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The influence of background winds and attenuation on the propagation of atmospheric gravity waves
Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 857 – 869
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The in(uence of background winds and attenuation on the propagation of atmospheric gravity waves F. Ding∗ , W. Wan, H. Yuan Wuhan Ionospheric Observatory, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, 430071, China Received 24 April 2002; received in revised form 16 December 2002; accepted 17 March 2003
Abstract The in(uence of background winds and energy attenuation on the propagation of atmospheric gravity waves is numerically analyzed. The gravity waves, both in the internal and ducted forms, are included through employing ray-tracing method and full-wave solution method. Background winds with di9erent directions may cause ray paths of internal gravity waves to be horizontally prolonged, vertically steepened, re(ected or critically coupled, all of which change the accumulation of energy attenuation along ray paths. Only the penetrating waves propagating against winds can easily reach the ionospheric height with less energy attenuation. The propagation status of gravity waves with di9erent periods and phase speeds is classi;ed into the cut-o9 region, the re(ected region and the propagating region. All the three regions are in(uenced signi;cantly by winds. The area of the re(ected region reduces when gravity waves propagate in the same direction of winds and expands when propagating against wind. In propagating region, the horizontal attenuation distances of gravity waves increase and the arrival heights decrease when winds blow in the same direction of gravity waves, while the attenuation distances decrease and the arrival heights increase when gravity waves propagate against winds. The results for ducted gravity waves show that the in(uence of winds on waves of lower atmospheric modes is not noticeable for they propagate mainly under mesosphere, where the wind ;eld is relatively weak. However, strong winds at thermospheric height lead to considerable changes of dispersion relation and attenuation distance of upper atmospheric modes. Winds against the wave propagating direction support long-distance propagation of G mode, while the attenuation distances decrease when winds blow in the same direction of the wave. The distribution of TIDs observed by HF Doppler array at Wuhan is compared with the simulation of internal gravity waves. The observation of TIDs shows agreement with our numerical calculations. c 2003 Elsevier Science Ltd. All rights reserved. Keywords: Internal gravity wave; Ducted gravity wave; Attenuation; Wind
1. Introduction It is now generally believed that both the background winds and energy attenuation have considerable in(uence on the propagation of atmospheric gravity waves. Hines (1960) has initially pointed out that when an up-going gravity wave meets a wind, the wave may be re(ected or refracted. Cowling et al. (1971) studied the in(uence of
∗
Corresponding author. Tel.: 86-027-8719-7797. E-mail addresses: [email protected], [email protected] (F. Ding).
background winds on gravity wave ray paths. They found that depending on the direction of the wind and the wave vector, three di9erent e9ects may occur. If the gravity waves propagate along wind, the intrinsic frequency is shifted downward. When the wave frequency equals zero, the wave energy is lost to the mean (ow, which is known as critical coupling. If the gravity waves propagate against wind, it is possible in this case for the waves to be re(ected. In either of the two cases, if the wind speed is low, the waves will penetrate thermosphere and cause TIDs at ionospheric height. Yeh and Webb (1972) analyzed the relation of winds and TIDs and found that winds have the directional ;ltering e9ect on gravity waves. Waldock and
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Jones (1984) included the diurnal variation of winds in ray-tracing method and showed that the domain direction of gravity waves rotates clockwise in north hemisphere, which is experimentally addressed by Yuan and Wan (1997). Dunkerton (1984) showed that lower frequency gravity waves experienced severe horizontal refraction in winter hemisphere, due mainly to the meridional shears in zonal winds. Zhong et al. (1995) added a simpli;ed tidal model to the CIRA-86 reference wind model and used ray-tracing method to examine the e9ect of spatially and temporally varying wind on gravity wave propagation. He indicated that the temporal variation of wind causes the observed frequencies and the critical levels of gravity waves to be temporal, and that the time-dependent variation of the propagation features becomes more obvious in winter than in summer. As is discussed above, few studies take into consideration the energy attenuation of internal gravity waves in ray-tracing method. In fact, atmosphere is a dissipative medium for gravity waves especially above mesopause. This is one of the main reasons that a large number of up-going gravity waves cannot reach ionospheric height. A study by Scroeberl (1985) revealed that azimuthally refracted waves were considerably attenuated in amplitude. Eckermann (1992) used wave action conservation equation to calculate the attenuation of wave amplitude. In his paper, signi;cant di9erences in the latitude-season structure of the mean wave amplitudes are simulated, and comparison between measured and simulated distributions of wave propagation azimuths reveals good agreement in some locations. At the same time, ducting mechanism for surface and quasi-surface gravity waves was widely analyzed. The consideration of ducted gravity waves originated from the fact that the observed large-scale gravity waves can travel horizontally for long distance with small energy attenuation. As Francis (1973) and Richmond (1978) addressed, the atmosphere should consist of a continuous spectrum of freely propagating internal gravity waves and a discrete spectrum of ducted gravity wave modes. In the presence of the solid ground, surface waves known as Lamb waves can exist in the atmosphere that have energy concentrated near the surface of the Earth. The sudden temperature rise in the low thermosphere can support surface waves propagating horizontally with energy concentrated near the sudden temperature rise. Thome (1968) initially used a simpli;ed temperature pro;le composed of two isothermal half-spaces to get the zero-rank mode of gravity waves (namely mode G0 ). Francis (1973) deduced in isothermal atmosphere the L0 mode of Lamb waves supported by the full re(ection of the Earth. Because of the existence of both the sudden temperature rise and the earth’s surface, the realistic modes must be G0 , L0 and some higher rank modes of the series of L and G. Francis (1973) also used a realistic continuous temperature pro;le to get these series of higher rank modes. These modes can guide acoustic-gravity waves for
thousands of kilometers along the earth, and most of them show agreement with TID observations. Liang et al. (1998) solved the Green’s function of Navier–Stokes equations on the basis of an analytic atmospheric temperature model. The analytical solutions of Liang et al. (1998) con;rmed the result of Francis that the major re(ection mechanism of the imperfectly ducted modes is closely related to the sudden rise of temperature at the base of thermosphere. Though the guided modes of large-scale gravity waves were widely studied, most of the researches focused on the role of sudden temperature rise, while background wind was seldom included. In fact, the realistic wind speed always reaches 200 m=s at thermosphere (Hedin et al., 1991a). Thus the in(uence of background winds cannot be ignored even for large-scale ducted waves. Hickey et al. (1998) found that an accurate description of winds is an essential requirement for a complete interpretation of the observed wave-driven airglow (uctuations. The purpose of this paper is to examine gravity wave characteristics under the in(uence of background winds in a dissipative atmosphere, which will aid in understanding the distribution of TIDs observed by our HF-Doppler array in Wuhan Ionospheric Observatory, China (Yuan, 1995). 2. Methods and background parameter proles In this paper, both the internal waves and the large-scale ducted waves are included. For internal waves, the “extended” ray-tracing method is used. The tracing method differs from previous treatments by including the energy attenuation along the ray path. And atmosphere-strati;ed method is adopted to obtain the full wave solutions of ducted waves, in which the zero-rank wind ;eld is included. The two methods are displayed in the appendix. The numerical results together with the observed TIDs are used to discuss the e9ect of winds on gravity waves in a dissipative atmosphere. The HWM93 wind model and MSISE90 atmospheric model (Hedin et al., 1991a, b) are adopted to obtain the wind and temperature pro;les. The two-dimensional wind and temperature pro;les at 12:00 and 22:00 local time are plotted in Fig. 1, ranging in height from the ground to 600 km. The inputting date is December 1, 1988 in Wuhan (114◦ E, 30◦ N). The tide is included in the wind pro;les in order to simulate the actual wind ;eld. As shown in Fig. 1a, the wind in upper atmosphere is much stronger at 22:00 than it is at 12:00. At 22:00, the wind positive to the west (referring to Fig. 1a) is rather weak below the height of 100 km, while it increases rapidly when above 150 km. There is a strong wind shear in the height between 50 and 100 km, varying from 40 to −10 m=s. The wind grows slowly above the height of 200 km, and the speed remains around 140 m=s. The wind speed is basically positive when above 100 km height. While at 12:00, there is a strong wind shear below mesopause, and the wind speed remains less than 40 m=s in upper atmosphere. The temperatures of the
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two times change signi;cantly above 120 km height. The change of temperature is not noticeable below that height. The coeKcients of viscosity and thermal conductivity are computed from DalGarno and Smith (1962):
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3.1. Wind e6ect on gravity wave ray paths Assume that a gravity wave is launched at 22:00 from a ;xed point (114◦ E, 30◦ N) with a speci;c initial phase speed and period. Fig. 2 presents gravity wave ray paths with initial period of 30 min. The left branch in Fig. 2 are waves propagating against winds and the right along them. A similar result was given by Cowling et al. (1971), whereas the energy attenuation was not considered. Three types of propagation of gravity waves are then displayed as: (1) some of the waves are re(ected between mesospause and low thermosphere, e.g. the 100-m/s wave in the left branch and the 240-m/s wave in the right; (2) some are critically coupled at low thermosphere, e.g. the 100-m/s wave in the right branch; and (3) others penetrate the atmosphere and reaching ionospheric height. The di9erences of wave propagation features are mainly caused by the variation of winds. For the waves propagating along winds, the frequencies decreased. This is obvious especially for the small-velocity waves. If the intrinsic frequency equal zero, the energy of the gravity wave is coupled to the mean (ow. Other waves with larger phase velocity are able to propagate higher. The ray paths of the penetrating waves are bent down and horizontally prolonged, which consequently causes the increase of wave attenuation accumulated on ray paths; when propagating
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against wind, ray paths with both the small phase velocity and the higher ones are re(ected back to the ground. In this case, ray paths of penetrating waves are vertically steepened. These waves can propagate much higher than those along winds do. The height pro;les of wave amplitude for these waves are illustrated in Fig. 3. Most of the energy attenuation occurs above the height of 140 km. This is because the viscosity and heat conductivity of the atmosphere increases with height and they are more in(uential when above that height. Moreover, the wave amplitudes experience more severe attenuation under the condition of propagating along winds than waves against winds do. Since the viscosity, heat conductivity and temperature pro;les are the same in both cases, the
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di9erence of amplitude pro;les is caused by winds. When the ray paths are horizontally prolonged by winds along wave direction, the attenuation of wave energy accumulated along the ray paths increases and prevents the waves to propagate higher. Contrary to the case along winds, when the ray paths are steepened by winds against wave direction, the ray paths of the waves is shortened, which makes the waves propagate rapidly to higher height with small energy attenuation. The 100-m/s wave propagating along winds in Fig. 2 shows an example of critical coupling. It is shown in Fig. 3 that the wave energy dissipates quickly above 140 km height. Unlike the ray path presented by Waldock and Jones (1984), the wave ray path in Fig. 2 does not propagate horizontally. This is because the calculation of wave energy becomes uncertain when the waves is fully critically coupled, due to the limitation of ray-tracing method. In this case, the intrinsic frequency vanishes, and wave energy is lost to the mean (ow. In this case, the dispersion relation (A.10) is not valid, neither is Eq. (A.17). The analysis indicates that winds with di9erent directions are likely to cause ray paths to be horizontally prolonged, vertically steepened, re(ected or critically coupled. All of these change the accumulation of energy attenuation along ray paths. Among these rays, only the penetrating waves propagating against winds can easily reach the ionosphere height with less energy attenuation. 3.2. Wind e6ect on internal gravity waves with arbitrary period and phase velocity In this section, only the gravity waves with phase speeds between 100 and 300 m=s and periods of up to 80 min are
considered. The simulation will be compared later with the HF observations of TIDs, the observation height of which is about 200 km. Manson et al. (1997) showed that the gravity waves with phase speed less than 100 m=s are common in lower atmosphere. However, most of these low speed waves become evanescent or re(ected below or near the mesopause due to strong wind shears and steep atmospheric temperature gradient. So, only waves with faster speed are likely to reach the height of 200 km and observed by HF-Doppler arrays (Yuan, 1995). We choose gravity waves ranging from 100 to 300 m=s and 6 to 80 min and trace them from troposphere up to ionosphere. The results are presented in Fig. 4 at 22:00 local time and Fig. 5 at 12:00 local time. Figs. 4(a) – (c) are, respectively, propagation results of gravity waves under the condition of propagating without winds, along winds and against winds at 22:00. The re(ected region is denoted by horizontal hatching and the cut-o9 region by vertical hatching. And the rest of the area in Fig. 4 refers to the propagating region. Two type of waves constitute the propagating region: (1) The waves propagating without being re(ected or cut o9; (2) The waves reaching the height of 200 km and then being re(ected at a certain height above 200 km. Type two is included in the propagating region for the reason that the observation height of our Doppler Arrays is around 200 km. So, the waves in part two can still be detected, though they are ;nally re(ected above the observation height. (1) Cut-o6 region: The cut-o9 region mainly covers the high velocity and short period area. The waves in this region are cut o9 in troposphere due to the restrain of gravity wave dispersion relation. The existence of cut-o9 region shows that the propagation of gravity waves has an upper limit of phase velocity and a lower limit of wave period. For the waves propagating without winds, the limit is about 270 m=s and 6 min of the period. The limit changes into 260 m=s and over 10 min for waves along winds, and it becomes 285 m=s and 7 min for those against winds. The area of cut-o9 region in Fig. 4(a) is much similar to the result of Hines (1960). However, it changes signi;cantly in Fig. 4(b) and (c). The changes are obviously caused by Doppler-shift of wave frequency under the in(uence of winds. (2) Re8ected region: The re(ected region is a transitional area between the cut-o9 region and the propagating region. In case of no winds (Fig. 4(a)), the re(ection is completely caused by the vertical increase of atmospheric temperature, which leads to the decline of Brunt–VOasOalOa frequency !b and accordingly causes gravity waves to be re(ected at a certain height where the wave frequency exceeds !b . When the gravity waves propagate along winds (Fig. 4(b)), the area of re(ected region obviously reduces because winds in the same direction of waves cause wave frequency to decline so as to be kept away from the buoyancy frequency. This makes it hard for gravity waves to be re(ected. On the contrary, winds against waves make the re(ected region expand into
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Fig. 4. (a) – (c) are propagation results of gravity waves under the condition of propagating without winds, along winds and against winds at 22:00 local time. x- and y-axis represent wave period and phase velocity, respectively. Re(ection region is denoted by horizontal hatching and cut-o9 region by vertical hatching. The solid and dashed lines show the contours of horizontal propagating distance and arrival height of gravity waves, with the value of each contour marked near every contour. The unit of value for contours is km.
short-period area (less than about 20 min), which shows that high-frequency waves are more easily in(uenced by winds. This is shown in Fig. 4(c). (3) Propagating region: As the energy attenuation accumulated along ray path increases with height, the waves in propagating region have a limited range of the arrival height and horizontal propagating distance, which are plotted in Fig. 4 in the form of contours. The solid and dashed lines denote the horizontal propagating distance and arrival height contours, respectively. When there is no wind, most of the waves in propagating region can reach the height of 200 km. The arrival height rises from 200 km at the speed of 100 m=s to over 320 km at 200 m=s, and the horizontal propagating distance increases from 500 km at about 18 min to 5000 km at 80 min. The increase of horizontal propagating distance with the increase of period is slow when the period is small, while it increases rapidly when the period is more than 50 min. When the gravity waves propagate along winds (Fig. 4(b)), the horizontal distance increases for several hundred kilometers.
F. Ding et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 857 – 869 800 G0
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The change is not so noticeable as compared to the average propagating distance for gravity waves. However, the arrival heights of all the waves fall down obviously by 50 –70 km. The drop of height is more obvious for higher velocity waves than for lower ones. In contrast, when the waves propagate against winds (Fig. 4c), the arrival heights rise remarkably by 80 –100 km and the horizontal propagating distances decrease. In short-period area (from 20 to 30 min), both the arrival height and the horizontal propagating distance increase rapidly with the increase of period, which indicates that short-period waves are greatly in(uenced by winds. Fig. 5 presents the simulation results at 12:00 local time, where Fig. 5(a) and (b) are for gravity waves propagating along winds and against winds. The wind ;eld is much weaker around 12:00. The area of cut-o9 region and re(ected region is consequently similar with those propagating without wind (Fig. 4(a)). Also the change of horizontal propagation distances and arrival heights are not noticeable both in Fig. 5a and b, as compared to Fig. 4(a). The results show that the propagation of gravity waves changes signi;cantly with time. Cowling et al. (1971) and Waldock and Jones (1984) had studied the e9ect of diurnal change of winds on the propagation of gravity waves. In their studies, only the gravity waves with several speci;c initial periods and phase velocities are considered. This paper provides a full view of propagation status for any arbitrary initial period and phase velocity in a certain range. However, the results in Figs. 4 and 5 lack the continuous change of winds with time. As the wind is almost strongest at 22:00 and weakest at 12:00, the diurnal change of wind is supposed to lead to diurnal change of propagation status of gravity waves, with the horizontal propagation distances and arrival heights varying between the values shown in Figs. 4 and 5.
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The ducted waves di9er from internal waves in that the energy of ducted waves concentrates at the height where the background parameters vertically change sharply. In this case, the horizontal attenuation distance is an important parameter to examine the validity of ducting mechanism. Fig. 6(a) plots the dispersion curves and attenuation distance curves of ducted gravity waves under the condition of propagating without winds. Fig. 7(a) shows the corresponding attenuation distance curves. And Fig. 8(a) shows the corresponding energy pro;les. In all of the ;gures, the curves fall into two categories: mode L and mode G (Francis, 1973). L0 mode has a ;xed phase velocity of 312 m=s. Its horizontal attenuation distance is over 106 km. While L1 mode has an almost ;xed phase velocity of 255 m=s, and the horizontal attenuation distance is over 10; 000 km, which is much shorter. The distributions of wave energy for L modes mainly locate in lower atmosphere, namely the height below
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Fig. 6. Dispersion relation curves of ducted gravity waves under the condition of propagating without winds (a), along winds (b) and against winds (c) at 22:00 LT. x- and y-axis denote wave period and phase velocity, respectively. The mode of each curve is marked nearby. In order to show clearly the changes of dispersion relation for short-period gravity waves, the x-axis is set to be logarithmic.
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F. Ding et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 857 – 869 107 106 gravity 105 5000 4000 3000
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100 km. In this height range, the in(uence of dissipation is so weak that gravity waves for mode L0 and L1 can propagate much farther than waves of mode G do. As Lindzen and Blake (1972) pointed out, if we took into account the leakage of wave energy up to higher atmosphere and the surface friction of the Earth, the realistic attenuation distance would not be so long. However, from our calculation we ;nd clearly that mode L0 and L1 , with relatively smaller velocity, play an important role in transmitting energy to long distance in lower atmosphere. As winds are usually weak in lower atmosphere, the in(uence of winds on L modes is not obvious. Figs. 6(b), 7(b) and 8(b) show gravity wave modes under the condition of propagating along winds. And Figs. 6(c), 7(c) and 8(c) show results against winds. Only the results at 22:00 local time are shown here. The winds in the same direction of wave make the velocity of mode L0 and L1 rise by about 10 m=s. The corresponding attenuation distances drop a little. On the contrary, winds against gravity waves make the velocity of mode L0 and L1 fall down by about 10 m=s and the attenuation distances increase a little.
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The upper atmospheric modes, on the other hand, have a wide distribution of velocities ranging from 200 to 800 m=s. The attenuation distances are now smaller than 5000 km. The slower the velocity is, the closer the gravity waves are able to travel. The energy pro;les in Fig. 8(a) show that the energy of G modes concentrates between the heights of 100 and 400 km. This implies that gravity waves tend to propagate along that sudden temperature rise at the base of thermosphere. As Francis (1973) reported, the in(uence of viscosity and thermal conductivity become severe as the height rises. Hence the waves of G modes endure greater energy attenuation than waves of L modes do. Another factor of the energy attenuation is coupling of ionosphere and the gravity waves, which is shown as TIDs. All these make the attenuation distance of mode G much closer than those of waves of mode L. Because the wind usually grows strong at the base of thermosphere, it is found to have appreciable e9ects on gravity waves of G modes. When propagating against winds, the velocities of mode G fall sharply by 50 –100 m=s, and the attenuation distances rise by several hundred kilometers. The rising of attenuation distance is more obvious for high-rank low-velocity short-period modes. As compared with wave modes without winds, the energy distribution is shifted toward lower atmosphere. The descendent of energy distribution makes the energy attenuation caused by viscosity and thermal conductivity drop down. This leads to the rise of attenuation distances. Hence winds against the gravity waves can eKciently support the long-distance propagation of low-velocity short-period modes. When the waves propagate along winds, the velocities of all the G modes rise considerably by 50 –100 m=s, whereas G0 mode is not presented in Fig. 6(b) because its velocity exceeds 800 m=s. The attenuation distances are now less than 1000 km. Among them, the minimum distance for G5 mode is only 100 km. The distance is much closer than that of internal waves shown in Fig. 2. Though the ascendant of energy pro;le is not noticeable in Fig. 8(b), the viscosity and thermal conductivity, which increase exponentially with height, can cause severe energy attenuation. That is why the attenuation distances of G mode fall sharply when propagating along winds. 5. TID observation and statistical results Frequently the ionosphere is considered as a passive tracer of the disturbances of atmosphere. Hooke (1970) found that the F region ionospheric response to gravity waves is highly anisotropic. He suggested that the traveling ionospheric disturbance statistics should show diurnal and seasonal variations corresponding to atmospheric gravity waves. A statistical study of TIDs observed with the EISCAT radar by Hocke et al. (1996) indicated that the average phase and amplitude behavior of the TID parameters show the typical ionospheric response to gravity waves, and that the observed phase and amplitude behavior con;rms predictions
of the gravity wave theory of Hines (1960). The observation of TIDs comes from HF-Doppler measurements, GPS observation and radio wave absorption measurements. Recently, LaPstovicka (2001) obtained gravity wave activities on the lower ionosphere through radio wave absorption P measurements; BoPska and Sauli (2001) examined the acoustic gravity waves detected from ionospheric sounding data and found signi;cant increase of the AGW activities during the passage of cold front. The HF-Doppler array in Wuhan (114◦ E, 30◦ N) was established in 1985. The array is composed of three receivers spaced by the distance of about 100 km and is ready to receive the HF radio wave signals transmitted by Shanxi Broadcasting Station (109◦ E, 35◦ N) (Yuan, 1995). Analyzing the Doppler-shift and arrival angle of HF radio waves could identify TID events. Wan et al. (1998) reported the observation during year 1985 –1990. About 5500 records of TIDs were obtained, most of which are medium-scale TIDs launched by internal gravity waves. In this paper, the statistical result is re-analyzed and is presented in Fig. 9, where y-axis is the subtraction of TID propagating azimuth wave and thermosphere wind azimuth wind , and x-axis is wave period (Fig. 9(a)) and phase velocity (Fig. 9(b)), respectively. The azimuths of winds at the time the TIDs were measured are obtained from HWM93 wind model. The comparison of the observation results in Fig. 9 with the simulation results in Figs. 4 and 5 are conducted. In Fig. 9, there are mainly two characteristics for the distribution of the observed TIDs: (1) The major part of TIDs propagating along winds covers the area of 10 –20 min, 140 –160 m=s. TIDs in this area is corresponding to the northeast propagating TIDs shown in Fig. 1 of Wan et al. (1998). It is seen from Figs. 4(b) and 5(a) that this range of TIDs lies in the propagating region derived from calculations. Short-period waves would be easy to be re(ected if there was no wind, for the wave period is close to Brunt–VOasOalOa period. However, the Doppler shift caused by winds blowing in the same direction of wave decreases the wave frequency. Hence the short-period waves propagating along wind are likely to avoid re(ection and reach ionospheric height. Figs. 4(b) and 5(a) show this Doppler shift e9ect. As gravity waves with the period of 10 –20 min and 140 –160 m=s phase velocity always belong to re(ected region in Figs. 4(c) and 5(b), these short-period TIDs are not likely to be detected when they propagate against winds. This indicates that short period waves tend to propagate along winds. Besides, the numerical results in Figs. 4(b) and 5(a) indicate that the waves with period being more than about 20 min and phase velocity being more than 170 m=s are also able to arrive the observation height. However, it is found in Fig. 9(a) that few TIDs are detected in longer-period area when they are propagating along winds. Under careful analysis, it can be seen from Fig. 9 that TIDs propagating in the same direction of wind are typical medium scale TIDs with the peak of distribution lying in
F. Ding et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 857 – 869 270 240 70
150
210
wave-wind (°)
180 150
80
120 70
90
40
60 30
120
0 -30 160
-60 -90 10
15
20
25
30
(a)
35
40
45
50
Period (min) 270 180
210
wave-wind (°)
180
In summary of the present work, the conventional Ray Theory for atmospheric gravity waves is extended so that the energy attenuation along ray paths can be obtained. And the full-wave solution method is employed to analyze the changes for the propagation parameters of ducted gravity waves under the in(uence of background winds in a two-dimensional dissipative atmosphere. The results are outlined as follows:
150
150
100
120 70
90 60 50
30 0
70
-30
12
-60
(b)
in Fig. 1 of Wan et al. (1998). The major part of TIDs propagating against winds appears at 20-min period, which is not shown in Fig. 1 of Wan et al. (1998). As the southeast propagating TIDs in Fig. 1 of Wan et al. (1998) covers a wide range of period (from 15 min to 1 h), the major part of TIDs propagating against winds also belongs to SE peaks. We ;nd in Figs. 4(c) and 5(b) that the propagating distances corresponding to 20 and 40-min peak are 500 –1000 km and 1500 –2000 km, respectively. The arrival height for these peaks is approximately 250 and 350 km. According to Figs. 4(c) and 5(b), strong winds against the wave propagation direction may cause high-frequency waves to be re(ected. This problem does not exist for low-frequency waves, which explains the distribution area of TIDs propagating against winds towards low-frequency area. 6. Conclusions
240
-90 100
865
120
140
160
180
200
220
240
260
280
300
Vp (m· s-1)
Fig. 9. Contours of observed TID occurrences with wave − wind against period (a) and phase velocity (b), respectively. The TIDs were observed by HF-doppler array located in Wuhan (114◦ E, 30◦ N), during year 1985 –1990. HWM93 wind model is used to obtain the directions of winds at the times when TIDs were observed.
about 14 min and 150 m=s. We can see that the propagation distance correspondent to the peak is around 700 km in Fig. 4(b) and 500 km in Fig. 5a. That implies the troposphere source of these waves may be about 500 – 700 km far away from the observation point. Then, if there were TIDs with longer period, the propagation distance exceeds 1000 km (Figs. 4(b) and 5(a)). The wave packets of the waves have propagated far away from the observation point before they arrive at the observation height of 200 km. In this case, the TIDs cannot be observed. (2) 60% of the TIDs propagate against winds. Among these TIDs, the maximum of the occurrence appears at the point where the angle of wave − wind is 210◦ . The angle has an o9set of 30 –180◦ . This may be caused by the system error of observation. Two peaks appear with 40 –45 min period and 20 min period. The peak with 40 –45 min period is corresponding to the southeast propagating TIDs shown
(1) Background winds with di9erent directions may cause ray paths of internal gravity waves to be prolonged, steepened, re(ected or critically coupled, all of which change the accumulation of energy attenuation along ray paths. Only the penetrating waves propagating against winds can easily reach the ionosphere height with less energy attenuation. Hence the wind ;eld, varying with position and time, leads to directional ;ltering e9ect of gravity wave distribution (Waldock and Jones, 1984). (2) The propagation parameters of upgoing gravity waves launched from troposphere are calculated. Based on the calculation, the propagation status of gravity waves with di9erent periods and phase speeds is divided into the cut-o9 region, the re(ected region and the propagating region, all of which are in(uenced signi;cantly by winds. The area of the re(ected region reduces when gravity waves propagate along winds and expands when propagating against wind. In propagating region, the attenuation distances of gravity waves increase and the arrival height decrease when winds blow in the same direction of gravity waves, while the attenuation distances decrease and the arrival height increase when gravity waves propagate against winds. (3) The in(uence of winds on gravity waves of lower atmospheric modes is not obvious for they propagate mainly under mesosphere, where the wind ;eld is relatively weak. However, strong winds at thermospheric height lead to considerable changes of the
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dispersion relation and attenuation distance for upper atmospheric modes. When propagating against wind, phase velocity of mode G decreases signi;cantly, and the heights of energy distribution obviously fall down, which causes wave attenuation distance to increase by several hundred kilometers. When the wave propagates along wind, the results are contrary to those propagating against winds. Hence winds against the wave propagating direction support long-distance propagation of mode G, while winds blowing in the same direction of the wave can e9ectively shorten the attenuation distance. The e9ect of winds is more obvious for small-scale waves than for large-scale ones. (4) TIDs observed by Doppler array at Wuhan are statistically analyzed. The results show that 60% the TIDs propagate against winds. Among these TIDs, the maximum of the occurrence appears where the angle of wave − wind is close to 180◦ . The rest 40% of TIDs, which propagate along winds, mainly range in period from 10 to 20 min. The distribution of TIDs can be successfully explained by numerical calculation of internal gravity waves. Acknowledgements The author is thankful to National Nature Science Foundation of China (49974039) and National Key Basic Research Science Foundation (G2000078407) for the ;nancial support to this study.
1
1 4 − − 3i 2 + 3 2 2i +
Navier–Stokes equations (Francis, 1973): D + ∇ · v = 0; Dt
Dv TT = g − ∇p + ∇ · S; Dt
(A.1)
R0 DT TT ∇v; = Q + ∇ · (∇T ) − p∇ · v + S: ( − 1) Dt where ; p; v; T are, respectively, density, pressure, particle velocity and temperature for atmosphere. Q is inputting quantity of heat. , , and R0 are thermal conductivity coeKcient, viscosity coeKcient and speci;c heat ratio and universal gas constant, D=Dt = @=@t + v · ∇, STT is the gas viscosity tensor which satis;es Sij = (@vi =@xj + @vj =@xi − 2=3ij ∇ · v); (i; j = 1; 3):
(A.2)
The variables in (A.1) are composed of two parts: the ambient quantities and the ;rst-rank perturbations. Higher rank perturbations are ignored. The variables can then be expressed as v = v 0 + v1 ;
p = p0 + p1 ;
T = T0 + T1 ; (A.3)
where v0 is neglected by Francis (1973). Under these consumptions, Eqs. (A.1) are subsequently linearized as follows:
− i
−1
0
− 3i + i v˙0x
1
− − 4i + 4 2 + 3
kx v1x =! 1 − + R kx v1z =! −1 = 0; · p1 =p0 0 T1 =T0 −i 1
(A.4)
where Appendix A. Ray-tracing method was widely employed to calculating the propagation of internal gravity waves. As to gravity waves with a horizontal range of several thousand kilometers, ducting mechanism is more suitable. To analyze the ducted waves, atmospheric-strati;ed method can be adopted to get full wave solution of Navier–Stokes equations. The two method used in this paper are brie(y introduced below.
2
= ! =gkx2 H;
= (kz + i=2H )=kx ; R = k 2 − i + 1; = 1=kx H;
= i! =3p0 ;
v = iT0 kx2 =!p0 :
(A.5)
! = ! − kx v0x ;
A.1. Atmosphere-strati9ed method for ducted gravity wave
where H is average scale height. The matrix equation (A.4) di9ers from what derived by Francis (1973) in that background zero rank wind v0x is included here. So we can calculate the in(uence of wind on the propagation of guided waves. We mark the coeKcient matrix in Eq. (A.4) with A, and get the polarization relation:
In our study, a two-dimensional Cartesian coordinate system is used, in which the movement of atmosphere satis;es
p1 T1 v1z v1x = = ; = Ap At Ax Az
(A.6)
F. Ding et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 857 – 869
where Ap = (A12 A24 A31 + A32 − A22 A31 − A24 A32 ) · p0 ; At = (A12 + A32 − A22 − A22 A31 ) · T0 ; Ax = (A22 − A12 A24 − A24 A32 ) · ! =kx ;
(A.7)
Az = (A24 + A24 A31 − 1) · ! =kx : On the other hand, Eq. (A.4) has a non-trivial solution so long as the determinant of matrix A is zero. From this we get the dispersion relation: C3 R3 + C2 R2 + C1 R + C0 = 0;
(A.8)
where
C1 =−[ 2 −2 2 (1+3 )]− (1+7 )=(−1)− ;
(A.9)
C0 = [ 2 − 2 2 (1 + 3 )]=( − 1) + 2 (1 + 3 ): For a given real frequency ! and complex horizontal wavenumber kx , Eq. (A.8) has three pairs of roots for kz : one pair for gravity wave, other two for dissipative waves. Each pair of kz corresponds to the upgoing and downgoing waves. In our calculation, only the root for gravity wave is included. If the viscosity and thermal conductivity are ignored, the dispersion relation (A.8) yields: 2 !2 !2 ! kz2 = 2 1 − a2 − kx2 1 − b2 ; (A.10) c ! ! where !a and !b are the Brunt–VOasOalOa frequency and sound cut-o9 frequency, respectively. The non-dissipative dispersion relation (A.10) will be used later in ray-tracing method. The procedure of the atmosphere-strati;ed method is outlined as follows: given a real frequency ! and a complex horizontal wavenumber kx , the complex vertical wavenumber for gravity waves is obtained though (A.8). We assume that the lower boundary is ground and the upper one is at the height of 600 km. At the lower boundary, the perturbations satisfy rigid surface condition (namely v1z = 0); By using the polarization relation, we get the initial value of the eight variables (namely downgoing and upgoing branches for , T , vx , vz ). In each layer, the eight variables vary in the form of exp(i!t − ikz z + z=2H ); The variables in adjacent layers are related by interface condition, which is shown as follows: + − v1z = v1z ;
+ v1z p0 4 @v1z 2 − ikx v1x p1 − − i! H 3 @z 3
− v1z p0 4 @v1z 2 = p1 − − − ikx v1x ; i! H 3 @z 3
where the variables with prime + and − represent values above and below the interface respectively. Eq. (A.11) shows continuity of the vertical velocity, and (A.12) indicates the conservation of momentum (ux. At the highest layer, the radiation condition must be satis;ed. We search the gravity waves in the range of 60 –180 min and 200 –800 m=s that satisfy the upper boundary condition, and calculate the corresponding attenuation distance 1=Im(kx ). The height of each layer is chosen to be 600 m. At last, the dispersion curves and attenuation distance curves are obtained. A.2. Ray-tracing method for internal gravity wave
C3 = −3 (1 + 4 ); C2 = 3 (1 + 4 )=( − 1) + (1 + 7 ) + 3 ;
867
(A.11)
(A.12)
Ray-tracing method was widely applied to the analysis of internal gravity waves under the in(uence of background winds for its feasibility and explicitness in calculation. In order to get the propagation features of internal gravity waves in a dissipative atmosphere, we should derive the complex ray equations that are based on theory of Fermi. However, these kinds of complex equations are not feasible in calculation. Hence the theory of complex ray needs to be simpli;ed so as to be more convenient in application. In this paper, ray theory under non-dissipative condition is used to trace the ray paths. At the same time, energy attenuation expression is derived to calculate the wave energy dissipation. The assumption fails, however, when the gravity waves are critically coupled. In this case, the ray theory is not suitable for the simulation of gravity wave propagation. Compared with the range of time and space of propagation of gravity waves, atmosphere is a non-transient, low-dissipative media, in which the gravity wave can be simpli;ed as quasi-monochromatic plane wave. According to Generalized Ray Theory, we assume that the packet of grav* * ity wave propagates in generalized space ( k ; !; r ; t) along a path *. If the dissipation of atmosphere could be ignored, the propagation parameters should meet with the dispersion relation (11). Under this restriction, the phase of the wave can be expressed as * * += [! dt − k · d r ]: (A.13) *(t)
Among the rays that can satisfy (A.13), only those have steady phase are valid in propagation. This requires that the variation of phase equals zero: dx + = ! − kx · dt *(t)
* dz = 0: (A.14) + kz (kx ; ky ; !; r ; t) · dt According to Variation Principle, the generalized ray-tracing equations of gravity waves can be derived and
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shown as follows: d! = − d kx =
@kz @(! − kx vx ) dt; @t @kz
@kz @(! − kx vx ) dt; @x @kz
dx =
@kz @(! − kx vx ) dt; @kx @kz
dz =
@(! − kx vx ) dt: @kz
(A.15)
The vertical vector kz in (A.15) is produced by dispersion relation (A.10). Given the initial parameters ! and kx , the ray path of gravity waves can be obtained from Eq. (A.15). The change of logarithmic wave amplitude ln A along the ray path * can be expressed as *
d(ln A) = −kzi (kx ; ! − kx vx ; r ; t) · d z:
(A.16)
The change of amplitude along ray path can be obtained by integrating (16): z A(z)=A(z0 ) = exp − kzi · d z ; (A.17) z0
in which z0 and z are, respectively, the heights of wave packet at the launched time and time t. The exponential increase of amplitude is not included. Eq. (A.17) implies that the variation of relative amplitude of gravity wave can be obtained through integrating kzi along the ray path. In our calculation, ray equations (A.15) are used to calculate the gravity wave propagation paths, and expression (A.17) is used to obtain the energy attenuation along ray paths. In this paper, only the upgoing waves are included. The wave energy decreases with the increase of height, and so do the value calculated by Eq. (A.17). We de;ne that, besides the re(ection and critically coupling of the waves, the propagation is valid until the wave energy cannot satisfy A(z)=A(z0 ) ¿ 1=e.
References P BoPska, J., Sauli, P., 2001. Observations of gravity waves of meteorological origin in the F-region ionosphere. Physics and Chemistry of the Earth 26, 425–428. Cowling, D.H., Webb, H.D., Yeh, K.C., 1971. Group rays of internal gravity waves in a wind-strati;ed atmosphere. Journal of Geophysical Research 76, 200–213. DalGarno, A., Smith, F.J., 1962. The thermal conductivity and viscosity of atomic oxygen. Planetary and Space Science 9, 1–6. Dunkerton, T.J., 1984. Inertia-gravity waves in the stratosphere. Journal of Atmospheric Science 41, 3396–3404. Eckermann, S.D., 1992. Ray-tracing simulation of the global propagation of inertia gravity waves through the zonally
averaged middle atmosphere. Journal of Geophysical Research 97, 15849–15866. Francis, S.H., 1973. Acoustic gravity modes and large scale traveling ionospheric disturbances of a realistic dissipative atmosphere. Journal of Geophysical Research 78, 2278–2301. Hedin, A.E., et al., 1991a. Extension of the MSIS thermospheric model into the middle and lower atmosphere. Journal of Geophysical Research 96, 1159–1172. Hedin, A.E., et al., 1991b. Revised global model of thermosphere winds using satellite and ground-based observations. Journal of Geophysical Research 96, 7657–7688. Hickey, M.P., Taylor, M.J., Gardner, C.S., et al., 1998. Full-wave modeling of small-scale gravity waves using airborne lidar and observations of the Hawaiian airglow (ALOHA-93) O(1 S) images and coincident Na wind/temperature Lidar measurements. Journal of Geophysical Research 103, 6439–6453. Hines, C.O., 1960. Internal atmospheric gravity waves at ionospheric heights. Canadian Journal of Physics 38, 1441–1481. Hocke, K., Schlegel, K., Kirchengast, G., 1996. Phase and amplitudes of TIDs in the high-latitude F-region observed by EISCAT. Journal of Atmospheric and Terrestrial Physics 58, 245–255. Hooke, W.H., 1970. The ionospheric response to internal gravity waves, 1. The F region response. Journal of Geophysical Research 75, 5535–5544. LaPstovicka, J., 2001. E9ects of gravity and planetary waves on the lower ionosphere as obtained from radio wave absorption measurements. Physics and Chemistry of the Earth 26, 381–386. Liang, J., Wan, W., Yuan, H., 1998. Ducting of acoustic-gravity waves in a nonisothermal atmosphere around a spherical globe. Journal of Geophysical Research 103, 11229–11234. Lindzen, R.S., Blake, D., 1972. Lamb waves in the presence of realistic distributions of temperature and dissipation. Journal of Geophysical Research 77, 2166–2176. Manson, A.H., Meek, C.E., Zhan, Q., 1997. Gravity wave spectra and direction statistics for the mesosphere as observed by MF radars in the Canadian Prairies (49◦ N–52◦ N) and at TromsH (69◦ N). Journal of Atmospheric and Terrestrial Physics 59, 993–1009. Richmond, A.D., 1978. Gravity wave generation, propagation, and dissipation in the thermosphere. Journal of Geophysical Research 83, 4131–4145. Scroeberl, M.R., 1985. A Ray-tracing model of gravity wave propagation and breakdown in the middle atmosphere. Journal of Geophysical Research 90, 7999–8010. Thome, G.D., 1968. Long-period waves generated in the polar ionosphere during the onset of magnetic stroms. Journal of Geophysical Research 73, 6319–6336. Waldock, J.A., Jones, T.B., 1984. The e9ects of neutral winds on the propagation of medium scale atmospheric gravity waves at mid-latitudes. Journal of Atmospheric and Terrestrial Physics 46, 217–231. Wan, W., Yuan, H., Ning, B., Liang, J., 1998. Travelling ionosphere disturbances associated with tropospheric vortexes around Qinghai–Tibet plateau. Geophysical Research Letters 25, 3775–3778.
F. Ding et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 857 – 869 Yeh, K.C., Webb, H.D., 1972. Evidence of directional ;ltering of travelling ionospheric disturbance. Nature (London), Physical Science 235, 131–133. Yuan, H., 1995. The propagation parameters analysis and tracing of ionospheric disturbance. Ph. D. Thesis, Wuhan Institute of Physics and Mathematics, Wuhan, P.R. China.
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www.elsevier.com/locate/jastp
The in(uence of background winds and attenuation on the propagation of atmospheric gravity waves F. Ding∗ , W. Wan, H. Yuan Wuhan Ionospheric Observatory, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, 430071, China Received 24 April 2002; received in revised form 16 December 2002; accepted 17 March 2003
Abstract The in(uence of background winds and energy attenuation on the propagation of atmospheric gravity waves is numerically analyzed. The gravity waves, both in the internal and ducted forms, are included through employing ray-tracing method and full-wave solution method. Background winds with di9erent directions may cause ray paths of internal gravity waves to be horizontally prolonged, vertically steepened, re(ected or critically coupled, all of which change the accumulation of energy attenuation along ray paths. Only the penetrating waves propagating against winds can easily reach the ionospheric height with less energy attenuation. The propagation status of gravity waves with di9erent periods and phase speeds is classi;ed into the cut-o9 region, the re(ected region and the propagating region. All the three regions are in(uenced signi;cantly by winds. The area of the re(ected region reduces when gravity waves propagate in the same direction of winds and expands when propagating against wind. In propagating region, the horizontal attenuation distances of gravity waves increase and the arrival heights decrease when winds blow in the same direction of gravity waves, while the attenuation distances decrease and the arrival heights increase when gravity waves propagate against winds. The results for ducted gravity waves show that the in(uence of winds on waves of lower atmospheric modes is not noticeable for they propagate mainly under mesosphere, where the wind ;eld is relatively weak. However, strong winds at thermospheric height lead to considerable changes of dispersion relation and attenuation distance of upper atmospheric modes. Winds against the wave propagating direction support long-distance propagation of G mode, while the attenuation distances decrease when winds blow in the same direction of the wave. The distribution of TIDs observed by HF Doppler array at Wuhan is compared with the simulation of internal gravity waves. The observation of TIDs shows agreement with our numerical calculations. c 2003 Elsevier Science Ltd. All rights reserved. Keywords: Internal gravity wave; Ducted gravity wave; Attenuation; Wind
1. Introduction It is now generally believed that both the background winds and energy attenuation have considerable in(uence on the propagation of atmospheric gravity waves. Hines (1960) has initially pointed out that when an up-going gravity wave meets a wind, the wave may be re(ected or refracted. Cowling et al. (1971) studied the in(uence of
∗
Corresponding author. Tel.: 86-027-8719-7797. E-mail addresses: [email protected], [email protected] (F. Ding).
background winds on gravity wave ray paths. They found that depending on the direction of the wind and the wave vector, three di9erent e9ects may occur. If the gravity waves propagate along wind, the intrinsic frequency is shifted downward. When the wave frequency equals zero, the wave energy is lost to the mean (ow, which is known as critical coupling. If the gravity waves propagate against wind, it is possible in this case for the waves to be re(ected. In either of the two cases, if the wind speed is low, the waves will penetrate thermosphere and cause TIDs at ionospheric height. Yeh and Webb (1972) analyzed the relation of winds and TIDs and found that winds have the directional ;ltering e9ect on gravity waves. Waldock and
c 2003 Elsevier Science Ltd. All rights reserved. 1364-6826/03/$ - see front matter doi:10.1016/S1364-6826(03)00090-7
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F. Ding et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 857 – 869
Jones (1984) included the diurnal variation of winds in ray-tracing method and showed that the domain direction of gravity waves rotates clockwise in north hemisphere, which is experimentally addressed by Yuan and Wan (1997). Dunkerton (1984) showed that lower frequency gravity waves experienced severe horizontal refraction in winter hemisphere, due mainly to the meridional shears in zonal winds. Zhong et al. (1995) added a simpli;ed tidal model to the CIRA-86 reference wind model and used ray-tracing method to examine the e9ect of spatially and temporally varying wind on gravity wave propagation. He indicated that the temporal variation of wind causes the observed frequencies and the critical levels of gravity waves to be temporal, and that the time-dependent variation of the propagation features becomes more obvious in winter than in summer. As is discussed above, few studies take into consideration the energy attenuation of internal gravity waves in ray-tracing method. In fact, atmosphere is a dissipative medium for gravity waves especially above mesopause. This is one of the main reasons that a large number of up-going gravity waves cannot reach ionospheric height. A study by Scroeberl (1985) revealed that azimuthally refracted waves were considerably attenuated in amplitude. Eckermann (1992) used wave action conservation equation to calculate the attenuation of wave amplitude. In his paper, signi;cant di9erences in the latitude-season structure of the mean wave amplitudes are simulated, and comparison between measured and simulated distributions of wave propagation azimuths reveals good agreement in some locations. At the same time, ducting mechanism for surface and quasi-surface gravity waves was widely analyzed. The consideration of ducted gravity waves originated from the fact that the observed large-scale gravity waves can travel horizontally for long distance with small energy attenuation. As Francis (1973) and Richmond (1978) addressed, the atmosphere should consist of a continuous spectrum of freely propagating internal gravity waves and a discrete spectrum of ducted gravity wave modes. In the presence of the solid ground, surface waves known as Lamb waves can exist in the atmosphere that have energy concentrated near the surface of the Earth. The sudden temperature rise in the low thermosphere can support surface waves propagating horizontally with energy concentrated near the sudden temperature rise. Thome (1968) initially used a simpli;ed temperature pro;le composed of two isothermal half-spaces to get the zero-rank mode of gravity waves (namely mode G0 ). Francis (1973) deduced in isothermal atmosphere the L0 mode of Lamb waves supported by the full re(ection of the Earth. Because of the existence of both the sudden temperature rise and the earth’s surface, the realistic modes must be G0 , L0 and some higher rank modes of the series of L and G. Francis (1973) also used a realistic continuous temperature pro;le to get these series of higher rank modes. These modes can guide acoustic-gravity waves for
thousands of kilometers along the earth, and most of them show agreement with TID observations. Liang et al. (1998) solved the Green’s function of Navier–Stokes equations on the basis of an analytic atmospheric temperature model. The analytical solutions of Liang et al. (1998) con;rmed the result of Francis that the major re(ection mechanism of the imperfectly ducted modes is closely related to the sudden rise of temperature at the base of thermosphere. Though the guided modes of large-scale gravity waves were widely studied, most of the researches focused on the role of sudden temperature rise, while background wind was seldom included. In fact, the realistic wind speed always reaches 200 m=s at thermosphere (Hedin et al., 1991a). Thus the in(uence of background winds cannot be ignored even for large-scale ducted waves. Hickey et al. (1998) found that an accurate description of winds is an essential requirement for a complete interpretation of the observed wave-driven airglow (uctuations. The purpose of this paper is to examine gravity wave characteristics under the in(uence of background winds in a dissipative atmosphere, which will aid in understanding the distribution of TIDs observed by our HF-Doppler array in Wuhan Ionospheric Observatory, China (Yuan, 1995). 2. Methods and background parameter proles In this paper, both the internal waves and the large-scale ducted waves are included. For internal waves, the “extended” ray-tracing method is used. The tracing method differs from previous treatments by including the energy attenuation along the ray path. And atmosphere-strati;ed method is adopted to obtain the full wave solutions of ducted waves, in which the zero-rank wind ;eld is included. The two methods are displayed in the appendix. The numerical results together with the observed TIDs are used to discuss the e9ect of winds on gravity waves in a dissipative atmosphere. The HWM93 wind model and MSISE90 atmospheric model (Hedin et al., 1991a, b) are adopted to obtain the wind and temperature pro;les. The two-dimensional wind and temperature pro;les at 12:00 and 22:00 local time are plotted in Fig. 1, ranging in height from the ground to 600 km. The inputting date is December 1, 1988 in Wuhan (114◦ E, 30◦ N). The tide is included in the wind pro;les in order to simulate the actual wind ;eld. As shown in Fig. 1a, the wind in upper atmosphere is much stronger at 22:00 than it is at 12:00. At 22:00, the wind positive to the west (referring to Fig. 1a) is rather weak below the height of 100 km, while it increases rapidly when above 150 km. There is a strong wind shear in the height between 50 and 100 km, varying from 40 to −10 m=s. The wind grows slowly above the height of 200 km, and the speed remains around 140 m=s. The wind speed is basically positive when above 100 km height. While at 12:00, there is a strong wind shear below mesopause, and the wind speed remains less than 40 m=s in upper atmosphere. The temperatures of the
F. Ding et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 857 – 869 600
600 12:00
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Fig. 1. Wind (a) and temperature (b) pro;les calculated by HWM93 wind model and MSISE90 atmospheric Model. The positive value in (a) is to the west. The solid and dashed lines in the ;gures refer to wind speed and temperature at 22:00 and 12:00 LT, respectively. Date: 12-1-1988; Site: Wuhan (114◦ E, 30◦ N).
two times change signi;cantly above 120 km height. The change of temperature is not noticeable below that height. The coeKcients of viscosity and thermal conductivity are computed from DalGarno and Smith (1962):
= 6:71 × 10
−4
350 300
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= 3:34 × 10
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3.1. Wind e6ect on gravity wave ray paths Assume that a gravity wave is launched at 22:00 from a ;xed point (114◦ E, 30◦ N) with a speci;c initial phase speed and period. Fig. 2 presents gravity wave ray paths with initial period of 30 min. The left branch in Fig. 2 are waves propagating against winds and the right along them. A similar result was given by Cowling et al. (1971), whereas the energy attenuation was not considered. Three types of propagation of gravity waves are then displayed as: (1) some of the waves are re(ected between mesospause and low thermosphere, e.g. the 100-m/s wave in the left branch and the 240-m/s wave in the right; (2) some are critically coupled at low thermosphere, e.g. the 100-m/s wave in the right branch; and (3) others penetrate the atmosphere and reaching ionospheric height. The di9erences of wave propagation features are mainly caused by the variation of winds. For the waves propagating along winds, the frequencies decreased. This is obvious especially for the small-velocity waves. If the intrinsic frequency equal zero, the energy of the gravity wave is coupled to the mean (ow. Other waves with larger phase velocity are able to propagate higher. The ray paths of the penetrating waves are bent down and horizontally prolonged, which consequently causes the increase of wave attenuation accumulated on ray paths; when propagating
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500 1000 1500 2000 2500 x (km)
Fig. 2. Ray paths of internal gravity waves propagating to the west (right branch) and to the east (left branch). The initial wave period is 30 min. The x- and y-axis represent and horizontal and vertical propagating range of ray paths. Each ray represents a path of the gravity wave with a speci;c phase velocity, which is marked near the ray with the unit of m/s. The upper-atmosphere wind direction is also shown with an arrow.
against wind, ray paths with both the small phase velocity and the higher ones are re(ected back to the ground. In this case, ray paths of penetrating waves are vertically steepened. These waves can propagate much higher than those along winds do. The height pro;les of wave amplitude for these waves are illustrated in Fig. 3. Most of the energy attenuation occurs above the height of 140 km. This is because the viscosity and heat conductivity of the atmosphere increases with height and they are more in(uential when above that height. Moreover, the wave amplitudes experience more severe attenuation under the condition of propagating along winds than waves against winds do. Since the viscosity, heat conductivity and temperature pro;les are the same in both cases, the
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z (km)
150 240 200 150 100 100
0 1/e
2/e Relative Amplitude
1
Fig. 3. Wave amplitudes as a function of propagation height for internal gravity waves shown in Fig. 2. Solid and dashed lines denote gravity waves propagating against and along winds. The re(ected waves are not included. The phase velocity is marked near the curves with the unit of m/s.
di9erence of amplitude pro;les is caused by winds. When the ray paths are horizontally prolonged by winds along wave direction, the attenuation of wave energy accumulated along the ray paths increases and prevents the waves to propagate higher. Contrary to the case along winds, when the ray paths are steepened by winds against wave direction, the ray paths of the waves is shortened, which makes the waves propagate rapidly to higher height with small energy attenuation. The 100-m/s wave propagating along winds in Fig. 2 shows an example of critical coupling. It is shown in Fig. 3 that the wave energy dissipates quickly above 140 km height. Unlike the ray path presented by Waldock and Jones (1984), the wave ray path in Fig. 2 does not propagate horizontally. This is because the calculation of wave energy becomes uncertain when the waves is fully critically coupled, due to the limitation of ray-tracing method. In this case, the intrinsic frequency vanishes, and wave energy is lost to the mean (ow. In this case, the dispersion relation (A.10) is not valid, neither is Eq. (A.17). The analysis indicates that winds with di9erent directions are likely to cause ray paths to be horizontally prolonged, vertically steepened, re(ected or critically coupled. All of these change the accumulation of energy attenuation along ray paths. Among these rays, only the penetrating waves propagating against winds can easily reach the ionosphere height with less energy attenuation. 3.2. Wind e6ect on internal gravity waves with arbitrary period and phase velocity In this section, only the gravity waves with phase speeds between 100 and 300 m=s and periods of up to 80 min are
considered. The simulation will be compared later with the HF observations of TIDs, the observation height of which is about 200 km. Manson et al. (1997) showed that the gravity waves with phase speed less than 100 m=s are common in lower atmosphere. However, most of these low speed waves become evanescent or re(ected below or near the mesopause due to strong wind shears and steep atmospheric temperature gradient. So, only waves with faster speed are likely to reach the height of 200 km and observed by HF-Doppler arrays (Yuan, 1995). We choose gravity waves ranging from 100 to 300 m=s and 6 to 80 min and trace them from troposphere up to ionosphere. The results are presented in Fig. 4 at 22:00 local time and Fig. 5 at 12:00 local time. Figs. 4(a) – (c) are, respectively, propagation results of gravity waves under the condition of propagating without winds, along winds and against winds at 22:00. The re(ected region is denoted by horizontal hatching and the cut-o9 region by vertical hatching. And the rest of the area in Fig. 4 refers to the propagating region. Two type of waves constitute the propagating region: (1) The waves propagating without being re(ected or cut o9; (2) The waves reaching the height of 200 km and then being re(ected at a certain height above 200 km. Type two is included in the propagating region for the reason that the observation height of our Doppler Arrays is around 200 km. So, the waves in part two can still be detected, though they are ;nally re(ected above the observation height. (1) Cut-o6 region: The cut-o9 region mainly covers the high velocity and short period area. The waves in this region are cut o9 in troposphere due to the restrain of gravity wave dispersion relation. The existence of cut-o9 region shows that the propagation of gravity waves has an upper limit of phase velocity and a lower limit of wave period. For the waves propagating without winds, the limit is about 270 m=s and 6 min of the period. The limit changes into 260 m=s and over 10 min for waves along winds, and it becomes 285 m=s and 7 min for those against winds. The area of cut-o9 region in Fig. 4(a) is much similar to the result of Hines (1960). However, it changes signi;cantly in Fig. 4(b) and (c). The changes are obviously caused by Doppler-shift of wave frequency under the in(uence of winds. (2) Re8ected region: The re(ected region is a transitional area between the cut-o9 region and the propagating region. In case of no winds (Fig. 4(a)), the re(ection is completely caused by the vertical increase of atmospheric temperature, which leads to the decline of Brunt–VOasOalOa frequency !b and accordingly causes gravity waves to be re(ected at a certain height where the wave frequency exceeds !b . When the gravity waves propagate along winds (Fig. 4(b)), the area of re(ected region obviously reduces because winds in the same direction of waves cause wave frequency to decline so as to be kept away from the buoyancy frequency. This makes it hard for gravity waves to be re(ected. On the contrary, winds against waves make the re(ected region expand into
F. Ding et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 857 – 869 300
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Fig. 5. Same as Fig. 4, but for propagation results of gravity waves under the condition of propagating along winds (a) and against winds (b) at 12:00 LT.
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Fig. 4. (a) – (c) are propagation results of gravity waves under the condition of propagating without winds, along winds and against winds at 22:00 local time. x- and y-axis represent wave period and phase velocity, respectively. Re(ection region is denoted by horizontal hatching and cut-o9 region by vertical hatching. The solid and dashed lines show the contours of horizontal propagating distance and arrival height of gravity waves, with the value of each contour marked near every contour. The unit of value for contours is km.
short-period area (less than about 20 min), which shows that high-frequency waves are more easily in(uenced by winds. This is shown in Fig. 4(c). (3) Propagating region: As the energy attenuation accumulated along ray path increases with height, the waves in propagating region have a limited range of the arrival height and horizontal propagating distance, which are plotted in Fig. 4 in the form of contours. The solid and dashed lines denote the horizontal propagating distance and arrival height contours, respectively. When there is no wind, most of the waves in propagating region can reach the height of 200 km. The arrival height rises from 200 km at the speed of 100 m=s to over 320 km at 200 m=s, and the horizontal propagating distance increases from 500 km at about 18 min to 5000 km at 80 min. The increase of horizontal propagating distance with the increase of period is slow when the period is small, while it increases rapidly when the period is more than 50 min. When the gravity waves propagate along winds (Fig. 4(b)), the horizontal distance increases for several hundred kilometers.
F. Ding et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 857 – 869 800 G0
700 600
Vph (m/s)
The change is not so noticeable as compared to the average propagating distance for gravity waves. However, the arrival heights of all the waves fall down obviously by 50 –70 km. The drop of height is more obvious for higher velocity waves than for lower ones. In contrast, when the waves propagate against winds (Fig. 4c), the arrival heights rise remarkably by 80 –100 km and the horizontal propagating distances decrease. In short-period area (from 20 to 30 min), both the arrival height and the horizontal propagating distance increase rapidly with the increase of period, which indicates that short-period waves are greatly in(uenced by winds. Fig. 5 presents the simulation results at 12:00 local time, where Fig. 5(a) and (b) are for gravity waves propagating along winds and against winds. The wind ;eld is much weaker around 12:00. The area of cut-o9 region and re(ected region is consequently similar with those propagating without wind (Fig. 4(a)). Also the change of horizontal propagation distances and arrival heights are not noticeable both in Fig. 5a and b, as compared to Fig. 4(a). The results show that the propagation of gravity waves changes signi;cantly with time. Cowling et al. (1971) and Waldock and Jones (1984) had studied the e9ect of diurnal change of winds on the propagation of gravity waves. In their studies, only the gravity waves with several speci;c initial periods and phase velocities are considered. This paper provides a full view of propagation status for any arbitrary initial period and phase velocity in a certain range. However, the results in Figs. 4 and 5 lack the continuous change of winds with time. As the wind is almost strongest at 22:00 and weakest at 12:00, the diurnal change of wind is supposed to lead to diurnal change of propagation status of gravity waves, with the horizontal propagation distances and arrival heights varying between the values shown in Figs. 4 and 5.
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The ducted waves di9er from internal waves in that the energy of ducted waves concentrates at the height where the background parameters vertically change sharply. In this case, the horizontal attenuation distance is an important parameter to examine the validity of ducting mechanism. Fig. 6(a) plots the dispersion curves and attenuation distance curves of ducted gravity waves under the condition of propagating without winds. Fig. 7(a) shows the corresponding attenuation distance curves. And Fig. 8(a) shows the corresponding energy pro;les. In all of the ;gures, the curves fall into two categories: mode L and mode G (Francis, 1973). L0 mode has a ;xed phase velocity of 312 m=s. Its horizontal attenuation distance is over 106 km. While L1 mode has an almost ;xed phase velocity of 255 m=s, and the horizontal attenuation distance is over 10; 000 km, which is much shorter. The distributions of wave energy for L modes mainly locate in lower atmosphere, namely the height below
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4. Inuence of winds on ducted gravity waves: results by atmosphere-stratied method
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Fig. 6. Dispersion relation curves of ducted gravity waves under the condition of propagating without winds (a), along winds (b) and against winds (c) at 22:00 LT. x- and y-axis denote wave period and phase velocity, respectively. The mode of each curve is marked nearby. In order to show clearly the changes of dispersion relation for short-period gravity waves, the x-axis is set to be logarithmic.
Attenuation Distance (km)
F. Ding et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 857 – 869 107 106 gravity 105 5000 4000 3000
L0
waves, the x-axis is set to be logarith L 1
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Fig. 8. Vertical distributions of wave energy for ducted waves shown in Fig. 7. The x- and y-axis denote wave energy and height (in km), respectively. Figure (a) – (c) present results of ducted waves propagating without winds, along winds and against winds.
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Fig. 7. Attenuation distance against wave period for ducted waves propagating without winds (a), along winds (b) and against winds (c) at 22:00 LT. The x- and y-axis are logarithmic.
100 km. In this height range, the in(uence of dissipation is so weak that gravity waves for mode L0 and L1 can propagate much farther than waves of mode G do. As Lindzen and Blake (1972) pointed out, if we took into account the leakage of wave energy up to higher atmosphere and the surface friction of the Earth, the realistic attenuation distance would not be so long. However, from our calculation we ;nd clearly that mode L0 and L1 , with relatively smaller velocity, play an important role in transmitting energy to long distance in lower atmosphere. As winds are usually weak in lower atmosphere, the in(uence of winds on L modes is not obvious. Figs. 6(b), 7(b) and 8(b) show gravity wave modes under the condition of propagating along winds. And Figs. 6(c), 7(c) and 8(c) show results against winds. Only the results at 22:00 local time are shown here. The winds in the same direction of wave make the velocity of mode L0 and L1 rise by about 10 m=s. The corresponding attenuation distances drop a little. On the contrary, winds against gravity waves make the velocity of mode L0 and L1 fall down by about 10 m=s and the attenuation distances increase a little.
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F. Ding et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 857 – 869
The upper atmospheric modes, on the other hand, have a wide distribution of velocities ranging from 200 to 800 m=s. The attenuation distances are now smaller than 5000 km. The slower the velocity is, the closer the gravity waves are able to travel. The energy pro;les in Fig. 8(a) show that the energy of G modes concentrates between the heights of 100 and 400 km. This implies that gravity waves tend to propagate along that sudden temperature rise at the base of thermosphere. As Francis (1973) reported, the in(uence of viscosity and thermal conductivity become severe as the height rises. Hence the waves of G modes endure greater energy attenuation than waves of L modes do. Another factor of the energy attenuation is coupling of ionosphere and the gravity waves, which is shown as TIDs. All these make the attenuation distance of mode G much closer than those of waves of mode L. Because the wind usually grows strong at the base of thermosphere, it is found to have appreciable e9ects on gravity waves of G modes. When propagating against winds, the velocities of mode G fall sharply by 50 –100 m=s, and the attenuation distances rise by several hundred kilometers. The rising of attenuation distance is more obvious for high-rank low-velocity short-period modes. As compared with wave modes without winds, the energy distribution is shifted toward lower atmosphere. The descendent of energy distribution makes the energy attenuation caused by viscosity and thermal conductivity drop down. This leads to the rise of attenuation distances. Hence winds against the gravity waves can eKciently support the long-distance propagation of low-velocity short-period modes. When the waves propagate along winds, the velocities of all the G modes rise considerably by 50 –100 m=s, whereas G0 mode is not presented in Fig. 6(b) because its velocity exceeds 800 m=s. The attenuation distances are now less than 1000 km. Among them, the minimum distance for G5 mode is only 100 km. The distance is much closer than that of internal waves shown in Fig. 2. Though the ascendant of energy pro;le is not noticeable in Fig. 8(b), the viscosity and thermal conductivity, which increase exponentially with height, can cause severe energy attenuation. That is why the attenuation distances of G mode fall sharply when propagating along winds. 5. TID observation and statistical results Frequently the ionosphere is considered as a passive tracer of the disturbances of atmosphere. Hooke (1970) found that the F region ionospheric response to gravity waves is highly anisotropic. He suggested that the traveling ionospheric disturbance statistics should show diurnal and seasonal variations corresponding to atmospheric gravity waves. A statistical study of TIDs observed with the EISCAT radar by Hocke et al. (1996) indicated that the average phase and amplitude behavior of the TID parameters show the typical ionospheric response to gravity waves, and that the observed phase and amplitude behavior con;rms predictions
of the gravity wave theory of Hines (1960). The observation of TIDs comes from HF-Doppler measurements, GPS observation and radio wave absorption measurements. Recently, LaPstovicka (2001) obtained gravity wave activities on the lower ionosphere through radio wave absorption P measurements; BoPska and Sauli (2001) examined the acoustic gravity waves detected from ionospheric sounding data and found signi;cant increase of the AGW activities during the passage of cold front. The HF-Doppler array in Wuhan (114◦ E, 30◦ N) was established in 1985. The array is composed of three receivers spaced by the distance of about 100 km and is ready to receive the HF radio wave signals transmitted by Shanxi Broadcasting Station (109◦ E, 35◦ N) (Yuan, 1995). Analyzing the Doppler-shift and arrival angle of HF radio waves could identify TID events. Wan et al. (1998) reported the observation during year 1985 –1990. About 5500 records of TIDs were obtained, most of which are medium-scale TIDs launched by internal gravity waves. In this paper, the statistical result is re-analyzed and is presented in Fig. 9, where y-axis is the subtraction of TID propagating azimuth wave and thermosphere wind azimuth wind , and x-axis is wave period (Fig. 9(a)) and phase velocity (Fig. 9(b)), respectively. The azimuths of winds at the time the TIDs were measured are obtained from HWM93 wind model. The comparison of the observation results in Fig. 9 with the simulation results in Figs. 4 and 5 are conducted. In Fig. 9, there are mainly two characteristics for the distribution of the observed TIDs: (1) The major part of TIDs propagating along winds covers the area of 10 –20 min, 140 –160 m=s. TIDs in this area is corresponding to the northeast propagating TIDs shown in Fig. 1 of Wan et al. (1998). It is seen from Figs. 4(b) and 5(a) that this range of TIDs lies in the propagating region derived from calculations. Short-period waves would be easy to be re(ected if there was no wind, for the wave period is close to Brunt–VOasOalOa period. However, the Doppler shift caused by winds blowing in the same direction of wave decreases the wave frequency. Hence the short-period waves propagating along wind are likely to avoid re(ection and reach ionospheric height. Figs. 4(b) and 5(a) show this Doppler shift e9ect. As gravity waves with the period of 10 –20 min and 140 –160 m=s phase velocity always belong to re(ected region in Figs. 4(c) and 5(b), these short-period TIDs are not likely to be detected when they propagate against winds. This indicates that short period waves tend to propagate along winds. Besides, the numerical results in Figs. 4(b) and 5(a) indicate that the waves with period being more than about 20 min and phase velocity being more than 170 m=s are also able to arrive the observation height. However, it is found in Fig. 9(a) that few TIDs are detected in longer-period area when they are propagating along winds. Under careful analysis, it can be seen from Fig. 9 that TIDs propagating in the same direction of wind are typical medium scale TIDs with the peak of distribution lying in
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In summary of the present work, the conventional Ray Theory for atmospheric gravity waves is extended so that the energy attenuation along ray paths can be obtained. And the full-wave solution method is employed to analyze the changes for the propagation parameters of ducted gravity waves under the in(uence of background winds in a two-dimensional dissipative atmosphere. The results are outlined as follows:
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in Fig. 1 of Wan et al. (1998). The major part of TIDs propagating against winds appears at 20-min period, which is not shown in Fig. 1 of Wan et al. (1998). As the southeast propagating TIDs in Fig. 1 of Wan et al. (1998) covers a wide range of period (from 15 min to 1 h), the major part of TIDs propagating against winds also belongs to SE peaks. We ;nd in Figs. 4(c) and 5(b) that the propagating distances corresponding to 20 and 40-min peak are 500 –1000 km and 1500 –2000 km, respectively. The arrival height for these peaks is approximately 250 and 350 km. According to Figs. 4(c) and 5(b), strong winds against the wave propagation direction may cause high-frequency waves to be re(ected. This problem does not exist for low-frequency waves, which explains the distribution area of TIDs propagating against winds towards low-frequency area. 6. Conclusions
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Vp (m· s-1)
Fig. 9. Contours of observed TID occurrences with wave − wind against period (a) and phase velocity (b), respectively. The TIDs were observed by HF-doppler array located in Wuhan (114◦ E, 30◦ N), during year 1985 –1990. HWM93 wind model is used to obtain the directions of winds at the times when TIDs were observed.
about 14 min and 150 m=s. We can see that the propagation distance correspondent to the peak is around 700 km in Fig. 4(b) and 500 km in Fig. 5a. That implies the troposphere source of these waves may be about 500 – 700 km far away from the observation point. Then, if there were TIDs with longer period, the propagation distance exceeds 1000 km (Figs. 4(b) and 5(a)). The wave packets of the waves have propagated far away from the observation point before they arrive at the observation height of 200 km. In this case, the TIDs cannot be observed. (2) 60% of the TIDs propagate against winds. Among these TIDs, the maximum of the occurrence appears at the point where the angle of wave − wind is 210◦ . The angle has an o9set of 30 –180◦ . This may be caused by the system error of observation. Two peaks appear with 40 –45 min period and 20 min period. The peak with 40 –45 min period is corresponding to the southeast propagating TIDs shown
(1) Background winds with di9erent directions may cause ray paths of internal gravity waves to be prolonged, steepened, re(ected or critically coupled, all of which change the accumulation of energy attenuation along ray paths. Only the penetrating waves propagating against winds can easily reach the ionosphere height with less energy attenuation. Hence the wind ;eld, varying with position and time, leads to directional ;ltering e9ect of gravity wave distribution (Waldock and Jones, 1984). (2) The propagation parameters of upgoing gravity waves launched from troposphere are calculated. Based on the calculation, the propagation status of gravity waves with di9erent periods and phase speeds is divided into the cut-o9 region, the re(ected region and the propagating region, all of which are in(uenced signi;cantly by winds. The area of the re(ected region reduces when gravity waves propagate along winds and expands when propagating against wind. In propagating region, the attenuation distances of gravity waves increase and the arrival height decrease when winds blow in the same direction of gravity waves, while the attenuation distances decrease and the arrival height increase when gravity waves propagate against winds. (3) The in(uence of winds on gravity waves of lower atmospheric modes is not obvious for they propagate mainly under mesosphere, where the wind ;eld is relatively weak. However, strong winds at thermospheric height lead to considerable changes of the
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F. Ding et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 857 – 869
dispersion relation and attenuation distance for upper atmospheric modes. When propagating against wind, phase velocity of mode G decreases signi;cantly, and the heights of energy distribution obviously fall down, which causes wave attenuation distance to increase by several hundred kilometers. When the wave propagates along wind, the results are contrary to those propagating against winds. Hence winds against the wave propagating direction support long-distance propagation of mode G, while winds blowing in the same direction of the wave can e9ectively shorten the attenuation distance. The e9ect of winds is more obvious for small-scale waves than for large-scale ones. (4) TIDs observed by Doppler array at Wuhan are statistically analyzed. The results show that 60% the TIDs propagate against winds. Among these TIDs, the maximum of the occurrence appears where the angle of wave − wind is close to 180◦ . The rest 40% of TIDs, which propagate along winds, mainly range in period from 10 to 20 min. The distribution of TIDs can be successfully explained by numerical calculation of internal gravity waves. Acknowledgements The author is thankful to National Nature Science Foundation of China (49974039) and National Key Basic Research Science Foundation (G2000078407) for the ;nancial support to this study.
1
1 4 − − 3i 2 + 3 2 2i +
Navier–Stokes equations (Francis, 1973): D + ∇ · v = 0; Dt
Dv TT = g − ∇p + ∇ · S; Dt
(A.1)
R0 DT TT ∇v; = Q + ∇ · (∇T ) − p∇ · v + S: ( − 1) Dt where ; p; v; T are, respectively, density, pressure, particle velocity and temperature for atmosphere. Q is inputting quantity of heat. , , and R0 are thermal conductivity coeKcient, viscosity coeKcient and speci;c heat ratio and universal gas constant, D=Dt = @=@t + v · ∇, STT is the gas viscosity tensor which satis;es Sij = (@vi =@xj + @vj =@xi − 2=3ij ∇ · v); (i; j = 1; 3):
(A.2)
The variables in (A.1) are composed of two parts: the ambient quantities and the ;rst-rank perturbations. Higher rank perturbations are ignored. The variables can then be expressed as v = v 0 + v1 ;
p = p0 + p1 ;
T = T0 + T1 ; (A.3)
where v0 is neglected by Francis (1973). Under these consumptions, Eqs. (A.1) are subsequently linearized as follows:
− i
−1
0
− 3i + i v˙0x
1
− − 4i + 4 2 + 3
kx v1x =! 1 − + R kx v1z =! −1 = 0; · p1 =p0 0 T1 =T0 −i 1
(A.4)
where Appendix A. Ray-tracing method was widely employed to calculating the propagation of internal gravity waves. As to gravity waves with a horizontal range of several thousand kilometers, ducting mechanism is more suitable. To analyze the ducted waves, atmospheric-strati;ed method can be adopted to get full wave solution of Navier–Stokes equations. The two method used in this paper are brie(y introduced below.
2
= ! =gkx2 H;
= (kz + i=2H )=kx ; R = k 2 − i + 1; = 1=kx H;
= i! =3p0 ;
v = iT0 kx2 =!p0 :
(A.5)
! = ! − kx v0x ;
A.1. Atmosphere-strati9ed method for ducted gravity wave
where H is average scale height. The matrix equation (A.4) di9ers from what derived by Francis (1973) in that background zero rank wind v0x is included here. So we can calculate the in(uence of wind on the propagation of guided waves. We mark the coeKcient matrix in Eq. (A.4) with A, and get the polarization relation:
In our study, a two-dimensional Cartesian coordinate system is used, in which the movement of atmosphere satis;es
p1 T1 v1z v1x = = ; = Ap At Ax Az
(A.6)
F. Ding et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 857 – 869
where Ap = (A12 A24 A31 + A32 − A22 A31 − A24 A32 ) · p0 ; At = (A12 + A32 − A22 − A22 A31 ) · T0 ; Ax = (A22 − A12 A24 − A24 A32 ) · ! =kx ;
(A.7)
Az = (A24 + A24 A31 − 1) · ! =kx : On the other hand, Eq. (A.4) has a non-trivial solution so long as the determinant of matrix A is zero. From this we get the dispersion relation: C3 R3 + C2 R2 + C1 R + C0 = 0;
(A.8)
where
C1 =−[ 2 −2 2 (1+3 )]− (1+7 )=(−1)− ;
(A.9)
C0 = [ 2 − 2 2 (1 + 3 )]=( − 1) + 2 (1 + 3 ): For a given real frequency ! and complex horizontal wavenumber kx , Eq. (A.8) has three pairs of roots for kz : one pair for gravity wave, other two for dissipative waves. Each pair of kz corresponds to the upgoing and downgoing waves. In our calculation, only the root for gravity wave is included. If the viscosity and thermal conductivity are ignored, the dispersion relation (A.8) yields: 2 !2 !2 ! kz2 = 2 1 − a2 − kx2 1 − b2 ; (A.10) c ! ! where !a and !b are the Brunt–VOasOalOa frequency and sound cut-o9 frequency, respectively. The non-dissipative dispersion relation (A.10) will be used later in ray-tracing method. The procedure of the atmosphere-strati;ed method is outlined as follows: given a real frequency ! and a complex horizontal wavenumber kx , the complex vertical wavenumber for gravity waves is obtained though (A.8). We assume that the lower boundary is ground and the upper one is at the height of 600 km. At the lower boundary, the perturbations satisfy rigid surface condition (namely v1z = 0); By using the polarization relation, we get the initial value of the eight variables (namely downgoing and upgoing branches for , T , vx , vz ). In each layer, the eight variables vary in the form of exp(i!t − ikz z + z=2H ); The variables in adjacent layers are related by interface condition, which is shown as follows: + − v1z = v1z ;
+ v1z p0 4 @v1z 2 − ikx v1x p1 − − i! H 3 @z 3
− v1z p0 4 @v1z 2 = p1 − − − ikx v1x ; i! H 3 @z 3
where the variables with prime + and − represent values above and below the interface respectively. Eq. (A.11) shows continuity of the vertical velocity, and (A.12) indicates the conservation of momentum (ux. At the highest layer, the radiation condition must be satis;ed. We search the gravity waves in the range of 60 –180 min and 200 –800 m=s that satisfy the upper boundary condition, and calculate the corresponding attenuation distance 1=Im(kx ). The height of each layer is chosen to be 600 m. At last, the dispersion curves and attenuation distance curves are obtained. A.2. Ray-tracing method for internal gravity wave
C3 = −3 (1 + 4 ); C2 = 3 (1 + 4 )=( − 1) + (1 + 7 ) + 3 ;
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(A.11)
(A.12)
Ray-tracing method was widely applied to the analysis of internal gravity waves under the in(uence of background winds for its feasibility and explicitness in calculation. In order to get the propagation features of internal gravity waves in a dissipative atmosphere, we should derive the complex ray equations that are based on theory of Fermi. However, these kinds of complex equations are not feasible in calculation. Hence the theory of complex ray needs to be simpli;ed so as to be more convenient in application. In this paper, ray theory under non-dissipative condition is used to trace the ray paths. At the same time, energy attenuation expression is derived to calculate the wave energy dissipation. The assumption fails, however, when the gravity waves are critically coupled. In this case, the ray theory is not suitable for the simulation of gravity wave propagation. Compared with the range of time and space of propagation of gravity waves, atmosphere is a non-transient, low-dissipative media, in which the gravity wave can be simpli;ed as quasi-monochromatic plane wave. According to Generalized Ray Theory, we assume that the packet of grav* * ity wave propagates in generalized space ( k ; !; r ; t) along a path *. If the dissipation of atmosphere could be ignored, the propagation parameters should meet with the dispersion relation (11). Under this restriction, the phase of the wave can be expressed as * * += [! dt − k · d r ]: (A.13) *(t)
Among the rays that can satisfy (A.13), only those have steady phase are valid in propagation. This requires that the variation of phase equals zero: dx + = ! − kx · dt *(t)
* dz = 0: (A.14) + kz (kx ; ky ; !; r ; t) · dt According to Variation Principle, the generalized ray-tracing equations of gravity waves can be derived and
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F. Ding et al. / Journal of Atmospheric and Solar-Terrestrial Physics 65 (2003) 857 – 869
shown as follows: d! = − d kx =
@kz @(! − kx vx ) dt; @t @kz
@kz @(! − kx vx ) dt; @x @kz
dx =
@kz @(! − kx vx ) dt; @kx @kz
dz =
@(! − kx vx ) dt: @kz
(A.15)
The vertical vector kz in (A.15) is produced by dispersion relation (A.10). Given the initial parameters ! and kx , the ray path of gravity waves can be obtained from Eq. (A.15). The change of logarithmic wave amplitude ln A along the ray path * can be expressed as *
d(ln A) = −kzi (kx ; ! − kx vx ; r ; t) · d z:
(A.16)
The change of amplitude along ray path can be obtained by integrating (16): z A(z)=A(z0 ) = exp − kzi · d z ; (A.17) z0
in which z0 and z are, respectively, the heights of wave packet at the launched time and time t. The exponential increase of amplitude is not included. Eq. (A.17) implies that the variation of relative amplitude of gravity wave can be obtained through integrating kzi along the ray path. In our calculation, ray equations (A.15) are used to calculate the gravity wave propagation paths, and expression (A.17) is used to obtain the energy attenuation along ray paths. In this paper, only the upgoing waves are included. The wave energy decreases with the increase of height, and so do the value calculated by Eq. (A.17). We de;ne that, besides the re(ection and critically coupling of the waves, the propagation is valid until the wave energy cannot satisfy A(z)=A(z0 ) ¿ 1=e.
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