## 1. Introduction

Gravity waves are atmospheric buoyancy oscillations that transport energy and horizontal momentum vertically throughout the atmosphere (McLandress 1998). The vertical propagation and dissipation of gravity waves are important as the carried energy and momentum are deposited wherever these waves break, affecting the mean flow. Gravity waves and their dissipation have long been recognized to be important in middle atmosphere dynamics (Fritts 1989). Important gravity wave sources are mountains, convection, unbalanced flow, and wind shear (Fritts and Alexander 2003).

The increasing computational resources available to atmospheric scientists have recently made it practical to resolve a majority of the gravity wave spectrum in high-resolution simulations. Realistic, fully nonlinear, and nonhydrostatic simulations down to 1-km resolution are now relatively common. This recent capability presents a unique opportunity to study gravity waves from a variety of sources embedded within these more physically realistic simulations.

There have been many numerical studies of mountain- and convection-generated gravity waves. Much of this work has been idealized, where terrain and/or background states were specified and rotation neglected (e.g., Klemp and Lilly 1978; Fovell et al. 1992). In these types of simulations, gravity wave perturbations may be easily defined relative to the specified background state, which then may be used to quantitatively study their dynamics. Jet- or imbalance-generated gravity waves have also been studied within idealized numerical simulations (Plougonven and Snyder 2007; Lin and Zhang 2008), which develop in synoptic-scale baroclinic systems. In these domains, synoptic-scale quasigeostrophic variations in fields (e.g., pressure) may obscure gravity wave perturbations. Realistic simulations have also been used to study real mountain wave events (e.g., Doyle et al. 2005; Jiang et al. 2013), which may also have synoptic-scale variations. A major objective here is to construct a method to isolate gravity wave perturbations from these synoptic-scale variations in realistic mesoscale fields.

The simplest way to define gravity wave perturbation fields in realistic output is to subtract the mean (e.g., Doyle et al. 2005). This method may be appropriate for small domains, but gravity wave perturbations become increasingly dominated by synoptic-scale variations as domain size increases. A more sophisticated approach is to apply a box moving-average filter to the original field (e.g., Jiang et al. 2013). This effectively low passes the original field, allowing perturbations to be defined relative to the moving-averaged background field. The effective cutoff length scale of the box moving-average method is not isotropic, however. Recently, investigators have employed 2D spectral filtering to isolate gravity wave perturbations in mesoscale (Lin and Zhang 2008) and global (Wu and Eckermann 2008, their Fig. 10) domains. This method allows the spectral response and cutoff length scale of the filter to be explicitly specified. Yet another method was used by Shutts and Vosper (2011), where low- and high-resolution domains were subtracted to isolate wave perturbations. This method assumes the coarse domain does not resolve the gravity waves resolved in the fine-resolution domain and that the two solutions have not significantly diverged. The variety of methods makes meaningful quantitative comparisons of fundamental gravity wave quantities impossible.

Here we propose a method to quantify standard gravity wave quantities that does not depend on model, configuration, resolution, domain location, gravity wave orientation, or gravity wave source. The method is similar to that used by Lin and Zhang (2008). It uses isotropic spectral high-pass filtering to isolate gravity wave perturbations. The main difference is that a smooth-edged response function in wavenumber space is used to prevent Gibbs phenomenon oscillations. This study goes beyond previous work by identifying many quadratic gravity wave diagnostic quantities, allowing many characteristics of gravity waves to be diagnosed and robustly quantified.

The diagnostic method is tested in deep (80 Pa, ~45-km top) realistic Weather Research and Forecast (WRF) Model simulations forced by ECMWF grids. High-resolution simulations nested to 2-km resolution were completed for a deep mountain wave event, a shallow mountain wave event, and a convection-generated gravity wave event. Additionally, a 6-km continuous simulation was completed for the 2014 austral winter (24 May–31 July) over the New Zealand region, and a jet-generated gravity wave event was also chosen within this simulation for further analysis. Gravity wave characteristics in these four events are compared and summarized. Further details on the WRF Model setup are provided in appendix A.

The proposed method and quadratic diagnostic quantities are discussed in section 2. A discussion of the physical justification for such a method is provided in section 3. In section 4, the method is tested in two mountain wave events. Further analysis of mountain waves in the 2- and 6-km-resolution simulations is provided in section 5, while jet-generated and convection-generated nonorographic gravity wave events are presented in section 6. Conclusions are provided in section 7.

## 2. Diagnostic method description

The proposed four-step method transforms a complex 2D field of dynamic variables into a simpler field of wave diagnostic quantities. It combines several standard signal processing algorithms. The starting point is a set of gridded 2D velocity, pressure, and temperature fields from a numerical simulation or interpolated observational dataset. Ideally, these fields are on a level surface and a uniform square grid as the method assumes constant horizontal resolution. The first two steps (deplaning and high-pass spatial filtering) split the field into a smoothly varying background state and small-scale perturbations. The third step takes pointwise products of the perturbation quantities to form quadratic diagnostic quantities (e.g., momentum flux). The last step involves low-pass spatial filtering to smooth the field and reduce noise, effectively “regionalizing” the diagnostics. These steps are described in the following subsections.

Note that limited area models often make use of map projections leading to nonuniform grids on the earth. In the presented WRF simulations, the grid resolution in earth distance varied by less than 3% in all domains and the grids were taken to be uniform. However, grid resolution may vary significantly across a domain depending on size and projection. Regridding may be necessary over a smaller area of interest to ensure the fields are on a uniform grid or nearly so. This method would have to be revised if working with very large or global domains.

### a. Deplaning fields

The original 2D field, *a*(*x*, *y*), is deplaned by subtracting a least squares best-fit plane,

Deplaning is performed because the following step involves the discrete Fourier transform (DFT) and high-pass filtering. Many fields (e.g., pressure in Fig. 1a) have synoptic-scale variations and are aperiodic. Performing the DFT on an aperiodic field introduces high-wavenumber spectral artifacts (Denis et al. 2002). These high-wavenumber artifacts are retained in high-pass filtering, which manifest along domain edges and are proportional to the original aperiodicity. Deplaning reduces aperiodicity and edge artifact amplitude. An alternative method for addressing aperiodicity was proposed by Errico (1985) using edge values only, but this method may generate high-wavenumber artifacts in the interior of the field that can be confused with gravity waves [e.g., compare Figs. 1 and 3 of Denis et al. (2002)].

While edge artifact amplitude is reduced via deplaning, edge artifacts are not eliminated (e.g., Fig. 1d). Edge artifacts decay away from the boundaries with a decay length scale proportional to the cutoff length scale, *L*. By 0.55*L* from the boundaries, edge artifacts are negligible (see appendix C). This decay length scale does not depend on deplaning. If the outer portions of the domain are not of interest, deplaning is optional.

### b. High-pass filtering

The deplaned field still contains a broad spectrum of variations, with synoptic-scale variations often dominating. These variations are unrelated to gravity waves. Two dimensional spectral high-pass filtering is employed to remove synoptic-scale variations and isolate the gravity wave–scale perturbations (see appendix B).

Figure 1 gives an example of partitioning a realistic 2D pressure field from a 2-km-resolution WRF simulation. The original pressure field, *p*, at 4-km MSL is shown in Fig. 1a. Deplaning this 2D field produces the full perturbation field, *L* = 400 km are shown in Figs. 1c and 1d, respectively. Orographic pressure perturbations are obvious in the

### c. Quadratic quantities

The perturbation quantities from the high-pass filter may be analyzed directly or used to compute quadratic diagnostic quantities. Table 1 contains many variance (e.g.,

A partial list of variance and covariance gravity wave diagnostic quantities that may be computed with the proposed spatial filtering method. Depending on the application, it may be useful to “regionalize” these diagnostic qualities by low-pass filtering or area integrating/averaging, indicated by brackets.

The simplest diagnostic for gravity waves is probably the perturbation vertical velocity,

*η*. In an environment with a well-defined lapse rate, temperature anomalies can be approximated by

*γ*is the background lapse rate and

*f*= 0), the regional potential and kinetic energy may exhibit equipartition (PE = KE). The total wave energy density is ED = PE + KE.

The vertical heat flux,

Other potentially useful quadratic quantities such as pseudomomentum are not considered in this paper [see Durran (1995)].

Not all useful diagnostics are simple products of perturbation quantities. Horizontal divergence computed from the full velocity field is usually an indicator of gravity waves as quasigeostrophic flows are nearly divergence free. Wave dissipation can also generate mesoscale (i.e., *f* = 0) Boussinesq potential vorticity [

### d. Low-pass filtering

When computed pointwise, the quadratic quantities in Table 1 may be noisy. It may even be difficult to determine the dominant sign of the quantity. Low-pass filtering or areal averaging of diagnostic quantities is usually helpful in this regard. Low-pass filtering objectively simplifies these fields and allows the dominant sign of the diagnostic quantity to be determined. In this paper, low-pass filtering of quadratic quantities uses the same Gaussian spectral filter described in appendix B.

The *L* = 150 km.

The need for smoothing quadratic quantities is not limited to cases with complex terrain or multiple wave sources. It is needed even in the simplest case. An example is the classic solution for stratified flow over a single smooth hill (Queney 1948) (see appendix D).

## 3. Physical justification for spatial filtering and quadratic diagnostics

Here we address the question of whether the identification of gravity waves with spatial filtering and quadratic diagnostic quantities has a physical basis.

One justification for a spatial filter would be the existence of a spectral gap between gravity wave and synoptic-scale motions. An early search for a spectral gap in horizontal wind speed (Van der Hoven 1957) found a wide gap in the frequency domain between the “weather cycle” with a period of four days and the “turbulent eddy cycle” in the boundary layer with a period of 1 min. This gap is in the frequency domain, however, and provides no support for the spatial filtering proposed here. Galmarini and Thunis (1999) estimated the error in Reynolds averaging associated with using a running mean to separate mean and perturbation quantities. They concluded, however, that “apart from vertical motion, no distinct scale separation appears in the atmospheric frequency distribution of energy” and that such a gap does not exist. The difficulty in isolating gravity waves with simple filtering is characterized by Bretherton (1969): “The decomposition of velocity and temperature structure … can only be achieved by a very sophisticated analysis (if indeed a truly objective distinction is, in principle, possible).”

A careful attempt to separate gravity waves from quasi-2D turbulent motions in aircraft data was carried out by Cho et al. (1999). Their method used multiple variables such as the full wind vector and temperature. They were handicapped by not having perturbation pressure as we have now (e.g., Smith et al. 2008).

A systematic approach to gravity wave identification in numerical model output is described by Lane and Zhang (2011). They examine the joint frequency and wavenumber spectra of disturbances and identified those that satisfied the dispersion relation for gravity waves. To apply such a method to horizontal wavenumber spectra alone requires extra assumptions, as shown below.

*N*cannot be caused by buoyancy forces and frequencies less than |

*f*| are prevented by the Coriolis force. If perturbations are wavelike (i.e., nearly periodic), the intrinsic frequency can be written

As an example, let ^{−1} so that the Coriolis cutoff in the midlatitudes *L* = 1257 km, high-pass filtering retains only waves with

At the other end of the gravity wave spectrum, there is the buoyancy cutoff at *N* in the troposphere. Thus, any well-resolved wave will satisfy

This discussion makes it clear that high-pass spatial filtering is at best a crude approximation to a Coriolis cutoff and that gravity waves cannot generally be distinguished spectrally from larger-scale phenomena.

Covariance quadratic quantities are key in this method, as they allow the phase between velocity, temperature, and pressure to be exploited. While there might not be a spectral gap for an individual variable, there may be a cospectral gap in covariance quantities between synoptic and mesoscale motions (see section 4a). With a cospectral gap, filtering and covariance quadratic quantities allow mesoscale quantities to be distinguished from synoptic-scale quantities. Further, gravity wave and convective mesoscale motions can be distinguished via covariance quadratic quantities like

## 4. Method verification

In this section, the filter-based identification of gravity waves is tested a number of ways. First, gravity wave energy and momentum fluxes are shown to be insensitive to the cutoff length scale over a considerable range. Second, the relationship between wave energy and momentum fluxes is shown to be consistent with linear theory. Third, we exhibit the robust property of mountain waves to propagate upwind. Finally, fluxes derived from the proposed filtering method are compared with an alternative method.

These tests are applied to 2-km WRF simulations of two mountain wave events on 14 and 24 June 2014. The winds at 4-km MSL, just above the highest terrain of New Zealand, are shown for these events in Fig. 3. The 14 June event (Fig. 3a) was a relatively weak event, with strongest winds near 20 m s^{−1} traversing the southern South Island. The 24 June event (Fig. 3b) was stronger, with ≈30 m s^{−1} winds more broadly affecting the entire South Island.

### a. Sensitivity to cutoff length scale

The cutoff length scale *L* is the key parameter in isolating mesoscale perturbations. Ideally, covariance or flux quantities will be insensitive to *L* in some range between mesoscales and synoptic scales. To check for this sensitivity, the South Island area-integrated *L* from 10 to 2000 km on 2D fields at a single time and height (12 km) during the 14 June event. This analysis was repeated every 3 h producing the curves shown in Fig. 4. As *L* increases from 10 to 300 km, fluxes increase as more of the gravity wave spectrum is included. For wavelengths from 300 to 900 km, the fluxes plateau; covariations of *L*. Within this cospectral gap region, standard deviations of

The same analysis was performed for *L* = 400 km was used when high-pass filtering to isolate gravity wave–scale perturbations.

While there are clear cospectral gaps for

### b. Relationship between energy and momentum fluxes

The pointwise E–P relation is tested in Fig. 6. The filtering method was used to produce

*z*= 7 and

*z*= 15 km in both events (Figs. 7b and 7f), where strong winds and shear were present (Figs. 7a and 7e),

*z*= 7 km and above

*z*= 15 km, the waves are nonlinear and neither Eqs. (9) nor (11) are valid.

### c. Upwind propagation

A known property of mountain waves is their tendency to propagate upwind to balance the advection of their energy downwind. In the case of 2D terrain and waves, wave ray paths tilt downwind if the waves are nonhydrostatic with an intrinsic frequency just slightly less than *N*

In 3D terrain, the situation is less definite as some laterally propagating waves can be launched by the complex terrain (Smith 1980) or refracted by lateral shear (Jiang et al. 2013). The most energetic waves are launched with their horizontal wavenumber vector pointed into the wind, however. This property is used in the design of wave drag parameterization schemes (McLandress 1998).

The proposed method captures the upwind propagation in both events at 12 km (Fig. 8). The background wind direction, approximately parallel to the smoothed isobars, differed between the two events. The vectors in the diagrams represent the horizontal energy flux vector, *L* = 150 km. These vectors point into the wind, indicating that these waves are propagating into the flow. The location of the

### d. An alternative “difference” method

The difference method is simpler than the proposed filtering method but has significant disadvantages. For example, the difference method requires two identical simulations, which becomes very computationally expensive when resolving most of the gravity wave spectrum. Also, nonorographic gravity waves will not be retained!

## 5. Mountain waves

Mountain waves are easily visualized using the proposed method, illustrating many 3D characteristics. For example,

*z*. The two parameters

*a*are functions of

*z*. The radius of the

*a*constant, if the chosen isosurface value

Time series of

An additional useful application of the filter method is to examine how it depicts the role of low tropospheric wind speed on wave generation. Both linear and nonlinear mountain waves may be associated with the deformation of low-altitude winds by terrain. To test this relationship, the South Island–averaged

## 6. Nonorographic gravity waves

The proposed method may also be used to quantitatively investigate nonorographic gravity waves as well. Jet-generated and convection-generated gravity wave events within 6- and 2-km-resolution WRF simulations (appendix A), respectively, are presented in this section.

### a. Jet-generated gravity waves

Gravity waves events are quite apparent and frequent in midstratospheric analyses of vertical velocity and temperature over the Southern Ocean south of New Zealand. This is a particularly stormy region, with frequent midlatitude cyclones and high-amplitude upper-tropospheric waves. Within the complex jet structures in the upper troposphere, flow imbalances may act as a source of gravity waves (Lin and Zhang 2008). One such nonorographic event occurred on 11 July 2014 and is illustrated in Fig. 15. A wave packet with northwest–southeast-oriented phase lines is apparent to the southwest of New Zealand. The packet moved northeast with time, with the more intense perturbations evident for at least 9 h.

The ^{−2}

More details of the vertical structure of the wave packet are illustrated in Fig. 17, which shows an east–west vertical cross section of the wave packet along 52°S. The wave packet was vertically coherent, originated near 15 km, and had a westward phase tilt consistent with the vertical propagation shown in Fig. 15. Subsequent analyses (not shown) show this coherent wave packet translating east, retaining its vertical structure. This suggests a single phase speed for the whole packet, which was found to be 15.7 m s^{−1} toward 52° east of north.

Average energy and zonal momentum flux profiles are shown in Figs. 18b and 18c, respectively. The packet had increasing vertical energy flux until the upper sponge layer, which started near 34 km, effectively damped the waves. This increase in energy flux is likely due to wave interaction with the mean shear, similar to mountain waves [Eq. (11)]. The kinks in the profiles of

### b. Convection-generated gravity waves

Wintertime convection is frequent over the Tasman Sea region to the northwest of New Zealand. Convection may generate gravity waves (Clark et al. 1986; Kuettner et al. 1987; Fovell et al. 1992), which may also be analyzed with the proposed method. A convectively active event over this region occurred on 10 July 2014 associated with a dissipating surface low pressure system moving southeast. This event was simulated with WRF at 2-km resolution and analyzed with the proposed method.

Multiple regions of convection-oriented roughly north–south occurred within the simulation, visualized via

Area-averaged profiles of quadratic diagnostics are used to quantitatively describe the convection and stratospheric waves within the boxed region in Fig. 19b. The heat flux profile (Fig. 20e) suggests this convection was deep, with positive heat fluxes extending to the tropopause at 9 km. Within the troposphere,

At and just above the tropopause (9–11 km), there was strong vertical shear. The convective region along 160°E moved east at ≈9.5 m s^{−1}, similar to the mean tropospheric eastward wind speed. The shear at and above the top of the convection produced increasing wind speed relative to the convection. The tops of the convection may act as obstacle to the faster flow just above, producing upstream propagating gravity waves via the “moving obstacle” mechanism (e.g., Beres et al. 2002). In this region, both

The quadratic diagnostics in the stratosphere are consistent with vertically propagating gravity waves. Between 11 and 16 km, *u*, suggesting wave-/mean-shear interaction. The quantity ^{−1}. With this phase speed, a critical level was present near 18 km (Fig. 20a). Just below this level, negative

The stratospheric gravity waves in this event, again, had many similarities with the steady mountain wave events; these waves contained positive

## 7. Conclusions

A method for quantifying gravity waves in mesoscale fields was proposed in this paper. This method combines 2D spectral filtering with a selection of quadratic diagnostic quantities. Covariant diagnostic quantities (e.g., vertical energy flux) allow differentiation between synoptic and mesoscale quantities with the existence of a cospectral gap. Diagnostics also allow different types of mesoscale motion to be distinguished. This method may be used to quantify both orographic and nonorographic gravity waves and to compare them between models.

These ideas were tested in realistic 2-km mesoscale WRF simulations. Filter-derived mountain wave fluxes were relatively insensitive to the high-pass filter cutoff length scale in the range of 300–700 km, providing evidence for a cospectral gap and robust mesoscale flux estimates (Figs. 4 and 5). The resulting wave fields quantitatively satisfied relations predicted from idealized linear gravity wave theory (e.g., Fig. 6) and reproduced many known gravity wave characteristics. Filter-derived fluxes agreed within 20% of those from an alternative difference method where perturbations were defined relative to a no-mountain simulation.

The filtering method was applied to realistic 2-km event simulations and a 6-km winter-long WRF simulation over the New Zealand region to illustrate how this method might be used. Vertical energy flux time series (Fig. 13) revealed the episodic nature of the mountain wave events throughout the 2014 winter and identified deep wave propagation events. Vertical energy fluxes increase with increasing wind speed (Fig. 14), but nonlinear processes appear to be important in mountain wave generation over New Zealand.

Four events from the 2-month period were chosen for detailed analysis using the filter method: 1) a deep mountain wave, 2) a shallow mountain wave, 3) a deep jet-generated gravity wave, and 4) a convection-generated gravity wave. The two nonorographic wave events 3 and 4 had nonzero phase speeds but, in other respects, resembled the mountain wave events 1 and 2. Analogous attributes among all four wave events include positive energy flux, upwind-oriented fluid-relative horizontal energy flux, wave-/mean-shear interaction, and nonlinear dissipation when dissipation occurred. In addition, generation regions in events 1, 2, and 3 had relatively high nonlinearity ratios.

In events 2 and 4, the waves were strongly dissipated in the low stratosphere and do not reach the upper stratosphere. In both of these events, the waves encountered slower mean winds in the altitude range 15–22 km. The slower winds make mountain waves become nonlinear, promoting wave breaking and dissipation. The hypothesis that this low-wind “valve layer” controls deep wave propagation will be evaluated in a future paper.

The general applicability and robustness of the proposed method suggests that it might feasible to compare the gravity waves captured in gridded mesoscale fields from both models and observations.

## Acknowledgments

This work was supported by the National Science Foundation under Grant NSF-AGS-1338655. High-performance computing support from Yellowstone (ark:/85065/d7wd3xhc) was provided by NCAR’s Computational and Information Systems Laboratory, sponsored by the National Science Foundation. We would like to acknowledge Andreas Dörnbrack for providing the ECMWF grids to force the WRF simulations, Johannes Wagner for assistance with WRF and ECMWF grids, and Simon Vosper for facilitating difference method and model intercomparisons of the proposed diagnostics. Additionally, we acknowledge the testing and implementation of energy and momentum flux diagnostics in COAMPS by Alex Reinecke and in NIWA’s version of the UKMO Unified Model by Michael Uddstrom and Stuart Moore. Jeffrey Park’s time series course was helpful in developing the proposed method. Finally, we would like to acknowledge David Fritts, James Doyle, Steve Eckermann, and Michael Taylor for their help in organizing the DEEPWAVE field project in New Zealand, which motivated this work.

## APPENDIX A

### WRF Model Setup

Two types of WRF (v3.6.0) simulations were completed: high-resolution simulations with a 6-km outer domain and 2-km inner nest and a 6-km-resolution continuous simulation over Austral winter (24 May–31 July). Initial and boundary conditions were provided by ECMWF analysis (valid at 0000, 0600, 1200, and 1800 UTC) and 3-h forecast (valid at 0300, 0900, 1500, and 2100 UTC) grids. The high-resolution simulations were completed with 150 vertical levels extending from the surface to 80 Pa, with the highest vertical resolution near the surface decreasing to Δ*z* = 470 m at the domain top. The vertical resolution in the winter-long simulation also decreased with height, with the coarsest vertical resolution of Δ*z* = 1115 m. A 10-km upper damping layer was specified in all simulations. A time step of 25 s was chosen for the 6-km domains, scaled appropriately for the 2-km nests. Parameterization selections are given in Table A1.

Physical parameterizations selected in all domains of all presented WRF simulations.

## APPENDIX B

### 2D Spatial Filtering

Two dimensional spatial filtering is performed in three steps. The discrete Fourier transform (DFT) is performed on the original field,

*L*is the cutoff wavelength. This Gaussian low-pass response function is isotropic; it is maximized (i.e., unity) at

*κ*increases.

The three filtering steps may be repeated to produce both high- and low-passed parts of some 2D field; however, since

## APPENDIX C

### Edge Artifact Extent

Aperiodicity in 2D fields produces edge artifacts after high-pass filtering. Periodic Fourier basis functions reproduce aperiodic edge discontinuities by soliciting primarily high-wavenumber Fourier components (Denis et al. 2002). These components are retained in high-pass filtering, producing edge artifacts (e.g., Fig. 1d). The objective here is to find a length scale beyond which edge artifacts are negligible.

Low-pass filtering (appendix B) is equivalent to convolving the inverse DFT of the low-pass response function with the original data series. In 1D, the continuous inverse Fourier transform of the low-pass response function, *A* some constant. The high-pass response function in physical space is

The width of *L*. The quantity

To test this artifact length scale, high-pass filtering of 1D single saw tooth data series (linear series beginning at *L* was varied between 5Δ*x* and *D*/2, where Δ*x* was the domain resolution and *D* was the domain width. At a distance of 0.55*L* from the domain edges, edge artifact magnitude decreased to 1.4% of the value at the edge. Beyond 0.55*L*, edge artifacts are negligible. Within 0.55*L* of domain edges, filtered quantities are suspect, except for filtered quantities like

## APPENDIX D

### Energy and Momentum Fluxes in the Queney Solution

*U*> 0) and positive, respectively, and independent of height.

*a*, and

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