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In the present study, 12 TC cases are selected according to the best-track data from the Japan Meteorology Agency (JMA) for exploring the sensitivity of TC intensities on the SST forcing errors. The 12 TC cases are chosen according to the following criteria: (1) the TCs which greatly influenced China, (2) the lifetimes of TCs were longer than 5 days and stayed over the ocean during the overwhelming majority of the simulation period, and (3) the TC whose tracks were minimally altered when the SST forcing was modified, with the largest deviation of this subset being less than 60 km when the SST forcing is modified, which almost excludes the impacts of the bias of TC tracks on TC intensities. The brief information concerning the selected 12 TCs is listed in Table 1. The geometric centers of the TCs are identified as the locations with the minimum sea level pressures (MSLPs), which are utilized to represent the TC intensities hereafter. The NFSV represents a special type of SST forcing error, with which the simulated TC intensities depart the most from, as compared to without it, as Eq (5) describes.
Name No. Start time (h; UTC) End time (h; UTC) Time of TC mature phase (h; UTC) TC intensity at mature phase Soulik 201307 0000 Jul 08 0000 Jul 13 0000 Jul 10 925 hPa; 50 m s−1 Utor 201311 0000 Aug 10 0000 Aug 15 1200 Aug 11 925 hPa;53 m s−1 Soudelor 201513 0000 Aug 02 0000 Aug 07 1800Aug 03 900 hPa;57 m s−1 Rammasun 201409 0000 Jul 13 0000 Jul 18 0600 Jul 18 935 hPa; 45 m s−1 Chanhom 201509 0000 Jul 06 0000 Jul 11 1800 Jul 09 935 hPa; 45 m s−1 Muifa 201109 0000 Jul 29 0000 Aug 03 1800 Jul 30 930 hPa; 48 m s−1 Bolaven 201215 0000 Aug 23 0000 Aug 28 1200 Aug 25 910 hPa; 50 m s−1 Sanba 201216 0000 Sep 12 0000 Sep 17 1800 Sep 13 900 hPa; 55 m s−1 Goni 201515 0000 Aug 15 0000 Aug 20 0600 Aug 17 935 hPa; 48 m s−1 Halong 201411 0000 Aug 02 0000 Aug 07 1200 Aug 02 920 hPa; 53 m s−1 Man-Yi 200704 0000 Jul 11 0000 Jul 16 1200 Jul 12 930 hPa; 48 m s−1 Noul 201506 0000 May 07 0000 May 12 0000 May 10 920 hPa; 55 m s−1 Note: The numbers (No.) and intensities are from the best-track data of JMA (Japan Meteorological Agency), the latter of which include the minimum sea level pressure and maximum surface wind speed at the corresponding time. Table 1. Twelve TC cases of investigation
Here, T represents the SST that forces the WRF model during the simulation period and updates every 6 hrs. δT denotes the SST forcing error that is superimposed to T.
$\left\| {\delta {{T}}} \right\| \leqslant 1$ , where$\left\| {\delta {{T}}} \right\| = \sqrt {\displaystyle\sum {\delta {{{T}}^2}(i,j)/N} } $ constrains the standard deviation of superimposed SST forcing errors not larger than 1 K, which is determined by the mean anomaly of SST in the western North Pacific region [i.e. 10º−35ºN, 100º−150ºE; (i, j) is the grid point in this region] 5 days before and after a storm passes,$P({{T}},t)$ and$P({{T}} + \delta {{T}},t)$ represent the MSLP at time t without and with the forcing error δT, respectively. For simplicity, we refer to the former as an unperturbed run and the latter as a perturbed run. For each TC case, the MSLPs are calculated at 3 hrs intervals during the 120 hrs (i.e. 5 days) simulation. Thus, there are 41 MSLPs calculations in each run. The cost function J represents the sum of the deviations of these MSLPs in the perturbed run from those in the unperturbed run measured by absolute values, which indicates the total error of the TC intensity simulation during the 120 hrs. Here, the PSO algorithm is utilized to solve Eq. (5) and the NFSV ($\delta {{T}}^*$ ) denotes the SST forcing error that has the largest effect upon TC intensity, which is referred to as NFSV-type SST forcing errors hereafter for simplicity.The NFSV-type SST forcing errors are calculated for the predetermined 12 TC cases and plotted in Fig. 1. It is shown that, although the tracks of these 12 TC cases differ a lot from each other, the NFSV-type SST forcing errors are always along the TC tracks. This indicates that the SST errors in the areas along the TC tracks, compared with those in the areas away from the TCs, are likely to exert a greater influence the TC intensities. Although the NFSV-type SST forcing errors are located along the TC tracks, they display the largest anomalies in different time periods (e.g. from 24 h to 48 h for Soulik, from 60 h to 96 h for Rammasun, and so on; see Fig. 1) of the different TCs. Recalling the definition of the aforementioned NFSV scheme, the NFSV-type SST forcing errors represent the forcing errors that result in the largest TC intensity simulation errors over the 120 hrs. Then the distribution of the NFSV-type SST forcing error along the TC track may indicate that the TC intensity simulations that are significantly sensitive to the SST forcing errors occurring in a particular time period of TC movement.
To clarify the above sensitivity of the NFSV-type SST forcing errors, we propose the following experiment. It is known that the ensemble spread is often used to measure the sensitivity. As such, we select 22 SST forcing perturbations randomly from the predetermined 198 perturbations (see section 2.3) to form an ensemble and scaled them to have an amplitude of 1 K, measured by
$\left\| {\delta {{T}}} \right\| = $ $ \sqrt {\displaystyle\sum {\delta {{{T}}^2}(i,j)/N} } $ [see Eq. (5)], which is the same amplitude as in the NFSV-type forcing errors, for evaluating the sensitivity. Interestingly, when we selected more SST forcing perturbations in this experiment, the result did not change. Therefore, we just use these pre-determined 22 SST forcing perturbations to describe the result. The 22 SST forcing perturbations are superimposed upon the unperturbed SST forcing fields (used in the unperturbed run) for TCs during the period of [t0, t0+6], with t0 being 0 h, 6 h, 12 h, ..., 114 h and t0+6 indicating the period lengthy being 6 hours, respectively. For each t0, after the WRF model is integrated with the unperturbed SST forcing from 0 h to t0 h, the 22 SST forcing perturbations are then superimposed during the interval, [t0, t0+6]. Then, the differences of the TC intensities between the unperturbed run and 22 perturbed runs during the interval, [t0, t0+6] are obtained. The ensemble spread of the 22 perturbed runs in the TC intensity represents the sensitivity of the TC intensity during the interval, [t0, t0+6] for the 22 SST forcing perturbations. With a changing t0, it is conceivable that the ensemble spread in each 6 hrs can reveal the period of the TC movement, or evolution, that exhibits the strongest sensitivity of TC intensity to the 22 SST forcing perturbations. The spreads in each 6 hrs of all 12 TCs are plotted in Fig. 2. Clearly, the largest spreads occur in different time periods for different TCs. This indicates that the sensitivity of TC intensity to the SST forcing perturbations is dependent on the TCs themselves. Nevertheless, it is found that such sensitivity dependence upon TCs fits the distribution of the NFSV-type SST forcing errors well along the TC track (see Fig. 3). The details are as follows.Figure 2. The spread of TC intensity simulations perturbed by 22 SST forcing perturbations during [t0, t0+6] with t0 being 0 h, 6 h, 12 h, ..., 114 h and the TC intensity of the unperturbed run. Red lines denote the TC intensity [indicated by minimum sea level pressure (MSLP); units: hPa] of the unperturbed run. The blue bars represent the spread of the 22 perturbed runs of the TCs during [t0, t0+6] (units: hPa).
Figure 3. The regionally-averaged NFSV-type SST forcing errors at t0 as a function of the spreads of TC intensity simulations perturbed by the 22 SST forcing perturbations during [t0, t0+6] with t0 being 0 h, 6 h, 12 h, ..., 114 h. The red line represents the linear regression line between NFSV-type SST forcing errors and spreads.
The regionally-averaged NFSV-type SST errors of the selected 12 TCs, within a radius of 300 km, centered at the central location of the TC, and at t0 are respectively calculated, which is plotted in Fig. 3 as a function of the spread of the 22 perturbed runs in TC intensity for each TC during the interval, [t0, t0+6]. With the change of t0, the linear regression between the regionally-averaged NFSV-type SST forcing errors and the ensemble spread is calculated (see the line in Fig. 3). It follows that the regionally-averaged NFSV-type SST forcing errors are significantly correlated with the spread of the intensity in the perturbed runs of the TCs, which demonstrates significance at the 0.01 level when subjected to a t-test. That is to say, the larger the ensemble spread during one time period of TC, the larger the corresponding NFSV-type SST forcing errors. It is therefore obvious that the NFSV-type SST forcing errors can identify the time period when the TC intensity simulations are highly sensitive to the SST forcing errors.
The 22 randomly-selected SST forcing perturbations described above are further-superimposed on the unperturbed SST fields to force the TCs for the whole simulation period of 120 hrs. Then the total error of the TC intensity in each perturbed run, during the 120 hrs, is calculated for each TC. After this is done, the correlation coefficients are calculated at each grid point, for each TC between the total errors of the TC intensities in the perturbed runs and the corresponding member of the 22 SST forcing perturbations (see Fig. 4). Figure 1 shows that the distributions of the correlation coefficients are very similar to those of the corresponding NFSV-type SST errors. More specifically, the spatial correlation coefficients between them can be as large as 0.64, on average, for the 12 TC cases (the details can be seen in Table 2), indicating that the larger the ensemble spread particular to a TC location, the larger the NFSV-type errors there. It can also be seen that the correlation coefficients are much larger along the TC track, which implies that the total error of the TC intensity is especially sensitive to the SST forcing errors along the TC track. The similarity between the distributions of the correlation coefficients and the corresponding NFSV-type SST errors indicates that the NFSV-type SST errors link the sensitivity of the TC intensity simulation errors to the SST forcing errors in space. Furthermore, this relationship confirms that the SST errors in the areas along the TC tracks, especially those during the time period when the NFSV-type SST forcing errors attain large values, may significantly influence the TC intensities.
Figure 4. The spatial correlation coefficients (shaded) between the 22 SST forcing perturbations and the sum of the absolute values of intensity errors during the whole simulation period of 120 hrs for the selected 12 TC cases. The blue, green, yellow, red and purple dots represent the simulated TC intensity within [980, 1000], [970, 980], [960, 970], [950, 960] and [900, 950] (hPa).
Cases Soulik Utor Soudelor Rammasun Chanhom Bolaven Sanba Muifa Goni Halong ManYi Noul Cor. 0.82 0.48 0.41 0.58 0.82 0.55 0.64 0.60 0.59 0.78 0.72 0.72 Table 2. The spatial correlation coefficient between intensity errors and the 198 SST perturbations for 12 TC cases.
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It is clear now that the NFSV-type SST forcing errors can identify the particular time period of the TC movement when the TC intensities exhibit high sensitivity to the SST forcing errors; but such time periods are dependent on the individual TC cases. The issue then becomes, whether or not these different time periods for different TCs correspond to common physical and environmental states of TCs. That is to say, what physical and environmental factors determine the sensitivity displayed by the NFSV-type SST forcing errors? Since the NFSV-type SST forcing errors cause the largest simulation errors of the TC intensity, we choose to explore the contributing factors of the NFSV-type SST errors by addressing which states of TCs are favorable for the SST forcing errors that yield large TC intensity errors. We go on to explain the physical processes responsible for the TC state that is consistent with the formation of the NFSV-type SST forcing errors.
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We classify the lifetimes of TC movement into two categories of time periods according to the sensitivities of the simulated TC intensities in perturbed runs to the SST forcing perturbations. Specifically, for the selected 22 SST forcing perturbations and the time periods [t0, t0+6] as in section 3, if a time period possesses a spread (among the 22 perturbed runs in TC intensities) greater than 1.5 hPa (i.e. the mean value of the ensemble spreads during [t0, t0+6] with the changing t0), this time period is categorized as a relatively high sensitivity period (referred to as “H-Sen” hereafter); conversely, the other time periods are categorized as relatively low sensitivity periods (referred to as “L-Sen” hereafter).
For the environmental factors, the SST, relative humidity (RH), vertical wind shear (VWS), and translation speed are considered. During the period [t0, t0+6] (noting that t0 changes), all of the above environmental factors are calculated at t0, t0+3 hrs, and t0+6 hrs for the unperturbed run, respectively. At each of these timings, the SST is regionally-averaged in a circular domain with a radius of 300 km centered at the simulated TC center; it is then further averaged consistent with the three timings. This averaged SST represents the environmental SST of the TCs in the period [t0, t0+6]. The VWS is similarly calculated, but denotes the difference of the regionally-averaged horizontal wind between 200 hPa and 850 hPa; and the RH is vertically averaged in the layers between 1 km and 6 km in the aforementioned circular domain. The translation speed, is calculated by taking the difference of the centers of the TC at both t0 and t0+6 hrs and then dividing this distance by the 6 hrs time interval. All of the above calculations are classified according to H-Sen and L-Sen and the results are shown in Table 3. It is found that the TCs in the H-Sen periods, compared with those in the L-Sen periods, have much more humid inner-cores, move profoundly slower over warmer SSTs, and are accompanied by stronger VWS. In previous studies, all of these environmental factors were shown to benefit the intensification of TCs (Mei et al., 2012; Zhang and Tao, 2013; Tao and Zhang, 2014; Walker et al., 2014; Torn, 2016; Zhao and Chan, 2017). Nevertheless, when we calculate the correlation coefficients between the spread of the 22 perturbed runs in TC intensity during the period [t0, t0+6], with the changing t0, and the environmental factors of SST, RH, VWS, and translation speed in the unperturbed run, they are shown to be very low and thus weakly correlated [see Fig. 5 (a−d)]. This suggests that the environmental factors of TCs cannot be responsible for the relatively high sensitivity in the H-Sen period of TCs.
SST RH VWS Speed MSLP W Vr I2 H-Sen 302.0 66.0 8.10 19.82 957.9 0.26 −5.9 0.0042 L-Sen 301.2 53.0 6.50 26.42 965.9 0.17 −4.2 0.0029 P-value 0.0005 0.02 0.0104 0.0007 0.0016 1.8×10−6 2.1×10−5 1.1×10−5 Note: MSLP represents minimum sea level pressure (units: hPa) of the TCs; W, Vr, and I2 denotes corresponding vertical velocity (units: m s−1), radial velocity (units: m s−1), and inertial stability (units: s−2), respectively; and SST, VWS, Speed, and RH represents the absolute value of sea surface temperature (units: K), vertical wind shear (units: m s−1), translation speed of TC (units: m s−1), and the relative humidity (%). P-value indicates the significance level of the differences between the values in H-Sen and those in L-Sen. Table 3. TC states and their environmental factors during H-Sen and L-Sen
Figure 5. The scatter plots between spreads of TC intensity simulation yielded by the 22 SST forcing perturbations and corresponding (a) SST forcing (units: K), (b) RH (units: %), (c) translation speed (units: km h−1), (d) VWS (units: m s−1), (e) vertical velocity (units: m s−1), (f) inflow (units: m s−1), (g) inertial stability (units: 10−3 s−2) and (h) MSLP (units: hPa). The red lines represent the linear regression. The red dots are for the H-Sen period and the black dots are for the L-Sen periods. The SST, RH, translation speed, VWS, vertical velocity, inflow, inertial stability and MSLP are calculated as in Table 3 (see section 4).
The characteristics of the TCs themselves in the H-Sen and L-Sen periods are also examined and associated inflow, vertical velocity, and inertial stability influencing TC intensity are calculated, respectively. Specifically, the inflow is calculated by taking the regionally-averaged radial component of the horizontal wind in a circular area centered at the simulated TC center, with the radii between 50 km and 300 km and vertically-averaged from 0 km to 1 km; the vertical velocity is estimated by a similar scheme but estimates the vertical wind by taking the vertically- and regionally-averaged component from 0 km to 15 km within a circular area centered at the simulated TC center with the radii between 50 km and 150 km; and the inertial stability is calculated by the formula
${I^2} = (f_c^{} + \xi)({f_c} + 2v/r)$ (where fc is the Coriolis parameter,$ v $ represents the tangential wind velocity,$\xi $ denotes the relative vorticity, and$ r $ is the radius), which is then vertically and regionally-averaged from 0 km to 15 km and in a circular area, centered at the simulated TC center, with a radius of 100 km. The above results are also listed in Table 3 according to H-Sen and L-Sen. It is shown that the TC intensity in the H-Sen period is about 957.9 hPa on average, which is stronger than the average of 965.9 hPa in the L-Sen, and is found to be significant at the 0.0016 level using a t-test. This indicates that the TC intensities show higher sensitivity to the SST forcing errors when they are much stronger. We also note similar results from Table 3 with respect to the inflow, vertical velocity, and inertial stability; that is, these variables are much larger in H-Sen period, which is consistent with the presence of relatively strong TCs in the H-Sen periods. Furthermore, we calculate the correlation coefficients between the spread of the 22 perturbed runs in TC intensity during the period [t0, t0+6] with the changing t0 and corresponding inflow, vertical velocity, inertial stability, and MSLP of the unperturbed run. We find that they, compared to those for the environmental factors, are more significantly correlated. This indicates that the TC intensities are much more sensitive to the SST forcing perturbations when the TCs exhibit strong intensity, larger inflow, large vertical velocity, and strong inertial stability (see Fig. 5). Moreover, from Fig. 2, it is shown that the TC cases tend to have the largest forecast spread when the TCs exhibit relatively strong intensity but are still intensifying, just prior to being fully mature, with the notable exception of TC Noul. To facilitate the discussion, such spatio-temporal period is hereafter called the Critical Time and Phase (CTP).When we further examine the corresponding secondary circulation and inertial stability, it is found that the CTP period shows strong secondary circulation but the highest inertial stability for TCs. As an example, Fig. 6 plots the evolution of inertial stability, inflow, and vertical velocity of the TC Soulik (201307). It is shown that the inertial stability and the vertical velocity reach the largest values during the time period from 24 h to 48 h, while the inflow is still increasing in this period. Here, the time period from 24 h to 48 h fits the CTP of TC Soulik (also see Fig. 2). Therefore, we conclude that the strongest sensitivity of TC intensities to SST forcing perturbations occurs during the CTP of the TCs, especially with a TC state of the highest inertial stability and the strong secondary circulation. Such time periods can also be seen in Fig. 1, i.e. the period with the deepest color in the shaded area, during which the SST errors associated with the NFSV are of the largest assigned values. Particularly, the time period with the deepest color in the shaded area for the TC Soulik is about from 24 h to 48 h, which coincides with the CTP of the TC Soulik Therefore, the NFSV-type SST forcing errors locating at the TC track not only describe the sensitivity of the TC intensities to the SST forcing errors (see section 3) but also capture the time period of the TC movement when the TC intensity presents the strongest sensitivity to the SST forcing errors. Based upon these results, we advance the hypothesis that the TC states of high inertial stability and the strong secondary circulation are responsible for the NFSV-type SST forcing errors being dominated by the errors occurring during the CTP.
Figure 6. The evolution of (a) inertial stability (units: 10−3 s−2), (b) inflow (units: m s−1) and (c) vertical velocity (units: m s−1) for the TC Soulik. The inertial stability is regionally-averaged in a round area centered at the simulated TC center within 50 km; the inflow is an azimuthal mean of radial velocity, which is vertically-averaged in the layers between 0 km and 1.5 km; and the vertical velocity is vertically-averaged in the layers between 3 km to 7.5 km. The black lines in (b) and (c) denote the RMW at 2.0 km.
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The eleven TC cases with the strongest sensitivity in their respective CTP are used to address the nature of the physical mechanisms which explain the enhanced sensitivity. To facilitate the discussion, we take TC Soulik as an example. It has been previously mentioned that the distribution of the NFSV-type SST forcing errors reflects the sensitivity of TC intensities to the SST forcing errors; and the NFSV-type SST forcing errors tend to have the largest anomalies during the time period from 24 h to 48 h of TC movement when the TC is during the CTP (see the NFSV-type SST forcing error pattern for the TC Soulik in Fig. 1). This time period is coincident with the strongest sensitivity of TC intensity on the SST forcing errors and is particularly associated with a strong secondary circulation and very highest inertial stability of the unperturbed run. The issue of concern is whether or not the along-track SST forcing errors that occurred during this unique and relative stage and timing in TC evolution, significantly differ from the response of TCs that exist at another timings in the TC growth cycle.
To address this concern, we conduct another group of experiments for intensity simulations of TC Soulik. We superimpose an artificial SST forcing error to the unperturbed SST forcing field at each grid point, where the SST error at each grid point is of uniform magnitude of −1.0 K, which has the same amplitude as in the NFSV-type SST forcing errors calculated by
$\left\| {\delta {{T}}} \right\| = \sqrt {\displaystyle\sum {\delta {{{T}}^2}(i,j)/N} } $ [see Eq. (5)]. We integrate the WRF for 120 hrs and obtain a perturbed simulation of the intensity of TC Soulik, hereafter as “UN-all” simulation. In addition, we also superimpose the above SST forcing error only during the H-Sen period of the TC Soulik (exactly, as its identified from 24 h to 96 h by the criterion determining the H-Sen; see section 4.1 and hereafter as “UN-Sen”) and then, only during the L-Sen periods (i.e. the period from 0 h to 24 h and from 96 h to 120 h; hereafter as “UN-non-Sen”), respectively. In this way, we may obtain two additional perturbed runs of the TC Soulik intensity forecast. Comparison is then made among these three perturbed runs to determine whether or not the SST forcing errors in H-Sen period causes a significantly larger intensity error compared to that in L-Sen period.The errors of the TC intensities in the experiments of UN-all, UN-Sen, and UN-non-Sen with respect to the unperturbed run are shown in Fig. 7a. It can be seen that the error of the TC intensity in the UN-Sen experiment is significantly larger than that in the UN-non-Sen experiments and accounts for a forecast error of almost 80% compared to that in the UN-all experiment (Fig. 7b). This indicates that the total error of TC intensity in the UN-all experiment is mainly caused by the SST forcing errors during the H-Sen period. In addition, we can notice from Fig. 7a that the growth rate of the TC intensity error in the UN-Sen experiment is much larger than that in the UN-non-Sen experiment. Specifically, during the time periods from 0 h to 24 h and from 96 h to 120 h of the L-Sen periods in the UN-non-Sen experiment, the intensity errors are constrained to be than 6 hPa; while in the UN-Sen experiment, the intensity errors increase by about 10 hPa during the time period from 24 h to 48 h of the H-Sen period, which is the equivalent time interval of 24 hrs as the former two periods(from 0 h to 24 h and from 96 h to 120 h, in the L-Sen period). Furthermore, the time period from 24 h to 48 h of the H-Sen period coincides with the interval when the NFSV-type SST forcing error of the TC Soulik achieves the largest values and the TC intensities are most sensitive to the SST forcing perturbations. The rapid increase of the intensity error during the time period from 24 h to 48 h can also be seen in the UN-all experiment if one uses the slope of the error evolutionary curve to measure the error growth (see Fig. 7a). In particular, the time period from 24 h to 48 h shows the fastest growth of intensity error for the TC Soulik in the UN-all experiment, therefore, contributing the most to the total error of the TC intensity during the entire simulation period. This may explain why the NFSV-type SST forcing errors occur in the H-Sen period, particularly during the time period from 24 h to 48 h in the case of TC Soulik.
Figure 7. (a) The evolutionary curves of TC intensity of unperturbed run (black line) for the TC Soulik and its error evolutionary curve of TC intensity perturbed by the NFSV-type SST forcing error (orange); and the error evolutionary curves of TC intensity in the UN-all (blue), UN-Sen (purple), and UN-non-Sen (red) experiments; (b) the total error of the intensity [see Eq. (5)] for the TC simulation perturbed by the NFSV-type SST forcing error (orange), and the TC simulation in the UN-all (blue), UN-Sen (purple), and UN-non-Sen (red) experiments. Note that the maximum intensity error measured by MSLP forced by NFSV-type SST forcing errors can reach up to about 35 hPa during 72−96 h, which is comparable with the 23 hPa of the annual mean of root mean square errors (RMSEs) for TC center pressure forecasts with lead time 72 hrs.
Next, we will explore how the SST forcing errors in the H- and L-Sen periods influences TC intensity by perturbing the processes influencing TC intensity. To facilitate the calculation, we select only the 18 h, 114 h, and 42 h of the unperturbed run as representative of two L-Sen and one H-Sen periods to calculate the TC processes influencing TC intensity and associated simulation errors, where these three timings are all relative to the initial time of their respective time periods for 18 hrs.
We plot in Fig. 8 the azimuthal mean of surface latent heat flux errors, water vapor errors, and diabatic heating errors at 18 h and 114 h in the UN-non-Sen experiments, and at 42 h in the UN-Sen experiments. It is shown that the surface latent heat fluxes at all three timings yield negative errors, which are certainly caused by the negative SST forcing error of magnitude of −1 K. Nevertheless, the surface latent heat flux error measured by its absolute value at 42 h in the UN-Sen experiment is larger than those at 18 h and 114 h; noting that especially large errors at 42 h occur closer to the inner core, which indicates that the significant decreases of surface latent heat flux in the UN-Sen experiments are due to the effects of the negative SST forcing errors and that the primary area of decrease occurs close to the inner core. According to the theory of wind induced surface heat exchange (WISHE; Emanuel et al., 1994), the surface latent heat flux is related to the wind speed associated with the TC intensity. Therefore, for the same magnitude of SST forcing errors, the stronger intensity of the TC in the unperturbed run will enhance the surface latent heat flux errors induced by the SST forcing errors, which explains why the surface latent heat flux errors in the UN-Sen experiments are larger than those in the UN-non-Sen experiments. The larger decrease of the surface latent heat flux in the UN-Sen experiments results in reduced water vapor in the boundary layer (especially closer to the inner core) of TC area by error, which will reduce the water vapor in the upper layers by the advective vertical term in the model,
$\overline w q'$ ($\overline w $ indicates the vertical velocity of the unperturbed run and$q'$ denotes the negative water vapor error in lower layer, i.e. the reduced amount of water vapor in lower layer). Then the stronger secondary circulation of the TC in the unperturbed run during the H-Sen period will go on to favor a decrease of water vapor in the upper layers in the UN-Sen experiments, which reduces the diabatic heating compared to that in the UN-non-Sen experiments. As a result, the secondary circulation of the TC in the UN-Sen experiments is weaker at 42 h than that in the UN-non-Sen experiment due to the reduction of the diabatic heating [see Fig. 8 (d−f)], which, together with the weakened vertical velocity closer to the inner core, causes a large decreases of subsidence in the inner core, especially in the upper layer. The reduced subsidence by errors in the UN-Sen experiment indicates that the entry of high entropy air parcels into the warm core will be inhibited, which will cause considerable negative errors regarding the potential temperature in the upper layers in the UN-Sen experiment compared to that of the UN-non-Sen experiment. It is well known that there is a significant inverse correlation between TC warm core strength measured by the potential temperature in the upper layers and MSLP amplitude (indicating TC intensity); consequently a stronger warm core corresponds to stronger TC intensity (Chang and Wu, 2017). Thus, the TC intensity, as indicated by MSLP, is much lower in the UN-Sen experiment than in the UN-non-Sen experiments [see Fig. 8 (e)], indicating that the SST forcing error, with magnitude of −1 K at each grid point in the UN-Sen experiment, tends to cause much larger simulation errors of the intensity of TC Soulik. This mechanism may explain why the SST forcing errors in the H-Sen period influence the TC intensity more significantly, and shed light on the physical reason behind the formation of the NFSV-type SST forcing errors.Figure 8. TC Soulik: azimuthal mean of surface latent heat flux errors (black lines, units: W m−2), water vapor errors (shaded, units: g kg−1), and diabatic heating errors (red contours, units: K h−1, contour interval: 0.5 K h−1) are plotted at (a) 18 h, (b) 42 h, and (c) 114 h; azimuthal mean of potential temperature errors (shaded, units: K), vertical velocity errors (red contours, units: m s−1, contour interval: 0.1 m s−1), and inflow errors (blue contours, units: m s−1, contour interval: 1 m s−1) are shown at (d) 18 h, (e) 42 h, (f) 114 h.
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The results in Table 3 and Fig. 5 show that the greatest sensitivities of TC intensity on SST forcing errors are associated with conditions of high inertial stability. We then pose the following two questions. How does the inertial stability affect TC intensities and through what mechanism does the high inertial stability in H-Sen period cause the SST forcing errors to yield such large intensity errors? Here, we show that the high inertial stability in the H-Sen period is favorable for the large growth of errors in TC intensity that is caused by the SST forcing errors. Furthermore, we suggest that this is caused by the response of the secondary circulation to the heat forcing. In the developments to follow, we address this issue by analyzing the Sawyer-Eliassen (SE) equation (Montgomery et al., 2006; Chen et al., 2018) through sensitivity experiments of the unperturbed run associated with TC Soulik with respect to the TC state in the UN-Sen experiment.
The application of the Boussinesq approximation, hydrostatic equilibrium and gradient wind balance, to the SE equation yields the transverse streamfunction which is written as follows (Montgomery et al., 2006; Chen et al., 2018).
here, r and z represent the radius and height, respectively; the overbar denotes the azimuthal mean;
$\psi $ is streamfunction related to radial and vertical velocities with$u = - \left({1/r} \right)\left({\partial \psi /\partial z} \right)$ and$w = (1/r)(\partial \psi /\partial r)$ , respectively; the coefficients A, B and C describe static stability, baroclinity, and inertial stability, which representing TC stability, and are given as follows.where
$\overline \theta $ ,$\overline v $ and$\overline \eta $ are azimuthal mean potential temperature, tangential velocity and absolute vertical vorticity, and g and${\theta _{\rm{0}}}$ are the gravitational acceleration and the reference potential temperature (300 K), respectively;$\overline {{N^2}} $ is the azimuthally averaged Brunt-Vaisala frequency and$\overline \xi = {f_c} + 2\overline v /r$ is the local Coriolis parameter. The heating and momentum forcing terms on the rhs of Eq. (6) are defined as:where the prime is the deviation from azimuthal mean;
$\dot \theta $ is diabatic heating rate. The parameters A, B, C,$\overline Q $ and$\overline F $ , are calculated according to the output of the WRF. Obviously, when we know the TC stability [i.e., A, B, and C in the Eqs. (7)−(9)] and the heating$\overline Q $ and momentum sources$\overline F $ , the SE equation can be solved by standard successive over-relaxation (SOR; Press et al., 1992), where the momentum sources,$\overline F $ , is the term associated with asymmetric advection, friction, subgrid-scale processes, and interpolation errors and so on.The TC state at 42 h in the UN-Sen experiment (see the last sub-section) is used as input for the SE equation with the intent of diagnosing the role of inertial stability in response to the secondary circulation that is driven by diabatic heating. We replace the coefficient C, originally specific to the inertial stability of the unperturbed run, with that which is specific to the UN-Sen run. This adjustment yields resultant vertical and radial velocities that are shown in Fig, 9. It is shown that the vertical velocity in the convection area and the inflow and outflow measured by the radial velocity are all weaker, with the vertical velocity being especially weaker. In addition, it can be seen from Fig. 9 that the subsidence in the inner core becomes weaker. Collectively, these factors indicate that the secondary circulation becomes much weaker with the replacement of C (denoting the inertial stability) in the UN-Sen experiment while holding the TC states of static stability A, baroclinity B, heating forcing
$\overline Q $ , and momentum sources$\overline F $ unchanged. We further note that, replacing coefficient A (static stability) has little effect on the secondary circulation, and replacing coefficient B (baroclinity) only slightly affects the response of the secondary circulation in the boundary layer (corresponding figures are omitted here). We conclude that the inertial stability of the unperturbed run modulates the response of secondary circulation to the diabatic heating; and that the higher the inertial stability of the unperturbed run, the stronger the secondary circulation responds to the diabatic heating. To further clarify, high inertial stability will enhance the perturbation, regardless of whether it is positive or negative. In the last sub-section, we have shown that the secondary circulation in the UN-Sen experiments for TC Soulik becomes weaker due to the effect of negative SST forcing errors. From the results shown here, it is inferred that the high inertial stability of the unperturbed run in the H-Sen period enhances the weakening of secondary circulation in the UN-Sen experiments and a reduction of vertical velocity, which will cause considerable errors favoring for weaker subsidence which then leads to much greater negative errors regarding the potential temperature in the warm core. Ultimately, this results in a positive error of the MSLP which leads to an under-estimation of simulated TC intensity. Consequently, the high inertial stability of the unperturbed run during the H-Sen is favorable for the growth of the TC intensity errors caused by the SST forcing errors that occurred during this period, which is the suggested mechanism that contributes to the formation of the NFSV-type SST forcing errors.Figure 9. The azimuthal mean of (a) vertical velocity (units: 0.06 m s−1) and (c) inflow (units: m s−1) in the SE equation for the unperturbed run TC Soulik at time 42 h; the differences of the SE solution in (b) vertical velocity (units: 0.006 m s−1) between the run in (a) and that with the coefficient C in the UN-Sen experiment and in (d) the inflow (0.1 m s−1) between the run in (a) and that with the coefficient C in the UN-Sen experiment.
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In section 4, we showed that the NFSV-type errors occur during the time period (i.e. the H-Sen period above) when the TCs present strong intensities,strong secondary circulations and high inertial stability. Moreover, the NFSV-type SST forcing errors tend to have the largest anomalies when the TCs are in the CTP, which identifies the time period when the intensity of TCs exhibit the strongest sensitivity to the SST forcing errors. In sections 4.2 and 4.3, we analyze TC Soulik as an example to explain why the aforementioned TC states are favorable for the growth of the intensity errors which are caused by the NFSV-type SST forcing error and we furthermore advance a physical mechanism which logically explains the the response of the secondary circulation to the NFSV-type SST forcing error. All evidence leads to the conclusion that the strongest sensitivity of TC intensity to the SST forcing errors along the TC track during the period when the TCs are of the CTP state. Therefore, if we manage to implement additional along-track SST observations during this particular time period, it would help to obtain a much more accurate SST forcing field for the WRF model, ultimately, reducing the simulation uncertainties of the TC intensity. If one uses a coupled model to simulate the TC, the along-track SST in this time period should be better simulated so as to greatly improve the TC intensity simulation skill.
In the present section, we continue to use the TC Soulik to explore how the NFSV-type SST forcing errors perturb the TC intensities by influencing the processes associated with TC intensification. From Fig. 7a, it is shown that, from about 24 h to 48 h, the intensity error caused by the NFSV-type SST forcing error produces larger growth rates (measured by the slope of the error evolutionary curve) than that in the UN-all experiment, which then causes larger intensity errors during the mature phase (i.e. from 48 h to 96 h) of TC Soulik despite the SST forcing errors in the UN-all experiment - having the same amplitude as in the NFSV-type SST forcing errors. We have known that the time period from 24 h to 48 h corresponds to the one when the NFSV-type SST forcing error possesses the largest anomalies and that the TC intensities are most sensitive to the SST forcing errors, which, as revealed in section 4, can explain why the NFSV-type SST forcing errors cause much larger intensity errors of a TC. We now pose the question, how does the NFSV-type SST forcing error influence the processes associated with the TC intensity and finally perturb the TC intensity?
Since we use the minimum MSLP to measure the intensity of TC, it is required to figure out the mechanism of the NFSV-type SST forcing error resulting in the change of the MSLP of the TC. As mentioned above, the MSLP of the TC is strongly correlated with the potential temperature in the upper layers of the TC. Therefore, we derive the potential temperature (PT) error tendency equation by examining the difference between the PT tendency equation component of the WRF model associated with unperturbed run and perturbed run mentioned above. The equation is given in Eq. (12):
Here, the overbar denotes the azimuthal mean, the star signifies the deviation from the azimuthal mean, the prime indicates the error caused by SST forcing error, and R is the residual term associated with unresolved processes as in section 4. However, for the residual term R, we cannot exactly separate the role of each of its inclusive processes. That is to say, the total effect of R is not of clear physics. For simplicity, we only considered the role of the terms with clear physics and do not analyze R here. The meanings of the other terms are listed in Table 4.
Term Description I The tendency of the azimuthal mean of PT error; II The azimuthal mean diabatic heating error; III The radial advection of the azimuthal mean of PT by the azimuthal mean of radial velocity error; IV The radial advection of the azimuthal mean of PT error by the azimuthal mean of the perturbed radial velocity; V The azimuthal mean of the eddy component error of radial advection; VI The vertical advection of the azimuthal mean of PT error by the azimuthal mean of the perturbed vertical velocity; VII The vertical advection of the azimuthal mean of PT by the azimuthal mean of the vertical velocity error; VIII The azimuthal mean of the eddy component error of vertical advection. Table 4. The meaning of the terms in the PT error tendency equation
With the NFSV-type SST forcing error disturbed, we calculate the terms of Eq. (12) (confined in the eye region of TC within a round area centered at the simulated TC center with a radius of 50 km) for the TC Soulik. We find that the terms II, V and VII are much larger, which indicates that the diabatic heating error, eddy process error, and vertical advection of the PT by the vertical velocity error play an important role in generating PT error. These three terms are plotted in Fig. 10. It is found that the term VII (i.e. the vertical advection of the azimuthal mean of PT by the azimuthal mean of vertical velocity error) is negative especially in the upper layers (i.e. about 9 km above the surface for TC Soulik). Out of the three terms, Term VII is the largest when measured according to absolute value. This indicates that the term VII plays a dominate role in affecting PT, further noting that the vertical advection denoted by the term VII contributes to suppressing PT growth, especially when the TC is still intensifying just prior to the mature phase [i.e. from 30 to 48 h, which is within the window of the most sensitive period from 24 h to 48 h identified by the NFSV-type SST error]. Since the positive direction of the vertical PT gradient vector is upward, the negative contribution of the term VII to the PT growth mainly results from the positive error of vertical velocity (see Fig. 10f). As we know, in the eye region of TC, the vertical velocity is generally downward. Thus, the positive vertical velocity error reduces downward advection, and limits the amount of high entropy air parcels that enter the warm core which results in a decrease of TC intensity, ultimately yielding a negative error of TC intensity. Compared to term VII, term V, the error of eddy component of radial PT advection is much smaller, which indicates the SST forcing error has a negligible effect upon the asymmetric structure of the vortex associated with TC. Term 2, the diabatic heating error, is also relatively small, which may be due to the latent heat release occurring in the convection region instead of the eye region. In fact, as shown in section 4, the important effect of diabatic heating error, in the convection region, upon TC intensity simulation is uncertain. Therefore, the change of diabatic heating induced by the NSFV-type SST forcing error affects the warm core of TC in an indirect way.
Figure 10. The time-height cross section of terms in potential temperature (PT) error tendency equation. PT error is caused by NFSV-type SST errors. All the terms are calculated by regional average within a round area centered at the simulated TC center with a radius of 50 km. (a) the term I; (b) the sum of rhs of Eq. (12) except the residual term; (c) the term II; (d) the term V; (e) the term VII; (f) the vertical velocity error (units: 0.1 m s−1) caused by NFSV-type SST errors.
We plot in Fig. 11a the evolution of the absolute value of negative latent heat flux errors caused by the NFSV-type SST forcing error for TC Soulik. It is obvious that the latent heat flux error increases from 0 h to 48 h, especially within the interval of 24 h to 48 h due to the increase in surface wind consistent with the strong intensity of the TC during this period. Correspondingly, the water vapor error shows similar evolutionary behavior, with the largest reduction during the same sensitive period, from 24 h to 48 h. Then the water vapor advected into the inner core and upper layers is reduced as a consequence of the error, which causes large negative diabatic heating errors in the upper layers especially during the mature phase (i.e. from 48 h to 96 h) of TC Soulik, where the strong secondary circulation of the unperturbed run, as clarified in section 4, enhances the diabatic heating error in the upper layers induced by the water vapor error. Fig. 11b shows negative diabatic heating errors and vertical velocity errors forced by diabatic heating. It is shown that both the vertical velocity and the secondary circulation weaken, due to the effect of negative diabatic heating error. Despite the fact that the maximum of the negative diabatic heating errors do not occur during the most sensitive period of TC Soulik (see Fig. 11b), the strongest inertial stability of the unperturbed run during this period (see Fig. 6a) triggers a response of the secondary circulation to the diabatic heating and significantly increases the errors occurring in the secondary circulation, ultimately decreasing the vertical velocity, to the largest extent, during the most sensitive period from 24 h to 48 h. Particularly, we can see from Fig. 6a that the inertial stability is highest during the period from 24 h to 48 h and correspondingly, the vertical velocity error is the largest during this period. In response, the subsidence in the inner core is increased by a positive error (see Fig. 11c), especially from 24 h to 48 h. This results in reducing potential temperature of the upper layers of the inner core which leads to negative errors in potential temperature which accumulate rapidly from 24 h to 48 h (see Fig. 11c).The potential temperature then exhibits oscillatory behavior during the mature phase (i.e. from 48 h to 96 h) before dropping quickly from 96 h to 120 h, which coincides with the evolutionary behavior of the TC intensity error caused by the NFSV-type SST forcing error shown in Fig. 7a. Thus, the above mechanism interprets how the NFSV-type SST forcing errors perturb the intensity error.
Figure 11. Evolution of (a) surface latent heating errors (black line; units: W m−2) and water vapor errors (shaded; units: g kg−1) regionally-averaged in a round centered at the simulated TC center within a radius of 300 km, (b) vertical velocity errors (red contours; units: m s−1, contour interval: 0.1 m s−1) and diabatic heating errors (shaded; units: 10 K h−1) regionally-averaged in a ring area centered at the simulated TC center with the radii between 50 km and 150 km, and (c) subsidence errors (shaded; units: 0.1 m s−1) and potential temperature errors (blue contours; units: K, contour interval: 2 K) regionally-averaged in a round centered at the simulated TC center within a radius of 50 km, which are all yielded by NFSV-type SST forcing errors.
Apart from the TC Soulik, we also explore the other ten TC cases with the strongest sensitivity occuring in the CTP of TCs. They have mechanisms similar to TC Soulik in the NFSV-type SST forcing errors affecting the TC intensity, except that some cases show signs opposite to those of the TC Soulik in the NFSV-type SST forcing errors and their resultant TC intensity uncertainties. Therefore, we can summarize the physical mechanisms as follows. When a NFSV-type SST forcing error with negative anomalies (or positive anomalies, depending on TC cases) occurs, it causes the surface latent heat flux and water vapor in low layers to decrease (increase). This further leads to a reduction (an increase) in diabatic heating which forces a weaker (stronger) secondary circulation and causes the downward vertical velocity in the eye region to decrease (increase). Eventually, the warm core of the TC becomes weaker (stronger) and the TC intensity tends to be under- (over-) estimated. In this process, the unperturbed run of the TC tends to present relatively strong intensity with the highest inertial stability and strong secondary circulation during the time period when the NFSV-type SST forcing error occurs. This property of the unperturbed run greatly enhances the latent heat flux errors in the low layers due to the effect of SST forcing error which goes on to increase the diabatic heating error, thereby, promoting a significant response from the secondary circulation before finally increasing the TC intensity error, most notably during the CTP.
Name | No. | Start time (h; UTC) | End time (h; UTC) | Time of TC mature phase (h; UTC) | TC intensity at mature phase |
Soulik | 201307 | 0000 Jul 08 | 0000 Jul 13 | 0000 Jul 10 | 925 hPa; 50 m s−1 |
Utor | 201311 | 0000 Aug 10 | 0000 Aug 15 | 1200 Aug 11 | 925 hPa;53 m s−1 |
Soudelor | 201513 | 0000 Aug 02 | 0000 Aug 07 | 1800Aug 03 | 900 hPa;57 m s−1 |
Rammasun | 201409 | 0000 Jul 13 | 0000 Jul 18 | 0600 Jul 18 | 935 hPa; 45 m s−1 |
Chanhom | 201509 | 0000 Jul 06 | 0000 Jul 11 | 1800 Jul 09 | 935 hPa; 45 m s−1 |
Muifa | 201109 | 0000 Jul 29 | 0000 Aug 03 | 1800 Jul 30 | 930 hPa; 48 m s−1 |
Bolaven | 201215 | 0000 Aug 23 | 0000 Aug 28 | 1200 Aug 25 | 910 hPa; 50 m s−1 |
Sanba | 201216 | 0000 Sep 12 | 0000 Sep 17 | 1800 Sep 13 | 900 hPa; 55 m s−1 |
Goni | 201515 | 0000 Aug 15 | 0000 Aug 20 | 0600 Aug 17 | 935 hPa; 48 m s−1 |
Halong | 201411 | 0000 Aug 02 | 0000 Aug 07 | 1200 Aug 02 | 920 hPa; 53 m s−1 |
Man-Yi | 200704 | 0000 Jul 11 | 0000 Jul 16 | 1200 Jul 12 | 930 hPa; 48 m s−1 |
Noul | 201506 | 0000 May 07 | 0000 May 12 | 0000 May 10 | 920 hPa; 55 m s−1 |
Note: The numbers (No.) and intensities are from the best-track data of JMA (Japan Meteorological Agency), the latter of which include the minimum sea level pressure and maximum surface wind speed at the corresponding time. |