Quantifying the Rightward Bias Extent of Tropical Cyclones' Cold Wakes

Although the cold wake of tropical cyclones (TCs) is recognized to be typically rightward biased relative to TC track (in the North Hemisphere), it remains unclear to what extent the rightward bias should be for specified TCs. Based on numerical simulations and observational statistics, this study shows that the distance of cold wake rightward bias is closely related to latitude, translation speed and radius of maximum wind (RMW) of a TC, but is independent from maximum surface wind and radial wind profile outside the RMW, namely TC intensity and size. A semi‐empirical equation is derived to represent the rightward bias extent of cold wakes. For TCs with slower translation speed, smaller RMW, or at higher latitudes, the cold wakes are less asymmetric relative to the TC track. Atmospheric simulations also show that the TC tends to be weaker and more asymmetric as the cold wake is closer to TC center.

Another important feature of the cold wake is its asymmetry relative to the TC track, which is typically rightward biased in the Northern Hemisphere but leftward biased in the Southern Hemisphere. Price (1981) illustrated that the near-resonance between the turning rate of wind stress vector and that of the inertial motion is largely responsible for such a phenomenon. This has been the theoretical foundation for the cold wake asymmetry relative to the TC track (e.g., Huang & Oey, 2015). Recently, Pun et al. (2018) noted that the right bias in sea surface cooling induced by Supertyphoon Megi (2010) would be 60% less if it were not for Megi's size increase, implying that the cold wake asymmetry could also be related to TC wind feature. Although the rightward bias of cold wakes has been a consensus of the community given horizontal uniform oceanic environment, it remains unclear how asymmetric the cold wake would be for a specified TC case. Intuitively, the maximum cooling of cold wake should be located right to the TC center, but it certainly cannot be outside the TC regime. What factors specifically control the rightward bias of the cold wake? How is a cold wake rightward-biased given specified TC and local environment? These issues are still open.
The azimuthal asymmetries are inherent characteristics of TCs throughout their life cycles (e.g., Krishnamurti et al., 2005;Lee & Wu, 2018;Nolan et al., 2007;Persing et al., 2013;Qin et al., 2016;Zhang et al., 2021). Wu et al. (2005) noted that the symmetric and asymmetric components of cold wakes can contribute to TC intensity and asymmetry effectively. Another question this study aims to address is whether the asymmetric structures of TCs is sensitive to the rightward bias extent of cold wakes. To address these issues, the asymmetry of cold wake as well as its feedback to TC asymmetry is examined in this study. The data and methodology are introduced in Section 2. Section 3 presents the main results, with the conclusions provided in Section 4.

Oceanic Simulations
Numerical simulations are conducted to evaluate parameters setting the asymmetry of cold wakes, using the three-dimensional Price-Weller-Pinkel (3DPWP) model (Price et al., 1994). The model is configured with dimensions of 200 × 200 and horizontal resolutions of 5 km × 5 km. A total of 45 levels are placed in the vertical direction, with 20 layers in the upper 100 m. The temperature and salinity fields are set horizontally homogeneous as default, in which the salinity is also vertically constant. The model is set on an f plane with the latitude set uniformly at 20°N unless otherwise mentioned. A modified Rankine vortex with a maximum surface wind of 60 m s −1 at a radius of 60 km is assigned. The radial wind profile is given by: where V is tangential wind, V m is maximum surface wind, r is radius, and α is a ratio of −0.75 that modulates wind profile outside the radius of maximum wind (RMW). The vortex is set to translate northward from the bottom of the domain at a speed of 6 m s −1 . The surface drag coefficient levels off at high winds (larger than 25 m s −1 ) for the calculation of surface wind stress. A total of 53 sensitivity experiments are conducted to examine the sensi tivity of cold wake asymmetry to TC characteristics in terms of intensity, translation speed, RMW, and radial wind profile outside the RMW, and environmental characteristic in terms of latitude. More details are listed in Supporting Information S1 (Table S1).
To quantify the cold wake asymmetry, the distance of rightward bias is calculated as follows: the location of maximum surface cooling is first searched and recorded every 6 hr; since the sea surface cooling induced by TCs reaches its maximum 1-2 days after the TC passage on average (e.g., Dare & McBride, 2011;Lloyd & Vecchi, 2011;Ma et al., 2017), a temporal averaging of 48 hr is taken to derive the rightward bias extent of the cold wake.

Observational Data Set
Observational data samples are also used for a further examination. The TC best track data in terms of 6-hr TC location and translation speed are obtained from the International Best Track Archive for Climate Stewardship (IBTrACS), using the World Meteorological Organization source. The daily SST product is based on the optimally interpolated microwave and infrared Remote Sensing Systems product, which is at a resolution of ∼0.09°, available since June 2002. The time period from 2002 to 2021 is chosen to identify TCs' cold wakes in the Northern Hemisphere. The data samples with the maximum cooling occurred in right of TC tracks are gathered, with a total of 44,538 data samples.

Atmospheric Simulations
The Cloud Model 1 (CM1) release 20.1 is used to simulate cold wake feedback to TC intensity and asymmetry with a fixed cold wake pattern but differing cold wake positions (Bryan & Fritsch, 2002). The model is configured with 800 × 800 dimensions with horizontal resolutions of 3 km × 3 km. A total of 49 levels are placed as default in the vertical direction. The details are introduced in Supporting Information S1. A total of four experiments are conducted ( Table S2 in Supporting Information S1): the first one integrates for a total of 240 hr with a constant SST of 29°C, denoted as CONST; the experiment with the SST anomaly same as the 3DPWP simulation is considered as the control run, denoted as CONTROL; the experiment with the cold wake artificially shifted leftward, namely right below the storm center, is denoted as CENTER; the experiment with the cold wake further shifted rightward, with the distance of rightward bias being doubled, is denoted as DRIGHT. The position of SST anomaly in each cold wake experiment is adjusted every 6 hr according to modeled storm centers, which are calculated directly by the CM1 model.

Theoretical Deduction
As pointed out by Price (1981), the rightward bias of TC's cold wake is primarily attributed to the wind stress rotating in the same direction as the inertial current in right of the TC track. In light of this illustration, the location where the wind stress turning rate is resonant to the inertial motion should be prone for inducing the strongest sea surface cooling. An illustration of the resonance between anticyclonic rotation of storm wind vector and inertial motion is provided in Supporting Information S1 (Text S1; Figures S1-S3). Briefly, a storm is assumed to translate northward at a speed of u in a short time of t. The distance of storm translation can be simply expressed as ut.
Assuming that the wind fields are symmetrically cyclonic, the turning angle of the wind stress at a distance of D from the storm center can be calculated by = arctan , where θ ≈ as t → 0, corresponding to a period of 2π . The inertial period of oceanic current at this location is 2 , where f is the Coriolis parameter. If the wind stress turning is resonant to the inertial current, 2π should be equal to 2 . Therefore, the distance of rightward bias from the storm center where the resonance between wind stress and inertial motion occurs can be expressed as: = . (2) From Equation 2, the asymmetry of cold wakes relies on the translation speed and Coriolis parameter. Specifically, the faster the storm translates or the lower the latitude is, the more rightward biased the cold wake will be. Besides, this equation also implies that the asymmetry of the cold wake is independent from the maximum surface wind, namely storm intensity.
Another potential contributor to the asymmetry of cold wake is the radial wind profile. As well reported in previous studies, the TC-induced vertical mixing tends to be stronger for larger surface wind. Since the wind fields of TCs are not horizontally homogeneous, the location of the largest surface wind should be prone to induce the strongest cooling, namely the RMW. Supposing a symmetric vortex moving northward without resonance effect, by neglecting the surface wind asymmetry induced by storm translation speed, there would be two peak cooling locations on both sides of the track, below the RMW. It is the clockwise turning of inertial current that suppresses surface cooling on left side of the track while enhances it on right side of the track, rendering the maximum SST anomaly occurring on right side of the track near to the RMW. Pun et al. (2018) modulated the RMW together with other storm size parameters, including radii of 64, 50, and 34 kt, simultaneously. They noted a positive correlation between the radial wind profiles and the rightward bias of cold wake. These results support our preliminary deduction. Figure 1a shows the SST anomaly caused by storm wind forcing, exhibiting a typical cold wake feature. An elongated sea surface cooling is formed immediately following the vortex with the maximum cooling in the 10.1029/2023GL104578 4 of 8 range of 2-3°C. The wake is remarkably rightward biased by ∼50 km. The other sensitivity experiments show similar patterns but with different cooling magnitudes and rightward bias extents (not shown; Lee et al., 2021;Price, 1981;Wu et al., 2016).

Cold Wake Asymmetry
The magnitudes of TC-induced SST anomaly tend to be larger for more intense, larger-size, or slower-moving TCs, and decrease slightly with increasing the latitude. Results are not shown here because the focus of this study is the asymmetry of cold wakes. The rightward bias extents of cold wakes as functions of translation speed, latitude, RMW, and maximum surface wind are displayed in Figures 1b-1e. Overall the rightward bias extent increases steadily from ∼40 to ∼70 km with increasing the translation speed, suggesting that the faster the storm moves, the farther the cold wake shifts from the storm center. This phenomenon was also found in Zedler (2009) and Mei and Pasquero (2013). The storm translation could contribute to the wind asymmetry of TCs (e.g., Holland, 1980). A comparison between the experiments with and without addition of translation speed on surface wind pattern indicates that the cold wake tends to be more rightward biased due to the effect of translation speed on surface wind asymmetry. Since Uhlhorn et al. (2014) observed no notable change in the asymmetry amplitude of surface wind with varying translation speeds, this effect has not been considered in other experiments. On the other hand, the rightward bias extent decreases with increasing the latitude, indicating that TCs at higher latitudes tend to produce cold wakes closer to their eye centers. Mei and Pasquero (2013) found no relation between the latitude and the cold wake asymmetry, which may because the role of latitude change is not salient relative to other factors such as the background oceanic environment in real world. The wake's positive relation to translation speed but negative relation to latitude support well the theoretical relationship reflected in Equation 2. Apart from the translation speed and Coriolis force, the cold wake asymmetry depends closely on the RMW (Figure 1d). The rightward bias extent increases steadily from ∼30 to over 90 km with increasing the RMW from 20 to 100 km, evincing that the RMW plays a vital role in determining the asymmetry of the cold wake. Another interesting feature is that the rightward bias shows no detective relationship with the maximum surface wind. These suggest that the resonance effect and RMW act together to determine the cold wake asymmetry, which, however, is independent from the TC intensity.
We also conducted additional series of sensitivity experiments with a fixed RMW but different radial wind profiles via modulating α in Equation 1. The magnitudes of sea surface cooling increase as the storm size increases, but the rightward bias gives no dependence on the storm size ( Figure 2). This indicates that unlike the close dependence of SST decrease on storm size (Pun et al., 2018), the cold wake asymmetry is irrelevant to the radial wind profiles outside the RMW, namely the TC size. Figure 2d shows the rightward bias extents of cold wakes as a function of the RMW with varying latitudes. The rightward biases of cold wakes overall increase with decreasing the latitude. For each group of experiments with a fixed latitude, the rightward bias extent increases quasi-linearly with increasing the RMW, suggesting that the positive dependence of cold wake asymmetry on the RMW is relatively robust.
Above analysis indicates that both the resonance effect and the RMW of TCs are crucial in determining the rightward bias of TC's cold wakes, which, however, is independent from the intensity and outer size of TCs. This is different from the extent of TC-induced sea surface cooling, which is positively dependent on TC intensity and size, but is inversely related to TC translation (e.g., Mei & Pasquero, 2013;Pun et al., 2018). To quantify the rightward bias extent of cold wakes, the resonance effect and the RMW are combined to obtain a linear fitting relationship, as shown in Supporting Information S1 ( Figure S4). The simulated rightward bias extents of cold wakes and RMW × u/f correlate well, with R 2 = 0.64. On the basis of this linear fitting, a semi-empirical equation is derived to represent the rightward bias extent of cold wakes: = 4 × 10 −6 m −1 RMW + 31 km, where m and km are units; Y is the rightward bias distance of cold wake from the storm track; both RMW and Y are in units of km. Since the cold wake asymmetry is irrelevant to the specific strength of turbulent mixing, this relationship may not rely on the model or parameterization scheme chosen, but care should be taken when it is applied in spatially varying oceanic environment.
To consolidate the findings based on numerical simulations, observational data samples are also used for a further examination. Figure 3 shows the distance of cold wake rightward bias categorized by u/f. The average distances of rightward bias increase quasi-linearly with increasing u/f from less than 5 × 10 4 m to over 20 × 10 4 m. This supports well the relationship reflected in Equation 2, an indication of remarkable effect of resonance on determining the cold wake asymmetry in the real world. Figure 4a shows the temporal evolution of the simulated maximum surface wind speed. The cold wake experiments produce evidently weaker TCs than the CONST experiment, with large discrepancies existing among CENTER, CONTROL, and DRIGHT due to their different cold wake positions ( Figure  S5 in Supporting Information S1). The TCs in these three experiments experience an intensification period from 24 to ∼96 hr, during which the eyewalls contract first to ∼72 hr and then keeps quasi-steady in the remainder of the simulations ( Figure S6 in Supporting Information S1). A comparison of three cold wake experiments indicates that the TC intensification is sensitive to the cold wake position, with the intensification rate being slower as the cold wake is moved closer to the storm center.

TC Asymmetry
The spatial distributions of surface enthalpy flux are effectively affected by the asymmetry of cold wakes (Figures 4b-4d). Relative to CONST, the CENTER experiment produces smaller surface enthalpy fluxes under  the storm center as well as in the rear of the TC; the flux difference between CONTROL and CENTER shows larger values on the left side but smaller values on the right side due to the rightward shifting of SST anomaly; the difference between DRIGHT and CONTROL gives similar patterns with further rightward shifting of negative flux. As a consequence, the inner-core enthalpy flux tends to be smaller as the cold wake is closer to the TC center, leading to slower intensification of TCs. The changes in surface enthalpy flux distribution also give feedback to TC internal structure ( Figure S7 in Supporting Information S1), with the asymmetry rate tending to be stronger as the cold wake is closer to the storm center ( Figure S8 in Supporting Information S1).

Conclusions
It has been known for decades that the cold wake is typically rightward biased relative to the TC track in the Northern Hemisphere given relatively straight translation, as a result of the resonance between anticyclonic turning of wind stress on the right side of TC track and oceanic inertial motion (Price, 1981). However, it remains unclear how rightward biased of a TC's cold wake should be. An analysis of the resonance between wind stress turning rate and inertial motion suggests that the rightward bias of a cold wake is proportional to the translation speed of a TC but inversely proportional to the Coriolis parameter (Equation 2). This indicates that the cold wake asymmetry is dependent on the translation speed and the latitude of a TC, but is independent from the TC intensity. Another deduction is that since the SST anomaly is positively related to surface wind strength and a TC possesses its peak wind at the RMW, the cold wake asymmetry should also be positively correlated to the RMW. A total of 53 sensitivity experiments using the 3DPWP model confirm the above theoretical deduction. The cold wake asymmetry also relies closely on the RMW, but is independent from the radial wind profile outside the RMW. A semi-empirical equation is derived to quantitatively characterize the rightward bias distance of cold wake from the TC center, as listed in Equation 3. A combination of TC best track data and SST satellite data set consolidates these findings, revealing an increasing trend of rightward bias extent with increasing u/f. The influence of cold wake asymmetry on the intensity and asymmetric structures of TCs is further investigated by comparing sensitivity experiments with varying cold wake locations relative to the TC track. As the cold wake is closer to the TC eye, the surface enthalpy flux becomes smaller in the inner core; consequently, TC intensification tends to be slower and the asymmetry becomes stronger.