Fetch effects on air-sea momentum transfer at very high wind speeds

ABSTRACT Accurate estimates of momentum flux through the air-sea interface at very high wind speeds are important for predicting tropical cyclone intensities. To estimate the air-sea momentum flux under the long-fetch condition (20 m fetch) at very high wind speeds using a laboratory tank, a simple momentum flux measurement method using only four water-level gauges is conducted based on the momentum budget method. The air-water momentum flux under long-fetch conditions at very high wind speeds was measured in a typhoon simulation tank at the Research Institute of Applied Mechanics, Kyushu University. The verification was performed using previous values estimated by the eddy correlation method in a typhoon simulation tank at Kyoto University, Japan. The results showed good correlation between the values of momentum flux measured by the present momentum budget method and the other methods. The drag coefficient at very high wind speeds (41 m/s) under long-fetch conditions leveled off, as well as under the short-fetch condition (4.5 m and 6.5 m fetch). Moreover, a very weak relationship was found between the drag coefficient and the fetch. Since the maximum fetch in the present laboratory experiments is 20 m, the future field observation with the longer fetch condition will be needed for applying the results to oceans.


Introduction
The accurate prediction of the development and decay of tropical cyclones (TC) is of great importance for minimizing the loss of human life and damage to the living environment.Precise estimation of the air-sea heat and momentum transfer is essential for reliable prediction of the intensity of TCs, as the development of TCs depends on the balance between the supply of thermal energy from the ocean and the loss of kinetic energy caused by the drag acting on the sea surface (e.g.Miyamoto, Fudeyasu, and Wada 2022;Richter and Wainwright 2023).
The value of the drag coefficient (C D ), which represents the magnitude of the air-sea momentum transfer, is defined as where, τ w is the sea surface wind stress, u* is the friction velocity of air, ρ the density of air, and U 10 the wind speed at 10 m above the sea surface.In the 1960s, laboratory and field experiments (e.g.Hawkins and Rubsam 1968;Kunishi and Imasato 1966) showed that C D increased monotonically with U 10 from low to very high wind speeds.However, recent field experiments (Powell, Vickery, and Reinhold 2003) using global-positioning system sondes in tropical cyclones have shown that C D decreases with an increase in U 10 at very high wind speeds.Furthermore, recent laboratory experiments using a typhoon simulation tank (TST) have shown almost constant values of C D under strong winds of U 10 = 30-68 m/s (Curcic and Haus 2020;Donelan et al. 2004;Takagaki et al. 2012;Troitskaya et al. 2012).This change (regime shift) in the wind velocity dependency occurs at approximately U 10 = 30-35 m/s (Donelan et al. 2004;Takagaki et al. 2012;Troitskaya et al. 2012) and U 10 = 25 m/s (Curcic and Haus 2020).Takagaki et al. (2012) directly measured C D using the eddy correlation method (ECM), and specifically found that C D may be saturated by interface slipping and flattening due to intense wave breaking.Subsequently, Takagaki, Komori, and Suzuki (2016a) and Takagaki et al. (2016b) demonstrated that the equilibrium range constant and peak enhancement factor of the wind-wave spectrum are strongly correlated with the inverse wave age and wind speed at very high and normal wind speeds.They discovered that distinctive wave breaking occurred at very high wind speeds, causing C D saturation.In addition, regime shifts have been reported to occur in air-sea heat and mass transport (Iwano et al. 2013;Komori et al. 2018;Krall et al. 2019;Troitskaya et al. 2020).However, these studies were conducted under fetch conditions shorter than 10 m and distinctive wave breaking and drag saturation under long-fetch conditions have not yet been observed.Furthermore, the wind stress and wave height etc. measurements in high wind speed region were very difficult in field.Even in laboratory experiments, breaking waves and splash droplets significantly affected the measurements.
This study examined the use of the momentum budget method (MBM), that can indirectly measure friction velocity u* which is a parameter of wind stress.In addition, the effects of fetch on the momentum transfer across the breaking sea surface were investigated for very high wind speeds under long-fetch conditions using two TST.Note that present experiments are conducted with the fetch ranging from 4.5 m to 42.5 m.And the definition of "short fetch" and "long fetch" are used for the condition under 10 m or over 10 m in the present laboratory experiments, respectively.

Equipment and measurement methods
We used two TSTs, one for the construction of the momentum budget method (MBM) and the other for the examination of the measurements in the long fetch.Figure 1 shows schematic diagrams of the two TSTs.The first TST (custom-built Masatoyo) was established at Kyoto University (Figure 1a), and the second TST (custom-built West Japan Fluid Engineering Laboratory Co., Ltd.) at the Research Institute of Applied Mechanics, Kyushu University (Figure 1b).
The Kyoto tank (Figure 1a) consisted of a centrifugal fan, settling chamber, glass test section, and outlet duct.The test section was 15.0 m long, 0.8 m wide, and 1.6 m high (0.8 m filled with water).The inlet of the air-side test section was 0.8 m wide and 0.8 m high.The liquid-side of the test section was filled with filtered tap water.Air flowing in the test section generated wind waves on the air-liquid interface at U 10 = 7-68 m/s.The wave absorbers at the inlet and outlet of the test section suppressed wave reflection.The x, y, and z axes corresponded to the flow, span, and vertical directions, respectively.The origin of the measurement coordinate system was at the edge of the entrance slope.The water level and pressure were measured at three fetches: x = 2.5, 6.5, and 10.5 m.The water-level fluctuations and mean wind-velocity were measured at x = 4.5, and 6.5 m.
The Kyushu tank (Figure 1b) had the same structure as the Kyoto tank.The Kyushu tank consisted of an axial fan, test section, and outlet duct.The test section was 54 m long, 1.5 m wide, and 2 m high (1.26 m filled with water).The inlet of the airside test section was 1.5 m wide and 0.7 m high.The sidewall of the test section was partially composed of clear glass.The x, y, and z axes corresponded to the flow, span, and vertical directions, respectively.The origin of the measurement coordinate system was at the edge of the entrance slope.The maximum wind speed in the air-side test section (U ∞ ) was 24.0 m/s and maximum U 10 is 41.0 m/ s.The mean wind-velocity fluctuation and water- In the Kyoto TST, three measurement instrument: a sonic anemometer (Gill Windsonic), laser Doppler anemometer (Dantec Dynamics LDA), and phase Doppler anemometer (Dantec Dynamics PDA) were used to measure the wind velocity (see the details in Takagaki et al. 2012).A total of 37 measurement points were employed at arbitrary intervals of Δz at x = 6.5 m and z = 0-0.61m.The sampling time and frequency were 1000 s and 240 Hz, respectively.In the Kyushu TST, a standard Pitot tube (Okano Works, LK-0), precision differential manometers (DeltaOHM; HD402T2L), and data logger (GRAPHTEC; GL900) were used to measure the wind velocity.A total of 39 measurement points were employed at arbitrary intervals of Δz at x = 20 m and z = 0-0.45m.The sampling time and frequency were 60 s and 10 Hz, respectively.
In Kyoto, water-level fluctuations were measured using resistance-type wave gauges (Kenek CHT4-HR60BNC).The resistance wire was placed vertically across the air-liquid interface, and the instantaneous electrical resistance proportional to the water level was recorded at 500 Hz for 600 s using a digital recorder (Sony EX-UT10).Each wind wave was detected using the zero-up-crossing method.The wave height (H S ) and period (T S ) of the significant wind waves were defined as the mean wave height and length of the largest one-third of the waves.Here, subscript s indicates a significant wave.In the Kyushu TST, water-level fluctuations were measured using custom-built resistance-type wave gauges.A resistance wire was placed in the water, and the electrical resistance at the instantaneous water level was recorded at 200 Hz for 120 s using a logger.From this, H S and T S were estimated.
A custom-built water-level gauge was used to measure the mean water level.The water-side of the waterlevel gauge was connected to the bottom wall of the TST using a vinyl tube.The air-side of the water-level gauge was connected to the Pitot tube set on the top wall of the TST with a vinyl tube in the case of the two water-level gauges in the Kyoto TST (Figure 2a and 2b) or kept open in the case of the four water-level gauges in the Kyushu TST (Figure 2c and 2d).The water column in the water-level gauge, which fluctuated with changing water levels in the TST, was captured using a digital single-lens reflex camera (Nikon D1×) in the Kyoto TST, or the water level was measured using a laser displacement meter (Omron, ZX1-LD300A61) in the Kyushu TST.The sampling time and frequency in Kyoto were 600 s and 1/30 Hz, respectively.Water Capital "A" denotes the connection tube to a pressure gauge, "B" denotes the connection tube to a differential pressure gauge, "C" the wave gauge, and "D" the pitot tube, respectively.The subscripts x and y in the variable h wxy denote fetches in liquid-side and air-side; numbers of 0, 1, 2, 3, and 4 denote the atmosphere release, the liquid-side upstream tube connection, liquid-side downstream tube connection, air-side upstream tube connection, and air-side downstream tube connection, respectively.In (b), two water levels are shown as right to left h w13 , and h w24 . in (d), four water levels are shown as right to left h w13 , h w24 , h w10 , and h w20 .
level was analyzed by motion tracking with Photron FASTCAM Analysis (PFA).The sampling time and frequency in Kyushu were 60 s and 10 Hz, respectively.
The total number of measurement cases was 33, with cases 1-16 performed in the Kyoto TST and cases 17-33 in the Kyushu TST (Table 1).MBM was performed in all 33 cases.Values measured by the ECM from previous study (Takagaki et al. 2017) are also listed as cases 34-44 in Table 1.In cases 34-44, the loop-type wave-generation method (LTWGM) was used for the fetch extension in the Kyoto TST.Consequently, cases 1-16 show the experiments at extreme high wind speeds at short fetch conditions, and cases 17-33 show the experiments at relatively high wind speeds at long fetch conditions.It should be noted that the values of tanh(2πD/L S ) were greater than 0.99, where L S is the wavelength of significant wind waves and D the depth of water, because the maximum value in the Kyoto TST of L S was approximately 1.83 m.This means that the effects of the bottom wall on L S and phase velocity of the significant waves (C P ) were smaller than 1% (see Lamb 1932) and considered negligible in the present study.

Momentum budget method
Surface wind stress τ w is important for estimating the future intensity of hurricanes, as discussed in Section 1. Generally, the air friction velocity (u*) represents the magnitude of the momentum transfer across the airliquid interface, as given by Eq. ( 1).In Addition, the surface boundary layer was regarded as a turbulent boundary layer, and the wind velocity profile over the rough boundary was expressed by the following logarithmic law: Table 1.Wind and wind-wave properties at the Kyoto TST (MBM, x = 4.5 and 6.5 m; ECM, x = 6.5-42.5 m) and Kyushu TST (MBM, x = 20 m).L X : distance (L X = x 2 -x 1 ); x: fetch, N: fan rotational number; U ∞ : freestream wind speed; u*: friction velocity of air; U 10 : wind speed at 10 m above the sea surface; C D : drag coefficient relative to U 10 ; H s : significant wave height; T s : significant wave period.The values of U 10 , and C D in the Kyoto TST at x = 4.0 m were estimated from u* using the empirical curves from Iwano et al. (2013).

Case Facility Metod
Wave generating method where U(z) is the wind speed at elevation z, z 0 is the water surface roughness length, and k is the von Karman constant ( = 0.4).Several techniques for measuring u* have been developed in the field of fluid dynamics, such as, the load-cell method, profile method (PM) (e. g.Kunishi and Imasato 1966), eddy correlation method (ECM) (e. g.Takagaki et al. 2012), and momentum budget (balance) method (MBM) (e. g.Donelan et al. 2004).The load-cell method is a direct technique for measuring wind shear on a rigid wall surface using a load cell embedded in a rigid wall.However, this method cannot be applied to measure wind shear on a mobile free surface.The PM is better for measuring the wind shear when the surface is smooth.It is difficult to apply PM to measure the wind shear on a rough surface because the thickness of the log-law region is less than the height of the roughness elements.The ECM is a better technique for measuring wind shear stress precisely.However, it requires high-frequency measurements for capturing turbulent eddies, and has the weakness of measuring wind shear stress in particle-laden turbulent fields.The MBM is a simple indirect method for measuring wind shear stress, and will be detailed later in this section.
The weakness of the MBM is the difficulty in evaluating the differential momentum flux; however, its advantage is its wide applicability; i.e. the MBM can be used in particle-laden turbulent fields over mobile free surfaces without high-frequency measurement equipment.Therefore, the MBM is best for measuring wind shear stress over an intensive breaking wind-wave turbulent field (e.g.Donelan et al. 2004;Kukulka and Hara 2005).In this study, the MBM was used in two TSTs located in Kyoto and Kyushu (cases 1-33), and u* values measured by ECM (cases 34-44) are listed in Table 1 for verification.
When the MBM was applied to the wind-wave turbulent field, the control volume was set for both the air and liquid sides of the wind-wave turbulent field (see Figure 3).Sections D and G show the air-side and liquid-side control volumes, respectively.The value of I 1 is the inflow of the liquid-side momentum flux upstream, I 2 is the outflow of the liquid-side momentum flux downstream, J 1 is the inflow of the air-side momentum flux upstream, J 2 is the outflow of the airside momentum flux downstream, T a is the momentum flux from the air side to the liquid side across the wavy air-liquid interface, T b is the momentum flux absorbed at the bottom wall, T c is the momentum flux absorbed at the top wall.Here, the momentum flux absorbed at side walls T d is similar to T b , which is known to be negligible small (Donelan et al. 2004).Thus, we neglected T d in present MBM.For ease of measuring and analysis, the origin of the vertical direction height z was set at the bottom wall of the tank (Figure 3).The following six flow-field conditions were assumed: (1) the wind-wave turbulent field is a steady and two-dimensional flow; (2) the air-side pressure changes in the streamwise direction, not in the vertical direction; (3) the slope of the water surface is due to the pressure gradient in the streamwise direction on the air side; (4) although a surface wave is present, only the radiation stress is considered; (5) the amount of water in the tank does not change; and (6) the vertical velocity is small.The flow field was governed by the Reynolds-averaged Navier-Stokes equation on the liquid side: where Eqs. ( 3) and (4) refer to the x and z directions, respectively.In Eqs. ( 3) and (4), U and W are streamwise and vertical mean velocity, respectively, P is pressure, ρ w is the water density, and g is the gravity acceleration.The three τ are stresses, with τ xx being the normal stress acting on the plane perpendicular to the x-axis, τ zz the normal stress acting on the plane perpendicular to the z-axis, and τ xz the tangential stress acting on the plane perpendicular to the z-axis.By considering the previously mentioned fluid-dynamic assumptions and the negligible terms, Eqs. ( 3) and ( 4) are transformed, respectively, into: Thus, the pressure P at x and z (P = P(x, z)) was estimated using Eq. ( 6) and is given by where h(x) represents the water level at fetch x, P(x) represents the pressure at fetch x, and P 0 represents the gauge pressure at the air-water interface.Here, the definition of P(x, h(x)) = P(x)+ P 0 was used.Then, using Eq. ( 7), Eq. ( 5) can be transformed into the following equation: where τ xz (x, z) can be rewritten as τ(x, z).The first spaceintegrated term on the left-hand side of Eq. ( 8) can be transformed as follows: According to the Leibniz integral rule: Eq. ( 9) is transformed into follows: where the relation: is obtained using the surface boundary condition (kinematic balance law).Because the vertical momentum flux (second term in the integral of the third equation in Eq. 11) is negligibly smaller than the streamwise momentum flux (the first term in the integral of the third equation in Eq. 11), Eq. ( 11) is transformed into Here, the streamwise momentum flux across the crosssection at x is defined as: The first space-integrated terms on the right-hand-side of Eq. ( 8) can be transformed as follows: The second space-integrated term on the right-handside of Eq. ( 8) can be transformed as follows: The third space-integrated term on the right-hand-side of Eq. ( 8) can be transformed as follows: The forces per unit width (T a and T b ) are defined as Þdx, respectively.Thus, using Eqs.( 13), ( 15), ( 16), and ( 17), the spaceintegrated Eq. ( 8) can be transformed as follows: where the wave radiation stress S on the left-hand-side of Eq. ( 18) is added in the same manner as in Donelan et al. (2004).Generally, the pressure and water level change linearly with fetch x (e.g.Donelan et al. 2004).Thus, Eq. ( 18) can be transformed as follows: where � h is the mean water height � L X , and L x (= x 2 -x 1 ) is the distance between x 1 and x 2 .Moreover, using the four relations of ΔP = P 2 -P 1 , ΔS = S 2 -S 1 , ΔM = M 2 -M 1 , and T a = τ w L x , Eq. ( 19) can be transformed as follows: where the relation of T b = 0 is used because the forces per unit width acting on the bottom wall T b is negligibly smaller than the forces per unit width acting on the water surface T a (see Donelan et al. 2004).In addition, the forces per unit width acting on the side walls T d is negligibly smaller than T a , since T d is estimated to be double T b .Of note, Eq. ( 20) is similar to Eq. ( 5) of Donelan et al. (2004), except for the fourth term on the right-hand-side of Eq. ( 20).Eq. ( 20) shows that the surface shear stress τ w is balanced with the air-side pressure gradient (first term), water-level change (second term), change of radiation stress (third term), and momentum flux due to the liquid-side kinetic energy (fourth term in Eq. 20).Since present two wind-wave tanks are closed tanks in which the water is conserved, we treated ΔM = 0 according to Donelan et al. (2004): In this study, two momentum budget methods were used: the MBM using two water-level gauges (Figure 2a and 2b) and the MBM using four waterlevel gauges (Figure 2c and 2d).In the MBM with two water-level gauges (Figure 2a and 2b), two pressure gauges and two water-level gauges were used to measure the pressure and water level at x 1 and x 2 .Subsequently, three values of ΔP, � h, and s were estimated.In the Kyoto TST, the differential radiation stresses ΔS are measured by two wave gauges too.Thus, τ w can be estimated using Equation ( 21).
In the MBM that used four water-level gauges (Figure 2c and 2d), the water levels at x 1 and x 2 in the wind-wave tank were defined as h 1 and h 2 , respectively, and the water levels at the first, second, third, and fourth water-level gauges were defined as h w13 , h w24 , h w10 , and h w20 , respectively (see Figure 2d).Using four water levels, h w13 , h w24 , h w10 , and h w20 along with the initial water levels in the wind-wave tank (h 0 ) and water-level gauge (h w0 ), the pressures, slope of the water level, and mean water level were calculated as follows: Using Eqs.(22-25), Eq. ( 21) is transformed as follows: The present MBM using four water-level gauges can estimate the fluid force acting over the wave-breaking air-water interface by microdisplacement measurements.

Results and discussion
Figure 4 shows the vertical distributions of the streamwise mean wind velocity measured in the Kyoto TST at x = 6.5 m (cases 9-16).The velocities were measured using a sonic anemometer, laser Doppler anemometer (LDV), or phase Doppler anemometer (PDA).In the cases 9-12, the velocity boundary layers and free-stream regions are clearly observed in the velocity profiles measured using the LDV and sonic anemometer.However, in cases 13-16, it was difficult to measure the mean wind velocity using LDV and sonic anemometer, because large droplets were dispersed from the surface of the intensively breaking waves.To overcome this problem, the mean wind velocities were measured using the PDA in the same manner as in our previous studies (Komori et al. 2018;Takagaki et al. 2012).In cases 13-16 in the figure, the velocity boundary layers are clearly observed in the velocity profiles measured by the PDA.Moreover, the velocities measured by the LDV, PDA, and sonic anemometer corresponded to each other, except in case 16.Therefore, the wind velocity measured by LDV are used to estimate C D and U ∞ at cases 9-12.The wind velocity measured by PDA is used to estimate C D and the wind velocity by LDV are used to estimate U ∞ at case 13-16.In case 16, the free-stream wind speeds measured by the LDV and sonic anemometer did not correspond.This may be due to the measurement error of the sonic anemometer caused by water droplets wetting the body of the anemometer.
Figure 5 shows the wind velocity dependence of the differential pressure and water level between the two measurement locations in the Kyoto TST (cases 1-16).To verify the magnitudes of the momentum terms defined in Eq. ( 1), the differential variance of the water-level fluctuation and the differential values of the momentum terms are shown in Figure 5. From Figure 5 (a -c), the differential pressure, differential water level, and differential variance of water-level fluctuation between two measurement locations increase with increasing the wind speed U ∞ and measurement distance L x .However, at U ∞ = 4.7 m/s (case 1), the values of the differential pressure, and differential water level at L x = 4.0, 8.0 m correspond to each other, respectively.This might be due to the low measurement precision of the water level and differential pressure gauges.For comparing the total differential momentum term ΔI (= T a ) to each differential momentum term on estimation of the wind stress on the water surface τ w , three differential momentum terms (ΔI a , ΔI b , and ΔS) are described against the wind speed U ∞ in Figure 5 and ΔS (= S 2 À S 1 ) are normalized by the total differential momentum term ΔI (= T a ).Thus, Eq. ( 21) can be transformed into: From Figure 5(d), ΔI a /ΔI has positive values, since the water level increases with increasing the fetch.In contrast, ΔI b /ΔI has negative value, since the pressure decrease with increasing the fetch.Interestingly, two absolute values of ΔI a /ΔI and ΔI b /ΔI have similar magnitude.This might cause the low measurement precision for estimating the wind stress on the water surface τ w since τ w is estimated by the sum of ΔI a and ΔI b .The value of ΔS is negligible compared to the values of ΔI a and ΔI b , which is same trend reported in Donelan et al. (2004).Figure 6 shows the relationship between the friction velocity u* and U ∞ in the Kyoto TST (x = 4.5 m: cases 1-8; x = 6.5 m: cases 9-16).The solid curve shows the best-fit curve obtained from the values measured using ECM in Takagaki et al. (2012).The dotted curve shows the extrapolation of the solid curve at U ∞ <19.3 m/s.The dotteddashed curve represents the friction velocity on a solid surface (Schlichting 1955).Generally, the friction velocity in the low-wind-speed region is expected to show a value similar to that of the solid plate model, because wind waves develop in the low-wind-speed region minimally.Thus, the Schlichting's curve was drown here.From the figure, present values of u * measured by MBM change the velocity trend at U ∞ ~19.3 m/s.The trend is proposed by several previous studies (e.g.Donelan et al. 2004;Takagaki et al. 2012;Troitskaya et al. 2012), and is called a "regime shift."This regime shift is caused by intense wave breaking (e.g.Takagaki et al., 2016b;Troitskaya et al. 2018), and also occurs in mass/heat transport at high wind speeds (e.g.Iwano et al. 2013;Komori et al. 2018;Krall et al. 2019;Troitskaya et al. 2020).The u* measured in x = 4.5 m with L x = 4.0 m corresponds well to u* in x = 6.5 m with L x = 8.0 m except for the lowest wind speed cases (cases 1 and 9).This indicates that u* measured here is not affected by the measurement distance L x and fetch x.At the lowest wind speeds, the values of u* in cases 1 and 9 were scattered because of low measurement precision.In particular, u* in case 9 was highly scattered from the Schlichting's curve (dotted-dashed curve), too.At high wind speeds (U ∞ >19.3 m/s), the present values of u* measured by MBM are higher than the previous values measured by ECM.This may be due to a systematic error between the two measurement methods (MBM and ECM).
Figure 7 shows the fetch dependence of the pressure and water level in the Kyushu TST when U ∞ is 17. 1 and 24.0 m/s at x = 20 m.The pressure decreased with an increase in fetch.In contrast, the water level increased with fetch.This result indicates that the turbulent field is a typical pressuredrop flow field.Therefore, it was shown that the MBM can be used in the Kyushu TST because the pressure and water level at the fetches showed linear changes from 16 to 24 m of the measurement positions used in this study.
In Figure 8, the vertical distributions of the streamwise mean wind speed measured at x = 20 m (cases 17-33) in the Kyushu TST are plotted.The logarithmic profile is clearly shown at z = 0.05-0.3m at all wind-speed conditions.However, the profile is a pseudo-logarithmic profile (wake region), which  has been explained in several previous studies (e.g.Takagaki et al. 2022).Because the surface boundary layer thickness δ is ~0.35 m, the logarithmic law region is provided at z/δ < 0.15 in an aerodynamic wind tunnel (Hinze 1959) or z/δ < 0.3 in the windwave turbulent field (Zavadsky and Shemer 2012).
The water levels h w13 , h w24 , h w10 , and h w20 , measured using the four water-level gauges at x = 20 m (cases 17-33) in the Kyushu TST are plotted in Figure 9.Because the h w24 value (downstream side) is larger than the h w13 value (upstream side), the results indicate that the water level within the tank increases with the fetch.Furthermore, the h w10 (or h w20 ) values (water-level gauges are open to the atmosphere) were greater than the h w13 (or h w24 ) values (water-level gauges connected to both the air and water sides of the tank), indicating that the tank was pressurized.Moreover, four water levels gradually change with increasing U ∞ without water-level fluctuation due to wave fluctuation in TST.Consequently, the momentum flux through the air-water interface can be indirectly calculated using Equation ( 26) of the momentum budget method using four measured water levels (see Section 3 for more information).Therefore, the friction velocity was calculated using MBM.Note, the term of the differential radiation stress was neglected here (see Figure 5d).
Figure 10 shows the relationship between the friction velocity u* and U ∞ in the Kyushu TST (x = 20 m, cases 17-33).These results are similar to those reported by Takagaki et al. (2012).In the high-wind-speed region, the tendency of the friction velocity changed from sharply increasing to gradually increasing with wind speed from 19.3 m/ s, which is the regime shift, as explained in Figure 6.Thus, the friction velocity (and momentum flux) are considered to be independent of the fetch because the present result of the long fetch ( = 20 m) in the Kyushu TST showed values similar to those of Takagaki et al. (2012) for the short fetch (x = 6.5 m).At the low -middle wind speed region (3.0 < U ∞ <8.0 m/s, cases 19-21), the friction velocity shows the slightly higher value than Takagaki et al. (2012).The cause may be a measurement error because the water level difference between the two measurement points is very small, and the wave height of the wind waves in the low -middle wind speed region is small.At very low wind speed region (U ∞ <3.0 m/s, cases 17 and 18), u* shows the value close to the Schlichting (1955), who proposed a momentum transfer model for the solid plate.Generally, the friction velocity in the low-windspeed region is expected to show a value similar to that of the solid plate model, because wind waves develop in the low-wind-speed region minimally.The error bar is small enough to measure u*   in whole wind speed region.Moreover, in Figure 9, four water levels gradually change with increasing U ∞ without water-level fluctuation due to wave effects.Therefore, the MBM can be measured with high accuracy using the four water-level gauges.
To calculate the drag coefficient C D , the wind speed at a height of 10 m above the sea surface was calculated using the logarithmic law of the wind profile.First, we calculated the roughness length z 0 by Equation ( 2) with the wind speed U (z min ) at the lowest measurement height z min .Secondly, using Equation (2), we calculated the wind speed U 10 at the height of 10 m above the sea surface.Finally, using Equation (1), the drag coefficient was calculated using the friction velocity and U 10 .
The relationship between the drag coefficient C D and wind speed at 10 m height above sea surface U 10 for the Kyoto TST (x = 4.5, 6.5 m: cases 1-16) and Kyushu TST (x = 20 m: cases 17-33) data sets together with data sets from previous field measurements (Bell, Montgomery, and Emanuel 2012;Hawkins and Rubsam 1968;Johnson et al. 1998;Mitsuyasu and Nakayama 1969;Powell, Vickery, and Reinhold 2003;Takagaki et al. 2012), laboratory experiments (Curcic and Haus 2020;Donelan et al. 2004;Takagaki et al. 2017), and numerical study (Kurose et al. 2016;Takagaki et al. 2015) are plotted in Figure 11.The plots in cases 34-44 were obtained from Takagaki et al. (2017).The solid and dashed lines represent the models of Takagaki et al. (2012) and WRF version 3 (Skamarock et al. 2008), respectively.All of this study's results (cases 1-44) show values close to those of Takagaki et al. (2012) and small variations in the drag coefficient, which showed a constant value with an increase in U 10 >33.6 m/s.The regime shift is considered to have occurred at U 10 >33.6 m/s.The field measurement data-sets show a large variation in the drag coefficient compared to the laboratory experimental data-sets.However, field measurement data-sets are affected by swelling and wind speed variation etc.However, it is considered that the laboratory experimental data-sets show the phenomenon at purely high wind speeds because of the stable flow field.Considering the effects of swell and wind speed fluctuations etc. on the laboratory experiment, the results of the laboratory experiment may show a similar variation in the drag coefficient as the results of the field measurements.Consequently, this study's results are considered applicable to oceans because the momentum flux (friction velocity and drag coefficient) is not dependent on the fetch.Furthermore, the relationship between the drag coefficient C D and fetch x was investigated by classifying the range of wind speeds U 10 measured at each fetch (15 < U 10 <20, 30 < U 10 <35, and 40 < U 10 <45 m/s).Figure 12 shows the relationship between drag coefficient and fetch, where the values are normalized by values at x = 6.5 m in the Kyoto tank.The equation for the linear regression line is as follows: with a correlation coefficient R 2 = 0.112.Thus, this regression analysis clearly shows a very weak relationship between C D and x.Consequently, the result are considered applicable to oceans because the momentum flux (friction velocity and drag coefficient) is not dependent on the fetch.
Changes in wind waves at high wind speeds were investigated.Figure 13 shows the windwave spectra at wind speeds U 10 = 19.3,32.0, 42.0 m/s.Spectra are described in the fetch ranging from x = 4.5 m to 42.5 m, and these spectra are offset for the clarity of this figure.Note that some wind-wave spectra observed in laboratory experiment were compared to those in field experiments (Donelan, Hamilton, and Hui 1985) and found good agreement of two spectra (Takagaki et al. 2016b).In addition, Figure 14 shows the wind speed dependence of the significant wave height H S and significant wave period T S .In Figure 13, the −4 power spectral slope is found in a frequency region higher than the significant frequency for all wind speeds and fetch conditions.Moreover, as the fetch increased, the peak frequency decreased (Figure 13), and the significant wave period and significant wave height increased (Figure 14).This suggests that wind waves develop with the fetch.Interestingly, C D is independent of the fetch when the fetch is triple at 40 < U 10 <45 m/s, as shown in Figure 12, although H S develops twice with a triple fetch in Figure 14.

Conclusion
We used a momentum budget method that can indirectly measure u*, a wind stress parameter, at high wind speeds.A simple measurement method (present momentum budget method) for wind stress using only four waterlevel gauges is proposed based on the momentum budget method.The friction velocity of air calculated by the present momentum budget method showed good agreement with the friction velocity of air calculated by the previous momentum budget and eddy correlation methods (Takagaki et al. 2012).Moreover, we investigated the fetch effects on momentum transfer across the winddriven breaking air -water interface at extremely high wind speeds using two typhoon simulation tanks.Laboratory measurements showed that the drag coefficients increase with increasing wind speed at the height of 10 m above the sea surface U 10 and become constant at short and long fetches at extremely high wind speeds of U 10 >33.6 m/s.This wind speed tendency is similar to that reported by Takagaki et al. (2012) and is called the regime shift of the transfer coefficient.In addition, our laboratory measurements show that the drag coefficients become constant at short and long fetches at extremely high wind speeds (U 10 >33.6 m/s) despite the significant wave height increasing with an increase in the fetch.Although the maximum fetch in the present laboratory experiments is 20 m, the maximum fetch in the ocean is over 10 km.At normal wind speeds (U 10 <33.6 m/s), there are many laboratory and field experiments for evaluating the fetch dependency of the drag coefficient at normal wind speeds (e. g.Drennan et al. 2003;Toba and Ebuchi 1991).Although the drag coefficient increases with increasing of the wave age at young wave ages (e. g.Toba and Ebuchi 1991), it decreases with increasing of the wave age at old wave ages (e. g.Drennan et al. 2003).Thus, it will be important to verify these new laboratory correlations under long-fetch conditions through future field experiments at extremely high wind speeds.Moreover, it is also important to verify the effects of the surfactant contaminating the water (e.g.Matsuda et al. 2023) and numerous large obstacles (e.g.Horinouchi and Mitsuyuki, 2023) on the air-sea momentum transfer at extremely high wind speeds because surfactant and obstacles significantly change the wave development.

Figure 2 .
Figure 2. Schematic diagram of experiment, (a), (b) MBM in Kyoto University, (c), (d) MBM in Kyushu.Capital "A" denotes the connection tube to a pressure gauge, "B" denotes the connection tube to a differential pressure gauge, "C" the wave gauge, and "D" the pitot tube, respectively.The subscripts x and y in the variable h wxy denote fetches in liquid-side and air-side; numbers of 0, 1, 2, 3, and 4 denote the atmosphere release, the liquid-side upstream tube connection, liquid-side downstream tube connection, air-side upstream tube connection, and air-side downstream tube connection, respectively.In (b), two water levels are shown as right to left h w13 , and h w24 . in (d), four water levels are shown as right to left h w13 , h w24 , h w10 , and h w20 .

Figure 3 .
Figure 3. Schematic diagram of momentum budget method in wind-wave turbulent field.

Figure 4 .
Figure 4. Vertical distributions of streamwise mean wind speed in the Kyoto TST at x = 6.5 m (cases 9-16).

Figure 6 .
Figure 6.Air-side friction velocity u* versus U ∞ in the Kyoto TST (x = 4.5 m: cases 1-8; x = 6.5 m: cases 9-16).Solid curve shows the best-fit curve obtained from values measured by ECM in Takagaki et al. (2012).Dashed curve shows the extension of the solid curve at U ∞ <19.3 m/s.

Figure 7 .
Figure 7. Relationship between fetch x and water level h and upstream pressure P in the Kyushu TST.Black: U ∞ = 17. 1 m/s and x = 20 m; red: U ∞ = 24.0m/s and x = 20 m.

Figure 11 .
Figure 11.Drag coefficient against wind speed U 10 at a height of z = 10 m.Symbols show present and previous data obtained from laboratory and field experiments.Solid and dashed lines show the models of Takagaki et al. (2012) and WRF version 3 (Skamarock and Coauthors 2008), respectively.Color bars show the present fetch.

Figure 14 .
Figure 14.Wind-speed dependences of (a) significant wave height H S , (b) significant wave period T S .