## 1. Introduction

Gravity waves transport momentum and energy vertically, causing secondary circulations and heating in the stratosphere and above (Eliassen and Palm 1960; Bretherton 1969; Lilly and Kennedy 1973; Lindzen 1981, 1988; Holton 1982; Palmer et al. 1986; McFarlane 1987; Fritts and VanZandt 1993; Lu and Fritts 1993; Fritts and Alexander 2003; Smith et al. 2008; Alexander et al. 2010; McLandress et al. 2012; Placke et al. 2013; Geller et al. 2013). While many clever methods have been devised to observe gravity waves [e.g., balloon soundings, vertically pointing lidar and frequency-modulated continuous-wave (FMCW) radar, and limb and nadir infrared detection from satellites], they usually observe only one or two physical variables. For example, recent advances in superpressure balloon technology (Vincent and Hertzog 2014) provide good horizontal structure of pressure and wind, but vertical air motion must be inferred and temperature is not available.

Compared to other methods, aircraft observations, obtained by flying horizontally though a field of waves, provide the best available multivariable observations with good horizontal scale information. Wind components *u*, *υ*, and *w*, pressure *p*, and air temperature *T* can all be determined at high resolution along a flight leg. Aircraft data analyzed by Jasperson et al. (1990) showed that the zonal wind spectrum with respect to horizontal wavelength

Other authors have examined the spectra of Earth’s terrain. Bretherton (1969) found a power law relating terrain variance and wavenumber with a log–log slope of −5/2. Young and Pielke (1983) and Steyn and Ayotte (1985) found exponents between −2 and −5/2, with some dependence on region and transect orientation. These exponents roughly match the wind and temperature exponents mentioned above.

In comparison to these earlier projects, the recent Deep Propagating Gravity Wave Experiment (DEEPWAVE) campaign over New Zealand in 2014 (Fritts et al. 2016; Smith et al. 2016, hereinafter S16) provides an improved dataset for gravity wave spectral studies over mountains. All the standard physical variables (e.g., *u*, *υ*, *w*, *p*, and *T*) were measured independently and redundantly. The Southern Alps of New Zealand are surrounded by ocean and are therefore compact. Spectral and physical analyses are easier if the disturbance is compact. The Southern Alps have rapid tectonic uplift and erosion rates (Williams 1991) and one of the most rugged terrains in the world. Small-scale relief exceeds 1 km in the high mountain areas (Korup et al. 2005). This roughness broadens the terrain spectrum and the associated wave spectra found in the atmosphere. The DEEPWAVE project design included longer flight legs flown repetitively over a small set of predefined flight tracks. This repetition allows multiple spectral analyses within each wave event.

Typically, regionwide or project-total gravity wave spectra are broad and smooth, motivating a broad spectral approach to momentum flux description (Bretherton 1969; Bannon and Yuhas 1990; Fritts and VanZandt 1993; Lu and Fritts 1993; Wright and Gille 2013). Alternatively, individual remote sensing– or aircraft-derived spectra may show uneven, peaked, or even nearly monochromatic spectra. When this occurs, a common type of gravity wave analysis uses the monochromatic assumption. If one or two physical variables are measured along a horizontal leg, other variables may be inferred using the linearized monochromatic “dispersion relation,” “impedance relation,” or “polarization relation” (Gossard and Hooke 1975; Gill 1982; Nappo 2002; Ern et al. 2004). These relations are physically based formulae that assume that all the physical variables (e.g., *u*, *υ*, *w*, *p*, and *T*) take the form of a single plane wave. The monochromatic assumption is also used in writing the “saturation condition” for marginally breaking waves (Lindzen 1981, 1988). With broad spectra, however, these relations may be invalid, as different physical variables may be dominated by different part of the spectrum. In this study, we examine mountain-wave events that mostly violate the monochromatic assumption. We consider whether the monochromatic approach has validity in mountain-wave analysis.

## 2. Broad mountain-wave spectra

*u*,

*υ*,

*w*,

*p*, and

*T*variance and covariance power spectra. To illustrate this property, we imagine a two-dimensional steady flow of a stratified hydrostatic atmosphere with streamlines and air parcels oscillating up and down about a horizontal reference level. The vertical displacement of air parcels as a function of horizontal distance is given by

*w*) and horizontal (

*u*) velocity:

*w*power, momentum flux (MF) power, and

*u*power respectively. Expressions (3)–(5) differ primarily in the wavenumber weighting in the integrand (i.e.,

*k*

^{2},

*k*

^{1}, and

*k*

^{0}, respectively). This difference arises from the fact that the ratio of vertical to horizontal wind perturbations (i.e.,

*k*. In two dimensions, the continuity equation is

*m*independent of

*k*, the ratio

*k*. It follows that, for a single wavenumber, the variances satisfy

*k*in the numerator and denominator, respectively, these variance spectra are blue and red shifted, respectively, relative to the

*U*and stability

*N*, the vertical displacement

*z*is related to the underlying terrain

*m*is real, the phase shift factor exp(

*imz*) cancels out from (2)–(5) when

*k*-squared weighting with respect to the displacement field (

*w*power only). Type K1 has a single

*k*weighting (e.g., MF and EFz). Type K0 has no

*k*weighting (e.g.,

*u*power,

*T*power,

*p*power, and EFx).

*k*weightings. For example, using (4) and (8), drag (type K1) and temperature variance (type K0) are related by

Examples of narrow and broad terrain spectra are given in Figs. 1–3. Each figure shows the terrain (or parcel displacement) shape and the three different *k*-weighted spectra discussed above: types K0, K1, and K2. The spectra are plotted with linear scales, so the area under each curve is the total variance. Figures 1 and 2 are the cosine–parabola terrain discussed in the next section with well-defined analytical properties. The first spectrum is narrow, and the second is broad. Figure 3 is an actual transect across the Mt. Cook section of the Southern Alps with 1.4-km sampling and 4-point smoothing. Similar to Fig. 2, it has a broad spectrum. The terrain-height spectrum (K0) is dominated by the small-wavenumber “volume mode.” As the *k* weighting is increased (K1 and K2), the higher-wavenumber “roughness mode” becomes evident. With K2 weighting, the volume mode is barely seen. According to (2)–(11), these curves are predictions for the mountain-wave spectra of *w* power (type K2); MF and EFz (both type K1); and displacement, *u* power, *p* power, *T* power, and EFx (all type K0).

Idealized terrain with a broad spectrum: (a) terrain shape, (b) terrain power spectrum (type K0), (c) *k*-weighted power spectrum (type K1), and (d) *k*-squared weighted power spectrum (type K2). Terrain parameters are *u*-power, momentum flux, and *w*-power spectra, respectively.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Idealized terrain with a broad spectrum: (a) terrain shape, (b) terrain power spectrum (type K0), (c) *k*-weighted power spectrum (type K1), and (d) *k*-squared weighted power spectrum (type K2). Terrain parameters are *u*-power, momentum flux, and *w*-power spectra, respectively.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Idealized terrain with a broad spectrum: (a) terrain shape, (b) terrain power spectrum (type K0), (c) *k*-weighted power spectrum (type K1), and (d) *k*-squared weighted power spectrum (type K2). Terrain parameters are *u*-power, momentum flux, and *w*-power spectra, respectively.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

As in Figs. 1 and 2, but for a Mt. Cook section.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

As in Figs. 1 and 2, but for a Mt. Cook section.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

As in Figs. 1 and 2, but for a Mt. Cook section.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

## 3. Idealized complex terrain

*d*is the half-width;

*β*is the relative amplitude of the cosine. This formula drops the mean level below the envelope parabola by an amount

The parabola–cosine shape given by (13) allows one to vary both the width and the depth of the valley structures. It does not capture the randomness of real terrain, but it allows us to do prototype calculations with a broad-spectrum hill. The four terrain parameters could be chosen to match various New Zealand cross sections by a rough visual comparison or by matching quantitative measures, such as volume, variance, slope variance, or wave drag coefficient. A visual match for the Mt. Cook section in Fig. 3 could give *h*_{m} = 1800 m, *d* = 100 km, *β* = 0.3, and *n* = 11. Their spectra can be compared between Figs. 2 and 3.

The drag spectra in (4) for terrain given by (13) is shown in Fig. 4 for *h*_{m} = 1800 m, *d* = 100 km, *β* = 0.3, and *n* = 3–17. The area under each curve is the total wave drag. For any *n* increases.

The superposed drag spectra for eight different terrains [(13)] with *n*. The hydrostatic roughness-mode peak shifts right and increases its amplitude with *n*. The nonhydrostatic drag decreases as the dominant roughness wavenumber *n* > 15, the roughness drag is zero.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

The superposed drag spectra for eight different terrains [(13)] with *n*. The hydrostatic roughness-mode peak shifts right and increases its amplitude with *n*. The nonhydrostatic drag decreases as the dominant roughness wavenumber *n* > 15, the roughness drag is zero.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

The superposed drag spectra for eight different terrains [(13)] with *n*. The hydrostatic roughness-mode peak shifts right and increases its amplitude with *n*. The nonhydrostatic drag decreases as the dominant roughness wavenumber *n* > 15, the roughness drag is zero.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

*n*because shorter waves carry more flux per unit distance. With the modes well separated, the total drag in (4) can be accurately approximated as the sum of the two modes:

*u*power) is also the sum of the two modes if

*n*. For the value of

*u*power (Fig. 2b). This result is important as the perturbation

*u*field usually dominates the breakdown of gravity waves by stagnating the flow, by overturning the flow, or by shear instability. The volume mode will increase its

*u*-power dominance even further when we consider nonhydrostatic effects in the next section.

*w*power (Fig. 2d).

*w*power to

*u*power. For a monochromatic hydrostatic wave, this ratio,

## 4. Nonhydrostatic effects

*k*, (20) reduces to its hydrostatic form

*k*approaches

*m*vanishes. When

*u*′ and

*w*′ are in quadrature, and the momentum flux is zero. For the present purposes, however, we are interested in the cases when

*m*is small. In this case, the disturbance penetrates vertically very well, but the amplitude of the horizontal velocity perturbations decreases with

*m*according to

*u*′ perturbations.

*N*,

*U*, and width

*d*constant and increase the number of bumps

*n*in the terrain. According to hydrostatic theory, the roughness drag increases linearly with

*n*[(14b)]. Using (20), however, as the roughness wavenumber increases and approaches the buoyancy cutoff wavenumber

*n*and vanishes for

*d*, as

*n*increases, the

*U*= 20 m s

^{−1},

*N*= 0.01 s

^{−1}, and

*d*= 100 km, the cutoff value of

*n*= 15 and 16. Implicit in (21) is that, when the roughness wavenumber exceeds

Wave drag as a function of wind speed for complex terrain [(13)] with *n* = 11 and *β* = 0.3. Shown is the full hydrostatic drag (blue), the volume drag alone (black), the full nonhydrostatic drag (brown), and the expression (21) (green). Beyond the buoyancy cutoff, at

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Wave drag as a function of wind speed for complex terrain [(13)] with *n* = 11 and *β* = 0.3. Shown is the full hydrostatic drag (blue), the volume drag alone (black), the full nonhydrostatic drag (brown), and the expression (21) (green). Beyond the buoyancy cutoff, at

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Wave drag as a function of wind speed for complex terrain [(13)] with *n* = 11 and *β* = 0.3. Shown is the full hydrostatic drag (blue), the volume drag alone (black), the full nonhydrostatic drag (brown), and the expression (21) (green). Beyond the buoyancy cutoff, at

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

The influence of wind speed on drag is shown in Fig. 5 for *n* = 11. Two reference curves are given: the full hydrostatic drag in (14b) and the volume drag alone [first term in (14b) or (21)]. The full nonhydrostatic drag, computed from the analytical spectrum in (A5), is compared with the modal equation [see (21)]. The agreement is excellent. The total drag rises rapidly with *U* and is dominated by the roughness drag. Eventually, however, the high wind speed makes the short waves hit the cutoff. The total drag plummets, and only the smaller volume drag remains.

This drop in the roughness drag due to nonhydrostatic effects is further illustrated in Fig. 6 for a case with strong winds. These curves are computed using a fast Fourier transform incorporating (1), (4), (6), and (20). Shown are *w*′(*x*), *u*′(*x*), and accumulated MF(*x*) at 12 km, the altitude of the aircraft data in section 5. When the nonhydrostatic effect in (20) is included, the *w* field is shifted slightly downwind as a result of dispersion, but its amplitude remains large. The disturbance easily reaches the 12-km level in both cases. The roughness part of the *u* field is nearly zero as *u*′ oscillations, the drag is much reduced. The roughness contribution to *u* power vanishes when *u* field has long-wave power while the *w* field does not.

An example of nonhydrostatic *u*′ and drag reduction for mountain waves near the buoyancy cutoff. Three wave fields at *z* = 12 km from terrain [(13)] are shown: (a) *w*′(*x*), (b) *u*′(*x*), and (c) accumulated MF(*x*). The nonhydrostatic effect shifts the *w* field slightly downwind and reduces the magnitude of the *u*′ oscillations and momentum flux, compared to the hydrostatic approximation. Parameters are as follows:

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

An example of nonhydrostatic *u*′ and drag reduction for mountain waves near the buoyancy cutoff. Three wave fields at *z* = 12 km from terrain [(13)] are shown: (a) *w*′(*x*), (b) *u*′(*x*), and (c) accumulated MF(*x*). The nonhydrostatic effect shifts the *w* field slightly downwind and reduces the magnitude of the *u*′ oscillations and momentum flux, compared to the hydrostatic approximation. Parameters are as follows:

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

An example of nonhydrostatic *u*′ and drag reduction for mountain waves near the buoyancy cutoff. Three wave fields at *z* = 12 km from terrain [(13)] are shown: (a) *w*′(*x*), (b) *u*′(*x*), and (c) accumulated MF(*x*). The nonhydrostatic effect shifts the *w* field slightly downwind and reduces the magnitude of the *u*′ oscillations and momentum flux, compared to the hydrostatic approximation. Parameters are as follows:

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Sections 2–4 contain many predictions for wave characteristics over mountains, like those in New Zealand, that have both volume and roughness. First, the wave spectrum is expected to be broad, perhaps spanning a factor of 10 or more in scale, as it comprises both the volume and roughness modes. Second, the power spectra for K0, K1, and K2 quantities will be very different from each other. Third, different power spectra will be dominated by different parts of the wavelength spectrum. The volume mode will dominate the *u*-power, *p*-power, and *T*-power spectra (i.e., type K0), while these three spectra may not show the roughness mode. Conversely, the roughness mode will dominate the *w*-power spectra (i.e., type K2). The MF and EFz spectra may have contributions from both modes. Fourth, near the buoyancy cutoff wavelength, we expect to find waves with strong penetrating *w* power but little *u* power or MF because of nonhydrostatic effects. For still shorter waves, evanescence (i.e., exponential decay aloft) may prevent detection at flight level.

## 5. Power spectra from aircraft transects over New Zealand

During the DEEPWAVE project in 2014, the NSF/NCAR Gulfstream V research aircraft flew 14 missions over New Zealand comprising 97 cross-terrain legs (Fig. 7). The legs were mostly flown at an altitude of *u*, *υ*, and *w*, pressure *p*, and temperature *T*, so several different variance and covariance spectra can be computed. To illustrate this type of data, we show nine legs from research flight 05 (RF05) in Fig. 8. Each trace shows the vertical parcel displacement computed from ^{−2} for the seven 12.1-km legs and 3.1 and 1.7 W m^{−2} for the two 13.8-km legs. The reduction of EFz aloft is probably caused by nondissipative wave energy absorption in the negative shear above the jet stream core.

The South Island of New Zealand with the DEEPWAVE cross-mountain flight legs indicated. The box is the domain of variance integration for WRF Model output. The 2-km-resolution WRF Model terrain is shown in color shading.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

The South Island of New Zealand with the DEEPWAVE cross-mountain flight legs indicated. The box is the domain of variance integration for WRF Model output. The 2-km-resolution WRF Model terrain is shown in color shading.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

The South Island of New Zealand with the DEEPWAVE cross-mountain flight legs indicated. The box is the domain of variance integration for WRF Model output. The 2-km-resolution WRF Model terrain is shown in color shading.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Aircraft-derived patterns of vertical air displacement for nine legs over Mt. Cook during RF05 on 16 Jun 2015. Seven of the legs were flown at *z* = 12.1 km (black) and two at *z* = 13.7 km (red). Note the lack of disturbance upstream and the multiple scales in the wave field. The mountain is to scale but offset vertically. Airflow is from left to right.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Aircraft-derived patterns of vertical air displacement for nine legs over Mt. Cook during RF05 on 16 Jun 2015. Seven of the legs were flown at *z* = 12.1 km (black) and two at *z* = 13.7 km (red). Note the lack of disturbance upstream and the multiple scales in the wave field. The mountain is to scale but offset vertically. Airflow is from left to right.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Aircraft-derived patterns of vertical air displacement for nine legs over Mt. Cook during RF05 on 16 Jun 2015. Seven of the legs were flown at *z* = 12.1 km (black) and two at *z* = 13.7 km (red). Note the lack of disturbance upstream and the multiple scales in the wave field. The mountain is to scale but offset vertically. Airflow is from left to right.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

As shown by S16, these observed waves over New Zealand have many characteristics of steady linear mountain waves, such as positive EFz, negative MFx, and small nonlinearity ratio (i.e., *w* power was dominated by the shorter waves, while the *u* power was dominated by the longer waves. The momentum and vertical energy fluxes had contributions from both wave scales, with the short waves dominating in the stronger events.

For a more detailed discussion, we select four representative legs (Table 1) from four different research flights (i.e., RF05, RF08, RF09, and RF16). These examples are chosen to be representative of characteristics seen in the full dataset. For each chosen leg, several other similar legs can be found in the full dataset. A cutoff wavelength is computed for each leg from

Four analyzed DEEPWAVE aircraft legs, all during 2014.

In Figs. 9–12, we compare the power spectra from these flight legs over New Zealand. The abscissa has a log scale covering almost three orders of magnitude in wavelength. The total variance is the area under the curves. All signals have had their means and trends removed prior to spectral analysis. Of the five spectral quantities plotted in Figs. 9–12, the two most similar are *u* power and *p* power. These two fields approximately satisfy the steady unsheared linearized integrated horizontal momentum equation

One common characteristic is seen in all four examples: a strong *w*-power peak in the range *λ* = 8–40 km. In each case, the shorter part of this range, *λ* = 8–15 km, nearly coincides with the buoyancy cutoff and has little MFx, *u* power, or *p* power. Following sections 3 and 4, we interpret this feature as a highly nonhydrostatic part of the roughness peak. The longer part of the *w*-power roughness peak, *λ* = 15–40 km, has significant MFx, *u*-power, and *p*-power spectra. All four legs (Figs. 9–12) have significant *u* power, *p* power, and *T* power from 40 to 300 km.

In Fig. 9, the MFx, *u* power, and *p* power are distributed widely from *λ* = 15–200 km, thus including both volume and roughness modes. In Fig. 10, the MFx, *u*-power, and *p*-power peaks at *λ* = 26 km give a more monochromatic appearance. In Fig. 11, a strong MFx peak at *λ* = 23 km is supplemented by significant peaks at *λ* = 50–200 km. In Fig. 12, all the variances have strong nearly monochromatic roughness peaks at *λ* = 26 km. This last case was the largest MFx leg in the entire DEEPWAVE project (S16).

Aircraft-derived spectra for Mt. Cook flight RF05, leg 11 on 16 Jun 2014: (a) *w* power, (b) MFx, (c) *u* power, (d) *p* power, and (e) *T* power. The vertical blue line is the estimated buoyancy cutoff for this leg. Note the shift in dominant wavelength between (a) and (c)–(e).

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Aircraft-derived spectra for Mt. Cook flight RF05, leg 11 on 16 Jun 2014: (a) *w* power, (b) MFx, (c) *u* power, (d) *p* power, and (e) *T* power. The vertical blue line is the estimated buoyancy cutoff for this leg. Note the shift in dominant wavelength between (a) and (c)–(e).

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Aircraft-derived spectra for Mt. Cook flight RF05, leg 11 on 16 Jun 2014: (a) *w* power, (b) MFx, (c) *u* power, (d) *p* power, and (e) *T* power. The vertical blue line is the estimated buoyancy cutoff for this leg. Note the shift in dominant wavelength between (a) and (c)–(e).

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Aircraft-derived spectra for Mt. Aspiring flight RF08, leg 7 on 20 Jun 2014: (a) *w* power, (b) MFx, (c) *u* power, (d) *p* power, and (e) *T* power. The vertical blue line is the buoyancy cutoff wavelength for that leg. Note the shift in dominant wavelength between (a) and (c)–(e).

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Aircraft-derived spectra for Mt. Aspiring flight RF08, leg 7 on 20 Jun 2014: (a) *w* power, (b) MFx, (c) *u* power, (d) *p* power, and (e) *T* power. The vertical blue line is the buoyancy cutoff wavelength for that leg. Note the shift in dominant wavelength between (a) and (c)–(e).

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Aircraft-derived spectra for Mt. Aspiring flight RF08, leg 7 on 20 Jun 2014: (a) *w* power, (b) MFx, (c) *u* power, (d) *p* power, and (e) *T* power. The vertical blue line is the buoyancy cutoff wavelength for that leg. Note the shift in dominant wavelength between (a) and (c)–(e).

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Aircraft-derived spectra for Mt. Aspiring flight RF09, leg 11 on 24 Jun 2014: (a) *w* power, (b) MFx, (c) *u* power, (d) *p* power, and (e) *T* power. The vertical blue line is the buoyancy cutoff wavelength for that leg. Note the shift in dominant wavelength between (a) and (c)–(e).

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Aircraft-derived spectra for Mt. Aspiring flight RF09, leg 11 on 24 Jun 2014: (a) *w* power, (b) MFx, (c) *u* power, (d) *p* power, and (e) *T* power. The vertical blue line is the buoyancy cutoff wavelength for that leg. Note the shift in dominant wavelength between (a) and (c)–(e).

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Aircraft-derived spectra for Mt. Aspiring flight RF09, leg 11 on 24 Jun 2014: (a) *w* power, (b) MFx, (c) *u* power, (d) *p* power, and (e) *T* power. The vertical blue line is the buoyancy cutoff wavelength for that leg. Note the shift in dominant wavelength between (a) and (c)–(e).

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Aircraft-derived spectra for Mt. Aspiring flight RF16, leg 1 on 4 Jul 2014: (a) *w* power, (b) MFx, (c) *u* power, (d) *p* power, and (e) *T* power. Unlike Figs. 6–8, these spectra are quite narrow with a peak at

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Aircraft-derived spectra for Mt. Aspiring flight RF16, leg 1 on 4 Jul 2014: (a) *w* power, (b) MFx, (c) *u* power, (d) *p* power, and (e) *T* power. Unlike Figs. 6–8, these spectra are quite narrow with a peak at

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Aircraft-derived spectra for Mt. Aspiring flight RF16, leg 1 on 4 Jul 2014: (a) *w* power, (b) MFx, (c) *u* power, (d) *p* power, and (e) *T* power. Unlike Figs. 6–8, these spectra are quite narrow with a peak at

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

There is certainly no single “dominant wavelength” for these fields of waves. It depends on which diagnostic quantity one uses. This property verifies the predictions from sections 2 and 3 concerning K0, K1, and K2 spectral types. The lack of *u* power and MFx power near the buoyancy cutoff, in spite of the *w*-power peak, verifies the nonhydrostatic dynamics there. The lack of any signal with wavelength shorter than the cutoff verifies the idea of evanescence.

A key question concerns the existence of a volume mode indicating airflow lifting over the whole New Zealand massif. We should look for this signature in the K0 spectral types, such as *u* power, *p* power, and *T* power. This feature is clearly seen in Figs. 9 and 11 with wavelengths from 200 to 400 km but is mostly absent in Figs. 10 and 12. Its identification is compromised slightly by the limited length of the aircraft transects (e.g., 350 km) and the presence of large-scale nonorographic disturbances. Note that these long waves contribute to the momentum flux but have small *w* power.

These data allow us to speculate about the relationship between wave breaking, *u* power, and momentum flux. In RF09, (Fig. 11 and Table 1) there is considerable *u* power, suggesting that wave breaking aloft may be possible. Indeed, as shown in S16, this case had stagnation-induced wave breaking at *z* = 13 km. In contrast, RF16 has a much larger MFx but smaller *u* power. The aircraft found no evidence of wave breaking in this case. Thus, the validity of the saturation momentum flux concept is called into question (see section 8).

We end this section by summarizing the 92 best aircraft transects across New Zealand in Table 2. We use a double boxcar filter as described by S16 to divide the flight-level data into the volume and roughness mode with a *λ* = 60 km dividing wavelength. The filtered records are used to compute normalized *w* power, MFx, and *u* power. All units are square meters per square seconds. The long-wave volume mode dominates the *u* power. The two modes contribute equally to MFx. The short-wave roughness mode dominates the *w* power. These results qualitatively agree with the linear theory of broad-spectrum mountain waves in section 3 (e.g., Figs. 2, 3) and the spectral analysis of aircraft data (Figs. 9–12). Note that the modal variances in Table 2 do not exactly sum to the total variance because of filter damping in the region of spectral overlap.

Volume and roughness contributions to 92 DEEPWAVE aircraft leg variances (m^{2} s^{−2}).

## 6. Flow fields and spectra from numerical simulation

For another perspective on broad-spectrum mountain waves, we use numerical simulation. Numerically simulated waves have imperfect accuracy (Reinecke and Durran 2009) but allow full 3D wave analysis and area averaging superior to the narrow leg averages from aircraft. Numerical simulation allows us to include several fluid dynamical aspects that are missing from the theory in section 3, especially unsteadiness, nonlinearity, variable wind and stability with height, boundary layer dynamics, and terrain three-dimensionality.

Five 2-km-resolution full-physics Weather Research and Forecasting (WRF) Model simulations of 3D time-dependent airflow over New Zealand (Table 3) were recently completed by Kruse et al. (2016) to understand wave generation and attenuation in the lower stratosphere. The WRF domain was nested into the global ECMWF analysis. Model terrain is shown in Fig. 7. The 2-km grid will capture most of the roughness mode and the entire volume mode. The model predictions were compared with balloon and aircraft data with remarkable success. For the present purpose, we focus on two events: 24 June (Figs. 13, 14) and 4 July 2014 (Figs. 15, 16).

The five New Zealand wave events simulated with 2-km WRF runs. Dominant mode is either volume (Vol) or roughness (Rgh).

In Fig. 13, we show the WRF wind and temperature fields on 24 June 2014 along an east–west cross section through Mt. Cook (43.6°S) and the spectra at *z* = 12 km. In Fig. 13a, we see that the low-tropospheric wind speed slows upstream and then increases over the mountain. There is a hint of plunging flow over the massif, with the axis of fast winds descending. Aloft at *z* = 13 km, there is a large-scale stagnation and streamline steepening that will cause wave dissipation (Kruse et al. 2016). The vertical velocity in Fig. 13b shows a few convective updrafts upwind and weak roughness oscillations and general descent over the mountains.

WRF 2-km simulation of flow over Mt. Cook at 1200 UTC 24 Jun 2014 (RF09 case). (a) The wind and theta fields; (b) the vertical velocity and theta fields. Note convective updrafts upwind of New Zealand and dominant descent over the terrain. Spectral analysis is in Fig. 14.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

WRF 2-km simulation of flow over Mt. Cook at 1200 UTC 24 Jun 2014 (RF09 case). (a) The wind and theta fields; (b) the vertical velocity and theta fields. Note convective updrafts upwind of New Zealand and dominant descent over the terrain. Spectral analysis is in Fig. 14.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

WRF 2-km simulation of flow over Mt. Cook at 1200 UTC 24 Jun 2014 (RF09 case). (a) The wind and theta fields; (b) the vertical velocity and theta fields. Note convective updrafts upwind of New Zealand and dominant descent over the terrain. Spectral analysis is in Fig. 14.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

A spectral analysis of the flow in Fig. 13 is shown in Fig. 14. Here we extract data from the WRF run along a 400-km cross-mountain transect at *z* = 12 km to simulate an east–west aircraft leg. In Fig. 14, two *w*-power peaks are seen at *λ* = 12 and 20 km. Because of the lack of *u* power there, neither contributes to MFx. Smaller but significant *w* power is seen at long wavelengths *λ* = 50, 90, and 300 km. All the other spectra (i.e., *u* power, MFx, *p* power, and *T* power) are dominated by these long waves. Their spectra are red, and the volume mode dominates, associated with the plunging flow in Fig. 13.

Spectral analysis of 2-km WRF-simulated airflow at *z* = 12 km over Mt. Cook on 24 Jun 2014 (RF09 case): (a) *w* power, (b) MFx, (c) *u* power, (d) *p* power, and (e) *T* power. The vertical reference line is the buoyancy cutoff. A 1200-km transect is used to capture long waves. See flow field in Fig. 13.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Spectral analysis of 2-km WRF-simulated airflow at *z* = 12 km over Mt. Cook on 24 Jun 2014 (RF09 case): (a) *w* power, (b) MFx, (c) *u* power, (d) *p* power, and (e) *T* power. The vertical reference line is the buoyancy cutoff. A 1200-km transect is used to capture long waves. See flow field in Fig. 13.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Spectral analysis of 2-km WRF-simulated airflow at *z* = 12 km over Mt. Cook on 24 Jun 2014 (RF09 case): (a) *w* power, (b) MFx, (c) *u* power, (d) *p* power, and (e) *T* power. The vertical reference line is the buoyancy cutoff. A 1200-km transect is used to capture long waves. See flow field in Fig. 13.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

The WRF run for 4 July is shown in Figs. 15 and 16. In contrast to the 24 June case, Fig. 15a shows weaker speed variation on any scale but strong roughness oscillations in the theta lines. In Fig. 15b, we see dramatic columns of positive and negative vertical velocity over the terrain. Approximately 10 updrafts and 10 downdrafts lie within a 150-km distance, giving a *λ* = 10–30-km wavelength. Without strong *u* power aloft, wave breaking is delayed but finally occurs at *z* = 15 km in small regions.

WRF 2-km simulation of flow over Mt. Cook at 1200 UTC 4 Jul 2014 (RF16 case). (a) The wind and theta fields; (b) the vertical velocity and theta fields. Note the short-wavelength up- and downdrafts over the rugged terrain. Spectral analysis is in Fig. 16.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

WRF 2-km simulation of flow over Mt. Cook at 1200 UTC 4 Jul 2014 (RF16 case). (a) The wind and theta fields; (b) the vertical velocity and theta fields. Note the short-wavelength up- and downdrafts over the rugged terrain. Spectral analysis is in Fig. 16.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

WRF 2-km simulation of flow over Mt. Cook at 1200 UTC 4 Jul 2014 (RF16 case). (a) The wind and theta fields; (b) the vertical velocity and theta fields. Note the short-wavelength up- and downdrafts over the rugged terrain. Spectral analysis is in Fig. 16.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Spectral analysis of 2-km WRF-simulated airflow at *z* = 12 km over Mt. Cook on 4 Jul 2014 (RF16 case): (a) *w* power, (b) MFx, (c) *u* power, (d) *p* power, and (e) *T* power. The vertical reference line is the buoyancy cutoff. A 1200-km transect is used to capture long waves. See flow field in Fig. 15.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Spectral analysis of 2-km WRF-simulated airflow at *z* = 12 km over Mt. Cook on 4 Jul 2014 (RF16 case): (a) *w* power, (b) MFx, (c) *u* power, (d) *p* power, and (e) *T* power. The vertical reference line is the buoyancy cutoff. A 1200-km transect is used to capture long waves. See flow field in Fig. 15.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Spectral analysis of 2-km WRF-simulated airflow at *z* = 12 km over Mt. Cook on 4 Jul 2014 (RF16 case): (a) *w* power, (b) MFx, (c) *u* power, (d) *p* power, and (e) *T* power. The vertical reference line is the buoyancy cutoff. A 1200-km transect is used to capture long waves. See flow field in Fig. 15.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

The spectra in Fig. 16 for 4 July has strong *w*-power peaks at *λ* = 10 and 30 km. A small peak is seen at 80 km. The *λ* = 10-km peak does not appear in MFx, *u* power, or *p* power, probably because it coincides with the buoyancy cutoff. The *λ* = 30-km peak dominates all of the other spectra and gives all except *T* power a monochromatic appearance with roughness mode dominance. This case was unique as the strongest MF case in the project. The model spectra in Fig. 16 agree well with the aircraft spectra (Fig. 12) for this event.

These spectra suggest that 2-km WRF simulations capture the broad spectrum of New Zealand disturbance, including the volume and roughness modes. The shorter part of the roughness mode makes no contribution to the drag because of the effect of the nonhydrostatic cutoff. The longer part of the roughness mode and the volume mode contribute to the drag. Another view of this pattern is seen from model statistics in the next section.

## 7. Bandpass analysis of waves from numerical simulation

The previous section indicated that the numerical simulation of airflow over New Zealand captures the breadth of the mountain-wave spectra. Therefore, we can use it to paint a fuller picture of the role of the volume and roughness modes in mountain-wave dynamics. We separate the two modes by applying a spectral bandpass filter described by Kruse and Smith (2015). Our volume-mode filter retains wavelengths from 60 to 600 km. Our roughness-mode filter retains wavelengths shorter than 60 km. These spectral cutoffs are not sharp, but, with broad spectra, they should give meaningful estimates of the two modes. We average over the large box shown in Fig. 7 with an area of about 711 000 km^{2}.

The WRF runs for five events are summarized in Table 3, and two of these events are shown in Figs. 17 and 18. In Table 3, we give the maximum momentum flux at *z* = 12 km and corresponding 4-km winds for each event. Also given is the dominant mode for each variance type. The two figures show the time development of low-tropospheric wind speed and direction at *z* = 4 km, as well as *w* power, MF, and *u* power in the lower stratosphere at flight level *z* = 12 km. The component of MF in the direction of the regionally averaged 4-km wind is shown. The two variances are normalized to units of square meters per square seconds, while MF is multiplied by air density to obtain units of millipascals. Generally speaking, the variances and the wind speed reach their peaks at about the same time, but there are exceptions.

WRF simulated area-averaged variances at *z* = 12 km for 24–25 Jun (RF09 case) over New Zealand: (a) *w* power, (b) MF, and (c) *u* power. (d) Wind speed and direction at *z* = 4 km. In (a)–(c), the total (black) and the contributions from volume mode (blue) and roughness mode (red) are shown. The volume mode dominates the wave drag in this case.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

WRF simulated area-averaged variances at *z* = 12 km for 24–25 Jun (RF09 case) over New Zealand: (a) *w* power, (b) MF, and (c) *u* power. (d) Wind speed and direction at *z* = 4 km. In (a)–(c), the total (black) and the contributions from volume mode (blue) and roughness mode (red) are shown. The volume mode dominates the wave drag in this case.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

WRF simulated area-averaged variances at *z* = 12 km for 24–25 Jun (RF09 case) over New Zealand: (a) *w* power, (b) MF, and (c) *u* power. (d) Wind speed and direction at *z* = 4 km. In (a)–(c), the total (black) and the contributions from volume mode (blue) and roughness mode (red) are shown. The volume mode dominates the wave drag in this case.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

WRF-simulated area-averaged variances at *z* = 12 km for 2–5 Jul (RF16 case) over New Zealand: (a) *w* power, (b) MF, and (c) *u* power. (d) Wind speed and direction at *z* = 4 km. In (a)–(c), the total (black) and the contributions from volume mode (blue) and roughness mode (red) are shown. The roughness mode dominates the wave drag in this case.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

WRF-simulated area-averaged variances at *z* = 12 km for 2–5 Jul (RF16 case) over New Zealand: (a) *w* power, (b) MF, and (c) *u* power. (d) Wind speed and direction at *z* = 4 km. In (a)–(c), the total (black) and the contributions from volume mode (blue) and roughness mode (red) are shown. The roughness mode dominates the wave drag in this case.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

WRF-simulated area-averaged variances at *z* = 12 km for 2–5 Jul (RF16 case) over New Zealand: (a) *w* power, (b) MF, and (c) *u* power. (d) Wind speed and direction at *z* = 4 km. In (a)–(c), the total (black) and the contributions from volume mode (blue) and roughness mode (red) are shown. The roughness mode dominates the wave drag in this case.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

In all five events, the volume mode dominates the *u* power, and the roughness mode dominates the *w* power, indicating broad spectra (Table 3) In four of the five events, the roughness mode dominates the MF. For 24–25 June (Fig. 17), however, the volume-mode drag slightly exceeds the roughness drag. The 3–5 July simulation (Fig. 18) is more typical, with the roughness mode dominating MF. This event has the largest winds and fluxes of the DEEEPWAVE project, as seen on both the numerical simulation and the aircraft observations (RF16).

We can understand the variation in partitioning between volume and roughness modes by examining the triads of figures: Figs. 13, 14, and 17 for the 24 June event and Figs. 15, 16, and 18 for the 4 July event. In Fig. 13, the dominance of the volume mode can be seen by the plunging flow and the weakness of the small-scale up- and downdrafts over terrain. In contrast, Fig. 15 has strong roughness-driven vertical motion. This contrast is also seen in the spectral diagrams (Figs. 14, 16) and in the bandpass diagrams (Figs. 17, 18).

All five weather events (Table 3) are summarized in Fig. 19, where all area-averaged, hourly output, 4-km wind, and *z* = 12-km bandpassed wave drag values are combined into scatterplots. Overall, the roughness-mode drag exceeds the volume-mode drag by roughly a factor of 2. In general, wave drag is small until the wind speed at 4 km exceeds 12 m s^{−1}. Beyond this speed, however, the drag magnitude is chaotic with little correlation with wind speed. A similar chaotic variation was reported by S16 on the basis of the aircraft data alone.

Area-averaged wave drag over New Zealand from five 2-km WRF event simulations (Table 2). (a) Volume mode

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Area-averaged wave drag over New Zealand from five 2-km WRF event simulations (Table 2). (a) Volume mode

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

Area-averaged wave drag over New Zealand from five 2-km WRF event simulations (Table 2). (a) Volume mode

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0297.1

One can convert these area-average drag values (Figs. 17–19) to drag values per ridge length (Fig. 5) by multiplying by the averaging area

## 8. Saturation momentum flux with broad spectra

*u*power

*C*prescribes the degree of flow stagnation required for wave breaking. Combining these expressions gives

*u*power in (15) and eliminating

*d*is the half-width of the parabola and

*n*and

*u*power.

The WRF event runs (Table 3 and Fig. 19) also support the hypothesis that wave fields dominated by the roughness mode can transport more momentum than wave fields dominated by the volume mode. We use parameters *C* = 0.07 so that (22) matches the maximum volume mode drag (i.e., controlling the *u* power) in Fig. 19a. When the roughness mode is added (Figs. 19b,c), the maximum total drag is increased by a factor of 2–3, in rough agreement with the enhancement estimate in the square brackets above. Unfortunately, this test has several uncertainties, and much more work would be required to prove the hypothesis. For instance, we have not demonstrated that the wave drag in Fig. 19 is limited by a wave saturation process. Still, the comparison illustrates that the volume mode that controls the *u* power carries only a fraction of the momentum flux.

## 9. Conclusions

The goal of this paper is to combine theory, aircraft data, and numerical simulation to explain the striking difference between the spectra of different physical variables in New Zealand mountain waves. We used three wavenumber weightings (*k*^{0}, *k*^{1}, and *k*^{2}) to identify important short- and long-wave components. According to linear hydrostatic theory, these particular weightings (i.e., types K0, K1, and K2) correspond to physically important spectra, such as *u* power, *p* power, *T* power, and horizontal energy flux (type K0); zonal momentum flux and vertical energy flux (type K1); and *w* power (type K2). These spectra have different shapes, with K0 quantities often being red, while K1 is white and K2 is blue.

We showed that an idealized complex terrain constructed from a parabola with *n* embedded cosine waves generates a broad wave spectrum composed of two modes: the volume mode and roughness mode. The volume mode represents smooth flow rising and descending over the main mountain range. The roughness mode represents airflow into and out of interior valleys. With *w* power) is dominated by the roughness mode. The momentum flux and vertical energy fluxes (type K1) typically have contributions from both modes: the volume mode with strong

Nonhydrostatic waves near the buoyancy cutoff wavelength *w* power in the lower stratosphere. There is little *u* power, however, and thus it shows little tendency to transport momentum, stagnate the flow, trigger instability, or cause wave breaking. This mode is generated by the approximate

The analysis of selected DEEPWAVE aircraft data verified most of the above predictions. Notable is the strong roughness mode of *w* power from *λ* = 8 to 40 km. The shorter part of this range is nearly centered on the buoyancy cutoff wavelength (i.e., *λ* = 8–15 km). This part has little *u* power or momentum flux. The longer part of this range (i.e., *λ* = 15–40 km) carries most of the momentum flux. Using *u*-power, *p*-power, and *T*-power spectra, we identified the volume mode with *λ* = 200–400 km, caused by airflow rising over the whole Southern Alps massif. While not dominant, it does contribute to the total momentum flux. Statistics from 92 DEEPWAVE legs confirmed the relative contributions of the volume and roughness modes.

Evidence of broad wave spectra was seen in five 2-km-resolution WRF event simulations. Similar to the aircraft data, they show a strong band of *w* power that we interpret as the roughness mode. The shorter portion of this band had no *u* power and no momentum flux. The longer portion of the roughness mode carried most of the momentum flux. In one event only (case RF09), the volume mode dominated the flux. In the other four events, including the strongest flux event (case RF16), the roughness mode dominated the MF. This variation in mode dominance from event to event probably arises from changes in wind direction, moist stability, and valley-air heat budgets controlling the upstream blocking, moist convection, large-scale plunging, and valley flushing.

In one respect, our numerical results are consistent with a case study of mountain waves over New Zealand by Lane et al. (2000). With a nested single-sounding model run, they found an area-averaged

The short scale of the dominant roughness mode puts a constraint on numerical simulation of waves and wave drag. While it may be permissible to truncate the nonhydrostatic *w*-power peak at *λ* = 8–15 km, the MF peak at *λ* = 15–40 km should be accurately represented in numerical models. We found a grid spacing of 2 km was satisfactory, but spacing of 5 km or larger may not be. The model should also be skillful at estimating airflow blocking and valley penetration in order to accurately predict the volume and roughness modes.

The current results concerning broad wave spectra pose a challenge for observing mountain waves. Measurements of *T* power, such as those from satellite passive IR or Rayleigh lidar, are seeing the K0 spectra dominated by the volume mode. The use of (12) to estimate momentum flux will give poor results. The same is true for measurements of *u* power, such as radiosondes, constant-level balloons, and conical scanning Doppler lidar. In a broad-spectrum wave field, these instruments too will mostly see the volume mode, while the roughness mode may carry the majority of the momentum flux. These problems are aggravated for satellite remote sensing systems that may not fully resolve the wave structures.

The prediction and parameterization of wave breakdown and momentum deposition are profoundly influenced by the breadth of the mountain-wave spectrum. With a broad spectrum, the usual monochromatic gravity wave relationships between different variables (e.g., *u*, *υ*, *w*, *p*, and *T*) are not even approximately satisfied because these different variables are controlled by different parts of the spectrum. In general, it is the *u* power that controls wave breakdown by flow stagnation, shear instability, or other mechanisms. It is dominated by the volume mode. The momentum flux typically gets contributions from a broad spectrum, including roughness and volume modes. We see a slight dominance of the roughness-mode contribution to MF in the aircraft data and the WRF Model output.

Here is the dilemma then: the long waves control wave breaking, while the shorter waves carry the momentum and vertical energy fluxes. The common monochromatic assumption used to derive saturation conditions for wave drag parameterization does not capture this important complexity. In the future, if satellite data interpretation and wave drag parameterization are to be quantitatively useful, we may have to drop the popular “monochromatic” simplification in favor of a binned or continuous wave spectrum. For New Zealand, we have introduced a “binned” approach, where we divide the spectrum into two categories: a volume mode and a roughness mode. Interior continental terrains may require a spectral continuum.

## Acknowledgments

The grant to Yale University was NSF-AGS-1338655. The NSF/NCAR Gulfstream V aircraft was operated by the National Center for Atmospheric Research. The high quality of the airborne data was essential for this project. Additional collaboration came from the Naval Research Laboratory, German DLR, the New Zealand MetService, and NIWA. The WRF computing was performed on the Yellowstone supercomputer provided by NCAR’s Computational and Information Systems Laboratory sponsored by the NSF. Special thanks to the DEEPWAVE project PIs, Dave Fritts, James Doyle, Steve Eckermann, Mike Taylor, Andreas Dörnbrack, Michael Uddstrom, and the Yale team: Alison D. Nugent, Campbell Watson, Azusa Takeishi, Christine Tsai, Larry Bonneau, and Sigrid R.-P. Smith. Conversations with DLR students Martina Bramberger and Tanja Portele are appreciated. Discussions with Fuqing Zhang and Ulrich Schumann regarding short waves in the atmosphere were valuable. Three anonymous reviewers made helpful comments.

## APPENDIX

### Mathematical Properties of the Parabola–Cosine Hill

*d*, and embedded waves of relative amplitude

*n*= 1, 3, 5, 7, … so that an odd integer number of peaks fits into the terrain. This choice reduces the slope discontinuity at

*h*

_{m}= 1800 m,

*d*= 100 km,

*n*= 11, and

*n*= 0. A parabola–cosine with valley bottoms at

*w*variance in (3) occurs at a slightly shorter wavelength:

*k*comes outside the integral in (4) and

*k*space, the two drag contributions [(A7) and (A9)] are additive. Furthermore, additional roughness modes could be factored into (13), each with their own

*n*. Their drags will be additive too, as long as their spectra do not overlap. As seen in Fig. 4, even roughness patterns with adjacent odd

*n*values (e.g.,

*n*= 9 and

*n*= 11) have little spectral overlap. Thus, (13) and (19) can be easily generalized to a broad family of rough terrains.

The variances and covariances determined above using the analytical

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