# On the Tropical Cyclone Integrated Kinetic Energy Balance

Arthur Avenas, Alexis Mouche, J. Knaff, Xavier Carton, Bertrand Chapron

► To cite this version:

Arthur Avenas, Alexis Mouche, J. Knaff, Xavier Carton, Bertrand Chapron. On the Tropical Cyclone Integrated Kinetic Energy Balance. Geophysical Research Letters, 2024, 51 (16), e2024GL108327 (10p.). 10.1029/2024GL108327. hal-04777762

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Distributed under a Creative Commons Attribution 4.0 International License
# Geophysical Research Letters®

## RESEARCH LETTER

10.1029/2024GL108327

**Key Points:**

* High-resolution spaceborne synthetic aperture radar measurements inform on the tropical cyclone kinetic energy balance

* The tropical cyclone integrated kinetic energy balance is controlled by the surface wind decay and thermodynamical characteristics

* Accumulating high-resolution surface wind measurements shall allow to better assess trends in the tropical cyclone destructive potential

**Supporting Information:**

Supporting Information may be found in the online version of this article.

**Correspondence to:**

A. Avenas,
arthur.avenas@ifremer.fr

**Citation:**

Avenas, A., Mouche, A., Knaff, J., Carton, X., & Chapron, B. (2024). On the tropical cyclone integrated kinetic energy balance. *Geophysical Research Letters*, *51*, e2024GL108327. https://doi.org/10.1029/2024GL108327

Received 18 JAN 2024

Accepted 26 JUL 2024

**Author Contributions:**

**Conceptualization:** A. Avenas

**Formal analysis:** A. Avenas

**Investigation:** A. Avenas

**Methodology:** A. Avenas

**Software:** A. Avenas

**Supervision:** A. Avenas

**Validation:** A. Avenas

**Visualization:** A. Avenas

**Writing – original draft:** A. Avenas, A. Mouche, J. Knaff, X. Carton, B. Chapron

**Writing – review & editing:** A. Avenas, A. Mouche, J. Knaff, X. Carton, B. Chapron

## On the Tropical Cyclone Integrated Kinetic Energy Balance

A. Avenas¹ ID, A. Mouche¹ ID, J. Knaff² ID, X. Carton¹, and B. Chapron¹ ID

¹Laboratoire d'Océanographie Physique et Spatiale (LOPS), Ifremer, University Brest, CNRS, IRD, IUEM, Plouzané, France, ²NOAA/NESDIS Regional and Mesoscale Meteorological Branch, Fort Collins, CO, USA

<figure><img src="image_2.png" /></figure>

**Abstract** Current global historical reanalyzes prevent to adequately examine the role of the near-core surface wind structural properties on tropical cyclones climate trends. Here we provide theoretical and observational evidences that they are crucial for the monitoring of integrated kinetic energy. The kinetic energy balance is reduced to a simple rule involving two parameters characterizing the surface wind structure and directly suggested by the governing equations. The theory is uniquely verified with a database of high-resolution ocean surface winds estimated from all-weather spaceborne synthetic aperture radar. Such measurements provide indirect estimates of a multiplicative constant modulating the kinetic energy balance and associated with the system thermodynamics. Consequently, accumulated high-resolution acquisitions of the ocean surface shall allow to better monitor the integrated kinetic energy and provide new means to tackle climatological studies of tropical cyclones destructiveness.

**Plain Language Summary** Studying the long-term climate trends of tropical cyclones is challenging because the historical data is not always reliable. One particular issue concerns the accurate reporting of surface wind properties near the core of these storms in past and present records. This study uses both theory and high-resolution surface wind observations from satellite radar to highlight the importance of investigating these properties, specifically for monitoring the total energy, which is a measure of a storm destructive potential. Two spatial scales describing the tropical cyclone wind structure are identified and may be efficiently measured thanks to the high-resolution sensor. The storm energy equilibrium is shown to be controlled by these two spatial scales, in both theory and observations. This equilibrium is also influenced by the temperature characteristics of a storm, which are themselves modulated by environmental and climatological conditions. Consequently, future high-resolution observations from the satellite radar should help better understanding the dependence of integrated kinetic energy with space and time.

## 1. Introduction

Expressing the combined effect of intensity and size, Integrated Kinetic Energy (IKE) measures the tropical cyclone (TC) destructive potential (Powell & Reinhold, 2007). Understanding the fundamental physics governing this integrated quantity, to better anticipate its evolution in a global warming context, is thus of major importance. Until this day, research studies focused on examining the climate-dependence of intensity (K. Emanuel, 2005; K. Emanuel, 2021; Kossin et al., 2020; Kossin, 2017; Patricola & Wehner, 2018; Sobel et al., 2016; Webster et al., 2005; Wang & Toumi, 2021) and more recently, of size (Chavas & Emanuel, 2010; Knaff et al., 2014; Wang & Toumi, 2021, 2022). Modulated by climate change, the sea surface temperature and the atmospheric temperature and humidity vertical profiles control both TC intensity (Done et al., 2022; Gilford et al., 2017; K. Emanuel, 2007; Wing et al., 2015; Strazzo et al., 2015) and size (Chavas et al., 2016; Lin et al., 2015). While a few methods have been tested to assess past and future IKE trends (Kozar & Misra, 2014; Kreussler et al., 2021; Misra et al., 2013; Morris & Ruf, 2017; Wang & Toumi, 2016, 2021), less is known about how oceanic and atmospheric parameters affect IKE and its variations. This lack of knowledge may be problematic for both operations and research, especially if the TC vitals were to fail capturing parameters that are critical to assess IKE.

In steady-state theories describing axisymmetric TCs, kinetic energy gained through the heat source is hypothesized to balance that lost through the dissipation source (Anthes, 1974; Golitsyn, 2008; K. A. Emanuel, 1986; Kalashnik, 1994; K. A. Emanuel, 1995; Ooyama, 1982; Pearce, 2004; Riehl, 1963). Analytical criteria expressing this steady-state balance may then be derived provided further assumptions on the outflow and inflow layer of TCs. For instance, Riehl (1963) assumes conservation of absolute angular momentum in the upper outflow of TCs. Momentum losses then solely occur in the surface inflow. Without reliable surface wind speed estimates,

<footnote>© 2024. The Author(s).
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.</footnote>
one way to express momentum losses is to assume potential vorticity (PV) conservation in the inflow, which leads to (Riehl, 1963):

$$C_d r v^2 = \text{cst} \qquad (1)$$

With $C_d$, r and v drag coefficient, radius (i.e., distance from TC center) and tangential velocity, respectively. Under hydrostatic and cyclostrophic balances, the heat source, expressed as the vertical gradient of atmospheric temperature, may be related to the gradient-level wind structure. While the accuracy of Equation 1 in TCs remains to be substantiated, steady-state balance can still be temptingly assessed using surface wind estimates only.

The justification of an overall PV conservation was first facilitated by aircraft data (Riehl, 1963; Riehl & Mal-kus, 1961) and later by numerical modeling capacities (K. A. Emanuel, 1986; Ooyama, 1982). Observational and experimental research efforts then concentrated on a better characterization of the $C_d$ parameter under high wind speed conditions (Black et al., 2007; Bell et al., 2012; Curcic & Haus, 2020; M. Donelan et al., 2004; Jarosz et al., 2007; M. A. Donelan, 2018; Powell et al., 2003; Soloviev et al., 2014) following Emanuel's steady-state theory (K. A. Emanuel, 1986; K. A. Emanuel, 1995). In such a context, Synthetic Aperture Radar (SAR) has emerged as a promising satellite technology capable of producing fine-scale, wide-swath TC boundary-layer process data in nearly all-weather conditions (A. A. Mouche et al., 2017; A. Mouche et al., 2019). SAR surface wind estimates provide an unprecedented opportunity to examine the TC radial wind structure (Avenas et al., 2023; Combot et al., 2020) and complete existing theories for steady IKE balance.

In the present study, we aim at understanding the fundamental laws governing the steady IKE and its relationship to the TC surface winds. Starting from existing theoretical developments (Charney & Eliassen, 1964; Kalashnik, 1994; Riehl, 1963), we reduce the steady IKE balance to a simple rule that involves two parameters describing the surface wind structure, all measurable with a high-resolution wind profile estimate. This theory is then tested across an extended database of SAR high-resolution observations (178 cases), and discussed with respect to IKE variations estimates from best-track data. The relationship between the SAR-derived surface wind structure parameters and thermodynamic quantities that are most relevant to IKE balance is emphasized. Consequences of Equation 1 on the drag coefficient are also examined through the lens of the SAR measurements. Our investigation suggests that systematic knowledge of the wind structure parameters, especially if they were included in TC vitals, would not only help assessing the IKE balance, but also improve future climatological IKE studies.

## 2. Preliminary SAR Diagnostic

Spaceborne SAR allows for high spatial resolution estimates of the TC surface wind speeds (see Text S1 in Supporting Information S1). From the 178 SAR surface wind field estimates, Figure 1a displays that of TC Lane on 23 August 2018, while Figure 1b displays that of TC Meranti on 12 September 2016. Both the outer-, near- and inner-core regions of TCs are well captured by SAR observations. The resulting axisymmetric profiles (green curves in Figures 1c and 1d) show that both the axisymmetric maximum intensity ($V_{max}$) and radius of maximum wind ($R_{max}$) may be accurately retrieved (Combot et al., 2020). The system center can also be precisely located (Vinour et al., 2021), whose latitude provides the Coriolis parameter (f).

Controlling both the momentum losses and the amplitude of vertical velocities at the top of the boundary layer, the surface wind decay is critical to the IKE balance (see below). It may be quantified in terms of an effective Holland $B_s$ parameter (Holland, 1980), once a Holland parametric wind profile is adjusted (purple curves in Figures 1c and 1d) to the SAR axisymmetric wind profiles estimates (see Text S2 in Supporting Information S1). Note that the wind decay could be characterized using quantities derived from other adjusted parametric wind profiles, for example, the exponent of a modified Rankine vortex. Such alternative quantities are expected to be well correlated to $B_s$, especially in the near-core region, and thus the results of the present study shall not be affected by this arbitrary choice. Seemingly, Lane and Meranti (Figures 1c and 1d) had substantially different wind decays (hereafter $B_s$ values) while similar TC vitals (i.e., $V_{max}$ and $R_{max}$ values). The question arises how this difference impacts the IKE balance.
<figure><img src="image_3.png" /><figcaption>Figure 1. SAR wind speed estimates for (a) Lane and (b) Meranti. Corresponding axisymmetric wind profile (green) and adjusted Holland parametric wind profile (purple) for (c) Lane and (d) Meranti.</figcaption></figure>

## 3. Deriving the IKE Balance Rule

### 3.1. Structural Parameters: Definition and Analysis

Assuming a constant air density, the steady-state balance between momentum sink and heat source writes (see Text S3 in Supporting Information S1):

$$ \int_{0}^{R_0} [C_d r v^3]_{z=0} dr = U_c^2 \left[ \frac{C_d r v^2}{\omega_z + f} \right]_{z=0, r=R_+} \quad (2) $$

with $\omega_z = \frac{\partial}{\partial r}(rv)$ the vertical component of relative vorticity and $U_c$ a constant which depends on the ther-
modynamics of the system. Note, $z = 0$ refers to the top of the boundary layer. $R_+$ and $R_0$ are two radii characteristic of the IKE balance. The former defines the region of significant upward motions, while the latter defines the integration volume. The amplitude of vertical motions at the top of the boundary layer due to Ekman pumping are expressed by

$$w_E(r) = \frac{1}{r} \frac{d}{dr} \left( \frac{C_d r v^2}{\omega_z + f} \right) \quad (3)$$

Considering a slow numerator variation $C_d r v^2 \approx \text{cst}$, $\omega_z$ decreases with $r$, and $w_E$ becomes close to zero for radii where $\omega_z$ is of the same order of the Coriolis parameter $f$. Conversely, significant upward motions occur in a region where $\omega_z$ is at least a few times higher than $f$. With $\omega_z$ monotonically decreasing from a maximum near the TC core to the outermost radii, $R_+$ may be defined as

$$\omega_z(R_+) = 5f \quad (4)$$

With this definition, the characteristic radius ($R_+$) and the corresponding surface wind speed ($V_+$) can be directly estimated using a SAR axisymmetric wind profile (Figures 1c and 1d).

Specifying the integration volume, $R_0$ is introduced as a natural characteristic radius because of the assumption of absolute angular momentum conservation in the outflow layer (Riehl & Malkus, 1961). Accordingly, if $R_0$ is defined as the radius where the outflow velocity vanishes, it is directly related to $Ro_{max} := \frac{V_{max}}{fR_{max}}$, the Rossby number evaluated at $R_{max}$ via

$$\sqrt{2Ro_{max}} = \frac{R_0}{R_{max}} \quad (5)$$

This radius ($R_0$) and the corresponding surface wind speed ($V_0$) can thus be directly estimated from the SAR surface wind speeds (Figures 1c and 1d).

The two characteristic radii $R_0$ and $R_+$ are controlled by the wind structure parameters $Ro_{max}$ and $B_s$ (see Text S4 in Supporting Information S1). Hence, in what follows we propose to reduce the steady-state balance (e.g., Equation 2) to a relationship involving these structural ($B_s$ and $Ro_{max}$), in addition to a thermodynamical ($U_c$) parameter.

## 3.2. The IKE Balance Rule

Equation 2 involves quantities from the inflow layer, so that recalling the argument of PV conservation (Equation 1), it is tempting to divide each side of Equation 2 by $C_d r v^2$. Equivalently, this corresponds to consider a constant drag coefficient $C_d$ and a relation of the kind $rv^2 \approx \text{cst}$, consistent with the view of K. A. Emanuel (1986) for typical air-sea temperature differences.

Figure 2 shows how the normalized ratio $\kappa_* := \frac{rv^2}{R_*V_*^2}$ varies as a function of the normalized radius $r_* := \frac{r}{R_*}$ for all the wind profiles of the SAR database. On average (solid black curve), the normalized ratio varies slowly with radius, confirming the approximation $rv^2 = \text{cst}$. Note that these slow radial variations reach a maximum at $r = R_+$. While Equation 4 has been quite arbitrarily, yet reasonably, defined, $R_+$ appears to correspond to the radius that maximizes, on average, the integrand of the IKE over the SAR database. This local maximum provides information on the radius and area where a TC is the most efficient, that is, where heating is maximum and momentum sink is minimum. This a posteriori justifies our definition for $R_+$.

For single cases, variations of this normalized ratio $\kappa_*$ with radius mostly stress the variations in $B_s$ and $Ro_{max}$. Deviations from the average $\kappa_* \approx 1$ have opposite signs at $R_{max}$ and $R_0$. For instance, high $B_s$ and $Ro_{max}$ values (red profiles covered by the shaded green area) result in $\kappa_*(R_{max}) > 1$ and $\kappa_*(R_0) < 1$, suggesting that errors inside $R_+$ compensate those outside $R_+$ when considering $rv^2 \approx \text{cst}$ and simplifying the integral in Equation 2.

Following these results, Equation 2 can be rewritten as
<figure><img src="image_4.png" /><figcaption>**Figure 2.** SAR-derived $\kappa_*$ profiles colored by adjusted Holland $B_s$ parameter (blue to red) and average $\kappa_*$ profile (black) estimated from the SAR database. Shaded green area denotes the standard deviation interval inside which SAR cases satisfy $Ro_{max} > 50$.</figcaption></figure>

$$ \int_{0}^{R_0} v(z=0) \, dr = \frac{U_c^2}{6f} \qquad (6) $$

Now, the aim is to further express Equation 6 in terms of parameters that can be estimated from SAR data, that is, $V_{max}$, $B_s$ and $Ro_{max}$. Here and after it is assumed that the SAR surface wind speed estimates and the corresponding parameters are close to their boundary layer top counterparts. We also recall that, lacking reliable high resolution ocean surface wind vectors data, these parameters were computed using the total wind speed (as provided by SAR) rather than its tangential component. The integral in Equation 6 is approximated as

$$ \int_{0}^{R_0} v(z=0) \, dr \approx \frac{V_{max}R_0}{2\sqrt{B_s}} \qquad (7) $$

Leading to

$$ \frac{V_{max}R_0}{\sqrt{B_s}} \approx \frac{U_c^2}{3f} \qquad (8) $$

With use of Equation 5, a TC with steady-state conditions should then obey the following rule:

$$ V_{max}^2 = \frac{U_c^2}{3\sqrt{2}} \sqrt{B_s Ro_{max}} \qquad (9) $$

Which expresses the IKE balance, now reduced to structural parameters ($B_s$ and $Ro_{max}$) that we can estimate from SAR data. Before testing this rule across the entire SAR database, the degree of IKE steadiness for each observational case must be estimated.
# 4. Observational Assessment of the IKE Balance Rule

## 4.1. Best-Track Estimates of IKE Variations

Equation 9 assumes that the TC is in steady-state, that is, that the partial time derivative of azimuthal wind speed and potential temperature can be neglected at each radii. The current spatio-temporal sampling of TC surface observations, including SAR measurements, prevents a direct estimation of the time evolution of these quantities. Yet, neglecting time variations in the potential energy equation, a TC departs from the steady-state assumptions when the absolute temporal variation of IKE is large. Temporal evolution of IKE is given by:

$$ \frac{\partial \mathrm{IKE}}{\partial t} = \frac{\partial}{\partial t} \left( \int_{0}^{H} \int_{0}^{R_{0}} \bar{\rho}_{0} r v^{2} dr dz \right) \qquad (10) $$

Building on our observational knowledge on the $κ_*$ ratio (Figure 2), the double integral in Equation 10 can be simplified by considering the constant $rv^2$ at a fixed relevant radius. To evaluate $\frac{\partial \mathrm{IKE}}{\partial t}$, we would then need partial time derivative estimates of $v$ and $r$ at this fixed radius. Because of the limited temporal sampling of SAR data, temporal evolutions of these parameters must be evaluated using best-track (Knapp et al., 2010) reanalyzes. However, while $V_{\max}$ best-track uncertainty is rather low (Landsea & Franklin, 2013; Torn & Snyder, 2012), $R_{\max}$ best-track estimates have been shown to be often inconsistent with SAR $R_{\max}$ estimates (Combot et al., 2020). Indeed, $R_{\max}$ is not systematically reanalyzed (unlike $V_{\max}$). From best-track data, the most reliable size parameter is the radius of gale $R_{34}$, that is, the maximum radial extent of the 34-knots winds (Knaff et al., 2021). Thus, we use the following approximation:

$$ \frac{\partial \mathrm{IKE}}{\partial t} \approx \frac{\partial}{\partial t} \left( \int_{0}^{H} \bar{\rho}_{0} R_{34}^{2} V_{34}^{2} dz \right) \qquad (11) $$

where $V_{34}$ is the azimuthal surface wind speed at $r = R_{34}$. Limited by observational capabilities, the dependence of $R_{34}$ in $z$ is unknown. The integral on the vertical component is thus simplified by a multiplication by $H$. Following the zero PV approximation, we assume that surfaces of constant potential temperature and absolute angular momentum coincide, so that in steady-state $H$ scales as (Shutts, 1981)

$$ H = \frac{V_{\max}^{2}}{g \frac{\Delta\theta}{\theta_{0}}} \qquad (12) $$

where $g$ is standard gravity and $\Delta\theta$ the difference between the potential temperature at the vortex center and its environmental value noted $\theta_0$. Finally, Equation 11 writes

$$ \frac{\partial \mathrm{IKE}}{\partial t} = \frac{\bar{\rho}_0 V_{34}^2}{g \frac{\Delta\theta}{\theta_0}} \frac{\partial}{\partial t} (V_{\max}^2(t) R_{34}^2(t)) \qquad (13) $$

where $\bar{\rho}_0 \approx 1.15 \text{ kg.m}^{-3}$ and $\frac{\Delta\theta}{\theta_0} \approx 10^{-2}$ are assumed constant in time. Now we may test the IKE balance rule (Equation 9) across the observational database.

## 4.2. Testing the IKE Balance Rule Across the SAR Database

In Equation 9, we can assume that $U_c$ does not vary too much across different TCs, especially for the present analysis which was restrained to tropical and sub-tropical latitudes (see Text S1 in Supporting Information S1). Figure 3 then shows the SAR observations in a ($V_{\max}^2$, $\sqrt{B_s R_{\max}}$) log-linear plane. The corresponding values of absolute partial time derivative of IKE (Equation 13) evaluated using best-track data (see above and Text S5 in Supporting Information S1) are also represented (colors). Cases with the lowest third absolute IKE time derivative estimates (black stars) satisfy the relation of proportionality suggested by Equation 9, as modeled by the least squares regression (dashed black curve, $R^2 = 0.64$), with a variance probably due to both observational errors and variations in the characteristic velocity $U_c$.
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Wind structure parameters estimated from SAR observations (stars) in the plane suggested by Equation 9 (y-axis is in logarithmic scale), and colored by absolute value of $\frac{\partial \mathrm{IKE}}{\partial r}$. Dashed lines denote best-fit linear regressions to cases with the lowest third (black), or highest two thirds (orange/red) absolute IKE time derivative estimates, using a fixed zero intercept.</figcaption></figure>

The velocity $U_c$ characterizes the steady IKE balance in Equation 9. Using known values of thermodynamic constants and assuming that heating occurs in the lowest layers of the TC, we find that $U_c \sim 32$ m/s from its definition (see Text S3 in Supporting Information S1). Based on the steady IKE linear regression slope (black dashed curve in Figure 3), the SAR observations lead to $U_c \sim 27$ m/s, which is close to the theoretical value. We remind readers that this characteristic velocity is that of an average situation. In nature, $U_c$ certainly varies from one TC to another.

Remarkably, cases with the highest two thirds absolute IKE time derivative estimates (orange/red stars) may also be modeled by a least squares regression (dashed orange curve) in the ($V_{\mathrm{max}}^2, \sqrt{B_s R o_{\mathrm{max}}}$) plane, although with more variance ($R^2 = 0.52$) than their steady counterparts, certainly associated with neglected unsteady terms during the derivation of Equation 9. For such cases undergoing significant unsteady IKE transitions, the characteristic velocity $U_c \sim 31$ m/s based on the regression slope is higher than that found for cases with a steady IKE. Here, the steady IKE rule and the resulting characteristic velocity are practical to interpret typical IKE changes across the different groups of cases.

Characterizing the surface winds, the structural parameters ($B_s$ and $R o_{\mathrm{max}}$) are critical to determine $U_c$ and assess the IKE balance. As an illustration, TC Lane, well captured by a SAR observation (Figures 1a and 1c), had a relatively small IKE variation ($\sim -0.7$ PJhr$^{-1}$), corresponding to small changes of

$V_{\mathrm{max}}$ and $R_{34}$ in best-track data. In contrast, for TC Meranti (Figures 1b and 1d), the high (positive) temporal variation of IKE ($\sim 24.3$ PJhr$^{-1}$) corresponds to a high temporal variation of $R_{34}$ ($\sim 4.4$ kmhr$^{-1}$) due to an eyewall replacement cycle, while $V_{\mathrm{max}}$ stayed relatively stable ($\sim 0.3$ ms$^{-1}$ hr$^{-1}$). While Lane and Meranti had similar $R o_{\mathrm{max}}$ values (90 and 100, respectively) at the time of their acquisitions, their $B_s$ values differed significantly (2.2 and 1.3, respectively), especially considering the typical range of possible $B_s$ values (between 1 and 2.5, see also Avenas et al. (2023)) encountered in the SAR database. Characterizing the effective area where significant energetic exchanges occur, $B_s$ and the corresponding near-core surface wind distribution were recently found to control the short-term evolution of the system (Avenas et al., 2024). Necessary to accurately estimate $B_s$ and $R o_{\mathrm{max}}$, measurements of the TC wind structure at high-resolution, for instance using SAR sensors, are thus crucial for future IKE studies.

## 5. Concluding Remarks

Based on existing steady-state theories describing axisymmetric TCs and a PV conservation argument in the inflow, the steady IKE balance was reduced to a simple rule involving two structural parameters suggested by theory and measurable on high-resolution surface data. The derived rule is shown to be consistent with high-resolution SAR observations. In contrast to previous studies (e.g., K. A. Emanuel (1986)), the maximum intensity ($V_{\mathrm{max}}$) that a TC with steady conditions achieves in our work does not involve the exchange coefficients of enthalpy and momentum, whose values at high wind speeds are still actively debated, but rely on an accurate knowledge of the surface winds ($B_s$ and $R o_{\mathrm{max}}$) and a scalar quantity ($U_c$). With only one scalar unknown as degree of freedom, the proposed framework thus allows to efficiently assess the TC dynamical state (i.e., both intensity and IKE balance) from the surface winds only.

In Riehl (1963) conceptual framework, the momentum losses are characterized by an assumption of PV conservation close to the surface. Equation 1 was thus considered in the inflow layer and $C_d$ further assumed constant. The SAR database analysis reveals that the distribution of the regions where this assumption is valid depends on the Rossby number $R o_{\mathrm{max}}$. Indeed, in Figure 2, SAR cases with $R o_{\mathrm{max}} < 50$ (curves outside the green shaded area) have a $\kappa_*$ ratio that increases with $r$, so that if Equation 1 was valid, $C_d$ would decrease with $r$ and thus increase with $v$, in agreement with the reported literature (M. Donelan et al., 2004; Powell et al., 2003) for wind speeds below hurricane force winds ($\sim 33$ m/s). At higher Rossby numbers $R o_{\mathrm{max}} > 50$ (curves inside the green shaded area), the decrease of $\kappa_*$ with $r$ between $R_+$ and $R_{\mathrm{max}}$ suggests that $C_d$ decreases/saturates. For such TCs, hurricane force winds are largely exceeded in this area and the suggested decrease/saturation of $C_d$ agrees with reported
estimates under hurricane conditions (Black et al., 2007; Bell et al., 2012; Curcic & Haus, 2020; M. Donelan et al., 2004; Jarosz et al., 2007; M. A. Donelan, 2018; Powell et al., 2003; Soloviev et al., 2014). At radii greater than $R_+$, the assumption expressed by Equation 1 certainly breaks down because the $\kappa_*$ ratio decreases with $r$. A relation of relative angular momentum conservation $rv \approx cst$ is approached, so that friction is presumably small in this region. As a result, vertical velocities at the top of the boundary layer are close to zero, because both $C_d \rho v^2$ and $\omega_z$ are small in Equation 3. Further understanding how $R_+$, or more generally the wind decay, is related to $C_d$ increase/decrease with wind speed is beyond the scope of this study, but SAR observations may be instrumental to help better determining the spatial distribution of $C_d$.

Assuming steady conditions for the system, Equation 9 can be used to indirectly estimate the characteristic velocity $U_c$ from a given SAR observation. Strongly influencing the IKE balance (Equation 9), $U_c$ depends on both oceanic and atmospheric parameters. As a consequence, our understanding of the basin- and climate-dependence of $U_c$, and in turn the IKE, should benefit from the increasing number of spaceborne SAR sensors (e.g., the recently launched Radarsat Constellation Mission) and the corresponding accumulation of $U_c$ estimates. In the absence of SAR, the developed theory suggests that the knowledge of the near-core wind decay for example, with $B_y$ and the maximum Rossby number with $Ro_{max}$ (or equivalently $R_0$ and $R_+$), along with the maximum intensity $V_{max}$, should be sufficient to estimate the TC characteristic velocity $U_c$ and assess the IKE balance. Presently, while the maximum intensity may be one of the most reliable parameters in the TC vitals, accurate estimates of the near-core wind decay and the maximum Rossby number are not systematically available. Consistently including reliable estimates of these two structural parameters in operational and historical records would allow, in combination with theory, to better monitor short- and long-term changes in TCs destructive potential.

## Data Availability Statement

Data sets for this research are freely available online at https://cyclobs.ifremer.fr/app/tropical using the steps described at https://cyclobs.ifremer.fr/app/docs/.

## Acknowledgments

This work was financially supported by the ERC Synergy project 856408-STUOD. 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. We also acknowledge an anonymous reviewer for helpful comments.

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