Impact of Consistent Semi-Lagrangian Trajectory Calculations on Numerical Weather Prediction Performance

Transcription

1 OCTOBER 2017 H U S A I N A N D G I R A R D 4127 Impact of Consistent Semi-Lagrangian Trajectory Calculations on Numerical Weather Prediction Performance SYED ZAHID HUSAIN AND CLAUDE GIRARD Atmospheric Numerical Prediction Research Section, Meteorological Research Division, Environment and Climate Change Canada, Dorval, Quebec, Canada (Manuscript received 15 May 2017, in final form 14 July 2017) ABSTRACT Inconsistencies may arise in numerical weather prediction models that are based on semi-lagrangian advection when the governing dynamical and the kinematic trajectory equations are discretized in a dissimilar manner. This study presents consistent trajectory calculation approaches, both in the presence and absence of off-centering in the discretized dynamical equations. Both uniform and differential off-centering in the discretized dynamical equations have been considered. The proposed consistent trajectory calculations are evaluated using numerical experiments involving a nonhydrostatic two-dimensional theoretical mountain case and hydrostatic global forecasts. The experiments are carried out using the Global Environmental Multiscale model. Both the choice of the averaging method for approximating the velocity integral in the discretized trajectory equations and the interpolation scheme for calculating the departure positions are found to be important for consistent trajectory calculations. Results from the numerical experiments confirm that the proposed consistent trajectory calculation approaches not only improve numerical consistency, but also improve forecast accuracy. 1. Introduction The majority of operational global numerical weather prediction (NWP) models employ the semi- Lagrangian method for advection, which was pioneered by Robert (1981, 1982). Staniforth and Temperton (1986) have shown that a semi-implicit treatment of the linear terms in the dynamical equations responsible for the fast waves pertaining to gravitational oscillations permits the use of long time steps with semi-lagrangian models. This helped the wide adoption of the semi-implicit semi- Lagrangian approach for operational NWP purposes. The basic idea is to compute the flow characteristics associated with the air parcels arriving at fixed model grid points at every time step by tracking back their positions in the previous time step. This approach thus allows combining the advantages of the Eulerian fixed Denotes content that is immediately available upon publication as open access. Corresponding author: Syed Zahid Husain, syed.husain@canada. ca computational grid with Lagrangian treatment of advection. To track the air parcels, models based on the semi- Lagrangian approximation for advection require calculations of the parcel trajectories to determine the meteorological fields at the departure positions. It involves an iterative process at each time step and requires numerical approximation of a velocity integral along the trajectories at every iteration. Once the departure positions are known, the velocity field needs to be interpolated to estimate its value at the departure positions. Depending on the particular discretization associated with a model, extrapolation of the velocity field in time may be required before the spatial interpolation. At the end of the iterative trajectory calculations for a given model time step, the semi- Lagrangian advection is applied to the model prognostic equations, which again requires integration of the forcing terms in the discretized dynamical equations along the trajectory as well as interpolation of the advected fields and the forcing terms at the estimated departure positions. The choice of numerical approximation for the integrals along the trajectories and interpolation techniques at the departure positions for the prognostic DOI: /MWR-D For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy ( PUBSReuseLicenses).

2 4128 M O N T H L Y W E A T H E R R E V I E W VOLUME 145 dynamical equations can affect both the accuracy and stability of model integrations. Tanguay et al. (1992) have demonstrated the advantages of replacing midpoint trajectory evaluation for the nonlinear terms within the prognostic model equations by averages along the trajectory. Similarly, the approximation of velocity integral and spatial interpolation for the trajectory calculations is expected to have important consequences. White (2003) has discussed the theoretical foundation of consistency between the discretizations applied to the prognostic dynamical equations and the kinematic trajectory equations from a dynamical equivalence perspective. However, systematic investigation of the impact of consistency between the numerical approximations associated with the semi- Lagrangian advection and trajectory calculations, particularly in the context of NWP applications, is largely absent in the existing literature. A critical vulnerability of the semi-implicit semi- Lagrangian approach was demonstrated by Rivest et al. (1994). They have demonstrated that long time steps with semi-lagrangian schemes can lead to a numerical instability due to a spurious resonant response attributable to the presence of orographic forcing. They proposed off-centered averaging of the source terms along the trajectory, where additional weight is placed on the implicit components of the discretized system, as a means to address the orographic resonance problem. Although off-centering is very effective in controlling orographic resonance and suppresses computational noise by imparting numerical dissipation, it generally reduces the order of truncation error associated with temporal discretizations. Ritchie and Tanguay (1996) have later proposed a spatially averaged Eulerian (SAE) treatment of orographic terms in the prognostic equations as an alternative approach to address orographic resonance. Further study by Lindberg and Alexeev (2000) revealed that an SAE treatment of orographic forcing considerably inhibits the growth of resonant response although the singularities are not entirely removed. They proposed small off-centering along with horizontal diffusion to obtain more acceptable results. Some operational NWP models have adopted different forms of Eulerian treatment of orographic forcing to control orographic resonance for the semi-implicit semi-lagrangian approach (Bénard et al. 2010; ECMWF 2016). Other models, including the Global Environmental Multiscale (GEM) atmospheric model, used operationally by Environment and Climate Change Canada (ECCC), however, continue to use off-centering. Off-centering in the discretized dynamical equations poses another source of potential numerical inconsistency when the trajectory calculations are not adjusted accordingly. Staniforth et al. (2003) have extended the work of White (2003) to include the presence of off-centered averaging in the dynamical equations to determine possible consistent discretizations for the trajectory equations to preserve the so-called dynamical equivalence. The possible impact of off-centered averaging of the velocity integral within the trajectory calculations in the presence of off-centering within the discretized dynamical equations, however, has not been studied elaborately in the context of actual NWP models and warrants a systematic investigation. The present study explores the impact of consistency in the numerical discretization employed between the dynamical equations for semi-lagrangian advection and the corresponding trajectory calculations in the context of ECCC s GEM model. Two numerical experimental setups are utilized to derive useful conclusions covering both hydrostatic and nonhydrostatic scenarios. On the one hand, the hydrostatic experiments are conducted using the GEMbased Global Deterministic Prediction System (GDPS), which is based on a Yin Yang grid (Qaddouri and Lee 2011). The nonhydrostatic tests, on the other hand, are based on the theoretical case of two-dimensional mountain wave simulations proposed by Schär et al. (2002). This case is selected for this study as it has been found by others to be an important test bed for exploring consistency issues between the discretizations for semi-lagrangian advection and the other components of the dynamical kernel of an NWP model (Schär et al. 2002; Klemp et al. 2003; Girard et al. 2005; Melvin et al. 2010). Relevant background information about the GEM model, including the trajectory calculations and the different approaches around off-centering, is briefly described in section 2. The two experimental setups to study the impact of consistency in trajectory calculations are presented in section 3. The pertinent results are presented and discussed in section 4, while the conclusions of the study are summarized in section The GEM model a. Vertical coordinate The vertical coordinate for the current nonhydrostatic version of the GEM model is of loghydrostatic-pressure type (Girard et al. 2014) that follows the concept of the generalized hydrostaticpressure-type hybrid coordinate proposed by Laprise (1992), and has the form z 5 lnp 2 Bs, (1)

3 OCTOBER 2017 H U S A I N A N D G I R A R D 4129 where z defines the vertical coordinate, p denotes the hydrostatic pressure, B is a metric term (to be defined below), and s 5 lnp s 2 z S with p s being the hydrostatic pressure at the surface and z S 5 ln(p ref ), where the reference pressure p ref hpa. The term B is defined as z 2 x zt B 5, (2) z S 2 z T with z T 5 ln( p T ), where p T refers to pressure at the model top, and x is a variable exponent providing control over the stretching of the coordinate with height over orography, defined as x 5 x max 2 (x max 2 x min )[(z 2 z T )/z S 2 z T ]. The rate of flattening of the coordinate with height can be adjusted by properly selecting the values of x max and x min. The vertical coordinate (1) can be expressed in terms of nonhydrostatic pressure p as z 5 lnp 2 q 2 Bs, (3) where q 5 ln( p/p) is the nonhydrostatic logarithmic pressure perturbation. In addition to having a linear dependence on lnp, the chosen definition of z greatly simplifies linearization of model equations as both q and s appear as model variables (shown in section 2b). b. Governing equations The GEM model equations are derived from the shallow-atmosphere nonhydrostatic primitive equations comprising five prognostic equations (Newton s second law for momentum, the second law of thermodynamics, and mass continuity) along with a diagnostic equation, namely, the ideal gas law. After transforming the basic equations to the log-hydrostatic-pressure-type terrainfollowing vertical coordinate z, followed by perturbations around a stationary and isothermal basic state, the governing equations can be expressed as dv H 1 f k 3 V dt H 1 RT= z (Bs 1 q) 1 (1 1 m)= z f 0 5 F H, (4) dw d T ln 2k(Bs 1 q) dt T * dt 2 gm 5 F, (5) w 2k_z 5 F T, (6) d dt [Bs 1 ln(1 1 z s)] 1 = z V H 1 (1 1 z ) _z 5 0, (7) df 0 dt 2 RT * _ z 2 gw 5 0. (8) In the above, (4) (8) represent the equations for horizontal momentum, vertical momentum, thermodynamic, continuity, and vertical velocity, respectively, where V H denotes the horizontal wind, f is the Coriolis parameter, w is the vertical velocity, T is the virtual temperature, R is the gas constant for dry air, T * is the constant basic-state temperature (generally, T * K), f 0 is the deviation of the geopotential from the basic state f * 52RT * (z 2 z S ), k 5 R/c p with c p being the specific heat at constant pressure, _z 5 dz/dt, and z 5 / z. Thetermm, definedas m 5 ( p/ p) 21, is the nonhydrostatic index such that m 5 0 reduces the equations to those for a hydrostatic system. Furthermore, the terms F H, F w,andf T stand for various subgrid-scale physical forcing terms that are represented through a combination of physical parameterizations. The unknowns V H, w, T, _z, s, and q are determined prognostically by solving (4) (8), while the remaining unknowns, m and f 0, are obtained from two diagnostic relations given by T 1 e q f/rt z * 5 0, and (9) T * 1 1 z Bs 1 1 m 2 e q z q 5 0, (10) 1 1 z Bs which come from the definition of hydrostatic pressure p and nonhydrostatic index m, respectively. In addition to time t, the three other independent variables related to space are given by the position vector x, where x 5 (r, z) 5 (l, u, z) with l and u denoting the coordinates in longitude and latitude, respectively. c. The Yin Yang grid system The global forecast system in GEM is based on a Yin Yang grid (Qaddouri and Lee 2011) that utilizes the Schwarz method for domain decomposition (Cai 2003). It comprises two perpendicular and overlapping quasiuniform limited-area model (LAM) latitude longitude grids that together form a global grid. Unlike the global latitude longitude grid system, dividing the global domain into two LAM subdomains allows the Yin Yang approach to avoid polar singularities while not suffering from the computationally expensive issue of grid convergence near the poles (Kageyama and Sato 2004). The governing equations are discretized and resolved over the two subdomains with an iterative process for the solutions to converge within the overlapping regions. The boundary conditions for each LAM domain is thus provided by the solution from the other LAM domain. Further details on GEM over the Yin Yang grid are provided by Qaddouri and Lee (2011). d. Spatiotemporal discretization The model employs identical numerical discretizations in space and time irrespective of the scope of the

4 4130 M O N T H L Y W E A T H E R R E V I E W VOLUME 145 computational domain (i.e., same discretization for both global and LAM integrations). The system of governing equations (4) (10) are augmented with the appropriate boundary conditions and are first resolved in the absence of the physical forcing terms (F H, F w, and F T ). The unresolved subgrid-scale physical processes are then accounted for through parameterized physical tendencies using a splitting method. Any individual equation belonging to the dynamical system, in the absence of the physical forcing, can be generally expressed as df 1 G 5 0, (11) dt where F represents the advected quantity and the source term G contains the remaining linear and/or nonlinear terms. In space, the governing equations as well as the boundary conditions are discretized horizontally over an Arakawa C grid (Arakawa 1988; C^oté et al. 1998; Yeh et al. 2002) and vertically over a Charney Phillips grid (Charney and Phillips 1953; Girard et al. 2014). Further discussions regarding the details of spatial discretization in GEM can be found in the existing literature (Yeh et al. 2002; Qaddouri and Lee 2011; Girard et al. 2014). The temporal discretization in GEM is based on an implicit two-time-level semi-lagrangian method (C^oté et al. 1998), and when applied to any equation of the form (11), results in (F A 2 F D ) 1 ð t1dt t Gdt5 0, (12) where the superscripts A and D, respectively, denote the arrival and departure positions in the space time coordinate, given by (r A, z A, t 1Dt) and (r D, z D, t). As a result, for every model grid cell, the line AD denotes the parcel trajectory in the space time coordinate. This approach assumes that at each new time step (t 1Dt), parcels departing from departure positions (r D, z D )attimet arrive at the model grid cells located at (r A, z A ). In other words, any scalar F, when advected along the fluid trajectory AD in the space time coordinate, is conserved in the absence of any other forcing. This fundamental concept associated with semi-lagrangian advection is ensured by the first term in (12). The second term in (12) represents an integral of the source term G along the trajectory AD. This integral can be evaluated using different numerical approximations. The midpoint rule and the trapezoidal rule are two of the most widely used approaches in this regard. In GEM, the trapezoidal rule is used to approximate these integrals, which leads to F A 2 F D 1 1 Dt 2 (GA 1 G D ) 5 0. (13) Rivest et al. (1994) have demonstrated that a Lagrangian treatment of orographic forcing leads to a spurious resonant response with semi-implicit discretization. Clancy and Lynch (2011) have shown that the semi-lagrangian advection scheme coupled with the Laplace transform scheme does not suffer from such resonance. Although GEM uses an iterative implicit semi-lagrangian approach, it is found to suffer from resonance issues similar to the semiimplicit models. The SAE treatment of orographic forcing proposed by Ritchie and Tanguay (1996), when implemented within the current version of the GEM model, fails to improve the orographic resonance characteristic (not shown). As a result, in order to control spurious orographic resonance, the GEM model, in its present state, continues to utilize off-centering while approximating the integral of the source term along the parcel trajectories. Equation (13) in the presence of off-centering then becomes F A 2 F D Dt 1 1 b 1 2 GA b 2 GD 5 0, (14) where b is the off-centering parameter. It is worth noting that off-centering reduces the order of truncation error while adding significant numerical dissipation that is not scale selective, and therefore, large values of b are detrimental to solution accuracy. When all the dynamical equations are subjected to the same level of off-centering the underlying approach may be referred to as uniform off-centering. The operational version of GDPS, at present, uses uniform off-centering for producing global forecasts, with b for all the dynamical equations. The general implementation of off-centering in the GEM model is, however, based on a differential approach (i.e., the different dynamical equations are allowed to have different levels of off-centering ). The equations are grouped into three different categories with a specific off-centering parameter assigned to each of these groups as follows: 1) off-centering parameter b M for the horizontal momentum equations (4); 2) off-centering parameter b H for the continuity, thermodynamic, and vertical velocity equations given by (6) (8); and 3) off-centering parameter b NH for the vertical momentum equation (5), which is only available for nonhydrostatic situations.

5 OCTOBER 2017 H U S A I N A N D G I R A R D 4131 e. Trajectory calculations Based on the initial discussions about the semi- Lagrangian scheme, it is evident that the arrival positions (r A, z A ) along parcel trajectories at time (t 1Dt) coincide with the positions of the model grid cells. The departure positions (r D, z D ) that are valid at time t are, however, unknown and need to be calculated as a part of the semi-lagrangian method through trajectory calculations. These calculations involve a numerical approximation of the solution of the following kinematic displacement equations: dr dt 5 V (r, z, t), (15a) H dz dt 5 _z(r, z, t). (15b) These kinematic equations can be discretized in the same manner as the dynamical equations, resulting in r A 2 r D 5 z A 2 z D 5 ð t1dt t ð t1dt t V H (r, z, t) dt, _z(r, z, t) dt, (16a) (16b) where integrals on the right-hand side are approximated numerically. A common approach in this regard, as has been the case for GEM, is to use the midpoint rule, which gives r A 2 r D 5DtV H r M, z M, t 1 Dt 2 ffi Dt[V H (r M, z M, t 1Dt) 1 V H (r M, z M, t)]/2, (17a) z A 2 z D 5Dt_z r M, z M, t 1 Dt 2 ffi Dt[_z(r M, z M, t 1Dt) 1 _z(r M, z M, t)]/2, (17b) where the superscript M denotes the midpoint in the trajectory AD. Rearranging these equations leads to the following discretized kinematic relations: r A 2 r D Dt [VM H (t 1Dt) 1 VM H (t)], (18a) z A 2 z D 5 1 Dt 2 [ _z M (t 1Dt) 1 _z M (t)]. (18b) When compared to the general form of the discretized dynamical equations (13), where the integral over the source term G is approximated using the trapezoidal rule, (18) appears inconsistent. This inconsistency can be removed by using a similar trapezoidal rule for the trajectory calculations, resulting in r A 2 r D 5 1 Dt 2 (VA H 1 VD H ), (19a) z A 2 z D 5 1 Dt 2 ( _z A 1 _z D ). (19b) In the presence of differential off-centering in the discretized dynamical equations, the more consistent trajectory discretizations are expected to have the form r A 2 r D 1 1 bv 5 Dt 2 z A 2 z D 1 1 bz 5 Dt 2 V A H b VV D H, (20a) 2 _z A b z _z, D (20b) 2 where the appropriate values of the off-centering parameters b V and b z will be determined later in this paper with the help of numerical experiments. f. Interpolation Interpolation is a critical component of the semi- Lagrangian approach for advection. Interpolation is necessary in determining both the departure wind fields for the trajectory equations and the advected fields along with the source terms within the dynamical equations. The computation of the advected fields and the source terms at the departure positions for the discretized dynamical equations generally requires higher-order schemes for spatial interpolation. In this regard, cubic interpolation based on Lagrange polynomials has been found to be optimum in balancing the accuracy requirements and the associated computational cost (Staniforth and C^oté 1991). Compared to linear interpolation, McDonald (1984) has shown cubic interpolation to be considerably less damping for the smallest scales. Furthermore, cubic interpolation is found to provide a better phase representation than even the more expensive quartic interpolation (McDonald 1984). As a result, cubic interpolation has been widely adopted for interpolating the advected fields and the associated source terms. Contrastingly, Staniforth and C^oté (1991) had found linear interpolation to be sufficient for interpolation within the trajectory calculations. Many NWP models, including GEM (until recently), continue to use linear interpolation for trajectory calculations. This, however, leads to another possibility of inconsistency in the numerical approach between the kinematic trajectory calculations and the solutions to the dynamical equations. To study the impact of this potential inconsistency, cubic interpolation has been introduced as an option in the GEM model for trajectory calculations.

6 4132 M O N T H L Y W E A T H E R R E V I E W VOLUME Numerical experiment and evaluation setup In the previous section, three potential sources of numerical inconsistencies associated with the trajectory calculations have been identified, which are as follows: 1) the use of midpoint rule instead of the trapezoidal rule for the numerical approximation of the velocity integral along the trajectory in (16), 2) the use of linear interpolation for the wind field instead of cubic to determine the departure positions, and 3) the absence of any form of off-centering in the averaging within the trajectory calculations when the dynamical equations use off-centered averaging along the trajectories for the source terms. For convenience of notation, the different modifications in the trajectory calculations for numerical experimentations are henceforth denoted as follows: 1) ML: midpoint averaging and linear interpolation, 2) MC: midpoint averaging and cubic interpolation, 3) TL: trapezoidal averaging and linear interpolation, and 4) TC: trapezoidal averaging and cubic interpolation. The above configurations involving the trapezoidal averaging do not include any form of off-centering for numerical approximation of the velocity integral. In the presence of off-centered averaging in the trajectory equations, the names of the associated experimental configurations will have _OC added as asuffix. As mentioned earlier, two setups covering both hydrostatic and nonhydrostatic scenarios have been used for numerical experiments in this study to evaluate the impact of these inconsistencies on NWP performance. Some brief information pertaining to these setups is provided below. a. The nonhydrostatic Schär mountain case A theoretical two-dimensional test case, originally presented by Schär et al. (2002), has been used in this study to investigate the impact of trajectory calculations in nonhydrostatic situations. It describes stably stratified dry flow over an idealized mountain defined by x 2 z s 5 z 0 exp 2 a cos 2 px l x, (21) where z s is the surface height, z m is the maximum mountain height, a 5 5 km is the mountain envelop half-width, and l x 5 4 km is the mountain wavelength. The results to be presented in this paper correspond to an upstream flow conditions defined by N 5 0:01 s 21 (Brunt Väisälä frequency), U 5 10 m s 21 (upstream wind speed in the x direction), T K (upstream surface temperature), and p hpa (upstream surface pressure). All other conditions associated with this test case are identical to those used by Schär et al. (2002). Schär et al. (2002) have presented a time-invariant analytical solution to the corresponding linearized problem. The analytical solution is characterized by large-scale hydrostatic waves that propagate deep into the vertical along with small-scale nonhydrostatic mountain-generated waves that rapidly decay with height. Integration in time by the model is carried out for a sufficient number of time steps to reach a quasi steady state. Model outputs are then compared qualitatively against the analytical solution to assess the impact of the different trajectory calculation approaches. It is important to note that the Schär tests are carried out using the same GEM dynamical kernel. To do so, the twodimensional theoretical problem is converted into a threedimensional problem with slab symmetry. b. Global hydrostatic forecast Two series of 25-km global hydrostatic forecasts, involving 44 cases of winter and summer (for the Northern Hemisphere), have been carried out using the GDPS (i.e., GEM over the Yin Yang grid) that uses a time step of 720 s. Each GDPS case consists of a 5-day forecast and the consecutive cases for both periods are initialized 36 h apart. The first summer case was initialized at 0000 UTC 25 June 2014, whereas the first winter case was initialized at 0000 UTC 19 December Forecast scores are then computed for the individual cases associated with each season for the different forecast lead times by comparing the model output against upper-air radiosonde observations at different pressure levels in terms of bias and standard deviation of error (SDE). This is followed by the computation of the average error profile for all the 44 cases for each seasonal period again for different forecast lead times. The statistical significance of the computed average bias and SDE at the different pressure levels are then determined by applying the t test and F test, respectively. In addition to the upper-air scores, comparisons are also made in the spectral space to study the impact of the different trajectory calculation approaches on the spectral variance of the different meteorological fields at different pressure levels. 4. Results and discussion This section presents the results associated with the different numerical experiments that have been conducted during this study using the setups described in the previous section. The primary objective is to identify the consistent approaches for trajectory calculations in

7 OCTOBER 2017 H U S A I N A N D G I R A R D 4133 FIG. 1. Simulated vertical velocity for the Schär mountain case with different lengths of time step: (a) Dt 5 32 s (Courant number, C o 5 0:76); (b) Dt 5 16 s, (C o 5 0:38), (c) Dt 5 8s, (C o 5 0:19), and (d) Dt 5 4s (C o 5 0:095). Simulations were carried out without any off-centering in the discretized dynamical equations as well as the trajectory calculations. The solid and dashed lines indicate positive and negative values, respectively, with a contour interval of 0.1 m s 21. the absence and presence of off-centering. Another objective is to assess the impact of different trajectory calculations on NWP accuracy. Figure 1 shows the quasi-steady numerical solution of vertical velocity obtained for the Schär mountain case for different lengths of model time step where the trajectory calculations are based on midpoint averaging and linear interpolation. These simulations have been carried out without any off-centering applied to the dynamical equations. When compared to the analytical solution presented by Schär et al. (2002), the model solutions obtained with both 32- and 16-s time steps (Figs. 1a and 1b) show significant distortions in the large-scale vertically propagating hydrostatic waves. Even with the smallest time step of 4 s (Fig. 1d), these wave perturbations are not completely eliminated. It should be noted that the Courant numbers (C o )fordt 5 32 s and 4 s are 0.76 and 0.095, respectively, and therefore even the largest time step used in these tests falls well within the acceptable range for semi-lagrangian models. Although the convergence test presented in Fig. 1 show considerable improvement in the simulated mountain waves with significant reduction in the time step, the distortions for the larger time steps resemble the well-documented consequence of various numerical inconsistencies associated with the choice of numerical details for this test case (Klemp et al. 2003; Girard et al. 2005; Melvin et al. 2010). For example, Schär et al. (2002) have also experienced similar distortions and proposed a new type of vertical coordinate, namely, the smooth level vertical (SLEVE), as a possible fix where the impact of the finescale orographic variations on the coordinate with increasing altitude are quickly reduced. However, for the same model with three-time-level scheme and height-based vertical coordinate, Girard et al. (2005) have later identified the inconsistency in the calculation of departure height as the primary reason for the distorted waves. Their proposed resolution to the problem is to use a semi-lagrangian discretization for the relation between the actual vertical velocity and its counterpart in the transformed vertical coordinate. This

8 4134 M O N T H L Y W E A T H E R R E V I E W VOLUME 145 FIG. 2. Simulated vertical velocity for the Schär mountain case for different combinations of approaches for trajectory calculations with Dt 5 32 s: (a) midpoint averaging and linear interpolation (ML), (b) midpoint averaging and cubic interpolation (MC), (c) trapezoidal averaging and linear interpolation (TL), and (d) trapezoidal averaging and cubic interpolation (TC). All other conditions are as in Fig. 1. makes the numerical discretization consistent and, as a result, eliminates the distortions in the simulated mountain waves. Melvin et al. (2010) demonstrated the benefit of such a discretization for a model with a centered two-time-level approach. Furthermore, for a conventional hybrid terrain-following height-based vertical coordinate, Klemp et al. (2003) have demonstrated the importance of numerical consistency in the approximated horizontal derivatives, attributable to the presence of the metric terms resulting from coordinate transformation, in order to obtain a distortion-free solution. Unlike the Eulerian models, a major advantage of the semi-lagrangian approach is its ability to work for large Courant numbers (C o. 1). The severe restriction on C o, as found from the results in Fig. 1, is therefore unacceptable and warrants identification and correction of the possible inconsistencies related to the numerical discretizations within the dynamical kernel of the model. As has been stated in the previous section, for the pressure-based vertical coordinate used in GEM and in the absence of off-centering, two potential sources of numerical inconsistency have been identified in the trajectory calculations that are related to the averaging for approximating the velocity integral and the interpolation for determining the departure positions. The impact of the different combination of approaches related to averaging and interpolation in the trajectory calculations on the simulated mountain waves for the Schär case is presented in Fig. 2. All the tests pertaining to this figure have been conducted with Dt 5 32 s, which corresponds to the largest time step used for the tests associated with Fig. 1. Figure 2 clearly demonstrates that only replacing linear interpolation by cubic interpolation (MC, Fig. 2b) or midpoint averaging by trapezoidal (TL, Fig. 2c) isnotsufficienttoeliminatethewavedistortions, and therefore, not a sufficient condition for numerical consistency. The simulated mountain waves are found to be free of the erroneous distortions only when a combination of trapezoidal averaging and cubic interpolation is used (TC, Fig. 2d). The results

9 OCTOBER 2017 H U S A I N A N D G I R A R D 4135 presented in Fig. 2 thus confirm the origin of the distortions in the mountain waves to be numerical inconsistency in the trajectory calculations. To be numerically consistent, the trajectory calculations should employ the same discretizations in the approximation of velocity integral and interpolation to obtain the departure positions as those utilized in the discretized dynamical equations for semi-lagrangian advection and the computation of the advected and the source terms at the departure positions. This conclusion regarding the importance of the averaging method conforms to the consistent trajectory calculation approach proposed by White (2003) whose objective was to discretize kinematic trajectory equations in a manner that preserves the dynamical equivalence between momentum and angular momentum formulations of dynamical equations. White (2003) has shown that the discretized form of the trajectory equations based on trapezoidal averaging, as presented in (18), ensures consistency between the evolution of the velocities obtained from the momentum and the trajectory equations. However, in addition to demonstrating the importance of consistency in the averaging for approximating the integrals in (13) and (19), results presented in Fig. 2 show that the consistency in the choice for the interpolation schemes between semi- Lagrangian advection and trajectory calculations is equally important. All the test results presented in Figs. 1 and 2 correspond to no off-centering in the discretized dynamical equations. In the full three-dimensional GEM model, off-centering has however remained critically important from the perspective of forecast scores since its first introduction. Figure 3 shows the comparative global forecast scores for GEM with and without off-centering in terms of the vertical profiles of bias and SDE of zonal wind, wind speed, temperature, and geopotential height obtained from the average of the 44 winter cases for the Northern Hemisphere. In this figure, the blue lines represent the control case involving standard off-centering (i.e., b M 5 b H 5 0:2), whereas the red lines indicate the results without any off-centering. Another important thingtonoteregardingthesefiguresisthenumbers(in percentage) shaded with blue or red color placed along the left and right edges of the individual figure panels. These color-shaded numbers denote the confidence in the statistical improvements obtained by the model configuration associated with the shading color in terms of bias and SDE over the other configuration. Significance is computed for bias and SDE by applying the t test and the F test, respectively. Therefore, in the context of this particular figure, the numbers shaded in blue (red) color indicate statistically significant improvements obtained with (without) off-centering. The 120-h forecast scores presented in Fig. 3 clearly reveals statistically significant deterioration, particularly in terms of the SDE, for all meteorological variables when off-centering is removed from the discretized dynamical equations. The results shown in this figure are obtained with the TC configuration. However, the impact is qualitatively similar with the ML configuration and is also similar for the summer cases. Although the primary purpose of off-centering in GEM and other models is to control spurious resonant response due to orographic forcing, it also adds substantial numerical dissipation. The level of dissipation attributable to off-centering can be determined by comparing the spectral variance of the modelgenerated meteorological fields at different pressure levels for different wavenumbers (or length scales) for different model configurations. Variance spectra for any global meteorological field at a given vertical level can be computed by decomposing the field using spherical harmonics. For any meteorological field from two experiments, the ratio of variance for different horizontal wavenumbers can help in analyzing the important characteristics of the pertinent experiments (Husain et al. 2014). For a discrete horizontal wavenumber n, variance ratio for any meteorological field C canbecomputedas Var C ratio (n) 5 VarC Exp (n), (22) Var C Ctl (n) where the subscripts Ctl and Exp refer to the control and modified experiments, respectively. A variance ratio less (greater) than 1.0 for a given wavenumber thus indicates reduced (increased) variance at the associated length scale with Exp as compared to Ctl. The variance ratio between the 120-h GDPS outputs averaged over the 44 winter cases, with and without off-centering, is presented in Fig. 4 for temperature, kinetic energy, and vertical velocity at two different pressure levels (200 and 700 hpa). The figure shows substantial dissipation for a wide range of length scales resulting from off-centering in the GEM model. This additional dissipation resulting from offcentering leads to significant smoothing in the predicted meteorological fields and is presumably an important contributor to the improved forecast scores shown in Fig. 3. Particularly for the diagnostic vertical velocity, off-centering is found to not only affect the smaller scales (wavenumber. 100) but also the largest scales

10 4136 M O N T H L Y W E A T H E R R E V I E W VOLUME 145 FIG. 3. Impact of off-centering in the discretized dynamical equations on the 120-h 25-km GDPS forecast scores of zonal wind (UU), wind speed (UV), temperature (TT), and geopotential height (GZ) obtained by comparing model outputs against radiosonde observations. The blue and red lines correspond to model forecasts using standard off-centering (blue lines, b M 5 b H 5 0:2) and no off-centering (red lines, b M 5 b H 5 0), respectively. The dashed and solid lines, respectively, indicate bias and standard deviation of error (SDE). The scores are obtained by averaging over 44 Northern Hemisphere winter cases. The red and blue shaded numbers along the left (right) vertical axes within each panel indicate the confidence in percentage in the statistically significant improvements in bias (SDE) for the model configuration associated with each color. Significance for bias and SDE are computed using the t test and F test, respectively. (wavenumber, 10). Although variance in the smaller scales for the vertical velocity is orders of magnitude smaller than the horizontal wind, it can nevertheless have substantial impact on meteorologically important physical processes including convection. As elimination of off-centering leads to substantial deteriorations in the GDPS scores, all the results presented for the full three-dimensional GDPS forecasts hereafter unless it is mentioned otherwise explicitly will include uniform off-centered averaging in the discretized dynamical equations (i.e., b M 5 b H 6¼ 0). Figures 5 and 6 demonstrate the impact of the different trajectory calculation approaches MC, TL, and TC on the predicted temperature and geopotential height for the 120-h GDPS forecasts averaged over the winter and summer cases, respectively. The results clearly show statistically significant improvements in the forecast scores, particularly in SDE, with the TC approach, which utilizes trajectory calculations (19) that are consistent with the discretized dynamical equations for semi-lagrangian advection (13). The figures also reveal that the improvements are primarily due to trapezoidal averaging during winter, whereas during summer cubic interpolation also leads to moderate improvements in SDE of both temperature and geopotential height within the upper troposphere. However, during winter, cubic interpolation (MC) is found to adversely affect the SDE of both temperature and geopotential height. Cubic interpolation is found to result in cooling of the air in the lower troposphere, eventually leading to some deterioration in the geopotential height bias. Within the upper troposphere and stratosphere, the TC

11 OCTOBER 2017 H U S A I N A N D G I R A R D 4137 FIG. 4. Spectral variance ratio [(22)] for 120-h forecasts of (top) temperature (TT), (middle) kinetic energy (KE), and (bottom) diagnostic vertical velocity (WW) between model configurations involving standard off-centering (b M 5 b H 5 0:2) and no off-centering (b M 5 b H 5 0). Results are presented for the average of over 44 winter cases at (left) 200 and (right) 700 hpa. Any value of the ratio less than 1.0 for a given wavenumber indicates lack of variance for the standard off-centering configuration for the corresponding length scale and vice versa. approach is found to improve the bias in both geopotential height and temperature, which is particularly attributable to trapezoidal averaging. It is important to note that any change in bias can often be adequately addressed through some recalibration of the different physical parameterizations, whereas any increase in SDE is a considerably more difficult problem to solve. Therefore, consistent trajectory

12 4138 M O N T H L Y W E A T H E R R E V I E W VOLUME 145 FIG. 5. Comparison of 120-h 25-km GDPS forecast scores for geopotential height (GZ) and temperature (TT) obtained with the different modifications in the trajectory calculation approach. The blue lines in all the figures represent the ML (midpoint averaging and linear interpolation) configuration, whereas the red lines correspond to the (top) MC, (middle) TL, and (bottom) TC approach. For all configurations, standard off-centering is used in the discretized dynamical equations (b M 5 b H 5 0:2). The scores represent an average over 44 Northern Hemisphere winter cases. calculations based on the TC approach can be considered to have a significant net positive impact on the forecast accuracy of both temperature and geopotential height. The forecast scores for the zonal and horizontal wind with respect to the different modifications in trajectory calculation are presented in Figs. 7 and 8 for the same winter and summer cases. The overall

13 OCTOBER 2017 H U S A I N A N D G I R A R D 4139 FIG. 6. As in Fig. 5, but for the summer period. influence of the different approaches for trajectory calculations are qualitatively very similar to those found for temperature and geopotential height. Trapezoidal averaging (TL) is found to reduce SDE of both fields during both seasonal periods, whereas cubic interpolation has an overall negative impact on SDE during the winter period. The cumulative impact of the TC approach is found to result in statistically significant improvements in SDE during both periods with some increase in bias attributable primarily to trapezoidal averaging. Overall, the results presented in Figs. 5 8 reveal that the trajectory calculations based on the TC approach is not only important from the consistency perspective but also

14 4140 M O N T H L Y W E A T H E R R E V I E W VOLUME 145 FIG. 7.AsinFig. 5, but for zonal wind (UU) and wind speed (UV). leads to considerable improvements in forecast accuracy. Variance ratios of the meteorologically important fields for the different modifications in trajectory calculations (MC, TL, and TC) with respect to the original ML configuration can provide some valuable insight. Figures 9 and 10 show the variance ratios for temperature, kinetic energy, and vertical velocity associated with the different variations in the trajectory calculations for the winter and summer periods, respectively. The first thing to note is that the responses of the different trajectory calculations are very similar during both seasonal periods at both pressure levels. Cubic interpolation (MC, blue dots) is found to

15 OCTOBER 2017 H U S A I N A N D G I R A R D 4141 FIG. 8. As in Fig. 7, but for the summer period. add substantial variance in the intermediate (50, wavenumber, 100) to small (wavenumber. 100) length scales. This is also evident when the variance ratio associated with the TC and TL configurations are compared, and is also true during both winter and summer periods. The impact of cubic interpolation in trajectory calculations is thus similar to the increase in small-scale variance it generates when used to compute the advected fields and source terms within the dynamical equations. Conversely, comparing TL to ML we find that trapezoidal averaging leads to considerable dissipation, particularly at the smallest length scales (wavenumber. 500). The arrival and departure points are further apart in both space and

16 4142 M O N T H L Y W E A T H E R R E V I E W VOLUME 145 FIG. 9. Spectral variance ratio for 120-h forecasts of (top) temperature (TT), (middle) kinetic energy (KE), and (bottom) the diagnostic vertical velocity (WW) between trajectory calculations involving the ML approach and the other modifications that include: MC (blue dots), TL (green dots), and TC (red dots). Results are presented for the average of the 44 winter cases at (left) 200 and (right) 700 hpa. time in the case of trapezoidal averaging than with midpoint averaging. As a result, the small-scale smoothing effect of trapezoidal averaging is not unexpected. However, the increase in variance resulting from the use of cubic interpolation is so substantial that an overall increase in small-scale variance is still seen when the TC approach is compared to the ML approach (everywhere except for temperature at 200 hpa). The fact that the TC approach leads to statistically significant improvements in the SDE not

17 OCTOBER 2017 H U S A I N A N D G I R A R D 4143 FIG. 10. As in Fig. 9, but for the summer period. only over TL but over both ML and MC also implies that this additional variance is beneficial from a NWP perspective for the GEM model. During both the winter and the summer periods, cubic interpolation (both MC and TC) is found to increase the variance of vertical velocity substantially at the smaller scales (wavenumber. 100). Even the largest scales are found to be somewhat affected. This is contrary to the tremendous dissipating impact of off-centering on vertical velocity seen in Fig. 4, and may have important consequences for many subgrid-scale physical processes including parameterized convection. Although off-centering in the discretized dynamical equations leads to substantial improvements in

18 4144 M O N T H L Y W E A T H E R R E V I E W VOLUME 145 FIG. 11. Simulated vertical velocity for the Schär mountain case as a function of different levels of uniform offcentering (i.e., b M 5 b H 5 b NH 5 b) used in the discretized dynamical equations: (a) b 5 0.2, (b) b 5 0.1, (c) b5 0.04, and (d) b The TC approach is used for the trajectory calculations without any off-centering applied to approximate the velocity integral in (16) (i.e., b V 5 b z 5 0). All other conditions are as in Fig. 2. the forecast scores, it adds a potential source of inconsistency when a similar approach is not adopted for averaging the velocity integral along the trajectory to compute the departure positions. For the full three-dimensional simulations, as in the case of the GDPS, the significant dissipation resulting from off-centered averaging may mask any downside associated with this possible inconsistency. Therefore, it is useful to systematically investigate the impact of off-centering in the context of the theoretical Schär mountain problem. The results for this theoretical case with off-centering in the discretized dynamical equations and the TC approach for trajectories are presented in Fig. 11. The figure clearly shows that the introduction of standard off-centering (b 5 0:2, Fig. 11a) leads to the reemergence of the distortions in the large-scale mountain waves that disappeared with the TC approach in the absence of off-centering. Although the amplitude of the distortions is reduced with reduced off-centering, the results indicate a possible inconsistency between the discretization for semi-lagrangian advection and trajectory calculations in the presence of off-centering. Staniforth et al. (2003) extended the work of White (2003) to include offcentering in the dynamical equations and proposed standard centered trapezoidal averaging for the trajectory calculations (Option 1A in their paper) as a possible approach that preserves dynamical equivalence. However, the results presented in Fig. 11 show that this approach fails to produce distortion-free mountain waves for this theoretical test case. For full three-dimensional problems, as revealed in the GDPS forecast scores (Figs. 5 8), utilization of this approach nevertheless leads to statistically significant improvements in the different meteorological fields, presumably due to the large dissipation effect of off-centered averaging in the dynamical equations. Figure 12 shows the impact of differential offcentering. As can be seen in this figure, the distortions remain in the model solution even when off-centering is removed from the discretized momentum equations (Fig. 12a), but disappear when off-centering is only applied to the vertical momentum equation (Fig. 12b). This implies that irrespective of the

19 OCTOBER 2017 H U S A I N A N D G I R A R D 4145 FIG. 12. The impact of differential off-centering on the simulated vertical velocity for the Schär mountain case: (a) no off-centering in the momentum equations (b M 5 0, b H 5 b NH 5 0:2), and (b) off-centering applied only to the vertical momentum equation (b M 5 b H 5 0, b NH 5 0:2). All other conditions are as in Fig. 2. impact of the damping of the nonhydrostatic vertical velocity on the model solutions, any nonzero value of b NH does not introduce any numerical inconsistency between the discretized dynamical and trajectory equations, whereas any nonzero value of either b M or b H makes the discretization inconsistent when the discretized trajectory calculations are based on standard centered averaging (i.e., when b V 5 b z 5 0). In the presence of uniform off-centering in the discretized dynamical equations (b M 5 b H 5 b NH 5 b, where b 6¼ 0), numerical consistency may be achieved by simply using the same value for off-centering in the discretized trajectory equations given by (20) (i.e., b V 5 b z 5 b). Results presented in Fig. 13 confirm this hypothesis. The figure shows that the use of the same level of off-centering in the trajectory calculations leads to distortion-free mountain waves. However, as expected, the solution with larger off-centering (Fig. 13a, b 5 0:2) shows considerable large-scale smoothing when compared to solution with reduced off-centering (Fig. 13d, b 5 0:02). It is important to note that the smoothing effect of b NH is negligible (see Fig. 12b), and, therefore, most of the dissipation results from b M and b H. Staniforth et al. (2003) have proposed x A 2 x D 5 j(bv A 1 av D ), (23) Dt as another form of the discretized trajectory equations (Option 1 in their paper) that preserves dynamical equivalence. In (23), x is the position vector, V stands for the velocity vector, j is a scalar defined as j 5 (av A 1 bv D ) 2 /(bv A 1 av D ) 2, and a and b are off-centering parameters with a 5 (1 1 b)/2 and b 5 (1 2 b)/2. Unlike (20), the time weights denoted by a and b are in opposite order in (23). This implies that when off-centering makes the dynamical equations more time implicit, the corresponding off-centered averaging in the trajectory calculations should be made more time explicit in order to achieve dynamical equivalence. If the Courant number smaller than 1, then for smaller values of b, the scalar j can be assumed to approach unity. From numerical experiments for such a scenario (i.e., when j 1), trajectory calculations based on (23) are found to generate considerable distortions in the simulated mountain waves even for smaller values of b (not shown). Numerical experiments were later carried out to ascertain the appropriate values of b V and b z with respect to the values of b M and b H. Figure 14 shows the test results for a case where b NH. b H. b M. The outcome of these experiments shows that large-scale distortions from the simulated mountain waves are only removed when b V and b z are replaced by b M and b H, respectively. Based on the test results presented in Fig. 14, the consistent discretizations of the trajectory equations, when differential off-centering in GEM is taken into account, will have the following form: r A 2 r D Dt z A 2 z D Dt bm bh 2 V A H b MV D H, (24a) 2 _z A b H _z. D (24b) 2 As off-centering reduces the order of truncation error, its implementation in the trajectory calculations although being more numerically consistent can have adverse impacts on the solution accuracy. Figures 15