The wave parameters consist of the speed of gravity waves, gH, the wave number and wave mode, k and n, the wave amplitude, A, and the wave type. Any given combination of these parameters completely specifies a unique wave using the expressions in Eqs. ()–(6). We consider all other parameters, including the spatiotemporal resolution and the form of diffusion/viscosity terms, to be modeling choices left to the developers. This approach is aimed at testing the models in their intended mode of operation. However, as noted in , different choices of the form of diffusion/viscosity terms correspond to different sets of equations and may not converge to the same solutions.

The choice of gravity-wave speed gH is inspired by the observed speed of gravity waves of the baroclinic modes in the atmosphere. In practice, we keep g fixed to Earth's gravitational acceleration and set the speed of gravity waves by letting H=30 m, which is within the range of observed equivalent depths in the equatorial atmosphere . As mentioned in Sect. , the analytic solutions obtained by on the equatorial β-plane are only accurate approximations of the SWEs on the sphere in the asymptotic limit of low gravity-wave speeds. The above value was found to be sufficiently accurate, using trial and error, in the sense that it yields stable solutions for at least 100 wave periods in the simulations described in Sect. .

In addition to the speed of gravity waves, the accuracy of Matsuno's solutions also depends on the wave number and wave mode. For a given value of gH, these solutions become asymptotically accurate in the limits ks, n→0 (but ks≠0) . Also, the spatial variability and the required spatial resolution both increase with the wave number or wave mode, so both of these considerations suggest that reasonable choices for the wave number and wave mode consist of small to moderate values. The proposed wave number and wave mode are ks=5 (i.e., k=5/a m-1) and n=1, i.e., within the range of dominant values observed in the equatorial atmosphere , but other choices may work just as well provided ks and n are not too large.

The proposed test case is based on the solutions of the linear SWEs but is intended to be used in non-linear models. Therefore, the wave-amplitude should be sufficiently small so as to satisfy the linearization condition. The proposed amplitude of v^ in Eq. (6) is A=10-5 m s-1, chosen by trial and error so as to enable stable solutions for at least 100 wave periods in the simulations of Sect. .

In general, there are two qualitatively different wave types, Rossby and IG, which differ in the magnitude of their divergence and vorticity fields. The former is more solenoidal (non-divergent), whereas the latter is more irrotational. In order to assess the performance of the models in these two qualitatively different limits we suggest using one of each. As Rossby waves are exclusively westward propagating, we choose the EIG wave of the two IG waves as the second case to cover the two directions of longitudinal propagation.

For these chosen values of gH, k and n, the wave periods (T=2πω) are T=18.5 d for the Rossby wave and T=1.9 d for the EIG wave.

For sufficiently small wave amplitudes we expect the spatiotemporal structure of the simulated solutions to be that of zonally propagating waves, i.e., q=q^(ϕ)ei(kλ-ωt) (where q stands for any of the dependent variables u, v or Φ), with frequency and latitude-dependent amplitudes corresponding to the initial wave. In this case, it is desirable to assess the accuracy of the zonal and meridional structures of the waves independently. A fast and simple way of doing so is using Hovmöller diagrams, where the temporal change in any direction is isolated by intersecting the fields along a fixed value of the other direction. This results in the following two diagrams: i.

A time–longitude diagram is obtained by intersecting the fields at a certain latitude. The contour lines in the time–longitude plane are the set of points satisfying kλ-ωt= constant (for some real constant). Thus, the expected pattern for this diagram is that of straight lines with slopes that equal the inverse of the wave's phase speed k/ω. In order to avoid small fluctuations in the vicinity of latitudinal zero crossings, we recommend using latitudinal intersects at or near local extrema.

Likewise, for zonally propagating waves it is also desirable to isolate the errors in the phase speed and spatial structure. As discussed in , the frequently used spherical l2 error entangles the two, and is therefore of lesser use for assessing the accuracy with which the model simulates a propagating wave. Thus, for a more quantitative assessment we suggest using the relative difference between the root-mean-square of the analytic solution and the simulated solutions, i.e., 8I[q2]-I[qa2]I[qa2], where the quantities q and qa (which are scalars in the case of geopotential and vectors in the case of velocity) correspond to the simulated and analytic solutions, respectively, and where 9I[q]=14π∫02π∫-π/2π/2q(λ,ϕ)cos⁡ϕdϕdλ. Henceforth we refer to the quantity in Eq. () as the “structure error” as, in contrast to the l2 error, it is unaffected by phase speed errors (i.e., phase shifts in λ).

The model is a spherical version of the Cartesian model used in , in which the integration forward in time is carried out using the conservation form of the SWEs ∂U∂t+1acos⁡ϕ∂∂λU2h+1a∂∂ϕUVh10a-2UVtan⁡ϕah-2Ωsin⁡ϕV=-g2acos⁡ϕ∂h2∂λ∂V∂t+1acos⁡ϕ∂∂λUVh+1a∂∂ϕV2h-(U2-V2)tan⁡ϕah10b+2Ωsin⁡ϕU=-g2a∂h2∂ϕ10c∂h∂t+1acos⁡ϕ∂U∂λ+∂(Vcos⁡ϕ)∂ϕ=0, where U=hu, V=hv and h is the total layer thickness. The numerical scheme employs a standard finite-difference shallow-water solver in which the time-differencing follows a leapfrog scheme (center differencing in both time and space). The computations were carried out on an Arakawa C-grid. The model contains provisions for a temporal Robert–Asselin filter, but the filter's coefficient was set to zero in the simulations of the present section. In addition, the model includes no diffusion/viscosity terms.

The computational domain is -180∘≤λ≤180∘ and -30∘≤ϕ≤30∘. The boundary conditions are periodicity at the zonal boundaries λ=±180∘ and vanishing meridional velocity at the channel's boundaries ϕ=±30∘. The corresponding values of h and U at the boundaries are determined by the differential equations. For the chosen wave parameters the amplitude of the meridional velocity v^ in Eq. (6) has an e-folding latitude of 11∘, and its amplitude at ϕ=±30∘ decays to 4×10-3 of its maximal value; therefore, the velocity outside the computational domain can be comfortably neglected. The grid spacing and time step are Δλ=Δϕ=0.5∘ and Δt=600 s, which were found to yield stable solutions for at least 100 wave periods.

Figure  shows the initial u, v, Φ, ξ, and δ fields (where ξ and δ are the relative vorticity and divergence, respectively) of the chosen Rossby wave mode (Fig. 1a–e), and the resulting latitude–time (Fig. 1f–j) and time–longitude (Fig. 1k–o) Hovmöller diagrams of the simulated solution. The initial fields were obtained using the analytic expressions from Sect.  and the wave parameters from Sect. . The simulated solutions were obtained using the abovementioned equatorial channel model. The chosen intersects used in the calculation of the Hovmöller diagrams are indicated by white dashed lines superimposed on the initial fields, and are also provided in the figure's caption. For the sake of legibility the time domain shown in each panel is only the last wave period of the simulation, i.e., 99T≤t≤100T, where T is the wave period. The fields are normalized on their global maximum at t=0. Thus, white regions correspond to times at which the simulated solution exceeds the initial wave amplitude, momentarily. With this in mind, recall that the patterns in the latitude–time diagrams are similar to those in latitude–longitude diagrams, and can therefore be used for comparison with the initial fields. In general, the initial wave structure is preserved and the dominant slope in the time–longitude diagrams corresponds to the analytic slope indicated by dashed white lines (Fig. 1k–o). There are, however, some noticeable deviations: a slight east–west tilt can be observed in the latitude–time diagrams (Fig. 1f–j), but most egregiously, the divergence field is less regular than the other four. We return to this last point at the end of Sect. 4.3. The phase of the simulated patterns in the latitude–time diagrams fit the expected patterns considering the westward propagation of the Rossby mode at λ=-18∘ in 1 wave period after 99 wave periods.

Similarly, Fig.  shows the initial u, v, Φ, ξ, and δ fields of the chosen EIG wave mode (Fig. 2a–e), and the resulting latitude–time (Fig. 2f–j) and time–longitude (Fig. 2k–o) Hovmöller diagrams of the simulated solution. Note that under the normalization used here the initial v field is independent of the wave type and is therefore identical to the one in Fig. . In contrast to Fig.  the patterns in the latitude–time diagrams of the simulated solutions are noticeably out of phase. However, considering the agreement between the dominant slope in the time–longitude diagrams and the analytic slope indicated by the dashed white lines (Fig. 2k–o), it is reasonable to say that this phase shift only results from a small phase speed error that accumulates over time. In addition, in contrast to the Rossby wave in Fig. , the divergence field in this case is just as regular as the other four fields.

The structure error defined in Eq. () is shown in Fig.  for both Rossby (Fig. 3a) and EIG (Fig. 3b) waves as a function of time. In both cases the structure error fluctuates about a mean value of less than 1 % and there is no visible trend throughout the simulation time of 100 wave periods. Recall that the structure error defined in Eq. () is insensitive to phase differences.

To demonstrate the applicability of the Matsuno wave as a test case for global-scale models we use the Geophysical Fluid Dynamics Laboratory's (GFDL's) spectral transformed shallow-water model which uses spherical harmonics as its basis functions (https://www.gfdl.noaa.gov/idealized-spectral-models-quickstart/). The chosen spectral resolution was T85, i.e., a triangular truncation where both the highest retained wave number and the total wave number equal 85. The chosen time step was Δt=600 s, as in the equatorial channel model. The model contains provisions for hyper-diffusion terms as well as a temporal Robert–Asselin filter, but the coefficients of both were set to zero for the simulations described below.

Figure  shows the initial u, v, Φ, ξ, and δ fields (where ξ and δ are the relative vorticity and divergence, respectively) of the chosen Rossby wave mode (Fig. 4a–e), and the resulting latitude–time (Fig. 4f–j) and time–longitude (Fig. 4k–o) Hovmöller diagrams of the simulated solution. The initial fields were obtained using the analytic expressions from Sect.  and the wave parameters from Sect. . The simulated solutions were obtained using the abovementioned GFDL global-scale spectral model. The chosen intersects used in the calculation of the Hovmöller diagrams are indicated by white dashed lines superimposed on the initial fields, and are also provided in the figure's caption. For the sake of legibility the time domain shown in each panel is only the last wave period of the simulation, i.e., 99T≤t≤100T, where T is the wave period. The fields are normalized on their global maximum at t=0. Thus, white regions correspond to times at which the simulated solution exceeds the initial wave amplitude, momentarily. With this in mind, recall that the patterns in the latitude–time diagrams are similar to those in latitude–longitude diagrams, and can therefore be used for comparison with the initial fields. Indeed, the patterns in the latitude–time diagrams of the simulated solutions agree quite accurately with those of the initial wave structure, but are noticeably out of phase. Nevertheless, considering the agreement between the dominant slope in the time–longitude diagrams and the analytic slope indicated using the dashed white lines (Fig. 4k–o), it is reasonable to say that this phase shift results from a small phase speed error that accumulates over time. In addition, the divergence field is less regular than the other four fields. We return to this point at the end of Sect. 4.3.

Similarly, Fig.  shows the initial u, v, Φ, ξ, and δ fields of the chosen EIG wave mode (Fig. 5a–e), and the resulting latitude–time (Fig. 5f–j) and time–longitude (Fig. 5k–o) Hovmöller diagrams of the simulated solution. Note that under the normalization used in the present paper the initial v field is independent of the wave type and is therefore identical to the one in Fig. . As in Fig. , the patterns in the latitude–time diagrams of the simulated solutions are noticeably out of phase, but the dominant slope in the time–longitude diagrams (Fig. 5k–o) agrees well with the analytic slope, indicating that the observed phase shift results from a small phase speed error that accumulates over time.

Finally, the structure error in Fig.  fluctuates about a mean value of less than 1 % and there are no visible trends throughout the 100 wave period simulations. Recall that the structure error defined in Eq. () is insensitive to phase differences.

In this section we examine the generation of small-scale features and the stability of the proposed wave solutions in a similar fashion to that used in for the original Rossby–Haurwitz wave number 4.

In , the generation of small-scale features and the potential enstrophy cascade is observed by examining the potential vorticity field, which generates tongues that wrap up around themselves and break the initial east–west symmetry. For the small wave amplitude A=10-5 m s-1 used in the present work, the potential vorticity is dominated by the planetary vorticity which is 5–6 orders of magnitude (depending on the wave) larger than the relative vorticity. Thus, instead of the potential vorticity we examine the relative vorticity (as well as the geopotential). Figures –, as well as Figs. –, show the evolution of these two fields between t=99T and t=100T, where T is the wave period in each case. Clearly, both fields remain regular throughout the simulations and do not develop small-scale features such as those observed in . Recall that the simulations in the present work were carried out without any diffusion/viscosity terms. Thus, the simulations remain stable for at least 100 wave periods with no need to remove potential enstrophy at the grid scale.

In order to examine the stability of the chosen initial waves we repeat the simulations of the previous section with an added perturbation (white noise) to the initial fields. We demonstrate the stability of the waves using only the global-scale model, which was found to yield more stable results when adding the perturbation.

Figures  and  show the initial fields of the perturbed Rossby (Fig. 7a–e) and EIG waves (Fig. 8a–e), respectively, and the resulting latitude–time (Figs. 7f–j and 8f–j) and time–longitude (Figs. 7k–o and 8k–o) Hovmöller diagrams of the simulated solution, obtained using the GFDL global-scale spectral model. The initial perturbation in these figures consist of a uniformly distributed random white noise with an amplitude of 5 % of the field's amplitude added to each of the fields u, v, and Φ. Specifically, let q stand for any of the variables u, v or Φ, then the initial perturbation is given by 11q=qa+0.05max⁡λ,ϕqa2R-1, where qa is the analytic solutions obtained as in Sect. , and R is a uniformly sampled random array with elements in (0,1) whose dimensions are the same as qa (in the present work a different R was drawn for each of the three variables).

Overall, the perturbed waves seem to be stable. The u, v and Φ fields are almost as regular as those of the non-perturbed waves, except for the zero contour. The small-scale features in the vorticity field of the perturbed Rossby wave smooth out with time, in contrast to the potential vorticity field of the Rossby–Haurwitz wave number 4. Conversely, the perturbed Rossby wave divergence field is completely eroded. The vorticity and divergence fields of the perturbed EIG wave are not as regular as those of the non-perturbed wave. However, they too become smoother with time and the initial wave remains the most dominant wave throughout the entire 100 wave period simulation. The structure error in Fig.  is similar to the previous ones in Figs.  and . These results are quite surprising, as we would expect a sufficiently large perturbation to excite other modes, regardless of the waves' stability.

Both the non-perturbed Rossby wave in Sect.  and , and the perturbed Rossby wave in the present section indicate that the divergence field is more sensitive than the other four fields of the Rossby wave. An immediate suspect in this regard is the divergence field amplitude, which is small for the chosen Rossby wave. For reference, the meridional wind amplitude for the chosen waves parameters (of both the Rossby and EIG waves) is 6.4×10-6, whereas the Rossby wave divergence field amplitude is 2.6×10-12. Conversely, the divergence field amplitude is only 1 order of magnitude smaller than the vorticity field amplitude, which is 2.7×10-11. Regardless of the cause, the fact that all other four fields remain quite regular while the divergence field is completely eroded suggests that the small but significant divergence field described by is in fact a small and insignificant divergence field.

In addition to the test cases proposed by a resolution convergence test of linearized SWEs in which the simulations are compared to higher-order simulations is also useful for ensuring that the errors decrease with the increase in resolution. In this section we demonstrate that Matsuno's analytic wave solutions can be used for this purpose. We use the equatorial channel model which can be easily turned into a linear shallow-water model.

Figure  shows the structure error in absolute value as a function of the grid spacing Δ=Δλ=Δϕ, from Δ=2.5∘ to Δ=0.25∘ every 0.25∘. For each resolution, the initial non-perturbed waves were integrated for 100 wave periods. As an estimate of the structure error at each resolution we use the time-series averages (indicated by dots). The error bars were estimated using the standard deviations of the entire time series. As the resolution increases from Δ=2.5∘ to Δ=0.25∘, the structure error time-series average decreases from about 2 % to less than 1 %, and the standard deviation decreases from about 2 % to about 0.5 %. The time step across all resolutions in this figure was held fixed at Δt=100 s. Note that all of the results of the previous sections were obtained for Δ=0.5∘ and Δt=600 s. This time step was found to yield convergent results for Δ=0.5∘ in the sense that decreasing the time step by a factor of 2 yields no improvements. Nevertheless, for the convergence test in the present section we have further decreased the time step to Δt=100 s in order to allow a further increase in the spatial resolution by another 0.25∘. This has also enabled a comparison with the results of the previous sections, thus ensuring that the simulations remain stable. Needless to say, for any time step one can expect to encounter numerical instabilities at some (high) spatial resolution.