Maximum wind speed derived from HLS approximation in comparison with the best track data of hurricane Rita.
Spiral cloud-rain bands (SCRBs) are some of the most distinguishing features inherent in satellite and radar images of tropical cyclones (TC). The subject of the proposed research is the finding of a physically substantiated method for estimation of the TC’s intensity using SCRBs’ configuration parameters. To connect a rainband pattern to a physical process that conditions the spiraling feature of a rainband, it is assumed that the rainband’s configuration near the core of a TC is governed primarily by a streamline. In turn, based on the distribution of primarily forces in a TC, an analytical expression as a combination of hyperbolic and logarithmic spirals (HLS) for the description of TC spiral streamline (rainband) is retrieved. Parameters of the HLS are determined by the physical parameters of a TC, particularly, by the maximal wind speed (MWS). To apply this theoretical finding to practical estimation of the TC’s intensity, several approximation techniques are developed to “convert” rainband configuration to the estimation of the MWS. The developed techniques have been tested by exploring satellite infrared imageries and airborne and coastal radar data, and the outcomes were compared with in situ measurements of wind speeds and the best track data of tropical cyclones.
- hyperbolic-logarithmic spiral
- tropical cyclone
- spiral cloud-rain bands
- maximum wind speed
The issue addressed in this chapter relates to methods for estimating the intensity of a tropical cyclone (TC) from the characteristics of its cloud-rain field (CRF) structure. In general, these methods are empirical and semiempirical, i.e., they are based on the correlation of the structural features of the CRF with the intensity of the TC found from observations from satellites and radars. The most widely used method in operational practice is the Dvorak method [1, 2]. This chapter relates to the exploring of one of the most pronounced structural elements of the CRF, which are spiral cloud-rain bands (SCRBs). Attention was first attracted to these bands by Wexler  based on aircraft observations. It was suggested that SCRBs indicate the mature cyclone and its organization follows the streamlines. The authors of  described the SCRBs observed on the radar and suggested to use a modified logarithmic spiral to express its configuration in mathematic form. Although SCRBs have been studied for a long time, there is currently no consensus about their origin and mechanism of generation. Reviews of proposed hypotheses are given in [5, 6, 7, 8]. In the paper of Lahiri , a first attempt was undertaken to estimate the effect of the TC intensity on the geometric characteristics of SCRBs. In this study, a simple model of a TC in which the low-level streamlines were described by logarithmic spirals was suggested. It was shown that the rate of generation of a latent heat in the model was proportional to the crossing angle that is a parameter of the spiral. (As follows from , the “crossing angle” of a SCRB of a finite width is the angle at which the longitudinal axial curve of the SCRB crosses the concentric circle (centered at the center of the cyclone) at a given arbitrary point belonging to the axial curve; that is, this is the angle between the tangents to the axial curve and the said circle at this point). As a result, crossing angle decreases as cyclones matured since the latent heat is very small in this stage. However, no relationship between the maximum wind speed (MWS) and the crossing angle was obtained. Moreover, the description of SCRBs by a logarithmic spiral only is an internally contradictory approximation due to the following circumstances. On the one hand, the main property of the logarithmic spiral is the constancy of the crossing angle for any of its points. Therefore, in particular, the alternative name of this spiral is an equiangular spiral. On the other hand, it was found that the crossing angle is sensitive to wind speed. But wind speed is not constant along the SCRB. Therefore, the crossing angle cannot have a constant value along the SCRB as well. Thus, the main feature of the logarithmic spiral (constancy of the crossing angle) is not consistent with the physics of the process. In the modified logarithmic spiral , the experimentally observed dependence of the crossing angle on distance is taken into account by introducing a radius of so-called inner limiting circle, at which the crossing angle is zero. However, the relationship of this radius with the intensity of the cyclone and the radius of the maximum wind (RMW) has not been established. It should be noted that the “spiralness” of the SCRB is also used in the empirical Dvorak method, although only at the qualitative level, by estimating the sector occupied by the spiral structure (“count the tenths” method), which is approximated by a logarithmic spiral with the crossing angle of 10°. However, as it was stated in , this spiral does not have a physical basis. In our papers [11, 12, 13, 14], the assumption of authors [3, 4, 9] that rainbands are well arranged along streamlines was also used. In general, this assumption is confirmed by a comparison of radar and aircraft data (e.g., [15, 16]). The same orientation of the principal rainband along the jet is demonstrated in . Further, the expression for spiral streamline has been derived in  in the closed form as hyperbolic-logarithmic spiral (HLS). The most advantage feature of the HLS is the dependence of one of its parameters on the MWS. Unlike the previous studies, the HLS was not assumed but was accurately derived based on physical considerations. At the same time, it turned out that only the peripheral portion of the HLS is similar to the logarithmic spiral. The size of this portion and the corresponding crossing angle are determined by the parameters of the cyclone, including the MWS. Unlike the modified logarithmic spiral, the change in the tangent of the crossing angle in the HLS is governed not by the distance weight function (equal to zero at the distance equal to the radius of the inner limiting circle and approaches unity at the outer edge of the cyclone), but by the physical parameters of the cyclone. However, it is not possible to determine the MWS by the crossing angle of the logarithmic section of the HLS only, as well, due to multifactor influence on the crossing angle. On the other hand, approximation of a SCRB by the HLS allows determining the MWS based on the known characteristics of the cyclone in cases when the SCRB is sufficiently long and markedly different from the logarithmic spiral. A discussion of the physical basis of the proposed method, methodology, and results of its application is the subject of this chapter. In particular, the derivation of the HLS based on the distribution of forces affecting the cyclone, methods for approximating the SCRB, the results of estimating the MWS from radar, and satellite data in comparison with the data of direct measurements and final conclusions of the corresponding meteorological services (Best Track reports) are provided.
2. Hyperbolic-logarithmic model of a streamline in a cyclone
2.1 Streamline equation in polar coordinates
As per Batchelor , radial (
Considering , where
It should be noted that the sum of inflow and crossing angles is the right angle (Figure 1).
2.2 Inflow angle from the balance of forces in a cyclone
To express the streamline Eq. (4) through the physical parameters of a cyclone, the inflow angle should be evaluated based on the balance of forces in a cyclone. The diagram of forces is depicted in Figure 2 as follows from .
The balance condition called Guldberg-Mohn balance  is expressed as follows:
is the gradient force (
is the balancing force containing
which is the centrifugal force;
which is the deflecting force (is the Coriolis parameter, ω is the angular speed of the Earth rotation, and
which is the frictional force, where
2.3 The streamline equation as a function of physical parameters of a cyclone
Let us assume for definiteness that the speed of wind
where is the normalized polar radius and coefficient
The first and second terms of Eq. (17) can be named, respectively, as hyperbolic and logarithmic components of the streamline equation of wind in a TC in the polar coordinates. Accordingly, Eq. (17) can be named the hyperbolic-logarithmic spiral.
2.4 Alternate representations of HLS streamline
Because , the expression (17) can be transformed to the exponent-logarithmic form
Using the Maclaurin series expansion for an exponential function and combining the coefficients at the argument of the first power, Eq. (19) can be written as a polynomial:
As follows from Eq. (20), the HLS contains linear and nonlinear parts in regard to the logarithm of the relative polar radius (ln
where is the Coriolis speed at distance
2.5 Graphical representations of HLS streamline
To illustrate the dependence of HLS primary features versus the MWS and the friction factor, the graphical diagrams in polar coordinates (Figure 3) and semilogarithmic coordinates (Figure 4) for three values of MWS and two values of the friction factor are provided below.
Figure 3 reflects the known feature of a rainband, as it has been summarized by Willoughby . A rainband of a cyclone from the periphery to its center is curved, due to the increase in the angular velocity, and therefore can be transformed into a logarithmic spiral. As the intensity of a cyclone increases, the slope of the twist decreases, and the band configuration becomes like a circular arc. Indeed, the HLS is similar to a logarithmic spiral, but only on the periphery of a cyclone, as shown in Eq. (20), where the normalized polar radius (
3. HLS approximation techniques
The task of using the analytical expression of the streamline (17) or (19) considered in the previous section is to use it to determine the MWS in a TC. The base for resolving this task is the expression (18) for the coefficient
Thus, the MWS can be estimated by approximating the SCRB of the TC with the HLS Eq. (19). Under this approximation, values of
3.1 Estimation of coefficients of the HLS by the least squares method
3.1.1 Relationships for calculation
Taking the exponentially logarithmic form of the HLS (19) and denoting ln
Performing the routine LSM procedure, one gets the calculation relationships for the HLS coefficients estimates which are provided below in the Gauss designation
The error estimates of the coefficients are equal:
is the residual variance, where
3.1.2 The condition of stationarity under the application of the LSM for approximation of a SCRB with the HLS
As follows from Eq. (18), coefficient
3.1.3 An example of HLS approximation with LSM
As an example of the application of the above approach to the approximation of the SCRB by the HLS, we consider the estimate of the MWS from an image from the geostationary satellite GMS-5 in the visible range of TC Mitag, at 01:31 UTC on March 5, 2002 (Figure 5).
The corresponding logarithmic normalized profile of the annotated signature is shown in Figure 6.
As can be seen from Figure 6, the “regularity” of the profile is broken in the range lny = (− 0.6)–(− 0.7). Apparently this is due to not quite correct annotation of the final segment of the SCRB. It would have to be carried out along the steeper contrast border observed in this area. Therefore, the main part of the profile in the range lny = 0–(− 0.7) was subjected to approximation, only. The main characteristics of the signature were radius of the most distant starting point of the signature
The average values of the coefficients
3.2 Assimilation technique
As follows from the previous section, for the HLS approximation of SCRB by the LSM with the subsequent checking for stationarity, a lot of operations are needed to be performed. This requires considerable time and, therefore, is unlikely to have a prospect for use in operational practice. In addition, annotation of the spiral structure on the image is, to a certain extent, a subjective process as was shown by the example of Figure 6. Moreover, sometimes, as was shown in , the LSM approximation leads to nonphysical values of HLS coefficients due to incorrect annotation of the spiral structure. The main problem of the HLS approximation with the LSM is that, by its nature, this technique is applicable mainly to so-called thin or clearly depicted spiral structures, which have a small width or sharp contour in satellite or radar images, and are suitable for its uniquely annotation. However, the same thin spiraling bands very often turn out to be squall lines, which are not related to streamlines. These circumstances stimulated the search for another technique that would be applicable to typical SCRBs, having a noticeable width and fuzzy contours. This technique, called the assimilative technique, is discussed below.
3.2.2 A principle of assimilative technique
When choosing a technique for determining the HLS coefficients (3) and (4), it should be noted that a spiral rainband observable on a radar image has a finite width and is a mapping of the resultant involvement of cloud-rain particles in the region affected by the streamline. In this case, parameters of the HLS describing all possible streamlines within the rainband should be considered as “equally possible.” In this technique, an approach based on the allocation of HLSs “fitting” into geometric boundaries of a rainband and determining the “expected” (mean) and “modal” HLSs was used. The fitting spirals were designated as “signatural” HLSs. The signatural HLSs have coefficients
4. Application of HLS assimilation technique for assessment of the maximum wind speed from satellite and radar data
4.1 HLS assimilation approximation of spiral rainbands applied to satellite IR images
4.1.1 Typhoon Phanfone (2002)
The first attempt to apply the HLS assimilation approximation to spiral rainbands on satellite IR images was undertaken by exploring data from the GMS-5 geostationary satellite during the monitoring of typhoon Phanfone existing in the Pacific Northwest in August 2002 . The source of satellite data was the archive of IR images from the Naval Research Laboratory (NRL, USA; http://www.nrlmry.navy.mil/sat_products.html). Meteorological data were taken from the Navy/Air Force Joint Typhoon Warning Center (JTWC-WP; Hawaii, USA) and Regional Specialized Meteorological Center (RSMC-JMA; Tokyo, Japan). IR spiral rainband patterns pertain to the top of cyclone’s cloud system (280–230 hPa, altitude 10–12 km). The rationale for comparison of the MWS resulted from the HLS processing with the best track data is the same as mentioned above.
Twenty-one images were processed which corresponded to the period from 21:31 UTC on August 14, 2002, till 12:31 UTC on August 18, 2002. Their results were subsequently united to six estimates. The length of the period covered by the analysis was 84 h. The estimates of the MWS are used to identify the stages of intensification and weakening of TC which are satisfactorily synchronized with the respective data of the Navy/Air Force Joint Typhoon Warning Center (JTWC). The correlation with data of the Regional Specialized Meteorological Center (RSMC-JMA) is observed only for the stage of intensification. Data of JTWC did not correlate either with RSMC data at the stage of TC weakening. In terms of the absolute value of the MWS, JTWC data exceed RSMC data for the moment of maximum TC intensification. It should be noted that data of the mentioned meteorological services differ from each other by 33% in terms of maximum TC intensity. The estimates of the MWS based on the HLS approximation occupy intermediate position between the above data. A detailed description of this study is given in .
4.1.2 Hurricane Rita (2005)
In the current paper, an example of application of the HLS enhanced assimilation technique under processing IR images of a TC is provided exploring observation data of Hurricane Rita (AL182005) from geostationary satellite GOES-10 when the hurricane was in Gulf of Mexico. An image acquired at 6:30 UTC 23 September 2005 has been processed (Figure 8).
The image contains two spiral rainbands. They are the outer impressive band that starts from the northwest corner of the image and the inner one that starts from the north. Both bands reach the core of the cyclone close to its south sector. The radius of maximum wind has been assessed to be 70 km as a distance from the eye’s center to the middle of a convective eyewall. Illustrations of the HLS approximation of both rainbands are provided in Figure 9. Resulted maximum wind speeds in comparison with the best track data  are presented in Table 1.
|Rainband||Maximum wind speed, kts|
|Modal MWS derived from HLS approximation||Best track *|
|Inner band||117.7 ± 8.7||115|
|Western sector of outer band||114.1 ± 13.6|
Following the HLS technique feature that presumes to use for approximation a signature closest to the TC core (where a streamline impact on cloud organization is most pronounced), the western sector of the outer rainband was approximated only. As follows from Table 1, the approximation of both rainbands results in the modal maximum wind speeds close to the best track speed with the acceptable accuracies (less than 12%). It should be noted that among other things, this illustrative example also shows the possibility of increasing the reliability of the HLS approximation by combining multiband estimates. In particular, the combined weighted estimate of MWS based on the data provided in Table 1 is 116.6 ± 7.4 kts.
4.2 HLS assimilation approximation of spiral rainbands applied to airborne and coastal radar images
4.2.1 Comparison of operational and HLS estimates of the intensity of TC Irma (AL112017) based on the airborne and the best track data
Maximum wind speeds in Hurricane Irma (AL112017) were estimated in  using its rainband radar signatures acquired by the NOAA Hurricane Research Division during routine aircraft missions into the hurricane. Most appropriate radar and other accompanied data were taken from the NOAA Hurricane Research Division (HRD) archive acquired for Hurricane Irma (AL112017) that existed in the Atlantic basin from August 30 to September 12, 2017, and reached category 5 intensity. The radar data from airborne radars were taken from the HRD’s Atlantic Oceanographic and Meteorological Laboratory (AOML) website (http://www.aoml.noaa.gov/hrd/data_sub/radar.html). The best track and aircraft data were taken from the National Hurricane Center’s Tropical Cyclone Report . The data pertain to one of eight aircraft missions to the cyclone. This mission was conducted from morning to afternoon September 5, 2017. The numerical outcomes of the HLS approximation are listed in Table 2.
|Time point no.||Altitude, m||UTC||Observation time, hours||Error |
|m s−1||kts||m s−1||kts|
The data of the first two time points were acquired at altitude from 5 km to 6 km, the six others from 2 km to 3 km. At the higher altitude, the MWS was approximately from 80 kts to 100 kts; at the lower level, the mean modal wind speed was about 161 ± 5 kts. This estimate is for a middle time point at approximately 10:45 UTC September 5 and manifests the maximum TC intensity that was estimated by the HLS approximation of all period of the HLS application from 21:16 UTC September 3 till 23:51 UTC September 8. As follows from , the wind speed of 164 kts measured directly by the aircraft at the flight level occurred approximately 10 h later. The best track MWS, which is for 10 m altitude, was about 150 kts. As per the best track data, the maximum intensity was 154 kts from noon September 5 to approximately 18:00 UTC September 6. The HLS estimates for this period are provided in  and amounted to 158–146 kts. These results indicate the satisfactory agreement of the HLS approach with in situ data (compare 161 kts and 164 kts). Lower speeds evaluated at high levels (two first time points in Table 2) follow the contemporary understanding of the vertical tangential wind profile in a TC (e.g., ) that presumes the decreasing of the wind speed up from the level of the maximum wind at altitude approximately 1–1.3 km (850–900 mb). The average error of the HLS approximation for the entire observation time (∼ 130 h) of the comparative analysis with the best track data given in  was no more than 5%.
4.2.2 The maximum wind speed in TC Irma (2017) by the HLS approximation of the rainband signatures from the coastal San Juan radar in comparison with best track data
During the passage of the TC Irma near the island of Puerto Rico, the cyclone was in the survey zone of the weather radar WSR-88D installed in the city of San Juan. The base for processing was the reflectivity image at 21:15 UTC 6 September 2017 that is shown in Figure 10 (left image) when the TC center was located at 18.9° N/65.4° W . The image refers to the beginning of the cyclone weakening stage and has the double eyewall structure that indicates a double wind maximum. Under the HLS approximation, the RMW was selected within the outer eyewall, as follows from findings in [27, 28], and estimated to be about 62 km.
The modal and mean maximum wind speeds estimated from the HLS approximation of the northwest rainband (Figure 10, right image) were 74 m s−1 (143.9 kts) and 75.4 m s−1 (146.6 kts), respectively. Accordingly, these speeds are approximately by 6 kts and 3 kts lower than the best track speed 150 kts at this time.
A new approach to use the characteristics of spiral cloud-rain bands of a tropical cyclone, observed by ground-based and aircraft radars, as well as satellites in visible and infrared wavelengths, is considered. The physical substantiation of the proposed approach is (1) the assumption about the orientation of the SCRBs mainly along the streamlines and (2) the analytically derived streamline equation in the form of the hyperbolic-logarithmic spiral. It is shown that the logarithmic spiral usually used to describe the configuration of SCRB is only a special case of the HLS. Unlike the empirically applying logarithmic spiral, the HLS coefficients depend on the MWS and the friction factor. The analysis of changes in the configuration of the HLS depending on the intensity of a TC is conducted. An explanation for experimentally observed phenomenon of the “rounding” of the SCRB (i.e., a decrease in the crossing angle) with increasing intensity of the TC as a whole, as well as with approaching to the radius of maximum wind, is proposed. The similarity of the SCRB configuration to the logarithmic spiral in some cases is interpreted also. The technique for approximation of a SCRB in the form of the HLS based on the least squares method and on the assimilation procedure was developed for obtaining MWS estimates at the height of the SCRB location. Testing of the proposed approach was performed based on literature data from ground-based coastal and aircraft radars, data of regular aircraft reconnaissance missions in the TC, and satellite data available via the Internet. A certain disadvantage of the method is its applicability, as a rule, for mature tropical cyclones, where its cloudy field manifests well-defined SCRBs and a clearly defined circulation center (eye center). On the other hand, the physically based configuration of a SCRB as the hyperbolic-logarithmic spiral allows one to develop the method for estimating the position of the circulation center with the eye covered with clouds, examples of which are given in [11, 13]. The results obtained suggest that the development and improvement of the proposed approach will make it possible to use the radar and satellite information more fully to assess the physical characteristics of a TC. The HLS approach to retrieve the TC’s intensity is particularly benefited for ground-based coastal radar probing of a TC before its landfall and the absence of aircraft reconnaissance missions.