Arthur Avenas
email: arthuravenas@ifremer.fr

Alexis Mouche

Pierre Tandeo

Jean-Francois Piolle

Dan Chavas

Ronan Fablet

John Knaff

Bertrand Chapron

Reexamining the Estimation of Tropical Cyclones Radius of Maximum Wind from Outer Size with an Extensive Synthetic Aperture Radar Dataset

The radius of maximum wind (𝑅 𝑚𝑎𝑥 ), an important parameter in tropical cyclones (TCs) ocean surface wind structure, is currently resolved by only a few sensors, so that, in most cases, it is estimated subjectively or via crude statistical models. Recently, a semi-empirical model relying on an outer wind radius, intensity and latitude was fit to best-track data. In this study we revise this semi-empirical model and discuss its physical basis. While intensity and latitude are taken from best-track data, 𝑅 𝑚𝑎𝑥 observations from high-resolution (3 𝑘𝑚) spaceborne synthetic aperture radar (SAR) and wind radii from an inter-calibrated dataset of medium-resolution radiometers and scatterometers are considered to revise the model coefficients. The new version of the model is then applied to the period 2010-2020 and yields 𝑅 𝑚𝑎𝑥 reanalyses and trends more accurate than best-track data. SAR measurements corroborate that fundamental conservation principles constrain the radial wind structure on average, endorsing the physical basis of the model.

Observations highlight that departures from the average conservation situation are mainly explained by wind profile shape variations, confirming the model's physical basis, which further shows that radial inflow, boundary layer depth and drag coefficient also play roles. Physical understanding will benefit from improved observations of the near-core region from accumulated SAR observations and future missions. In the meantime, the revised model offers an efficient tool to provide guidance on 𝑅 𝑚𝑎𝑥 when a radiometer or scatterometer observation is available, for either operations or reanalysis purposes.

Introduction

Estimating tropical cyclone (TC) ocean surface wind structure is challenging but crucial for several applications. In particular, TC surface wind spatio-temporal distributions are used as input of surface wave studies [START_REF] Wright | Hurricane directional wave spectrum spatial variation in the open ocean[END_REF][START_REF] Young | A review of parametric descriptions of tropical cyclone wind-wave generation[END_REF][START_REF] Kudryavtsev | Self-similarity of surface wave developments under tropical cyclones[END_REF], storm surge studies (Irish et al. 2008;[START_REF] Takagi | Maximum wind radius estimated by the 50 kt radius: improvement of storm surge forecasting over the western north pacific[END_REF], or the upper ocean responses to TC passages [START_REF] Price | Upper ocean response to a hurricane[END_REF][START_REF] Ginis | Tropical cyclone-ocean interactions[END_REF][START_REF] Kudryavtsev | A simplified model for the baroclinic and barotropic ocean response to moving tropical cyclones: 2. model and simulations[END_REF]Combot et al. 2020b). In such studies, the radius of maximum winds (hereafter 𝑅 𝑚𝑎𝑥 ) is a critical parameter that significantly affects the wave developments, surges estimates, sea surface height, temperature and salinity variations within the TC wakes. Most parametric surface wind fields, often used for those applications, assume that 𝑅 𝑚𝑎𝑥 is known [START_REF] Holland | An analytic model of the wind and pressure profiles in hurricanes[END_REF][START_REF] Willoughby | Parametric representation of the primary hurricane vortex. part ii: A new family of sectionally continuous profiles[END_REF]). Thus, 𝑅 𝑚𝑎𝑥 errors cascade into errors for the entire spatial distribution of wind speeds. Figure 1 shows a comparison between two Rankine profiles for two different 𝑅 𝑚𝑎𝑥 values

𝑉 𝑅𝑎𝑛𝑘𝑖𝑛𝑒 (𝑟) =        𝑉 𝑚𝑖𝑛 + (𝑉 𝑚𝑎𝑥 -𝑉 𝑚𝑖𝑛 )( 𝑟 𝑅 𝑚𝑎𝑥 ) if 𝑟 ≤ 𝑅 𝑚𝑎𝑥 𝑉 𝑚𝑖𝑛 + (𝑉 𝑚𝑎𝑥 -𝑉 𝑚𝑖𝑛 )( 𝑅 𝑚𝑎𝑥 𝑟 ) if 𝑟 > 𝑅 𝑚𝑎𝑥 (1)
TC Lane, a North Eastern Pacific hurricane that reached category 5 on the Saffir-Simpson scale in 2018, provides an example of such a situation. TC Lane's wind speeds were estimated by a swath of satellite-based SAR observation on 23 August at 0437 UTC. From the SAR wind speeds, the azimuthally-averaged wind profile can be derived (dashed green curve in Fig. 1). The inferred 𝑅 𝑚𝑎𝑥 is 15 𝑘𝑚, about 2 to 3 times smaller than the 37 𝑘𝑚 value interpolated to the SAR aquisition time in the best-track data [START_REF] Knapp | The international best track archive for climate stewardship (ibtracs) unifying tropical cyclone data[END_REF]; hereafter IBTrACS). Such a mismatch between best-track and SAR 𝑅 𝑚𝑎𝑥 estimates is representative of what has been reported in the literature (Combot et al. 2020a). In the present case (Fig. 1), this discrepancy results into a Mean Absolute Error (MAE) as high as 28 𝑚𝑠 -1 near the eyewall region when using subsequent Rankine profile estimates.

To date, airborne Stepped Frequency Microwave Radiometer (SFMR) surface winds [START_REF] Uhlhorn | Hurricane surface wind measurements from an operational stepped frequency microwave radiometer[END_REF]) provide means to estimate 𝑅 𝑚𝑎𝑥 . Yet, airborne measurements have limited azimuthal coverage, and are operated over only few ocean regions and events. From a satellite perspective, high spatial resolution estimates of TC ocean surface wind field are now more systematically carried out, especially from SAR dedicated acquisitions [START_REF] Mouche | Combined co-and cross-polarized sar measurements under extreme wind conditions[END_REF]Combot et al. 2020a). More UTC. Rankine profiles are defined with SAR 𝑅 𝑚𝑎𝑥 (15 𝑘𝑚, solid green) or IBTrACS 𝑅 𝑚𝑎𝑥 (37 𝑘𝑚, solid blue) and the same 𝑉 𝑚𝑎𝑥 (54 𝑚𝑠 -1 ) and 𝑉 𝑚𝑖𝑛 (7 𝑚𝑠 -1 ), consistently with the SAR azimuthally-averaged profile (dashed green). MAE between the two Rankine profiles is shaded in red.

reliable 𝑅 𝑚𝑎𝑥 estimates are then obtained for all ocean basins, though with limited spatio-temporal sampling. Presently, the most often available spaceborne observing systems, capable of probing the ocean surface during TC conditions, are the combined capabilities from active scatterometers and passive radiometers [START_REF] Quilfen | The potential of quikscat and windsat observations for the estimation of sea surface wind vector under severe weather conditions[END_REF]). Compared to radiometers, scatterometers generally have an improved medium spatial resolution. Yet, the strong gradients of the surface wind existing at scales of a few kilometers may still be smoothed to precisely locate the wind maxima, and the position of the center [START_REF] Quilfen | Observation of tropical cyclones by high-resolution scatterometry[END_REF]). In addition, scatterometers, especially those operating at Ku-band and higher microwave frequencies, can suffer from rain contamination. Signal sensitivity at high winds, above hurricane force wind (33 𝑚𝑠 -1 ), has also been questioned [START_REF] Donnelly | Revised ocean backscatter models at c and ku band under high-wind conditions[END_REF][START_REF] Mouche | Copolarized and cross-polarized sar measurements for high-resolution description of major hurricane wind structures: Application to irma category 5 hurricane[END_REF]. Radiometer measurements may be less impacted by rain, especially those operating at L-band [START_REF] Reul | Smos satellite l-band radiometer: A new capability for ocean surface remote sensing in hurricanes[END_REF][START_REF] Reul | A new generation of tropical cyclone size measurements from space[END_REF], and demonstrated to be still highly sensitive above hurricane force winds. However, actual spaceborne radiometers operating at L-or C-band have a lower spatial resolution. High wind speed gradients near the 𝑅 𝑚𝑎𝑥 region for most intense TCs are then generally indistinct. Direct estimates of 𝑅 𝑚𝑎𝑥 using scatterometers or radiometers are thus difficult to perform, possibly limited to particular large storm cases.

More indirect means to infer 𝑅 𝑚𝑎𝑥 were also considered. Both [START_REF] Mueller | Objective estimation of tropical cyclone wind structure from infrared satellite data[END_REF] and [START_REF] Kossin | Estimating hurricane wind structure in the absence of aircraft reconnaissance[END_REF] used geostationary infrared satellite data. For the cases where a clear eye is welldefined on the infrared image, using linear regression to estimate 𝑅 𝑚𝑎𝑥 results in a MAE of only ∼5

𝑘𝑚 when compared to aircraft-based estimates. Under less favourable conditions, 𝑅 𝑚𝑎𝑥 can still be estimated via multiple linear regression in combination with a principal components analysis, but leads to a degraded MAE of ∼20 𝑘𝑚. Notably, for the clear-eye case, [START_REF] Tsukada | Strong relationship between eye radius and radius of maximum wind of tropical cyclones[END_REF] trained the linear regression with available SAR 𝑅 𝑚𝑎𝑥 estimates and improved the method, decreasing the MAE to ∼2 𝑘𝑚.

In the absence of infrared data, a rough 𝑅 𝑚𝑎𝑥 estimate can also be obtained, considering the storm intensity and latitude known, as evidenced by [START_REF] Willoughby | Parametric representation of the primary hurricane vortex. part ii: A new family of sectionally continuous profiles[END_REF] and [START_REF] Vickery | Statistical models of holland pressure profile parameter and radius to maximum winds of hurricanes from flight-level pressure and h* wind data[END_REF]. Indeed, following the angular momentum conservation, 𝑅 𝑚𝑎𝑥 must decrease when the intensity increases. On average, such a physical constrain agrees well with observations (see for instance Fig. 9 in Combot et al. (2020a)). In addition, it is also known that 𝑅 𝑚𝑎𝑥 increases with latitude (e.g. [START_REF] Willoughby | Parametric representation of the primary hurricane vortex. part i: Observations and evaluation of the holland (1980) model[END_REF]), another consequence of angular momentum conservation along with the decrease of intensity with latitude. Solely using intensity and latitude to predict 𝑅 𝑚𝑎𝑥 yields a root-mean square error of the order ∼20 𝑘𝑚. Results from [START_REF] Vickery | Statistical models of holland pressure profile parameter and radius to maximum winds of hurricanes from flight-level pressure and h* wind data[END_REF] show that in several cases, the observed 𝑅 𝑚𝑎𝑥 is inconsistent with the general principle of angular momentum conservation. This suggests that 𝑅 𝑚𝑎𝑥 natural variability can hardly be captured by such simple statistical models.

More recently, Chavas and Knaff (2022) -hereafter CK22 -suggested to use information on the TC outer-size in combination with latitude and intensity. In CK22 framework, 𝑅 𝑚𝑎𝑥 is estimated from the TC intensity 𝑉 𝑚𝑎𝑥 , the radius of gale 𝑅 34 (i.e the maximum radial extent of the 34-knots winds) and the Coriolis parameter, defined as 𝑓 = 2Ω𝑠𝑖𝑛(𝜙), where Ω = 7.292𝑥10 -5 𝑠 -1 is the Earth angular velocity and 𝜙 is the latitude of the TC center. Such an approach is practical, especially because 𝑅 34 is well estimated by satellite scatterometers and radiometers [START_REF] Brennan | The operational use of quikscat ocean surface vector winds at the national hurricane center[END_REF]Chou et al. 2013;[START_REF] Reul | A new generation of tropical cyclone size measurements from space[END_REF]. In fact, 𝑅 34 estimates are routinely produced for every TC and included in IBTrACS.

The CK22 framework is based on physical understanding of the radial wind structure [START_REF] Emanuel | Tropical cyclone energetics and structure[END_REF][START_REF] Emanuel | Self-stratification of tropical cyclone outflow. part i: Implications for storm structure[END_REF] and phrased in terms of absolute angular momentum 𝑀 (𝑟) = 𝑟𝑉 + 1 2 𝑓 𝑟 2 , where 𝑓 , 𝑟 and 𝑉 are the Coriolis parameter, the radius and the tangential wind speed of an air parcel, respectively. If the ratio 𝑀 𝑚𝑎𝑥 𝑀 34 := 𝑀 (𝑟=𝑅 𝑚𝑎𝑥 ) 𝑀 (𝑟=𝑅 34 ) is prescribed, one can then estimate 𝑅 𝑚𝑎𝑥 provided estimates for the 3 above-mentioned parameters using:

𝑅 𝑚𝑎𝑥 = 𝑉 𝑚𝑎𝑥 𝑓 √ 1 + 2 𝑓 𝑀 𝑚𝑎𝑥 𝑉 2 𝑚𝑎𝑥 -1 (2)
CK22 fitted a log-linear regression model to estimate the ratio 𝑀 𝑚𝑎𝑥 𝑀 34 with the two predictors 𝑋 (1) 34 := (𝑉 𝑚𝑎𝑥 -17.5𝑚𝑠 -1 ) and 𝑋 (2) 34 := (𝑉 𝑚𝑎𝑥 -17.5𝑚𝑠 -1 )( The quality of 𝑅 34 best-track estimates has also already been questioned [START_REF] Sampson | Tropical cyclone gale wind radii estimates for the western north pacific[END_REF]).

This parameter is reanalyzed and compiled in IBTrACS since 2004 for North Atlantic and North Eastern Pacific and since 2016 for North Western Pacific [START_REF] Knaff | Estimating tropical cyclone surface winds: Current status, emerging technologies, historical evolution, and a look to the future[END_REF]). Yet, surveying specialists who produce best-tracks in the Atlantic ocean [START_REF] Landsea | Atlantic hurricane database uncertainty and presentation of a new database format[END_REF] are on average much less confident in their wind radii estimates (∼25-50% of relative uncertainty) than in their intensity estimates (∼10-20%).

In addition, best-tracks may also suffer from temporal and spatial heterogeneities [START_REF] Schreck | The impact of best track discrepancies on global tropical cyclone climatologies using ibtracs[END_REF][START_REF] Wang | Recent tropical cyclone changes inferred from ocean surface temperature cold wakes[END_REF]. Indeed, the reanalysis methodology depends on the available data at each reanalysis time: best-track estimates of TC events covered by aircraft data are for instance more trustworthy [START_REF] Landsea | Atlantic hurricane database uncertainty and presentation of a new database format[END_REF]. Reanalysis is also subjective, each agency or Regional Specialized Meteorological Center (RSMC) specialist conducting his own weighting of the available observations. Lastly, best-tracks are finalized annually and not updated with evolving reanalysis methodology, creating a temporal discontinuity in the final IBTrACS database.

Finally, a possible limitation of the CK22 approach is the arbitrary choice of the outer wind radius 𝑅 34 . Indeed, their model could well be trained using 𝑅 50 or 𝑅 64 . In CK22, the choice of 𝑅 34 was motivated by the fact that best-track estimates of 𝑅 50 and 𝑅 64 are generally more uncertain than 𝑅 34 estimates. With more reliable 𝑅 50 and 𝑅 64 estimates, possibly obtained from radiometer or scatterometer observations, one could assess whether using these wind radii would improve the CK22 model.

The physical basis for wind structure relationships such as CK22 is a long-running issue. The assumption that an outer wind radius partly constrains the wind structure dates back to [START_REF] Riehl | Some relations between wind and thermal structure of steady state hurricanes[END_REF]. [START_REF] Riehl | Some relations between wind and thermal structure of steady state hurricanes[END_REF] used a two-layer conceptual model constrained by an angular momentum conservation

in the outflow and a potential vorticity (PV) conservation in the inflow layer. [START_REF] Riehl | Some relations between wind and thermal structure of steady state hurricanes[END_REF] could then derive a relationship between 𝑅 𝑚𝑎𝑥 , 𝑉 𝑚𝑎𝑥 , 𝑓 , and an outer radius 𝑅 𝑜𝑢𝑡 , corresponding to a distance where the outflow velocity vanishes. Later, [START_REF] Kalashnik | On the maximum wind velocity in the tropical cyclone[END_REF] considered the Holland parametric profile [START_REF] Holland | An analytic model of the wind and pressure profiles in hurricanes[END_REF]) within a theoretical framework, to analyze the dependence of the near-core wind structure on the wind profile. [START_REF] Emanuel | Self-stratification of tropical cyclone outflow. part i: Implications for storm structure[END_REF] also derived an analytical solution for the near-core wind profile based on an assumption on the outflow temperature.

While these studies offer theoretical guidance, these theoretical inferences of 𝑅 𝑚𝑎𝑥 are difficult to apply in practice. Indeed, most actual sensors fail to capture the wind profile shape used in [START_REF] Kalashnik | On the maximum wind velocity in the tropical cyclone[END_REF], while the model of [START_REF] Emanuel | Self-stratification of tropical cyclone outflow. part i: Implications for storm structure[END_REF] relies on parameters that are difficult to evaluate. Following [START_REF] Riehl | Some relations between wind and thermal structure of steady state hurricanes[END_REF], the theoretical outer radius 𝑅 𝑜𝑢𝑡 is unknown and cannot be specified to correspond to a given surface wind speed.

Building on the above considerations, the aim of this study is twofold. First, the CK22 model is revised using SAR 𝑅 𝑚𝑎𝑥 estimates, different wind radii (referring hereafter to 𝑅 34 , 𝑅 50 , and 𝑅 64 ) estimated on inter-calibrated radiometers and scatterometers, and intensity and latitude best-track estimates. Second, the physical basis of the CK22 model is further assessed through an examination of conservation equations and a thorough analysis of the SAR database.

The data used in the present work are introduced in section 2 and further analysed in section 3.

Then, the CK22 model is revised and its performance assessed in section 4. Finally, the physical basis of the model is discussed with respect to SAR observations in section 5. Concluding remarks and possible routes for future investigations are provided in the last section.

Data

In the present work, different radiometer and scatterometer data ( We used different radiometer and scatterometer missions to constitute the most extensive dataset of 𝑅 𝑚𝑎𝑥 reanalyses. These sensors rely on different physical principles (passive or active sensors), and have different frequencies (L-band, C-band or Ku-band) and spatial resolutions. In order to ensure homogeneity of the wind radii estimates, we used radiometer and scatterometer winds inter-calibrated by [START_REF] Portabella | Maxss: Algorithm theoretical baseline document for sfmr-based satellite-derived extreme wind recalibration[END_REF].

A thorough analysis of this database revealed that the wind profiles issued from Ku-band scatterometer data barely exceed 64 knots, even for most intense TCs, as shown in appendix A. Thus, we chose to remove Ku-band scatterometers from the present analysis.

a. Radiometer missions

Because both the foam coverage and bubble surface layer thickness increase with surface wind speed [START_REF] Reul | A model of sea-foam thickness distribution for passive microwave remote sensing applications[END_REF], passive microwave measurements have long been known to display very high sensitivity under extreme wind conditions. With large ∼1000 𝑘𝑚 swaths, satellite-borne radiometers are well suited to monitor TCs. However, they have nominally low spatial resolutions (∼40 𝑘𝑚) that generally prevent accurate retrieval of the extreme surface wind speeds associated with the inner-core of most intense TCs. The radiometer wind products used in this work are at 50

𝑘𝑚 spatial resolution with a 25 𝑘𝑚 grid spacing [START_REF] Portabella | Maxss: Algorithm theoretical baseline document for sfmr-based satellite-derived extreme wind recalibration[END_REF]).

In the present study, four different sources of radiometer data were used. Among them, the L-band (1.4 GHz, 21 cm wavelength) radiometers from the European Space Agency (ESA) Soil

Moisture and Ocean Salinity (hereafter SMOS) mission and the National Aeronautics and Space Administration (NASA) Soil Moisture Active Passive (hereafter SMAP). The ability of L-band radiometers to retrieve ocean surface wind speeds under TCs has been discussed both in the case of SMOS [START_REF] Reul | Smos satellite l-band radiometer: A new capability for ocean surface remote sensing in hurricanes[END_REF][START_REF] Reul | A revised l-band radio-brightness sensitivity to extreme winds under tropical cyclones: The five year smos-storm database[END_REF] and SMAP [START_REF] Yueh | Smap l-band passive microwave observations of ocean surface wind during severe storms[END_REF][START_REF] Meissner | Capability of the smap mission to measure ocean surface winds in storms[END_REF]. [START_REF] Reul | A new generation of tropical cyclone size measurements from space[END_REF] demonstrated that SMOS, SMAP, as well as AMSR-2 can be used to estimate wind radii.

The Japan Aerospace Exploration Agency launched the Advanced Microwave Scanning Radiometer 2 (hereafter AMSR-2) onboard the Global Change Observation Mission Water 1 satellite in 2012. This instrument is still operating today and uses 7 different frequencies (6.93,7.3,10.65,18.7,23.8,36.5 and 89.0 GHz. For TCs,the first 3 channels (6.93,7.3,and 10.65 GHz) 6.8, 10.7, 18.7, 23.8 and 37.0 GHz). To minimize heavy precipitation impacts, the C-band 6.8 and the X-band 10.7 GHz channels are used for TC wind retrieval algorithms. Again, changes in the respective contribution of wind and rain to the signal measured by each channel can be used to better infer and discriminate both quantities [START_REF] Klotz | Improved stepped frequency microwave radiometer tropical cyclone surface winds in heavy precipitation[END_REF]. Heavy precipitation is still found to complicate surface wind speed retrieval with this sensor [START_REF] Quilfen | The potential of quikscat and windsat observations for the estimation of sea surface wind vector under severe weather conditions[END_REF], and more recent studies addressed this issue [START_REF] Meissner | Tropical cyclone wind speeds from windsat, amsr and smap: Algorithm development and testing[END_REF][START_REF] Manaster | Tropical cyclone winds from windsat, amsr2, and smap: Comparison with the hwrf model[END_REF].

b. Scatterometer missions

Scatterometers are active sensors that emit a pulse and measure the signal backscattered by the rough ocean surface with different viewing angles. Because backscatter signals are dependent upon both wind speed and wind direction, ocean surface wind vectors can be retrieved. The achieved nominal spatial resolution (up to ∼25 𝑘𝑚) is higher than satellite-borne radiometers.

Actual scatterometers operate at different frequencies (C-band or Ku-band).

The Meteorological Operational satellite programme is a series of 3 satellites (Metop-A, -B and -C) launched by ESA (in 2006ESA (in , 2012ESA (in and 2018, respectively) , respectively) which include scatterometers (ASCAT, for "Advanced Scatterometer") operating at 5.3 GHz (C-band). With 3 antennas oriented at 45°, 90°and 135°with respect to the satellite track, the wind direction can be retrieved. ASCAT instruments have 2 sub-swaths, each having a width of ∼550 𝑘𝑚. At C-band, the signal may be influenced by very heavy rain. Backscatter signals also tend to saturate at high winds [START_REF] Donnelly | Revised ocean backscatter models at c and ku band under high-wind conditions[END_REF], and ASCAT measurements progressively lose sensitivity under high wind speeds [START_REF] Soisuvarn | Cmod5. h-a high wind geophysical model function for c-band vertically polarized satellite scatterometer measurements[END_REF][START_REF] Polverari | On high and extreme wind calibration using ascat[END_REF]. The ASCAT wind product used in the present study is at 25 𝑘𝑚 spatial resolution with a 12.5 𝑘𝑚 grid spacing [START_REF] Stoffelen | The cmod7 geophysical model function for ascat and ers wind retrievals[END_REF][START_REF] Portabella | Maxss: Algorithm theoretical baseline document for sfmr-based satellite-derived extreme wind recalibration[END_REF].

Scatterometers operating at Ku-band (∼ 13.5 GHz) usually have larger swaths (∼1000 𝑘𝑚) than C-band scatterometers, but suffer more contamination in heavy rainfall conditions (see [START_REF] Quilfen | The potential of quikscat and windsat observations for the estimation of sea surface wind vector under severe weather conditions[END_REF] for more details). The Ku-band scatterometer wind products used in [START_REF] Portabella | Maxss: Algorithm theoretical baseline document for sfmr-based satellite-derived extreme wind recalibration[END_REF] were finally removed (see appendix A). They include the China National Space Administration (CNSA) Haiyang missions (hereafter HSCAT), the Indian Space Research Organisation (ISRO)

OceanSat-2 and SCATSat-1 satellites (hereafter OSCAT), and the NASA RapidScat (hereafter RSCAT) onboard the International Space Station (Table 1). Table 1. The radiometer and scatterometer data used in [START_REF] Portabella | Maxss: Algorithm theoretical baseline document for sfmr-based satellite-derived extreme wind recalibration[END_REF]. The period, spatial resolution, and pixel spacing rows refer to the wind product. The same data were used for the present work, except the Ku-band scatterometers, which were removed from the analysis.

RADIOMETER

c. SAR missions

The SAR data used here come from three different missions: ESA Sentinel-1A and Sentinel-1B (hereafter S1A and S1B, respectively), and the Canadian Space Agency (CSA) Radarsat-2 (hereafter RS2). The SAR instrument onboard these three missions is an active sensor operating at 5.4 GHz (C-band). By analysing the received signal in both co-and cross-polarization, wind speeds can be inferred under TC conditions including at very high wind speeds [START_REF] Mouche | Combined co-and cross-polarized sar measurements under extreme wind conditions[END_REF][START_REF] Mouche | Copolarized and cross-polarized sar measurements for high-resolution description of major hurricane wind structures: Application to irma category 5 hurricane[END_REF]. Convincing comparisons with passive radiometers have been performed [START_REF] Zhao | Direct comparison between active c-band radar and passive l-band radiometer measurements: Extreme event cases[END_REF]. The ability of SAR-derived wind speeds to accurately capture the TC ocean surface wind structure, including 𝑅 𝑚𝑎𝑥 , has further been demonstrated and discussed by Combot et al. (2020a).

Today, SAR wide-swath acquisitions cannot be continuously performed over oceans. Based on track forecasts, it is still possible to best anticipate when the sensor will overpass a TC, and to trigger a SAR acquisition. ESA started the Satellite Hurricane Observation Campaign (SHOC) in 2016, resulting in more than ∼500 acquisitions over TCs. The derived wind products [START_REF] Mouche | Combined co-and cross-polarized sar measurements under extreme wind conditions[END_REF]) are further interpolated on a regular polar grid based on the TC center (see appendix in [START_REF] Vinour | Observations of tropical cyclone inner-core fine-scale structure, and its link to intensity variations[END_REF]). The product has a 3 𝑘𝑚 spatial resolution, with a 1 𝑘𝑚 grid spacing. This spatial resolution approximates a 1-minute wind speed as a 50 𝑚𝑠 -1 wind moves 3 𝑘𝑚 in a minute.

In this study, a certain number of SAR cases have been discarded on a qualitative basis, e.g. when the detected TC center was judged to be wrong, or when the SAR file contained corrupted pixel values.

SAR S1A S1B RS2 Period 2016-2021 2016-2021 2012-2021 Spatial resolution 3 𝑘𝑚 3 𝑘𝑚 3 𝑘𝑚 Pixel spacing 1 𝑘𝑚 1 𝑘𝑚 1 𝑘𝑚 Frequency C-band C-band C-band
Table 2. The SAR data used in the present study. The period, spatial resolution, and pixel spacing rows refer to the wind product.

d. Best-tracks

Here, IBTrACS were used for several purposes: the storm centers (latitude and longitude) allowed to azimuthally average the radiometer and scatterometer wind fields, while the wind radii (𝑅 34 , 𝑅 50 , and 𝑅 64 ) were compared to satellite-based wind radii. Both IBTrACS latitude (to compute 𝑓 ) and maximum sustained wind speed (𝑉 𝑚𝑎𝑥 ) were used in the CK22 framework, and the distance to closest land (from the TC center) enabled filtering of the dataset. These parameters were extracted for the period 2010-2020.

In IBTrACS, some storm tracks are given on a six-hourly basis, while others are interpolated and thus given on a three-hourly basis. To account for this varying sampling time, all tracks and their associated parameters were interpolated to an hourly basis with a monotonic cubic interpolation.

Lastly, because of varying definitions of the maximum sustained wind speed across the different agencies, we selected only USA agencies (i.e National Hurricane Center, Joint Typhoon Warning Center, and Central Pacific Hurricane Center) which all provide the 1-minute maximum sustained wind speed.

e. Data filtering

To further restrain the analysis to well-formed systems, i.e. for which 𝑅 𝑚𝑎𝑥 can be well determined from the axisymmetric mean profile, and to best ensure consistency with CK22 for further comparison, the following filters have been applied to our dataset:

1. 𝑉 𝑚𝑎𝑥 > 20 𝑚𝑠 -1 ;

2. 𝑅 𝑚𝑎𝑥 < 150 𝑘𝑚;

3. Any wind radius must be > 5 𝑘𝑚;

4. Absolute latitude < 30°;

5. Distance to closest land > 𝑅 34 .

Unlike CK22, we didn't apply any filter on longitude. Therefore, the method presented here applies in every basin and does not depend on the availability of aircraft analysis.

Methods and data analysis a. Estimation of the CK22 predictors

In order to apply the CK22 framework to the inter-calibrated dataset of radiometer and scatterometer data, estimates of the predictors (𝑉 𝑚𝑎𝑥 , 𝑅 34 , 𝑓 ) were needed for every satellite file.

Regarding the wind radii, an azimuthally-averaged wind profile was first computed for every satellite file using the corresponding IBTrACS center linearly-interpolated to the acquisition time.

For each of the three speed values of interest (i.e 34, 50 and 64 knots), we then selected the radius where the outer-profile matches this value to the closest kilometer. Should there be more than one radius value, the wind radius was defined as the smallest of the radii.

Unlike the wind radii, 𝑉 𝑚𝑎𝑥 and 𝑓 cannot be accurately estimated from radiometer and scatterometer data, especially for intense small TCs, but both parameters are systematically reanalyzed in the best-tracks. However, IBTrACS 𝑉 𝑚𝑎𝑥 definition does not strictly coincide with the axi-symmetric view adopted here. In particular, the analysis (appendix A) highlighted that 𝑉 𝑚𝑎𝑥 estimated using SAR azimuthally-averaged profiles were, on average, lower than IBTrACS 𝑉 𝑚𝑎𝑥 . This can be modeled by applying a linear regression (dashed grey line in Fig. A1) to IBTrACS 𝑉 𝑚𝑎𝑥 estimates.

The resulting intensity estimates are denoted by 𝑉 𝑅𝐸𝐺 𝑚𝑎𝑥 and were used (instead of the raw IBTrACS 𝑉 𝑚𝑎𝑥 ) to ensure the consistency with the wind radii defined on azimuthally-averaged wind profiles.

The pair of parameters (𝑉 𝑅𝐸𝐺 𝑚𝑎𝑥 , 𝑓 ) was then linearly-interpolated to the satellite acquisition time for every file.

b. Quality assessment of radiometer and scatterometer wind radii estimates

To assess the quality of the satellite-based wind radii, comparisons were performed with IBTrACS wind radii. A strict comparison cannot be achieved because of varying definitions. In IBTrACS, wind radii are relative to the geographical quadrants and correspond to the maximum radial extent of the associated wind speed in each of the four quadrants. To make IBTrACS values as close as possible to the satellite-based wind radii, the nonzero IBTrACS values were averaged over all the quadrants. Furthermore, both the methodologies and the available observational data can vary across the IBTrACS dataset. Here, the adopted strategy was to compare the whole IBTrACS wind radii dataset (including non-USA agencies for this section) to the satellite-based wind radii.

Accounting for the differences between the specialists and agencies is beyond the scope of this study. Finally, after removing the Ku-band sensors (see appendix A), we separated radiometer wind radii from the C-band scatterometer wind radii to further investigate possible discrepancies between the remaining sensors.

Figure 2 Again, an overall consistency emerges between both data sources for all wind radii. RSD values and 𝑅 2 scores are comparable to the previous comparisons between radiometer and IBTrACS.

Data and methodology are thus consistent with IBTrACS (which is expected since radiometer and scatterometer data are often used during the reanalysis process), but it also shows that there is a good consistency between the various sensors in terms of wind radii. Consistency between scatterometer and IBTrACS also improves for both 𝑅 34 and 𝑅 50 over the same period (Figs. 4a and4b). The positive ME for 𝑅 64 in Fig. 3 likely corresponds to the improving quality of IBTrACS over the years. Mentioned in the introduction, wind radii best-track values

were not necessarily reanalyzed depending on the year and the basin. Similar conclusions were obtained with radiometer data (not shown).

To summarize, the comparison between IBTrACS and the inter-calibrated dataset shows that radiometers and scatterometers provide reliable wind radii estimates. Thus, for every radiometer or scatterometer acquisition, we can extract a corresponding set of predictors constituted by a satellite-based wind radius along with IBTrACS 𝑉 𝑅𝐸𝐺 𝑚𝑎𝑥 and 𝑓 estimates.

c. Collocations of radiometers and scatterometers with SAR

In order to fit the CK22 model, we also needed an estimate of the predictand (𝑅 𝑚𝑎𝑥 ) for each set of predictors. The latter cannot be directly evaluated from radiometer and scatterometer data, but is well observed on SAR data by taking the location of the wind profile maximum. Thus, we looked for collocations between SAR and radiometer or scatterometer TC overpasses. Two images were considered to be collocated if their absolute acquisition time difference is less than 90 minutes.

Regarding radiometer data (table 3, first four columns), this procedure resulted in a total of 269 collocations, which further reduced to 145 collocations after applying filters presented in section 3e. Notably, no collocation was found between any of the 3 SAR missions (S1A, S1B, RS2) and AMSR-2.

Regarding scatterometer data, no collocation was found between SAR and ASCAT (

Results

a. Fitting CK22 model

As explained in the introduction, the CK22 model relies on the estimation of the ratio 𝑀 𝑚𝑎𝑥 𝑀 34 via a log-linear regression model, using (𝑋 (1) 34 , 𝑋 (2) 34 ) as input. While CK22 used 𝑅 34 in their study, this method is in fact agnostic from the choice of wind radius. Therefore, the ratio 𝑀 𝑚𝑎𝑥 𝑀 50 can also be estimated using 𝑋 (1) 50 := 𝑉 𝑚𝑎𝑥 -25.7𝑚𝑠 -1 and 𝑋 (2) 50 := (𝑉 𝑚𝑎𝑥 -25.7𝑚𝑠 -1 )( 1 2 𝑓 𝑅 50 ) as input (or 𝑀 𝑚𝑎𝑥 𝑀 64 using 𝑋 (1) 64 := 𝑉 𝑚𝑎𝑥 -32.9𝑚𝑠 -1 and 𝑋 (2) 64 := (𝑉 𝑚𝑎𝑥 -32.9𝑚𝑠 -1 )( 1 2 𝑓 𝑅 64 ) as input).

CK22 estimated the coefficients of the log-linear regression model based solely on IBTrACS rather than direct observational estimates, and only for the ratio 𝑀 𝑚𝑎𝑥 𝑀 34 . In the present work, we use observational data not only to obtain improved estimates of the predictors in the CK22 model framework, but also to obtain improved estimates of the model coefficients that relate the parameters to one another. We also extend the CK22 model for the ratios 𝑀 𝑚𝑎𝑥 𝑀 50 and 𝑀 𝑚𝑎𝑥 𝑀 64 . A loglinear regression model was fitted for each of the three ratios using the SAR-radiometer collocation dataset previously presented. The following relationships were obtained:

𝑀 𝑚𝑎𝑥 𝑀 34 = 0.531𝑒 -0.00214(𝑉 𝑅𝐸𝐺 𝑚𝑎𝑥 -17.5𝑚𝑠 -1 )-0.00314(𝑉 𝑅𝐸𝐺 𝑚𝑎𝑥 -17.5𝑚𝑠 -1 ) ( 1 2 𝑓 𝑅 34 ) (3)
𝑀 𝑚𝑎𝑥 𝑀 50 = 0.626𝑒 0.00282(𝑉 𝑅𝐸𝐺 𝑚𝑎𝑥 -25.7𝑚𝑠 -1 )-0.00724(𝑉 𝑅𝐸𝐺 𝑚𝑎𝑥 -25.7𝑚𝑠 -1 ) (

With these formulas, 𝑅 𝑚𝑎𝑥 can then be estimated using the steps presented in the introduction (eq. 2). Subsequent estimates will be referred to as 𝑅 𝐶𝐾22-𝑅 34

𝑚𝑎𝑥

, 𝑅 𝐶𝐾22-𝑅 50

𝑚𝑎𝑥

, or 𝑅 𝐶𝐾22-𝑅 64 𝑚𝑎𝑥 depending on which wind radius is used.

b. Assessment of the resulting 𝑅 𝑚𝑎𝑥 estimates

To check the fitting procedure, we compared 𝑅 𝐶𝐾22-𝑅 34

𝑚𝑎𝑥

estimates and SAR 𝑅 𝑚𝑎𝑥 (Fig. 5a). The consistency between both is reasonably good, with a 𝑅 2 -score of 0.41 and a RSD of 10.6 𝑘𝑚. A low ME of 3.7 𝑘𝑚 is observed, which can be related to the distribution of 𝑅 𝐶𝐾22-𝑅 34 𝑚𝑎𝑥 being slightly skewed toward higher 𝑅 𝑚𝑎𝑥 values compared to SAR. Because 𝑅 50 and 𝑅 64 are closer to 𝑅 𝑚𝑎𝑥 than 𝑅 34 , using one or the other wind radii thresholds should improve the quality of the ratio estimate compared to 𝑅 34 . Ideally, an estimate of 𝑅 𝑚𝑎𝑥 should be performed with 𝑅 64 if available. If 𝑅 64 is not defined (i.e if 𝑉 𝑚𝑎𝑥 is less than 33 𝑚𝑠 -1 ), 𝑅 50 should be used. 𝑅 34 should only be used if both 𝑅 64 and 𝑅 50 were not defined. Following this procedure, we further estimated 𝑅 𝑚𝑎𝑥 using the "best" available wind radius.

Figure 5b shows a comparison between these estimates (hereafter 𝑅 𝐶𝐾22-𝐵𝑅

𝑚𝑎𝑥

) and SAR 𝑅 𝑚𝑎𝑥 .

The 𝑅 2 -score increased to 0.63 and the ME decreased to 0.9 𝑘𝑚 compared to the 𝑅 𝐶𝐾22-𝑅 34 𝑚𝑎𝑥 methodology, while RSD decreased from 10.6 𝑘𝑚 to 8.8 𝑘𝑚. Therefore, using wind radii closer to 𝑅 𝑚𝑎𝑥 does improve the estimate quality. In addition, such a low RSD demonstrates the efficiency of the fitted CK22 relationships (eqs. 3-5) to provide reliable 𝑅 𝑚𝑎𝑥 estimates.

In their paper, the 𝑅 𝑚𝑎𝑥 predicted by CK22 had a systematic bias that could be bias-adjusted in post-processing to improve the model. Here we find that our model does not require a bias adjustment, which may be an indication of the benefit of using direct observational data for 𝑅 𝑚𝑎𝑥 (SAR).

While the method is successful on average, it is remarkable that errors can be large (more than ∼10 𝑘𝑚), even for cases where 𝑅 64 predictors are used (see for instance Kong-Rey and Mangkhut in Fig. 5b). Before discussing how to explain these large uncertainties, a single TC life cycle was chosen to illustrate the potential of the present methodology.

c. Application to TC Kilo life cycle

Producing 𝑅 𝐶𝐾22-𝐵𝑅 𝑚𝑎𝑥 estimates every time a radiometer or a scatterometer TC overpass is available can be an efficient tool for characterizing the time evolution of 𝑅 𝑚𝑎𝑥 for any given TC. Figure 6 shows TC Kilo 𝑅 𝑚𝑎𝑥 and 𝑉 𝑅𝐸𝐺 𝑚𝑎𝑥 time series between 27 August and 10 September 2015, a period over which 𝑉 𝑅𝐸𝐺 𝑚𝑎𝑥 was larger than 20 𝑚𝑠 -1 . TC Kilo evolved in the Pacific ocean, reaching category 4 on the Saffir-Simpson scale. It intensified from 20 to 49 𝑚𝑠 -1 between 27 August and 30 August before entering a weakening phase. In the meantime, 𝑅 𝑚𝑎𝑥 first varied between 55 and 15 𝑘𝑚 according to IBTrACS, then stagnated at 37 𝑘𝑚 between the 30 August and the 2 September, before varying again after these date. Stagnation phases of 𝑅 𝑚𝑎𝑥 from IBTrACS are likely not physical according to the 𝑉 𝑅𝐸𝐺 𝑚𝑎𝑥 variations during that time interval (see section 5). 𝑅 𝐶𝐾22-𝐵𝑅 𝑚𝑎𝑥 estimates show much more pronounced variations during those phases, with an increasing trend between 30

August and 8 September. This particular phase corresponds to an overall decrease of 𝑉 𝑅𝐸𝐺 𝑚𝑎𝑥 and an overall increase of 𝑅 64 in our data (not shown), both of which would be expected to be associated with an increase in 𝑅 𝑚𝑎𝑥 .

For reference, 3 SAR 𝑅 𝑚𝑎𝑥 estimates were available during TC Kilo's life cycle (green stars). The first SAR 𝑅 𝑚𝑎𝑥 (10 𝑘𝑚) on 27 August, doesn't match with our first estimate of 𝑅 𝑚𝑎𝑥 (35 𝑘𝑚). This illustrates the limitations of our proposed methodology and is discussed hereafter. The second and third SAR 𝑅 𝑚𝑎𝑥 estimates are in better agreement with 𝑅 𝑚𝑎𝑥 estimates, especially if we account for the overall 𝑅 𝑚𝑎𝑥 trend given by our estimates.

Notably, there is more spread in the CK22 estimates on the last two days of the study period.

Despite this increasing uncertainty, the increase of 𝑅 𝑚𝑎𝑥 is well depicted, suggesting 𝑅 𝑚𝑎𝑥 to significantly increase much before 8 September in contradiction with the IBTrACS trend.

In summary, every time a radiometer or scatterometer wind profile is available, a subsequent 𝑅 𝐶𝐾22-𝐵𝑅 𝑚𝑎𝑥 estimate can be obtained, using the proposed objective method. In such a way, one can estimate 𝑅 𝑚𝑎𝑥 trends that are more realistic than IBTrACS, less impacted from spatial or temporal heterogeneities. Such a framework could also be used operationally.

Discussion

The previous section demonstrated the potential of the CK22 model fitted with SAR, when used in combination with inter-calibrated medium-resolution radiometer and scatterometer data. Still, 𝑅 𝐶𝐾22-𝐵𝑅 𝑚𝑎𝑥 estimates can display rather large uncertainties, despite the expected improved use of 𝑅 64 as predictor. To better understand the sources of such uncertainties, three other case studies (cyan circles on Fig. 5a) were considered before examining theoretical aspects and drawing a picture of the average situation.

a. Case studies from the SAR-radiometer dataset

The first case (Fig. 7, left column) is TC Olivia in 2018, an Eastern Pacific ocean hurricane that reached category 4 on the Saffir-Simpson scale. It reached a first intensity peak (∼56 𝑚𝑠 -1 ) on 5 September, then weakened before restrengthening (∼59 𝑚𝑠 -1 ) during the night between 6 and 7 September. On 8 September, both RS2 at 1510 UTC and Windsat at 1533 UTC overflew Olivia (Figs. 7a and7d). Its eyewall, depicted by the high-resolution SAR observation, was clearly defined though asymmetric. With its rather low spatial resolution, the radiometer failed to map the inner core areas with high wind speed gradients, and eyewall asymmetries. From the SAR observation, Olivia's 𝑅 𝑚𝑎𝑥 was 30 𝑘𝑚 at that time, with a 𝑉 𝑚𝑎𝑥 of 32 𝑚𝑠 -1 (Fig. 7g). Notably, Windsat failed to estimate 𝑉 𝑚𝑎𝑥 correctly, with a negative bias of almost 10 𝑚𝑠 -1 , which is largely attributable to sensor spatial averaging effects. In fact, the entire azimuthally-averaged wind profile is negatively biased, leading to an underestimation of 𝑅 34 , further reflected in 𝑅 𝐶𝐾22-𝐵𝑅 𝑚𝑎𝑥 . This case illustrates how wind radii uncertainties translate into 𝑅 𝐶𝐾22-𝐵𝑅 𝑚𝑎𝑥 uncertainties.

The second case (Fig. 7, middle column), Mangkhut, was a super typhoon (category 5 on Saffir-Simpson scale), causing considerable damages in the Western Pacific ocean in 2018. It reached its peak intensity (∼80 𝑚𝑠 -1 ) on 12 September. On 11 September, both S1B at 2048 UTC and Windsat at 2126 UTC overflew Mangkhut (Figs. 7b and7e). According to the SAR observation, Mangkhut had a clearly-defined symmetric eyewall at that time. Note that the eyebrow shape near the eyewall (Fig. 7b) is probably due to rain contamination (for discussion about such a feature see [START_REF] Mouche | Copolarized and cross-polarized sar measurements for high-resolution description of major hurricane wind structures: Application to irma category 5 hurricane[END_REF]). The extent of high winds was seemingly well captured by the radiometer sensor, but the eye was not resolved. Nevertheless, a very good agreement between S1B and Windsat wind outer-profiles is obtained for this case (Fig. 7h), with only ∼3 𝑘𝑚 error between 𝑅 64 estimates from the two sensors. Still, the estimate given by 𝑅 𝐶𝐾22-𝐵𝑅 𝑚𝑎𝑥 (∼30 𝑘𝑚) largely overestimates the actual SAR 𝑅 𝑚𝑎𝑥 (∼20 𝑘𝑚). With its large 𝑅 64 and small 𝑅 𝑚𝑎𝑥 at that time, Mangkhut illustrates the high variability that occurs in nature. Such a case is likely to depart from any statistical relationship (like CK22) that links a wind radius to 𝑅 𝑚𝑎𝑥 .

The last case study (Fig. 7, right column), Kong-Rey, in 2018, was a super typhoon reaching category 5 on the Saffir-Simpson scale, also evolving over the Western North Pacific ocean.

Following a ∼72 𝑚𝑠 -1 peak intensity on 2 October Kong-Rey experienced an eyewall replacement cycle (ERC) and entered its weakening phase. Kong-Rey was captured on 2 October by both S1A at 2111 UTC and SMAP at 2133 UTC (Figs. 7c and7f). The SAR observation depicts a well-defined symmetric eyewall, with a secondary ring of maximum winds further out from the TC center. In fact, Kong-Rey exhibited two eyewalls in 89 GHz imagery at this time (not shown). These two high wind regions were not well captured by the radiometer. The radiometer wind profile saturates in the 80 𝑘𝑚 inner-part of the TC, while the SAR wind profile exhibits two wind speed local maxima (Fig. 7i). Despite the inability of the radiometer sensor to capture the duel wind maxima observed at this time, the outer-part of the azimuthally-averaged wind profiles match well, both yielding a 𝑅 64 estimate of ∼128 𝑘𝑚. Though, 𝑅 𝐶𝐾22-𝐵𝑅 𝑚𝑎𝑥 is 42 𝑘𝑚, far from the 14 𝑘𝑚 of SAR 𝑅 𝑚𝑎𝑥 . However, it is noteworthy that 𝑅 𝐶𝐾22-𝐵𝑅 𝑚𝑎𝑥 lies between the two SAR wind maxima. The complex shape of Kong-Rey during its ERC is the main cause to explain such a huge discrepancy. Indeed, the 𝑅 64 estimate is pushed to an outer radius due to the existence of secondary wind maxima.

b. Structural aspects

From these examples, we see that neither the use of high quality data (SAR) to train the algorithm nor the use of a radius that is very close to 𝑅 𝑚𝑎𝑥 (i.e 𝑅 64 ) precludes large uncertainties of 𝑅 𝑚𝑎𝑥 estimates using the CK22 framework. Underlying CK22, the use of an outer wind radius (e.g 𝑅 34 1) to estimate 𝑅 𝑚𝑎𝑥 is justified by the angular momentum conservation principle: an air parcel, advected from the outer radii to the innermost radii, must lose angular momentum due to surface friction. The ratio 𝑀 𝑚𝑎𝑥 𝑀 34 thus represents the ability for an air parcel to keep its angular momentum 1In this section we chose 𝑅 34 as outer wind radius for clarity, but the reasoning well applies to any other wind radius (e.g 𝑅 50 and 𝑅 64 ). while being advected from 𝑅 34 to 𝑅 𝑚𝑎𝑥 . In the log-linear framework, this ratio solely depends on 𝑉 𝑚𝑎𝑥 , 𝑅 34 , and 𝑓 .

The use of these three parameters to estimate 𝑀 𝑚𝑎𝑥 𝑀 34 was discussed in Chavas et al. (2015) and [START_REF] Chavas | A model for the complete radial structure of the tropical cyclone wind field. part ii: Wind field variability[END_REF]. In these studies, the ability of a radial parametric wind profile to represent the variability of observational data was tested. In brief, the radial parametric profile geometrically merges an inner-part profile with an outer-part profile, previously anticipated from theoretical studies [START_REF] Emanuel | Self-stratification of tropical cyclone outflow. part i: Implications for storm structure[END_REF][START_REF] Emanuel | Tropical cyclone energetics and structure[END_REF]). [START_REF] Chavas | A model for the complete radial structure of the tropical cyclone wind field. part ii: Wind field variability[END_REF]

with 𝜋(𝑥) := ( 1 2 𝑥)

1 2-𝑥 a monotonically increasing function (see their eq. 38). This relationship assumes the TC is in steady-state and the Richardson number in the outflow is slightly below one. The latter implies the outflow is self-stratified by small-scale turbulence. Using numerical simulations that resolved convection, [START_REF] Emanuel | Self-stratification of tropical cyclone outflow. part i: Implications for storm structure[END_REF] showed that such an assumption was satisfied in an outflow region near 𝑅 𝑚𝑎𝑥 . This assumption might then not hold true further out. Chavas et al. (2015) suggested that the optimal merging radius between the inner-and outer-part of the model was ∼ 2 -3𝑅 𝑚𝑎𝑥 when fitting the complete parametric profile to observational data.

While not strictly corresponding to the region where the theoretical developments of [START_REF] Emanuel | Self-stratification of tropical cyclone outflow. part i: Implications for storm structure[END_REF] could remain valid, it identifies the region where the inner-part of the model is most likely to apply to the observations. When writing eq. 6, one assumes that the model of [START_REF] Emanuel | Self-stratification of tropical cyclone outflow. part i: Implications for storm structure[END_REF] is still valid at 𝑅 34 , which largely exceeds 3𝑅 𝑚𝑎𝑥 in nature. This might be a strong approximation, but it offers an instructive relationship between the rate of conservation of angular momentum (left-hand side) to a function of 𝐶 𝑘 𝐶 𝑑 , characterizing the balance between energy generation and friction loss (right-hand side). Most importantly, 𝐶 𝑘 𝐶 𝑑 controls the shape of the parametric radial wind profile, with higher values corresponding to more peaked profiles. In practice, unlike 𝐶 𝑘 𝐶 𝑑 values, this shape of the near-peak radial wind profile is more easily quantifiable using SAR data.

To highlight these considerations, we present TC cases that have the same CK22 predictors (𝑉 𝑚𝑎𝑥 , 𝑅 64 , 𝑓 ) but different wind profile shapes near their peak intensities. Comprehensively, the CK22 model cannot fully adjust to peculiar local wind profiles. To quantify the wind profile shapes, a Holland parametric profile [START_REF] Holland | An analytic model of the wind and pressure profiles in hurricanes[END_REF]) was adjusted to each SAR azimuthally-averaged wind profile:

𝑉 𝐻𝑜𝑙𝑙𝑎𝑛𝑑 (𝑟) = 𝑉 𝑚𝑖𝑛 + √ (𝑉 𝑚𝑎𝑥 -𝑉 𝑚𝑖𝑛 ) 2 ( 𝑅 𝑚𝑎𝑥 𝑟 ) 𝐵 𝑒 1-( 𝑅𝑚𝑎𝑥 𝑟 ) 𝐵 + ( 𝑟 𝑓 2 ) 2 - 𝑟 𝑓 2 (7)
This parametric formulation is useful to quantify variations in the shape of observed wind profiles. In particular, the empirical B parameter controls the rate of radial decay of the tangential winds, with higher (smaller) values corresponding to narrower (broader) vortices. In addition, this parameter was found to be sensitive to TC intensity and size while independent of 𝑅 𝑚𝑎𝑥 [START_REF] Knaff | Simple diagnosis of tropical cyclone structure via pressure gradients[END_REF].

Note, Holland's profiles were designed for gradient-level wind and are not necessarily suited for surface wind profiles with nonzero wind speeds at the TC center, well captured using SAR observations. A complementary degree of freedom (𝑉 𝑚𝑖𝑛 ) was thus included in eq. 7 to cope with the existence of nonzero minimum wind speeds.

Using the full extent of the wind profile, a solution for 𝑉 𝑚𝑖𝑛 , 𝑉 𝑚𝑎𝑥 , 𝑅 𝑚𝑎𝑥 and 𝐵 can be estimated via least squares. Applied to TC Rammasun and Maria, the fitting procedure results in two different 𝐵 values, ∼2.1 and ∼1.7, respectively (Fig. 8). Such a difference quantifies the remaining variability of the near-core wind profile for comparable outer-core wind profiles.

c. Analysis framework

The shape of the near-core wind profile is generally associated with the radial gradient of absolute angular momentum and thus the loss of angular momentum when an air parcel is advected from 𝑅 34 to 𝑅 𝑚𝑎𝑥 . To guide the analysis, we recall the equation of angular momentum conservation for an axi-symmetric vortex: with 𝜏 𝜃𝑠 ≈ 𝐶 𝑑 𝜌𝑉2 the surface stress, 𝐶 𝑑 a drag coefficient and 𝑉 the tangential surface wind component. Assuming 𝑤(𝑧 = 0) = 0 and the use of the continuity equation (eq. 9), the second term of the left hand-side in eq. 10 is integrated by parts, following developments presented by [START_REF] Kalashnik | On the maximum wind velocity in the tropical cyclone[END_REF], to obtain

𝜕 𝑀 𝜕𝑡 + 𝑢 𝜕 𝑀 𝜕𝑟 + 𝑤 𝜕 𝑀 𝜕𝑧 = 𝑟 𝜌 𝜕𝜏 𝜃𝑧 𝜕𝑧 (
𝑟𝑉 2 ≈ - ℎ𝑢 𝐶 𝑑 𝑑𝑀 𝑑𝑟 (13)
where 𝑑𝑀 𝑑𝑟 is the radial gradient of absolute angular momentum at the top of the boundary layer.

Assuming the latter closely related to its value at the surface, eq.13 then explicitly links the shape of the wind profile 𝑑𝑀 𝑑𝑟 to 𝑟𝑉 2 .

Using SAR measurements, both quantities can be accurately estimated, and the validity of eq. 13 assessed. Figure 9a Moreover, the space spanned in the (𝑅 34 𝑉 2 34 , 𝑅 𝑚𝑎𝑥 𝑉 2 𝑚𝑎𝑥 )-plane is still apparently large, even at constant 𝐵. From eq. 13, this increased variability is possibly associated with the factor ℎ𝑢 𝐶 𝑑 .

Overall, these results suggest that the variability encountered in nature does not solely depends on the three predictors (𝑉 𝑚𝑎𝑥 , 𝑅 34 , 𝑓 ). To further illustrate this diagnosis, Fig. 9b displays the same (𝑅 34 𝑉 2 34 , 𝑅 𝑚𝑎𝑥 𝑉 2 𝑚𝑎𝑥 )-plane, but using the radiometer and scatterometer database, and corresponding 𝑉 𝑅𝐸𝐺 𝑚𝑎𝑥 , 𝑅 34 and 𝑅 𝐶𝐾22-𝐵𝑅 𝑚𝑎𝑥 estimates. As expected, the variability captured by using 𝑅 34 or 𝑅 50 to estimate 𝑅 𝑚𝑎𝑥 via CK22 is poor. While using 𝑅 64 increases this variability, the overall spread is reduced compared to Fig. 9a, suggesting that the variability of the wind profile shapes associated with the 𝑅 𝐶𝐾22-𝐵𝑅 𝑚𝑎𝑥 estimates is low.

Note, the average situation 𝑟𝑉 2 ≈ constant which is depicted in our study thanks to the SAR database has already been discussed by [START_REF] Riehl | Some relations between wind and thermal structure of steady state hurricanes[END_REF] when he argued that PV is conserved within the inflow layer. PV conservation implies the vertical component of the curl of the frictional force to be zero, or

𝑟 𝜌 𝜕𝜏 𝜃𝑧 𝜕𝑧 = constant (14)
Integrating this equation over the boundary layer height yields (assuming constant density):

𝑟𝜏 𝜃𝑠 𝜌 = 𝐶 𝑑 𝑟𝑉 2 = constant (15)
Thus, for a constant or slowly varying drag coefficient 𝐶 𝑑 , PV conservation leads to 𝑟𝑉 2 ≈ constant [START_REF] Riehl | Some relations between wind and thermal structure of steady state hurricanes[END_REF]. Mentioned above, such a relationship is, on average, consistent with the SAR estimates. However, for this relationship, the only source of variability comes from 𝐶 𝑑 . From arguments raised above (eq. 13), ℎ and 𝑢 should also be further considered.

Lastly, one limitation of our observational analysis is that SAR 𝑉 𝑚𝑎𝑥 is an estimate of the maximum total wind speed rather than the maximum tangential wind speed. Knowing how the total wind speed is distributed between its tangential and radial component near the eyewall region would allow to better estimate the impact of 𝑢 on PV conservation and its variability. For further comparison, we computed 𝑅 𝑚𝑎𝑥 estimates from 𝑅 34 on the radiometer and scatterometer data using eq. 7 of [START_REF] Chavas | A simple model for predicting the tropical cyclone radius of maximum wind from outer size[END_REF]. Finally, the density contours of the radiometer and scatterometer dataset with 𝑅 𝐶𝐾22-𝐵𝑅 𝑚𝑎𝑥 span a larger space than IBTrACS in the (𝑉 𝑚𝑎𝑥 , 𝑅 𝑚𝑎𝑥 )-plane. This shows that the former captures more variability than best-track data. This is likely due to the use of 𝑅 64 in the regression, a result already suggested by Fig. 9b. Even though the datasets don't have the same 𝑉 𝑚𝑎𝑥 distributions, Fig. 10 also suggests that the radiometer and scatterometer density contours span less space than SAR observations in the (𝑉 𝑚𝑎𝑥 , 𝑅 𝑚𝑎𝑥 )-plane. While this is consistent with the above analysis, more SAR cases are needed to properly interpret Fig. 10.

Conclusions and perspectives

Understanding TC intensity changes certainly remains an observationally challenging problem.

As expressed during the Tenth International Workshop on Tropical Cyclones (IWTC-10, recommendation 4), both the operational and research communities recognize the need for more homogeneous and standardized datasets for TC wind structure parameters, such as 𝑅 𝑚𝑎𝑥 and the wind radii. The fact that 𝑅 34 was not systematically reanalyzed in all basins, and that 𝑅 𝑚𝑎𝑥 is still not reanalyzed today (best-track 𝑅 𝑚𝑎𝑥 value typically stems from its operational estimate) hampers the consolidation of such a dataset. Systematic and standardized wind radii are needed when using, and further improving, a semi-empirical model such as CK22. Although satellite sensors have their limitations, especially regarding the inter-calibration of different missions and sensors, resulting multi-modal observations shall serve for such a systematic and global approach, at least for wind radii estimation.

More specifically and thanks to high-resolution (SAR) data, it is now possible to more systematically estimate 𝑅 𝑚𝑎𝑥 . Fitted with SAR estimates and used in conjunction with the closest wind radius to 𝑅 𝑚𝑎𝑥 , our study proposed a revised CK22 model. It is shown to be an efficient tool to provide improved reliable estimates, with an average uncertainty of ∼ 9 𝑘𝑚. Because outer-core wind radii can be estimated from radiometer or C-band scatterometer data, the developed framework thus allows to produce a more extensive dataset of reanalyzed 𝑅 𝑚𝑎𝑥 estimates. The resulting time series are generally more realistic than those obtained from best-track 𝑅 𝑚𝑎𝑥 estimates. The method can also be used to provide operational guidance on the location of the maximum intensity every time a radiometer or C-band scatterometer overflies the TC, as long as its intensity and location are also estimated, noting that such estimates are routinely available from operational centers. Furthermore, the proposed method could also be used to guide the best-tracking process when no reliable 𝑅 𝑚𝑎𝑥 observation is available.

The efficacy of the semi-empirical CK22 model stems from fundamental conservation principles.

Indeed, the high-resolution SAR database highlights that TCs, on average, conserve their PV, with a resulting approximation 𝑟𝑉 2 ≈ constant. Accordingly, the use of CK22 to retrieve 𝑅 𝑚𝑎𝑥 , based on an outer-radius wind observation coupled with an intensity estimate is, on average, justified.

Single cases can still depart considerably from the PV conservation assumption, especially those at very high intensity (𝑉 𝑚𝑎𝑥 ) or with large inner-(𝑅 𝑚𝑎𝑥 ) or outer-size (𝑅 34 ). And, to first order, those deviations are well explained by variations of the observed wind profile shapes.

While the use of 𝑅 64 can account for some of the deviations due to the radial gradient of absolute angular momentum, the CK22 model seems to fail to capture the remaining variability observed in the SAR database. Large variability is apparently still occurring near the TC core.

To further advance our understanding, there continues to be a need for spaceborne SAR and airborne SFMR sensors as these are the only tools that resolve surface winds in this area. Both sensors however suffer from a lack of spatio-temporal sampling, and airborne measurements suffer from a lack of azimuthal coverage. The future is bright with the recently launched RADARSAT Constellation Mission (RCM) operated by CSA, which should improve the satellite SAR spatiotemporal sampling. RCM has already proved useful by providing significantly more 𝑅 𝑚𝑎𝑥 estimates than anticipated for the 2022-2023 season. And, increasing the number of available SAR cases will certainly allow to better understand how absolute angular momentum gradients are constrained in the near-core region.

Furthermore, the integrated equations show that both the boundary layer depth (ℎ), the average radial inflow (𝑢), and the drag (𝐶 𝑑 ) also impact the relationship between PV conservation and the near-core wind profile shape. While the 𝐶 𝑑 behaviour under very high winds is still actively debated [START_REF] Powell | Reduced drag coefficient for high wind speeds in tropical cyclones[END_REF][START_REF] Bell | Air-sea enthalpy and momentum exchange at major hurricane wind speeds observed during cblast[END_REF], measurements of both ℎ and 𝑢 may be facilitated by the Doppler-based motions derived from the Imaging Wind and Rain Airborne Profiler (IWRAP) instrument [START_REF] Sapp | Processing of high-resolution hurricane ida boundary layer winds from the iwrap instrument on the noaa wp-3d aircraft[END_REF]. For the radial inflow, improved estimates at the surface, in the nearcore region, shall be made possible with the future Harmony mission (ESA 2022), the ESA Earth Explorer 10. This mission will augment Sentinel 1D observations with two satellite companions, providing azimuth diversity from these bi-static observations. In addition, the Second Generation Meteorological Operational satellite programme (Metop-SG) will operate in both co-and crosspolarization. Unlike the current spaceborne instruments, ASCAT, which have only co-polarization measurements, the higher sensitivity of cross-polarized signals to ocean breaking waves may thus improve the ocean surface wind vectors measured by scatterometers, approaching the TC core regions. Also, the coming Copernicus Imaging Microwave Radiometer (CIMR) promises to offer large swath with improved resolution, low uncertainty observation capabilities, combining L-, Cand X-band frequencies . The presence of 1.4 GHz L-band channel on board CIMR will open up the possibility to further interpret the high-resolution C-and X-band measurements, to provide improved surface wind vector estimates under extreme conditions [START_REF] Kilic | Expected performances of the copernicus imaging microwave radiometer (cimr) for an all-weather and high spatial resolution estimation of ocean and sea ice parameters[END_REF].

Finally, in the absence of high-resolution observations, the shape of the near-core wind profile may also be indirectly estimated. Given the relation 𝑟𝑉 2 ≈ constant under a steady-state assumption, a departure from this relation can help understand the temporal variations of absolute angular momentum. Estimates of these temporal variations may then be used to evaluate how much the near-core wind profile shape departs from the average relationship. The wind profile shape is also linked to the drag coefficient (see for instance the steady-state view of [START_REF] Emanuel | Self-stratification of tropical cyclone outflow. part i: Implications for storm structure[END_REF]), which modulates asymmetries in the boundary layer response [START_REF] Shapiro | The asymmetric boundary layer flow under a translating hurricane[END_REF][START_REF] Kepert | The dynamics of boundary layer jets within the tropical cyclone core. part i: Linear theory[END_REF]. Asymmetries possibly captured by medium-or low-resolution observations (scatterometers or radiometers), may thus help to infer boundary layer frictional drag terms, and to quantify the resulting shape of the wind profile.

Fig. 1 .

 1 Fig. 1. Comparison between two Rankine profiles inspired by the SAR acquisition over TC Lane on 23 August 2018 at 0437

  Fig. 2. (Top) Comparison between radiometer (y-axis) and corresponding IBTrACS (x-axis) wind radii. Coefficient of determination (𝑅 2 ), Mean Error (ME) and Residual Standard Deviation (RSD) are displayed. (Bottom) Corresponding distributions and averages.

Figure 3

 3 Figure3shows comparisons between C-band scatterometer wind radii and IBTrACS values.

Fig. 3 .

 3 Fig. 3. Same as Fig. 2, but for the C-band scatterometer wind radii.

Fig. 4 .

 4 Fig. 4. Same as Fig. 3, but only for the 3-year period 2018-2020.

Fig. 5 .

 5 Fig. 5. Comparison between 𝑅 𝑚𝑎𝑥 estimates using the CK22 model and SAR 𝑅 𝑚𝑎𝑥 (top) and corresponding distributions (bottom) for 𝑅 𝐶𝐾22-𝑅 34 𝑚𝑎𝑥

Fig. 6 .

 6 Fig. 6. Kilo (2015) time series of IBTrACS 𝑅 𝑚𝑎𝑥 (left axis, dashed blue), radiometer-and scatterometer-based 𝑅 𝐶𝐾22-𝐵𝑅 𝑚𝑎𝑥

Fig. 7 .

 7 Fig. 7. Comparison of SAR and radiometer wind fields (top and middle rows, TCs are translating toward the top of each panel) and corresponding wind profiles (bottom row) for Olivia (left column), Mangkhut (middle column) and Kong-Rey (right column).

Fig. 8 .

 8 Fig. 8. SAR wind profiles for Rammasun (solid red) and Marie (solid blue) and associated Holland best-fit profiles (dotted curves) fitted on 0 ≤ 𝑟 ≤ 500𝑘𝑚.

  8)with 𝑢 and 𝑤, the radial and vertical velocities, 𝜏 𝜃𝑧 a tangential stress component, and 𝜌 the density. The continuity equation links 𝑢 and 𝑤 as condition, eq. 8 can be integrated from the surface to a boundary layer height, ℎ, where the stress vanishes:

  we can approximate the integral ∫ ℎ 0 𝑢𝑀 𝑑𝑧 ≈ ℎ𝑢𝑀 | 𝑧=ℎ and rewrite the continuity equation 𝑤| 𝑧=ℎ = -ℎ 𝑟 𝑑 𝑑𝑟 (𝑟𝑢). Rearrangement finally yields:

Fig. 9 .

 9 Fig. 9. Evaluation of the PV conservation assumption in the SAR dataset (a) and for Kilo's life cycle (b) using 𝑅 34 estimated on radiometer and scatterometer data along with corresponding 𝑅 𝐶𝐾22-𝐵𝑅 𝑚𝑎𝑥

Fig. 10 .

 10 Fig. 10. Density contours of (𝑉 𝑚𝑎𝑥 , 𝑅 𝑚𝑎𝑥 ) joint distribution for the SAR dataset (shaded green, 𝑉 𝑚𝑎𝑥 based on IBTrACS), for the dataset based upon radiometers and scatterometers with 𝑅 𝐶𝐾22-𝐵𝑅 𝑚𝑎𝑥

table 1

 1

	) over the period 2010-

  shows a comparison between radiometer wind radii and IBTrACS values (top) and their corresponding distributions (bottom). While radiometer wind radii look well correlated with IBTrACS values, with 𝑅 2 -scores ranging from 0.4 to 0.5, large discrepancies arise, with a Residual Standard Deviation (RSD) as high as 56.7 𝑘𝑚 for 𝑅 34 . The RSD decreases to 37.3 𝑘𝑚 for 𝑅 50 , and further to 24.1 𝑘𝑚 for 𝑅 64 , reflecting the decrease of the mean wind radius, i.e. 181 𝑘𝑚 for 𝑅 34 to 51 𝑘𝑚 for 𝑅 64 in IBTrACS. In terms of relative uncertainties, this leads to ∼31%, ∼36%, and ∼41%for 𝑅 34 , 𝑅 50 , and 𝑅 64 , respectively. Interestingly, the Mean Error (ME) is negative for both 𝑅 34 and 𝑅 50 , showing that, on average, these wind radii are lower when extracted from azimuthally-averaged radiometer profiles than from IBTrACS. This is likely the result of the differing definition of the

wind radii in the satellite data and in IBTrACS. Indeed, on average, a wind radius extracted from an azimuthally-averaged profile is expected to be smaller than the maximum radial extent of the same wind speed. Biases due to the differing definition are lower for 𝑅 50 and 𝑅 64 than for 𝑅 34 , because these radii are smaller on average. This definition effect is illustrated on the distribution for 𝑅 34 , where the radiometer 𝑅 34 distribution is biased toward lower values compared to IBTrACS.

Table 3 ,

 3 penultimate column).In what follows, we thus refers to the dataset obtained by this collocation

		SMOS SMAP AMSR-2 Windsat ASCAT TOTAL
	Before filtering	106	63	0	100	0	269
	After filtering	67	33	0	45	0	145

procedure as the "SAR-radiometer collocation dataset". It consists in predictors estimated on radiometer data (wind radii) or corresponding IBTrACS values (𝑉 𝑅𝐸𝐺 𝑚𝑎𝑥 and 𝑓 ), and predictands estimated on SAR (𝑅 𝑚𝑎𝑥 ).

Table 3 .

 3 Number of collocations between SAR and the inter-calibrated dataset (radiometer and ASCAT).

  concluded that the ratio 𝑀 𝑚𝑎𝑥 𝑀 0 between the angular momentum at 𝑅 𝑚𝑎𝑥 and at an outer-radius 𝑅 0 solely depends on four parameters: 𝑉 𝑚𝑎𝑥 , 𝑓 𝑅 0 , 𝐶 𝑘 𝐶 𝑑 , and 𝑊 𝑐𝑜𝑜𝑙 𝐶 𝑑 , where 𝐶 𝑘 and 𝐶 𝑑 are the heat and momentum exchange coefficients, while 𝑊 𝑐𝑜𝑜𝑙 models the radiative-subsidence rate in the free troposphere of the outerpart model. Considering 𝑅 0 = 𝑅 34 , a log-linear dependence of 𝑀 𝑚𝑎𝑥 𝑀 34 on (𝑉 𝑚𝑎𝑥 , 𝑅 34 , 𝑓 ) thus neglects the variations of both 𝐶 𝑘 𝐶 𝑑 and 𝑊 𝑐𝑜𝑜𝑙 𝐶 𝑑 . Besides, the axisymmetric and steady-state theory of Emanuel and Rotunno (2011) invokes a direct relationship between 𝑀 𝑚𝑎𝑥 𝑀 34 and 𝐶 𝑘 𝐶 𝑑 , that can be stated as

	𝑀 𝑚𝑎𝑥	= 𝜋(	𝐶 𝑘	)
	𝑀 34		𝐶 𝑑

  represents 𝑅 𝑚𝑎𝑥 𝑉 2 𝑚𝑎𝑥 (y-axis) as a function of 𝑅 34 𝑉 2 34 (x-axis) and colored by the fitted 𝐵 values2. On average, i.e 𝐵 ≃ 1.8, a relationship emerges when comparing 𝑅 𝑚𝑎𝑥 𝑉 2 𝑚𝑎𝑥 and 𝑅 34 𝑉 2 34 . Departures from a one-to-one relationship, related to conservation of the 𝑟𝑉 2 parameter, are seemingly well explained by 𝐵 values. Large 𝐵, corresponding to very peaked wind profiles near 𝑉 𝑚𝑎𝑥 , leads to larger 𝑅 𝑚𝑎𝑥 𝑉 2 𝑚𝑎𝑥 for a given 𝑅 34 𝑉 2 34 . For broader wind profiles, corresponding to smaller 𝐵, smaller 𝑅 𝑚𝑎𝑥 𝑉 2 𝑚𝑎𝑥 are generally found.

  𝑅 𝑚𝑎𝑥 , while, in fact, this is just a consequence of the lack of SAR data at these intensities.Nevertheless, and more importantly for high surface winds, discrepancies in 𝑅 𝑚𝑎𝑥 densities are observed. Indeed, on average IBTrACS density contours are centered on a higher 𝑅 𝑚𝑎𝑥 (∼30 𝑘𝑚)

	d. Comparison of 𝑅 𝐶𝐾22-𝐵𝑅 𝑚𝑎𝑥	with existing 𝑅 𝑚𝑎𝑥 estimates
	With these results in mind, we assessed how much 𝑅 𝐶𝐾22-𝐵𝑅 𝑚𝑎𝑥	estimates improved existing 𝑅 𝑚𝑎𝑥
	than SAR (progressively decreasing to ∼20 𝑘𝑚). Confirming the efficacy of the revised model,
	radiometer-and scatterometer-based density contours display an average 𝑅 𝐶𝐾22-𝐵𝑅 𝑚𝑎𝑥	(∼20 𝑘𝑚) that
	is consistent with SAR 𝑅 𝑚𝑎𝑥 . Depicted by the 𝑅 𝑚𝑎𝑥 density curves (right panel), for low 𝑅 𝑚𝑎𝑥 ,
	IBTrACS density is lower than both SAR and 𝑅 𝐶𝐾22-𝐵𝑅 𝑚𝑎𝑥	values.

estimates. Figure

10

displays density contours of (𝑉 𝑚𝑎𝑥 , 𝑅 𝑚𝑎𝑥 ) joint distributions using IBTrACS 𝑅 𝑚𝑎𝑥 (dashed blue) or 𝑅 𝐶𝐾22-𝐵𝑅 𝑚𝑎𝑥 estimates (solid black). For comparison, the same density contours are shaded for the SAR dataset (green).

We remind readers that SAR wide-swaths acquisitions cannot be continuously performed over the ocean. As a consequence, not only does the SAR dataset contain much fewer cases, it is also biased towards higher intensities. Indeed, acquisition orders are most often requested to observe higher intensity systems. Thus, for the lowest 𝑉 𝑚𝑎𝑥 (less than ∼ 30 𝑚𝑠 -1 ), possible inconsistencies in 𝑅 𝑚𝑎𝑥 densities arise when comparing SAR to radiometer and scatterometer or IBTrACS. The density contours suggest that both IBTrACS 𝑅 𝑚𝑎𝑥 and 𝑅 𝐶𝐾22-𝐵𝑅 𝑚𝑎𝑥 estimates are larger than SAR

𝐵, as a scalar value, was used instead of a criterion based on 𝑑𝑀 𝑑𝑟 to describe the shape of the wind profile

Acknowledgments. This work was financially supported by the ERC Synergy project 856408-STUOD, the ANR projects OceaniX and ISblue, and the ESA Marine Atmosphere eXtreme Satellite Synergy project (MAXSS). The radiometer and scatterometer dataset used in this report is part of the MAXSS project. The SAR database was obtained from IFREMER/CyclObs and produced with the SAR wind processor co-developed by IFREMER and CLS. J. Knaff thanks NOAA/Center for Satellite Applications and Research for providing the time work on this subject.

The views, opinions, and findings contained in this report are those of the authors and should not be construed as an official National Oceanic and Atmospheric Administration or U.S. government position, policy, or decision.

Data availability statement. The data used in this study are freely available online for both the dataset of radiometer and scatterometer winds (https://www. odatis-ocean.fr/donnees-et-services/acces-aux-donnees/catalogue-complet/ #/metadata/6c56bcde-050f-42eb-92b8-8e882e1f4db9) and the SAR database (https://cyclobs.ifremer.fr/). APPENDIX A

Scatterometer wind speed estimates

As explained in section 2, the wind speed estimates from different radiometer and scatterometer sensors have been inter-calibrated prior to our study. During this process, the C-band ASCAT missions were calibrated using a 25 𝑘𝑚 resolution, while the Ku-band scatterometer sensors were calibrated using a 50 𝑘𝑚 resolution. Spatial resolution was already demonstrated to impact how well TCs intensities are resolved in numerical models [START_REF] Davis | Resolving tropical cyclone intensity in models[END_REF] and observations [START_REF] Quilfen | Observation of tropical cyclones by high-resolution scatterometry[END_REF]). Here, we expect discrepancies between the C-and Ku-band observational wind products.

To quantify this resolution effect, SAR wind fields were degraded to both 25 and 50 𝑘𝑚 spatial resolution and then azimuthally-averaged. The 𝑉 𝑚𝑎𝑥 values estimated from these degraded wind profiles were then compared to IBTrACS 𝑉 𝑚𝑎𝑥 , as represented by the green (25 𝑘𝑚) and red (50 𝑘𝑚) stars of Fig. A1. Here, SAR 𝑉 𝑚𝑎𝑥 refers to the maximum found in an azimuthally-averaged wind profile. We thus expect slight discrepancies with IBTrACS 𝑉 𝑚𝑎𝑥 , whose definition does not strictly coincide with a wind profile maximum. The comparison between SAR azimuthal means and IBTrACS is indicated by the grey stars and modelled by a linear fit (grey dashed line in Fig. A1) which defines 𝑉 𝑅𝐸𝐺 𝑚𝑎𝑥 :

The green and red scatters in


S Jankov

A Domiciano De Souza

F Vakili

R G Petrov

S Robbe-Dubois

New Interferometric Constraints on the Physics of Active OB-Stars

In the recent years the optical interferometry proved its ability to provide important observational constraints on the physics of active OB stars by spatially resolving not only circumstellar environments but also central stars.

 in the context of the Circum-Stellar Envelope formation of Be stars in combination with their fundamental property of rapid rotation.

We also discuss new interferometric observational constraints from direct measures of Be star photospheres which should prove whether these objects ro-tate close, to a few percent, of their critical velocity or not, with a profound impact on the dynamical models of Be stars disk formation from rapid rota-tion combined to mechanisms like non-radial pulsation, radiation pressure of photospheric hot spots or expelled plasma by magnetic flares.

) which provide a qualitatively new informations that cannot be obtained with classical techniques as photometry and spectroscopy.

Introduction

The classical primary goal of interferometry has been the determination of fundamental astrophysical parameters characterizing stars through angular size measurements. The combination of the angular diameter with the bolometric flux is the most direct method to obtain the effective temperature of the star. This is an important constraint for comparisons of model atmospheres with stellar spectra. With the advent of accurate parallaxes from the Hipparcos astrometry satellite linear radii can be determined precisely as well. Additionally, interferometry can provide a very precise determination of stellar masses from binary star orbital motion studies as well as other fundamental stellar parameters (e.g. Hummel et al. 1995) The primary observables of an interferometer are the amplitude and phase of the complex visibility function, which is the Fourier transform of the object intensity distribution (Zernike-Van Cittert theorem). In presence of atmospheric turbulence, usually only the squared visibility amplitudes and the triple products are accessible by optical and near-infrared interferometers. A triple product is the product of three complex visibilities corresponding to baselines that form a triangle. The phase of the triple product, the closure phase, is free of atmospheric phase noise. Thus, the measurement of triple products is a method to obtain phase information of the object intensity distribution, i.e. allows us to detect deviations from point symmetry.

However, stellar interferometry is not limited only to these classical applications. For example the exact behavior of photospheric limb darkening leads to better insight into stellar atmospheres theory through wavelength dependence of the uniform disk diameter (e.g. Quirrenbach et al. 1996, Wittkowski et al. 2001). In addition to visibility amplitudes and triple products, the phase of the complex visibility is an interferometric observable as well, if atmospheric and instrumental noise can be controlled or compensated. This can be achieved for example by phase referencing using the same source at another wavelength or a reference star, when the relative position of the object on the sky can be derived with an accuracy which is much higher than the diffraction. If the position of the photocenter of the object depends on the wavelength, such differential phase measurements can be used to derive information on the brightness distribution on the stellar surface. Other applications, as well as interferometric techniques, have been reviewed by Quirrenbach (2001), and here we give a non-exhaustive list updated with some more recent developments in the field.

Rotation

Achernar's pronounced apparent asymmetry (Fig. 1) as revealed by Domiciano de Souza et al. (2003), together with the fact that it is a Be star, raised the question of whether they observed the stellar photosphere with or without an additional contribution from a Circum-Stellar Envelope (CSE). Arguing that the observed asymmetry of Achernar reflects its true photospheric distortion with a negligible CSE contribution, they concluded that the usual Roche approximation does not apply to Achernar, where deviations from gravitational potential and differential rotation must be closely regarded with respect to the internal momentum distribution. One possible scenario is that of the shellular rotation law (Zahn 1992), in which the angular velocity increases toward the stellar center, implying the oblateness larger than that of a uniformly rotating star [START_REF] Maeder | [END_REF]. In this context their result on Achernar's surface distortion should also impact other internal mechanisms like meridional circulation, turbulence, transport and diffusion of the chemical elements and angular momentum, Figure 1. Fit of an ellipse over the observed squared visibilities V 2 translated to equivalent uniform disk angular diameters. Each V 2 is plotted together with its symmetrical value in azimuth. Large error bars correspond to the 66 m baseline while the small ones correspond to the 140 m baseline. The fitted ellipse results in major axis 2a = 2.53 ± 0.06 milliarcsec, minor axis 2b = 1.62 ± 0.01 milliarcsec. The distribution of measurements reveals an extremely oblate shape of Achernar, with a ratio 2a/2b = 1.56 ± 0.05. increase of mass loss with rotation as well as anisotropies in the mass ejection and wind density from rotating stars.

The rotation is the biggest unknown in our current understanding of stellar evolution and especially of hot stars (Maeder & Meynet 2000). However, the new era of long baseline interferometry allows to test models of interior structure and evolution for massive rotating stars (Endal & Sofia 1979, Meynet & Maeder 2000). Reporting a similar very high oblateness of Regulus (α Leonis A), and taking into account the effects of rapid rotation, McAlister et al. (2005) showed that the position of Regulus in this diagram indicate that the star is over-luminous for its mass when compared with the theory (Heger & Langer 2000, Meynet & Maeder 2000) since fresh hydrogen is mixed into the convective cores of rapidly rotating stars. If Regulus is placed on an isochrone without consideration of this excess luminosity, the age of the star is overestimated.

Further, the highly distorted photospheres of Achernar and Regulus poses the question of Be stars rotation rate and the formation of out-flowing disks from Be stars which remains their central puzzle. Under the Roche approximation the observed oblateness indicates that Achernar should rotate close to its critical velocity since the model solution which implies 96% of the critical velocity is more consistent with the data. Original vision of a critically rotating Roche star, ejecting material from its equator, has been discarded in the past by observing that Be stars rotate at most at 70% to 80% of their critical velocity. However, this statistically observed limit may be biased by the fact that close to or beyond such velocities the diagnosis of Doppler-broadened spectral lines fails to determine the rotation value due to gravity darkening. The direct measures of Be star photospheres by interferometry can overcome the challenge to prove whether these objects rotate close, to a few percent, of their critical velocity or not. This will have a profound impact on the dynamical models of Be stars disks formation from rapid rotation combined to mechanisms like pulsation, radiation pressure of photospheric hot spots or expelled plasma by magnetic flares.

The new interferometric methods very significantly extended the analysis initiated by Slettebak (1949) for rapidly rotating O and B stars which includes the effects of limb darkening, gravity darkening based on Roche models incorporating the von Zeipel Effect (von Zeipel 1924), and differential rotation. Domiciano de Souza et al. (2002) showed that the interferometric signal from an asymmetric star depends on the sky orientation of the baseline ξ and gravity darkening coefficient β and stellar inclination i (Fig. 2). This method has been applied successfully by Domiciano de Souza et al. (2005) Using spectrally resolved interferometry with VLTI/AMBER [START_REF] Petrov | Proc. of the SPIE[END_REF] the differential rotation can be studied as well (Domiciano de Souza et al. 2004). The shift of the stellar photometric barycenter (photocenter) in function of wavelength is very sensitive to the differential rotation coefficient α (Fig. 3) and stellar inclination (Fig. 4) allowing to deduce this two parameters in the approximation of solar-like latitudinal (l) angular differential rotational law Ω = Ω eq (1 -α sin 2 l).

Non-Radial Pulsations, Surface Inhomogeneities and Magnetic Fields

The significant line profile variability of O and B stars due to the non-radial pulsations (e.g. [START_REF] Rivinius | [END_REF], magnetic fields (Neiner et al. 2003) and associated inhomogeneities stimulated recently the development of methods in order to spatially resolve such phenomena on stellar surfaces.

In parallel to the development of interferometric techniques, the method of Doppler Imaging (Rice 1996 and references therein) allowed access to the spatial structure of non-resolved stars through observation and interpretation of temporal spectroscopic variability along the stellar rotational phase. Similarly to classical spectroscopy, the differential interferometry (Beckers 1982[START_REF] Petrov | Diffraction-Limited Imaging with Very Large Telescopes[END_REF]) makes it possible to measure the shift of the stellar photocenter of an unresolved star as a function of wavelength. The photocenter is the angular vectorial function and it provides the first order moment of the spatial brightness distribution, in addition to the zero order moment spectroscopic information, allowing better spatial resolution of stellar atmospheres when compared to the classical Doppler Imaging (Jankov et al. 2001). Their method of Interferometric-Doppler Imaging has been applied to non-radial stellar pulsations (Jankov et al. 2001[START_REF] Jankov | ASP CS[END_REF], while applications to surface inhomogeneities and magnetic fields have been studied by [START_REF] Jankov | Proc. of the SPIE[END_REF] and [START_REF] Rousselet-Perraut | [END_REF] respectively.

  to the determination of gravity darkening of Altair, while McAlister et al. (2005) deduced the same value β = 0.25 for Regulus.

Figure 2 .

 2 Figure 2. Complex visibility amplitudes |V | as a function of projected baseline (in meters) at selected values of orientation of the baseline with respect to the rotation axis of the star (ξ), inclination (i) and gravity darkening coefficient (β). Part of the first and the second |V | lobes are shown. The curves cover part of the first and the second |V | lobes for three different projected baseline orientations corresponding to ξ = 0 deg (solid), 45 deg (dashes) and 90 deg (dots).

Figure 3 .Figure 4 .

 34 Figure3. Left: The photocenter shift components, parallel (top) and orthogonal to the rotation axis of the star (middle), given in units of angular stellar radius. The flux spectrum normalized to the local continuum flux is shown at the bottom. The curves correspond to a stellar inclination i = 45 deg and differential rotation parameter α equal to -0.5 (solid), 0.35 (dots) and 0.6 (dashes). Right: The corresponding Fourier transform amplitudes. The line styles correspond to the curves at the left panel and note that they depend strongly on α. The solid gray curves are the reference analytical Fourier transforms of the rotation broadening functions for uniform rotation and a linear limb darkening parameter ε = 0.23. In the case of uniform rotation the photocenter shift components, parallel to stellar rotation axis is equal to zero, so that there is no gray curve in the upper right panel.