Atmospheric stability parameters and sea storm severity

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1 Coastal Engineering 81 Atmospheric stability parameters and sea storm severity G. Benassai & L. Zuzolo Institute of Meteorology & Oceanography, Parthenope University, Naples, Italy Abstract The preliminary results of a statistical analysis of the atmospheric stability effects on the growth of sea waves are given in this paper. In unstable conditions (air colder than water) an increase of vertical mixing is experienced due to turbulence which makes the wind profile more uniform with height (deviating it from the logarithmic law). This circumstance increases the wind stress on the sea surface and thus gives rise to higher wave energy. In order to quantify the wave energy increase, the analysis of wave, wind and temperature data was performed, on the basis of the SWAN (Sea WAve monitoring Network) data set regarding both the Tyrrhenian and Adriatic Sea. The statistical analysis was first performed in terms of relative deviation of non-dimensional wave energy as a function of the Bulk Richardson Number in terms of the wind velocity U 10. Then another stability parameter, the Monin- Obukhov length, was related to the friction velocity u. This approach needed the implementation of a numerical procedure to estimate the set of Monin- Obukhov parameters, which account for atmospheric stability. Keywords: wind and wave statistics, wave energy, atmospheric stability, energy growth curves, Monin-Obukhov theory. 1 Introduction Recent studies highlighted the influence of some apparently secondary effects on the wave growth rate such as atmospheric stability. In unstable conditions (air colder than water) an increase of vertical mixing in the lower boundary layer is experienced as heat fluxes across sea surface contribute to the increase of turbulence. This effect causes the wind profile to be WIT Transactions on The Built Environment, Vol 78, 005 WIT Press

2 8 Coastal Engineering more uniform with height (deviating from the logarithmic law), increasing the wind stress on the sea surface. This circumstance suggested correlating atmospheric stability parameters with the energy growth rate of wave field and to quantify the wave energy increase with respect to neutral conditions. Energy growth was given as a function of non-dimensional fetch in order to evaluate the relative deviation with respect to a reference energy growth curve, given in literature. A representative stability parameter is the Bulk Richardson Number g( TA TW ) Rb = (1) ztta( U / z) where T A and T W are air and water temperature in K, z t is the height at which temperature is measured, z the height at which wind speed is measured (generally 10 m). This parameter represents the ratio between the buoyant production term and the mechanical production term by wind in the Turbulence Kinetic Energy budget equation; R b is a measure of the atmospheric stability, in fact unstable condition are characterized by negative values of air-sea temperature difference corresponding to negative values of R b. In this study a statistical analysis of the influence of atmospheric stability on wave growth was done through the examination of the scatter ξ=(- m )/ m between non-dimensional wave energy obtained by data and the corresponding neutral energy obtained by the curve m (χ), as a function of the Bulk Richardson Number R b. The analysis was then more focused the physical process of energy transfer from wind to waves, so the friction velocity u (proportional to wind stress) was used instead of U 10 as scaling velocity. The theoretical framework to this approach was made through Monin-Obukhov theory which involves the latent and sensible heat fluxes. A comparison between the results obtained with u and those obtained with U 10 was also made and discussed. 1.1 Wind profile as a function of atmospheric stability In homogeneous, stationary and stable condition, the wind stress in the boundary layer is generally considered constant and, for wind speed greater than 3 m/s, completely driven by the turbulent components τ τ fluct = < u'w' > () where u,w are the fluctuation of wind velocity around the mean value. A usual parameterization assumes that wind stress depends only on wind profile through the eddy viscosity ν e, which depends on the shear flow through the mixing length l = kz proportional to the height by a factor k=0.4, the Von-Karman constant. u τ fluct = ν e z (3) u ν e = l z (4) WIT Transactions on The Built Environment, Vol 78, 005 WIT Press

3 Coastal Engineering 83 An alternative measure of wind stress is the friction velocity u, which is proportional to wind stress τ by a factor 1/ρ a τ u = (5) ρ a By combining eqns. (3), (4), and (5) and integrating with respect to the height the usual logarithmic wind profile is obtained u = z u ( z) ln (6) k z0 where z0 is the roughness length, that for Vv>4 m/s and non-plane surface, may be expressed through the Charnock parameter α c =0.014 α z cu 0 = (7) g Surface roughness may be equivalently expressed by drag coefficient C D κ (8) C D = z ln z0 which provides a convenient parameterization of wind surface stress in terms of mean wind speed components at a given height. This parameter is related to stress by the empirical relation τ= ρ a C D ( z) u( z) (9) In presence of stratification, the combination of the effects of temperature gradients and buoyancy forces may play an important role in the turbulent mixing processes since cold air on a relatively warm sea gives rise to heat fluxes through the sea surface, which lead to the formation of cold (denser) air layers, laying on warmer (less dense) layers: heavier air masses tend to sink enhancing air turbulence. The mathematical framework of this physical process is the Monin-Obukhov theory which correlates non-dimensional parameters obtained by characteristic variables of the coupled air-sea system (velocity, temperature, etc.) with a stability parameter ζ=z/l through the universal functions φ m and φ H ; z is the height and L is the Monin-Obukhov length (which is the height at which shear production and buoyant production of turbulence are equal). This length is related to latent and sensible heat fluxes H L and H S and to the virtual temperature T v =T(1+0.61q) (where q is the humidity) by: 3 ρ ac pu Tv u Tv L = = (10) kg( H L + H S ) κg < w' Tv ' > The parameter ζ represents a local criterion of hydrostatic stability: if ζ<0, heat flux is directed from the sea surface to the atmosphere and the stratification is unstable; if ζ>0 the heat flux is directed to the water and the stratification is stable; if ζ=0 heat fluxes are absent, the stratification is neutral and there are not buoyancy forces. The two universal functions φ m φ H can be expressed through the form proposed by Businger et al. [] and Dyer and Hicks [6]. WIT Transactions on The Built Environment, Vol 78, 005 WIT Press

4 84 Coastal Engineering In stable conditions (L>0) 5z ϕ m =ϕh = (11) L In unstable conditions (L<0) 1+ x 1+ x 1 π ϕ = ln + ln tan + x (1) m 1+ x ϕ = ln (13) H where z x = 1 16 L (14) In unstable conditions, the stress depends on wind and temperature profiles. ( κz) U T τ < w' T ' >= u T = ΦMΦT z z (15) where u e T represent two scale parameters. z U u = κ Φ M z (16) z T T = κ Φ T z (17) Integrating the two equations with respect to height, wind speed and temperature profiles are obtained. u z z u( z) = ln ϕ (18) m k z0 L t z z T ( z) = T + ln ϕ (19) 0 H k zt L where z t is the roughness parameter associated to thermal gradients whose value is proposed by Large and Pond [17]. 1 4 Figure 1: Boundary layer profiles for different Monin-Obukhov lengths. WIT Transactions on The Built Environment, Vol 78, 005 WIT Press

5 Coastal Engineering 85 The drag coefficient depends now on the roughness length and on atmospheric stratification through the function φ m. κ (0) CD = z ln ϕm z0 1. Selection of wave and atmospheric data The field dataset were collected during in Tyrrhenian and Adriatic Sea. Wave data from SWAN buoys, coupled with wind and air-sea temperature data were employed. The non-dimensional parameters regarding sea state duration, fetch, energy and peak frequency were evaluated ς = gt / U (1) χ = gχ / U () σ g = U (3) 4 f p U ν = (4) g where σ is the variance of surface elevation (significant wave height H s =4σ), t the duration of wind action, x the length of wind action (fetch), g gravity acceleration, f p peak frequency. χ ζ Figure : Data set split into duration limited and fetch limited. The scaling velocity U was first set to the measured value at 10 m height. The selection of data intended to analyze storm events which required the overcoming of significant wave height H s =1,5 m and the persistence of 1 hours at least. The second selection removed swell components in order to analyze only the wind sea waves, that is waves whose values of non-dimensional peak frequency ν is greater than WIT Transactions on The Built Environment, Vol 78, 005 WIT Press

6 86 Coastal Engineering ν f p U = g > 0,1 Finally, fetch limited sea states were selected through the CERC [4] equation, given in figure. ς = K exp A(ln χ) B ln χ + C 1/ + D ln χ (5) {[ ] } In this study only fetch dependent (stationary) sea states were considered on which the statistical analysis further described were performed. 1.3 Statistical analysis of wave energy growth In the first stage of the analysis, non-dimensional energy and peak frequency ν were reported as a function of non-dimensional fetch χ and compared to the experimental curve referred to neutral condition, obtained by Kahma and Calkoen [13] by a composite data set for non-dimensional energy, and the curve obtained by Kahma [1] (re-elaborating JONSWAP data) for peak frequency. 0.5 (.0 ± 0.3) χ ν = max (7) m (0.13 ± 0.) m = χ (6) Non-dimensional energy curve is valid for fetch limited conditions. Peak frequency curve is valid for the different stages of sea wave growth; for large values of non-dimensional parameter peak frequency approach to full developed sea showing constant values. Figure 3 shows that experimental values of energy are well dispersed around the standard curve, meaning that experimental points are normally fetch limited sea states, as full developed condition are hardly reached. Energy deviation was made non-dimensional through the neutral energy m. ξ m = (8) m and reported as a function of Bulk Richardson Number R b in figure 4. The values of R b belong to the range (-0.5, 0.5) which corresponds to the greatest variability of the parameter in agreement with literature (Young [3]). Although there is significant scatter in the data, a linear regression yields to the following equation: ξ = 5.3Rb 0.8 The comparison with the regression obtained by Young [3]: ξ = 1.Rb shows that the coefficients are of the same order of magnitude. The trend of the relative deviation ξ with respect to R b agrees with the results of literature highlighting significant positive deviation of energy in presence of atmospheric instability. WIT Transactions on The Built Environment, Vol 78, 005 WIT Press

7 Coastal Engineering 87 ν χ Figure 3: Standard curves (-χ) and (ν-χ) compared with experimental points. ξ R b Figure 4: Linear regression of ξ as a function of Rb. The sensitivity analysis of R b as a function of both wind speed and air-sea temperature gradient was investigated in figures 5a and 5b, respectively. Figure 5a shows the behaviour of R b as a function of air-sea temperature gradient. The data scatter is minimum when the gradient is close to zero (as expected), while the dispersion is much wider far from neutral conditions. In WIT Transactions on The Built Environment, Vol 78, 005 WIT Press

8 88 Coastal Engineering particular, for T A -T W <- (unstable conditions) a significant trend of R b towards negative value is experienced. In figure 5b, a significant deviation of R b was experienced for Vv<7 m/s, whereas the dispersion of experimental points was much less significant for values of Vv>15 m/s with R b values approaching zero (neutral behaviour). This circumstance suggests that the decrease of R b is experienced for significant but not very high wind velocity, leading to an increase of wave energy in intermediate growth conditions. R b T a -T w R b Vv (m/s) Figure 5: Sensitivity analysis of R b with respect to (a) T and to (b) the wind speed. The statistical analysis of wave energy growth has then proceeded substituting u to the wind velocity U 10 as scaling velocity. Several authors (Janssen et al. [10]) underline how the use of this scale velocity leads to a reduction of deviation between the two growth curves relative to stable and unstable conditions. The estimation of friction velocity was made by solving iteratively the following set of equations (Kahma and Calkoen, [13]) u z U ( z) = ln ϕ M ( ζ ) (9) κ z0 WIT Transactions on The Built Environment, Vol 78, 005 WIT Press

9 Coastal Engineering 89 R b zt zt ln ϕ H ζ z zt z = ζ zt z ln ϕ M ( ζ ) z0 (30) where z T is the height of air temperature measurement, z is the anemometric height, z t is the temperature roughness (given in literature by Large and Pond, 198), z 0 is the roughness length chosen in the Charnock [5] form; universal functions φ M e φ H are given by Businger et al. [] and Dyer and Hicks [6]. The first equation is the wind profile in presence of atmospheric stratifications according to Monin-Obukhov theory, the second equation gives the relationship between the Bulk Richardson number and the Monin-Obukhov length. The difference between the two analyses in terms of U 10 and in terms of u was considered through the regressions (χ) and ν p (χ) for stable (R b >0) and unstable conditions (R b <0). Discrepancies between linear regressions are particularly evident when the scaling velocity used is the wind speed at 10 meters U 10. The regressions present the following equations 8 3 = χ +.10 Stable 8 3 = χ Unstable A comparison in terms of energy highlights that for the same non-dimensional fetch and same wind speed (typical values χ=10 4 and Vv=10 m/s) an energy increase of 37% was obtained, which leads to an increase of H s(stable) =.13 m and H s(unstable) =.49 m. unstable stable = 0.37 stable The use of u as scale velocity considerably reduces the distance between the two regressions because the energy gain is transferred in a gain of friction velocity. In particular, regression equations become 5 3 = χ Stable 5 3 = χ Unstable Finally, drag coefficients C D calculated on experimental data were reported in figure 7 with respect to wind speed U 10. Different curves represent linear regressions of experimental data grouped according to selected ranges of temperature gradients: T<-8, -8< T<-4, -4< T<0, T>0. In agreement with scientific literature (for example Kato et al., [14]) regressions exhibit a strong dependence on atmospheric instability, in fact, the highest temperature gradients get the highest drag coefficients. Drag coefficient behaviour is in agreement with the theory, that is C D increases with increasing wind speed (as expected) but also with increasing atmospheric instability. WIT Transactions on The Built Environment, Vol 78, 005 WIT Press

10 90 Coastal Engineering χ ν χ Figure 6: Regressions (-χ) e (ν-χ) for unstable (triangle spot and dashed line) and stable (point spot and solid line) conditions. χ ν χ Figure 7: Regressions ( - χ ) e (ν - χ ) for unstable (triangle spot and dashed line) and stable (point spot and solid line) conditions. WIT Transactions on The Built Environment, Vol 78, 005 WIT Press

11 Coastal Engineering Conclusions In this study a statistical analysis of the influence of atmospheric stability on wave energy growth was carried out. The energetic anomaly with respect to neutral conditions was reported as a function of the Bulk Richardson Number. The linear regression showed an increase of the energy growth rate with atmospheric instability and was in good agreement with literature (in particular the order of magnitude is the same as the one given by Young [3]). The sensitivity analysis of R b showed that a significant deviation for Vv<4 m/s, whereas the dispersion was much less significant for Vv>15 m/s with R b values approaching zero (neutral behaviour): the decrease of R b is experienced for significant but not very high wind velocity, leading to an increase of wave energy in intermediate growth conditions. C d Vv (m/s) Figure 8: Drag coefficient with respect to wind speed. Different lines are linear regressions obtained for different air-water temperature scatters. The sensitivity analysis was also performed in terms of air-sea temperature gradient. The results show that the scatter is minimum when the gradient is close to zero (as expected), while the dispersion is much wider far from neutral conditions. In particular, for T A -T W <- (unstable conditions) a significant trend of R b towards negative value is experienced. The use of u as scaling velocity reduced the scatter between the regressions relative to stable and unstable conditions. In particular, the regressions obtained with U 10 showed that the energy gain is almost 37% for typical conditions. Finally, the sensitivity analysis of the drag coefficient revealed a strong dependence on wind speed (as expected) but also on air-water temperature differences showing an increase with wind speed and atmospheric instability, in agreement with the physical evidence. WIT Transactions on The Built Environment, Vol 78, 005 WIT Press

12 9 Coastal Engineering References [1] Atakturk S.S. - Katsaros K.B., Wind stress surface waves observed on Lake Washington. J.G.R. (9), pp , [] Businger J.A. et al., Flux profile relationship in the atmospheric surface layer. J. Atmospheric Sc. (8), pp , [3] Cavaleri L., Wind variability, Atmospheric stability, Dynamics and modelling of ocean waves, Cambridge University Press, pp , [4] CERC, Shore Protection Manual, US Army Coastal Engineering Research Center, volumes, 1977,1984. [5] Charnock H., Wind stress of water surface, Q. J. R. Meteorol. Soc. (81), pp , 1955 [6] Dyer A.J., Hicks B.B., Flux gradient relationships in the constant flux layer. Q.J. Royal Meteorology Soc. (96), pp , [7] Donelan M., Air-Sea interaction. The sea: Ocean Engineering Science, pp. 39-9, J. Wiley & Son Edition, [8] Geernaert G. L. et al., Surface waves and fluxes, vols., Kluwer, Dordrecht, [9] Hasselmann K. et al., Measurement of wind wave growth and swell decay during the Joint North Sea Wave Project (JONSWAP). Dent. Hidrogr. Zeit., (vol. A, 8, 1), 95 pp., [10] Janssen P.A.E.M. et al., Friction velocity scaling in wind-wave generation. Boundary Layer Meteor., (38), 9-35, 1987a. [11] Johnson H. K. et al., On the dependence of sea surface roughness on wind waves. J.P.O., (8), pp , [1] Kahma K., A study of the growth of the wave spectrum with fetch. Journal of Physical Oceanography, (11), pp , [13] Kahma K., Calkoen C., Reconciling discrepancies in the observed growth of wind-generated waves. J. P O, (), pp , 199. [14] Kato et al., An experimental study on the effect of air-water temperature difference on growth of water waves, Coastal engineering, 00. [15] Komen J.G. et al., Dynamics and modelling of ocean waves. Cambridge University Press, pp. 53, [16] Komen J.G., Janssen P.A.E.M., Atmospheric stability effects. The ocean surface, Y. Toba, H. Mitsuyasu (Editors), Riedel Publishing Company, pp , [17] Large W.G., Pond S., Sensible and latent heat flux measurement over the ocean, J. Physic. Oceanogr. (1), pp , 198. [18] Miles J.W., On the generation of surface waves by turbulent shear flow. J.F.M. (7), pp , [19] Monin A.S., Statistical fluid mechanics, MIT Press, 769 pp,1971. [0] Monin A.S., Obukhov A.M., Basic relationship of turbulent mixing in the surface layer of the atmosphere. Akad. Nauk. SSSR Trud. Geofiz.Inst., No. 4 (151), pp , WIT Transactions on The Built Environment, Vol 78, 005 WIT Press

13 Coastal Engineering 93 [1] Phillips O.M., The dynamics of the upper ocean. Cambridge University Press, 336 pp., [] Smith S. D., Coefficients for Sea surface wind stress, heat stress, and wind profiles as a function of wind speed and temperature. J.G.R., (vol.93, no.c1), pp , [3] Young I.R., Wind generated ocean waves. Ocean Engineering Series- Elsevier, 88 pp., WIT Transactions on The Built Environment, Vol 78, 005 WIT Press

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