Embed Size (px)
Citation preview
Powered by Editorial Manager® and ProduXion Manager® from Aries Systems Corporation
Generated using version 3.2 of the official AMS LATEX template
Testing of a cell-integrated semi-Lagrangian semi-implicit1
nonhydrostatic atmospheric solver (CSLAM-NH) with idealized2
orography3
May Wong∗
University of British Columbia, Vancouver, British Columbia, Canada
4
William C. Skamarock, Peter H. Lauritzen, Joseph B. Klemp
National Center for Atmospheric Research†, Boulder, Colorado, USA
5
Roland B. Stull
University of British Columbia, Vancouver, British Columbia, Canada
6
∗Corresponding author address: May Wong, University of British Columbia, Room 2020 – 2207 Main
Mall, Vancouver, BC Canada V6T 1Z4.
E-mail: [email protected]†The National Center for Atmospheric Research is sponsored by the National Science Foundation.
1
ABSTRACT7
A recently developed cell-integrated semi-Lagrangian (CISL) semi-implicit nonhydrostatic8
atmospheric solver (CSLAM-NH) for the fully compressible moist Euler equations in two-9
dimensional Cartesian geometry is extended to include orographic influences. With the10
introduction of a new semi-implicit CISL discretization of the continuity equation, the non-11
hydrostatic solver has been shown to ensure inherently conservative and numerically consis-12
tent transport of air mass and other scalar variables, such as moisture and passive tracers.13
The extended CSLAM-NH presented here includes two main modifications: transformation14
of the equation set to a terrain-following height coordinate to incorporate orography and15
an iterative centered-implicit time-stepping scheme to enhance the stability of the scheme16
associated with gravity wave propagation at large time steps. CSLAM-NH is tested for a17
suite of idealized 2D flows, including linear mountain waves (dry), a downslope windstorm18
(dry), and orographic cloud formation.19
1
1. Introduction20
Semi-Lagrangian semi-implicit (SLSI) schemes have been widely used in climate and nu-21
merical weather prediction (NWP) models since the pioneering work of Robert (1981) and22
Robert et al. (1985). The fully compressible nonhydrostatic equations permit fast-moving23
waves which limit the model time step size. The combination of a semi-Lagrangian advec-24
tion scheme with semi-implicit treatment of these waves allows for larger stable time steps,25
and therefore, increased computational efficiency. Conservative semi-Lagrangian advection26
schemes, also known as cell-integrated semi-Lagrangian (CISL) transport schemes, are finite-27
volume methods that inherently conserve mass by tracking individual grid cells each time28
step (Rancic 1992; Laprise and Plante 1995; Machenhauer and Olk 1997; Zerroukat et al.29
2002; Nair and Machenhauer 2002; Lauritzen et al. 2010). CISL transport schemes allow for30
locally (and thus globally) conservative transport of total fluid mass (such as dry air in the31
atmosphere) and constituent (i.e., moisture and tracer) mass.32
However, some CISL schemes lack consistency between the numerical representation of33
the total dry air mass conservation, which we will refer to as the continuity equation, and34
constituent mass conservation equations (Jockel et al. 2001; Zhang et al. 2008; Wong et al.35
2013). Numerical consistency in the discrete tracer conservation equation requires the equa-36
tion for a constant tracer field to correspond numerically to the discrete mass continuity37
equation; this consistency ensures that an initially spatially uniform passive tracer field will38
remain so.39
To allow for large advection time steps, Lauritzen et al. (2010) developed a CISL transport40
scheme called the CSLAM (‘Conservative Semi-LAgrangian Multi-tracer’) transport scheme.41
The CSLAM scheme has recently been implemented in the National Center for Atmospheric42
Research (NCAR) High-Order Methods Modeling Environment (HOMME) and was found43
to be an efficient and highly scalable transport scheme for atmospheric tracers (Erath et al.44
2012). To ensure consistent numerical representations of the continuity equation and other45
scalar conservation equations, Wong et al. (2014) proposed a new discretization of the semi-46
2
implicit CISL continuity equation using CSLAM. They showed that the new formulation47
can be straightforwardly extended to the scalar conservation equations in a fully consistent48
manner. Wong et al. (2013) also showed that any discrepancy between the numerical schemes49
can lead to spurious generation or removal of scalar mass. We refer to this nonhydrostatic50
atmospheric solver with conservative and consistent transport as CSLAM-NH.51
Idealized 2D benchmark test cases for a density current, gravity wave, as well as a squall52
line, using CSLAM-NH have been performed in Wong et al. (2014). These test cases used53
flat bottom boundary conditions for simplicity. In the real atmosphere, the bottom fluid54
boundary is often not flat. Mountains act as a stationary forcing and generate horizontally55
and vertically propagating internal gravity waves in the atmosphere. Under certain atmo-56
spheric conditions they can also induce highly nonlinear flows such as wave amplification and57
breaking. Numerical simulations of these mountain waves have been extensively studied by58
many (e.g. Klemp and Lilly 1978; Peltier and Clark 1982; Durran and Klemp 1983; Durran59
1986) and several of these cases have become benchmark tests in model development and60
intercomparison studies (e.g. Pinty et al. 1995; Bonaventura 2000; Xue et al. 2000; Doyle61
et al. 2000; Melvin et al. 2010). To further develop CSLAM-NH as a viable nonhydrostatic62
atmospheric solver, we have incorporated orography into the model and have conducted a63
suite of these mountain-wave cases documented in the literature. The test suite includes64
linear hydrostatic and nonhydrostatic dry mountain waves, a highly nonlinear dry mountain65
wave with amplification and overturning of the waves, and a moist mountain flow with cloud66
and rain formation.67
The paper is organized as follows. A model description of CSLAM-NH is given in section68
2. In section 3, simulations from the suite of idealized mountain wave tests are presented.69
Finally, a summary is given in section 4.70
3
2. Model description71
a. Governing equations72
The major modification to the model prognostic equations described in Wong et al. (2014)73
is the transformation of the vertical coordinate from geometric height to a terrain-following74
height coordinate. In addition to this modification, we have also included the treatment of75
the gravity-wave terms in the implicit solver. The previous version of CSLAM-NH solves76
the buoyancy terms in the vertical momentum equation explicitly using a two time-level77
extrapolation scheme. For a gravity-wave test originally proposed in Skamarock and Klemp78
(1994), the time-step limit was found to be restricted by the explicit treatment of these79
buoyancy terms (Wong et al. 2014). To circumvent this time-step restriction, an iterative80
approach is used to include these terms in the implicit solve. We will focus on the description81
of these two modifications and provide a basic description of the solver [readers are referred82
to Wong et al. (2014) for a more detailed description].83
A height-based coordinate is used to avoid the complication of a time-varying vertical84
coordinate system, as is the case with mass (pressure) coordinates. The use of terrain-85
following coordinates substantially simplifies the bottom boundary condition when topogra-86
phy is present. For cell-integrated semi-Lagrangian advection, in a geometric height coordi-87
nate, approximated departure cell boundaries may intersect the orography and create more88
complex cell configurations (e.g. more cell edges/vertices, which complicate the subgrid-cell89
reconstruction). On a computational grid defined by terrain-following vertical coordinates,90
however, the lowest cell boundaries will always remain at the surface.91
Following Gal-Chen and Somerville (1975), the 2D terrain-following height coordinate ζ92
is expressed using the linear transformation93
ζ = ztz − h(x)
zt − h(x),
where z(x, ζ) is the physical height, h(x) is the terrain profile, and zt is the height of the94
model top [with the bottom defined as z = h(x)].95
4
The 2D governing equations expressed in (x, ζ)-coordinates are:96
∂u
∂t+
u∂u
∂x
ζ
+ ω∂u
∂ζ= − 1
ρmγRdπ
∂Θ
m
∂x
ζ
+∂(ζxΘ
m)
∂ζ
+ Fu, (1)
97
∂w
∂t+
u∂w
∂x
ζ
+ ω∂w
∂ζ= − 1
ρm
γRdπ
∂(ζzΘm)
∂ζ− gρd
π
π+ gρm
+ Fw, (2)
98
∂Θm
∂t+ (∇ · vΘm)ζ = FΘ, (3)
99
∂ρd∂t
+ (∇ · vρd)ζ = 0, (4)100
∂Qj
∂t+ (∇ · vQj)ζ = FQj (5)
101
p = p0RdΘm
p0
γ, (6)
where v = (u,w) is the horizontal and vertical wind components, π = (p/p0)κ is the Exner102
function, p0 = 100 kPa is the reference pressure, Rd = 287 J kg−1 K−1 is the gas constant for103
dry air, cp = 1003 J kg−1 K−1 is the specific heat for dry air at constant pressure, cv = 717104
J kg−1 K−1 is the specific heat of dry air at constant volume, and the ratios κ = Rd/cp ≈105
0.286 and γ = cp/cv ≈ 1.4. Perturbation variables from a time-independent hydrostatically106
balanced background state are used to reduce numerical errors in the calculations of the107
pressure-gradient terms (Klemp et al. 2007). The hydrostatically balanced background state108
is defined as dp(z)/dz = −ρd(z)g. Flux-form variables are coupled to a scaled dry density109
adjusted to the transformed coordinate, ρd = ρd/ζz, i.e. Θm = ρdθm and Qj = ρdqj. The110
notations (ζx, ζz) refer to the spatial derivatives of ζ. Perturbation variables (primed) are111
defined via Θm = ρd(z)θ(z) + Θm, π = π + π, ρd = ρd(z) + ρd, and the moist density112
ρm = ρd(1 + qv + qc + qr), where qv, qc, and qr are the mixing ratios for water vapor,113
cloud, and rainwater, respectively. The modified potential temperature θm is defined as114
θm = θ(1 + aqv) where a ≡ Rv/Rd 1.61.115
Notations (·)ζ denote evaluation at constant ζ, and (∇ · vb)ζ = δx(ub) + δζ(ωb) for116
any scalar variable b. The variable ω = dζ/dt is the vertical motion perpendicular to the117
coordinate surface. For simplicity, we assume a non-rotating atmosphere. The terms Fu and118
5
Fw represent diffusion, and FΘ and FQj represent diffusion as well as any diabatic source119
terms from parameterized physics.120
The governing equations used in CSLAM-NH include the advective form of the mo-121
mentum equations [(1), (2)] so that we can use a traditional semi-Lagrangian discretization.122
The flux-form advection for potential temperature, density, and moisture/passive scalar vari-123
ables [(3), (4), (5), respectively] are solved using the conservative semi-Lagrangian scheme124
CSLAM. Pressure is a diagnostic variable given by the equation of state (6). Following125
Klemp et al. (2007), the pressure-gradient terms are written in terms of potential tempera-126
ture. The recasting allows for coupling of the implicit pressure-gradient terms with the flux127
divergence term in the potential temperature equation. The compressible nonhydrostatic128
equation set is still exact and no approximations have been applied.129
b. CSLAM — a cell-integrated semi-Lagrangian transport scheme130
When advection terms are evaluated using an Eulerian scheme, the model time step sizes131
are restricted by the well-known Courant stability condition. To allow for larger advective132
time steps, the nonhydrostatic solver uses a cell-integrated semi-Lagrangian (CISL) transport133
scheme called the CSLAM (Conservative Semi-LAgrangian Multi-tracer) transport scheme134
developed by Lauritzen et al. (2010). This inherently conservative (both locally and globally)135
transport scheme is used to solve the continuity and potential temperature equations, and136
for transport of any moist species or other tracers.137
The stability criterion for the CSLAM transport scheme is limited by the trajectory138
approximations of the grid-cell vertices. To ensure stability in traditional semi-Lagrangian139
schemes, the Lipschitz stability condition requires that, in 1D, no trajectories in the space-140
time domain should intersect one another (Smolarkiewicz and Pudykiewicz 1992). In the141
CSLAM scheme, the stability condition is slightly more lenient in that the trajectories of142
neighbouring vertices may cross, as long as the discrete departure cells remain non-self-143
intersecting. In all test cases presented here, linear trajectories as described in Wong et al.144
6
(2014) are assumed. Figure 1a shows a discrete arrival grid cell (white box) originating145
from a non-self-intersecting discrete departure cell (grey box) with straight edges that are146
computed using the approximated displacement over one time step (arrows). The trajectories147
from the ends of the left cell edge intersect, but as long as the departure cells remain non-self-148
intersecting, the scheme is stable and ensures global mass conservation. In Fig. 1b, a more149
distorted flow causes the departure cell to self-intersect. This ‘twisting’ of the departure150
cell causes adjacent departure cells to overlap. In such a case, the scheme is no longer151
mass-conserving and becomes unstable. The stability and accuracy of the CSLAM scheme152
in highly deformed flows may be improved by using higher-order trajectory approximations153
and/or higher-order approximations of departure cell boundaries. One such example is to use154
the parabolic (curved) departure cell edges that account for acceleration in the trajectory155
approximations developed by Ullrich et al. (2012). In the present study, we did not test156
any geometrical definitions other than quadrilateral departure cells, but the option could be157
explored in the future.158
c. Discretized momentum equations159
The momentum equations are solved in a traditional semi-Lagrangian semi-implicit man-160
ner, where the total derivatives du/dt and dw/dt are computed using a grid-point interpo-161
lation to the departure point. Bicubic Lagrange interpolation is used for all departure point162
evaluations. The two time-level discretizations of the momentum equations are163
un+1A = un
D +∆t(Fu)nD (7)
− ∆t
2
γRd
π
ρm
x(δxΘ
m)ζ + δζ(ζxΘ
mxζ)n
D
− ∆t
2
γRd
πn
ρnm
x(δxΘ
m)ζ + δζ(ζxΘ
mxζ)n+1
A
,
7
and164
(1 + µ∆t)wn+1A = wn
D +∆t(Fw)nD (8)
− ∆t
2
γRd
π
ρm
ζδζ(ζzΘ
m)
− 1
ρm
gρd
π
π− gρm
ζn
D
− ∆t
2
γRd
πn
ρnm
ζδζ(ζzΘ
m)
n+1
A− 1
ρnm
gρd
π
π− gρm
n+1
A
ζ,
where subscripts D and A denote evaluation at the departure and arrival grid points, re-165
spectively, and superscripts denote the time level. To reduce gravity wave reflection at the166
upper boundary, a Rayleigh damping term −µw is added to the vertical momentum equa-167
tion, where the damping coefficient µ is a function of height z and applied in the top layers168
of the domain. The spatial averaging operators are defined as169
(·)x =1
2
(·)i,k + (·)i+1,k
, and
170
(·)ζ = 1
2
(·)i,k + (·)i,k+1
,
and gradient operators as171
δx(·) =(·)i+1,k − (·)i,k
∆x, and
172
δζ(·) =(·)i,k+1 − (·)i,k
∆ζ.
The prognostic variable for vertical motion perpendicular to the terrain-following vertical173
coordinate ζ is174
ωn+1 =ζxun+1 + ζzw
n+1.
We use the following notations to combine the known rhs terms in the momentum equa-175
tions:176
RU ≡ unD +∆t(Fu)
nD
− ∆t
2
γRd
π
ρm
x(δxΘ
m)ζ + δζ(ζxΘ
mxζ)n
D
,
8
177
RW ≡ wnD +∆t(Fw)
nD
− ∆t
2
γRd
π
ρm
ζδζ(ζzΘ
m)
− 1
ρm
gρd
π
π− gρm
ζn
D
,
the implicit pressure-gradient terms:178
PGU = −∆t
2
γRd
πn
ρnm
x(δxΘ
m)ζ + δζ(ζxΘ
mxζ)n+1
A
,
179
PGW = −∆t
2
γRd
πn
ρnm
ζδζ(ζzΘ
m)
n+1
A
,
and the implicit half of the buoyancy term:180
BW =∆t
2
1
ρnm
gρd
π
π− gρm
n+1
A
ζ
. (9)
The subscripts in the notations denote the momentum equations to which the terms belong.181
Using (7), (8), and the notations above, the vertical momentum equation can be rewritten182
as:183
ωn+1 = ζxRU + ζz(1 + µ∆t)−1(RW +BW ) RΩ
+ζxPGU + ζz(1 + µ∆t)−1PGW . (10)
The notation RΩ is used here to represent all the known terms plus the implicit buoyancy184
term (9).185
Often, off-centering of the time-averaged terms is needed in semi-Lagrangian semi-implicit186
time stepping schemes to help eliminate computational noise, especially when orographic187
forcing is present and at large Courant numbers (e.g. Rivest et al. 1994). In CSLAM-NH,188
no off-centering was needed to attain the numerical stability in the solver for the test cases189
presented here.190
9
d. Conservative and consistent flux-form equations191
As noted by Lauritzen et al. (2006) and demonstrated in Wong et al. (2013) and Wong192
et al. (2014), when a numerical scheme different from the one used to evaluate the continuity193
equation is used to transport scalar variables, conservation of scalar mass will no longer be194
guaranteed, despite the use of a conservative transport scheme. The problem of numerical195
consistency in cell-integrated semi-Lagrangian schemes is resolved through the use of a new196
flux-form CISL continuity equation introduced in Wong et al. (2013) for the shallow-water197
equations and tested for a 2D nonhydrostatic atmosphere without topography (Wong et al.198
2014). The new flux-form CISL continuity equation allows for a straightforward implementa-199
tion of a CISL scalar transport scheme that ensures numerical consistency. Here, we further200
test the proposed formulation based on the CSLAM transport scheme for 2D idealized cases201
over mountains.202
The potential temperature, continuity, and scalar-mass conservation equations are all203
solved consistently using the same numerical scheme presented in Wong et al. (2014):204
Θn+1m = Θn+1
m,exp +∆t
2
∇eul · (vnΘn
m
δA∗
∆A(11a)
+∆tF nΘm
δA∗
∆A
and205
Θn+1m = Θn+1
m − ∆t
2
∇eul · (vn+1Θn+1
m ). (11b)
The flux divergence in terms of a corrective velocity v in the semi-implicit correction206
term is defined as:207
∇eul · (Θmv) =
1
∆x[Θ
xm(ur − Fr/∆ζ)−Θ
xm(ul − Fl/∆ζ)]
+1
∆ζ[Θ
ζm(ωt − Fl/∆x)−Θ
zm(ωb − F/∆x)],
where F = F(u,ω) are Lagrangian flux areas, computed as in Wong et al. (2014). The208
velocities ur, ul, ωt, and ωb are staggered velocities at the cell faces (Arakawa C grid).209
10
In the semi-implicit flux-form equation, instead of linearizing around a mean reference210
state, we utilize Θn+1m using the CSLAM transport scheme to ensure consistency of the211
semi-implicit correction term among all the scalar flux-form equations. Included in this212
CSLAM computation are all the terms to be integrated over the departure cell: the explicit213
conservative CSLAM solution (Θn+1m,exp), a predictor-corrector term (the flux divergence term214
at time level n), explicit diffusion, and diabatic tendency (the latter two are combined in215
FΘm). The diabatic tendencies are approximated using values at the previous time level.216
The resulting approximation Θn+1m (11a) is then used in (11b). The solution from (11b)217
is the solution from the dynamics, and prior to any adjustment to saturation by a moist218
microphysics scheme.219
Consistent formulations of the continuity equation and scalar mass conservation equations220
are straightforwardly discretized as:221
φn+1 = φn+1exp +
∆t
2
∇eul · (vnφn
δA∗
∆A(12a)
+∆tF nφ
δA∗
∆A,
and222
φn+1 = φn+1 − ∆t
2
∇eul · (vn+1φn+1)
, (12b)
where φ = ρd or Qj. Similar to (11a), (12a) combines the advected quantities in the explicit223
solution using the CSLAM scheme, the predictor-corrector flux divergence term, diffusion,224
and diabatic tendencies from the previous time level. The solution of the velocities at the225
new time level are used to compute the flux divergence in (12b).226
11
e. Helmholtz equation227
By eliminating vn+1 in the potential temperature equation (11b) using the horizontal228
and vertical momentum equations, we form the Helmholtz equation:229
Θn+1m +
∆t
2
δx(PGUΘn+1
m
x) + δζ
(ζxPGU
xζ+ ζz(1 + µ∆t)−1PGW )Θn+1
m
ζ=
Θn+1m − ∆t
2
δx(RUΘn+1
m
x) + δζ(RΩΘn+1
m
ζ), (13)
where PGU and PGW are, as defined earlier, the implicit pressure-gradient terms expressed230
as functions of Θn+1m . All the terms on the rhs of (13) are pre-computed at the beginning231
of each time step, and the implicit buoyancy term in RΩ is updated at each iteration of the232
Helmholtz equation solver (described next).233
f. Iterative centered-implicit time stepping scheme234
The compressible Euler equations permit fast horizontally- and vertically-propagating235
acoustic and gravity waves. To alleviate the time-step limit due to acoustic waves, in the236
previous version of CSLAM-NH (Wong et al. 2014), an implicit time stepping scheme was237
used to solve the pressure-gradient and mass-divergence terms. The remaining buoyancy238
terms were evaluated explicitly using a two time-level extrapolation scheme. The semi-239
implicit time integration scheme allowed the use of time steps much larger than those allowed240
in a classical explicit scheme, which would otherwise have been restricted by the speed of241
sound. The buoyancy terms responsible for gravity waves, however, imposed a restriction to242
the maximum stable time step.243
Instead of evaluating the gravity wave terms explicitly using time extrapolation, we use an244
iterative approach for a more accurate and implicit treatment of these terms. The solution245
procedure can be summarized in two main components as follows. First, the departure246
cell areas are approximated using backward trajectories from the arrival grid cell vertices.247
The forcing terms (RU , RΩ) and the explicit departure cell-averaged potential temperature,248
12
Θn+1m (using 11a) are evaluated and form the RHS of the Helmholtz equation (13). The249
implicit buoyancy term BW in RΩ is evaluated at time level n as an initial estimate. The250
explicit departure cell-averaged density ρn+1d (using 12a) is also pre-computed. The second251
component involves solving the linear Helmholtz equation for Θn+1m ; here we use a conjugate252
gradient residual solver (Skamarock et al. 1997). The solution Θn+1m is then back-substituted253
into the momentum equations (7) and (10) to get un+1 and ωn+1, respectively. Finally, the254
implicit buoyancy term BW (9) in RΩ is updated using (i) ρn+1m by evaluating (12b) and (ii)255
πn+1 directly from Θn+1m using the equation of state π ≡ (RdΘm/p0)R/cv . At the end of the256
second component, the trajectories and forcing terms (first component of the procedure) are257
recomputed using the latest solution of un+1 and ωn+1.258
Depending on the test case, two to four iterations of each component are performed. For259
the nonlinear flow tests, iterating more than twice did not further improve the maximum260
stable time step size. For the linear cases, the maximum time step can be further increased261
by performing more iterations (iterating more than four times does not further improve262
stability). At each iteration, the Helmholtz solver converges progressively faster (since the263
latest estimate of Θn+1m is used as the starting point). The iterative scheme is used for264
advancing the dry dynamics; after which, tracers are advected using (12b) and the moist265
physics are called (once at each time step).266
The use of an iterative centered-implicit scheme is found to substantially increase the267
stable time step size in CSLAM-NH at the expense of solving the Helmholtz equation more268
than once per time step. To demonstrate this behaviour, we conduct the gravity-wave test269
originally proposed in Skamarock and Klemp (1994), using CSLAM-NH as was done in Wong270
et al. (2014) with a grid spacing of ∆x = ∆z = 1 km and an imposed mean wind U = 20 m271
s−1. Wong et al. (2014) used an explicit treatment of the buoyancy terms and found that the272
maximum stable time step was restricted to ∆t = 38 s [at a nominal Courant number (Cr)273
of 0.76]. In the current version of the iterative centered-implicit CSLAM-NH, we have found274
that for the same simulation, the maximum stable time step increased to 100 s, roughly by275
13
a factor of 2.6 (Cr = 2). For comparison, the maximum stable time step for an Eulerian276
split-explicit third-order Runge-Kutta time stepping scheme was 60 s (Cr = 1.2) (Wong et al.277
2014).278
Similar iterative approaches were found to improve numerical stability in other semi-279
Lagrangian solvers. In the Canadian Global Environmental Multiscale (GEM) model, Cote280
et al. (1998) discretize the governing equations in a fully implicit manner and use an it-281
erative procedure to avoid solving a nonlinear Helmholtz equation. This procedure is also282
implemented in Melvin et al. (2010) for the vertical-slice nonhydrostatic solver using the283
‘Semi-Lagrangian Inherently Conserving and Efficient’ (SLICE) transport scheme. An alter-284
native predictor-corrector (thus, also iterative) approach was tested in the European Centre285
for Medium-Range Weather Forecasts (ECMWF) Integrated Forecast System (IFS) model286
by Cullen (2001). In that study, a positive improvement in accuracy was noticeable only287
when the advective velocities, in addition to the buoyancy terms, were iterated. Using288
an idealized analysis of acoustic modes in a 1D nonhydrostatic vertical column, Cordero289
et al. (2005) demonstrated the impact of using time-extrapolated and -interpolated trajec-290
tory computations on the numerical stability of a semi-Lagrangian centered-semi-implicit291
scheme. When extrapolation and large time steps were used for the trajectories, the vertical292
structure of the acoustic modes were found to be distorted (with spurious zeros forming293
with time). The time interpolation scheme on the other hand was found to be stable in all294
cases. The idealized analysis by Cordero et al. (2005) supports the findings in Cullen (2001)295
and the method used in Cote et al. (1998), with a recommendation for time-interpolated296
trajectory computations (e.g. by repeating the first component of the CSLAM-NH solution297
procedure).298
The disadvantage of this approach is that the linear Helmholtz equation for potential299
temperature Θm is solved a number of times with the buoyancy terms updated at the end of300
each iteration. However, the increased stability will allow a larger time step to be used and301
can help offset the added computational expense of solving the dry dynamics (calculated302
14
once at each time step). After the dry dynamics, the solver then advects passive tracers303
(once at each time step). With a larger time-step size, the total number of times tracers304
are advected during the entire simulation is reduced. Therefore, the overall execution time305
spent on scalar transport is also reduced. This reduction may have a significant impact on306
computational time, especially when the number of tracers used in chemistry applications is307
large.308
g. Boundary conditions309
Periodic-in-x and free-slip top and bottom boundary conditions are applied in all our310
tests. The vertical velocities at the top and bottom boundaries are set to ω = 0, and ensures311
no normal flux through them. The boundary conditions are implemented by extrapolating312
Θm, ρd, and u into the boundary.313
h. Implicit Rayleigh damping314
To prevent the reflection of vertically-propagating gravity waves along the rigid model315
top, a damping term, −µw, is added in the vertical momentum equation based on the316
scheme proposed in Klemp et al. (2008) and implemented in Melvin et al. (2010). The317
damping profile µ(z) proposed by Klemp and Lilly (1978) is used:318
µ(z) =
µmax sin2π2
z−zdzt−zd
if z > zd,
0 if z ≤ zd.
The profile is characterized by a gradual increase of viscosity with height, which is desirable319
to prevent any reflections that would otherwise occur from a sharp increase in viscosity. The320
damping layer starts from a user-specified height zd and extends to the top of the domain321
zt. The values of µmax are specified for each test case. Klemp et al. (2008) analyzed the322
reflection properties of this implicit Rayleigh damping layer. The scheme they proposed323
is slightly different from the one applied here; in particular, they proposed implementing324
15
the damping term as an adjustment step. The adjustment step approach includes an extra325
damping term that ressembles vertical diffusion, in addition to the effect of a damping326
term −µw added directly in the vertical momentum equation. For smaller (nonhydrostatic)327
horizontal scales, however, the effect of the damping term dominates and there is little328
difference between the two approaches. Klemp et al. (2008) also provided some guidance in329
selecting a suitable µmax value based on the reflection properties of the scheme. Based on330
their experimentation, smaller damping coefficients µmax should be used for smaller dominant331
horizontal wavelengths.332
3. Idealized test cases: results333
a. Linear mountain waves over bell-shaped mountain334
To test the response of the nonhydrostatic solver to orographic forcing, two adiabatic335
linear mountain-wave simulations are conducted first. Both cases assume a simple hill profile336
h(x) of a witch-of-Agnesi curve, defined as337
h(x) =hma2
x2 + a2,
with a small amplitude hm = 1 m but different half-widths, a. Gravity waves generated338
by flow moving over a wide hill under conditions where U/Na 1 are approximately339
hydrostatic and are vertically propagating (Smith 1979). We simulate flow with a constant340
upstream wind speed U = 20 m s−1, in an atmosphere initially in hydrostatic balance and341
isothermal at T = 250 K (equivalent to a stratification of N2 ≈ 3.83 × 10−4 s−2, where342
N2 = g2/cpT for an isothermal atmosphere). The mountain half-width is set at a = 10 km343
to give U/Na = 0.1, such that nonhydrostatic effects are small for this broad low hill. The344
physical domain is 120 km wide and 25 km deep. The simulation is run for a nondimensional345
time Ut/a = 30 (equivalent to t = 15000 s) to ensure the solution has reached steady state.346
16
The numerical domain has dimensions 120 × 100 (∆x = 1 km and ∆z = 250 m). A347
Rayleigh damping layer (µmax = 0.1 s−1) is implemented in the top 10 km of the domain348
(approximately 1.5 times the vertical wavelength, λz = 2πU/N = 6.4 km).349
Results from simulations using a small Courant number Cr = 0.2 and large Cr = 1.5350
are shown in Fig. 2. An upstream tilt of the phase lines is observed, corresponding to351
energy originating from the ground (the mountain) and propagating upwards. As expected,352
the amplitude of the vertical velocity also increases with height (∝ ρ−1/2), corresponding353
to the effect of wave amplification due to decreasing density at higher altitudes. The slight354
downstream tilt of the wave pattern with height is due to weak nonhydrostatic influences355
and is also observed in other nonhydrostatic models for the same test case (e.g. Melvin et al.356
2010).357
For a narrow mountain (U/Na ≈ 1), the mountain waves are strongly nonhydrostatic.358
These waves are highly dispersive, with shorter horizontal scales propagating farther down-359
stream with height, and scales less than 2πU/N becoming evanescent. To simulate such a360
flow, the half-width a of the mountain is reduced to 1 km, and the initial background state361
has a constant stratification of N2 = 1 × 10−4 s−2. The impinging flow has an upstream362
wind speed of U = 10 m s−1 (U/Na = 1). The domain is 144 km wide and 25 km deep. The363
numerical domain has dimensions 360 × 100 grid cells (∆x = 400 m and ∆z = 250 m). Even364
though the mountain is only marginally resolved at this model resolution, the configuration365
is kept for comparison purposes with Melvin et al. (2010). The Rayleigh damping layer is366
applied to the top 13 km of the domain (twice the length of λz) with µmax = 0.05 s−1.367
Results from CSLAM-NH are shown in Fig. 3 for two different time step sizes (Cr =368
0.125 and Cr = 1.5). As expected, far away and downstream of the mountain, the solutions369
exhibit a more pronounced downstream tilt of the phase from the nonhydrostatic component370
of the waves, and the waves also decay with height (note: only part of the domain is shown371
in the figure). The results compare well with the solutions presented in the literature, e.g.372
Xue et al. (2000) and Melvin et al. (2010).373
17
b. Downslope windstorm374
To test the nonhydrostatic solver in a highly nonlinear flow, a simulation of the famous375
downslope windstorm that occurred on 11 January 1972 in Boulder, Colorado (Lilly 1978) is376
conducted. Strong surface winds, gusting to 55 m s−1, were observed in Boulder, Colorado377
on that day. The windstorm has been a long-standing case for theory development and378
numerical model verification e.g. Klemp and Lilly (1978); Peltier and Clark (1979); Durran379
(1986). More recently, Doyle et al. (2000) carried out a model intercomparison study of 11380
different high-resolution models to assess their ability in numerically simulating the wave381
breaking process of this windstorm. Prior to Doyle et al. (2000), smoothed soundings were382
used to initialize the models; here, we use the same 12 UTC 11 January 1972 Grand Junc-383
tion, Colorado sounding as in Doyle et al. (2000), where they showed that a more realistic384
simulation of the windstorm was generated.385
The numerical set-up is based on Doyle et al. (2000). The mountain half-width is 10386
km with a height of 2 km. The domain is 240 km wide and 25 km deep. The numerical387
domain dimensions are 240 × 125 grid cells (∆x = 1 km and ∆z = 200 m). The time388
step sizes used in Doyle et al. (2000) ranged from 1 s to 12 s (the largest time step size389
of 12 s was run using DK83 that uses a time-splitting scheme with a small time step of 3390
s). To compare the results with these models, a similar time step size of 10 s is used in391
CSLAM-NH. A Rayleigh damping layer (µmax = 0.05 s−1) is applied only in the top 7 km392
of the domain (18 ≤ z ≤ 25 km) to prevent the damping of the physically significant wave393
breaking in the lower stratosphere. A fourth-order horizontal smoothing filter is applied394
with a coefficient KD = 1 × 109 m4 s−1, which smooths out any small-scale variations and395
helps maintain numerical stability in the model. Unlike most of the models in that study, no396
turbulence parameterization or any other explicit diffusion was used; turbulent dissipation397
is solely dependent on the hyperviscosity applied and any inherent numerical dissipation398
associated with the model discretization.399
In the results presented within Doyle et al. (2000), all models produced significant400
18
strengthening of the winds on the lee of the mountain and wave breaking in the upper tro-401
posphere and stratosphere at time 3 h. Despite using identical initial conditions, however,402
significant differences were found among the model results due to differences in the model403
formulations (e.g. spatial and temporal discretizations, type of explicit diffusion used), as404
well as the nonlinearity of the flow.405
The CSLAM-NH results at 3 h are presented in Fig. 4. The wave breaking regions in406
CSLAM-NH can be identified as the adiabatic (well-mixed) regions (Fig. 4a) and highly407
turbulent areas (Richardson number, Ri < 0.25) (Fig. 4c, where to be consistent with Doyle408
et al. (2000), the Richardson number sgn(Ri)|Ri|0.5 is plotted). The Richardson number409
used in Fig. 4c is the bulk Richardson number (dry),410
Ri =g/θ(∆θ/∆z)
(∆u/∆z)2.
For locally statically stable air (Ri > 0), the critical Richardson number at which wind shear411
is strong enough to sustain turbulence and overcome the damping by negative buoyancy412
is 0.25. The wave breaking regions appear to be in the vicinity of 12 ≤ z ≤ 16 km and413
17 ≤ z ≤ 20 km, comparable to the results in the intercomparison study. An initial critical414
level at z = 21 km (where U = 0) is also found to be damping in the CSLAM-NH model415
simulation, and traps the vertically-propagating gravity waves. The damping effect is evident416
in the smooth isentropes and lack of turbulence (large Ri, not contoured) above that height.417
The lateral position of the hydraulic jumps at time 3 h varied among the models given in418
Doyle et al. (2000), with several occurring over the leeslope and others farther downstream.419
The associated maximum leeslope winds from the 11 models were found to range from 43 m420
s−1 to 86 m s−1. In CSLAM-NH, the hydraulic jump feature is found on the leeslope, and421
the simulated maximum downslope wind speed at the surface (lowest model level) is located422
at 10.5 km downstream from the mountain crest at 56.6 m s−1 (Fig. 4b). Flow features aloft423
such as the flow reversal at 5 ≤ z ≤ 10 km that were present in many of the models in Doyle424
et al. (2000), is also present in the CSLAM-NH results. This weakening of the winds above425
the hydraulic jump was also observed in the aircraft flight data analysis (see, e.g., Fig. 2b426
19
in Doyle et al. 2000).427
The hyperviscosity coefficients used by the models in the model intercomparison study428
ranged from 1.1 × 108 to 5.0 × 109 m4 s−1. Time series of simulated maximum downslope429
wind speeds using different diffusion coefficients in CSLAM-NH are given in Fig. 5. Results430
from varying the horizontal smoothing coefficient from 1.5 × 108 to 1 × 109 m4 s−1 show431
slight variation in the simulated maximum downslope wind speed, with values at time 3432
h ranging from 53.1 m s−1 to 56.6 m s−1. The impact of using different magnitudes of433
horizontal smoothing is apparent once the waves begin to break, giving a maximum range434
of predicted downslope wind speeds of approximately 12 m s−1. The general trend of the435
downslope windstorm development, however, is similar with maximum surface winds of the436
simulation occurring at around 3 h, with weakening thereafter due to the limited horizontal437
extent of the domain and periodic lateral boundaries.438
The maximum stable time step in CSLAM-NH for this wave breaking case is 20 s (when439
KD = 5.0 × 109 m4 s−1 is applied). With a time step larger than 20 s, the errors of the440
linear trajectory approximations become large enough that the departure cells self-intersect441
as illustrated in Fig. 6. In this case, the flow is characterized by a strong horizontal shear442
of the vertical wind speeds as well as large vertical Courant number, Crz ≈ 3 (not shown).443
Higher-order cell-edge approximations have been explored by Ullrich et al. (2012), and may444
help alleviate the time step limit by increasing the accuracy of the area integration. Overall,445
CSLAM-NH is able to generate comparable results to the other models, and at a maximum446
time step size that is roughly double of those used in Doyle et al. (2000).447
c. Moist flow over a mountain in a nearly neutral environment448
The nonhydrostatic solver is tested for another nonlinear flow, but in this case, we also449
include the effects from moist processes. A simulation of saturated flow over a mountain450
in an initially nearly neutral environment is conducted. This test case also demonstrates451
the ability of the solver in producing realistic orographic precipitation. The simulation452
20
is based on the test cases presented in Miglietta and Rotunno (2005). Moisture in the453
atmosphere is an important factor in modifying flow over topography. Durran and Klemp454
(1983) studied the influence of moisture on mountain waves using numerical simulations.455
In both a linear mountain-wave test and a downslope-windstorm test, they found that the456
inclusion of upstream moisture can greatly reduce the amplitude of these waves relative to457
their dry analogs. As the mountain enhances lifting of the moist flow over the windward458
side, condensation commonly occurs, leading to clouds and precipitation. The downstream459
evaporation of these clouds and precipitation can reduce the static stability at these altitudes,460
and the air can become desaturated on the lee side of the mountain due to rainout processes461
and diabatic warming in the descent.462
For a nearly neutral flow, Miglietta and Rotunno (2005) simulated the transition of satu-463
rated air upstream to unsaturated air downstream due to diabatic warming in the downward464
motion on the lee. The inverse Froude number Nmhm/U is near zero, indicating that the465
resistance due to gravity is minimal and the flow can freely translate over the mountain.466
To include moisture effects when determining local static stability, Lalas and Einaudi467
(1974), and later verified by Durran and Klemp (1982), derived an expression for the moist468
Brunt-Vaisala frequency:469
N2m = gΓ
d ln θ
dz+
Lv
cpT
dqsdz
− g
1 + qw
dqwdz
, (14)
where T is the absolute temperature, qs is the saturated water vapour mixing ratio, qw is470
the total water mixing ratio, and471
Γ =1 + 1
qs+∂qs∂ ln θ
π
1 + LvcpT
∂qs∂ ln θ
π
, (15)
is the ratio of the moist to dry adiabatic lapse rates. (All other variables are as defined472
previously.) More details on the generation of the saturated neutral sounding for a specific473
moist static stability Nm and surface temperature are given in the appendix.474
Miglietta and Rotunno (2005) used a smallN2m = 3×10−6 s−2 to represent a nearly neutral475
troposphere due to the limitations of the single machine precision accuracy of their model.476
21
They found that using any smaller Nm led to solutions that were apparently convectively477
unstable. The CSLAM-NH solver has machine double precision accuracy, so for this moist478
neutral flow case, an initial Nm = 0 in the troposphere is applied. Simulations using a range479
of ‘small’ N2m ∼ 10−11 in the initialization step show solutions similar to applying Nm = 0480
and resemble those in Miglietta and Rotunno (2005) more than using their N2m = 3 × 10−6
481
s−2. An initial N2m = 4.84× 10−4 s−2 is used for the isothermal stratosphere.482
Two mountain cases with different heights, hm = 700 m and 2 km, were chosen from483
Miglietta and Rotunno (2005) for their distinct differences in orographic distribution of484
moisture. Both test cases are run using the Witch-of-Agnesi curve with a half-width of 10485
km. The same numerical domain that is 800 km wide and 20 km deep is used, and the grid486
dimensions are 400 × 80 grid cells (∆x = 2 km and ∆z = 250 m). In both cases, a mean487
wind U = 10 m s−1 is applied. The atmosphere is initially saturated (qv ≡ qs) with constant488
cloud-water mixing ratio qc = 0.05 g kg−1 set everywhere in the domain to prevent the489
atmosphere from becoming subsaturated due to the impulsive introduction of the mountain490
at initial time. The Rayleigh damping layer (µmax = 0.1 s−1) is applied in the top 5 km of491
the domain. Second-order filters in the horizontal and vertical directions are applied with492
coefficients 3000 and 3 m2 s−1, respectively. The Prandtl number is 3. This configuration is493
the same as that in Miglietta and Rotunno (2005).494
Both cases suggest a desaturation of the air downstream of the mountain with time.495
Miglietta and Rotunno (2005) noticed in their simulations that for intermediate mountain496
heights (500 m ≤ hm ≤ 1500 m) the unsaturated region downstream of the hill unexpectedly497
extends upstream as well. A later study by Keller et al. (2012) showed that this upstream498
extent of the subsaturated air is due to local adiabatic descent and warming caused by499
a transient upstream-propagating gravity wave, a fundamental feature of a two-layer two-500
dimensional atmosphere with topography introduced impulsively. The purpose of performing501
the test case as prescribed in Miglietta and Rotunno (2005) with a mountain 700 m high502
is to ensure that CSLAM-NH can generate comparable results to models used in the liter-503
22
ature, such as that in Miglietta and Rotunno (2005) who used the Weather and Research504
Forecasting (WRF) model v1.3.505
Fig. 7 shows the solution from CSLAM-NH [c.f. Fig. 5d of Miglietta and Rotunno506
(2005)] using a time step size of 20 s. The white region indicates subsaturated air, as507
described previously. Although the upstream region of the subsaturated air in Miglietta and508
Rotunno (2005) extends farther upstream (x = −100 km) than that found using CSLAM-509
NH, the solution from CSLAM-NH compares very well with that obtained in Miglietta and510
Rotunno (2005). The maximum stable CSLAM-NH time step size is 50 s with two iterations511
of both components in the iterative centered-implicit scheme.512
Fig. 8a shows the CSLAM-NH cloud-water mixing ratio at time 5 h 10 mins (10 mins513
after autoconversion of rain is permitted) of a simulation using ∆t = 20 s for the large-514
amplitude mountain (hm = 2 km) case (c.f. black contours in Fig. 8a in Miglietta and515
Rotunno (2005)). Similar to the results presented in Miglietta and Rotunno (2005), no516
upstream region of the subsaturated air is found. In addition, the formation of convective517
cells due to the reduction of local static stability downstream of the mountain is also detected518
in the CSLAM-NH simulation. The instability is found to be primarily associated with a519
hydraulic jump feature downwind. Fig. 8b shows the rainwater mixing ratio for the same520
simulation time as in Fig. 8a.521
Compared to the results in Miglietta and Rotunno (2005), CSLAM-NH indicates more522
rain spillover to the lee of the mountain (c.f. grey contours in Fig. 8a in Miglietta and523
Rotunno (2005)). Simulation of our case using an Eulerian split-explicit model similar to the524
one used in Miglietta and Rotunno (2005) shows virtually the same distributions of cloud-525
and rainwater as in the CSLAM-NH simulation (Fig. 8). The similarity of the CSLAM-526
NH solution to that of the second Eulerian model seems to suggest that the discrepancy527
is not specific to CSLAM-NH and may be related to certain aspects of the initialization528
procedure. Miglietta and Rotunno (2005) suggested that their simulations were sensitive to529
small changes to Nm ≈ 0. If there is a slight discrepancy between the value of Nm specified530
23
in the initialization than that used in Miglietta and Rotunno (2005), the flow dynamics may531
be altered. In this situation, the greater amount of waterloading to the lee of the mountain532
could imply a lower effective terrain height (Nmhm/U), i.e. lower stability and/or stronger533
winds, such that the advection of the hydrometeors happens at a faster time scale than the534
fallout of precipitation.535
4. Summary536
A nonhydrostatic atmospheric solver (CSLAM-NH) that uses a new discrete formula-537
tion of the semi-implicit continuity equation for cell-integrated semi-Lagrangian transport538
schemes is further tested for flows over idealized orography. Here, the solver using the539
Conservative Semi-LAgrangian Multi-tracer (CSLAM) transport scheme is tested against540
various idealized mountain-wave cases and exhibits accurate and stable behaviour under the541
influence of a terrain-following height coordinate.542
The new discrete semi-implicit continuity equation used in CSLAM-NH allows for a543
straightforward implementation of consistent flux-form equations for scalars in the model.544
This aspect of the model is important in ensuring inherent mass conservation of these scalars,545
such as moisture and chemical species, and may prove to be important in longer NWP and546
climate simulations. The time integration of both the gravity and acoustic waves are handled547
implicitly in the solver using an iterative centered-implicit scheme. The iterative scheme al-548
lows for larger maximum stable time step sizes at the expense of solving the linear Helmholtz549
problem more than once. In large climate and chemistry models, the computational cost550
associated with the parameterized physics and the transport of the many (order of 102)551
tracer species is likely to outweigh that associated with the dynamics. The solution proce-552
dure for the dry dynamics is carried out once at each time step, whereas scalar transport553
is computed for hundreds of tracers. The larger time step sizes allowable by the iterative554
scheme in CSLAM-NH reduce the number of tracer advection steps per simulation, which555
24
may help compensate for the added expense. An implicit Rayleigh damping layer is also556
implemented in this extended version of CSLAM-NH to help prevent unphysical reflection557
of vertically-propagating gravity waves at the model top.558
Three idealized test cases available from the literature were used to verify the stability559
and accuracy of the proposed solver over topography. Simulations of linear hydrostatic and560
nonhydrostatic mountain waves compared well with numerical solutions from the literature.561
The simulation of a highly nonlinear wave-breaking case of the 11 January 1972 Boulder562
windstorm highlighted the ability of the solver to handle highly-sheared flow at large time563
steps. Due to the strong nonlinearity of the flow, the simulations from the models used in the564
intercomparison study of Doyle et al. (2000) varied in their fine-scale features. Although there565
is limited predictability of the precision of these features, all models, including CSLAM-NH566
(the simulation of which is presented here), showed similar main features of the windstorm,567
such as the locations of the wave breaking regions and hydraulic jump downstream of the568
mountain. Finally, moist nearly neutral orographic flows based on Miglietta and Rotunno569
(2005) are tested. Two mountain profiles were used: a lower 700-m tall mountain and a much570
higher 2 km mountain. For the lower mountain case, CSLAM-NH shows comparable results571
with those in Miglietta and Rotunno (2005), including downstream and upstream regions of572
subsaturated air. For the higher mountain case, there is more rain spillover to the leeside573
of the mountain as compared to the results presented in Miglietta and Rotunno (2005).574
However, similar solutions are found using another comparison Eulerian split-explicit model,575
which suggests that certain aspects (e.g. initialization) of the model other than model576
formulation may be causing the discrepancy, and that the discrepancy is not specific to577
CSLAM-NH.578
In its current state of development, CSLAM-NH is a two-dimensional prototypical non-579
hydrostatic atmospheric solver in Cartesian geometry that has shown promising potential580
for weather and climate applications. Attractive features of this solver include the consistent581
formulation of the semi-implicit cell-integrated semi-Lagrangian continuity and scalar con-582
25
servation equations, in conjunction with the inherently conservative multi-tracer CSLAM583
transport scheme. Further development work (e.g. implementation of three-dimensional584
CSLAM transport, extension of the scheme to a sphere) remains for the solver to be imple-585
mented as a dynamical core in a full NWP and climate model.586
Acknowledgments.587
The initial research of this work was done during the first author’s visits to the National588
Center for Atmospheric Research through the Graduate Visitor Advanced Study Program.589
The authors would like to thank Dr. James Doyle for providing the sounding data used in590
the initialization of the 11 January 1972 Boulder windstorm case. This research is funded591
by the Canadian Natural Science and Engineering Research Council via a Discovery Grant592
to the last author.593
26
APPENDIX594
Generation of a moist neutral sounding595
A few more specifics regarding the generation of the moist neutral sounding that sup-596
plements the derivation presented in Miglietta and Rotunno (2005) are given. Following597
the procedure in Miglietta and Rotunno (2005), to generate the initial sounding of a spe-598
cific Nm, a first-order ordinary differential equation is solved. To create the initial nearly599
neutral sounding, the first-order ordinary differential equation for potential temperature is600
solved iteratively based on a specified surface temperature (15 C) and reference pressure601
(p0 = 100 kPa). To be consistent with Miglietta and Rotunno (2005), the Wexler’s formula602
for saturated vapour pressure (in hPa) is used:603
es(T ) = 6.11 exp
17.67
T − 273.15K
T − 29.65K
. (A1)
The definition for qs = es/(p− es), where = Rd/Rv is used to derive ∂qs/∂ ln θ|π in (15).604
First, differentiating qs with respect to ln θ at constant Exner function π gives605
∂qs∂ ln θ
π
=qses
p
p− es
∂es∂ ln θ
π
,
and differentiating (A1) (using T = πθ) with respect to ln θ at constant π gives the expression606
∂es∂ ln θ
π
= esT17.67(243.15)
(T − 29.65)2.
To find θ(z) for a specific Nm, (14) must be iterated to convergence (10−12) at each pressure607
level (or height) since qs = qs(π, θ) and Γ are also functions of the unknown. To get the608
pressure at each height, the hydrostatic equation is used:609
πj+1 − πj
∆z= − g
cp
1 + qwz
θzm
.
For each model level j, the discrete form of the ODE solving for θ at height z is610
ln θj+1 +Lv
cpT
z
(qs,j+1 − qs,j)− (qw,j+1 − qw,j)
1
(1 + qw)Γ
z
= ln θj +N2
m
gΓ
z
(zj+1 − zj).
27
[Note: Miglietta and Rotunno (2005) expresses this equation in terms of (T, p).] Other611
aspects of the Kessler microphysics scheme also require modification. Following Miglietta612
and Rotunno (2005), no autoconversion from cloud water to rain is permitted in the first613
five hours to allow for initial adjustment of the flow to the impulsive introduction of terrain.614
For a consistent definition of qs throughout the model, the production of cloud water due to615
saturation is also modified:616 dqsdt
=qv − qs
1 + Lvcp
∂qs∂T
|p,
where, based on the Wexler’s equation for es:617
∂qs∂T
p
=qsp
p− es
17.67(243.5)
(T − 29.65)2.
28
618
REFERENCES619
Bonaventura, L., 2000: A Semi-implicit Semi-Lagrangian Scheme Using the Height Coor-620
dinate for a Nonhydrostatic and Fully Elastic Model of Atmospheric Flows. J. Comput.621
Phys., 158, 186–213.622
Cordero, E., N. Wood, and A. Staniforth, 2005: Impact of semi-Lagrangian trajectories on623
the discrete normal modes of a non-hydrostatic vertical-column model. Q. J. R. Meteorol.624
Soc., 131, 93–108.625
Cote, J., S. Gravel, A. Methot, A. Patoine, M. Roch, and A. Staniforth, 1998: The opera-626
tional CMC-MRB global environmental multiscale (GEM) model. Part I: Design consid-627
erations and formulation. Mon. Wea. Rev., 126, 1373–1395.628
Cullen, M. J. P., 2001: Alternative implementations of the semi-Lagrangian semi-implicit629
schemes in the ECMWF model. Q. J. R. Meteorol. Soc., 127, 2787–2802.630
Doyle, J. D., et al., 2000: An intercomparison of model-predicted wave breaking for the 11631
January 1972 Boulder windstorm. Mon. Wea. Rev., 128, 901–914.632
Durran, D. R., 1986: Another Look at Downslope Windstorms. Part I: The Development633
of Analogs to Supercritical Flow in an Infinitely Deep, Continuously Stratified Fluid. J.634
Atmos. Sci., 43, 2527–2543.635
Durran, D. R. and J. B. Klemp, 1982: On the effects of moisture on the Brunt-Vaisala636
frequency. J. Atmos. Sci., 39, 2152–2158.637
Durran, D. R. and J. B. Klemp, 1983: A Compressible Model for the Simulation of Moist638
Mountain Waves. Mon. Wea. Rev., 111, 2341–2361.639
29
Erath, C., P. H. Lauritzen, J. H. Garcia, and H. M. Tufo, 2012: Integrating a scalable and640
effcient semi-Lagrangian multi-tracer transport scheme in HOMME. Procedia Comput.641
Sci., 9, 994–1003.642
Gal-Chen, T. and R. Somerville, 1975: On the use of a coordinate transformation for the643
solution of the Navier-Stokes equations. J. Comput. Phys., 17, 209–228.644
Jockel, P., R. von Kuhlmann, M. Lawrence, B. Steil, C. Brenninkmeijer, P. Crutzen,645
P. Rasch, and B. Eaton, 2001: On a fundamental problem in implementing flux-form646
advection schemes for tracer transport in 3-dimensional general circulation and chemistry647
transport models. Q. J. R. Meteorol. Soc., 127, 1035–1052.648
Keller, T. L., R. Rotunno, M. Steiner, and R. D. Sharman, 2012: Upstream-Propagating649
Wave Modes in Moist and Dry Flow over Topography. J. Atmos. Sci., 69, 3060–3076.650
Klemp, J. B., J. Dudhia, and A. D. Hassiotis, 2008: An Upper Gravity-Wave Absorbing651
Layer for NWP Applications. Mon. Wea. Rev., 136, 3987–4004.652
Klemp, J. B. and D. K. Lilly, 1978: Numerical Simulation of Hydrostatic Mountain Waves.653
J. Atmos. Sci., 35, 78–107.654
Klemp, J. B., W. C. Skamarock, and J. Dudhia, 2007: Conservative split-explicit time655
integration methods for the compressible nonhydrostatic equations. Mon. Wea. Rev., 135,656
2897–2913.657
Lalas, D. P. and F. Einaudi, 1974: On the correct use of the wet adiabatic lapse rate in658
stability criteria of a saturated atmosphere. J. Appl. Meteor., 13, 318–324.659
Laprise, J. and A. Plante, 1995: A class of semi-Lagrangian integrated-mass (SLIM) numer-660
ical transport algorithms. Mon. Wea. Rev., 123, 553–565.661
Lauritzen, P. H., E. Kaas, and B. Machenhauer, 2006: A mass-conservative semi-implicit662
30
semi-Lagrangian limited-area shallow-water model on the sphere. Mon. Wea. Rev., 134,663
1205–1221.664
Lauritzen, P. H., R. D. Nair, and P. A. Ullrich, 2010: A conservative semi-Lagrangian multi-665
tracer transport scheme (CSLAM) on the cubed-sphere grid. J. Comput. Phys., 229,666
1401–1424.667
Lilly, D. K., 1978: A severe downslope windstorm and aircraft turbulence event induced by668
a mountain wave. J. Atmos. Sci., 35, 59–77.669
Machenhauer, B. and M. Olk, 1997: The implementation of the semi-implicit scheme in670
cell-integrated semi-Lagrangian models. Atmos.-Ocean, 35 (special issue), 103–126.671
Melvin, T., M. Dubal, N. Wood, A. Staniforth, and M. Zerroukat, 2010: An inherently mass-672
conserving iterative semi-implicit semi-Lagrangian discretization of the non-hydrostatic673
vertical-slice equations. Q. J. R. Meteorol. Soc., 136, 799–814.674
Miglietta, M. and R. Rotunno, 2005: Simulations of Moist Nearly Neutral Flow over a Ridge.675
J. Atmos. Sci., 62, 1410–1427.676
Nair, R. and B. Machenhauer, 2002: The mass-conservative cell-integrated semi-Lagrangian677
advection scheme on the sphere. Mon. Wea. Rev., 130, 649–667.678
Peltier, W. and T. Clark, 1982: Nonlinear mountain waves in two and three spatial dimen-679
sions. Q. J. R. Meteorol. Soc., 109, 527–548.680
Peltier, W. R. and T. L. Clark, 1979: The Evolution and Stability of Finite-Amplitude Moun-681
tain Waves. Part II: Surface Wave Drag and Severe Downslope Windstorms. J. Atmos.682
Sci., 36, 1498–1529.683
Pinty, J. P., R. Benoit, E. Richard, and R. Laprise, 1995: Simple tests of a semi-implicit684
semi-Lagrangian model on 2D mountain wave problems.Mon. Wea. Rev., 123, 3042–3058.685
31
Rancic, M., 1992: Semi-Lagrangian piecewise biparabolic scheme for two-dimensional hori-686
zontal advection of a passive scalar. Mon. Wea. Rev., 120, 1394–1406.687
Rivest, C., A. Staniforth, and A. Robert, 1994: Spurious resonant response of semi-688
Lagrangian discretizations to orographic forcing: Diagnosis and solution. Mon. Wea. Rev.,689
122, 366–376.690
Robert, A., 1981: A stable numerical integration scheme for the primitive meteorological691
equations. Atmos.-Ocean, 19, 35–46.692
Robert, A., T. Yee, and H. Ritchie, 1985: A semi-Lagrangian and semi-implicit numerical-693
integration scheme for multilevel atmospheric models. Mon. Wea. Rev., 113, 388–394.694
Skamarock, W. C. and J. B. Klemp, 1994: Efficiency and accuracy of the Klemp-Wilhelmson695
time-splitting technique. Mon. Wea. Rev., 122, 2623–2630.696
Skamarock, W. C., P. K. Smolarkiewicz, and J. B. Klemp, 1997: Preconditioned conjugate-697
residual solvers for Helmholtz equations in nonhydrostatic models. Mon. Wea. Rev., 125,698
587–599.699
Smith, R. B., 1979: The influence of mountains on the atmosphere. Adv. Geophys., 21,700
87–230.701
Smolarkiewicz, P. and J. Pudykiewicz, 1992: A class of Semi-Langragian approximations for702
fluids. J. Atmos. Sci., 49, 2082–2096.703
Ullrich, P. A., P. H. Lauritzen, and C. Jablonowski, 2012: Some considerations for high-order704
‘incremental remap’-based transport schemes: edges, reconstructions, and area integra-705
tion. Int. J. Numer. Methods Fluids, 71, 1131–1151.706
Wong, M., W. C. Skamarock, P. H. Lauritzen, J. B. Klemp, and R. B. Stull, 2014: A com-707
pressible nonhydrostatic cell-integrated semi-Lagrangian semi-implicit solver (CSLAM-708
NH) with consistent and conservative transport. Mon. Wea. Rev. (in press).709
32
Wong, M., W. C. Skamarock, P. H. Lauritzen, and R. B. Stull, 2013: A cell-integrated710
semi-Lagrangian semi-implicit shallow-water model (CSLAM-SW) with conservative and711
consistent transport. Mon. Wea. Rev., 141, 2545–2560.712
Xue, M., K. K. Droegemeier, and V. Wong, 2000: The Advanced Regional Prediction System713
(ARPS) - A multi-scale nonhydrostatic atmospheric simulation and prediction model. Part714
I: Model dynamics and verification. Meteorol. Atmos. Phys., 75, 161–193.715
Zerroukat, M., N. Wood, and A. Staniforth, 2002: SLICE: A semi-Lagrangian inherently716
conserving and efficient scheme for transport problems. Q. J. R. Meteorol. Soc., 128,717
2801–2820.718
Zhang, K., H. Wan, B. Wang, and M. Zhang, 2008: Consistency problem with tracer advec-719
tion in the atmospheric model GAMIL. Adv. Atmos. Sci., 25, 306–318.720
33
List of Figures721
1 Discrete departure cells in CSLAM-NH are approximated using straight edges722
(shaded in grey). The departure cell vertices (black circles) are computed723
using backward-in-time trajectories (arrows) from the vertices (white circles)724
of the Eulerian arrival grid cell (white box). The CSLAM transport scheme is725
stable as long as the discrete departure grid cells are (a) non-self-intersecting,726
and becomes problematic if (b) the departure cell self-intersects since the727
scheme is no longer mass-conserving. 35728
2 CSLAM-NH simulation of a linear hydrostatic wave (U/Na ≈ 0.1) for a low729
wide mountain showing vertical velocity w (in m s−1) after T = 15000 s using730
(top) ∆t = 10 s (Cr = 0.2) and (bottom) ∆t = 75 s (Cr = 1.5). Contour731
interval is 5× 10−4 m s−1. Mean wind is from left to right. 36732
3 CSLAM-NH simulation of a linear nonhydrostatic wave (U/Na = 1) for a733
narrow mountain showing vertical velocity w (in m s−1) after T = 9000 s734
using (top) ∆t = 5 s (Cr = 0.125) and (bottom) ∆t = 60 s (Cr=1.5). Contour735
interval is 6× 10−4 m s−1. Mean wind is from left to right. 37736
4 CSLAM-NH simulation for the 1972 Boulder windstorm case (a) potential737
temperature θ (K) (with contour interval of 8 K), (b) horizontal velocity738
U (m s−1) (with contour interval of 8 m s−1), and (c) Richardson number739
sgn(Ri)|Ri|0.5 (−5 ≤ Ri ≤ 1 are plotted with contour interval of 0.5, following740
Doyle et al. (2000); grey contour shows negative values) at T= 3 h using a741
time step ∆t = 10 s. 38742
5 Time series of the simulated maximum CSLAM-NH downslope wind speeds743
(m s−1) for the 1972 Boulder windstorm case using different horizontal smooth-744
ing coefficients, KD (m4 s−1). 39745
34
6 A self-intersecting departure cell (highlighted in red with vertices marked by746
black circles) in CSLAM-NH when a large time step size of 25 s is used for the747
strongly sheared flow in the 1972 Boulder downslope windstorm case. Black748
circles indicate departure grid cell vertices, and white circles the Eulerian749
arrival grid cell vertices. Arrows symbolize the computed backward-in-time750
trajectories. Trajectories and the arrival grid cell associated with the self-751
intersecting departure cell are highlighted in red. 40752
7 CSLAM-NH cloud-water mixing ratio (g kg−1) at time 5 h from an initially753
saturated nearly neutral flow (with an initial qc = 0.05 g kg−1) over a 700 m754
hill. White region above ground indicates subsaturated air (qc ≡ 0). 41755
8 (a) CSLAM-NH cloud-water mixing ratio qc (g kg−1) at time 5 h 10 mins from756
an initially saturated nearly neutral flow (with an initial qc = 0.05 g kg−1)757
over a 2 km mountain. White region above ground indicates subsaturated air758
(qc ≡ 0 g kg−1) (b) CSLAM-NH rain-water mixing ratio qr (g kg−1) at the759
same simulation time. 42760
35
(a) Non-self-intersecting departure cell (b) Self-intersecting departure cell
z
x
Fig. 1. Discrete departure cells in CSLAM-NH are approximated using straight edges(shaded in grey). The departure cell vertices (black circles) are computed using backward-in-time trajectories (arrows) from the vertices (white circles) of the Eulerian arrival gridcell (white box). The CSLAM transport scheme is stable as long as the discrete departuregrid cells are (a) non-self-intersecting, and becomes problematic if (b) the departure cellself-intersects since the scheme is no longer mass-conserving.
36
he
igh
t (k
m)
0
5
10
!4
!2
0
2
4x 10
horizontal distance (km)
he
igh
t (k
m)
!40 !20 0 20 400
5
10
-3
w (m s-1)
w (m s-1)
Cr = 1.5
Cr = 0.2
Fig. 2. CSLAM-NH simulation of a linear hydrostatic wave (U/Na ≈ 0.1) for a low widemountain showing vertical velocity w (in m s−1) after T = 15000 s using (top) ∆t = 10 s(Cr = 0.2) and (bottom) ∆t = 75 s (Cr = 1.5). Contour interval is 5 × 10−4 m s−1. Meanwind is from left to right.
37
he
igh
t (k
m)
0
5
10
!5
!2.5
0
2.5
5x10
!3
horizontal distance (km)
he
igh
t (k
m)
!10 0 10 20 300
5
10
w (m s-1)
w (m s-1)
Cr = 0.125
Cr = 1.5
Fig. 3. CSLAM-NH simulation of a linear nonhydrostatic wave (U/Na = 1) for a narrowmountain showing vertical velocity w (in m s−1) after T = 9000 s using (top) ∆t = 5 s (Cr= 0.125) and (bottom) ∆t = 60 s (Cr=1.5). Contour interval is 6× 10−4 m s−1. Mean windis from left to right.
38
0 0
0
0
0
0
0
00
0
0
00
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
8
8
8
8
8
8
8
8
8
8
8
8
88
88
8
8
8
88
8
8
8
8
8
16
16
16
16
16
16
16
16
16
16
6
16
16
16 16
1616
16
16
16
16
24
24
24
24
24
24
24
24
24
24
24
24
24
32
32
32 32
32
32
32
32
32
32
32
40
40 40
40
40
48
48
48
56
5664
!16
!8
!8
!8
!8
!8
!8
!8
288 288
296
296
296
296
304
304
304
312
312
320
320
320 320
328
328
336
336
344
344
344352
352
352
352360
360
360
360
368
368
368
376 376
376
384 384
384384
392 392
392
392
400400
408 408
408
416416
416
424424
424
432432
432
440
440
440448
448
448456 4
56
456464464
464472
472
472
480
480
480
480488 4
88
488 488
496 496
496504
504504
512
512
512
520 520
520528
528528
536 536536
544 544 544
552
552552
560 560560
568 568568
576 576576
584 584584
592592
600600
608 608616 616624 624632 632640 640648 648656 656664 664672 672
Horizontal distance (km)
0
5
10
15
20
25
He
igh
t (k
m)
He
igh
t (k
m)
He
igh
t (k
m)
!100 !50 0 50 100
(c) sgn(Ri) |R
i|0.5
0
5
10
15
20
25
0
5
10
15
20
25
(a) !
0
(b) u
Fig. 4. CSLAM-NH simulation for the 1972 Boulder windstorm case (a) potential temper-ature θ (K) (with contour interval of 8 K), (b) horizontal velocity U (m s−1) (with contourinterval of 8 m s−1), and (c) Richardson number sgn(Ri)|Ri|0.5 (−5 ≤ Ri ≤ 1 are plotted withcontour interval of 0.5, following Doyle et al. (2000); grey contour shows negative values) atT= 3 h using a time step ∆t = 10 s.
39
0 0.5 1 1.5 2 2.5 3 3.5 410
20
30
40
50
60
70
Time (mins)
Ma
xim
um
do
wn
slo
pe
win
d s
pe
ed
(m s
-1)
KD = 1.5 ! 108
= 2.5 ! 108
= 5.0 ! 108
= 10 ! 108
KD
KD
KD
Fig. 5. Time series of the simulated maximum CSLAM-NH downslope wind speeds (m s−1)for the 1972 Boulder windstorm case using different horizontal smoothing coefficients, KD
(m4 s−1).
40
z
x
Fig. 6. A self-intersecting departure cell (highlighted in red with vertices marked by blackcircles) in CSLAM-NH when a large time step size of 25 s is used for the strongly shearedflow in the 1972 Boulder downslope windstorm case. Black circles indicate departure gridcell vertices, and white circles the Eulerian arrival grid cell vertices. Arrows symbolize thecomputed backward-in-time trajectories. Trajectories and the arrival grid cell associatedwith the self-intersecting departure cell are highlighted in red.
41
!100 !50 0 50 1000
5
10
15
Horizontal distance (km)
He
igh
t (k
m)
0
0.1
0.5
qc (g kg-1)
Fig. 7. CSLAM-NH cloud-water mixing ratio (g kg−1) at time 5 h from an initially saturatednearly neutral flow (with an initial qc = 0.05 g kg−1) over a 700 m hill. White region aboveground indicates subsaturated air (qc ≡ 0).
42
!40 !20 0 20 40 60 800
1
2
3
4
5
6
7
8
9
Horizontal distance (km)
He
igh
t (k
m)
0
0.2
0.6
1
!40 !20 0 20 40 60 800
1
2
3
4
5
6
7
8
9
Horizontal distance (km)
He
igh
t (k
m)
0
0.2
0.4
0.6
0.8
1
(a) qc (g kg-1) (b) q
r (g kg-1)
Fig. 8. (a) CSLAM-NH cloud-water mixing ratio qc (g kg−1) at time 5 h 10 mins from aninitially saturated nearly neutral flow (with an initial qc = 0.05 g kg−1) over a 2 km mountain.White region above ground indicates subsaturated air (qc ≡ 0 g kg−1) (b) CSLAM-NH rain-water mixing ratio qr (g kg−1) at the same simulation time.
43
