Airsea Interaction under Hurricane Wind Conditions

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Yuliya Troitskaya, Daniil Sergeev, Alexander Kandaurov and Vasilii Kazakov Institute of Applied Physics Russia

1. Introduction

One of the main characteristics appearing in the models of forecasting wind over the sea is the roughness of the sea surface determined by the parameters of the wind waves, quantitatively parameterised by the sea surface drag coefficient CD. To define it we introduce the turbulent shear stress or turbulent momentum flux far from the sea surface

where p is the air density, u* is the wind friction velocity. Wind is the turbulent boundary layer with the logarithmic mean velocity profile:

Similar to the resistance law of the wall turbulent flow the sea surface drag coefficient is introduced as follows:

paU10 U10

where U10 -the wind velocity at a standard meteorological height H10=10 m. which relate this coefficient to U10 are obtained either by generalizing empirical data (Garratt, 1977; Large & Pond, 1981, Taylor & Yelland, 2002; Fairall et al., 2003) or by numerical models (see, for example, Janssen, 1989; Janssen, 1991, Makin, 1994; Hara & Belcher, 2004). Numerous field measurements give increasing dependencies of CD on wind speed, which relates to increasing of wave heights with the wind.

The aerodynamic drag coefficient of the sea surface is a critical parameter in the theory of tropical hurricanes (Emanuel, 1995). To illustrate it we consider here the ideas of theory of energy balance in a tropical cyclone suggested by (Emanuel, 1986; Emanuel, 1995, Emanuel, 2003). According to this theory the mature tropical cyclone may be idealized as a steady, axisymmetric flow whose energy cycle is very similar to that of an ideal Carnot engine, where the hot reservoir is the ocean with the temperature TS and the cold reservoir is .the troposphere with the temperature To. The details of construction and operation of this heat engine are presented in (Emanuel, 1986; Emanuel, 1995, Emanuel, 2003), but one of the most important characteristics of a tropical cyclone, the maximum surface wind velocity, which determines its category, can be estimated without details from the Carnot theorem. According to the Carnot theorem, the maximum efficiency of the ideal heat engine is determined by the absolute temperatures of the hot and cold reservoirs:

Q T0

where Qs is the heat energy entering the system from the hot reservoir and W is the mechanical work done by the system. Heat energy support of the tropical cyclone comes from the ocean (heat flux from the sea surface) and mechanical energy dissipated in the marine turbulent boundary layer (Emanuel, 2003), the heat energy entering the system is the surface integral of the heat flux from the sea Fq and mechanical energy dissipation rate Fp:

Mechanical work done by the system compensates mechanical energy dissipation, then

The heat flux from the sea and the mechanical energy dissipation rate are determined by the bulk formula:

here ko, k are enthalpy at the sea level and in marine atmospheric boundary layer.

In (7)-(8) Ck is heat exchange coefficient (or the Stanton number), CD is surface drag coefficient, defined by equation (3).

Taking into account the Carnot theorem (4) and estimating integrals (5) and (6) yields estimate for the maximum surface wind velocity in a tropical cyclone as a function of ratio CD/Ck.:

Conventional bulk formulas, derived by generalizing experimental data (Garratt, 1977; Large & Pond, 1981, Taylor & Yelland, 2002; Fairall et al., 2003) obtained at wind velocities less than 30 m/s, overestimate the drag coefficient of the sea surface under hurricane winds. The estimates presented in (Emanuel, 1995) indicate that energy dissipation due to friction proves too high to explain the observed velocity of a hurricane wind for realistic sources of energy.

The problem of explaining high wind velocities during hurricanes can be resolved if the drag coefficient of the sea surface does not increase with increasing wind velocity. To explain high quantities of wind speeds observed in tropical cyclones Emanuel, 1995 suggested that the drag coefficient flattens and even decreases at high wind speed in contradiction with intuition, since, it follows then, that sea surface should be effectively smoothed under the hurricane conditions. However, in the late 90-th these dependencies were observed experimentally in the field and laboratory conditions (Powell et al, 2003).

2. Observations of the sea surface drag reduction

The effect of sea surface drag reduction under hurricane wind was discovered by (Powell et al, 2003) in their experiments on measurements of the wind velocity profiling in the marine atmospheric boundary layer associated with tropical cyclones by 331 Global Positioning System sondes dropped in 15 storms. The wind friction velocity u* can be easily retrieved from equation (2) and sea surface drag can be calculated from its definition (3). Analysis of these measurements in (Powell et al, 2003) showed that that the drag coefficient of the sea surface is much less than the extrapolation of data measured at "usual winds" and even decreases if the wind velocity exceeds 30-35 m/ s (see fig. 1a). More precisely, according to (Powell, 2007) surface drag depends significantly on the sector of the tropical cyclone, where it is measured.

The similar dependencies of the surface drag coefficient on the wind speed were retrieved from the measurements of the ocean currents driven by the tropical cyclone Andrew (Jarosz et al., 2007). As it was reported in (Jarosz et al., 2007) on 15 September 2004, the centre of Hurricane Ivan passed directly over current and wave/tide gauge moorings on the outer continental shelf in the north-eastern Gulf of Mexico. Analysis of the along-shelf momentum balance in the water column, when the current structure was frictionally dominated was made within the following equation:

(where U and V are depth-integrated along- and cross-shelf current velocity components, f is the Coriolis parameter, p is the water density, Tsx is the along-shelf wind stress component, H is the water depth, and r is the constant resistance coefficient at the sea floor). Equation (10) enables one to retrieve the wind stress Tsx and estimate the sea surface grad coefficient by of the equation (3) using independently measured wind velocity. The results produced from evaluation of this procedure presented in (Jarosz et al., 2007) show a decreasing trend of CD for wind speeds greater than 32 m s-1 (see So field measurements of the wind stress both from the atmospheric and ocean sides of the air-sea interface show that the sea surface drag coefficient is significantly reduced at hurricane wind speeds in comparison with the extrapolation of the experimental data obtained at "normal" wind speeds and even decreases for U10 exceeding 35 m/s. The similar effect was observed in laboratory experiments performed at the Air-Sea Interaction Facility at the University of Miami (Donelan et al., 2004). In that experiment the

Drag Coefficient Sea Surface Wind

Fig. 1. Sea surface drag coefficient via 10-m wind speed: (a) - from Powel, 2003, (b) -from science, 2007.

Fig. 1. Sea surface drag coefficient via 10-m wind speed: (a) - from Powel, 2003, (b) -from science, 2007.

aerodynamic resistance of the water surface was measured by three different methods: using the profile method (in which the vertical gradient of mean horizontal velocity is related to the surface stress), the Reynolds stress method, and the momentum budget method based on analysis of a momentum budget of water column sections of the tank (Donelan et al., 2004). In comparison with two others, the latter method is insensitive to droplets suspended in the airflow at high wind speeds. The wind speed was measured at 30 cm height in the tank and extrapolated to the standard meteorological height of 10 m using the well-established logarithmic dependence on height (Donelan et al., 2004). All methods were in excellent agreement, and the momentum budget method enabled Donelan et al 2004 to measure the wind stress and aerodynamic resistance coefficient of the water surface up to equivalent 10-m wind speeds about 60 m/s. Figure 2 from (Donelan et al., 2004)

Wind Driven Rain Drag Coefficient
Fig. 2. Laboratory measurements of the neutral stability drag coefficient (reproduced from Donelan etal, 2004)

demonstrates a remarkable levelling of the drag coefficient for the 10-m wind speed exceeding 33 m/s. The difference between CD dependencies on the wind speed in field and laboratory experiments is discussed in (Donelan et al., 2004). Possibly it is due to strong inhomogenity and non-stationarity of the wind in the hurricane eye walls, where the constant stress concept derives from the boundary layer Reynolds equations is not confirmed.

So it can be concluded both from field and laboratory data, that the growth of the aerodynamic roughness of the water surface with wind speed is significantly reduced at extremely high winds in spite of increasing of surface wave heights. Several theoretical models were suggested for explanation of this empirical fact.

3. Possible mechanisms of the sea surface drag reduction at extreme wind speeds

Among a number of possible theoretical mechanisms suggested for explanation of the effect of the sea surface drag reduction at hurricane winds two groups of the models can be specified. First, (Kudryavtsev & Makin, 2007) and (Kukulka et al., 2007) explain the sea surface drag reduction by peculiarities of the airflow over breaking waves, which determine the form drag of the sea surface. For example, in (Donelan et al., 2004), the stabilization of the drag coefficient during hurricane winds is qualitatively explained by a change in the shape of the surface elevation in dominant waves at wind velocities above 35 m/s, which is accompanied by the occurrence of a steep leading front. In this case, occurrence of flow separation from the crests of the waves is assumed. This assumption is based on the laboratory experiments by (Reul et al., 1999), where airflow separation was observed at the crests of breaking waves by the PIV method. According to hypothesis by Donelan et al, 2004, existence of the airflow trapped in the separation zone skips the portions of the water surface in the troughs of the waves and thus, in conditions of continuous breaking of the largest waves the aerodynamic roughness of the surface is limited. Besides, generation of small-scale roughness within the separation zone is reduced due to sheltering, which can also reduce the surface resistance. This effect is expected to be dominant for the case of young sea (or in laboratory conditions as in (Donelan et al., 2004), when wave breaking events are not rare even for energy containing part of the surface wave spectrum. Another approach more appropriate for the conditions of developed sea exploits the effect of sea drops and sprays on the wind-wave momentum exchange (Andreas & Emanuel, 2001, Andreas, 2004, Makin, 2005, Kudryavtsev, 2006). (Andreas & Emanuel, 2001) and (Andreas, 2004) estimated the momentum exchange of sea drops and air-flow, while (Makin, 2005) and (Kudryavtsev, 2006) focused on the effect of the sea drops on stratification of the air-sea boundary layer similar to the model of turbulent boundary layer with the suspended particles by (Barenblatt & Golitsyn 1974). Suspended heavy particles (drops) in the marine turbulent boundary layer create stable stratification suppressing the turbulence, and then decreasing the effective viscosity of the turbulent flow and the aerodynamic resistance. In the same time, there is another effect of sea drops, the particles injected from the water surface should be accelerated, and then they consume some momentum flux from the airflow, increasing the surface drag in the turbulent boundary layer.

In the paper by (Troitskaya & Rybushkina, 2008) the sea surface drag reduction at hurricane wind speed is explained by reducing efficiency of wind-wave momentum exchange at hurricane conditions due to sheltering, but sheltering without separation. This assumption is motivated by reports of eye-witnesses of strong ocean storms, who confirmed that the sea at hurricane wind is unexpectedly smooth and wave breaking is a relatively rare event (see references in (Andreas, 2004). Relatively smooth water surface presents at the video-films taken on board of the research vessel "Viktor Buinitsky'', when it passes a polar lo in the Laptev sea and the Kara sea in October 2007 (cruise within the project NABOS - Dr.Irina Repina private communication). These visual observations are also confirmed by the instrumental measurements by (Donnely et al., 1999),, who observed saturation of the C-band and Ku-band normalized radar cross-section (NRCS) for wind speed above 25-30 m/ s. Similar reduction of NRCS was observed in the laboratory tank experiments by (Donelan et al., 2004). Microwave power scattered from the water surface is formed by i) the Bragg scattering at short waves and ii) by reflection from wave breakers, i.e. the NRCS reduction supports evidence of smoothing of the sea surface by reducing both short wave roughness and wave breaking events. Mechanisms of unusual smoothness of the sea surface are unknown. (Andreas, 2004) suggested two possible explanations of this effect. One supposes the effect of bubbles on surface tension. Another possible explanation exploits the effect of spume drops torn from the wave crests by wind and then falling back as a kind of strong rain, which causes effective damping of surface waves according to a number of experiments (see ex. (Tsimplis, & Thorpe, 1975).

In spite of a number of theoretical hypotheses the problem of explanation of the effect of surface drag reduction at hurricane winds is not solved mostly due to the lack of experimental data.

4. Laboratory modelling of the air-sea interaction under hurricane wind

In this section we describe the results of new laboratory experiments devoted to modelling of air-sea interaction at extremely strong winds.

4.1 Experimental setup and instruments

The experiments were performed in the wind-wave flume built in the Large Thermostratified Tank of the Institute of Applied Physics. The centrifugal fan equipped with an electronic frequency converter to control the discharge rate of airflow produces the airflow in the flume with the straight part 10 m. The operating cross section of the airflow is 40*40 cm2, whereas the sidewalls are submerged at a depth of 30 cm. Wave damping beach made of a fine mesh is placed at the airflow outlet at the end of the flume. Aerodynamic resistance of the water surface was measured by the profile method at a distance of 7 m from the outlet. Wind velocity profiles were measured by the L-shaped Pitot tube intended for measuring flow velocities of up to 20 m/ s (the axis velocity in the flume 25 m/ s approximately corresponds to U10=50-60 m/ s). Simultaneously with the airflow velocity measurements, the 3-channel string wave gauge measured waves at the water surface. The experiment was accompanied by video shooting of the top view of the water surface.

4.2 Peculiarities of the profile method for measuring surface drag coefficient in aerodynamic tunnels

The classical profiling method of measuring surface drag coefficient is based on the property of steady wall turbulent boundary layer to conserve tangential turbulent stress u»2, then the average flow velocity is logarithmic and the wind friction velocity u* can be easily determined from (2), if the velocity profile is measured. However developing turbulent boundary layers are typical for the aerodynamic tubes and wind flumes, then three sublayers at different distances from the water can be specified: viscous sub-layer, layer of constant fluxes and "wake" part (see fig. 3a). The viscous sub-layer, where viscous effects are essential, exists over the hydrodynamically smooth surfaces at the distances less than 20^30 v/u* (v is the kinematic viscosity), for moderate winds it is about 1 millimetre. The "wake" part is the outer layer of the turbulent boundary layer, where the boundary layer flow transits to the outer flow in the tube. Its thickness 8 increases linearly from the outlet of the flume. The layer of constant fluxes is extended from the upper boundary of the viscous sub-layer to approximately 0.158. Only in the layer of constant fluxes the flow velocity profile is logarithmic and can be extrapolated to the standard meteorological height H10. Typically in wind flumes the constant layer thickness is less than 10 cm. Measuring of wind velocity profiles at the distance less than 10 cm from the wavy water surface at strong winds is a difficult problem mainly due to the effect of sprays blown from the wave crests. Fortunately, parameters of the layer of the constant fluxes can be retrieved from the measurements in the "wake" part of the turbulent boundary layer, because the velocity profile in the developing turbulent boundary layer is the self-similar "law of wake" (see Hinze, 1959). The self-similar variables for the velocity profile and vertical coordinates are z/8 and (Umax-U(z))/u*., where Umax is the maximum velocity in the turbulent boundary layer. The self-similar velocity profile can be approximated by the following simple equations (see Hinze, 1959):

- in the layer of constant fluxes

Fig. 3. Airflow velocity profiles in the aerodynamic flume over the waves for different airflow velocities (a); dashed curves are logarithmic approximations in the layer of constant fluxes. I - the layer of constant fluxes, II - the "wake" part. Air-flow velocity profiles measured at different wind speeds over waves in self-similar variables (b).

Fig. 3. Airflow velocity profiles in the aerodynamic flume over the waves for different airflow velocities (a); dashed curves are logarithmic approximations in the layer of constant fluxes. I - the layer of constant fluxes, II - the "wake" part. Air-flow velocity profiles measured at different wind speeds over waves in self-similar variables (b).

o in the "wake" part

Collapse of all the experimental points in one curve in self-similar variables occurred in our experiments (see fig.3b). The parameters in equations (11) (12) were obtained by the best fitting of the experimental data: a=1.5, P=8.5.

The parameters of the logarithmic boundary layer can be retrieved from the measurements in the wake part of the turbulent boundary layer, first, retrieving parameters of turbulent boundary layer (Umax and SS) by fitting experimental data by equation (12) and then calculating parameters of the logarithmic boundary layer by the following expressions:

Expression for CD via measured parameters u*, Umax and S follows from equations (13-14):

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