V top V 0 = V 1, (36) V surf V L = V L. (37) These conditions are used to express the top and bottom outermost values of various terms involving vertical derivatives of the horizontal wind ( V/ η) in nonlinear model equations such as (4) and (11), and also to express w s as for example in (16) Bottom boundary conditions for the vertical Laplacian operator For semi-lagrangian integrations, the original Eulerian rigid bottom boundary condition (56) in BHBG95 is replaced by a diagnostic Lagrangian boundary condition directly derived from (15), which can be written for a 2-TL scheme as [ g m ] (p π) η L = (w L )+ F (w L )0 O, (38) t where w L is given by (16). In the above expression and throughout the article, superscripts (+, 0, ) respectively denote values at times (t + t, t, t t). The subscripts O and F respectively denote the evaluation at the final (grid) point and the interpolation at the origin point of the SL trajectory. The quantity (w L )+ is evaluated through an explicit guess for SI schemes, and through the information available at the previous iteration for ICI time-discretization schemes (more details on these schemes are given below) Time discretization In order to have a relatively complete view of their properties, most of the schemes examined in B03 have been implemented in Aladin NH. However, because of their potentially doubled efficiency, the focus is put on 2-TL schemes rather than the less efficient 3-TL schemes. Only 2-TL schemes are therefore examined here. Similarly, although Eulerian versions for the transport scheme were implemented, only the comparatively more efficient semi-lagrangian versions will be described and discussed here. The 2-TL SL time discretization follows that of the IFS/ARPEGE model (Temperton et al., 2001) and uses the so-called SETTLS time extrapolation (Hortal, 2002). For Aladin, as for many other NWP models, in the case of a 2-TL SL SI scheme the semi-lagrangian trajectories are determined by using explicitly known winds extrapolated in time. However, as mentioned in Cullen (2001), in the case of 2-TL SL ICI schemes, using implicit winds available from the previously completed iteration is a better choice (see also Cordero et al., 2005). This approach is therefore implemented in Aladin NH. A time-decentring of implicit terms (Rivest et al., 1994) was initially implemented in the model, in order to control possible instabilities or flow-response distortion, but the practice in various NWP configurations showed that this feature is not needed, and therefore the impact of timedecentring is not examined here. The possible distortion of the SISL response to orographic forcing for flows with a CFL number greater than 1 is alleviated by using an Eulerian treatment of orographic forcing (Tanguay and Ritchie, 1996). Inthesameway,for2-TLSIschemesthedynamical source terms are evaluated as a time extrapolation from the current and previous time steps. For 2-TL ICI schemes, as suggested in Cullen (2001), the sources for the first iteration ofthetimeschemecanbeevaluatedatthecurrenttime step (without any time extrapolation), while for subsequent iterations the sources are evaluated as a time average between the current and future time steps. This non-extrapolating (NE) approach is retained in Aladin NH when a 2-TL ICI scheme is used. In the NE approach, the winds used in the trajectory computation in the first iteration are also nonextrapolated, while implicit winds are used for subsequent iterations as indicated above. Finally, among the various schemes implemented in Aladin NH, the following ones (as defined in B03) will be discussed here: the 2-TL SISL scheme; the 2-TL NE SL ICI schemes with various numbers of iterations (N iter ). As stated in B04a, the use of 2-TL constant-coefficients SI schemes for the EE system is made possible only by using a reference temperature Te for the vertical elastic wave terms that departs from the reference temperature T used for the other terms. In practice, in Aladin NH, Te is taken to be lower than T. In this article, ICI schemes are examined only for 2-TL schemes. Without entering into detail, we recall that when a 2-TL ICI scheme is used, the system is symbolically solved

7 Dynamical Kernel of the Aladin NH Model 161 under the form X +(k) X 0 ( X = (M L +(k 1) + X 0 ) ). t 2 ( X + L +(k) + X 0 ). + F(X 0 ), (39) 2 where X is the state vector, M thecompletedynamicalmodel operator, L the linear associated operator, F the physical sources operator and t thediscretetimeincrement.the superscript +(k) denotes the future time-level (t + t) obtained after the kth iteration of the ICI scheme. When the iterative algorithm is completed, the final future state at time (t + t) isofcoursedefinedbyx + = X +(Niter).Asoutlined in B03, the SI scheme can be viewed as an ICI scheme with no iteration (N iter = 0), through a proper definition of the starting point X +( 1) in (39). For extrapolating ICI schemes, X +( 1) is a time-extrapolated state from past states at t and (t t), and for non-extrapolating (NE) ICI schemes, X +( 1) is simply defined as X 0. Since an SI scheme is denoted by N iter = 0, the term ICI is restricted, in this article, to schemes with at least one actual iteration after the SI scheme, that is schemes with N iter 1. The zeroth iteration of an ICI scheme is thus equivalent to an SI scheme. Any additional iteration of the ICI scheme implies a computational overcost similar to that of a whole SI step of the dynamical kernel. Therefore, the potential advantage of any additional iteration should always be evaluated, taking into consideration the disadvantage of this overcost. For ICI schemes, the spectral horizontal diffusion of the Aladin model is applied at each iteration, whilst physics terms F, computed only with the state vector X 0,donot need to be recomputed during the iterative algorithm, as seen from the last term of (39). The particular time-discretization of the last left-handside term Ẋ in (25) is now discussed. In the symbolic form (39), this term pertains to M. Whendlisusedas a spectral prognostic variable, this term is time-discretized as discussed in section 7 of BMS05 in the context of 3-TL schemes. Namely, for a 2-TL SETTLS SISL scheme this term is simply written as Ẋ = X0 F X O. (40) t In the case of a 2-TL ICI scheme, for the computation of the zeroth iteration, i.e. k = 0 in (39), this source term (Ẋ) +( 1) is written as in (40), and for subsequent iterations, k 1 in (39), this source term (Ẋ) +(k 1) is computed as X +(k 1) F X 0 O t. (41) The treatment of the Ẋ term in (40) is only first-order accurate in time for the SI scheme. As a consequence, when the prognostic variable dl is used, at least one iteration (k 1) of the 2-TL ICI scheme is required in order to achieve second-order accuracy, as in usual centred time discretizations. When w is used as a grid-point prognostic variable and dl is used as a spectral prognostic variable, the term Ẋ is not explicitly required (as seen in (38)), but the transformation of the explicit guess of w to that of dl at the zeroth iteration of the ICI scheme (k = 0) amounts to a similar approximation, because an estimate of X at time (t + t)isrequiredforthis transformation Physical parameterizations and interface to dynamics The physical package used in this article is the one used for current operational NWP applications of the ARPEGE and Aladin models at Météo-France, and is briefly described below. Soil processes and exchanges through the surface are based on the ISBA scheme (Noilhan and Planton, 1989). The surface and atmospheric turbulent fluxes are described through a first-order closure scheme following Louis et al. (1981), using a flux-gradient K theory with a dependence on the Richardson number R i. Mixing is applied on total specific humidity, static energy and momentum, and two distinct Blackadar-type mixing lengths are defined for energy and momentum variables. Roughness lengths over sea are computed using the Charnock formula with an additional gustiness stability-dependent term. An antifibrillation scheme (Bénard et al., 2000) is applied to prevent instabilities linked to the stiffness of the coupled equations resulting from the scheme. The subgrid-scale shallow-convection scheme is part of the turbulence scheme, through a modification of the local Richardson number (Geleyn, 1987). The scheme for subgrid-scale deep convection originally follows Bougeault (1985). It is a mass-flux scheme, with a CISK-type closure and triggering and a Kuo-type closure for the vapour budget, but some modifications have been brought: use of mass-flux type computations for the implicit algorithm described in Geleyn et al. (1982), the introduction of a variation with height and cloud depth of entrainment rate, a distinction between liquid and ice phases, and downdrafts. The water is represented through five fully prognostic variables: water-vapour content, liquid and solid cloud water, liquid and solid precipitating water. The microphysical scheme describing the interactions between these fives species follows the one documented in Lopez (2002). To diagnose the total grid-box cloudiness and cloudcondensate content, both needed as input for the radiation scheme, two contributions are aggregated: (1) a fully resolved part, using the approach of Smith (1990), which specifies a triangular probability-density function for describing the subgrid fluctuations of a synthetic thermodynamic variable over each model grid box and (2) a deep convection part, deduced from convective precipitation fluxes. The radiation scheme operational in IFS until June 2007 is used. It includes the long-wave radiation transfer scheme RRTM (Mlawer et al., 1997; Morcrette et al., 2001) and a short-wave radiation scheme with six spectral intervals (Fouquart and Bonnel, 1980; Morcrette, 1993). A maximum-random-overlap assumption is used and the cloud fraction and cloud optical thickness are two separate quantities. The parameterization of the gravity-wave drag follows the scheme documented in Catry et al. (2008). This mainly includes four features for subgrid-scale orographically induced effects: basic linear processes, nonlinearities as trapping and resonant damping, surface anisotropy effects and blocking effects. The time-discretized physics/dynamics interface consists of evaluating diagnostic physical sources solely using prognostic variables at the past state, that is X 0 for 2-TL schemes. These physical sources are then interpolated at the departure point of SL trajectories. As a consequence, the

8 162 P. Bénard et al. set of parameterizations is by nature applied in a parallel way (Dubal et al., 2004). Another consequence is that when an ICI scheme is used, physical sources do not have to be recomputed at each ICI iteration, thus restricting the overcost of the ICI scheme to the iteration of the dynamical model itself. The heat source Q may be optionally applied at constant volume as for example in (5) (6), or at constant pressure (Catry et al., 2007). In this latter case, as discussed in BHBG95, the source in the T equation simply becomes (Q/Cp) and the source associated with heating vanishes in the ˆq equation. Applying the diabatic source at constant pressure is an approximation that constitutes a parameterization of the hydrostatic adjustment, and is justified when the horizontal size of the mesh boxes is much larger than their vertical size (Bannon, 1995). This simplifying assumption is implemented as an option in the model, and used here for all real-case results. 3. Sensitivities of the dynamical kernel for real-case experiments In the series of articles B03, B04a, BLVS04 and BMS05, the high sensitivity of the dynamical kernel response to various basic parameters was discussed. However these results were obtained in very simple idealized contexts, i.e. sufficiently simple to allow conventional linear stability analyses. The aim of the real-case simulations presented here is to examine how the conclusions drawn in these idealized contexts extend to meteorologically relevant contexts in a typical NWP medium-scale model. These real-case simulations consists of series of 30 forecasts starting at 0000 UTC for each day of June The reason for performing 30 successive forecasts was initially to eliminate possible poorly representative individual results that might have arisen if only one forecast were examined. However, the results appeared to be highly homogeneous for all 30 forecasts, which, for a given configuration, behaved qualitatively in the same way. Therefore, most of the time, individual results for a particular forecast will not be shown, and the focus will be placed on the description of the general behaviour for the set of forecasts. The practical configuration of the model used here (domain, resolution, physics, etc.) fully reflects the operational version of the French application of the Aladin limited-area forecast model, called Aladin-FRANCE (each member of the ALADIN consortium operates its own national application). The only difference between the experiments presented here and operational forecasts is of course that the dynamical kernel solves the EE system (instead of the operational HPE system) through a logical switch that enables the activation of the NH-specific parts of the code. Initial data for these simulations are taken from the Aladin-FRANCE operational 3D-variational assimilation cycle at relevant dates and hours. For NH simulations, initial conditions must also be provided for the two additional NH prognostic variables, and this information, of course not available from the (HPE) Aladin-FRANCE cycle, is obtained diagnostically as in BHBG95, assuming hydrostatically balanced vertical motion and a pressure field that ensures an initial state free of elastic perturbations. The lateral boundary conditions are the same as for Aladin-FRANCE, and are provided by the output of the French global operational NWP model ARPEGE, acting as a coupling model. The procedure for providing the coupling information for the two additional NH variables is the same as mentioned above for the initial state. The geometry (horizontal and vertical grids) and all other settings are identical to those of the HPE operational Aladin-FRANCE application. The collocation grid has points with a horizontal spacing of 9.4 km, and the vertical domain is represented through 60 levels (the highest geopotential full level is located at 0.1 hpa). Unless specified, the time step is the operational one ( t = 450 s) Sensitivity to the choice of SI reference temperature As discussed in B03, B04a and BLVS04, it appears that for constant-coefficients semi-implicit schemes, the EE system does not behave in the same way as the HPE system in the presence of nonlinear explicitly treated thermal residuals. In the case of the EE system, these residuals generally combine themselves through the coupled time-discretized system in a deleterious way, resulting in a very detrimental effect on stability. For 2-TL SI schemes, for example, the stability domain was found to simply vanish, making it impossible to solve the EE system with this otherwise attractive timediscretization if Te is not introduced into the system. For the reference temperature T,awarmvalueT = 350 K was recommended by Simmons and Temperton (1997) when using a 2-TL scheme in the HPE context. This value is used as the default one in this article, unless otherwise specified. In the idealized context of isothermal atmospheres, the stability domain for the EE system was shown to be related to the interval [Te, T ], the exact width of the stability domain being dependent on the particular time discretization chosen (see B04a). Results in idealized non-isothermal contexts suggested that Te should be lower than the likely coldest temperature value encountered in the whole domain over the whole duration of the forecast. The value for this parameter is initially taken here as very cold (Te = 50 K), in order to fulfil this constraint. The choices for the other parameters are as follows: (ˆq,dl) as NH prognostic variables, and a 2-TL SI SL time-scheme. As an illustration, the 500 hpa temperature field after a 48 h forecast, starting on 1 June 2008 at 0000 UTC, is depicted in Figure 1 for the hydrostatic (operational) forecast and in Figure 2 for the NH forecast. Unsurprisingly, at these resolutions the differences remain quite small. The experiment depicted in Figure 2 constitutes the reference NH experiment, around which sensitivities will be examined throughout this section. For sake of clarity, the main parameters of this experiment are summed up as follows: 2-TL SI SL scheme; ˆq and dl variables T = 350 K, T e =50K, t =450 s. The sensitivity of the forecast model to the specification of the additional reference temperature Te is now examined. With the value Te = 50 K, taken as a reference, all integrations in the sample are stable and behave similarly to their hydrostatic counterpart. The forecasts are then repeated with increasing values of Te.ForT e = 100 K, the forecasts remain stable and almost indistinguishable from the reference. For Te = 125 K, some evidence of instability begins to appear: six integrations out of 30 do not reach the final forecast range (+48 h). When setting Te = 150 K, all forecasts except one fail to reach the final time +48 h, and a significant portion

9 Dynamical Kernel of the Aladin NH Model W 0 5 E 10 E 15 E 55 N 55 N 50 N 50 N N 45 N N 40 N W 0 5 E 10 E 15 E Figure h Aladin HPE forecasts of the temperature field (in K) at 500 hpa, starting from 1 June 2008 at 0000 UTC, with a 2-TL SI SL scheme. 5 W 0 5 E 10 E 15 E 55 N 55 N 50 N 50 N N 45 N N 40 N W 0 Figure 2. Same as Figure 1, but for the NH version of Aladin. 5 E 10 E 15 E of the forecasts (16 out of 30) blow up before reaching +6 h (48 time steps). Finally, it is worth noting that a conventional SI scheme, which would not make use of adistinctvalueforte (i.e. for which T e = T = 350 K), would be extremely unstable in EE 2-TL SI mode: in this case all integrations fail in about six or seven time steps, implying huge growth rates for the most unstable modes. For a given forecasting configuration, there is a limiting range for Te above which the integration scheme becomes progressively unstable. This behaviour is reminiscent of and consistent with the one observed with the reference temperature T used in conventional SI schemes for HPE systems (Simmons et al., 1978; Simmons and Temperton, 1997). Unlike in the idealized isothermal case, the value of this limit may not be directly related to the lowest temperature in the integration domain, and a value of Te significantly colder than the lowest temperature seems to be necessary to achieve a stable scheme. Conversely, the 2-TL SI SL scheme is found to be quite robust for low values of Te :witht e =1K,all30 forecasts were stable. However, in SI schemes the choice of linearization coefficients as close as possible to their actual counterpart is always an implicitly adopted strategy, in order to minimize possible side-effects linked to unduly large nonlinear residuals. The practical impact of decreasing Te is not examined further, since it is beyond the scope of this article.

10 164 P. Bénard et al. It was mentioned in the conclusions of B04a that the possibility of choosing Te T can be extended straightforwardly to ICI schemes. However, the potential benefit in robustness when combining the two features (i.e. ICI and Te T ) was not investigated therein. There are various way to investigate the possible increase of robustness resulting from this combination; here we only examine whether the use of a 2-TL ICI scheme allows an increase of Te compared with the 2-TL SI scheme examined just above. The above experiments are therefore repeated, but now with a2-tlneicischemewithn iter =1. The starting value for Te is 150 K, for which almost all integrations failed. Setting N iter to 1 results in a stabilization of the integration scheme for all forecasts of the series. The integrations remain uniformly stable for Te =200 K (with no significant differences from the reference), but for Te =250 K all forecasts blow up after less than 20 time steps. The impact of using an ICI scheme with N iter 1 is therefore an extension of the stability domain related to Te, as could be logically expected. In the configuration used here, the increase of Te when using N iter=1 is of almost 100 K, resulting in a much smaller magnitude of nonlinear thermal residuals (T Te ). The detrimental effect of large magnitudes for nonlinear residuals is not very clear in these experiments, but a smaller magnitude might result is a slightly more accurate scheme. Although the quantitative values for ensuring stability cannot be drawn directly from the results of isothermal academic tests, the qualitative behaviour of the integration scheme with respect to this parameter Te is consistent with the general academic results obtained in B04a: (1) a low value of Te is able to increase dramatically the stability of the integration scheme; (2) using ICI schemes instead of SI schemes results in a significant extension of the stability domain for Te. Finally, the inclusion of T e is found to be a very important component, which, in our case, has indeed made possible the use of 2-TL SI SL schemes for successfully integrating the EE system Sensitivity to the choice of vertical momentum prognostic variable Idealized simulations and analyses (BMS05) have shown that, in the presence of nonlinear thermal residuals and orography, the use of the prognostic variable d may lead to an instability of the EE system for constant-coefficients SI schemes. To restore a satisfactory stability, it was suggested therein to use the prognostic variable dl, which results in a more implicit treatment of some orographic terms. In this section, the differences in results with these two prognostic variables are examined in the context of real-case experiments. The above reference experiment (with the following settings: 2-TL SI, N iter =0, Te =50 K, t=450 s) is repeated, but the prognostic variable dl is replaced by d. The integration appears to be very unstable most of the time, blowing up at about 4 6 time steps. Dramatically decreasing the length of the time step, down to t=50 s has no significant stabilizing effect (the integrations blow up after about time steps). This behaviour is fully consistent with conclusions drawn from theoretical examination. Faced with this kind of instability, one can never expect to perform stable integrations over a given range (here +48 h) by decreasing the time-step length, since the smaller growth rate is counterbalanced by the need to compute a larger number of time levels (Simmons et al., 1978). This emphasizes the need to seek numerical schemes that are robust by themselves, with a weak dependence on the time-step length. Since no hope can be expected from the SI scheme with the prognostic variable d, the experiments are repeated with an ICI scheme and N iter =1, still for t=50 s. With these settings, all integrations become stable. The deviations from experiments with the variable dl remain small and limited in space (not shown), although the integration methods are quite different. However, this configuration is not stable for larger time steps: all integrations become uniformly unstable as soon as t is set to 100 s. Better stability can be achieved for this combination (d, ICI with N iter =1) by decreasing the value of T (while modifying Te has very little impact). Stable integrations can then be obtained for t=400 s by setting T =280 K. However, for T =270 K, as well as for T =290 K, the scheme is stable only for values of time step smaller than t=300 s. The area with a slightly better robustness is thus too narrow for reasonable NWP use. It should be noted that the configuration (dl, ICI with N iter =1, t = 450 s) is found to be stable with no problems for values of T as low as 210 K, indicating that the range in T for stability is very wide. All these results confirm that globally the robustness of theschemeforthevariabled appear to be relatively poor, and that it cannot be completely restored by using an ICI scheme. The versions using w as prognostic variable in the gridpoint explicit system (the GP-w option described in section (2.2)) have been compared with the original versions using d or dl throughout the integration scheme. The results in terms of model response and robustness are discussed now. As outlined above, the nature of the computations for the vertical momentum equation are quite different from the original version when using the GP-w version. However, since the linear part of the computations remains totally unchanged, no significant change is expected in the overall stability of the scheme. This again is confirmed by the results obtained from these configurations. Figure3showsthesamefieldasinFigure2inthesame format, for the configuration with dl, 2-TL SI SL, GP-w. Comparison with Figure 2 shows that in spite of some small differences, the overall output of the forecast is not substantially modified. Most differences consist of relatively small-scale features restricted in a band spanning from the north-west corner of the domain to the south-east one. The same experiment repeated with the d prognostic variable leads to unstable forecasts. Using the GP-w option with d as prognostic variable is therefore not sufficient to stabilize the scheme. Here again, this result is fully consistent with those previously suggested by analytical studies: for a given choice of the prognostic variable (that is the variable used in the spectral part of the model computations for solving the implicit problem), peculiar variants in the treatment of nonlinear explicitly treated source terms have no dramatic impact on the stability of the scheme. The choice of the prognostic variable itself appears to be a much more crucial factor for stability. In agreement with theoretical studies, it appears from these real-case experiments that the variable dl globally results in more robust schemes than d, whateveroptions