1. Introduction
Within the atmospheric surface layer (ASL) that is directly adjacent to the air–sea interface, physical processes occurring at turbulent scales take place that are important to heat and momentum exchange between the ocean and directly overlying atmosphere. Dissipation of the turbulent kinetic energy within the atmospheric surface layer, which is a consequence of interfacial friction and near-surface vertical shear of the horizontal wind, results in momentum transfers to surface waves and surface currents, and a dissipative heat transfer to the atmosphere. Bister and Emanuel (1998, hereafter BE98) were the first to investigate dissipative heating, believing the term to be important to tropical cyclone (TC) maximum potential intensity. BE98 derived the term from the momentum and kinetic energy equations and inserted this term along with turbulent enthalpy flux to balance radial advection of equivalent potential temperature and vertical motion of mass associated with TC secondary circulation [BE98’s Eq. (11)]. BE98 assumed all of the energy arising from the dissipation of turbulent kinetic energy is released as heat to the atmosphere, and mathematically concluded the result is a 50% increase in the enthalpy transfer coefficient and subsequent 20% increase in maximum TC wind speed.
Following the suggestions of BE98, dissipative heating parameterizations were added into more sophisticated, finescale models to produce intensity forecasts for real TCs (e.g., Zhang and Altshuler 1999; Jin et al. 2007). The dissipative heating parameterizations in these and other numerical studies (e.g., Businger and Businger 2001) have been directly related to wind speed, usually proportional to a surface or reference wind speed cubed. Although including the dissipative heating increased the modeled TC maximum wind speed intensity and intensity prediction accuracy of Hurricanes Andrew (Zhang and Altshuler 1999) and Irene (Jin et al. 2007), respectively, recent studies (e.g. Curcic 2015; Kieu 2015) have challenged the treatment of dissipative heating by BE98 because of their assumptions that (i) all dissipated TKE is transferred to the atmosphere as heat and (ii) the dissipative heating acts as an extra heat source to the atmosphere, which increases the total energy in the atmosphere–ocean system.
We present the first reported measurements of dissipative heating in the explicitly nonhurricane atmospheric surface layer and compare them with aircraft measurements from the hurricane environment during the Coupled Boundary Layers Air–Sea Transfer Experiment (CBLAST; Zhang 2010, hereafter ZH10). The results of this comparison were a primary motivation for this work, as our dissipative heating magnitudes agree with ZH10 only when the TKE dissipation rate is explicitly used (after ZH10 method). Despite the substantially different wind speed range and atmospheric state present in the two studies, we observed a similar range of dissipative heating values to those in ZH10. Therefore, we challenge the notion that the wind speed is the physical process most directly governing and constraining both TKE dissipation and dissipative heating and we agree with prior findings (e.g., Kieu 2015) that the assumptions made by BE98 are inappropriate and result in the overestimate of dissipative heating. If however, the dissipation rate and dissipative heating are not a function of wind speed but are physically constrained, what is governing their behavior? We use high-resolution ship measurements from the Lagrangian Submesoscale Experiment (LASER) in the Gulf of Mexico during January–February 2016 to investigate how the dissipation rate and dissipative heating change and how they are constrained by both the atmospheric stability and the structure of the sea surface.
2. Theoretical background
a. Original derivation of BE98
b. Revised theory from recent studies
While the results from a comparison of dissipative heating measurements in explicitly different atmospheric surface layer environments is a motivating impetus for our work, the focus of our investigation is the dissipation rate of turbulent kinetic energy. Nevertheless, we must give a complete account of the dissipative heating, which includes the reassessment of the conclusions put forward by BE98 and our knowledge to date. For the full details, we direct the reader to Kieu (2015).
Briefly, Kieu (2015) argue that the BE98 representation of dissipative heating in tropical cyclones is questionable because it (i) treats dissipative heating as an external heat input that adds energy to the atmosphere and (ii) assumes a perfect efficiency of energy transfer purely to the atmosphere as dissipative heat. We explain both of these issues in the following matter, beginning with the problem of external heat input.
Both of the assumptions we mentioned here result in either inappropriate addition of or overestimation of energy being transferred to the atmosphere. ZH10 used high-resolution measurements of velocity within the hurricane boundary layer in CBLAST to determine the dissipative heating using the TKE dissipation rate, and demonstrate that (4b) from BE98 overestimates the dissipative heating. We will show later that our dissipative heating measurements, which come from an explicitly nonhurricane atmospheric surface layer, clearly corroborate the discrepancy outlined by ZH10.
c. Dissipative heating from TKE dissipation rate
3. Data
a. Field experiment background
LASER was conducted January–February 2016 by investigators from the Consortium for Advanced Research on Transport of Hydrocarbons in the Environment (CARTHE) group, as part of the Gulf of Mexico Research Initiative (GoMRI) to investigate marine, biological, and physical effects of the Deepwater Horizon (DWH) oil spill (2010). LASER was motivated by data and preliminary findings on near-surface ocean transport from large-scale currents and eddies to submesoscale fronts, collected during the Grand Lagrangian Deployment (GLAD) experiment in August 2012 (e.g., Olascoaga et al. 2013; Poje et al. 2014; Beron-Vera and LaCasce 2016). In particular, LASER was a coordinated effort to investigate crude oil transport near to the DeSoto Canyon region of the continental shelf. Figure 1a shows the study region overlain with the vessel track of the R/V F.G. Walton Smith (FGWS), operated by the University of Miami Rosenstiel School of Marine and Atmospheric Sciences (RSMAS). Data for this study were taken from instruments mounted onto and on board the FGWS, and are described in further detail in the next section.
b. Instrumentation and measurements
Surface-layer atmospheric and both in situ and remote surface ocean measurements were collected on the FGWS using a combination of fixed-mount digital sensors, direct water sampling instruments, and a marine Doppler radar (MDR). With the exception of port calls and localized maneuvering or small-boat operations as needed, the ship was under way in open ocean with nonzero forward speed
R/V F.G. Walton Smith bow instrumentation heights. For the center column “P” denotes port prow, “S” denotes starboard prow, and “CT” denotes center truss. For the UDMs, “-O” denotes “outboard.”
c. Data limitations and motion correction
Quality-control flags were implemented on raw data collected on the FGWS bow, including the turbulent velocity, temperature, pressure, and humidity measurements from the flux towers as well as the truss and prow UDM data. The resulting subsample of the original dataset only included 1) measurements where the difference of the mean wind direction and ship heading (HMWD) was within an arc of 20° to port or starboard of 0° (ship moving directly into the wind), and 2) reduced the five UDM array to a 3 UDM triangle based on elevation spectra error characteristics described in section 4c. Any flux data for which HMWD was greater than ±20° was treated as possibly subject to flow distortion and not included. As a secondary precaution, only turbulent velocity and sonic temperature data from the highest two IRGASON sonic anemometers (A1 and B1; see Fig. 1b) and the middle two RM Young sonic anemometers (A2 and B2) were used. Inclusion of turbulent measurements at minimum two different heights was required to compute vertical gradients in the turbulent kinetic energy budget [see (8)]. An overview of meteorological and oceanographic conditions during LASER is presented in Fig. 2.
4. Analysis methodology
a. Dissipation rate and dissipative heating
b. Eddy-covariance fluxes and atmospheric stability
Having sampled at high-frequency the various atmospheric variables, we capture turbulent motions and therefore make use of the eddy-covariance method (e.g., Burba and Anderson 2007) to compute the fluxes of sensible heat, latent heat, and momentum. This technique is also used to determine the eddy transport and pressure work terms in the TKE budget [terms 4 and 5 in (8)]. The quotient of shear and buoyancy production terms from the TKE budget, which also involve eddy covariance fluxes, is used to determine atmospheric stability. The eddy-covariance method involves a Reynolds decomposition of vertical velocity and a scalar quantity of interest, which are then both detrended. The covariance represented by the product of the perturbation of the two terms is then averaged over a suitable time window. We use a time average of 10 min, however averages of 15, 30, and 60 min are also commonly used. Time averages of greater than 60 min are not advisable for turbulent measurements because of the probable loss of stationarity (Large and Pond 1981).
As a result of the definition in (19a), shear production of TKE homogenizes the surface layer and works against existing density stratification in the atmosphere while buoyancy production either homogenizes or stratifies the atmospheric surface layer based on the sign of the perturbations of equivalent potential temperature defined by (19b).
c. Sea-state determination
Subjection of the sea surface to physical forcing by the wind, particularly in the open ocean where fetch may be large and uninterrupted, allows for potentially large energy transfer between the atmosphere and the ocean. Roughening of the waves on the sea surface, regardless of whether the waves are locally growing by wind or are swell may influence the flow of air in the surface layer as well as the rate of TKE dissipation. Furthermore, since previous studies have shown that the TKE budget terms can balance and contribute differently in swell as compared with wind sea (e.g., Sjöblom and Smedman 2002; Högström et al. 2009), we have segregated the sea states using water surface elevation spectra from the FGWS bow UDM array (C1–5; see Fig. 1b) using a three-step process. Each 10-min data segment of water surface elevation spectra was quality controlled, then the wavelet directional method (WDM; Donelan et al. 1996) was implemented, and finally the Pierson and Moskowitz (1964) criterion, cp/U10N > 1.2 for swell, was applied. Discrimination of the sea states using wind speed and a coefficient of 1.2 is demonstrated in Potter (2015, see their Fig. 1).
Quality control of the five UDMs began by eliminating spectrum data for frequencies less than 0.0667 Hz (a period of 15 s) as being an unlikely region to find the spectral peak considering the fetch of the Gulf of Mexico (W. M. Drennan 2017, personal communication). In order for the WDM to be implemented properly and to provide the most useful output, a minimum-of-three group of geometrically arranged signals was needed. The instruments also needed to be separated sufficiently in both along-bow and across-bow axes to provide accurate estimates of wave direction across a range of wave scales. Given their large (10.22 m) separation distance, the median of the spectra from the port and starboard prow UDM (C1 and C5), computed for each frequency bin, was used as a truth measurement of wave data not contaminated by signal cross-talk resulting from UDMs being too close together. We followed this by determining which UDMs in the center-truss triplet (C2–C4; see Fig. 1b) had “runaway error,” frequency by frequency for each UDM against the median spectra we took as truth. The runaway error term refers to increasing error rapidly at some (critical) frequency, such that the spectrum no longer follows the slope of the truth spectrum at frequencies beyond this critical frequency. UDMs with this behavior were flagged as suspect, and the two prow spectra and one of the center triplet of C2–C4 spectra were flagged as good for each 10-min segment of wave data.
Figure 4a shows the median water surface elevation spectra
5. Results
a. Dissipative heating measurements and a dissipation–drag relationship
We first present the fundamental and motivating result of dissipative heating measurements, which we have calculated using the original equation of BE98’s (4b), as well as the dissipation-rate-dependent equation put forth in ZH10’s (9). The dissipative heating values obtained with these equations are shown as a function of 10-m neutral wind speed in Figs. 5a and 5b, respectively. The 10-m wind speeds observed including our data with that of ZH10 span a range from 0.25–28.5 m s−1, from calm to full-gale Beaufort-scale conditions.
Dissipative heating generally increases with wind speed regardless of the computational method used; however, the behavior as a function of wind speed is noticeably different. In Fig. 5a, dissipative heating magnitude increases from 0 to 52.5 W m−2 and the curve follows a continuous power-law shape with an overall positive trend. Responding to the explicit dependence on wind speed of the BE98 dissipative heating equation, the dissipative heating magnitude increases very rapidly and is substantially larger in the hurricane environment of ZH10 than in the nonhurricane environment of LASER (2016), which we investigated. Conversely, in Fig. 5b, both sets of data measured dissipative heating values between 0 and 12.5 W m−2. Furthermore, the dissipative heating does not increase continuously throughout the range of observed wind speeds in the manner of Fig. 5a, but rather the dissipative heating reaches a peak of 10.25 and 20.5 W m−2 in the two datasets, respectively, and then decreases thereafter. We use the same moist air density ρ to calculate dissipative heating in the two equations, and the prescribed surface-layer height is a scalar that would only alter the magnitudes via linear multiples. Hence, the dissipation rate of TKE over the atmospheric surface layer is governing the dissipative heating magnitude in a nonlinear way and the amount of heating appears physically constrained. Although the comparison in Fig. 5 is limited by the maximum 10-m wind speeds observed during LASER (2016) and by ZH10, which do not reach the highest wind speeds seen in a TC (i.e., 96.11 m s−1 by Hurricane Patricia in 2015), it is next important to explain why the use of ZH10’s direct computation is more appropriate for determining the dissipative heating.
To do so, we return to the reevaluation of dissipative heating in TCs put forth by Kieu (2015), in which the author presents energy budgets for two atmospheric volumes that compare the effect of inclusion of the ASL. The first volume extends from the top of the ASL, denoted
BE98 used a gradient wind balance, radial momentum, and thermodynamic equation in their derivation of dissipative heating. Explicitly, the behavior of the angular momentum
b. Stability dependence of dissipation rates
While the efficiency of momentum transfer increases for local wind sea in roughening conditions, the atmospheric stability can suppress TKE production and its rate of dissipation. The Monin–Obukhov stability parameter ζ is defined as the ratio of shear and buoyancy production; however, shear always contributes to TKE gain so the eddy virtual potential temperature flux dictates whether stable or unstable conditions are present. Figure 7 presents a time series of the entire LASER experimental record (19 January–13 February) of the ratio of dissipative heating to sensible and latent heat flux (Fig. 7a), the atmospheric stability (Fig. 7b), and 10-m neutral wind speed (Fig. 7c). Although sensible and latent heat fluxes are usually much larger in magnitude than the dissipative heating term, we can see under certain circumstances that the dissipative heating and therefore the ratio in Fig. 7a may be large. Yellow-highlighted regions in the time series show areas where the ratio of dissipative heating and heat flux is greater than 0.10 (10%). Within these yellow regions, we can see that large ratio values (0.2–0.8+), or where dissipation-driven heating is large, the atmospheric surface layer is either neutrally stable or weakly stable. Furthermore, the dissipative heating to heat flux ratio does not appear to respond to changes in the 10-m neutral wind speed. We also remind the reader that dissipative heating is an internal consistency and ought to be parameterized at the top of the atmospheric surface layer as part of the overall enthalpy flux, not as an external heat source to the atmosphere but a consequence within the boundary layer of the loss of kinetic energy.
We can place the stability dependence of the dissipation rate in the context of the overall TKE budget by examining similarity relationships. Normalized structure functions, dependent on Monin–Obukhov stability parameter alone, describe the behavior of each term in stable, neutral, and unstable atmospheric surface layers (e.g., Wyngaard and Coté 1971), and are derived by taking the product of all right-hand side terms from (8) with
Partitioning the data into wind sea and swell, it is immediately clear that swell does not have any dependence on the Monin–Obukhov stability; however, interestingly observations in the stable regime were almost entirely comprised of swell conditions. In stable conditions, TKE production by shear balances dissipation, while the negative buoyancy suppresses TKE; hence, a net negative TKE production is observed without substantial upward transport. With vertical velocities small and near zero in near-neutral stability, we surmise that the consequential reduction in enthalpy fluxes explains the large dissipative heating to enthalpy flux ratios observed in such stability conditions. By contrast, during LASER we observed predominantly neutral or unstable conditions. Buoyancy production of TKE by convection-induced mixing of the surface layer is the strongest contributor to the energy budget in these conditions, although shear production does still occur. Turbulent transport and pressure work are nonnegligible as instability increases; the turbulent transport of TKE upward results in a local loss of TKE while the local loss of mass induces negative pressure perturbations and a positive pressure work term overall. Hence, the TKE dissipation rate is largest, that is, contributes most negatively to the amount of local TKE, in nonneutral stability away from zero. Additionally, we surmise that the large values of the ratio of dissipative heating to heat fluxes in Fig. 7a reflect small values of the eddy heat flux in near-neutral stability conditions. In such circumstances, the vertical velocities would be small or near zero.
c. Sea-state influence on dissipation rates
Efficient momentum transfer and nonneutral atmospheric stability have been shown to improve or encourage the rate of TKE dissipation over the open ocean during LASER (Figs. 6 and 7). However, delving further into the relationship between dissipation and drag, we specifically investigated the role of steepness and wave age on the TKE dissipation rate. In Fig. 9, we compare the wave steepness (Figs. 9a,b) and wave age (Figs. 9c,d) of observed wind-sea and swell conditions during LASER. Wave steepness
Given the proposed mechanism for enhanced TKE dissipation with steepening of waves, and the fact that the Pierson and Moskowitz (1964) criterion for wave age is a simple threshold value, an increase in wave age expectantly decreases TKE dissipation rates for both wind sea and swell. TKE dissipation rates for the wind-sea waves begin large (
d. Eddy flux behavior with increasing wave age
Turbulent momentum and heat fluxes act on mean properties of the atmosphere to produce or suppress TKE in the budget, aided or limited by the steepness and roughness of the sea and its connection to the local wind stress. Following Edson and Fairall (1998), the normalized dissipation was shown to have the largest negative magnitude, that is, highest rate of dissipation by loss of TKE in nonneutral conditions; therefore, nonzero eddy momentum and heat fluxes are important to the air–sea interaction processes contained within the TKE budget. Furthermore, the efficiency of eddy fluxes, specifically momentum flux via the drag coefficient, has been demonstrated to be sensitive to wave age in previous investigations (e.g., Nordeng 1991). Figure 10c shows how
e. Dissipation rates near and at surface ocean fronts
We include a brief mention of intriguing behavior observed during crossings of submesoscale fronts by the R/V F.G. Walton Smith. Near-surface current vectors constructed from marine Doppler radar imagery taken during LASER have shown considerable variability in the amount of shear where surface brightness qualitatively indicates the existence of a surface front. In some cases, strong near-surface current shear was present, and typically associated with a roughness feature slanting as opposed to aligned exactly north–south or west–east. Figure 11 shows such a slanting frontal crossing from 0945 to 1019 UTC 29 January. The cross-front dissipation rate and other properties are given as a function of distance from the front in Fig. 11a, while the near-surface current shear gradient derived from the current vectors is mapped in Fig. 11b. Strong near-surface current shear at the frontal axis may be associated with more coherent or collocated local maxima in dissipation rate. In some cases, stability and/or sea surface roughness
6. Conclusions
We have used high-frequency ship data from LASER to collect the first reported dissipative heating measurements explicitly in the low-wind, nonhurricane atmospheric boundary layer. The magnitude of the dissipative heating we measured is very similar to aircraft-derived measurements from a high-wind hurricane atmospheric boundary layer observed during the Coupled Boundary Layers Air–Sea Transfer (CBLAST) field experiment, but only when using a dissipative heating formulation directly involving dissipation rate (ZH10) rather than wind speed (BE98). We hence believe the traditional formula in which dissipative heating is proportional to the cube of the wind speed is inappropriate and overestimates the dissipative heating magnitude as suggested by recent studies. A relationship constraining the dissipative heating based on the physical structure of the air–sea interface and overlying atmospheric surface layer is much more consistent with our observations, and more readily reconciled with the frictional origin of the phenomenon.
Although roughening of the sea surface is instigated by the wind, by measuring the aerodynamic drag coefficient we demonstrate that the dissipation rate follows a power-law relationship with the drag coefficient of
By taking advantage of ship measurements at multiple, fixed vertical levels sampling the boundary layer atmosphere and near-surface ocean, we find dissipative heating can equal 20%–80% of the sensible and latent heat flux in near-neutral or weakly stable stability. Internal TKE loss to dissipative heating (with net-zero energy gain) is therefore competitive with these heat fluxes and not dependent directly upon wind speed. Although a lack of multiple-height pressure measurements only permitted a residual computation of the pressure work, we directly calculated all other TKE budget terms and their normalized structure functions following Edson and Fairall (1998). We determine here that shear (buoyancy) is the dominant mechanism of TKE production, and therefore dissipative heating, in the stable (unstable) atmospheric surface layers sampled during LASER. Nondimensional turbulent transport and pressure work become important as the surface layer becomes more unstable; the turbulent transport of TKE upward results in a local loss of TKE while the local loss of mass induces negative pressure perturbations.
We evaluated the sea state using wave steepness and wave age and found dissipation rates increase with the steepening of wind-sea waves and decrease with wave age overall. Swell waves did not demonstrate strong relationships to the roughness or structure of the sea in the low-wind conditions during LASER. Ultimately, we conclude that dissipative heating will be largest in long-fetch environments with nonneutral atmospheric stability characterized by large shear or buoyancy TKE production in stable and unstable conditions, respectively. Steeper wind-sea waves have a larger form drag, resulting in larger eddy momentum and heat fluxes, while slower swell waves have a phase speed closer to frictional stress in the surface layer that may aid their contribution to local shear and buoyancy production, resulting in more dissipative heating. A laboratory experiment to determine the full TKE, heat, and momentum budgets at wind speeds up to U10N = 90 m s−1 will be conducted to address the caveats of this study; particularly, we hope to capture the effects of larger swells on buoyancy and pressure terms in the TKE budget, which could be important to the energetics of the hurricane boundary layer.
Acknowledgments
We thank the Consortium for Advanced Research on Hydrocarbon in the Environment (CARTHE) and LASER investigators for collection and management of the raw ship data, Björn Lund for providing marine Doppler radar data, and David G. Ortiz-Suslow for the subsequent data postprocessing. Data are publicly available through the Gulf of Mexico Research Initiative Information and Data Cooperative (GRIIDC) at https://data.gulfresearchinitiative.org (Gulf of Mexico Research Initiative 2018).This work was supported in part by Grant GR009805 from the Gulf of Mexico Research Initiative (GoMRI), as well as by Grant NA14NWS4680028 from the National Oceanic and Atmospheric Administration (NOAA) and Grant AGS1822128 from the National Science Foundation.
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