j2iUN(i,48)

wj850

Here, N(i, n) denotes n neighborhoods of grid point i,Vj850 is a relative vorticity [1/s] at the pressure level of 850[hPa] at grid point j. H/000 is height field [m] at the pressure level of 1000[hPa] at grid point l. wj is velocity [m/s] at the pressure level j[hPa] at grid point k. Ajn = aj — 1 XjeN(;' n) aj stands for air temperature anomalies [k] from average of n neighborhoods of grid point i at the pressure level of j[hPa], where aj is air temperature [k] at the pressure level of j[hPa] at grid point i. If these conditions are satisfied with more than one and a half day long, the phenomenon is regarded as a tropical cyclone.

It is not appropriate to apply this method to data from high resolution numerical models. First, these conditions are empirical, so it needs a trial and error to obtain effective identification of tropical cyclone. Second, since the longitudinal grid intervals depend on the latitude, conditions of criteria are not uniform. Third, it needs large amount of computational time, since it has to check these conditions at all grid points. In the case of high resolution numerical model, it would be troublesome to use this method.

In the proposed method, we use coordinate transformation and streamline with enhanced curvature in order to achieve high speed detection of centers of tropical cyclones without empirical conditions. For preparation, we construct a database of velocity fields from the NCEP reanalysis dataset. In the database, horizontal velocities at arbitrary positions are obtained using linear interpolation at desired time and pressure level.

We first transform the coordinate. Traditionally, observational datasets are provided in the latitude-longitude grid system and these datasets are used to identify tropical cyclones in the conventional method. However, if we project this grid system to the 2-dimensional plane, there are large distortions especially at higher latitude. To avoid this distortion, we need to transform flow velocity field provided in the latitude-longitude grid to the 3-dimensional Cartesian coordinate. In the proposed method, since we use vector field of streamline and its curvature in 3-dimensional Cartesian coordinate, we equip a conversion and inversion from latitude-longitude coordinate to 3-dimensional Cartesian one, as shown in Fig. 1.

The transformation from latitude-longitude coordinate (f , C) to 3-dimentional Cartesian coordinate (x,y,z) with the origin at the center of the earth is written in the following equation.

where, hi stands for the length from the center of the earth to the pressure level i. Rx(f) is the rotation matrix of angle f around x axis,

Ry(C) is the rotation matrix of angle C around y axis,

Then the velocity vector (wx, wy, wz) is written in 3-D Cartesian coordinate,

z where, u and v are longitudinal and latitudinal velocity, respectively. The latitude f and longitude C are obtained from 3-D Cartesian coordinate vector (x, y, z) as follows,

Streamline

Streamline is one of the ways to search convergent points in the vector field [Vollmers 2001]. Since velocity vector field around the tropical cyclone is nearly concentric circles and there is almost no flow at the center of tropical cyclone, the convergent point of velocity is considered as the center of tropical cyclone. In addition, because of the surface friction, the velocity field tends to converge on the center of tropical cyclone near the ground pressure level. Thus, we use streamline to detect the center of tropical cyclone.

Starting with several initial positions we search for the center of tropical cyclone by iterating the streamline path downstream until the interpolated wind speed at 850 [hPa] or 1000[hPa] is close to zero. At the next step of iteration, the position is moved to, lci+1 = — + At-. (7)

Here itj and vv is the reference position and its velocity in 3-dimensional Cartesian coordinate system, respectively. ~Xi+1 stands for the position at the next step. In the present study, we set At at 21600[s]. From equation (7) and the fact that vv is perpendicular to ^, we obtain

Since above equation gives > |-ti|, if we use (7), the reference point goes away and away from the center of the earth. To avoid this, we convert -x>i+1 to latitude-longitude coordinate, and then we obtain the position at the corresponding pressure level in 3-dimensonal coordinate.

We decide centers of tropical cyclones using the following criterion, since there is almost no flow at the center of tropical cyclone.

Here we set e = 1000[m], after checking several values of e. This means that our searching finishes when the velocity is less than 1 km per 6 hour. This small value is effective to search convergent points precisely. We also set total iteration steps to be 100.

Since velocity field of tropical cyclone tends to form concentric circle as mentioned before, it is possible that the method of streamline (9) does not converge on the center of tropical cyclone. To avoid this, we make use of velocity vector W0 at the next position Xi = "Xi + Dtwi as shown in Fig. 2.

Xi+1

Namely, we enhance the curvature of streamline at the position "xi to assume that the curvature at the position Xi is almost equal to that at the position "xi. Since the path of the streamline with enhanced curvature bends to the inner circle, this method converges on the center of tropical cyclone in contrast to the method with simple streamline. In addition, as | x i+1 x i converge rapidly and accurately. We set the criterion of the end of searching to be the same as (9). Total iteration steps are also the same as the method with simple streamline.

Fig. 2 Example of streamline with enhanced curvature (10)

Results

First, we compare the efficiency of the two methods, simple streamline (7) and streamline with enhanced curvature (10). Here we use a velocity field data from NCEP at 1000[hPa] pressure level on 1800UTC July 10, 2006. We put 12 initial positions at grid points, {10°N, 22.5°N, 35°N} x {120°E, 130°E, 140°E, 150°Eg. Fig. 3 shows the result of the comparison. Black marks express observational points, and short black lines at these points denote velocity vectors. Long black lines show the results of simple streamline, and long white lines show those of the streamline with enhanced curvature. Black and white hexagons show convergent points of two methods which colors correspond to those of the methods.

Although some simple streamlines approach near to a tropical cyclone, named BILIS, they never converge on the center and do not satisfy the criteria (9) even after 100 iterations. On the other hand, streamlines with enhanced curvature converge on the center of the tropical cyclone BILIS, from 5 initial positions at 20 iterations on average. These results show the efficiency of the streamline method with enhanced curvature. In the next subsection we make a comparison of this method with the conventional method.

Fig. 3. Comparison of the two streamline methods

Fig. 3. Comparison of the two streamline methods

Comparison of Streamline Method to Conventional One

We compare the efficiency of the proposed method (10) with that of the conventional method, using the NCEP reanalysis dataset in 2006. The accuracy of the identifications of tropical cyclones near Japan is evaluated by the "best track'' data. We use conditions in [Bengtsson et al. 1995] as the conventional method, since the resolution is comparable to NCEP. Note that since conventional methods proposed by [Sugi et al. 2002, Oouchi et al. 2006] are for the data of higher resolution numerical models, the conditions of criteria, such as the values of neighborhood points or the values of criterion (3), are somewhat different from [Bengtsson et al. 1995]. In the proposed method, we put 12 initial positions, which are the same as subsection 3.1 but at 850[hPa] pressure level to compare with the conventional method. We also evaluate accuracy of detection using only criterion (1) of the conventional method, since the other criteria based on different data than wind vectors may remove some tropical cyclones. We focus on the tropical cyclones in the range of [0°N-50°N] x [110°E-170°E] (we call it ''near Japan'').

The results are summarized in Table 1. There are 23 tropical cyclones in 2006. For each tropical cyclone, ''lifetime'' shows a duration time between developed

Name |
Lifetime |
Max |
Streamline |
Criterion 1 |
All |

strength [kt] |
criteria | ||||

CHANCHU |
5/9(12)-18(18) |
95, 5/15(0) |
5/5(0) |
5/13(0) |
5/15(6) |

JELAWAT |
6/27(12)-28(18) |
40, 6/28(6) |
6/23(6) |
none |
none |

EWINIAR |
6/30(18)-7/10(6) |
100, 7/4(12) |
6/29(18) |
6/30(18) |
none |

BILIS |
7/9(6)-7/15(0) |
60, 7/12(21) |
7/4(12) |
7/7(12) |
7/8(6) |

KAEMI |
7/19(12)-7/25(18) |
80, 7/21(12) |
7/17(12) |
7/20(18) |
7/24(18) |

PRAPIROON |
8/1(6)-8/4(18) |
65, 8/2(12) |
7/25(0) |
8/1(18) |
8/2(18) |

MARIA |
8/5(18)-8/10(0) |
70, 8/6(18) |
8/4(0) |
none |
none |

SAOMAI |
8/5(12)-8/10(18) |
105, 8/9(12) |
8/5(0) |
none |
none |

BOPHA |
8/6(12)-8/9(0) |
55, 8/7(21) |
8/3(6) |
none |
none |

WUKONG |
8/13(0)-8/19(6) |
50, 8/15(12) |
8/10(0) |
8/10(18) |
8/11(18) |

SONAMU |
8/14(0)-8/15(0) |
35, 8/14(0) |
none |
none |
none |

IOKE |
8/27(6)-9/6(12) |
105, 8/30(0) |
8/30(12) |
8/31(6) |
8/31(6) |

SHANSHAN |
9/10(12)-9/18(9) |
110, 9/15(15) |
9/9(0) |
9/15(6) |
9/15(6) |

YAGI |
9/17(6)-9/25(0) |
105, 9/21(12) |
9/18(18) |
9/22(6) |
9/22(6) |

XANGSANE |
9/26(0)-10/1(18) |
80, 9/29(18) |
9/24(6) |
9/27(0) |
none |

BEBINCA |
10/3(0)-10/6(0) |
50, 10/4(18) |
9/28(18) |
10/2(0) |
10/2(6) |

RUMBIA |
10/3(6)-10/5(18) |
45, 10/4(0) |
10/2(18) |
10/2(18) |
10/5(12) |

SOULIK |
10/9(12)-10/16(6) |
75, 10/14(6) |
10/7(12) |
10/13(12) |
10/13(6) |

CIMARON |
10/27(6)-11/4(6) |
100, 10/29(6) |
10/25(6) |
none |
none |

CHEBI |
11/9(12)-11/13(6) |
100, 11/10(12) |
11/8(12) |
none |
none |

DURIAN |
11/26(12)-12/5(0) |
105, 11/29(12) |
11/25(0) |
none |
none |

UTOR |
12/7(18)-12/14(0) |
85, 12/12(12) |
12/7(0) |
12/9(12) |
none |

TRAMI |
12/17(12)-12/18(12) |
35, 12/17(12) |
none |
none |
none |

time and decayed time when maximum wind velocity exceeds 0 at first and at final, in the ''best track'' data, respectively. The maximum strength [kt] and its time for each tropical cyclone are also shown. We list the time of identifying tropical cyclones at first in the proposed and the conventional methods, in which we also include the case using only criterion (1). There are only 11 tropical cyclones found by the conventional method. Even when we only use criterion (1), 14 tropical cyclones are found. In contrast, the proposed method finds 19 tropical cyclones before the developed time, and other 2 at the time of developing. Only 2 tropical cyclones, which have very short lifetime, are missed.

Figure 4 shows the result on 1200UTC August 13, 2006. Black pentagons show the centers of tropical cyclones in ''best track'' data. A black diamond denotes a tropical cyclone identified by the conventional method. By the conventional method, only a tropical cyclone ''WUKONG'' is found. White hexagons are the centers of tropical cyclones detected by the proposed method. The proposed method finds

1. A decaying tropical cyclone, MARIA

2. A developing tropical cyclone, WUKONG

3. A strong vortex near to Japan.

However, a tropical cyclone SONAMU is missed by the proposed method. The reason is that there is no vortex at 850[hPa].

We emphasize another benefit of the proposed method. In this method, at each iteration we need only few data points around the reference point. Thus, in the case of huge climatology datasets of the high resolution numerical model, we only use small subset of the data to detect centers of tropical cyclones. Although there are several empirical conditions in the proposed method, such as initial positions, Af,

Fig. 4 Comparison enhanced streamline with conventional method

Fig. 4 Comparison enhanced streamline with conventional method

Table 2 The statistical average of identification on 2006 by the proposed method

Level [hPa] Initial points convergence all area near Japan

S50 1000 1000

12 12 20

0.7S

0.91

and e in the criterion (9), it is easy to test several values of these conditions because of small computational time.

We investigate the efficiency of the proposed method for different pressure levels (850[hPa] and 1000[hPa]) and initial positions (12 and 20). Table 2 shows statistical average of identification on 2006. Each column shows the average numbers for a time step throughout the year of: convergent streamlines, convergent points, convergent points in the range of [0°N-50°N] x [110°E- 170°E] (near Japan), and convergent points at the centers of tropical cyclones (TC), respectively. Although there are some initial points from which streamline do not converge after 100 iterations, we obtain better convergence and thus find more tropical cyclones in higher pressure level of 1000[hPa]. On the other hand, in the case of 20 initial points, there are not significant differences to detect tropical cyclones. Although we have checked several choices of initial positions, the results are qualitatively the same.

One thing we have to note is that the proposed method detects several strong cyclones. These strong cyclones are important for natural disaster at one side. However, if we want to discuss the effect of global warming on tropical cyclones, we have to omit these strong cyclones correctly. Once we find strong vortices in the huge datasets, it is a relatively easy task to remove these cyclones by adding several conditions, such as coherent vertical structure and warm core. Since there are only small numbers of convergent points (3.51-4.97), we can identify tropical cyclones applying, for example, traditional empirical conditions to each of the convergent points and its neighborhoods. This procedure saves us a large amount of computational time, since we do not need to check empirical conditions for all grid points. Note that there is no convergence at high pressure point. This is because of two main reasons. First, flow velocity of anticyclone tends to be weaker than that of cyclone. Second, the center of anticyclone at lower levels is not a convergent point but rather a divergence point.

In the present study, we proposed a non-empirical streamline method to detect the center of tropical cyclone automatically with less computational time. In the proposed method, first we solve the problem of distortion at high latitude in latitude-longitude grid system by use of coordinate transformation. We also bend the path of streamline successively to accelerate the convergence on the center of tropical cyclone. This method does not depend on empirical conditions which are adopted in the conventional method. Moreover, since the proposed method does not need to check conditions at all grid points, we achieved significant reduction of computational time.

To evaluate the effectiveness of the proposed method, we made several comparison experiments using the NCEP reanalysis datasets. Although the data is too coarse to define tropical cyclones, the proposed method automatically detects almost all tropical cyclones some of which are not identified by the conventional method. While our method is robust for initial positions, we could obtain better results using lower level (1000hPa). It is true that some tropical cyclones are not detected by the proposed method because of small vorticity, but these tropical cyclones have short lifetime.

As future work, we extend this method to 3-dimensional one, using the streamlines at all pressure levels in huge climatology data. Not only the extension will allow us to detect tropical cyclones more accurately, but will reveal 3-dimensional structure. This will be important for risk management. It would be interesting to predict course and development of the tropical cyclone by using a revealed 3-dimensional structure.

Acknowledgments This work was supported by Grant-in-Aids for the 21st Century COE "Frontier of Computational Science'' and the Global Environment Research Fund (RF-070) of the Ministry of the Environment, Japan. The numerical analysis was performed by Fujitsu HPC2500 super computer system at the Information Technology Center, Nagoya University.

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