Estimation of Turbulent Diffusivity over the Tokyo Metropolitan Area

from Constant-volume Balloons

by

Junji Sato

Meteorological Research Institute, Tokyo

(Received February 28, 1975)

Abstract

In order to study the characteristics of turbulent diffusion over an urban area, constant-volume balloons with transponders were released at altitudes of 200 and 400 m, and their behavior was observed over the Tokyo Metropolitan area.

The atmospheric diffusivities are derived from the Lagrangian proper- ties of the balloons, in relation to the temperature lapse rate and the tur- bulent intensity. The result shows that the lateral diffusivity, Ky, has no direct relation to the lapse rate or the turbulent intensity. However, the vertical diffusivity, IC, is closely related with the lapse rate as well as with the turbulent intensity.

1. Introduction

Recently, a high concentration of photochemical smog has been observed frequently in the urban arid suburban areas of Tokyo Metropolis. It is maintained that this occurs in regions exposed to the sea breeze (KA w AmuRA, 1974). Therefore, the advection and diffusion phenomena in these regions urgently need to be clarified. Some attempts to obtain the trajectory of an air parcel containing multiple pollutants have been carried out in the Tokyo area (FuKuoKA et al., 1974, and SATO et al., 1974).

The Lagrangian statistics can be computed from the motion of air parcel. However, because of the lack of relevant information, most of the recent studies on atmospheric diffusion have been concerned with estimating the Lagrangian statistics from more easily available Eulerian statistics.

Experiments using a balloon as tracer of air motion have been made over the years. The usefulness of such experiments had been re-emphasized, notably through the study of GIFFORD (1953). In his shudy, the neutral pibals were released and tracked with two theodolites.

It has been suggested by ANGELI, and PACK (1961, 1962) that super-pressured and con-stant-volume balloons ("Tetroons") provide first approximation to the 3-dimensional air flow.

As tracer to obtain the motion of an air parcel over an urban area, the constant-volume balloons with transponder were released in the summer of 1973 at the heights of 200m and 400 m, and their behaviour was observed over the Tokyo Metropolitan area.

The author will show that the constant-volume balloons provide a feasible and logical means of estimating the atmospheric diffusion, and serve as a means of estimating the turbulent diffusivity.

2. Theoretical considerations on turbulent diffusion

We consider the spatial displacement of a particle of diffusing substances subject to atmospheric turbulence.

Expressing the velocity component of this particle by v' (t) and v' (t+) at time t and t+e respectively, e being the lag time, the mean square displacement of the particle after travel time t, Y2 (t) , can be shown as

where R(e) is the Lagrangian velocity autocorrelation function emphasized by TAYLOR (1921), and represented as

R()=v' (t) v' (t+e)Iv'2( 2 )

As in the textbook, the diffusivity K is formulated as follows (e. g., PASQUILL, 1962; LUMLEY and PANOFSKY, 1964; SLADE, 1968):

The quantity foR dt defines a time-scale characteristic of the turbulence, called the

Lagrangian integral time scale, L:

This argument makes it reasonable that the Fickian theory, in which K is constant, should apply when the diffusion time, t, is large compared to L.

3. Procedures

The tracking system consists of a radar, a transponder attached to the constant-volume balloon, and a transponder reciever. The principle of operation is the same as reported by PACK (1962), the only difference being the balloon. The tetroon .has a tet- rahedral shape, while the constant-volume balloon we utilized is spherical. It is open to controversy, which of the two is superior in following the atmospheric turbulence.

The balloon we used is about 1.6 m in diameter, and the physical properties on this constant-volume balloon are reported by OHTA and ITO (1970).

Usually we used a rubber tow balloon to send up the constant-volume balloon as

quickly as possible to the required level. In this case, it is necessary to separate the tow balloon at that level. A blasting fuze was used for this purpose.

The constant-volume balloon launch site was settled on the branch office and factory of the Japan Radio Co., Ltd., in Yokohama city located sDuth of Metropolitan Tokyo,

because our study required a southerly wind (sea breeze). And the tracking radar site was located on Tama-hillock, where the trajectory of the balloon could be tracked sideways.

Fig. 1 shows the trajectories of the constant-volume balloons, on a simplified map of Tokyo Metropolitan and the surrounding area, where the positions of the balloons are indicated at 10 minute intervals.

Since the balloons could not be positioned continuously, and the intervals between the

positions were irregular, it was necessary to derive sets of continuous statistics. This was accomplished by determining the spatial coordinates x, y, and z of the balloon at 1-minute intervals by linear interpolation. Since the average trajectories were almost directed north-south, the x-axis was oriented north-south, the y-axis east-west and the z-axis ver-tically. Thus the time rate 4x/tit (4t equivalent to 1 minute) gives the longitudinal speed u', LlyIZIt the lateral speed v' and tlzbit the vertical speed w'. The 3-dimensional wind components as the basis for the analysis of the turbulent diffusion phenomena were obtained in this way.

Trajectories are picked out in which a) the balloons flow relatively straight; namely, there are no trends on the trajectories, b) the positions of the balloon are determined at 1-minute intervals.

Trajectories satisfying the above conditions amount to six out of 29 trajectories. The constant-volume balloon launching information is given in Table 1.

4. The constant-volume balloon fluctuations

On several occasions a helicopter was flown along with the balloon, so the balloon height could be obtained with some degree of accuracy.

Figs. 2, 3 and 4 show the constant-volume balloon height traces for typical flights. As seen in Figs. 2, 3 and 4, the balloon oscillates considerably in the vertical. Two factors may be responsible for this oscillation: the air current and the properties of the balloon itself.

OHTA and ITO (1970) discussed the statical and dynamical properties of the response of the balloon. Namely, in case of no air current, the balloon shows oscillations around the balanced level with a half-amplitude less than about 2 meters, which is comparable with the diameter of the balloon, and can be neglected in our study. Whereas, in an air current, if the motion of the balloon has the same period as Brunt oscillation, the

1975

vertical velocity of the balloon is equal to that of the air current, and the displacement of the balloon is also equal to that of the air current. If the period of the motion of

the balloon is not equal to that of Brunt oscillation, the amplitude of the motion of the balloon is less than that of the air current.

The density of the gas contained in the constant-volume balloon is nearly invariant, and its period of vertical oscillation is given by

where To is the ambient temperature in degree Kelvin, r, the dry adiabatic lapse rate,

r the lapse rate of the ambient atmosphere and y the gravity acceleration.

chit r

1975Estimation of Turbulent Diffusivity from the Lagrangian Statistics41

Eq. (5) shows that the period of vertical oscillation is dominated by the ambient tem-

perature lapse rate. HASS et al., (1967) comfirmed this relation and found out that the period of vertical oscillation has an immediate connection with lapse rate. However, accord- ing to our study, the period of vertical oscillation determined from the peak in the absolute spectral density, n Sw(n), of the vertical velocity does not agree with that of Eq. (5).

Fig. 5 shows an example of the spectral density of the vertical velocity (flight No. 1). There are two peaks, one at 5.9 minutes and the other at 2.0 minutes. On another flight, too, n Szu(n) has its peak between n=0.1 and n=0.2 cycles per minute. As mentioned. above, the period of vertical oscillation is not so much affected by the ambient temper- ature lapse rate. Meanwhile, as far as Table 1, Figs. 2 and 3 are concerned, the amplitude of vertical oscillation appears to be slightly affected by the temperature lapse rate.

5. The estimation of turbulent diffusivity

On the basis of vertical velocity derived from the constant-volume balloon displace-ment, we see there is a pronounced tendency for the r. m.s. vertical velocity, ,/w12, to increase with increase in the lapse rate, dTIdz. HASS et al., (1967) obtained the quadratic regression curve for ,V-W12 versus dTldz. Fig. 6 shows the relation betweenand dTldz. In Fig. 6, circles with a flight number indicate the result by the author, and dots and solid line refer to data due to HASS et al., (1967) at New York City. The rearession curve is represented by them as

whereis in m/sec, dTldz in°C/100m, and dTldz >0 shows the lapse condition.

Though the circles are scattered about the solid line, Eq. (6) seems to represent fairly well the relation under the lapse condition.

On the other hand, the r. m. s. lateral velocity, A,/v-- , has no pronounced relation with the lapse rate.

Before calculating the diffusivity by the use of Eq. (3), it is necessary to obtain the Lagrangian velocity autocorrelation function, R(e), which is calculated from Eq. (2).

Figs. 7, 8 and 9 show the Lagrangian velocity autocorrelation functions for typical constant-volume balloon flights. It is noticed here that the Lagrangian velocity autocor-relation fuction, R(e), of the vertical component approaches zero in shorter lag time than that of the lateral component. As known from Eq. (4), this means that the scale of turbulence in the lateral direction is larger than that in the vertical.

According to Eq. (3), the turbulent diffusivity can be obtained by multiplying the mean square of turbulent velocity by the scale of turbulence. In this study, the integra- tion in Eq. (3) is carried out to the point where R(e) becomes zero. As the sampling duration is about 100 minutes, it shows that the diffusion phenomena during this order of duration are dealt with here.

Fig. 10 shows the relation between the vertical diffusivity, Kz, and the temperature lapse rate. It is found that the vertical diffusivity tends to increase with increase in the lapse rate. Meanwhile, the lateral diffusivity, Ky, has no pronounced relation with the lapse rate.

In the statistical theory of turbulence, two foundamental quantities are defined: the scale mentioned previously, and intensity of turbulence. These concepts are basic in the development of such applications as the theory of turbulent diffusion. The intensity of turbulence, I, is defined as the square root of the ratio of the turbulent energy to the mean energy:

44

1975Estimation of Turbulent Diffusivity from the Lagrangian Statistics45

For this reason, the relation between the turbulent intensity and diffusivity is in-vestigated. The result is shown in Fig. 11. It can be found in Fig. 11 that the vertical diffusivity, K,, shows a fairly close relation to turbulent intensity. The associated regres-sion curve for K, versus -1/ wf2 /a is given by

K,=- 14057 ( w/2 //V.' (m2/sec)( 8 )

Whereas the lateral diffusivity, Ky, has no such relation to turbulent intensity as in the case of K,.

6. Concluding remarks

Information on the value of diffusivity is of practical importance in the study of air

pollution, especially, in the numerical simulation of concentration. However, at the present stage, no definite observational result seems to have been established because of the large scatter of observational data and lack of reliable data. In this study , though based on a small number of data, the diffusivities were determined in due consideration of the Lagrangian conception. It is found that the vertical diffusivity is closely related with the thermal stability and the turbulent intensity, and that the value of K, changes from the order of several m2/sec to near 100 m2/sec with the increase of the turbulent intensity.

Hereafter, it will be necessary to obtain more data, and to get the profiles of the diffusivity and difference of its profiles between urban and rural areas.

Acknowledgements:—The author especially wishes to express his hearty thanks to Dr. S. Sakuraba for many suggestions and to Dr. K. Takeuchi for discussions on this study.

References

ANGELL, J. K. and D. H. PACK, 1961: Estimation of vertical ,air motions in desert terrain from tetroon flights. Mon. Wea. Rev., 89, 273-283.

-------------------------------------------, 1962: Analysis of low-level constant volume balloon (tetroon) flights from Wallops Island. J. Atmos. Sci., 19, 87-98.

FUKUOKA, S. et al., 1974: Measurements of low level air trajectory with non lift balloon, Part I. J. Japan Soc. Air Pollu., 9, 263 p.

GIFFORD, F., 1953: A study of low level air trajectories at Oak Ridge, Tenn.. Mon. Wea. Rev., 81, 179-192.

HAss, W. A. et al., 1967: Analysis of low-level, constant volume balloon (tetroon) flights over New York City. Quart. J. R. Met. Soc., 93, 483-493.

KAWAMURA, T., 1974: Meteorological analysis on photochemical air pollution in the broad of Kanto Plain. J. Japan Soc. Air Pollu., 9, 562 p.

LUMLEY, J. L. and H. A. PANOFSKY, 1964: The structure of atmospheric turbulence. Interscience Publishers, New York, 180-184.

OHTA, S. and T. ITo, 1970: Super pressure balloon and precise pressure radio-sonde system for the research of vertical air motion. Pap. Met. Geophys., 21, 45-72.

PACK, D. H., 1962: Air trajectories and turbulent statistics from weather radar using tetroons and radar transponder. Mon. Wea. Rev., 90, 491-504.

PASQUILL, F., 1962: Atmospheric diffusion. Van Nostrand, London, 92-101. SATO, J. and M. MORIGUCHI, 1974: Measurements of low level air trajectory with non lift balloon,

Part II. J. Japan Soc. Air Pollu., 9, 264 p. SLADE, D. H., 1968: Meteorology and Atomic Energy. U. S. Atomic Energy Commision. Division

of Technical Information. Oak Ridge, Tennessee. TAYLOR, G. I., 1921: Diffusion by continuous movements. Proc. London Math. Soc., Ser. 2, 20,

196-211.

ラグランジュ法による乱流拡散係数の推定

佐　藤　純　次

　 近年,大 気汚染 濃度 を予測す るために大気拡散モデルに よる シミュ レーシ ョンが数多 く行 なわれ る ように

なってきた。 しか しなが ら,こ れ らの シ ミュレー ションへ のイ ンプッ ト・デ ータと しての拡散係数 の プロフ

ィールとそ の絶対値については,ま だ未知 の部分 が多い。 したがって拡散 の シミュレー ションを実行す る場

合,適 当 な数値お よび形の拡散係数 の プロフ ィールを仮定 し,そ れ をシ ミュレーシ ョンにイ ンプ ッ トす ると

い う方法 を取 るのが常であ る。 ここでは詳細 な拡 散係数 の プロフ ィールを得 る手掛 りとして過圧 ・定容積 バ

ル ーンの流跡線 データか ら乱流拡散 場を考 察す ることを試 みた。流跡線 データは都市上空におけ る複合大気

汚染気塊 の動態を得 るための トレーサーと して用 いたノ ン リフ ト・バル ー ン観測に基づいている。

　 ノン リフ ト・バルー ンのバランス設定高度は200お よび400mの2高 度であ り,こ の高度 におけ る水平お

よび鉛直拡散係数ky,Kzを 推定 して,こ れ らの値 と大気安定度や乱れの強 さとの関係 を調べ た 。

　 鉛直方 向の乱流速 度変動 のr.m.s.値 は気温減率 と関係づけ られ,特 に中立状態ではHAss　 et　al.(1967)

に よる結果 と一致 した。Kzは 乱流強度,wt2/uに よって特徴づけ られ るが,気 温 減 率 と の相関 もよ

い。

　一方Kyに ついては,　v'2/uお よび気温 の減率 の両者 との関 係は見 出 し難い。 これは,鉛 直方 向では

地表面 お よび逆転 層や 混合層高度な どのLIDに よる物理 的境界条件 で気層の両端 が固定 され,特 に気層の

上限 はdT/dzに よって支配 され るために高い相 関を もつ もの と考え られ る。