We use a normal-mode analysis to investigate the impacts of the horizontal
and vertical discretizations on the numerical solutions of the
quasi-geostrophic anelastic baroclinic and barotropic Rossby modes on a
midlatitude

The results of our normal-mode analyses for the Rossby waves overall support the conclusions of the previous studies obtained with the shallow-water equations. We identify an area of disagreement with the E-grid solution.

In a companion paper (Konor and Randall, 2018; hereafter
Part 1), we discuss the horizontal discretization of the linearized anelastic
equations on the Z, C, D, CD, (DC), A, E and B grids, and vertical
discretization on the L and CP grids. We introduced the DC grid in Part 1 to
test the hypothesis that the CD-grid (and DC-grid) solutions are dominated by
the corrector step and the grid used with it. Part 1 focuses on the
dispersion of nonhydrostatic inertia–gravity modes on an

We use the quasi-geostrophic and quasi-static equations in our analysis
because Rossby waves are not significantly influenced by ageostrophic or
nonhydrostatic effects. Furthermore, the quasi-hydrostatic equations produce
an exact solution on the

In Sect. 2, we present the continuous linearized anelastic equations with the
quasi-geostrophic and quasi-static approximations on the midlatitude

In this section, we derive the basic linearized equations with the quasi-geostrophic (and quasi-static) approximations, referring to the equations of Part 1 when possible, for brevity.

Following Arakawa and Konor (2009), we assume quasi-geostrophic (and
quasi-static) balance with the midlatitude

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.

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

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

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

Same as Fig. 1 but on the C grid.

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

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.

In these equations,

In this system, the divergence is a diagnostic variable, defined on the cell
corners. This is why the divergence

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.

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

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.

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

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

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

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

A summary of the continuous and discrete dispersion relations with various horizontal and vertical grids.

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

By replacing

By dropping

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

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

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

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.

Fortran codes that are used to compute and plot the
frequencies for the CD grid will be provided by the corresponding author upon
request. Related files can also be found in

The supplement related to this article is available online at:

The authors declare that they have no conflict of interest.

We are grateful to Bill Skamarock for his comments and suggestions to improve the manuscript. We thank the reviewer Almut Gaßmann and an anonymous reviewer for their constructive and helpful comments. This research was supported by the National Science Foundation (NSF) under AGS-1500187, the US Department of Energy Office of Science DE-SC07050 (SciDAC), DE-SC00016273 (ACME) and DE-SC00016305 (CMDV). Edited by: Paul Ullrich Reviewed by: Almut Gaßmann and one anonymous referee