A New Method for Calculating the Wind Speed

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culating the wind speed distribution of a moving tropical cyclone. Adv. Atmos. Sci., 27(1) ... (1987) set the maximum wind speed radius of air pres- sure formula in ...
ADVANCES IN ATMOSPHERIC SCIENCES, VOL. 27, NO. 1, 2010, 69–79

A New Method for Calculating the Wind Speed Distribution of a Moving Tropical Cyclone HU Banghui∗ 1,2 (胡邦辉), YANG Xiuqun1 (杨修群), TAN Yanke2 (谭言科), WANG Yongqing3 (王咏青), and FAN Yong3 (范 勇) 1 2

Institute of Severe Weather and Climate, Nanjing University, Nanjing 210093

Institute of Meteorology, PLA University of Science and Technology, Nanjing 211101 3

Nanjing University of Information Science and Technology, Nanjing 210044 (Received 14 January 2009; revised 12 May 2009) ABSTRACT

Based on gradient wind equations, including frictional force, and considering the effect of the movement of a tropical cyclone on wind speed, the Fujita Formula is improved and further simplified, and the numerical scheme for calculating the maximum wind speed radius and wind velocity distribution of a moving tropical cyclone is derived. In addition, the effect of frictional force on the internal structure of the tropical cyclone is discussed. By comparison with observational data, this numerical scheme demonstrates great advantages, i.e.: it can not only describe the asymmetrical wind speed distribution of a tropical cyclone reasonably, but can also calculate the maximum wind speed in each direction within the typhoon domain much more accurately. Furthermore, the combination of calculated and analyzed wind speed distributions by the scheme is perfectly consistent with observations. Key words: tropical cyclone, maximum wind speed radius, wind velocity distribution Citation: Hu, B. H., X. Q. Yang, Y. K. Tan, Y. Q. Wang, and Y. Fan, 2010: A new method for calculating the wind speed distribution of a moving tropical cyclone. Adv. Atmos. Sci., 27(1), 69–79, doi: 10.1007/s00376-009-7209-5.

1.

Introduction

China is one of the countries suffering from the devastation caused by typhoons along the coast of the northwest Pacific and South China Sea. Accordingly, great efforts have been made in researching, forecasting and improving disaster reduction of tropical cyclones in China. Previous research achievements and forecasting techniques of typhoons were systematically reviewed by Chen and Ding (1979), which is a milestone paper for typhoon research and operational application. However, as remarkable progress has been made in tropical cyclone dynamics over the past two decades, Chen et al. (2002) have since published a further review of the latest literature on tropical cyclones. In this review, the authors cover both theoretical and modelling-based research from the international science community, as well as systematically summarizing modern-day theories and views on tropical cyclone ∗ Corresponding

dynamics. Furthermore, Meng et al. (2002) summarized progress made in the few years following the 8th five-year program of China on tropical cyclone structure, numerical prediction, evaluation of operational numerical models, and tropical-cyclone-related heavy rain and disasters. Due to a deficiency in observational data from the oceans, wind calculations for tropical cyclones face challenges. For example, the lack of precision in the initial and boundary conditions leads to lower calculation accuracy. Therefore, researchers have attached great importance to simplifying the tropical cyclone pressure field and wind field models, such as Fujita and Myers’ pressure formula (Chen, 1981). Wang (1986) analyzed several pressure models and concluded that the Fujita Formula gives the better distribution of air pressure within a typhoon. Such circular vortex was also used to construct an artificial tropical cyclone through numerical simulations, i.e. to form a

author: HU Banghui, [email protected]

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METHOD FOR CALCULATING THE WIND SPEED OF A MOVING TC

normal symmetric typhoon model in the initial field together with asymmetric components depicting the tropical cyclone and environmental flow (Wang et al., 1996; Wang, 1998). In fact, asymmetric construction is the essential characteristic of a tropical cyclone (Luo, 1991; Chen and Luo, 1998). Systematic deviation was found in the numerical model consisting of an axial typhoon (Ueno, 1990). The asymmetric feature impacts the tropical cyclone movement greatly (Chen and Luo, 1995). Therefore, based on observation, Fang et al. (1987) set the maximum wind speed radius of air pressure formula in different directions and considered the impact of typhoon movement and friction. The results showed the typhoon wind field asymmetry was very obvious, i.e. the wind in the typhoon center was weak, and the wind speed of the right semicircle was higher than the left semicircle. This result is consistent with the horizontal features of the wind field within the core area (about three to six times the maximum wind speed radius from the centre) of a mature tropical cyclone, which were synthesized on the basis of data large dataset by Shea and Gray (1973). In calculating the maximum sea surface wind speed of a typhoon, Chen and Qin (1989) introduced extended pressure models into atmospheric motion equations and derived the formula for calculating maximum sea surface wind speed near the typhoon centre. On the basis of the frictional gradient wind equations, and considering the effect of the tropical cyclone movement on the radius in the inertial term of control equations, Hu et al. (1999) calculated the maximum wind speed radius with the observational maximum wind speed, and deduced the asymmetrical distribution of wind speed within a tropical cyclone. Therefore, it is necessary to determine the maximum wind speed radius firstly when using synopticclimatological methods to calculate the wind speed distribution of a tropical cyclone. In addition, it is also an essential factor to be considered in today’s popular approaches to the numerical simulation of typhoons, as well as in research on typhoon dynamics (Chen et al., 2002). Consequently, the calculation of maximum wind speed radius on the basis of available data is of great theoretical significance to research on typhoon dynamics, and is applicable in the forecasting of tropical cyclones and disaster reduction. Hu et al. (1999) used the maximum wind speed to calculate the maximum wind speed radius. In fact, the maximum wind speed itself needs to be calculated, and the observed maximum wind speed relative to the position of the typhoon center is unknown. Moreover, the selection of environmental pressure is largely determined by experience. Thus, the calculation of tropical cyclone wind speed needs further improvement. In sec-

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tion 2 of this paper, the Fujita Formula is revised and the scheme is explained for calculating the maximum wind speed radius. Section 3 describes the method for calculating the maximum wind speed radius. In section 4, data from Typhoon 9711 is used to calculate the maximum wind speed radius, the maximum wind speed, and their influencing factors. The result of the calculated wind field and the reanalysed wind field are presented in section 5, and then brief conclusions are drawn in section 6. 2.

The scheme for calculating the maximum wind speed radius

The gradient wind equations used by Wilson (1956) are employed for the moving tropical cyclone. When frictional force is considered and pressure distribution is assumed to be axially symmetrical, the horizontal motion equations of any air particle within the domain of the moving tropical cyclone in a polar coordinate system are as follows:  2 Vθ Vs sin α Vθ   + + f Vθ = A − Fr  r r  Vθ Vr Vr Vs sin α   + + f Vr = Fθ r r

(1)

where Vθ is the tangential wind speed; Vr is the radial wind speed; r is the distance from the air particle to the typhoon centre; Vs is the moving speed of the tropical cyclone; α is the included angle between the moving direction of the tropical cyclone and the line connecting the air particle and the centre of the cyclone, with the assumption that the counter clockwise direction is positive; f is the Coriolis parameter; A is the radial air pressure gradient force; Fr is the radial frictional force; and Fθ is the tangential frictional force. Let the frictional coefficient be k, the sea level friction be roughly expressed as kV, and the real wind direction deflect to the low pressure side of isobars by an angle β shown in Fig. 1. Then, the following relations are obtained:  Vθ = V cos β       Vr = V sin β  Fθ = kV cos(ϕ + β)      F = kV sin(ϕ + β)

(2)

r

where V is the scalar wind speed, and ϕ represents the deflection angle of the frictional force from the oppo-

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and the pressure gradient are unduly small. Hence, considering the effect of environmental air pressure, the Fujita Equation is revised as: " µ ¶2 # · ³ r ´2 ¸− 12 r (4) 1− P =P∞−(P∞ − P0 ) 1+2 R R∞

F ϕ β

β

V

r

Vθ

V Fig. 1. A schematic map of the basic relationship between the wind direction inner deflection angle and the real wind direction, where β is the wind direction inner deflection angle, ϕ is the deflection angle of the frictional force from the opposite direction of the real wind vector, V is real wind, Vθ is the tangential wind component, Vr is the radial wind component, and F (|F | = k|F |) is sea level friction.

site direction of the real wind vector, which is positive when the direction is counter clockwise. Substituting Eq. (2) into Eq. (1), the following equation sets are obtained:  2 V cos2 β V Vs sin α cos β   + + f V cos β    r r    = A − kV sin(ϕ + β)  V 2 sin β cos β V Vs sin α sin β   + + f V sin β    r r   = kV cos(ϕ + β) which can be further transformed into    V = A sin β k cos ϕ   F (β, r) = 0

(3)

where F = k cos ϕ[Vs sin α + r(k sin ϕ + f )] tan β − k 2 r cos2 ϕ + A sin2 β Wang (1986) analysed the popular pressure models and pointed out that the Fujita Formula can represent well the air pressure distribution within the entire domain of a typhoon. However, some applications of the equations indicate that the selection of environment air pressure in the Fujita Formula is slightly arbitrary, and that the values of both the peripheral air pressure

where P∞ is the environment air pressure; P0 is the air pressure at the centre of a tropical cyclone; R is the maximum wind speed radius; and R∞ is the distance from the centre of the tropical cyclone when P = P∞ . The corresponding pressure gradient is: · ³ r ´2 ¸− 32 ∂P = 2 (P∞ − P0 ) r 1 + 2 × ∂r R · ¸ 1 r2 1 + + (5) 2 2 R2 R 2 R∞ R∞ If Eqs. (4) and (5) are compared with the Fujita Formula it can be found that the values of air pressure and barometric gradient in the revised equations are both greater than those in the unrevised equation. However, under usual conditions R is only a few kilometers, while R∞ ranges from hundreds of kilometers to over a thousand kilometers, i.e. R∞ >> R. In this case, there is little difference between the air pressure and the pressure gradient in the core area of a tropical cyclone in the revised equations and those in the unrevised equation. However, in the periphery of a tropical cyclone, the values of air pressure and the barometric gradient in the revised equations are both significantly increased. When r = R∞ and P = P∞ , the revised barometric gradient is twice as large than in the unrevised version. From Eq. (5), together with the state equation, the radial pressure gradient force is derived: £ ¤− 3 2kc T (P∞ − P0 ) 1 + 2(r/R)2 2 r h i× A= −1 2 P∞ − (P∞ − P0 ) [1 + 2(r/R)2 ] 2 1 − (r/R∞ ) µ ¶ 1 r2 1 + 2 2 + 2 , R2 R R∞ R∞ where kc represents the gas constant, and T stands for sea level atmospheric temperature. Let Ar = A/r and substitute this into Eq. (3). Then, its derivative by r can be obtained:  kr cos ϕ ∂V tan β ∂Ar tan β ∂β    ∂r = A cos β ∂r − A ∂r + r  r r ± . (6)  ∂F ∂r ∂β   =− ±  ∂r ∂F ∂β When r = R, the wind speed reaches its maximum value, i.e. ∂V |r=R = 0 . ∂r

72

METHOD FOR CALCULATING THE WIND SPEED OF A MOVING TC

If the effect of R∞ is omitted, the following equations are derived from Eqs. (6) and (3):  Vs sin αt  2  − R − R  (1 + B)(1 + t2 )k cos ϕ      t2 1 + (1 + B)t2   =0,  AR 1 + t2 (1 + B)(1 + t2 )k 2 cos2 ϕ (7) Vs sin αt  2  R + R +    (k sin ϕ + f )t − k cos ϕ    2  t 1    AR =0 , 2 1+t k cos ϕ[(k sin ϕ+f )t−k cos ϕ] where AR = Ar (r = R) = B=

2kc T (P∞ − P0 )3−3/2 , P∞ − (P∞ − P0 )3−1/2

¯ 2(P∞ − P0 )3−3/2 ¯ , and t = tan β . r=R P∞ − (P∞ − P0 )3−1/2

Hence, R from Eq. (7), i.e. the maximum wind speed radius, can be resolved. 3.

Numerical calculation of the maximum wind speed radius

For a moving tropical cyclone Vs 6= 0. By eliminating R from Eq. (7), the following equation can be obtained: AR [(k sin ϕ + f )t+(1 + B)(k sin ϕ + f )t3+Bk cos ϕ]2 = (1 + B)(1 + t2 )Vs2 t2 k cos ϕ sin2 α × [(k sin ϕ + f )t + kt2 cos ϕ(1 + B) + Bk cos ϕ] (8) The above is a tricubic equation about t. With regard to the factors at the point of maximum wind speed radius of a tropical cyclone, the wind direction inner deflection angle is a relatively small positive value because the airflow rotates rapidly. Consequently, its numerical solution will be found according to the characteristics of the equation. When a tropical cyclone is motionless Vs = 0, and R and t are set equal to Rs and ts respectively. So, the left side of Eq. (8) is null, and the equation will be: (1 + B)(k sin ϕ+f )t3+(k sin ϕ+f )t+Bk cos ϕ = 0 , (9) Assume p= and q= then ∆=

1 1+B

B k cos ϕ , 1 + B k sin ϕ + f ³ q ´2 2

+

³ p ´3 3

> 0,

and Eq. (9) only has one real root, i.e. ´1/3 ³ q ´1/3 ³ q + − − ∆1/2 . ts = − + ∆1/2 2 2

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(10)

For a tropical cyclone, its wind-direction inner deflection angle is positive, i.e. ts > 0, meaning ( k>f k sin ϕ < −f, i.e . (11) f ϕ < − arcsin . k It is not hard to recognize that when the distribution of peripheral environment air pressure and that of the frictional force of a motionless tropical cyclone are even in each direction, its wind direction inner deflection angle, maximum wind speed radius, and maximum wind speed will all be axially symmetrically distributed with regard to the cyclone centre. Conversely, when the peripheral environment air pressure and the frictional force do not evenly distribute in each direction, its wind direction inner drift angle, maximum wind speed radius, and maximum wind speed will not be in a symmetrical distribution. As for a moving tropical cyclone, in its moving direction sin α = 0. This is the same as the motionless situation. Out of the moving direction of a tropical cyclone, the solution to Eq. (8) is symmetrical to the moving direction. Using the intersectional point of the curves derived from the left and right sides of Eq. (8) respectively, the value of t can be found out: (a) When f f −γ − arcsin p < ϕ < − arcsin , k k 1 + 4B(1 + B) and

1 cos γ = p , 1 + 4B(1 + B)

Eq. (8) has one positive numerical solution, named Solution 1. (b) When f , ϕ 6 −γ − arcsin p k 1 + 4B(1 + B) Eq. (8) has two positive numerical solutions, named Solution 2 and Solution 3, respectively. Under such circumstances, if the central air pressure, the position, the peripheral environment air pressure, and the moving speed of a tropical cyclone are given, and proper frictional force is selected, the wind direction inner deflection angle at the point of the maximum wind speed radius can be calculated. Then, if the result is substituted into Eq. (7), the maximum wind speed radius can be obtained, and finally the maximum wind speed can also be solved from Eq. (3).

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Table 1. The centre position and analytical records of satellite images of Typhoon 9711 from 11 August 1997 to 17 August 1997 (China Meteorological Administration, 1997). Date

Centre position ◦

11 12 13 14 15 16 17

4.

Air pressure at centre ◦

latitude ( N)

longitude ( E)

(hPa)

16.3 17.3 18.8 20.9 22.7 23.8 24.8

150.1 147.2 143.2 139.4 135.7 132.1 128.2

980 945 920 935 950 950 960

Calculation results for maximum wind speed radius, maximum wind speed, and an analysis of influencing factors

Typhoon 9711 (Winnie) is used as an example. Winnie was the strongest typhoon that landed on China in 1997. After its formation at 11.2◦ N, 158.9◦ E on 8 August 1997, it strengthened whilst steadily moving northwest. On 10 August 1997 it was upgraded to a tropical cyclone, and further developed into a typhoon on the next day. The moving direction and moving speed of the typhoon remained constant throughout, but its intensity increased rapidly with a change in wind speed near its centre from 33 m s−1 on 11 August 1997 to 60 m s−1 on 13 August 1997, indicating a super-strong typhoon. Its strength maintained for nine days until midnight on 18 August 1997, when it landed in Wenling, Zhejiang Province with a wind speed near its centre still reaching 40 m s−1 . At the time of landing, the storm tide coincided with the astronomical tide. As a result, a super-strong sea tide was initiated and daily rainfall reached up to 499 mm. The impact of the typhoon covered a vast area, from Fujian Province to Heilongjiang Province. Over 66 667 hm2 of land were devastated and over 25 million people were affected, with the death toll reaching 240. The direct economic loss was more than $4.8 billion. Table 1 illustrates the position and air pressure at the typhoon centre, the maximum wind speed, and analysis data from satellite cloud pictures from 0800 LST 11 August 1997 to 0800 LST 17 August 1997 (China Meteorological Administration, 1997). Figure 2 shows the distribution of sea level pressure on 15 August 1997. The relevant data are extracted from NCEP-NCAR reanalysis data (Kalnay et al., 1996). At that moment, the value of the near (Kalnay et al., 1996) circular closed isobar in the outermost periphery of Winnie was 1005 hPa; the air pressure at the centre was 950 hPa; the maximum speed near the centre was 45 m s−1 ; and the Typhoon was moving in a west-by-north direction at 15 km h−1 .

Maximum speed (m s

−1

33 50 60 50 45 45 40

)

Moving speed (km h−1 ) 15 17 13 15 15 13 20

Figure 2 shows that the influenced region of the typhoon was vast, covering 20 degrees of latitude from north to south and 30 degrees of longitude from east to west. 4.1

Maximum wind speed radius and maximum wind speed when the typhoon is motionless

Suppose the tropical cyclone is stationary, the peripheral environment air pressure P∞ =1005 hPa, and the sea level atmospheric temperature T = 27◦ C. Then, based on the position and central air pressure of the tropical cyclone given in Table 1 and a given proper frictional force, the wind direction inner deflection angle, the maximum wind speed radius and the maximum wind speed of the tropical cyclone can be solved. Figure 3 illustrates the relationships between the wind direction inner deflection angle, the maximum wind speed radius and the maximum wind speed and the frictional force. Evidently, if the wind speed inner deflection angle is to be positive, the frictional force must be deflected clockwise from the opposite direction of the wind vector, i.e. the frictional force deflects toward the centre of the tropical cyclone. The wind direction inner deflection angle varies with the frictional resistance. When the frictional coefficient is given, the smaller the angle of clockwise departure of

Fig. 2. Sea level pressure field at 0800 LST 15 August 1997. Interval of isolines: 2.5 hPa. Unit: hPa.

74

METHOD FOR CALCULATING THE WIND SPEED OF A MOVING TC

tn iec if fe oc la no it ci rF

mum wind speed radius, and hence affects the internal structure of the tropical cyclone. 4.2

(a)

tn iec if fe oc la no it ci rF tn ei ci ff eo cl an oti ci rF

VOL. 27

(b)

(c)

°

Friction deflection angle ( ) Fig. 3. Relationship of wind direction inner deflection angle (a, degree), maximum wind speed radius (b, km), and maximum wind speed (c, m s−1 ) with frictional resistance at the point of the maximum wind speed radius in the motionless tropical cyclone. The x-axis represents the deflection angle of frictional resistance from the opposite direction of the wind vector, in which the positive (negative) value indicates the departure toward the high (low) pressure side. The y-axis represents the frictional coefficient (×10−4 s−1 ), and the shadow area indicates the positive value of the wind direction inner drift angle.

frictional resistance from the opposite direction of the wind vector, and the greater the wind direction inner deflection angle. When the direction of friction is given, the greater the frictional coefficient (frictional resistance), and the smaller the wind direction inner deflection angle (Fig. 3a). The relationship between the maximum wind speed radius and friction is similar to that of the wind direction inner deflection angle (Fig. 3b). However, the maximum wind speed does not vary with friction. Instead, when the wind direction inner deflection angle is positive, the maximum wind speed is about 43.7 m s−1 ; and when the wind direction inner deflection angle is negative, the maximum wind speed is about −43.7 m s−1 (Fig. 3c). This implies that friction has little effect on the numerical value of maximum wind speed, but it does affect the wind direction inner deflection angle and the maxi-

Maximum wind speed radius and maximum wind speed when the tropical cyclone is moving

Again, let us take the typhoon on 15 August 1997 as an example. At that time, the typhoon was moving in a west-by-north direction. Suppose the environment air pressure and friction are evenly distributed, and assume the peripheral environment air pressure P∞ = 1005 hPa, the frictional coefficient k = 1.5 × 10−4 s−1 , and the deflection angle of friction from the opposite direction of the wind vector ϕ = −32◦ , then Solution 1 as well as the distributions of the wind direction inner deflection angle, the maximum wind speed radius and the maximum wind speed at the point of maximum wind speed radius can be obtained, as shown in Fig. 4. First, it should be noted that none of these typhoon parameters is axially symmetrical. The wind direction inner deflection angle is symmetrical in relation to the moving direction of the typhoon, and it reaches its maximum along the moving direction while its minimum value occurs on the left and right sides of the moving direction (Fig. 4a). The maximum wind speed radius is not in an axially symmetrical distribution either. Its maximum value occurs along the moving direction of the typhoon, and its minimum value occurs on the right side of the moving direction. The value on the left is greater than that on the right (Fig. 4b). But the maximum wind speed does not occur in the moving direction, and the wind speed on the right side is larger than on the left side (Fig. 4c) Specifying the deflection angle of friction from the opposite direction of the wind vector ϕ = −45◦ , then Solution 2 and Solution 3 (not shown) can be obtained. In Solution 2, the wind direction inner deflection angle is still symmetrical along the direction of the typhoon’s motion and the deflection angle is considerably smaller than that in Solution 1. Its maximum value occurs in the moving direction of the typhoon. The maximum wind speed radius is also smaller than that in Solution 1. The maximum wind speed radius on the right side of the moving direction is smaller than that on the left side and the distribution of the maximum wind speed is basically identical with that in Solution 1. Likewise, the wind direction inner deflection angle in Solution 3 is also symmetrical in relation to the moving direction. But the inner deflection angle vertical to the direction of the typhoon’s motion is larger than that along the direction of the typhoon’s motion. The distribution of the maximum wind speed radius is similar to that in Solution 2 but with its values a little larger. Further-

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75

HU ET AL. o

(a) 315

o

6.8

0

o

45

6.6 6.4 6.2

o

270

o

90

6

225

o

135 180

o

o

o

(b) o 315

40 0 38

o

45

36 34 32

o

270

90

30

225

o

2 Vs sin α + R(k sin ϕ + f ) Vmax =1− t. AR kR cos ϕ

°

135

o o

(c) o 315

180 o 0

48

o

45

45 42 39

o

270

o

90

36

o

135

225

gests both the value and the direction of friction can have a great impact on the internal structure of the typhoon. When other parameters all distribute axially symmetrically, the wind direction inner deflection angle at the point of maximum wind speed radius of the tropical cyclone is symmetrical along the direction of motion. The maximum wind speed radius is smallest on the straight right side of the moving direction, and largest on the straight left side. The maximum wind speed is largest on the straight right side of the moving direction and smallest on the straight left side. However, why does friction have little effect on the value of maximum wind speed? From Eqs. (2) and (3) the following equation can be deduced:

o

o

180

Fig. 4. The wind direction inner deflection angle (a, units: ◦ ), the maximum wind speed radius (b, unit: km), and the maximum wind speed (c, units: m s−1 ) when the frictional coefficient k = 1.5×10−4 s−1 and the deflection angle of frictional resistance from the opposite direction of the wind vector. ϕ = −32◦ .

more, the distribution of maximum wind speed is basically similar to that in Solution 1. By comparing the above three solutions, it can be concluded that for the moving tropical cyclone their maximum wind speeds are basically the same, though none of the above three solutions is axially symmetrical around the typhoon centre. This suggests that friction has little effect on the value of maximum wind speed. Instead, friction greatly affects the wind direction inner deflection angle and the maximum wind speed radius. In the above three solutions, the frictional coefficient is supposed to be constant, and only the direction of friction varies. The result shows that the larger the deflection angle of friction from the opposite direction of the wind vector, the smaller the wind direction inner deflection angle, and also the smaller the maximum wind speed radius. This is the same as what happens when the tropical cyclone is motionless, which sug-

The second term on the right side of the equation reflects the effects of motion, friction and the geostrophic parameter on maximum wind speed. But in comparison with the pressure gradient force, they are relatively small. Therefore, the motion of the atmosphere at the point of maximum wind speed radius basically follows the cyclostrophic wind equation, and as a consequence the effect of frictional resistance is minute. Now that the effect of friction on maximum wind speed has been established as insignificant, there are many opportunities to select the coefficient and the direction of friction. Thus, it is very convenient for us to calculate the maximum wind speed. Here, NCEP– NCAR sea level pressure data are used, and the environment air pressure is prescribed as the value of the near circular closed isobar in the outermost periphery of the tropical cyclone (the interval of the contour is 2.5 hPa). Take Solution 1 for example. Figure 5 shows the calculated and observed results for maximum wind speed when the Winnie was motionless or moving from 11 August 1997 to 15 August 1997. Figure 5 shows 70

)/s (md 60 ee ps 50 dni w mu 40 mix aM 30

Date motionless

mobile

observation

20 10

11

12

13

14

15

16

17

18

Fig. 5. Observed and calculated maximum wind speed when the tropical cyclone Winnie was motionless and when moving from 11 August 1997 to 17 August 1997.

76

METHOD FOR CALCULATING THE WIND SPEED OF A MOVING TC

that, except for 14 August 1997, the calculated result of maximum wind speed of the moving tropical cyclone is closer to the observed result when compared to the motionless tropical cyclone. The calculated result is smaller than the observed result for the motionless tropical cyclone. Since the distribution of the elements of the motionless tropical cyclone is axially symmetrical around the centre, its control equation is a gradient wind equation involving friction. By contrast, the moving tropical cyclone is asymmetrical, and thus is closer to reality. Its control equation takes account of the effect of motion in the frictional gradient wind equation and adds the effect of motion on wind speed into the inertia term. Therefore, it is more rational to consider the asymmetrical structure of the tropical cyclone in the calculation of wind speed, and the result will be closer to observation. In addition, these results also suggest that it is valid to assume the environment air pressure to be the value of a near circular closed isobar in the outermost periphery within the domain of the tropical cyclone. 5.

The scheme for calculating wind speed

After calculating the maximum wind speed radius, A can also be calculated for any point with a distance r from the centre of the tropical cyclone. Now the problem is how to solve the wind direction inner deflection angle β according to Eq. (4). Assume w = tan β, then Eq. (2) will be a cubic equation of w, which has one or three real roots. Hu et al. (1999) obtained the analytic solution, but which solution is to be used in a real application remains a problem. Since the low layer is a cyclonic convergence with high wind velocity within the domain of a tropical cyclone, the wind direction inner deflection angle should be a small positive value. Then:  G1 (w) = k cos ϕ [Vs sin α + r (k sin ϕ + f )] ×    w − k 2 r cos2 ϕ = 0 , 2    G2 (w) = − Aw = 0 . 1 + w2 Now only if the small positive value at the intersectional point of G1 and G2 is required, the solution w is obtained. The calculated result indicates that, given a certain precision, the numerical solution obtained with this method corresponds with the feasible solution in the analytic solution (relevant figure omitted). Take Typhoon 9711 at 0800 LST 15 August 1997 as an example again. The frictional coefficient at any point with distance r from the centre r A k = kR AR

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kR = k(r = R) = 1.5 × 10−4 s−1 and the deflection angle of friction from the opposite direction of the wind vector ϕ = −32◦ . The calculation method of R∞ is as follows: First, P∞ needs to be selected in the reanalysed field of sea level pressure (Fig. 2). Then the longitude and latitude of P∞ should be read at eight azimuths that are roughly in an even distribution. After R∞ is calculated at these azimuths, P∞ can be obtained at any azimuth by a three-order spline interpolation. The derived distributions of wind speed and air pressure field are shown in Fig. 6. In order to compare with the wind speed recorded in the Yearbook on Tropical Cyclones (China Meteorological Administration, 1997) (Fig. 6a, dotted line) conveniently, the calculated wind speed is only shown as level six (10.8 m s−1 ), level eight (17.2 m s−1 ), and level ten (24.5 m s−1 ) (Fig. 6a, solid line). It can be seen that the calculated level ten wind is very consistent with observation, but the ranges of level eight wind and level six wind are both smaller than observed. This suggests that the calculated wind speed is a valid reflection of wind structure within the core area of the typhoon, but it fails to reflect the characteristics of the outer wind field (as shown in Fig. 6b). Why so? Let us look at the pressure field (Fig. 6b). Obviously, since the maximum wind speed radius and R∞ are not axially symmetrical, the pressure field determined by the revised Fujita Formula is not axially symmetrical either. However, it is considerably weaker than the NCEP reanalysed field (Fig. 2) and the outer pressure gradient is much smaller too, which results in the ranges of both level six and level eight winds being relatively small. Why is the pressure gradient relatively small? In the periphery of the tropical cyclone, there exists an interaction between the tropical cyclone and surrounding systems, and the wind field in the periphery of the tropical cyclone is representative of the combined effect of multiple systems. However, only a mono tropical cyclone system was considered previously, ignoring its interaction with surrounding systems. However, it is very difficult to describe the surrounding systems of the tropical cyclone with an air pressure field. Figure 7 illustrates the surface wind speed reanalysed field. It is shown that the wind speed reanalysed field basically shows a small wind speed within the centre of the tropical cyclone. Outwards from the centre, the wind speed increases steadily and reaches a maximum of over 20 m s−1 at the point nearly three degrees of latitude from the centre. But this value is unduly smaller than the maximum wind speed of 45 m s−1 reported in the Yearbook on Tropical Cyclones (China Meteorological Administration, 1997). Further outwards the wind speed begins to decrease, but it is larger than the observed. Thus, the reanalysed field roughly describes

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(b)

Fig. 6. The calculated wind field (a, units: m s−1 ) and baric field (b, unit: hPa). In Fig. 6a the solid line denotes calculated wind, the dashed line illustrates level six, level eight and level ten winds from the Yearbook on Tropical Cyclones (China Meteorological Administration, 1997).

the weak wind area within the centre, but fails to reflect maximum wind speed. The peripheral wind speed is larger than observed. That is to say, the reanalysed field on tropical cyclones fails to reproduce the distribution of winds in the core area and the peripheral wind speed is too great. Hence, it can be deduced that the reanalysed data describe the large-scale characteristics of the wind field of a tropical cyclone and reflect the environment wind field of the tropical cyclone. Therefore, the reanalysed field is induced into the calculation of wind speed for a tropical cyclone, and the following equation is used to synthesize: ! Ã r r A A Vc = V t Va + 1− AR AR

where Vc is synthesized wind speed, Vt is the calculated wind speed, and Va is the reanalysed wind velocity. Obviously, except for the southern boundary of the tropical cyclone (where the wind speed of the reanalysed field is relatively large), the synthesized wind speed tallies perfectly with observation (Fig. 8). This indicates that the adopted synthetic method combines well the advantages of the reanalysed field and the calculated field, and the synthesized wind speed reflects reasonably the distributional characteristics of wind speeds in a tropical cyclone. In order to verify the method, data from tropical cyclone 9104 and tropical cyclone 8706 were used (Fig. 9) (China Meteorological Administration, 1991, 1987). The results showed that synthetic wind (solid

Fig. 7. The reanalyzed field of the surface wind speed of tropical cyclones. The solid line denotes NCEP–NCAR reanalyzed wind, the dashed line illustrates level six, level eight and level ten winds from the Yearbook on Tropical Cyclones (China Meteorological Administration, 1997). Units: m s−1 .

Fig. 8. The wind field synthesized from the calculated wind field and the reanalyzed field. The solid line denotes synthesized wind, and the dashed line illustrates level six, level eight and level ten winds from the Yearbook on Tropical Cyclones (China Meteorological Administration, 1997). Units: m s−1 .

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METHOD FOR CALCULATING THE WIND SPEED OF A MOVING TC

(a)

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(b)

Fig. 9. The wind field synthesized from the calculated wind field and the reanalyzed field for (a) tropical cyclone 9104 at 0800 LST 12 May 1991, and (b) for tropical cyclone 8706 at 0800 LST 28 July 1987. The solid line denotes synthesized wind, and the dashed line illustrates level six, level eight and level ten winds from the Yearbook on Tropical Cyclones (China Meteorological Administration, 1987, 1991). Units: m s−1 .

line) is in good agreement with observed wind (dashed line). Consequently, the method is valid for calculating the wind distribution in a tropical cyclone. 6.

Conclusions

On the basis of previous research, the Fujita air pressure formula was improved. At the same time, the effect of tropical cyclone movement on wind speed was considered in the inertia term of the gradient wind equation, which includes frictional force. After rational simplification and deduction, calculation schemes were obtained for the maximum wind speed radius, maximum wind speed, and the wind speed of a tropical cyclone. With these schemes, the effect of frictional force on the internal structure of a tropical cyclone was discussed, and the calculation results compared with observations. It was found that the schemes can reasonably describe the asymmetrical characteristic of the internal structure of a tropical cyclone and can calculate the maximum wind speed in each direction within the domain of the typhoon with greater accuracy. The synthetic result of the calculated wind field and the reanalysed wind field coincide perfectly with observation. The main conclusions are as follows: (1) The Fujita air pressure formula was improved and the calculation schemes for the maximum wind speed radius and wind speed distribution within the domain of the tropical cyclone were derived. (2) When the environmental air pressure and friction are given, the structure of a motionless tropical cyclone is axially symmetrical. When the frictional coefficient is given, the smaller the clockwise departure

of friction from the opposite direction of the wind vector, and the larger the wind direction inner deflection angle and the maximum wind speed radius. When the direction of friction is given, the larger the frictional coefficient (friction), and the smaller the wind direction inner deflection angle and maximum wind speed radius. (3) Supposing the environment air pressure and friction are evenly distributed, then the structure of the moving tropical cyclone is asymmetrical. The wind direction inner deflection angle is symmetrical in relation to the direction of movement, and the maximum wind speed radius on the straight right side is smaller than that on the straight left side. The maximum wind speed occurs on the straight right side of the moving direction of the tropical cyclone. (4) The value and direction of friction have basically little effect on the value of maximum wind speed of a tropical cyclone. However, they can cause huge impact on the maximum wind speed radius. This will affect the structure of the tropical cyclone and thereby influence the wind speed distribution within the domain of the tropical cyclone. (5) The example of calculating the maximum wind speed showed that it is reasonable to assume the environment air pressure to be the numerical value of the near circular closed isobar in the outermost periphery of a tropical cyclone. (6) The calculated peripheral wind field of a tropical cyclone is relatively weak, and its synthetic result with the reanalysed field corresponds perfectly with observation. Therefore, it reflects rationally the distributional characteristics of wind speed within the do-

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main of a tropical cyclone. Acknowledgements. This work was jointly supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 40425009 and 40730953. REFERENCES Chen, K., 1981: The typhoon pressure field and wind field models. Acta Oceanologica Sinica, 3(1), 44–55. (in Chinese) Chen, K., and Z. Qin, 1989: The calculation of sea-level maximum wind speed of a typhoon. Acta Oceanologica Sinica, 11(1), 31–41. (in Chinese) Chen, L., and Y. Ding, 1979: An Introduction to Typhoon over the West Pacific Ocean. Science Press, Beijing, 491pp. (in Chinese) Chen, L., and Z. Luo, 1995: Effect of the interaction of different-scale vortices on the structure and motion of typhoon. Adv. Atmos. Sci., 12(2), 207–214. Chen, L., and Z. Luo, 1998: Numerical study on function affecting tropical cyclone structure and motion. Acta Meteorologica Sinica, 12(4), 504–512. Chen, L., X. Xu, Z. Luo, and J. Wang, 2002: Introduction to the Tropical Cyclone Dynamics. China Meteorological Press, Beijing, 317pp. (in Chinese) China Meteorological Administration, Eds., 1987: Yearbook on Tropical Cyclones. China Meteorological Press, Beijing, 121pp. (in Chinese) China Meteorological Administration, Eds., 1997: Yearbook on Tropical Cyclones. China Meteorological Press, Beijing, 105pp. (in Chinese) China Meteorological Administration, Eds., 1991: Yearbook on Tropical Cyclones. China Meteorological Press, Beijing, 124pp. (in Chinese)

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