Horizontal discretization on different grids and discrete
dispersion equation
In this section, we discuss the discretization of the basic equations and
derive the discrete dispersion relation on each horizontal grid. At the end
of this section, we present an illustrative discussion of the dispersion
equations showing frequency plots that are similar to the ones presented in
Part 1.
Solutions for the Z grid
Baroclinic Rossby modes. We horizontally discretize Eqs. (1)–(3) on
the Z grid shown in Fig. 1a of Part 1 as
∂∂tωzi,j=-f0Di,j-βf0Pi+1,j-Pi-1,j2d,0=f0ωzi,j-1d2Pi+1,j+Pi-1,j+Pi,j+1+Pi,j-1-4Pi,j,
and
N2Di,j=∂2∂z2-12ρ0∂ρ0∂z2∂Pi,j∂t,
respectively. By using Eq. (16) of Part 1 in Eqs. (1)–(3), we obtain the
discrete dispersion relation as
ν=-βξ̃kξ2k2+η2ℓ2+f02N2m2+14H2,
where
ξ̃≡sinkdkd,ξ≡sin12kd12kdandη≡sin12ℓd12ℓd.
Barotropic Rossby modes. We horizontally discretize Eqs. (5) and (6)
on the Z grid as
∂∂tωzi,j=-βf0Pi+1,j-Pi-1,j2d,
and
0=f0ωzi,j-1d2Pi+1,j+Pi-1,j+Pi,j+1+Pi,j-1-4Pi,j,
respectively. Equations (12) and (14) can also be obtained by assuming D=0
in Eqs. (8) and (9), respectively. The discrete dispersion equation for the
barotropic modes is
ν=-βξ̃kξ2k2+η2ℓ2,
where ξ̃, ξ and η are given by Eq. (12), which have
the same definitions in Part 1. For d→0, both Eqs. (11) and (15) become
identical to their continuous counterparts given by Eqs. (4) and (7),
respectively. This confirms that the discrete solutions are consistent and
that they correspond to the solutions of the continuous equations. On the
other hand, as the zonal scale approaches the shortest resolvable zonal scale
(hereafter SRZS), i.e., kd→π and ξ̃→0, the discrete
modes lose their ability to recognize the β effect, and the frequency
of the modes becomes zero at the SRZS. This result has been derived using the
β-plane approximation. It is not immediately clear whether or not it
holds in true spherical geometry. This could be studied through a discrete
normal-mode analysis on the sphere and/or numerical integrations of the
linearized equations on the sphere.
As in Part 1, we present plots of the discrete dispersion of the Rossby modes
generated by using the Z, C, D, CD, A, E and B grids. The basic state and
plot design are the same as Part 1. We use β=1.62×10-11 m-1 s-1, which is typical for a midlatitude plane.
The dispersion plots for baroclinic and barotropic Rossby modes with the Z
grid are presented in Fig. 1. The most striking feature is that the
frequencies of all modes, for all vertical scales and horizontal grid
spacings, approach zero at the SRZS. We use k=ℓ to plot these results.
This is a consequence of the use of the centered finite difference to
approximate the zonal pressure gradient at cell centers. As a result, the
β effect cannot be recognized by any of the modes at the SRZS.
Consequently, a dynamically inert mode is generated. Again, it should be
checked whether or not this conclusion carries over to the linearized
equations on the sphere.
Plots of the absolute value of frequency of the baroclinic (red
lines) and barotropic (dashed red lines) Rossby modes obtained on the Z grid
for the grid spacings (a) 2 km, (b) 10 km,
(c) 25 km and (d) 100 km, and for the various vertical
wavenumbers. The thin blue and thick green dashed lines are the
corresponding true baroclinic and barotropic frequencies, respectively.
Same as Fig. 1 but on the C grid.
Solutions for the C grid
Baroclinic Rossby modes. We horizontally discretize Eqs. (1) and (2)
on the C grid shown in Fig. 1b of Part 1 as
∂ωzi+1/2,j+1/2∂t=-f014Di,j+Di+1,j+Di,j+1+Di+1,j+1-βf0P‾i+3/2,j+1/2-P‾i-1/2,j+1/22d,
where
P‾i+1/2,j+1/2≡14Pi,j+Pi+1,j+Pi,j+1+Pi+1,j+1,and0=f014ωzi+1/2,j+1/2+ωzi+1/2,j-1/2+ωzi-1/2,j+1/2+ωzi-1/2,j-1/2-1d2Pi+1,j+Pi-1,j+Pi,j+1+Pi,j-1-4Pi,j.
Equations (16), (17) and (10) complete the set of discrete equations for the
C grid. By using Eqs. (16) and (22) of Part 1, we obtain the discrete
dispersion relation as
ν=-μ2ξ̃βkξ2k2+η2ℓ2+μ2f02N2m2+14H2,
where ξ̃, ξ and η are given by Eq. (12), and
μ≡cos12kdcos12ℓd.
The definition of μ is identical to that used in Part 1.
Barotropic Rossby modes. By using D=0 in Eqs. (16) and (17) and
then using Eqs. (16) and (22) of Part 1, we obtain the discrete dispersion
relation for barotropic Rossby modes on the C grid as
ν=-μ2ξ̃βkξ2k2+η2ℓ2.
The discrete baroclinic and barotropic dispersion relations, Eqs. (18)
and (20), for the C grid include an averaging factor μ2. This is a
difference from their Z-grid counterparts, Eqs. (11) and (15). Averaging of
pressure term P from the cell centers to the corners in Eq. (17) leads to
the factor of μ2 in the numerators of Eqs. (18) and (20). A factor of
μ2 also appears in the inertia term at the denominator of Eq. (18),
due to the averaging of divergence and vorticity to each other's grid points.
Since μ and ξ̃ are both equal to zero at the SRZS,
dynamically inert modes exist for both the baroclinic and barotropic Rossby
modes on the C grid, similar to those that exist in the Z-grid solutions.
The C-grid solutions shown in Fig. 2 are qualitatively similar to the Z-grid
solutions, but the C-grid solution deviates slightly because the dispersion
relation for the C grid given by Eq. (18) contains an averaging factor
μ2 in the numerator. Since μ also approaches zero at the SRZS,
and ξ̃ approaches zero, the small-scale modes on the C grid
move or oscillate more slowly than on the Z grid. As mentioned above, at the
SRZS, a dynamically inert mode is generated with the C grid.
Solutions for the D grid
Baroclinic Rossby modes. We horizontally discretize Eqs. (1) and (2)
on the D grid shown in Fig. 1c of Part 1 as
∂ωzi,j∂t=-f014Di-1/2,j-1/2+Di+1/2,j-1/2+Di-1/2,j+1/2+Di+1/2,j+1/2-βf0Pi+1,j-Pi-1,j2d,
and
0=f014ωzi+1,j+1+ωzi,j+1+ωzi+1,j+ωzi,j-1d2P‾i+3/2,j+1/2+P‾i+1/2,j+3/2+P‾i+1/2,j-1/2+P‾i-1/2,j+1/2-4P‾i+1/2,j+1/2,
respectively. In Eq. (22), P‾i+1/2,j+1/2≡14Pi,j+Pi+1,j+Pi,j+1+Pi+1,j+1. By
adding the discrete version of Eq. (3) given by
N214Di-1/2,j-1/2+Di+1/2,j-1/2+Di-1/2,j+1/2+Di+1/2,j+1/2-∂2∂z2-12ρ0∂ρ0∂z2∂∂tPi,j=0
to Eqs. (21) and (22), we complete the discrete equations for the D grid. The
resulting discrete dispersion relation is
ν=-ξ̃βkξ2k2+η2ℓ2+f02N2m2+14H2.
Barotropic Rossby modes. By using D=0 in Eqs. (21) and (22), and
using Eqs. (16) and (22) of Part 1, we obtain the discrete dispersion
relation for the discrete barotropic modes as
ν=-ξ̃βkξ2k2+η2ℓ2.
The dispersion equation for the discrete baroclinic and barotropic Rossby
modes on the D grid is identical to that of the Z-grid solution. In the
linear system, every averaging introduces a factor μ. For nontrivial
solutions of Eqs. (21)–(23), the factors of μ cancel each other. As a
result, the dispersion equation is identical to that of the Z grid.
Figure 1 is effectively a plot of the frequencies for the D grid because the
dispersion equations for the Z grid given by Eqs. (11) and (15) are identical
to those for the D grid, as given by Eqs. (24) and (25), respectively.
Solutions for the CD grid
Baroclinic Rossby modes. By dropping the finite-difference time
derivatives of divergence and vertical velocity in Eqs. (32)–(42) of Part 1
and adding -i∼12τμβ/f0ξ̃kP^ and -i∼τβ/f0ξ̃kP^ to Eqs. (32) and (38) of
Part 1, respectively, we write the CD-grid equations for a midlatitude
β plane as
Predictor step on the C grid:
ω^z(*)=μω^zn-12τf0D^-i∼12τμβ/f0ξ̃kP^,0=f0w^z(n)+L2P^,0=-i∼m+12HP^+B^(n),B^(*)=B^n-12τN2w^,D^+i∼m-12Hw^=0.
Corrector step on the D grid:
ω^zn+1=ω^zn-τf0μD^-i∼τβ/f0ξ̃kP^,0=f0w^z(n)+μL2P^,0=-i∼m+12HP^+B^(*),B^n+1=B^n-τN2w^,μD^+i∼m-12Hw^=0.
In these equations, ξ̃ is given by Eq. (12), L2≡ξ2k2+η2ℓ2, ξ and η are given by Eq. (12) and
μ is given by Eq. (15).
In this system, the divergence is a diagnostic variable, defined on the cell
corners. This is why the divergence D^ is multiplied by the
averaging factor μ in Eq. (31) but not in Eq. (26). Using Scheme I, as
discussed in Part 1, we eliminate ω^z(∗) by using
Eq. (26) in Eq. (32) and eliminate B^(∗) by using Eq. (29)
in Eq. (33). Then Eq. (43) of Part 1 is used to obtain the real frequency and
amplification factor equations as follows:
e2νiτμ2N2L2+f02σm2sin(2νrτ)+12τe2νiτμ2N2βξ̃cos(2νrτ)+τeνiτμ2N2cosνrτβξ̃k+2f02eνiτμ2-1σm2sinνrτ+12τμ2N2βξ̃k=0,
and
eνiτ=-b+b2-4ac2a,
where
a≡μ2N2L2cos2νrτ-12τμ2N2βξ̃ksin2νrτ+f02σm2cos2νrτ,b≡2f02μ2-1σm2cosνrτ-τμ2N2βξ̃ksinνrτ,
and
c≡f021-2μ2σm2-μ2N2L2.
Barotropic Rossby modes. By eliminating the divergence, vertical
velocity and buoyancy in Eqs. (26)–(35), we obtain the two-part dispersion
equation for the barotropic modes as
0=L22-12τβξ̃k2sinνrτ+τβξ̃kL2cosνrτ,
and
eνiτ=L2L2cosνrτ-12τβξ̃ksinνrτ.
At the SRZS, for which ξ̃=0 and μ=0, the real frequency
νr becomes 0 in Eqs. (36) and (39), and the amplification factor
eνiτ becomes 1 in Eqs. (37) and (40). The Supplement
gives a more detailed derivation of the discrete equations.
The CD-grid solution shown by Fig. 3 is virtually identical to that for the
Z-grid solutions (and D-grid solutions) shown in Figs. 1 and 2, respectively.
Same as Fig. 1 but on the CD grid.
DC grid
As stated above, the CD grid behaves similarly to the D grid rather than the
C grid in the numerical solution of the Rossby waves on a midlatitude β plane. The normal-mode analysis of the Rossby waves with the DC grid
produce a solution that is very close to the C-grid solution. A detailed
discussion and frequency plots are presented in the Supplement.
This is consistent with the findings of Part 1 that the correction step
dominates the solutions with the CD and DC grids.
Solutions for the A grid
Baroclinic Rossby modes. We horizontally discretize Eqs. (1)–(3) on
the A grid shown in Fig. 1e of Part 1 as
∂∂tωzi,j=-f0Di,j-βf0Pi+1,j-Pi-1,j2d,
and
0=fωzi,j-14d2Pi+2,j+Pi,j+2+Pi,j-2+Pi-2,j-4Pi,j,
respectively. Similarly, we obtain the discrete dispersion relation for the
baroclinic Rossby modes as
ν=-βξ̃kξ̃2k2+η̃2ℓ2+f02N2m2+14H2,
where the definition of ξ̃ is given by Eq. (12) and
η̃≡sinℓdℓd.
The frequency becomes zero at the SRZS because ξ̃ is zero in
the nominator of Eq. (43). This indicates the existence of a non-moving and
non-oscillating computational mode. Moreover, the factor of
ξ̃2 in the denominator causes the frequency to behave badly
near the smallest resolvable horizontal scale.
Barotropic Rossby modes. By dropping f02/N2m2+1/4H2 in Eq. (43), the discrete
dispersion relation for the barotropic Rossby modes can be obtained as
ν=-βξ̃kξ̃2k2+η̃2ℓ2.
The frequency of the barotropic modes becomes strongly negative
(retrogressing) at the SRZS. This means that small-scale barotropic Rossby
modes can behave very badly. We discuss the behavior of these modes in
connection with the plots below.
Same as Fig. 1 but on the A grid.
Same as Fig. 1 but on the E grid.
Same as Fig. 1 but on the B grid.
Figure 4 shows the frequency of the Rossby modes obtained on the A grid. The
A grid produces very fast retrogression speeds of the barotropic mode at the
SRZS. The baroclinic modes with short vertical scales retrograde faster than
the true solution near the SRZS, but right at the SRZS, they do not move at
all.
Baroclinic and barotropic Rossby modes with the E grid
Part 1 discusses in detail the horizontal discretization on the E grid. There
it is pointed out that the E grid can be viewed as the superposition of the
two C grids, in which the cell centers of one C grid are placed at the
corners of a second C grid. It is also shown that, from the vorticity and
divergence point of view, the E grid can be viewed as a superposition of two
independent and non-interacting Z grids, as shown in Fig. 1f of Part 1. The
dispersion relation for the E grid is identical to that for the Z grid, but the
smallest resolvable zonal scale extends to kmax=2π/d (and
ℓmax=2π/d) for the E grid. Therefore, the dispersions of
baroclinic and barotropic modes on the E grid are governed by Eqs. (11)
and (15) with kmax and ℓmax as described above. Recall
that we use a grid spacing of 2d with the E grid to maintain the
same cell density as with the other grids.
The E grid produces the wildest solutions, as shown in Fig. 5. It is the only
grid that generates prograding Rossby modes. The modes with all vertical
scales and horizontal grid spacings used in the models generate prograding
solutions near the SRZS. The deeper the mode is, the faster the progradation
speed is. The prograding modes are generated near the SRZS because the factor
ξ̃ yields negative values for k>π/d. A interpretation is
that the finite-difference pressure gradient determined over the two-grid
distance is subject to aliasing errors for zonal waves with k>π/d, which
causes the system to recognize the pressure gradient with the wrong sign.
Solutions for the B grid
Baroclinic Rossby modes. We can obtain the equations for the B grid
by ignoring ∂D/∂t in Eq. (63) of Part 1, replacing f
with f0 and using Eqs. (8) and (9). Similarly, we obtain the discrete
dispersion relation for the baroclinic Rossby modes on the B grid as
ν=-ξ̃βkξ2k2+η2ℓ2-12d2ξ2k2η2ℓ2+f0N2m2+14H2,
where the factors ξ, η and ξ̃ are defined by
Eq. (12). The frequency becomes zero for the SRZS because ξ̃ is
zero. The Laplacian term
ξ2k2+η2ℓ2-12d2ξ2k2η2ℓ2
also approaches zero in the numerator of Eq. (46) as the zonal wavenumber
approaches the SRZS. This makes the frequency behave similarly to that of the
A grid.
Barotropic Rossby modes. By dropping f02/N2m2+1/4H2 in
Eq. (46), we obtain the discrete dispersion relation of the barotropic Rossby
modes as
ν=-ξ̃βkξ2k2+η2ℓ2-12d2ξ2k2η2ℓ2.
The denominator approaches zero at the SRZS, which yields an infinite
retrogression speed for these modes.
Figure 6 shows the frequency of the Rossby modes on the B grid. As with the
A-grid solutions, the B grid produces infinitely fast retrogression speeds for
the barotropic mode at the SRZS, and the shallow baroclinic modes retrograde
faster than the true solution near the SRZS and do not move at all at the
SRZS.
Plots of (a, d)
true and discrete frequencies for the baroclinic and barotropic Rossby modes
obtained on the (b, e) L
and (c, f) CP grids. The thick blue and green dashed lines on the
left panels indicate the true baroclinic and barotropic frequencies,
respectively. The thick red and red dashed lines on the center and right
panels indicate the discrete baroclinic and barotropic frequencies,
respectively. The upper and lower panels show the plots for the maximum
vertical integer wavenumbers of nmax=320 (δz=250 m) and
nmax=80 (δz=1 km), respectively.
As discussed in Sect. 3.8 of Part 1, the A, E and B grids generate multiple
(or non-unique) solutions and dynamically inert modes. Here, we see that the
impact of the dynamically inert modes on the short Rossby waves is very
severe.
The results of our normal-mode analysis of the nonhydrostatic anelastic
barotropic and baroclinic Rossby waves on a midlatitude β plane
the C, D, A, E and B grids overall agree with the results of
Dukowicz's (1995) normal-mode analysis with the shallow-water equations. An
exception is that we include the prograding modes with the E-grid solutions,
whereas Dukowicz (1995) excludes them as “inadmissible”.
A summary of the continuous and discrete dispersion relations with
various horizontal and vertical grids.
True
Z and D grid
Baroclinic modes:
Barotropic modes:
Baroclinic modes:
Barotropic modes:
ν=-βkk2+ℓ2+f02N2m2+14H2
ν=-βkk2+ℓ2
ν=-βξ̃kξ2k2+η2ℓ2+f02N2m2+14H2
ν=-βξ̃kξ2k2+η2ℓ2
m≡πn/zT for n=1,2,3,…
ξ≡sin12kd/12kd, η≡sin12ℓd/12ℓd,
ξ̃≡sinkd/kd, 0≤kd,ℓd≤π
C grid
E grid
Baroclinic modes:
Barotropic modes:
Baroclinic modes:
Barotropic modes:
ν=-μ2ξ̃βkξ2k2+η2ℓ2+μ2f02N2m2+14H2
ν=-μ2ξ̃βkξ2k2+η2ℓ2
Same as Z grid but for 0≤kd,ℓd≤2π
μ≡cos12kdcos12ℓd
CD grid
Baroclinic modes:
Barotropic modes:
e2νiτμ2N2L2+f02σm2sin2νrτ+12τe2νiτμ2N2βξ̃kcos2νrτ
0=L22-12τβξ̃k2sinνrτ+τβξ̃kL2cosνrτ
+τeνiτμ2N2cosνrτβξ̃k+2f02eνiτμ2-1σm2sinνrτ
eνiτ=L2/L2cosνrτ-12τβξ̃ksinνrτ
+12τμ2N2βξ̃k=0
L2≡ξ2k2+η2ℓ2, σm2≡m2+1/4H2, 0≤kd,ℓd≤π
eνiτ=-b+b2-4ac/2a for a, b and c; see Eqs. (38a)–(38c)
A grid
B grid
Baroclinic modes:
Barotropic modes:
Baroclinic modes:
Barotropic modes:
ν=-βξ̃kξ̃2k2+η̃2ℓ2+f02N2m2+14H2
ν=-βξ̃kξ̃2k2+η̃2ℓ2
ν=-ξ̃βkLB2+f0N2m2+14H2
ν=-βξ̃kLB2
η̃≡sinℓd/ℓd, 0≤kd,ℓd≤π
LB2≡ξ2k2+η2ℓ2-12d2ξ2k2η2ℓ2
L grid
CP grid
Baroclinic modes:
Barotropic modes:
Baroclinic modes:
Barotropic modes:
ν=-μz2βkμz2k2+ℓ2+f02N2ζ2m2+μz214H2
ν=-βkk2+ℓ2
ν=-βkk2+ℓ2+f02N2ζ2m2+μz214H2
ν=-βkk2+ℓ2
ζ≡sin12mδz/12mδz μz≡cos12mδz 0≤mδz=nπδz/zT≤π for n=1,2,3,…
Vertical discretization of the linear anelastic equations on the L
and CP grids and discrete dispersion equation
Part 1 presents a discussion on the vertical grids, including a historical
perspective, used in atmospheric models. Our purpose in this section is to
assess and compare the performance of the L and CP grids in simulating
Rossby modes on a midlatitude β plane through a normal-mode analysis.
The L grid
By replacing f with f0 in Eqs. (65) and (66) of Part 1, adding the
β term -β/f0∂Pk/∂x to the right-hand side of Eq. (65) of Part 1
and dropping ∂Dk/∂t and ∂wk+1/2/∂t in Eqs. (66) and (67) of Part 1, respectively, and using
Eq. (70) of Part 1, we obtain (after some manipulations) the discrete
dispersion relation for the baroclinic Rossby modes as
ν=-μz2βkμz2k2+ℓ2+f02N2ζ2m2+μz214H2,
where
ζ≡112mδzsin12mδzandμz≡cos12mδz.
By dropping f02/N2ζ2m2+μz2/4H2 in Eq. (48), we obtain
the discrete dispersion relation for the barotropic Rossby mode as
ν=-βkk2+ℓ2.
In Eq. (48), the numerator is proportional to μz2, which is zero
for the smallest resolvable vertical scale (SRVS), for which mδz=π.
This means that, for all horizontal scales, the modes with the SRVS cannot
propagate. They are dynamically inert (computational) modes. The pressure in
the β term cannot recognize the SRVS buoyancy perturbation in the
vertical velocity equation Eq. (67) of Part 1 with the quasi-static
assumption (∂wk+1/2/∂t≈0). The frequency of
the discrete barotropic mode given by Eq. (50) is identical to the true
frequency in Eq. (7), which is expected because the barotropic mode has no
vertical structure and therefore is not affected by the vertical
discretization.
The CP grid
We now derive the discrete dispersion relation for the baroclinic and
barotropic Rossby modes on the CP grid, following the same strategy used
with the L grid. The results are
ν=-βkk2+ℓ2+f02N2ζ2m2+μz214H2,
and
ν=-βkk2+ℓ2,
respectively. The dispersion equation for the baroclinic Rossby modes on the
CP grid given by Eq. (51) does not have an averaging factor in the numerator,
and therefore it does not allow a dynamically inert mode with zero frequency
at the SRZS.
Figure 7 shows the frequencies as functions of composite horizontal
wavenumber of barotropic and baroclinic Rossby modes obtained with the L and
CP grids. The true frequencies are also shown in separate panels of the
figure. The figure shows the results for two vertical wavenumbers (or number
of layers), namely nmax=320 and 80. We included additional
frequency lines corresponding to more vertical wavenumbers than were used in
the plots of Sect. 3 (indicated by thinner solid lines in the plots). In the
L-grid solutions shown in Fig. 7b and e, the frequency of the smallest
vertical resolvable mode, identified by nmax , deviates greatly from
the true frequency, which yields zero values. Similar to the case of the
inertia–gravity modes, as the vertical scale approaches the smallest
resolvable scale, the modes gradually lose their ability to recognize the
effects of buoyancy and therefore baroclinicity. For the mode with the
smallest scale, the buoyancy and baroclinicity are completely decoupled from
the wind field; for that mode, the buoyancy is dynamically inert. In
contrast, the frequency of the CP-grid solutions shown in Fig. 7c and f is
generally close to the true frequency but slightly higher.
Summary and conclusions
We have discussed the effects on the dispersion of middle-latitude Rossby
waves of the horizontal and vertical discretizations of the quasi-geostrophic
(quasi-static) linearized equations on the A, B, C, CD, (DC), D, E and Z
horizontal grids and the L and CP vertical grids. We present a summary of the
discrete dispersions of Rossby modes for the horizontal and vertical grids in
Table 1 for an easy comparison.
The Z, C, D and CD (DC) grids generate similar dispersion of the baroclinic
and barotropic Rossby modes. All have a dynamically inert mode at the
SRZS because
these scales cannot recognize the β effect. The dispersion equations
for the A and B grids give infinite frequencies at the SHZS. Among all
horizontal grids, the E grid produces the wildest solutions. The Rossby modes
of all vertical scales near the SHZS prograde, while the true modes
retrograde. The A, E and B grids generate multiple (non-unique) solutions,
including dynamically inert (computational) modes. The impact of the
computational modes on the short Rossby modes appears very severe on these
grids.
The results of our normal-mode analysis of the Rossby waves for the C, D, A,
E and B grids overall agree with the results of Dukowicz's (1995) normal-mode
analysis with the shallow-water equations. Dukowicz (1995) considers the
prograding modes with the E-grid solutions “inadmissible”, however, while
we include them.
The selection of the vertical grid impacts the dispersion of the Rossby
modes as much as the horizontal grid selection. The modes with the smallest
resolvable vertical scale on the L grid do not retrograde. The CP-grid
solutions are much more accurate than the L-grid solutions.