### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Site and Measurements
- 3. Theoretical Considerations
- 4. Data
- 5. Terms in the Turbulent Kinetic Energy Budget
- 6. Results for Near-Neutral Conditions
- 7. The Inertial Dissipation Method
- 8. Discussion
- 9. Conclusions
- Appendix A.: The Relation Between Sonic Temperature Flux and Virtual Temperature Flux
- Acknowledgments
- References

[1] The terms in the turbulent kinetic energy (TKE) budget have been analyzed according to stability, wave age, and wind speed, using long-term measurements over the sea. The measurements were performed at the island Östergarnsholm in the middle of the Baltic Sea. The results show that there is an imbalance between normalized production and normalized dissipation, also in neutral conditions, and that this imbalance depends not only on stability, which has been previously suggested, but also on wave age and wind speed. For small wave ages and high wind speeds, production is larger than dissipation at neutral conditions. For saturated waves and moderate wind speeds, the sea surface resembles a land surface, while for swell and low wind speeds, dissipation strongly exceeds production. The normalized pressure transport becomes significant during swell conditions, and is not balanced by the normalized turbulent transport. “Inactive” turbulence, where energy is being brought down to the surface from higher levels, is probably the reason for these high values of the pressure transport. The traditional “inertial dissipation method,” where the sum of the transport terms is assumed small and neglected, therefore needs to be corrected for an imbalance between production and dissipation.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Site and Measurements
- 3. Theoretical Considerations
- 4. Data
- 5. Terms in the Turbulent Kinetic Energy Budget
- 6. Results for Near-Neutral Conditions
- 7. The Inertial Dissipation Method
- 8. Discussion
- 9. Conclusions
- Appendix A.: The Relation Between Sonic Temperature Flux and Virtual Temperature Flux
- Acknowledgments
- References

[2] The energy exchange at the surface of the oceans plays a decisive role for the climate of the Earth. In modeling these energy fluxes, it is vital that the physics of the governing processes is correctly understood. Traditionally, results obtained over land are employed. Recent research (for references, see below) has, however, identified processes which depend critically on specific air-sea interaction mechanisms and which are not taken into account in current models. The effects are expected to depend on the wave state, i.e., the degree of wave development, including the effect of swell originating in distant areas.

[3] Most air-sea interaction experiments in the past are either offshore experiments of limited duration and with measurements at only one level or coastal measurements at nearshore sites. In this paper data are analyzed from several years worth of measurements on a small island, Östergarnsholm, in the Baltic Sea. The data include eddy correlation measurements at three levels and wave measurements outside the island. From previous analysis [*Smedman et al.*, 1999, appendix B] it is known that the measured fluxes at approximately 10 m at this site, are likely to be largely representative of deep sea conditions even during gale winds. The present analysis aims at studying the turbulent kinetic energy (TKE) budget in a wide range of conditions. Knowledge of the behavior of each term in the TKE-budget is fundamental for application of the widely used inertial dissipation method.

[4] Several methods for calculating fluxes over the sea are used today. The “bulk aerodynamical method” uses only mean meteorological parameters and is probably the most practical method for climatological purposes. The largest uncertainty in this method is the numerical values of the drag coefficients, but there is also a problem to determine a correct wind speed from a moving ship or buoy.

[5] The “eddy-correlation method” or “direct covariance method” is the most direct way to determine the fluxes since the correlation between fluctuations of measured quantities (for example u and w) can be measured directly, and from that the stress and the heat fluxes can be calculated. A serious problem, however, is that the instruments usually have to be mounted on a buoy or on a ship, and the frequencies of the movement of the buoy or ship are in the same interval as that of the flux itself. Thus it is very difficult to make corrections. The eddy-correlation method is also very sensitive to flow distortion [e.g., *Edson et al.*, 1991].

[7] The inertial dissipation method has been used for about 30 years as a method to measure wind stress over the ocean [*Smith et al.*, 1996], but the technique has had a reputation of being very “fragile and exotic,” relying on many assumptions and very fragile sensors [*Fairall et al.*, 1990]. Today more robust instruments can be used, and the algorithms have been improved.

[8] During HEXOS main field experiment HEXMAX [e.g., *Katsaros et al.*, 1987; *Smith et al.*, 1992; *DeCosmo et al.*, 1996], a detailed comparison of the inertial dissipation against the eddy correlation method was performed [*Edson et al.*, 1991; *Fairall et al.*, 1990]. The measurements took place on the Dutch offshore platform Meetpost Noordwijk in the North Sea in October and November 1986. The estimates with the inertial dissipation method agreed well with the direct covariance method, and the results gave an uncertainty of 10% for stress, 20% for sensible heat flux, and 25% for latent heat flux [*Fairall et al.*, 1990].

[9] The inertial dissipation method has many advantages: Only measurements in the inertial subrange are needed, and these high-frequency fluctuations can easily be separated from the disturbances caused by the movements of for example a ship. The wind speed needed for the calculations is the apparent wind speed, which is the wind speed actually measured on-board (i.e., the wind speed relative to the moving platform). Explicit measurements of the vertical velocity are in this case avoided. It is also clear that this method is superior to the direct covariance method in regions of severe flow distortion [*Edson et al.*, 1991].

[10] There are, however, disadvantages with the inertial dissipation method. It relies on several assumptions and choices of constants and these need to be further investigated for the method to be reliable under all conditions. It also builds on a correct parameterization of the terms in the TKE-budget.

[11] The energy budget has been quite well analyzed over land [e.g., *Wyngaard and Cote*, 1971; *Högström*, 1990], even if the results sometimes are inconclusive. There have not been as many measurements of the TKE-budget over sea, and there are still a lot of uncertainties regarding the different terms, and whether the results are the same over sea as over land. The problem with earlier measurements over sea is that all terms have not been determined directly, normally only production and dissipation were measured, and not the transport terms [e.g., *Edson and Fairall*, 1998].

[12] The response of the TKE-budget to different wave conditions has not been investigated thoroughly up till now. There are, however, some earlier studies of wave influence on the turbulent structures in the marine atmospheric surface layer. The influence of swell, that is waves traveling faster than the wind, was considered already in the 70s by, for example, *Kitaigorodskii* [1973], *Makova* [1975], *Volkov* [1970], and *Benilov et al.* [1974]. They discussed a “supersmooth” surface and how waves traveling faster than the wind would influence the measurements. How swell actually does influence the turbulence structure of the atmospheric boundary layer has been a controversial question in the last years, but nowadays the influence of swell is more or less accepted [e.g., *Smedman et al.*, 1999]. There is evidence that the influence of swell can reach up to a considerable height [*Makova*,1975; *Drennan et al.*, 1999b; *Smedman et al.*, 1994] in contrast to the influence of growing waves.

[13] During swell conditions, it has been argued that the traditional inertial dissipation method can not be used [e.g., *Donelan et al.*, 1997]. A major problem with the inertial dissipation method is that it is not possible to detect the direction of the flux. It automatically gives a positive value of the total stress (τ) [*Grachev and Fairall*, 2001], which is not always the case in swell conditions.

[14] The current investigation is a study of the TKE-budget using long-term measurements over the Baltic Sea. A data set with more than 2000 thirty-minute averages, more than 1000 of which also contain wave information, has been used. The measurements were taken at a site in the middle of the Baltic Sea called Östergarnsholm. The measurements and the site are described in section 2. The wave influence of the marine atmospheric surface layer is discussed in section 3 as well as the turbulent kinetic energy budget. Section 4 describes the data used in the study and the different terms in the TKE-budget are presented in section 5. In section 6 results for the near-neutral range of this study are discussed in detail and in section 7 the consequences for the inertial dissipation method are outlined. A discussion of these findings with reference to previous studies is presented in section 8, and in section 9 the final conclusions are presented.

### 6. Results for Near-Neutral Conditions

- Top of page
- Abstract
- 1. Introduction
- 2. Site and Measurements
- 3. Theoretical Considerations
- 4. Data
- 5. Terms in the Turbulent Kinetic Energy Budget
- 6. Results for Near-Neutral Conditions
- 7. The Inertial Dissipation Method
- 8. Discussion
- 9. Conclusions
- Appendix A.: The Relation Between Sonic Temperature Flux and Virtual Temperature Flux
- Acknowledgments
- References

[74] As shown above, the terms in the TKE-budget have a clear stability dependence, and there is a large difference between the stable and unstable side (Figures 567–8). But, it is of course possible that the waves will influence the measurements as well. The scatter of the individual data points might be an indication of this.

[75] The wave age, *c*_{0}/(*U*_{10} cos θ), was defined in section 3 as a parameter to describe the wave influence. The wave state can be divided into three different types according to wave age. Small wave ages, *c*_{0}/(*U*_{10} cos θ) < 0.50, where waves are growing; high wave ages, *c*_{0}/(*U*_{10} cos θ) > 1.2, is defined as swell, where the waves are traveling faster than the wind, and the waves are in a decaying state. Between these two wave states, i.e., 0.5 < *c*_{0}/(*U*_{10} cos θ) < 1.2, the waves are in a saturated condition, neither growing, nor decaying. As will be shown in this section, the sea surface then acts as a land surface on the atmospheric flow.

[76] In section 3 it was discussed that the total stress (i.e., sum of turbulent shear stress, wave induced stress and viscous stress), will be reduced or even negative during swell conditions (equation 2), since the wave induced part will be in opposite direction to the turbulent stress. This will then lead to a decreased value of *u*_{*} (equation 5). The small value of *u*_{*} during swell will also affect the stability parameter *z*/*L* (equation 4), since it will give a large value of |*z*/*L*|, even if the heat flux is small. This means that *z*/*L* does not only act as a stability parameter, but is also influenced by the waves. For some swell situations, this influence can be seen up to considerable heights [e.g., *Smedman et al.*, 1994].

[77] To be able to distinguish between the stability dependence and the wave influence, only near neutral data will be considered in this section, and the terms in the TKE-budget will be plotted as a function of two variables, stability (*z*/*L*) and wave age (*c*_{0}/(*U*_{10} cos θ)) or stability (*z/L*) and wind speed (*U*). The data has been bin-averaged in two dimensions, and the numbers inside the plot are the averaged values of the specific term. As earlier, positive values indicate a gain in energy and negative values a loss. The dashed lines in the figures are drawn subjectively.

[78] In Figure 9a, the normalized mechanical production (*PN*) is plotted as a function of wave age and stability. *PN* is approximately 1.0 at neutral conditions, but the values increase rapidly on the stable side (*z*/*L* > 0). *PN* decreases for increasing instability, and also to some degree with increasing wave age. Otherwise the influence of the wave age seems to be quite small.

[79] In Figure 9b*PN* is given as a function of stability (*z*/*L*) and mean wind speed (*U*). There is of course a strong correlation between wind speed and wave state, high wind speed (>8 m s^{−1}) being likely to be associated primarily with wind waves and low wind speed (<6 m s^{−1}) with swell. The decrease in *PN* with *z*/*L* is more pronounced for low wind speeds, again indicating the influence of swell. The stability dependence seen in both figures could also be seen in Figure 5, where the slope of the curve in the near-neutral range is quite steep.

[80] Figure 10 shows minus the normalized dissipation (−*DN*) as a function of stability and wave age. Notice that the sign is changed to positive (i.e., dissipation is by definition a loss in energy). −*DN* increases with increasing wave age for neutral conditions, but the minimum of −*DN* is for swell and unstable conditions. During stable conditions, −*DN* increases when stability and wave age increase, even if there is a maximum for −*DN* also at small wave ages.

[81] The two transport terms are shown in Figure 11. The normalized turbulent transport (*T*_{t}*N*) in Figure 11a has some scatter, but is clear that the values are small or very close to zero for all stabilities and wave ages. *T*_{t}*N* does not balance the normalized pressure transport (*T*_{p}*N*), which is shown in Figure 11b. The pressure transport has a clear dependency on both stability and wave age. For small wave ages the pressure transport is negative, it changes sign at *c*_{0}/(*U*_{10} cos θ) ≈ 0.50. Between *c*_{0}/(*U*_{10} cos θ) ≈ 0.50 and 1.2 the value is approximately 0.25, and for swell *c*_{0}/(*U*_{10} cos θ) > 1.2, *T*_{p}*N* becomes significant.

[82] *T*_{p}*N* ≈ 0.25 is in accordance with *Högström* [1996], who refers to an experiment over land at Laban's Mills in Sweden, where (*T*_{p}*N*) was found to be 0.25 and (*T*_{t}*N*) zero. This is explained by *Högström et al.* [2002] as a result of surface-layer-scale structures playing a decisive role in the momentum transport in the neutral atmospheric surface layer.

[83] Most of the imbalance, normalized production-normalized dissipation (Figure 12a), therefore originates from the pressure transport. The values are almost the same as the values for *T*_{p}*N*, but the sign is the opposite. For small wave ages, *c*_{0}/(*U*_{10} cos θ) < 0.50, production exceeds dissipation, energy is building up the waves rather than being dissipated into heat. For moderate wave ages 0.50 < *c*_{0}/(*U*_{10} cos θ) < 1.2, the imbalance resembles that found over land [e.g., *Högström*, 1996]. For swell *c*_{0}/(*U*_{10} cos θ) > 1.2, dissipation is much larger than production.

[84] The same structure is also found in Figure 12b, where the imbalance is plotted as function of stability and wind speed. At high wind speeds production exceeds dissipation, and at low wind speeds the dissipation is much larger than the production. The same stability dependence as in Figure 8 can also be seen in Figures 12a and 12b, with larger imbalance as we move away from neutral conditions.

[85] The results in this section have some important implications for the inertial dissipation method. This will be discussed in the next section.

### 7. The Inertial Dissipation Method

- Top of page
- Abstract
- 1. Introduction
- 2. Site and Measurements
- 3. Theoretical Considerations
- 4. Data
- 5. Terms in the Turbulent Kinetic Energy Budget
- 6. Results for Near-Neutral Conditions
- 7. The Inertial Dissipation Method
- 8. Discussion
- 9. Conclusions
- Appendix A.: The Relation Between Sonic Temperature Flux and Virtual Temperature Flux
- Acknowledgments
- References

[86] As described earlier, the inertial dissipation method builds on estimating *u** from the TKE-budget. The terms in the budget must then be known. Solving for *u** in equations 9, 12 and 15 gives:

It is then possible to calculate the momentum flux τ

This usually involves an iterative process, since *L* is needed to calculate *u**, and *u** is needed to calculate *L* (see equation 4). Problems with convergence in not uncommon by using this method. It is also a problem how to estimate the flux of virtual potential temperature (equation 4). *Dupuis et al.* [1997] suggested two methods, either to use of a bulk formula or to use dissipation rates for the temperature in each step. The last method can, however, be troublesome if a clear inertial subrange in the temperature spectra can not be found from the measurements, which is quite common due to instrumental problems.

[87] It should also be remembered that the inertial dissipation method only gives positive values of the flux, so some indicator for swell cases when the momentum transport is upward is also needed.

[88] The use of the traditional inertial dissipation method (assuming that the transport terms are small) has been questioned by some authors, and especially for cases with swell [*Donelan et al.*, 1997; *Drennan et al.*, 1999b]. *Drennan et al.* [1999b] how that flux measurements with the eddy-correlation method were approximately twice the calculated fluxes with the inertial dissipation method when strong swell was present. In pure wind sea, the agreement with the eddy-correlation method was excellent [*Donelan et al.*, 1997].

[89] An interesting fact is also that a wave influence is difficult to see in measurements with the inertial dissipation method. *Drennan et al.* [1999a] compared measurements of the drag coefficient made with the inertial dissipation method and the eddy-correlation method. It is clear that the measurements made with the eddy-correlation method showed a sea state dependence, while the inertial dissipation method did not. *Yelland and Taylor* [1996] claim that their measurements made with the inertial dissipation method show hardly any sea state dependence.

[90] The difference between the eddy-correlation method and the inertial dissipation method might also be seen for high wind speeds and low wave ages. *Janssen* [1999] discusses the lack of scatter in the drag coefficient obtained with the inertial dissipation method and found no wave influence, which is then in contrast to measurements performed with the eddy-correlation method. His explanation is that the pressure transport is not included in the inertial dissipation method. By using a wave model, he shows that pressure transport is important in the presence of wind generated ocean waves and high wind speeds (>15m s^{−1}), and that this might increase the surface stress with about 20%. *Taylor and Yelland* [2001] do not agree with *Janssen* [1999]. They claim that they see no wave age dependency in their data.

[91] An explanation for these seemingly contrasting results may be that the inertial subrange does not react in the same way as the whole spectrum. The influence of swell is contained in frequencies between 0.06 and 0.16 Hz [*Rieder and Smith*, 1998], and the inertial subrange is usually located at higher frequencies. This is in accordance with *Drennan et al.* [1999b], who show that the inertial subrange during swell follow the pure wind sea data well, while the lower frequencies do not.

[92] A. Smedman, U. Högstrom, and A. Sjöblom (A note on velocity spectra in marine boundary layer, submitted to *Boundary-Layer Meteorology*, 2002) also show that for some swell situations the inertial subrange is not affected by swell. During decreasing wind conditions (increasing wave ages), the frequency range where the wave influence can been seen is increasing toward higher frequencies. However, it will take some time before all of the inertial subrange will be affected. Hence, if we measure in the high-frequency part of the inertial subrange, it may not be certain that the swell influence has reached at that frequency yet. Therefore the measurements do not show the same wave influence that would be seen if the whole spectrum was affected.

[93] As discussed earlier, the different terms in the TKE-budget must be parameterized correctly if the inertial dissipation method should work during all conditions. The results in this study show that it is not possible to exclude the transport terms under most conditions, and that the terms are very sensitive to stability, wave age, and wind speed.

### 8. Discussion

- Top of page
- Abstract
- 1. Introduction
- 2. Site and Measurements
- 3. Theoretical Considerations
- 4. Data
- 5. Terms in the Turbulent Kinetic Energy Budget
- 6. Results for Near-Neutral Conditions
- 7. The Inertial Dissipation Method
- 8. Discussion
- 9. Conclusions
- Appendix A.: The Relation Between Sonic Temperature Flux and Virtual Temperature Flux
- Acknowledgments
- References

[94] It has been shown that the terms in the TKE-budget depend not just on stability, which has been previously suggested, but also on wave age and to some degree on wind speed. The transport terms are not negligible, not even during neutral conditions. It is mainly the normalized pressure transport term that is responsible for the imbalance. The normalized turbulent transport was found to be very small during all conditions.

[95] The use of the traditional inertial dissipation method (i.e., assuming that the sum of the transport terms is small) therefore seems to be inaccurate during most conditions encountered; correction functions need to be included to account for this imbalance.

[96] The wave age dependence can be divided into three regions, small wave ages (*c*_{0}/(*U*_{10} cos θ) < 0.5), moderate wave ages with saturated waves 0.5 < *c*_{0}/(*U*_{10} cos θ) < 1.2, and swell *c*_{0}/(*U*_{10} cos θ) > 1.2.

[97] For small wave ages or high wind speeds, the imbalance between normalized production and normalized dissipation was found to be positive for near neutral conditions (Figure 12). This is in accordance with *Edson and Fairall* [1998], who found that for slightly convective conditions, production exceeded dissipation. Their explanation was that energy is building up the waves rather than being dissipated into heat. Since the waves are growing at these small waves ages, this seems to be the case also here.

[98] For moderate wave ages, we have “saturated” conditions, where the waves are neither growing, nor decaying. The sea surface then seems to resemble a land surface, at least in near neutral conditions. The values for the normalized transport terms (*T*_{t}*N* ≈ 0 and *T*_{p}*N* ≈ 0.25) and the imbalance between normalized production and normalized dissipation (≈ −0.30), is close to values found over land in similar measurements [e.g., *Högström*, 1996].

[99] For swell dissipation is much larger than production and the normalized pressure transport becomes significant.

[100] The process of “inactive” turbulence can probably explain the high values of the pressure transport found during unstable conditions. Inactive turbulence arises in the upper part of the boundary layer and is being brought down to the surface by the pressure transport term. A possible source for inactive turbulence can be a low-level jet at the top of the boundary layer.

[101] This phenomenon is described by *Smedman et al.* [1994]. In that paper an analysis of airborne turbulence measurements throughout a slightly unstable marine boundary layer is presented. It is shown that, in that particular case, ordinary turbulent production at the surface is cut off. Instead, turbulence of boundary layer-size scale was observed to be produced in an elevated shear zone at the top of the boundary layer and brought down to the surface with the aid of the pressure transport mechanism. Although this case with zero turbulence production at the surface is believed to be rather special, the upside-down production mechanism is thought to be at work as soon as there is an elevated shear zone at the top of the boundary layer, a situation which is quite common in the case of a slightly unstable marine boundary layer.

[102] This “inactive” part of the turbulence does not contribute to the shearing stress, and does not interact with the “active” turbulence. The “inactive” turbulence, represented by the pressure transport, is being dissipated at the surface, whereas the production is only the “active” turbulence [*Högström*, 1990]. This explains why the dissipation is larger than the production.

[103] Note, however, that as explained by *Högström et al.* [2002], there is an additional process at work during neutral conditions, which will cause downward transport of momentum by the pressure transport term. This mechanism is, however, far from “inactive,” being in fact the major agent for momentum transport in the near-neutral surface layer. This process is of surface-layer scale, not of boundary layer scale, as is the inactive turbulence discussed above.

[104] The stress at the surface can be very small during swell conditions (see discussion above), implying that the “active” part of the turbulence is small. Only the “inactive” part will be left, since this does not contribute to the shearing stress, and energy will be brought down to the surface from higher levels [*Smedman et al.*, 1999] through the pressure transport. Therefore, “inactive” turbulence is likely to play an important role during swell conditions.

[105] As discussed by *Rutgersson et al.* [2001], inactive turbulence is not expected to occur during stable conditions. Here instead, the pressure transport term appears to be the result of upward transfer of momentum from the waves during swell, the magnitude of this term decreasing rapidly with height. This contrasts strikingly to the unstable case, which is characterized by observed constancy of *T*_{p}*N* with height, at least up to the highest measuring level, about 26 m [*Smedman et al.*, 1999], indicating that energy must also come from above and not only from the waves.

[106] For more convective conditions, however, ϕ_{m} is very sensitive to the wave conditions, since swell will give negative values for the wind gradient, and therefore negative values of ϕ_{m}. The normalized buoyancy will also give a significant contribution to the production during unstable conditions. This can be somewhat misleading since, as discussed above, it is mainly *u*_{*} that controls the normalized buoyancy, rather than the heat flux, which is usually very small.

[107] As discussed in Section 5.3, the value 0.52 was chosen for the Kolmogorov constant α. This value is based on numerous simultaneous measurements of dissipation and inertial subrange spectra. The analysis shows that for the saturated wave range (0.5 < *c*_{0}/(*U*_{10} cos θ) < 1.2), it gives results that are in agreement with corresponding results over land. When employing this value of α, however, the transport terms of the TKE-budget become nonzero. As α has been determined in the only physically correct way, this is an inevitable result and a physical reality. The use of an “apparent” Kolmogorov constant is a way to artificially eliminate the effect of the transport term in the TKE-budget, which has been used in some studies, but it is not to be recommended, as we have seen that the magnitude of the transport terms varies considerably with wave age and stability.