A GENERALIZED FAMILY OF SCHEMES THAT ELIMINATE THE SPURIOUS RESONANT
RESPONSE OF SEMI-LAGRANGIAN SCHEMES TO OROGRAPHIC FORCING
Jean Côté, Sylvie Gravel and Andrew Staniforth
Recherche en prévision numérique
Service de l'environnement atmosphérique
2121 Route Transcanadienne, porte 500
Dorval, Québec
CANADA H9P 1J3
March 1995
Submitted to Monthly Weather Review
ABSTRACT
The one-parameter three-time-level family of
-accurate
schemes, introduced in Rivest et al. (1994) to address the problem of the
spurious resonant response of semi-implicit semi-Lagrangian schemes at large
Courant number, has been generalized to a two-parameter family by introducing
the possibility of evaluating total derivatives using an additional time level.
The merits of different members of this family based on both theory and results
are assessed. The additional degree of freedom might be expected a priori to
permit a reduction of the time truncation errors while still maintaining
stability and avoiding spurious resonance. Resonance, stability and truncation
error analyses for the proposed generalized family of schemes are given. The
sub-family that is formally
-accurate
is unfortunately unstable for gravity modes. Sample integrations for various
members of the generalized family are shown. Results are consistent with
theory, and stable non-resonant forecasts at large Courant number are possible
for a fairly broad choice of values of the two free parameters. Of the two
methods proposed in Rivest et al. (1994) for computing trajectories, the one
using a piecewise-defined trajectory is to be preferred to that using a single
great-circle arc, since it is more accurate at large timestep for some members
of the generalized family.
Since Robert et al. (1985) first demonstrated a semi-implicit semi-Lagrangian
discretization of the hydrostatic primitive equations, this method has found
increasing favor at weather-forecasting and climate centers throughout the
world (e.g., Bates et al. 1993; McDonald and Haugen 1992; Purser and Leslie
1994; Ritchie and Beaudoin 1994; Ritchie et al. 1995; Tanguay et al. 1989;
Williamson and Olson 1994).
Although it has been known for some time (Kaas 1987; Coiffier et al. 1987;
Staniforth & Côté 1991; Tanguay et al. 1992) that
semi-implicit semi-Lagrangian schemes can experience difficulties at high
Courant number in the presence of orography, it is only recently that this
problem has been elucidated. In Rivest et al. (1994), hereafter referred to as
RSR94, its source was clearly identified and a solution proposed in a
two-time-level context. It consists of off-centering the
semi-implicitly-treated terms along the trajectory, and Rivest & Staniforth
(1995) have proposed a possible retrofit scheme for three-time-level-based
centered schemes. Héreil & Laprise (1995) have recently extended
the RSR94 analysis to the semi-implicit semi-Lagrangian discretization of
Tanguay et al. (1990) of the non-hydrostatic Euler equations.
The goals of the present study are to present a generalization of the RSR94
family of schemes, and to assess the merits of different members of this family
based on both theory and results.
The linearized one-dimensional shallow-water equations after a three-time-level
semi-implicit semi-Lagrangian discretization may be written as (cf. RSR94):
,
(2.1)
,
(2.2)
,
(2.3)
where
and
are respectively the fluid depth and orographic height multiplied by
,
the other symbols have their usual meaning, and for the generalized
family of schemes introduced here:
,
(2.4)
,
(2.5)
.
(2.6)
The basic-state quantities
and
are assumed non-zero, and the orographic geopotential
is both non-zero and time invariant. The linearized equations before
discretization may be recovered by redefining
and
as:
,
(2.7)
.
(2.8)
The above 2-parameter (
,
)
family of schemes is the most general
-accurate
scheme based on three time levels, and is a generalization of the 1-parameter (
)
family presented in RSR94. The parameter
here is exactly as in RSR94. It was introduced there to decenter the
2-time-level Crank-Nicolson semi-implicit semi-Lagrangian discretization, and
thereby address the problem of spurious orogrophically-induced resonance by
introducing a third time level to evaluate the time-averaged contributions
along the trajectory. A further new parameter
has been added here. It optionally introduces the use of a third time level
in the evaluation of the time derivatives along the trajectory. This
additional degree of freedom might be expected a priori to permit an
improvement to the RSR94 family of schemes by reducing the time truncation
errors [possibly to
]
while still maintaining stability and avoiding spurious resonance. This is
discussed further later. Special cases of the above generalized scheme for
particular values of
and
are summarized in Table 1.
d [[eps Scheme
ilon]
]
0 0 Crank-Nicolson using time levels t and t-[[Delta]]t
Crank-Nicolson using time levels t and
t-2[[Delta]]t
d 0 General RSR94 1-parameter (d) scheme
0 Particular RSR94 scheme, for which results were
shown
Backward implicit
Table
1: Particular cases of the generalized scheme as a function of
and
.
The complete solution to the linear system (2.1)-(2.3) can be written as the
sum of free and forced modes:
.
(2.7)
The free solutions satisfy (2.1)-(2.3) with the forcing
set identically to zero. Letting
,
(2.8)
each free mode (there are three for each wavenumber) then satisfies
,
(2.9)
where
,
(2.10)
,
(2.11)
,
(2.12)
,
(2.13)
and exact interpolation has been assumed. Setting
,
(2.14)
then gives the dispersion relation for
.
Note that for the exact solution of the linearized equations (with no
discretization), (2.11)-(2.12) may be replaced by the definitions
,
(2.15)
,
(2.16)
and the usual (free) Rossby and gravity-wave dispersion relations then result
from (2.14):
(2.17)
The forced solutions satisfy (2.1)-(2.3) in the absence of any time
variation (
),
and may be Fourier decomposed as
.
(2.18)
They then satisfy
.
(2.19)
Note that for the exact solution of the linearized equations (with no
discretization),
simplifies to
,
(2.20)
(cf. RSR94). Physical resonance then occurs when the determinant of
vanishes, i.e. when
,
(2.21)
which, as discussed in RSR94, corresponds to supersonic flow and is unlikely to
occur in a shallow-water model representative of the atmosphere at 500 hPa.
The existence or not of resonance is determined from (2.19), and a scheme's
stability from the solutions of the dispersion relation (2.14). Note that the
matrix
defined by (2.10) plays a determining role for both resonance and stability.
These latter are respectively discussed in the following two sub-sections.
For the discretized linear equations, whenever
,
(2.22)
the stationary forced gravity modes determined by (2.19) are resonant, and
these resonances may be spurious (RSR94). This leads to the following two
quadratic equations that govern the resonance of the stationary forced gravity
solutions:
,
(2.23)
where
,
(2.24)
.
(2.25)
Since
is real by definition, resonance is only possible if
lies on the unit circle, i.e. if
.
(2.26)
For the special case
,
the conditions (2.23) for resonance in the present 2-parameter (
)
family of schemes reduce to (15) of RSR94 for their 1-parameter (
)
family. The further special case
(cf. Table 1), corresponds to the conditions (9) of RSR94 for the spurious
resonance of centered 2-time-level semi-implicit semi-Lagrangian schemes. As
discussed in RSR94, spurious resonance occurs for certain combinations of large
Courant number and nondimensional wavenumbers of the orographic forcing, and
can be avoided by suitably off-centering the time discretization along the
trajectory. In the present study this is controlled by the
and
parameters.
Summarizing, any of the schemes considered here will be non-resonant provided
.
They may not however necessarily be stable, and this aspect of the
discretization is examined in the following sub-section.
Stability is determined from the dispersion relation obtained by solving
(2.14). Thus
,
(2.27)
where
,
(2.28)
,
(2.29)
and the scheme is stable provided
.
(2.30)
Note that (2.27) can be identified with the dispersion relation that would
result from applying the generalized family of schemes of the present study to
an oscillation equation, yielding the discretization
,
(2.31)
where
is real. This is equivalent to having performed a decomposition in terms of
the eigen-functions of the matrix
.
The leading-order term of the truncation error (placed on the right-hand side
of 2.31) is
.
(2.32)
For the quadratic associated with the Rossby modes (for which
),
(2.27) factors easily and
or
.
The first root corresponds to the physical Rossby mode and it is stable since
it satisfies (2.30). The other root corresponds to a computational Rossby mode
(introduced by discretizing a 1st-order-in-time equation over three time
levels), and it will satisfy (2.30) and be stable provided
.
Now
corresponds to a centered three-time-level discretization, and this value for
must be excluded since (see above and RSR94) the stationary forced gravity
modes would suffer from spurious resonance. Although satisfaction of the
strict inequality
is sufficient to stabilize the computational Rossby mode,
should be positive, otherwise it will have the undesirable property of changing
the sign of this computational mode at alternate timesteps (timestep
decoupling).
should not only be positive but small to ensure adequate damping of this
computational mode during the integration.
For the quadratic associated with the gravity modes (for which
),
(2.27) does not factor easily. It is nevertheless possible to get some insight
into the influence of
and
on the stability of the free gravity solutions by solving (2.27) asymptotically
for
for the physical mode when
is small. Taking its modulus then gives
.
(2.33)
From above, since
for stability of the computational Rossby solution, it is necessary to take
,
(2.34)
in order to both satisfy the stability condition (2.30) and to also remain off
resonance. This is a necessary condition for stability. From (2.32), it has
the further consequence that the only
-accurate member of the present generalized family of three-time-level schemes
(i.e.
)
is unstable and therefore inadmissible. This is unfortunate. By numerically
solving (2.27) when
for a broad range of values of the parameters
,
,
and
,
it is found that the valid domain for remaining both stable and off resonance
is
,
(2.35)
as suggested by the above analysis.
Summarizing, it is advisable to choose
and
from the more restricted domain
,
(2.36)
to avoid having a time-decoupled computational Rossby mode. It is also
advisable to use the smallest-possible values of
and
(consistent with being adequately off-resonance), to respectively minimize the
truncation errors (see 2.32) and to maximally damp the computational Rossby
mode.
The variable-resolution global shallow-water model described in
Côté et al. (1993) has been modified to include the present
generalized family of semi-implicit semi-Lagrangian time schemes. To evaluate
the proposed family of schemes and to provide some guidance on the choice of
the values for the parameters
and
,
the methodology introduced in RSR94 is adopted. The grid configuration and the
model orography used here are as depicted in their Figs. 4 and 5. The grid has
a uniform (
)
resolution window over N. America, and the orography is zero everywhere outside
this window. The experiments consist of 48-h forecasts initiated from the
500hPa height and wind analyses of 1200 UTC 12 February 1979 after an
implicit-normal-mode initialization. The initialized geopotential used in all
of the described experiments is displayed in Fig. 6 of RSR94.
Two methods for computing trajectories were described in RSR94, and for both of
these the upstream point at time
is obtained as in Côté et al. (1993) using winds extrapolated
forward from those at ess the problem of the spurious resonance of
semi-implicit semi-Lagrangian schemes at large Courant number, has been
generalized to a two-parameter three-time-level family by introducing the
possibility of evaluating total derivatives using an additional time level of
information. The RSR94 family can be considered to be the limiting case of the
generalized family that maximally simplifies the evaluation (using a
two-time-level difference) of the total derivatives. The other limiting case
is the "backward-implicit" scheme, where maximum simplification of the
evaluation (entirely at the arrival point) of the other terms occurs, and where
an additional (third) time level is employed to evaluate the total derivatives.
It has the virtues that the only upstream evaluations required are those
associated with the total derivatives, and that it does not require derivative
terms such as
and
to be evaluated at any time other than the present one (which is done
implicitly and only at arrival meshpoints). Both of these limiting cases are
off-centered discretizations, to avoid spurious resonance, and all schemes
examined are
accurate.
Resonance, stability and truncation-error analyses have been performed for the
proposed generalized family of schemes. Theory then leads to the following
conclusions:
a) to avoid instability and resonance the domain of validity for the parameters
is (
,
);
b) it is advisable to choose
and
from the more restricted domain (
,
),
to avoid having a time-decoupled computational Rossby mode;
c) it is also advisable to use the smallest-possible values of
and
(consistent with being adequately off resonance), to respectively minimize the
truncation errors and to maximally damp the computational Rossby mode;
d) the backward-implicit scheme has twice the damping rate of the RSR94 scheme,
even though they both have the same leading-order truncation error; and
e) although there is a sub-family of the generalized family of schemes that is
formally
-accurate
(
,
free), it is unstable for the gravity modes; this is unfortunate; starting from
a stable-but-resonant centered two-time-level scheme (
),
a degree of freedom can be added to address resonance (by using an additional
time level and thereby decentering either the evaluation of the total
derivatives or the other terms), but adding a further degree of freedom (while
still only using three time levels) does not permit increasing accuracy to
.
Sample integrations for various members of the generalized family were
performed. It was found that the results were consistent with the theory, and
that stable non-resonant forecasts at large Courant number are possible for a
fairly-broad choice of values for the
and
parameters. Within the RSR94 family of schemes, the scheme (
)
for which results were shown in RSR94 is a good choice but not an optimal one.
A smaller value of
than 1/2, but no smaller than 1/6, can reduce the leading-order term for the
truncation error by as much as a factor of 2 (see 2.32) and the damping rate by
as much as a factor of 3 (see 2.33), while still acceptably controlling
resonance. Finally, it was found that while either of the two methods
presented in RSR94 for computing the trajectories is acceptable for their
scheme, only one of them is acceptable at large timestep for some members of
the generalized family. The method that computes the trajectory in a piecewise
fashion is therefore to be preferred to the one that uses a single great-circle
trajectory.
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Fig. 1 The geopotential height (dam) for the 48-h "control" forecast using a
centered 2-time-level scheme (
)
with
=10 min.
Fig. 2 As in Fig. 1, but for the
"backward-implicit" scheme (
)
with
= 60 min.
Fig. 3 As in Fig. 1, but for an uncentered RSR94 scheme (
)
with
= 60 min.
Fig. 4 The rms ge242E-02FULLGRD11700
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THE TIME STEP 36 IS COMPLETED
THE TIME STEP 37 IS COMPLETED
THE TIME STEP 38 IS COMPLETED
THE TIME STEP 39 IS COMPLETED
THE TIME STEP 40 IS COMPLETED
THE TIME STEP 41 IS COMPLETED
THE TIME STEP 42 IS COMPLETED
THE TIME STEP 43 IS COMPLETED
THE TIME STEP 44 IS COMPLETED
THE TIME STEP 45 IS COMPLETED
THE TIME STEP 46 IS COMPLETED
THE TIME STEP 47 IS COMPLETED
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ECRIT(63) 289-GO P 2000 24 0 136 101 1 GEF772 6021293122 1800 48 Z 0 0 1 0 R16 1007926 3436
GO1826519139.287095 12752 121522.9932873652 9050 140595.4755748139FULLGRD3400
ECRIT(63) 290-GO P 3400 24 0 136 101 1 GEF772 6021293122 1800 48 Z 0 0 1 0 R16 1011376 3436
GO868474489.2723909 12616 56311.38878271090 9454 67709.04178724089FULLGRD6300
ECRIT(63) 291-GO P 6300 24 0 136 101 1 GEF772 6021293122 1800 48 Z 0 0 1 0 R16 1014826 3436
GO312658296.5220652 12610 19893.64884636139 9184 24871.66923459355FULLGRD9480
ECRIT(63) 292-GO P 9480 24 0 136 101 1 GEF772 6021293122 1800 48 Z 0 0 1 0 R16 1018276 3436
GO33213607.13354784 12748 1978.805145948684 9184 2724.236434006759FULLGRD11700
ECRIT(63) 293-GO P 11700 24 0 136 101 1 GEF772 6021293122 1800 48 Z 0 0 1 0 R16 1021726 3436
GO0.000000opotential height differences (m) of four schemes wrt the
control as a function of timestep (h). Solid - RSR94 (
);
dash/dot - RSR94 (
);
long dashed - backward implicit (
);
short dashed - (
).
)