1. Introduction
Mountain wave drag is a horizontal pressure force acting on the terrain associated with the generation of gravity waves. The equal and opposite reaction force on the atmosphere is applied at some higher altitude where the gravity waves dissipate their energy. Since the recognition that mountain wave drag could influence the general circulation of the atmosphere (Sawyer 1959; Blumen 1965; Bretherton 1969; Lilly 1972), there have been numerous theoretical attempts to predict this wave drag using regional environmental parameters.
The development of wave drag theory for complex terrain has a long history. Wurtele (1957), Crapper (1962), Blumen and McGregor (1976), and Smith (1980) examined mountain waves from ideally shaped isolated hills using linear theory. Phillips (1984) derived the wave drag on a smooth elliptical hill at various angles to the wind vector. He developed the idea of “transverse drag”: the component of wave drag perpendicular to the wind direction. Drags for random terrains were analyzed by Blumen (1965), Bretherton (1969), Bannon and Yuhas (1990).
In parallel with these theoretical studies, surface observations of pressure drag (Smith 1978) and aircraft observations of wave momentum flux (e.g., Lilly and Kennedy 1973; Bougeault et al. 1990; Smith et al. 2016) provided the motivation to include wave drag schemes in general circulation models (GCMs).
Interactive wave drag–predicting schemes (i.e., parameterizations) in GCMs were first introduced by Palmer et al. (1986) and McFarlane (1987). These methods were improved by Miller et al. (1989), Lott and Miller (1997), Gregory et al. (1998), and more recently by Scinocca and McFarlane (2000), Webster et al. (2003), Kim and Doyle (2005), and Vosper et al. (2016). The subject is reviewed by Kim et al. (2003), Geller et al. (2013), and Teixeira (2014). As our understanding of wave drag has advanced, two problems have been identified as the most challenging and important: terrain anisotropy and dynamical nonlinearity.
Miller et al. (1989) extended the wave drag formulation by including a nonlinear correction for flow blocking and used Phillips’ elliptical-mountain approach to anisotropy. Baines and Palmer (1990), Lott and Miller (1997), Gregory et al. (1998), Scinocca and McFarlane (2000), and Teixeira and Miranda (2006) followed a similar methodology with a nonlinear correction and Phillips’ “equivalent ellipses” approach to anisotropy.
Kim and Arakawa (1995) and Kim and Doyle (2005) took a new approach using numerical mesoscale models over a variety of hill shapes to derive wave drag laws. Anisotropy was treated by using a range of wind directions over specified hill shapes.
In the gravity wave “parameterization” literature, the authors worked not only to predict the gravity wave drag but also to predict how the associated momentum flux would be applied to the atmosphere above the rough terrain (e.g., Shutts and Gadian 1999; Teixeira and Miranda 2009; Teixeira and Yu 2014; Xu et al. 2012, 2017). In this paper, we undertake a simpler task. We look only at wave generation and wave drag on the terrain.
2. Uncertainties in wave drag prediction
The motivations for this paper are three uncertainties in existing wave drag prediction schemes: (i) the role of the terrain spectrum, (ii) the generality and efficiency of the Phillips anisotropy formula, and (iii) the best way to correct for nonlinearity.
Regarding the first issue, linear theory predicts that for waves of a given displacement amplitude, the momentum flux increases with the horizontal wavenumber as in (1). As real terrain typically has a broad power spectrum, it is inaccurate to use a monochromatic representation. Likewise, the terrain variance does not provide sufficient spectral information. Recent aircraft observations have suggested that high-drag states may result from a shift in the spectral peak rather than a change in average wave amplitude (Smith and Kruse 2017). The use of total terrain variance and a single wavenumber, as in (1), may not capture this sensitivity to the terrain spectrum.
The second issue relates to the generality of the Phillips formula for transverse drag. Some authors have proceeded as if the drag component formulas derived for a smooth elliptical hill may be valid only for that simple shape. To preserve validity, they imagined the terrain as composed of “equivalent ellipses.” We will show that the effect of terrain anisotropy can be treated for any complex terrain h(x, y) using a 2 × 2 symmetric drag matrix
The third uncertainty regards the nonlinear correction. The importance of wave drag nonlinearity is widely accepted in the drag parameterization literature (e.g., Lott and Miller 1997; Gregory et al. 1998; Scinocca and McFarlane 2000; Webster et al. 2003; Garner 2005). The standard representation is an “airflow-blocking algorithm.” The concept of mountain airflow blocking due to density stratification has been well developed by Kao (1965), Hunt et al. (1979), Snyder et al. (1985), Spangler (1987), Smith (1989), Miranda and James (1992), Smith and Grønås (1993), Ólafsson and Bougeault (1996), Baines (1997), Reinecke and Durran (2008), and others. These studies all found that when the nondimensional mountain height (hN/U) is large, the lower air layers will stagnate, split, and flow around isolated hills. Only the layers just below the hilltop will be able to rise over the top. There may also be an upstream blocking wedge of cold air that will produce a broader and gentler upslope ascent (Pierrehumbert and Wyman 1985).
In a standard blocking algorithm, it is assumed that in slow stratified flow, the air is unable to lift more than
While this conceptual model is clear enough for simple isolated hills, it is difficult to implement quantitatively in complex terrain (Lott and Miller 1997). The difficulty arises because the depth of the blocked layer depends in the terrain height. If mountain peaks with h = 1, 2, and 3 km exist in close proximity and Δz = 1 km, the blocked layer has depths of D = 0, 1, and 2 km in the same region. This pattern of flow is difficult to visualize.
There are at least three additional problems with this popular conceptual model of blocking. First is the lack of explicit treatment of the lateral extent of the blocked layer. Unless the blocked layer extends very far, it must taper off, producing some large-scale “effective” terrain. This effective terrain will still generate waves, albeit with larger horizontal wavelength. These longer waves cannot be ignored. Unless they exceeded a wavelength of 1000 km or so, their wave drag could be significant.
A second problem is that the blocking algorithm is not well designed to account for cold air trapped in valleys (e.g., Whiteman et al. 1999; Sheridan et al. 2014; Kiefer and Zhong 2015). While strong ambient winds may flush out this cold air, its depth and extent may not scale with regional variables.
The third problem with the blocking algorithm is its influence on the wave spectrum. During slow-wind events, with the lower part of the terrain removed, the blocking algorithm is likely to shift the spectra to shorter waves with smaller horizontal scales, as the retained higher parts of the terrain are usually sharp and rugged. The smoother lower valleys would be removed. This problematic tendency for shorter waves under slow winds goes opposite to the findings of the recent Deep Propagating Gravity Wave Experiment (DEEPWAVE) project over New Zealand in 2014 (e.g., Smith et al. 2016; Smith and Kruse 2017). From more than 90 transects across New Zealand, they found a tendency for the high-wind and high-drag events to have wave spectra shifted toward short waves. They concluded that it is the shift in spectral peak rather than an increase in wave amplitude that causes the increased wave drag.
A good example of this shift in spectral peak is the New Zealand wave event on 21 June 2014, which was observed during DEEPWAVE research flight 16 (RF16; Smith et al. 2016). This event had the strongest winds of the DEEPWAVE project and the highest wave drag. It was dominated by short wavelengths in the range of 20–40 km as opposed to weaker events with dominant wavelengths of 60–300 km. Both these wavelengths were in the hydrostatic range, so it is not the ability of the waves to propagate that matters; it is their generation.
In this paper, we focus on the terrain of the South Island of New Zealand. This terrain is high, rugged, highly anisotropic, and misaligned with the cardinal directions. It is surrounded by the featureless Tasmanian Sea and the Southern Ocean. It was the site of the 2014 DEEPWAVE project (Bossert et al. 2015; Fritts et al. 2016; Gisinger et al. 2017). The DEEPWAVE project provides a unique set of aircraft data (Smith et al. 2016; Smith and Kruse 2017) and a well validated set of high-resolution Weather Research and Forecasting (WRF) Model simulations (Kruse et al. 2016). These resources allow us to propose and test a new hypothesis regarding wave drag on complex terrain.
3. Describing New Zealand’s terrain
a. Volume and variance
Characteristics of South Island terrain with various smoothings.
b. Anisotropy
While the scalars volume and variance and matrices
c. Roughness and smoothing
The roughness of terrain is important for wave drag. A rough terrain may generate gravity waves with shorter scales and more momentum transport (Smith and Kruse 2017).
Terrain smoothing may be done for several reasons. A terrain dataset may be smoothed and subsampled to reduce the size of the dataset. It may be smoothed to match the grid size of a numerical model. Herein, we smooth to represent the nonlinear effect of stratification, weak winds, low-level blocking, and flow around steep terrain.
The effect of smoothing is to spread out the terrain onto the sea, decreasing the mountain peaks and filling in the valleys. The terrain volume in (2) is unchanged by smoothing. Other effects of Gaussian smoothing are given in Table 1. Generally, smoothing has little impact on the terrain orientation (
4. The WRF wave drag dataset for New Zealand
a. Data quality
The observational basis for the current study is a continuous full-physics WRF Model simulation of airflow over New Zealand, done for the DEEPWAVE project from June through August 2014. The mesoscale simulation was carried out with 6-km resolution with boundary conditions from MERRA, version 2 (MERRA-2), global simulations. This run was compared carefully with a prior run using the operational ECMWF analysis as boundary conditions (Kruse et al. 2016). The two runs compared very well. A detailed comparison with observations was carried out, including aircraft leg winds and temperatures and vertically smoothed balloon soundings. The agreement was excellent. The general meteorological conditions during this period are described by Gisinger et al. (2017).
b. Wave drag anisotropy
A key question is whether the WRF drag vectors in (18) show anisotropy arising from the anisotropy of the New Zealand terrain. Our WRF dataset has 735 regional wind and wave drag vectors, with values every 3 h for a 3-month period (Figs. 2 and 3). Values of wind and wave drag are computed at z = 4 km, averaged and summed over the South Island and surrounding waters, respectively.
The wind directions over New Zealand are shown in Fig. 2 by plotting U versus V. The looping character of this scatter diagram indicates the passage of frontal cyclones past NZ bringing a sequence of wind directions. The zonal wind component is usually positive (i.e., eastward), but the meridional component has both signs with nearly equal probability. Average values are
c. Wave drag nonlinearity
5. Linear theory drag matrix and anisotropy
These wave drag components can be computed using the 2D fast Fourier transform for any wind direction (Smith 1980). Equations (23)–(26) are similar to (6.2) in Phillips (1984), (18) in Gregory et al. (1998), and (19) in Teixeira and Miranda (2006).
To illustrate the properties of linear theory drag for complex terrain, we show in Fig. 5 the two components of wave drag for South Island as a function of wind direction (WD), with L = 10 km and
A similar drag matrix approach was suggested by Garner (2005). Garner computed a velocity potential using Fourier methods but computed the matrix elements in physical space, not using Parseval’s theorem [e.g., (26)]. His method would fail for hills with finite volume as the velocity potential integral does not converge.
The diagonal elements of
The elements of the drag matrix for South Island, its eigenvalues, and other metrics with different smoothings (
For South Island, the elements of the drag matrix are shown in Table 2. We do not show computations with smoothing scale less than L = 10 km as those terrains may generate nonhydrostatic waves, which violate our hydrostatic assumption. The off-diagonal
The drag anisotropy
Table 2 also shows how the drag matrix elements in (26) change with terrain smoothing. The matrix elements decrease strongly with smoothing, as smoothing eliminates the smaller terrain scales that contribute strongly to the total drag (Smith and Kruse 2017). The drag anisotropy
We could probably find a better fit than (34) for New Zealand, but it would not be universal. The form of (34) depends on the terrain power spectrum. Terrains with less small-scale roughness will have a slower decrease than inverse L.
6. Wave drag nonlinearity and smoothing
The importance of nonlinearity is widely accepted in the wave drag literature and supported by (20) for the WRF drag dataset. To avoid the problematic “airflow-blocking algorithm,” we implement a new variable smoothing approach. Stronger terrain smoothing is used at low wind speeds. At low wind speeds, upstream blocking and flow around isolated peaks tend to smooth the effective terrain shape. Cold air trapped in valleys has a similar smoothing effect. With stronger winds, the airflow is able to follow the rough terrain more accurately. It overcomes the smoothing effects of stratification. Winds can flow up steep slopes and flush out deep valleys. The smoothing algorithm is easy to apply and avoids any untreated blocked layer (section 2). Utilizing smoothing to implement drag nonlinearity finds support in aircraft and modeling data that show the largest drag events are those with lots of short gravity waves with wavelengths in the range from 20 to 60 km (Smith and Kruse 2017).
7. Comparing wave drag laws with WRF drag time series
The goal of this section is to compare the matrix drag law [(25), (26), (34), and (37)] against the WRF wave drag time series. To review, our drag law uses the instantaneous regional wind speed to compute the smoothing length L and then uses the corresponding drag matrix
In Figs. 8 and 9, we show the time series of zonal and meridional WRF drag components and the predicted drag components from the full matrix model. The agreement for both components appears good, and the Pearson correlation coefficients are CC = 0.91 and CC = 0.87, respectively, for
To see whether our treatments of anisotropy and nonlinearity significantly improve drag predictions, we degrade the full model (say, drag law 4) to three simpler drag laws (drag laws 1–3) in Table 3. Drag law 1 is linear and isotropic. Drag laws 2 and 3 include either anisotropy or nonlinearity. Drag law 4 has both. To linearize the model, we use the RMS wind speed for the full test period |U| = 14.4 m s−1 to determine a fixed value for smoothing length scale. From (37), this gives L = 33 km. To make the model isotropic, we imagine an isotropic South Island, with a “scalar” drag matrix
Error characteristics of four wave drag formulations for zonal (DX) and meridional (DY) components.
The Pearson CCs rise from CC = 0.75 and 0.71 for the degraded linear isotropic model (drag law 1) to CC = 0.91 and 0.87 for the full model (drag law 4). The mean absolute error (MAE) decreases from MAE = 11.9 and 10.0 GN to MAE = 7.8 and 7.3 GN. Likewise, the root-mean-square error (RMSE) decreases from model 1 to model 4. By all of these measures, the full nonlinear anisotropic model makes the best prediction.
8. Summary and conclusions
We have shown that a 2 × 2 drag matrix with variable elements
The proposed drag matrix formulation requires that we neglect the Coriolis force and nonhydrostatic effects. These assumptions are well supported by previous work and recent aircraft wave observations (Smith et al. 2016; Smith and Kruse 2017). The hydrostatic assumption can lead to large errors if applied to small-scale rough terrain. We avoid this problem by always smoothing the terrain into the hydrostatic range. For simplicity, we have also taken the stability frequency N and the air density ρ as constant. Perhaps it is surprising that the drag vector can be well forecast knowing only the regional wind vector. No meteorological information regarding fronts, vertical wind shear, static stability, moisture, or diurnal/seasonal solar insolation is needed.
Two mathematical properties of the wave drag matrix
The nonlinearity in wave generation is treated by smoothing the terrain to qualitatively take into account upstream blocking, flow around steep isolated peaks, and cold air trapped in valleys. Wind speeds of |U| = 10, 20, and 30 m s−1 give smoothing scales of L = 54, 27, and 18 km, respectively. At slow wind speeds, we find that a smoothing scale as long as L = 60 km is suitable. This eliminates wave drag contributions from short-scale waves. As wind speed increases, we reduce the smoothing scale to almost L = 10 km to retain rougher terrain and shorter waves. The elements of the drag coefficient matrix
The increased terrain smoothing at low wind speeds works to decrease the typical mountain height
For South Island, the drag matrix has principal axes oriented at about −42° to the cardinal eastward direction. This direction agrees with the NZ orientation determined from the second moments (
The quantitative success of the theory may be caused in part by the large area of South Island, over which we average. Wave drag on such a large area seems to be very predictable if U and V are known. Drag in a smaller area (e.g., a 20 km × 20 km cell) may be more difficult to predict as special types of terrain or wave dynamics may apply locally (Bacmeister and Pierrehumbert 1998; Rottman and Smith 1989). On the other hand, our use of a single averaged environmental wind vector (U, V) and drag vector
The current method could be applied to other mountainous regions. The first step is to perform two linear theory FFT hydrostatic drag calculations along cardinal directions to determine the three elements of the drag matrix (
The application of these results to wave drag parameterization design will vary depending on the grid size of the global model (Vosper et al. 2016). In general, the subgrid parameterized terrain [i.e., h(x, y)] will be the residual after the original terrain is smoothed to fit the GCM grid. This residual terrain may have zero volume and a reduced variance. The residual terrain will be used to derive the drag matrix
Finally, the wave drag predicted here is not the full representation of momentum transfer between Earth and the atmosphere. Pressure gradients from cold-air damming or from geostrophic flow can act on mountains but are not included in wave drag. Turbulent wind stress on small hills and vegetation is also not included.
Acknowledgments
This work was supported by Grant NSF-AGS-1338655 from the Physical and Dynamical Meteorology branch of the Geoscience division of the National Science Foundation. Terrain datasets were prepared by Laurent Bonneau. A discussion with Julio Bacmeister was helpful. Thanks also to David Fritts, James Doyle, Steve Eckermann, and Michael Taylor for their work in organizing the DEEPWAVE field project in New Zealand, which motivated this work. We acknowledge the high-performance computing from Cheyenne provided by the NCAR’s Computational and Information Systems laboratory sponsored by the National Science Foundation. Three anonymous reviewers offered extensive useful comments.
APPENDIX
Further Properties of the Drag Matrix
a. Axisymmetric terrain
If the terrain is axisymmetric so that
b. Proof of positive definiteness
To see why
c. Unphysical drag matrix
d. Rotated terrain
e. Adding terrains
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