Diurnal variation of mountain waves

Mountain waves could be modified as the boundary layer varies between stable and convective. However case studies show mountain waves day and night, and above e.g. convective rolls with precipitation lines over mountains. VHF radar measurements of vertical wind (1990–2006) confirm a seasonal variation of mountain-wave amplitude, yet there is little diurnal variation of amplitude. Mountain-wave azimuth shows possible diurnal variation compared to wind rotation across the boundary layer.


Introduction
Information on diurnal variation of mountain waves could be useful since the effect of, for instance, diurnal convection is uncertain.Convection could disrupt stable airflow of mountain waves (Ludlam, 1952), add to the mountain peaks forcing waves (Wallington, 1977), or modify wave modes (Ralph et al., 1997) and amplitudes (Georgelin et al., 1996).
Gravity waves above convection are usually categorised as convection waves, separate from mountain waves, and waves above orographic convection have also been interpreted as a type of convection wave (Rovesti, 1970;Bradbury, 1990;Hauf, 1993).However, waves above convective rolls over mountains (vertical wind tens of cm s −1 or more, on timescale of several hours, and disappearing with a turbulence layer for horizontal wind near zero) often appear typical of mountain waves (Worthington, 2002).
Correspondence to: R.M. Worthington (rmw092001@yahoo.com)1.1 Definitions of mountain waves Mountain waves could be defined as in standard theoretical models of wavelike air flow over a ridge, with the lowest streamlines usually following the mountain surface; similar waves modified by convection could be excluded from this category of classic or idealised mountain wave.Waves linked to orographic convection could have been wrongly identified as mountain waves in many studies using e.g.aircraft or radar data in the troposphere and stratosphere, without also measuring the boundary-layer structure.
However, terminology for waves above mountains is often less specific about the cause of the waves, and instead based on wave characteristics, e.g."standing wave" if remaining almost static, or "orographic wave" if associated with mountains.A typical definition of mountain wave as "an atmospheric gravity wave, formed when stable air flow passes over a mountain or mountain barrier" (American Meteorological Society, Glossary of Meteorology) does not exclude effects of convection, rotors and turbulence on mountain wave formation, or a wave launching height above the mountain surface (although "lee wave" could imply a mountain obstacle upwind, at the same height as the wave, and more directly causing the wave).
There are other variations on standard mountain-wave theory, such as a stagnant boundary layer absorbing waves instead of reflection at the ground (Smith et al., 2002;Jiang et al., 2006).Also there can be separate categories of mountain wave, such as "evening wave" (Roper and Scorer, 1952) for a mountain wave formed as convection stops and a stable boundary layer develops, maybe linked to katabatic wind.
This paper uses a definition of mountain wave as a standing gravity wave above mountains (excluding e.g.propagating gravity waves such as typical convection waves).The paper looks at mountain waves above convective rolls for two case studies in Sect.2, since extensive convective rolls could be expected to disrupt daytime mountain waves, and provide an example of diurnal variation linked to stable-convectiveresidual boundary layer development above mountains (e.g., Kalthoff et al., 1998).Section 2.1 also includes weather radar measurements of precipitation lines for comparison with e.g.Kirshbaum and Durran (2005a).Section 3 then uses thousands of hours of VHF radar data to check for diurnal and seasonal variations of mountain-wave amplitude, and Sect. 4 shows mountain-wave azimuth and compares VHF radar and satellite measurement methods.

Convective rolls, precipitation lines, and mountain waves
2.1 Case study, 1 October 2001 Miniscloux et al. (2001), Cosma et al. (2002) and Kirshbaum and Durran (2005a) ("MCKD") show weather radar measurements of along-wind precipitation lines above mountains.Numerical models imply these precipitation lines can be caused by convergence lines and convective rolls, triggered by mountains and mountain waves.Weather radar in Fig. 3 shows precipitation often also in lines, in the region of cloud lines above mountains in Fig. 2 (e.g.north and south of label "Birmingham").Average rain distribution is similar to orographic rain increasing above high ground (e.g., Bonell and Sumner, 1992).Some southwest-north-east rain lines to the east at ∼12:00-15:00 UT advect with the wind instead of remaining above mountains, and orographic rain to north, ∼53-56 • N, 1-4 • W, is less linear.There is also deeper convection in the cloud and rain lines, with thunder ∼11:00-12:00 UT at Birmingham, from a heavy rain area appearing near mountains of south Wales a few hours earlier.Occurrence of rain lines allows compari-son with MCKD, using VHF radar to measure the wave field (e.g., Röttger, 2000), and with convective rain as another factor in any diurnal effect of the convective boundary layer on mountain waves.
Figures 4a, c show bands of upward and downward vertical wind (W ) typical of mountain waves (e.g.Worthington, 2002, Figs. 8a, 11a), measured using a 46.5 MHz VHF radar near Aberystwyth (Fig. 1).Figures 4a, c, 7a use a vertical radar beam; symmetric 6 • beams show similar waves but are noisier above ∼16 km height.Vertical wavelength increases with jet wind speed as expected for mountain waves in Figs. 4a, b (e.g., Worthington et al., 2001).Therefore along-wind rain lines above mountains in Fig. 3 (MCKD) are occurring in a case study similar to other observations of mountain waves above convective rolls (Worthington, 2002(Worthington, , 2005)).
Convective rolls could raise the effective surface of the mountains causing mountain waves, for sheared airflow over shallow convection (Sinha, 1966); alternatively vertical air motion in mountain waves could trigger convective rolls downwind within the lowest region of wave flow (Kirshbaum and Durran, 2005a,b).Mountain waves and convective rolls could be difficult to separate as cause and effect; whether mountain wave or convection starts further upwind could be significant.However cloud streets in Figs.2b-d already start slightly upwind of the VHF radar measuring mountain waves in Fig. 4.
Figure 5 shows MODIS (Moderate-resolution Imaging Spectroradiometer) 250-m resolution images, with examples of wave cloud upwind of cloud streets near 62 • N, 7 • W (Fig. 5c); cloud streets possibly upwind of wave cloud then continuing over mountains higher than 1 km, 53 • N, 4 • W (Fig. 5f); and smooth cloud streets similar to lenticular wave cloud, 51.5 • N, 10 • W (Figs. 5a, b, d, e).In Figs.5a, b, d, e there also appear to be cloud streets above sea to south and west.Since convective rolls and mountain waves can often occur separately, or with either upwind, they could be described as separate processes which can coincide and interact, instead of one process causing the other.
For diurnal variation, Figs.2-4 show mountain waves and also cloud and rain lines occur both night and day on 1 October 2001, e.g.Fig. 2a at 02:26 UT.Sunrise and sunset are ∼06:15 and 17:55 UT.However along-wind cloud lines in Figs.2a and 2b-e could be of different types such as streaks and rolls (e.g., Young et al., 2002;Shun et al., 2003).

Case study, 19 February 2004
Figure 6 shows cloud streets starting upwind, continuing above and downwind, of mountains.There are variations in thickness of the cloud streets, with appearance similar to "knots in strings", above mountain areas of e.g.North York Moors (54.3 • N, 1 • W), Peak District (53 • N, 2 • W) and near Wales (52.5 • N, 3 • W) which are higher than e.g.Chiltern Hills in Tian et al. (2003)."Knot" spacing of ∼9 km is larger  than individual cumulus clouds, not caused by satellite scan lines, and positions of "knots" are aligned perpendicular to the cloud streets, suggesting perhaps a wave pattern with phase lines perpendicular to cloud streets (Bradbury, 1990;Hindman et al., 2004).
Vertical-beam spectral width corrected for beambroadening (Fig. 7c) shows a turbulent layer for over 10 h at 16-17 km height, where mountain waves disappear at a critical layer (e.g., Worthington and Thomas, 1996).High spectral width near 10 km is partly spurious, caused by low signal-noise ratio below the tropopause at ∼11 km, and problems of exactly removing a substantial beam-broadening component in >30 m s −1 horizontal wind.Since 16-17 km is above the regions of high jet-stream wind shear, Fig. 7 could show mountain-wave breaking not shear instability although horizontal wind speed is several m s −1 .Typical convection waves should propagate downwind, whereas the waves above cloud streets in Fig. 7a keep the same phase for hours above the VHF radar.One explanation could be if mountain waves can exist through the convective boundary layer (Winstead et al., 2002), keeping the wave pattern "anchored" to the mountains; then waves as in Fig. 7a not only look like mountain waves, but can be partly caused as in standard mountain wave theory, modified by convection.
Satellite images at 02:35, 04:16 and 21:07 UT show wave cloud instead of the cloud streets in Fig. 6, yet Fig. 7 shows mountain waves and a turbulence layer for most of 19 February 2004.Sunrise and sunset are at ∼07:25 and 17:35 UT.Sections 2.1 and 2.2 therefore show a lack of diurnal variation of mountain waves, when variation could be expected.

Diurnal and seasonal mountain-wave amplitude
Figure 8 shows diurnal and seasonal variation of surface weather (Figs.8a-d), and magnitude of vertical wind |W | (Figs.8f-i) as a more direct measure of mountain-wave   (Kuettner et al., 1987;Gage et al., 1989;Sato, 1992;Böhme et al., 2004) with periods of tens of minutes while retaining more static mountain waves.
Surface weather is included in Fig. 8 since if solar radiation, temperature and wind show minimal diurnal variation in winter at the VHF radar location, then diurnal variations of mountain waves might also be minimal.Sorting data for time of year shows also if any diurnal effect follows seasonal variation of sunrise and sunset time; Fig. 8 uses 30 intervals of ∼12.2 days.Figures 8a-d show some diurnal vari-ation through winter.Surface wind differs in Figs.8c and  d, since Fig. 8c is measured near the top of a low hill, and Fig. 8d in a valley, more sheltered from prevailing wind, except for increased afternoon wind speed from e.g.sea breeze channelled in valleys, slope and valley winds, and convective boundary-layer mixing.
Wind speed higher in the boundary layer could be more correlated to mountain-wave amplitude than surface wind.Nastrom and Gage (1984) report |W | more correlated to 700 mB than 850 mB wind.Correlation of wind profiles is also possible to e.g.airglow (Sukhodoyev et al., 1989) or orographic rain (Neiman et al., 2002).Figure 9 shows correlation of wind profiles to |W | at 1.7-2.5 km height where mountain waves are immediately above their source region and below possible critical layers higher in the atmosphere.Correlation to |W | at e.g.3-8 or 12-15 km instead of 1.7-2.5 km is more constant above ∼1 km.Correlation in Fig. 9 is only ∼0.35 or less, because of e.g.variations in horizontal position of mountain waves, and wave structure; also correlation profiles are altered by e.g.UHF data quality decreasing with height, and the horizontal separation of VHF  radar from radiosondes (typically tens of km; 50 km to their launch site at Aberporth).However, the height of maximum correlation is mostly ∼0.5-1 km, using all wind directions, or subsets as |W | increases with both westerly and easterly wind (Prichard et al., 1995).Figure 8e shows wind speed at 800 m height, with less diurnal variation, and faster wind speed in autumn and winter than Figs.8c, d.
|W | increases in autumn and winter, similar to Fig. 8e and seasonal variation of mountain-wave clouds (Cruette, 1976;Lester, 1978).Diurnal variation of |W | is much less than seasonal variation and appears fairly random.Figure 8 can use subsets of wind speed, wind direction, and/or surface weather; Fig. 8i is for 2-km wind >10 m s −1 , to check for diurnal variation in summer with faster wind speed.However, there is a pattern similar to Figs. 8f-h for Fig. 8i, and also for: low-level wind from north-west-south 180 • segment mostly over the sea, or north-east-south 180 • segment over land (Fig. 1) with different surface heating and boundary layer; using e.g.maximum |W | at any height 3-8 km instead of mean; or using variance, W 2 .Also, W probability distribution is nearly constant with time of day.
Despite lack of diurnal variation in Figs.8f-i, the boundary layer below mountain waves varies between stable and convective.However, even case studies in Figs. 4, 7 show little diurnal effect, above convective rolls.If mountain waves show almost no diurnal variation of amplitude, this could imply that mountain wave systems can have altered forcing mechanisms (stable and linear, or turbulent including convection) in the boundary layer without the wave field varying significantly above the boundary layer.(d, e) satellites, 1996-2006, and surface wind averaged from up to 26 sites (Fig. 1).Green, red and blue lines show yearly medians, for 00:00-24:00 UT and other time intervals.Vertical lines in (a, b, d, e) show last data in Worthington (1999bWorthington ( , 2001)).Dots in (c) are for every third data point.Mountain-wave azimuth is of horizontal wavevector for (a, b), and wave clouds for (d, e), measured clockwise from north.

Boundary-layer wind and wave azimuth
Another parameter to check for diurnal variation is azimuth of mountain-wave horizontal wavevector, on average between the surface and tropospheric wind azimuths (Worthington, 1999b(Worthington, , 2001)).Figures 10-12 are to check for diurnal variation in over 15 years of data, and compare results from VHF radar and satellites, using an improved method of measuring wave azimuth on satellite images.Other mountain-wave parameters could also be useful, such as any diurnal variation of horizontal phase speed from zero, for e.g.numerical models.
Figure 10 shows mountain-wave azimuth measured as in Worthington (1999aWorthington ( ,b, 2001)), on average clockwise from surface wind in Figs.10a, d, and anticlockwise from ∼2 km (1.7-2.3 km) wind in Figs.10b, e. Wave azimuth from VHF radar uses height-time intervals 3-8 km ×1 h, with |W |>0.05 m s −1 and azimuth error <20 • .Data to right of vertical lines in Figs.10a, b, d, e are more recent than Worthington (1999bWorthington ( , 2001)), to check that the wave and cloud azimuth results persist and do not disappear.Also Fig. 10c shows expected clockwise wind rotation with height in the boundary layer, to compare with Figs.10a, b, d, e. Liziola andBalsley (1997, 1998) use an alternative 3-radar method for measuring wave azimuth, which may give better results for propagating convection waves than for mountain waves (Carter et al., 1989).
2. set all areas to zero except mountain-wave or convective-roll cloud lines, above or near land since surface wind measurements in Fig. 1 are above land, also so mountain waves are above their source region, instead of being downwind lee waves.
3. take 2-D autocorrelation of each image using Fast Fourier Transform (and optionally subtract smoothed autocorrelation).
4. find the azimuth of the autocorrelation pattern, by rotating in steps of 1 • using cubic interpolation, and averaging north-south in the square region of Fig. 11.The north-south average shows maximum east-west variations from its median, when the autocorrelation pattern is rotated with its lines north-south.
5. cloud azimuths (∼5%) are discarded if there are problems from e.g.lines of cloud shadows, or multiple wave azimuths., 1996-2006.Wave azimuth from satellite is perpendicular to cloud lines, pointing upwind.Time separation between radar and satellite data is 0, 1 or 2 h.The diagonal line is for equal radar and satellite azimuths.
Figures 10d, e include cloud streets as in Worthington (2001), expected to be slightly clockwise from parallel to the surface wind, in checking if mountain-wave clouds are slightly clockwise from perpendicular.There are more data before 1999 in Figs.10d, e than Worthington (2001), since the autocorrelation method can also measure azimuth of patchy cloud lines.An average image rotation of 5 • clockwise from north is subtracted as in Worthington (2001).
Green, red and blue lines in Figs.10a-c show yearly medians of the difference between wave and wind azimuth, for 00:00-24:00 UT and other time intervals.Red lines in Figs.10a-c mostly show less positive or more negative azimuth differences than blue lines.Horizontal wind rotation from surface to 2 km height is less for the daytime convective boundary layer (Fig. 8j), so red and blue lines in Fig. 10c are for 12:00-17:00 UT and 23:00-04:00 UT.
Plots similar to Fig. 8j for difference of mountain-wave and wind azimuth are more variable than Fig. 8j, but possibly show wave azimuth is nearer to 2-km wind and further away from surface wind at e.g.15:00-20:00 UT compared to 05:00-10:00 UT (blue, red lines in Figs.10a, b).An explanation could be if, at the mountain-wave launching height (Shutts, 1997), the horizontal wavevector of mountain waves is on average parallel to horizontal wind (Worthington, 1999b); in the afternoon, with a developed convective boundary layer, mountain wave azimuth could be parallel to upper-boundary-layer wind; in the morning, with more stable residual-layer flow over a shallower convective boundary layer, mountain wave azimuth could instead be parallel to lower-boundary-layer wind, with much variability from profiles of e.g.wind shear and temperature lapse rate.Occurrence of boundary-layer mountain-wave clouds at night, and convective clouds in daytime, could be consistent with a higher wave launching height instead of mountain waves ceasing in daytime., e).Measurements are in the same hour, or ±1, ±2 h to provide more data.VHF radar and satellite measurements of wave azimuth agree fairly well, with median difference <5 • for Fig. 12, despite being limited to occurrence of VHF aspect sensitivity and wave cloud.

Conclusions
Mountain waves near 52.4 • N, 4.0 • W show seasonal variation of amplitude, but much less diurnal variation despite the effects of boundary-layer convection.This negative result is however useful, in studying the effects of boundary layers on mountain waves.
Rain lines above mountains (Miniscloux et al., 2001;Cosma et al., 2002;Kirshbaum and Durran, 2005a,b) can occur in convective rolls beneath typical mountain waves observed by VHF radar.Convective rolls can start upwind of mountain waves, not only triggered downwind.
Horizontal wavevector of mountain waves is between surface and tropospheric wind direction, both day and night, but possibly nearer to surface than 2-km wind azimuth in the morning, compared to evening.
Figures 1-4 show a case study on 1 October 2001 with mountain waves above lines of convection and rain.Figures 2a, c, d, e are from NOAA AVHRR (National Oceanic and Atmospheric Adminstration, Advanced Very High Resolution Radiometer), and Fig. 2b from Landsat.There are cloud lines south-west to north-east, near parallel to the south-westerly surface wind, above mountains ∼52-53 • N, 3-4 • W.

Fig. 11 .
Fig. 11.Example satellite images of (a) mountain waves, (b) convective cloud streets, and autocorrelation of the unshaded areas of cloud.North is at the top.Diagonal lines show cloud azimuth from the region of autocorrelation marked by a square.

Fig. 12 .
Fig. 12. Mountain-wave azimuths from VHF radar and satellite images, 1996-2006.Wave azimuth from satellite is perpendicular to cloud lines, pointing upwind.Time separation between radar and satellite data is 0, 1 or 2 h.The diagonal line is for equal radar and satellite azimuths.