1. Introduction
While acoustic modes are present in the compressible atmosphere, they typically play an insignificant role in atmospheric processes of physical interest. This recognition has led to several different approaches in designing the numerics for nonhydrostatic simulation models. One technique is to remove acoustic modes from the governing equations by solving a modified anelastic set of equations, which requires the solution of a 3D Poisson equation using an implicit solver at each time step (e.g., Ogura and Phillips 1962; Wilhelmson and Ogura 1972; Lipps and Hemler 1982; Kurowski et al. 2014). A second approach is to solve the compressible equations using a semi-implicit scheme, resulting in a 3D Helmholtz equation for pressure that also must be solved using implicit numerics at each time step (e.g., Tapp and White 1976; Tanguay et al. 1990; Staniforth and Côté 1991). This approach stabilizes the acoustic modes by essentially retarding their frequencies. A third alternative numerical procedure treats the compressible equations by solving the terms responsible for vertical acoustic propagation using semi-implicit numerics, while solving for the terms accommodating horizontal acoustic propagation using split-explicit or horizontally explicit vertically implicit (HEVI) schemes (e.g., Klemp and Wilhelmson 1978; Klemp et al. 2007; Lock et al. 2014). This approach avoids the need for computationally expensive 3D implicit solvers and does not intermingle higher-frequency acoustic modes with lower-frequency modes of physical interest. However, care must be taken with this approach to ensure that energy does not artificially accumulate in these acoustic modes due to initialization or physics imbalances, nonlinear interactions, or model numerics. In this paper, we focus on techniques to filter acoustic modes through 3D divergence damping for this latter category of schemes, in which horizontally propagating acoustic modes are treated with explicit numerics. HEVI/split-explicit schemes are widely used in research and operational atmospheric models, and most of these models employ 3D divergence damping to control acoustic noise [e.g., MM5 (Dudhia 1993), COAMPS (Hodur 1997), ARPS (Xue et al. 2000), WRF (Klemp et al. 2007), Nonhydrostatic Icosahedral Atmospheric Model (NICAM; Satoh et al. 2008), LM/COSMO (Doms and Baldauf 2015), Model for Prediction across Scales (MPAS; Skamarock et al. 2012), and Icosahedral Nonhydrostatic GCM (ICON; Zängl et al. 2015)]. We are particularly interested in the design of acoustic filtering for the split-explicit numerics in MPAS, which employs a variable-resolution global mesh based on centroidal Voronoi tessellations.
In an early implementation of split-explicit numerics for integrating the compressible equations of motion, Klemp and Wilhelmson (1978) solved the vertically implicit terms using a centered Crank–Nicolson scheme that does not create numerical diffusion. Recognizing the need to filter acoustic modes, Durran and Klemp (1983) introduced a small forward centering of the vertically implicit numerics and verified that the damping achieved on the small acoustic time steps had little effect on the gravity wave modes of interest. This means of acoustic filtering is not entirely effective, however, as it does not affect acoustic modes that have little vertical structure. To more effectively damp acoustic modes in split-explicit solvers, Skamarock and Klemp (1992, hereafter SK92) proposed an explicit filter on the full 3D divergence to augment the filtering of vertical acoustic modes provided by off-centering the vertically implicit numerics. Analyzing the linear compressible Boussinesq equations [see Durran (2010), p. 409], they demonstrated that the explicit damping of 3D divergence can provide effective attenuation of acoustic modes with negligible effect on the gravity wave modes. Acoustic filters based on the SK92 approach have been implemented in the horizontal momentum equations in the WRF Model (Skamarock et al. 2008) and MPAS (Skamarock et al. 2012) and have proven effective in controlling acoustic noise.
Gassmann and Herzog (2007, hereafter GH07) evaluated the compressible linear 2D acoustic/gravity wave equations in analyzing the divergence-damping characteristics for the numerics implemented in the split-explicit German Weather Service Lokal–Modell (LM) model (see also Gassmann 2005). Their analysis suggested that divergence damping for the compressible equations should be applied in both the horizontal and vertical momentum equations to avoid undesirable effects on the gravity wave modes (i.e., phase errors in the gravity wave frequencies). Therefore, we begin in section 2 by documenting the suitability of employing divergence damping for the fully compressible system only in the horizontal momentum equations, provided the divergence to be filtered is suitably defined. In recent work with MPAS, some issues have arisen with regard to the divergence damping, particularly in applications on variable-resolution meshes, that have prompted a redesign of our techniques for horizontal acoustic-mode filtering. In section 3, we propose a new approach for horizontal damping of the full divergence in a numerically consistent manner and document its impact on both the acoustic and gravity wave modes in conjunction with the off-centering employed in the vertically implicit portion of the solver. For simplicity, the analysis of these numerical filters will be conducted for HEVI schemes to avoid the added complication of the split-explicit numerics. In section 4, we discuss how this acoustic filter is implemented in MPAS and present results from MPAS simulations in section 5 to illustrate its behavior.
2. Filtering horizontally propagating acoustic modes




















































While applying divergence damping with
3. Acoustic filtering in the numerical finite-difference equations
In constructing a divergence damping term in the horizontal momentum equation, it is beneficial to have a consistent numerical representation of the divergence, such that the divergence in the damping term in (7) has the same numerical form as the divergence in the





More recent adaptations of MPAS, however, have exposed deficiencies in employing divergence damping as expressed in (19). The current version of MPAS has the capability to run a time step for the dynamics that is smaller than that for the rest of the model system (scalar transport, physics, etc.). In this configuration, there might typically be two acoustic steps per dynamic time step and three dynamics steps per scalar/physics step. Using third-order Runge–Kutta for the dynamics time step, a total of four small time steps are then computed per dynamics time step, but only one of them (the second time step in the third Runge–Kutta stage) has







Notice that combining (21) and (25) recovers the same equation as (7), and now the divergence term has the same numerical form as the divergence in (24). Thus, one can also think of this adjustment as being the first step in the next acoustic time step, applied in a manner that maintains a consistent finite difference form for the divergence. We will refer to this form of the divergence damping as the time-adjusted acoustic filter. Adding the divergence damping as an adjustment at the end of the time step also has the computational advantage that, rather than constructing the full divergence for use in (25), it can be readily recovered using






















Amplification factors
Citation: Monthly Weather Review 146, 6; 10.1175/MWR-D-17-0384.1
Further analysis of (28) reveals that for
As suggested by the amplification factor (30), for

Amplification factor
Citation: Monthly Weather Review 146, 6; 10.1175/MWR-D-17-0384.1
The damping characteristics of the full acoustic/gravity wave amplification equation (28) are illustrated in Fig. 3 for

Amplification factors
Citation: Monthly Weather Review 146, 6; 10.1175/MWR-D-17-0384.1









The real part of the gravity wave frequency

Normalized gravity wave frequencies for the divergence-damping acoustic mode filter in a HEVI integration scheme with the damping term applied either at the beginning of the time step or as an adjustment at the end of the time step for
Citation: Monthly Weather Review 146, 6; 10.1175/MWR-D-17-0384.1


















4. Implementation of the acoustic filter in MPAS
















In applying a dimensionless acoustic filter coefficient
5. Acoustic filter in MPAS global forecasts
In real-data applications of the global MPAS over a wide range of scales, acoustic filtering plays a crucial role not only in the numerical efficiency and stability, but also in the quality of the analysis and forecast. Fast-moving, small-scale acoustic modes can interact with uncertainties in the initial state to accelerate the error growth and ultimately limit the predictability of the simulated flow (Hohenegger and Schär 2007). Acoustic noise can be particularly detrimental in conjunction with ensemble data assimilation, as high-frequency noise in the background forecast can be significantly amplified by sampling error in the cycled ensemble analysis. Unless it is effectively suppressed during the model integration, such spurious noise remaining in the forecast can lead to a noisy analysis, which in turn can deteriorate the quality and the numerical stability of the ensuing forecast. The noise can continuously accumulate through cycles and speed up the forecast error growth at each cycle (and may eventually cause the forecast to crash). As Ha et al. (2017) demonstrated, without the effective noise filtering, even if the ensemble analysis was produced in a variable-resolution MPAS mesh, the forecast error grew quickly to counteract the benefit of the local refinement (e.g., forecasting in a high-resolution mesh).
The surface pressure tendency reflects the vertically integrated mass divergence and is known to be especially sensitive to the presence of noise (McPherson et al. 1979). Previous studies of initialization schemes have evaluated the time evolution of surface pressure or the mean absolute tendency of surface pressure to measure the global noise levels (Williamson and Temperton 1981; Temperton and Williamson 1981; Lynch and Huang 1992; Huang and Lynch 1993). To test the behavior of the new acoustic filtering technique in MPAS as described in the previous section, global MPAS forecasts are initialized with the 1° × 1° NCEP Final (FNL) global analyses at 0000:00 UTC 15 June 2012 on three different meshes: 120- and 30-km quasi-uniform meshes and a 120–30-km variable-resolution mesh in which a 120-km global resolution is refined to a 30-km mesh spacing over the contiguous United States (CONUS). All the experiments have 55 vertical levels with the model top at 30 km, and the model integration time steps are 720, 180, and 180 s for the simulations on 120-, 120–30-, and 30-km meshes, respectively. For comparison, simulations were also conducted with the previous version of the acoustic filter in MPAS, described at the beginning of section 3.
To compare the high-frequency noise resulting from a cold-start model initialization, Fig. 5 shows the area-weighted global mean absolute tendency of surface pressure (Pa s−1) at every time step for 3-day forecasts using the two acoustic-filtering methods. In simulations labeled with “old” in Fig. 5, the divergence damping is applied by forward extrapolating the pressure gradient in the horizontal momentum equation as in (19), while the runs with the new acoustic filter applied as a final adjustment step as in (36) are simply shown by the mesh names. Both methods use the same filter coefficient

Time series of the area-weighted global average of the absolute value of surface pressure tendency (Pa s−1). Each color represents an MPAS 3-day global forecast on different meshes using two different acoustic filtering methods, initialized with the same FNL analyses valid at 0000:00 UTC 15 Jun 2012. The 120–30-km variable-resolution mesh is configured with the local 30-km refinement over the CONUS domain. The divergence damping is applied using the old method (dotted lines) and the new method (solid lines) as explained in the text.
Citation: Monthly Weather Review 146, 6; 10.1175/MWR-D-17-0384.1
The new acoustic filter has been further tested on a much larger number of cases by running 120-km mesh MPAS forecasts initialized at 0000 and 1200 UTC on the first day of each month every month for 5 years (120 cases). For these cases, the average amplitude of the surface pressure tendency for the 6-h forecasts is found to be about 0.015 Pa s−1 with little variance across cases. This asymptotic value for the MPAS forecasts is comparable to the ones presented in Lynch and Huang (1992) in their digital filter initialization (DFI) study. By applying a digital filter in the High-Resolution Limited-Area Model (HIRLAM) simulations at half-degree resolution, they obtained asymptotic values of mean absolute surface pressure tendency of ~0.01 Pa s−1 in a 6-h forecast. Note that no initialization schemes such as DFI are applied in our MPAS runs to further reduce the noise caused by initial dynamical imbalances.
Figure 6 illustrates the horizontal distribution of the surface pressure tendency at three different forecast lead times for the 30-km uniform mesh using the old filtering technique. With the old acoustic filtering, spurious small-scale noise gradually decreases with time, but a significant amount still remains everywhere in the 6-h forecast. One interesting aspect of this case is the ring of large pressure tendencies emanating from the vicinity of the southern Philippine Sea. This disturbance is caused by the presence of Typhoon Guchol, which had category 2 intensity around the initialization time, but was poorly resolved in the 1° NCEP FNL analysis. The acoustic character of these high-frequency waves is confirmed by the radial expansion of the disturbance ring, with an estimated velocity greater than 305 m s−1. A second, weaker disturbance ring is also visible in Fig. 6a, centered off the southwest coast of Mexico. This acoustic noise is caused by Hurricane Carlotta, which was a category 2 storm around this time. These kinds of initial dynamic imbalances are a natural consequence of initializing a model forecast from a coarse-resolution analysis.

Horizontal distribution of surface pressure tendency (Pa s−1) in the old filtering approach for the 30-km uniform mesh (e.g., “30 km_old”) at (a) 2-, (b) 4-, and (c) 6-h forecast times, starting from the initialization time at 0000:00 UTC 15 Jun 2012.
Citation: Monthly Weather Review 146, 6; 10.1175/MWR-D-17-0384.1
On the other hand, the new acoustic filtering method efficiently eliminates such high-frequency acoustic modes, as illustrated by the significantly reduced noise levels at 2 h shown in Fig. 7. (Comparable plots at 4 and 6 h are not shown, as the noise levels at 2 h are already so much lower than those obtained with the old filtering approach at those later times.) Because mesoscale analysis/forecast cycling typically runs at the frequency of 6 h (or shorter), the noise remaining in the 6-h forecast can be closely associated with the forecast reliability and skill throughout the cycles. Therefore, the efficient suppression of high-frequency oscillations is particularly important at such short forecast lead times in mesoscale applications.

As in Fig. 6a, but for the new acoustic filtering method with the 30-km mesh at 2 h.
Citation: Monthly Weather Review 146, 6; 10.1175/MWR-D-17-0384.1
Figure 8 depicts vertical velocity (m s−1) at the 6-h forecast time in a vertical cross section over the Himalayas along the 30°N latitude line, as marked in Figs. 6 and 7. In comparing the two filtering methods, it is evident that vertical velocity perturbations are significantly smaller with the new approach (Fig. 8b), with the maximum amplitudes reduced by a factor of 2–3. Evaluating the time evolution of these perturbations (not shown), two main features are found: (i) the stronger upward motions (in orange) are triggered over the high topography around 5 h into the forecast, about the time the acoustic waves originating from the tropical storm traverse this high terrain area, and (ii) the waves appear to be quasi stationary, oscillating slowly over the next 6 h with maximum amplitudes in the range of 1–3 m s−1. Thus, it appears that acoustic noise caused by dynamical imbalances in the model initialization is artificially amplifying atmospheric waves that would be of physical interest in this real-data forecast. With the new acoustic filter (Fig. 8b), the excitation of perturbations over the Himalayas is much weaker, with the maximum vertical velocities remaining below 0.5 m s−1 over the first 12 h.

Cross sections of vertical velocity (m s−1) at 6 h along the 30°N line marked in Figs. 6 and 7 in the 30-km uniform mesh simulations with (a) old and (b) new acoustic filtering. Minimum and maximum values are shown within brackets in the upper-right corner. Contours range from −1.15 to 1.15 m s−1 at 0.1 m s−1 interval. Downward motions are contoured in dashed lines.
Citation: Monthly Weather Review 146, 6; 10.1175/MWR-D-17-0384.1
It should be also mentioned that when the acoustic filter is turned off altogether, the rapid growth of instabilities from the initial imbalances causes the model simulation to blow up in just under a 5-h integration time. Uncontrolled small-amplitude acoustic waves imposed on the initial conditions are quickly excited while propagating throughout the entire atmosphere and nonlinearly interacting with gravity (and inertia-gravity) waves to reach amplitudes that exceed the numerical stability limit.
6. Summary
In filtering acoustic modes by damping the full 3D divergence, the characteristics of the filter will depend on how the divergence is defined. For the adiabatic compressible atmosphere, based on the form of the pressure equation as expressed in (1), we have found that the expression for divergence that is most specific to the acoustic modes is given by
In numerically computing the horizontal divergence damping terms, we find that it is important that the numerical representation of the divergence used in the damping terms is numerically the same as the numerical form of the divergence contained in the
As illustrated by the real-data MPAS simulations in section 5, the model forecasts at early times may be severely contaminated by high-frequency oscillations as a result of dynamic imbalances in the initial conditions, which are typically interpolated from coarse-resolution analyses in cold-start simulations. With the acoustic filter applied as a final adjustment in the acoustic time step as indicated in (36), this high-frequency noise is rapidly attenuated and asymptotes to low residual values that do not appear to be sensitive to the mesh configuration. With the old acoustic filter, applied as in WRF based on (19), this acoustic noise is damped out much more slowly and asymptotes to higher values that are more sensitive to the specific mesh configuration.
It is not unexpected that the horizontal divergence damping technique used in the WRF Model is not sufficient to control acoustic noise in MPAS. As mentioned above, in the split-explicit numerics, the horizontal filter (19) is only applied after the first acoustic time step, which limits its application in the current configuration of MPAS that employs only two acoustic steps per dynamic time step. In addition, the WRF Model is less dependent on a 3D divergence filter since it also contains an external mode filter, as described by Skamarock et al. (2008), that effectively damps external modes that have large vertical wavelength. These external modes can be excited in the WRF Model as a result of its pressure-based sigma coordinate used in conjunction with a constant pressure boundary condition along the top of the model domain. The global MPAS uses a terrain-following height coordinate with a rigid-lid upper boundary, which does not support external modes, so this additional filter is not present in the MPAS numerics.
The National Center for Atmospheric Research is sponsored by the National Science Foundation, and funding for this research was provided through support from the National Science Foundation under Cooperative Support Agreement AGS-0856145.
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