Role of Near-Inertial Internal Waves in Subthermocline ... - AMS Journals

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Role of Near-Inertial Internal Waves in Subthermocline Diapycnal Mixing in the Northern Gulf of Mexico ZHAO JING Department of Oceanography, Texas A&M University, College Station, Texas, and Qingdao Collaborative Innovation Center of Marine Science and Technology, Ocean University of China, Qingdao, China

PING CHANG Department of Oceanography, and Department of Atmospheric Sciences, Texas A&M University, College Station, Texas, and Qingdao Collaborative Innovation Center of Marine Science and Technology, Ocean University of China, Qingdao, China

STEVEN F. DIMARCO Department of Oceanography, and Geochemical and Environmental Research Group, Texas A&M University, College Station, Texas

LIXIN WU Qingdao Collaborative Innovation Center of Marine Science and Technology, Ocean University of China, Qingdao, China (Manuscript received 14 November 2014, in final form 20 September 2015) ABSTRACT Moored ADCP data collected in the northern Gulf of Mexico are analyzed to examine near-inertial internal waves and their contribution to subthermocline diapycnal mixing based on a finescale parameterization of deep ocean mixing. The focus of the study is on the impact of near-inertial internal waves generated by an extreme weather event—that is, Hurricane Katrina—and by month-to-month variation in weather patterns on the diapycnal mixing. The inferred subthermocline diapycnal mixing exhibits pronounced elevation in the wake of Katrina. Both the increased near-inertial (0.8–1.8f, where f is the Coriolis frequency) and superinertial (.1.8f ) shear variances contribute to the elevated diapycnal mixing, but the former plays a more dominant role. The intense wind work on near-inertial motions by the hurricane is largely responsible for the energetic near-inertial shear variance. Energy transfer from near-inertial to superinertial internal waves, however, appears to play an important role in elevating the superinertial shear variance. The inferred subthermocline diapycnal mixing in the region also exhibits significant month-to-month variation with the estimated diffusivity in January 2006 about 3 times the values in November and December 2005. The subseasonal change in the diapycnal mixing mainly results from the subseasonal variation of the near-inertial wind work that causes intensification of the near-inertial shear in January 2006.

1. Introduction Diapycnal mixing in the ocean plays an important role in transport of heat, freshwater, dissolved gases, nutrients, and pollutants, such as oil spills. Understanding its

Corresponding author address: Zhao Jing, Department of Oceanography, Texas A&M University, MS 3146 TAMU, College Station, TX 77843-3146. E-mail: [email protected] DOI: 10.1175/JPO-D-14-0227.1 Ó 2015 American Meteorological Society

spatial and temporal variation is the key to improving model representation and prediction of large-scale ocean circulation and climate (Richards et al. 2009; Saenko and Merrifield 2005; Wunsch and Ferrari 2004), as well as marine environment hazards, such as hypoxia (e.g., DiMarco et al. 2012) and oil spills (e.g., Adalsteinsson et al. 2013). Much of the diapycnal mixing away from boundaries occurs through internal wave breaking. Energy is input into the internal wave field primarily by the tides and winds (Wunsch and Ferrari 2004). The

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total tidal dissipation rate in the open ocean is estimated to be around 1 TW (Egbert and Ray 2001; Jayne and St. Laurent 2001), implying that tidal energy may account for only 50% of the energy required for sustaining diapycnal mixing below the thermocline (Munk and Wunsch 1998). It has been conjectured (Wunsch and Ferrari 2004) that the wind work on oceanic near-inertial motions, estimated to be between 0.5 and 1.4 TW (Alford 2003; Watanabe and Hibiya 2002; Jiang et al. 2005), may be crucial in closing the energy budget. As such, it can have a pronounced influence on the variation of subthermocline diapycnal mixing. However, the significant contribution of near-inertial internal waves to diapycnal mixing inferred from the energy budget argument is subject to debate. Part of the wind work on near-inertial motions is inevitably dissipated within the mixed layer. Recent observational and modeling studies reveal that the wind-generated nearinertial energy flux into the deep ocean is relatively weak. For example, Alford et al. (2012) analyzed the mooring data at Ocean Station Papa and found the downward near-inertial energy flux within 200–800 m to be less than 33% of the near-inertial wind work. Based on numerical model experiments, Zhai et al. (2009) showed that more than 70% of near-inertial wind work is lost to turbulent mixing within the top 200 m, confirming earlier findings reported by Furuichi et al. (2008). Given the relatively small downward energy flux, it remains uncertain whether the wind-generated nearinertial internal waves can significantly influence diapycnal mixing below the thermocline. To help address this controversy, an examination of the relation between mixing strength and near-inertial shear on various time scales is necessary but still lacking because of the limited observations. Most of wind work on near-inertial motions occurs when moving windstorm systems pass over the ocean. The oceanic response may be conveniently divided into two stages as a result of the relatively long time scale of baroclinic geostrophic adjustment compared to the wind forcing time scale (Price et al. 1994). The first stage is considered as a ‘‘forced stage,’’ when storms pass over and energy is rapidly deposited to surface mixed layer. The second stage, regarded as a ‘‘relaxation stage,’’ corresponds to the wake of a storm when energy is dispersed as near-inertial waves in different vertical modes. Gill (1984) examined the second stage by projecting the slab mixed layer velocity onto vertical modes and showed that the downward radiation of energy is achieved by phase separations among different vertical modes. As demonstrated by Gill (1984), the expected modal partition of the energy depends on the mixed layer depth. In the real ocean, most of the energy is typically projected

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onto a few leading modes (e.g., the first 10 modes), implying that near-inertial currents below the thermocline are characterized by vertical wavelengths of O(100–1000) m. This has been supported by previous observations (Alford et al. 2012; Brooks 1983; D’Asaro and Perkins 1984; Shay and Elsberry 1987; Shay et al. 1989). For example, phase delay among velocity records at different depths indicates that the vertical scale of near-inertial internal waves generated by Hurricane Allen is about 1000 m (Brooks 1983). Because shear, rather than velocity and energy, is crucial in destabilizing flows, these large-vertical-scale near-inertial internal waves are not very effective in generating turbulence. Evidence for breaking of these large-scale near-inertial waves is limited. It has been conjectured that their energy might be first transferred to smaller-scale higher-frequency internal waves, which subsequently break and cause diapycnal mixing (Alford and Pinkel 2000; Polzin et al. 2014). However, direct observational evidence is still lacking to support this conjecture. This study presents an attempt to test this hypothesis using moored ADCP data collected in the wake of a hurricane when energetic large-scale nearinertial waves were generated. Hurricanes are rapidly rotating storm systems characterized by a low-pressure center and strong winds. Energetic near-inertial waves are excited after the hurricanes pass (Price 1981, 1983; Shay and Elsberry 1987; Shay et al. 1989). The response of near-inertial currents is more pronounced to the right of a hurricane track as both wind stress vector and inertial oscillations rotate anticyclonically so that near-inertial currents are effectively accelerated (Price 1981, 1983). The infamous category 5 Hurricane Katrina occurred in the Gulf of Mexico during 23–30 August 2005 (Fig. 1). The three Minerals Management Service (MMS) moorings (Cox and EvansHamilton, Inc., 2011) happened to be deployed to the right of Katrina’s track and thus were able to capture the energetic near-inertial internal waves generated by Katrina (Jaimes and Shay 2010). In this paper, we will analyze the relationship between diapycnal mixing and near-inertial internal waves in the wake of Katrina (i.e., the relaxation stage) using a finescale parameterization method (Gregg et al. 2003; Kunze et al. 2006) and the interaction between near- and superinertial internal waves. The result will provide insight into short-time variation of diapycnal mixing induced by near-inertial internal waves under extreme weather conditions. Furthermore, the extended deployment duration of the mooring data makes it possible to evaluate longer time-scale influences of near-inertial internal waves on diapycnal mixing, which is associated with subseasonalto-seasonal atmospheric variability. The paper is organized as follows. Data and methodology are given in

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TABLE 1. Summary on the measurements of the three MMS moorings.

Depth (m)

Equipment

Measured parameter

Sampling interval (h)

75 112 150 187 225 300 400 509b

Sea-Bird Hugrun Sea-Bird Hugrun Sea-Bird Hugrun Hugrun ADCP

C, T, P a T C, T, P T C, T, P T T T, P, u, y

0.5 0.5 0.5 0.5 0.5 0.5 0.5 1

a b

FIG. 1. Depth contours (m) and geographic distribution of the three MMS moorings (M1/M2/M3: red/blue/green triangles, respectively). Black dots represent the locations of CTD casts taken during deployment cruises. Thick colored lines are the tracks of Hurricane Katrina and Rita with dashed circles denoting their radii. Here, the radius is defined as the distance at which the wind speed decreases to 1/e of its maximum value and is computed from the H*wind analyses (Powell et al. 2010).

section 2. Results and analyses are presented in section 3. Conclusions and discussions are summarized in section 4.

2. Data and methodology a. Mooring data In January 2005, three MMS moorings (M1–M3) were deployed in the northern Gulf of Mexico (Cox et al. 2010; Cole and DiMarco 2010; Fig. 1). The bottom depths at M1–M3 are 2600, 2570, and 2812 m, respectively. Each mooring was equipped with a Rowe–Deines Instruments (RDI) broadband 75-kHz long-range acoustic Doppler current profiler (ADCP) with 8-m bin spacing to measure horizontal velocity in the upper 50–500 m of the water column. There were also three Sea-Bird MicroCAT 37SM CTD sensors at ;75, ;150, and ;225 m and four Hugrun temperature sensors at ;112, ;187, ;300, and ;400 m. A summary of their measured parameters and sample intervals is given in Table 1. The data were collected, processed, and quality controlled according to standard oceanographic protocols (Cox et al. 2010). Additionally, ADCP parameters, such as instrument tilt, acoustic echo intensity, beam correlation, and percent good, were investigated to ensure that sensors were operating properly during the collection period. Particular attention was paid to the time of the hurricane passage, to assure the data are of high quality. The instruments were deployed for one year and underwent a service and maintenance recovery after

C for conductivity, T for temperature, and P for pressure. Depth listed is for ADCP on M1. Depth for M2 (M3) is 520 (487) m.

7 months. Only the data collected during the second deployment were used here, covering the period from 22 August 2005 to 24 January 2006. Additionally, the CTD casts collected during the service and maintenance of the moorings were also used (Fig. 1). Altogether, there were 28 CTD casts, which were averaged into halfmeter bins during the postcollection data processing using manufacturer processing software. Two category 5 hurricanes occurred during the second deployment. Hurricane Katrina formed over the Bahamas on 23 August 2005 and dissipated on 30 August after landfall in the north-central Gulf of Mexico. Hurricane Rita formed on 18 September 2005 in the similar location and dissipated on 26 September 2005 inland near the Texas–Louisiana border. Near-inertial internal waves at the moorings were elevated in the wake of both Katrina and Rita. However the elevation was much more pronounced in the wake of Katrina (Table 2). This might be in part due to the longer distance from the mooring locations to Rita’s path than to Katrina’s path (Fig. 1). According to stationary wave theory (Lighthill 1978), wave energy density is inversely proportional to the distance from the generation site. In fact, the nearinertial kinetic energy in the wakes of Katrina and Rita becomes much closer if rescaled by the respective distance between the mooring location and the hurricane’s path. In this paper, we will focus on the near-inertial internal waves and diapycnal mixing in the wake of Katrina, because the stronger near-inertial wave response makes it a better case study for the hurricaneinduced diapycnal mixing. Henceforth, the wake of Katrina is defined as the period between 30 August and 18 September, which is the interval between the two hurricanes. To examine the elevation of near-inertial internal waves and diapycnal mixing, background values are required. These are obtained from a period free from hurricane influences. Here the period between

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TABLE 2. Mean values of various variables within 150–470 m during different time periods: K is diffusivity, « is dissipation rate, N2 is squared vertical mean buoyancy frequency, S2s is superinertial shear variance, S2i is near-inertial shear variance, Q2s is squared superinertial horizontal velocity, and Q2i is squared near-inertial horizontal velocity. Values in the parentheses represent the standard deviations of each quantity during corresponding periods.

K (m2 s21) « (m2 s23) N2 (s22) S2s (s22) S2i (s22) Q2s (cm2 s22) Q2i (cm2 s22)

Wake of Katrina

Wake of Rita

Hurricane-free period

11.7 (21.9) 3 1025 13.8 (26.4) 3 1029 2.3 (0.1) 3 1025 16.9 (3.2) 3 1026 5.8 (3.7) 3 1026 17.2 (8.7) 167.2 (125.0)

2.9 (5.1) 3 1025 3.4 (4.5) 3 1029 2.4 (0.2) 3 1025 16.3 (3.0) 3 1026 4.1 (1.0) 3 1026 10.2 (3.4) 62.4 (26.3)

1.8 (2.5) 3 1025 2.0 (2.6) 3 1029 2.4 (0.5) 3 1025 15.3 (2.8) 3 1026 3.3 (1.2) 3 1026 8.0 (2.5) 24.0 (12.4)

12 October 2005 and 24 January 2006 was used; henceforth, this will be referred to as the hurricane-free period.

b. Near- and superinertial signals During 22 August 2005–24 January 2006, the rotary frequency spectra of horizontal velocity and its vertical shear exhibit pronounced peaks near the negative Coriolis frequency 2f (Fig. 2), consistent with the polarization preference of near-inertial internal waves. The relatively broad near-inertial peak may be partly due to the time-varying Doppler shift and shift of f toward the effective Coriolis frequency, feff 5 f 1 §/2, where § is the relative vorticity (Kunze 1985; Jaimes and Shay 2010). Throughout this study, we applied a bandpass filter to the horizontal velocity to isolate the nearinertial signals in the frequency range between 0.8f and 1.8f. A sensitivity test indicates that a change in the filtering bandwidth—for example, 0.9f–1.2f or 0.8f–2.0f— does not make a substantial impact on the results hereinafter. The term ‘‘superinertial’’ is defined for frequencies higher than 1.8f, including semidiurnal tides

and the continuum of internal waves (Garrett and Munk 1979). As the ADCP moorings were located around 288N, the diurnal signals project strongly onto the near-inertial band (Zhang et al. 2009). The near-inertial currents in this study thus include the contribution from diurnal tides. The magnitude of the barotropic diurnal tides in the Gulf of Mexico, however, is less than 1 cm s21 (DiMarco and Reid 1998; Egbert and Erofeeva 2002). Therefore, the baroclinic diurnal tides are expected to be much weaker than the wind-generated nearinertial waves.

c. Finescale parameterization method In this study, the diapycnal mixing is inferred from the finescale parameterization (Gregg et al. 2003; Kunze et al. 2006). It is based on the idea that weakly nonlinear interactions among internal waves act to steadily transfer energy from large (vertical) scales, at which they are generated, to small scales, where they break as a result of shear or convective instabilities. The turbulent kinetic

FIG. 2. Three-mooring-mean rotary frequency spectra at 150-m depth for (a) horizontal velocity Z 5 u 1 iy and (b) its vertical shear. Red (blue) line represents the clockwise (counterclockwise) rotating component with its 95% confidence interval (shaded) computed using a chi-square approach. Vertical line denotes the local inertial frequency.

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dissipation rate can be parameterized in terms of finescale shear as 2

« 5 «0

N hS2 i2 h (R )j( f /N) , N02 hS2GM i2 1 v

(1)

where «0 5 6.7 3 10210 m2 s23, N is the vertical mean buoyancy frequency, hS2GM i is the shear variance derived from the Garrett–Munk (GM) model spectrum (Gregg and Kunze 1991) treated in the same way as the observed shear variance hS2 i, and j( f /N) and h1 (Rv ) are defined as f a cosh(N/f ) , j( f /N) 5 f30 a cosh(N0 /f30 ) 3(R 1 1) h1 (Rv ) 5 pffiffiffi vpffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Rv Rv 2 1

and

(2) (3)

respectively, where f30 5 f(308), N0 5 5.2 3 1023 rad s21, and Rv represents the shear–strain variance ratio. The method of computing the dissipation rate basically follows Kunze et al. (2006) (see appendix A for details). The diapycnal diffusivity can be expressed in the form of K5G

« N

2

,

(4)

where G is the mixing efficiency and is typically taken to be 0.2 (Osborn 1980; Oakey 1982; Moum 1996). Gregg et al. (2003) and Polzin et al. (1995) analyzed the dissipation rate inferred from (1) and found that the estimates agreed with those derived from microstructure measurements within a factor of 2 in the open ocean, including areas characterized by strong near-inertial waves. A direct comparison between the inferred dissipation rate from (1) with a constant Rv value of 7 and microstructure measurements near the mooring location in the northern Gulf of Mexico (Z. Wang 2014, personal communication) reveals an error of a factor of 2.5, which is slightly greater than the estimated error in the open ocean. Therefore, the inferred dissipation rate should be treated with caution. In this study, we will focus on the variability of the diapycnal mixing and its relationship to shear variance and stratification rather than the exact numbers of the dissipation rate and diffusivity. As profiles of potential density are not available to resolve finescale strain, it is difficult to directly estimate Rv. In our analysis a fixed value of Rv 5 7 is used. This is justified based on Kunze et al. (2006), who found that the measured shear–strain variance ratio is given by Rv 5 7 6 3 for N . 4.5 3 1024 s21 using 3500 lowered ADCP/CTD profiles from the Indian, Pacific, North Atlantic, and Southern Oceans. It should be noted that

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the value of Rv is expected to vary and is probably larger than 7 in the wake of Katrina as a result of the energized near-inertial internal wave activity. However, h1(Rv) does not change significantly for Rv greater than 7, and in fact it only changes less than by a factor of 2 for Rv varying from 4 to 10 and less than by a factor of 3 for Rv varying from 4 to 20. As demonstrated in appendix B, this range of uncertainty in estimating Rv is not significant enough to have a substantial impact on the major conclusions of this study.

d. Stratification The vertical mean buoyancy frequency N within 150– 470 m is needed to compute the dissipation rate in (1). The main difficulty in computing N lies in the lack of salinity measurements below 225 m for the mooring data. To overcome this difficulty, we use the 28 CTD casts taken during the maintenance of the moorings to derive a temperature–salinity relationship for the region and then apply it to the mooring data. The CTD data reveal a tight relationship between salinity and temperature in the region. Below 150 m, the correlation coefficient reaches a value of above 0.99 (Fig. 3a), strongly suggesting a linear salinity–temperature relationship, that is, S 5 aT 1 b, where the slope a and intercept b can be determined by a least squares fit (Figs. 3b,c). We estimate that using this inferred salinity will lead to a relative error of ;0.5% in N and ;2% in diffusivity. To ensure that the salinity–temperature relationship derived before the passage of Katrina is applicable to the entire period, we further tested the time dependency of a and b. We computed a and b at depths of 75, 150, and 225 m using the mooring data during the entire time period. The discrepancy between the coefficients derived from the CTD casts and the mooring data decreases as depth increases (Figs. 3b,c). It is evident that the coefficients agree well with each other at 225 m. Therefore, we conclude that the salinity–temperature relationship can be regarded as time independent below 225 m. The coarse vertical resolution of mooring hydrographic sensors may lead to additional uncertainties in the estimation of N. Figure 3d displays N computed from the 0.5-m potential density data derived from the 28 CTD casts and that from the subsampled potential density data according to the depths of mooring hydrographic sensors. Before the passage of Katrina, the northern Gulf of Mexico is characterized by a strong thermocline centered at 60 m. The surface mixed layer is not well developed probably because of the strong surface heating by shortwave radiation. Below 70 m, N decreases smoothly as depth increases. At each sensor depth, the values of N computed from these two different methods agree well with a discrepancy less than

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FIG. 3. (a) Correlation coefficient between salinity and temperature at different depths derived from the CTD casts. (b) Linear slope in the temperature and salinity relationship. Gray line represents the values derived from the CTD casts, while the short colored lines denote those derived from the mooring data with their widths representing the 95% confidence intervals. (c) As in (b), but for the intercept. (d) Background buoyancy frequency derived from the 28 CTD casts (gray solid) and its 95% confidence interval computed from the bootstrap method (gray shaded). Red circles denote the values computed from the subsampled CTD cast data according to the locations of mooring hydrographic sensors.

10%. In particular, the vertical mean buoyancy frequency N within 150–470 m differs less than 1% between the two methods, suggesting that the estimated N using the mooring hydrographic sensors is reliable.

3. Results a. Energetic near-inertial internal waves in the wake of Katrina Near-inertial internal waves become much more energetic in the wake of Katrina. The mean near-inertial kinetic energy within 150–470 m in the wake of Katrina is about 7 times of that during the hurricane-free period (Table 2) and the difference is significant at the 5% significance level based on a Wilcoxon rank-sum test (Lehamann and D’Abrera 1975). Correspondingly, the near-inertial shear variance is also significantly elevated, although the elevation is somewhat weaker compared to the near-inertial kinetic energy (Table 2). The wavenumber spectrum of near-inertial shear indicates that the elevated near-inertial internal waves in the wake of Katrina are more evident in small vertical wavenumbers (Fig. 4). The elevation becomes less evident with the increased wavenumber (Fig. 4). This suggests that near-inertial internal waves generated by Katrina in this region are dominated by low-order vertical modes. Variations of near-inertial kinetic energy and shear variance do not always follow each other. On one hand, the elevated near-inertial kinetic energy between

30 August and 3 September 2005 was associated with weak near-inertial shear variance (Figs. 5a–c). On the other hand, the pronounced near-inertial shear variance after 10 September 2005 did not correspond to the elevation of near-inertial kinetic energy (Figs. 5a–c). This may be related to the decreasing vertical scale of the nearinertial waves as time increases. The vertical p scale of ffiffiffiffiffiffiffiffiffiffiffiffi near-inertial waves can be measured as Di 5 2p Q2i /S2i , where 0:5Q2i and S2i represent the near-inertial kinetic energy and shear variance, respectively. For a pure plane wave, Di should be equal to the wavelength. In the

FIG. 4. Vertical wavenumber spectra for buoyancy-frequencynormalized near-inertial (solid) and superinertial (dashed) shear in the wake of Katrina (red) and during the hurricanefree period (blue).

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FIG. 5. Temporal evolution of various quantities in the wake of Katrina at (from left to right) M1–M3. p(a)–(c) ffiffiffiffiffiffiffiffiffiffiffiffi Squared near-inertial velocity Q2i (blue) and near-inertial shear variance S2i (green) averaged within 150–470 m, (d)–(f) Di 5 2p Q2i /S2i (blue solid) and WKBnormalized Di (red dashed), and (g)–(i) diffusivity inferred from (4). Vertical gray dashed lines mark the landfall time of Katrina.

environment, the observed near-inertial waves have far more complex structure than a plane wave. Therefore, Di can only be considered as a rough estimate of the vertical scale but is sufficient for our qualitative analysis here. From Figs. 5d–f, it is evident that Di is much larger in the early part of the records and decreases as time increases. The decreasing Di with time cannot be simply ascribed to changes in background stratification, as the Wentzel–Kramers–Brillouin (WKB)-normalized Di exhibit very similar variations. The dispersion of the different baroclinic modes excited by Katrina may be only possible cause for the decreasing vertical scales with time. Because smaller horizontal group velocities are associated with the higher-order modes, large verticalscale near-inertial waves are expected to arrive at the mooring site first, followed by small vertical-scale nearinertial waves. Indeed, numerical simulations indicate that energetic near-inertial internal waves propagate into the mooring location from the west in the wake of Katrina with low-order modes arriving first followed by high-order modes (not shown). The decrease in vertical scale with time has an important impact on diapycnal mixing, as it is the shear variance rather than kinetic energy of near-inertial internal waves that contributes to diapycnal mixing. For instance, the near-inertial waves that arrived before 3 September were associated with strong horizontal currents but had little vertical shear due to their large vertical scales. Correspondingly, the diffusivity remained at its background level (Figs. 5g–i). In contrast, near-inertial waves that arrived in the later stage had smaller vertical scales and produced stronger shear. They were more effective in enhancing diapycnal mixing within 150–470 m. Finally, we note that the near-inertial responses vary significantly among different moorings (Figs. 5a–c). For

instance, the near-inertial shear variance in M2 is stronger than those in M1 and M3. The reasons for these differences remain unknown. Possible candidates include the different distance from different mooring sites to Katrina’s track, as the mooring (i.e., M2) recording the strongest near-inertial shear was closest to Katrina’s track. Another possible cause might be related to the spatial variations of planetary vorticity and geostrophic vorticity, which can affect energy propagation (ray path) of near-inertial waves (Kunze 1985). Identifying the causes requires further analyses but is beyond the scope of this study.

b. Inferred diapycnal mixing in the wake of Katrina Both the diapycnal diffusivity K in (4) and turbulent kinetic dissipation rate « in (1) are significantly elevated in the wake of Katrina within 150–470 m (Table 2). The inferred mean diffusivity in the hurricane-free period is about 1.8 3 1025 m2 s21, while this value increases to 11.7 3 1025 m2 s21 in the wake of Katrina. As mentioned earlier, Rv is likely to be larger in the wake of Katrina as a result of the energetic near-inertial internal waves, which may cause an overestimation of the diffusivity value using a constant value of Rv 5 7. However, the inferred diffusivity value difference for Rv varying from 4 to 20 is within a factor of 3. This range of uncertainties in the diffusivity estimate is small when compared to the diffusivity difference between the Katrina wake period and the hurricane-free period, which amounts to a factor of 6–7. Therefore, it appears unlikely that the elevated diapycnal mixing observed in the wake of Katrina is caused by uncertainties in estimating Rv. Diapycnal mixing strength is affected by both back2 ground stratification N and shear variance S2. In the wake of Katrina, the latter plays a much more important

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FIG. 6. Scatterplots of (a) shear variance, (b) squared buoyancy frequency, (c) superinertial shear variance, and (d) near-inertial shear variance vs diffusivity. Variable R represents the correlation coefficient. Values shown are the mean values within 150–470 m derived from M1 (blue), M2 (green), and M3 (red) in the wake of Katrina.

role. The correlation between K and S2 has a value of about 0.75, which is significant at the 5% significance level (Fig. 6a). But the value decreases to 0.12 for the 2 correlation between K and N (Fig. 6b) and is not significant at the 5% significance level. The dependence of 2 « on S2 and N is similar. We estimate that more than 85% of the shear variance comes from the internal waves in the wake of Katrina. The correlation between the total shear variance and that produced by internal waves is at 0.93 (significant at the 5% significance level). The results indicate that the elevated diapycnal mixing in the wake of Katrina is primarily due to the energetic internal waves. The negligible role of stratification in modulating the diapycnal mixing is partly due to its much weaker variability compared with the shear variance. The variability of stratification or shear variance can be conveniently measured by the corresponding standard deviation normalized by the mean value. In the wake of Katrina, 2 the variability of S2 is 6 times the value for N . Another plausible reason is the nonlinear dependence of the diapycnal mixing on shear variance. As shown in Fig. 6a, diffusivity increases roughly as S8. A small increase in shear variance can lead to a pronounced elevation of diapycnal mixing. Both the near-inertial shear variance S2i and superinertial shear variance S2s are elevated in the wake of Katrina (Table 2), raising the question, which of these quantities are more important to the enhanced diapycnal mixing in the wake of Katrina? The correlation between the near-inertial shear variance and diffusivity is at 0.79 (significant at the 5% significance level), while it is only at 0.29 (significant at 5% significance level) for superinertial shear variance (Figs. 6c,d). In particular, the elevated diffusivity is always associated with the appearance of strong near-inertial shear variance, but it is not so with the superinertial shear variance. We note that the tight correlation between the near-inertial shear variance and diffusivity is not an artifact of using a

constant Rv (see appendix B). Therefore, we conclude that it is the near-inertial internal waves that play a dominant role in enhancing the diapycnal mixing in the wake of Katrina. There are two reasons accounting for the dominant contribution of near-inertial shear to the elevated mixing in the wake of Katrina. First, the variation of shear variance in the wake of Katrina is dominated by nearinertial internal waves. The variance of S2i is 1.35 3 10210 s24, about 34% larger than the variance (1.01 3 10210 s24) for S2s . Second, the diffusivity increases more rapidly than linearly with shear variance, so that vigorous shear events can have a more dramatic impact on diapycnal mixing. Compared to S2s , the probability density function (PDF) of S2i is highly skewed and characterized by a longer tail (Fig. 7a), suggesting that there are more vigorous shear events for near-inertial shear than superinertial shear. For instance, the probability for S2i to be 1.0 3 1025 s22 greater than its median value is 3.4%, while it is only about 0.35% for S2s . Correspondingly, about 44.3% of the extreme diapycnal mixing events, on average, are accompanied with extreme near-inertial shear events, while this number is much lower (only about 12.2%) for extreme superinertial shear events (Fig. 7b). Here, extreme events are defined as those with values larger than their 90th percentile. Using a different definition of extreme events does not alter the result (Fig. 7b). Therefore, we conclude that the elevated diapycnal mixing in the wake of Katrina is mainly due to the vigorous shear variance associated with the elevated near-inertial wave activity by Katrina.

c. Evidence for energy transfer from near-inertial internal waves to superinertial small-scale internal waves Although it is expected that there are energetic nearinertial internal waves in the wake of Katrina (Jaimes and Shay 2010), since the wind stress vector to the right

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FIG. 7. (a) PDF of shear variance minus its median. (b) Probability of extreme diapycnal mixing events accompanied by extreme shear events for various definitions of extreme events. Values shown are in the wake of Katrina. In all the plots, red (blue) represents near-inertial (superinertial) motions.

of the hurricane track rotates anticyclonically, inputting a large amount of near-inertial energy into the ocean (Price 1981, 1983), it is less clear how superinertial internal waves are energized. There are at least two plausible causes. The first may be the wind work on superinertial motions, which was probably also increased during the passage of Katrina. It is thus possible that the elevated superinertial internal waves were directly generated by Katrina. The second plausible cause is a nonlinear energy cascade in the frequency domain from near-inertial waves to superinertial waves. In the following discussion, we attempt to examine the relevance of these two candidates to the extent that the available data permit. The wavenumber spectrum of superinertial shear exhibits significant elevation from the lowest resolved wavenumber to at least 0.1 rad m21 in the wake of Katrina (Fig. 4). The elevation peaks around 0.06 rad m21, corresponding to a wavelength of about 100 m. If these energetic superinertial internal waves were generated directly by superinertial wind forcing associated with Katrina, then their vertical wavelengths would be related to the horizontal scale of Katrina. Following Gill (1984), we can estimate the horizontal scale required to generate such superinertial internal waves based on vertical mode analyses. The frequency of hydrostatic internal waves satisfies the dispersion relation: v2n 5 f 2 1 c2n k2H ,

(5)

where vn is the wave frequency for the nth vertical mode, kH is the horizontal wavenumber determined by wind stress, and cn is the eigenvalue, representing gravity wave speed, determined by the Sturm–Liouville problem:

d 1 dFn 1 1 2 Fn 5 0, and dz N 2 dz cn dFn 50 dz

z 5 0, 2H ,

(6) (7)

where H represents the entire ocean depth and the rigid lid approximation is used as the barotropic mode is not considered. The buoyancy frequency in (6) is computed based on the CTD casts taken during the service and maintenance of the moorings. Although the temperature/salinity profiles in the wake of Katrina may differ from those sampled by these CTD casts, no substantial changes in the eigenvalues are expected. So uncertainties in the estimated stratification should not affect the results significantly. Internal waves with vertical wavelength around 100 m are dominated by vertical modes with mode numbers higher than 15 (not shown) and the corresponding eigenvalues cn are smaller than 0.2 m s21. As the superinertial internal waves are defined as those with frequencies higher than 1.8f, their horizontal wavelengths should be no larger than 12 km according to (5), which is much smaller than the horizontal scale of Katrina, suggesting that Katrina is not effective in generating superinertial internal waves with vertical wavelengths of 100 m or shorter. Next we examine the energy transfer from nearinertial to superinertial internal waves, which can be estimated as follows. As near-inertial internal waves are characterized by a vertical-to-horizontal aspect ratio much smaller than that of superinertial internal waves, the energy transfer should be dominated by the vertical shear production rate:

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FIG. 8. (a) Mean shear production rate as a function of depth in the wake of Katrina (red) and during the hurricane-free period (blue). (b) As in (a), but for the near-inertial shear variance.

P 5 2us ws

›Ui ›V 2 y s ws i , ›z ›z

(8)

where us (components us, ys, and ws) and Ui (components Ui and Vi) represent the velocity vectors associated with super- and near-inertial waves, respectively. The superinertial vertical velocity is computed as ws 5 2(›bs/›t)/N2, where bs is the superinertial buoyancy. Pronounced energy transfer from near-inertial waves to superinertial waves occurs in the wake of Katrina (Fig. 8a). At each depth, the shear production rate is an order of magnitude larger than that during the hurricanefree period. The elevation of the shear production rate is particularly evident in the upper 200 m, where energetic near-inertial shear developed in the wake of Katrina (Fig. 8b), indicating the important role of the near-inertial shear in the energy cascade. We further examine the relation of the shear production rate and superinertial shear variance in the wake of Katrina at 112 m, where the energy transfer is most energetic (Fig. 8a). Despite the significant differences of the shear production rate and the superinertial shear variance among the three moorings, energetic superinertial shear variance at all the mooring sites was observed to occur when the shear production rate showed pronounced elevation (Fig. 9), lending further support to the argument. In summary, we argue that Hurricane Katrina was inefficient in directly generating superinertial small-scale internal waves and that the energy transfer from near-inertial

internal waves seemed to play an important role in the elevation of superinertial shear in the wake of Katrina.

d. Subseasonal variation of diapycnal mixing The abovementioned analysis makes it clear that extreme meteorological events, such as hurricanes, can substantially enhance subthermocline diapycnal mixing in the northern Gulf of Mexico. How does this eventdriven change in the diapycnal mixing then compare to subseasonal-to-seasonal variation of the diapycnal mixing in the region? Are similar physical mechanisms responsible for both these changes? Previous studies have revealed distinct low-frequency (subseasonal and longer) variation of diapycnal mixing below the thermocline and attributed it to changes in the wind work on near-inertial oscillations (Jing and Wu 2010; Wu et al. 2011; Qiu et al. 2012). However, velocity measurements were not available to examine the wind-generated nearinertial internal waves in these previous studies. Therefore, the reasons for the low-frequency variation of diapycnal mixing remain uncertain. In this section, we will examine several plausible physical factors—that is, stratification, nearinertial, and superinertial internal waves—in modulating the subseasonal-to-seasonal variability of diapycnal mixing. Only data in the hurricane-free period between November 2005 and January 2006 were used in the following analysis. Consistent with previous studies, the inferred diapycnal mixing in the Gulf of Mexico also exhibits pronounced

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FIG. 9. (top) Shear production rate and (bottom) superinertial shear variance at 112 m for different moorings. Dashed lines show the low-pass filtered time series with a cutoff period of 3 days. Note that the low-pass-filtered time series is multiplied by a factor of 20 (3) for the shear production rate (superinertial shear variance) for easy viewing.

month-to-month variation in the subthermocline depth range (Fig. 10), although the amplitude is considerably weaker than that during Katrina. The diffusivity in January is about 3 times that in November and December (Fig. 10a), accompanied by moderate variations in internal wave field during November 2005– January 2006 (Figs. 10e–h). Since Rv is unlikely to change dramatically during this period, this subseasonal variation of the inferred diapycnal mixing should be reliable. In the following, we attempt to diagnose the dominant mechanisms responsible for the subseasonal variation of the diapycnal mixing, so that

we can make comparisons to those for Katrina-induced diapycnal mixing enhancement. Stratification can play a role in modulating the subseasonal variation of diapycnal mixing. Indeed, a weak 2 negative correlation between K and N was found between November 2005 and January 2006 (Fig. 11a) with a value of 20.22 (significant at the 5% significance level). In these 3 months, the weakest stratification occurred in January (Fig. 10c), contributing to the elevated diffusivity (Fig. 10a). However, the 2 weakness of the correlation between K and N also implies that stratification is most likely not the major

FIG. 10. Monthly variations of (a) diapycnal diffusivity, (b) turbulent kinetic dissipation rate, (c) squared buoyancy frequency, (d) surface eddy kinetic energy, (e) superinertial shear variance, (f) near-inertial shear variance, (g) squared superinertial velocity, and (h) squared near-inertial velocity. Apart from the surface eddy kinetic energy, all the quantities are derived from the mooring data and averaged within 150–470 m.

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FIG. 11. Scatterplots of (a) squared buoyancy frequency, (b) superinertial shear variance, and (c) near-inertial shear variance vs diffusivity. Variable R represents the correlation coefficient. Values shown are averaged within 150–470 m and derived from all the moorings for November (blue) and December (green) 2005 and January (red) 2006.

factor responsible for the elevated mixing in January 2006. The other candidate is shear. While S2i is highly correlated to K throughout November 2005–January 2006, the correlation between K and S2s is much less significant (Fig. 11b,c). In particular, the subseasonal variation of S2i is in phase with diffusivity, while S2s exhibits little variation (Figs. 10e,f). The difference between the PDF of S2i in January and those in November and December indicates that the elevated near-inertial shear variance is mainly due to increases of strong shear events in January (Fig. 12a). As shown in Fig. 6a, the diffusivity increases roughly as S8, suggesting that the vigorous shear events may be responsible for the pronounced elevation of

diapycnal mixing. Indeed, more than 85% of the strong mixing events with diffusivity larger than 1024 m2 s21 occurred in January of 2006, while only less than 15% occurred in November and December of 2005. Therefore, the elevated diapycnal mixing in January 2006 is mainly due to the vigorous near-inertial shear in that month. The near-inertial shear variance exhibited a dominant downward radiation from November 2005 to January 2006 (Fig. 13), suggesting that the elevated near-inertial internal waves in January 2006 are mainly forced by the winds. To verify this conjecture, we compute the wind work on near-inertial motions in the mixed layer using a slab mixed layer model (Pollard and Millard 1970).

FIG. 12. (a) Probability density distribution of near-inertial shear variance averaged within 150–470 m during November (blue) and December (green) 2005 and January (red) 2006. (b) Time series of near-inertial wind work computed from the slab model.

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4. Discussion and conclusions The contribution of near-inertial internal waves to diapycnal mixing below the thermocline in the northern Gulf of Mexico is examined using the moored observations in combination with a finescale parameterization of ocean mixing. Several important findings emerge from the study.

FIG. 13. Frequency-wavenumber spectrum (m s21) of vertical shear during November 2005–January 2006. Positive (k . 0) and negative (k , 0) vertical wavenumbers correspond to the upward and downward phase propagation, respectively.

The governing equation for velocity components u and y of a slab mixed layer reads dZ T 1 (r 1 if )Z 5 , dt r0 Hmix

(9)

where Z 5 u 1 iy is the mixed layer current, r0 is the seawater density, Hmix is the mixed layer depth, T 5 t x 1 it y is the wind stress, and r is the frequencydependent damping parameter (Alford 2003). The wind stress data are taken from NCEP-2 with a temporal resolution of 6 h (http://www.esrl.noaa.gov/psd/data/ gridded/data.ncep.reanalysis.surfaceflux.html; Kanamitsu et al. 2002). The mixed layer depth is assigned a value of 60 m for November, 70 m for December, and 75 m for January according to Mendoza et al. (2005). Equation (9) is solved using the spectral method following Alford (2003). The near-inertial wind work is computed as W 5 RefZT*g, where * denotes the conjugate (Alford 2003). The wind work on near-inertial motions in the mixed layer exhibited the highest value in January of 2006. The estimated mean wind work within 27.58–29.58N, 888– 868W is about 1.6 mW m22 in January, significantly higher than those in November (0.9 mW m22) and December (1.1 mW m22) of 2005. Furthermore, most of the episodic energetic events also occurred in January (Fig. 12b), consistent with the highly skewed PDF of the near-inertial shear variance in this month. In other words, the underlying mechanism for the elevated diapycnal mixing in January 2006 is similar to that in the wake of Katrina. Both are principally due to the energetic near-inertial shear variance generated by strong near-inertial wind forcing.

1) Energetic near-inertial internal waves were observed below the thermocline in the wake of Katrina. The elevation of near-inertial internal waves was much more pronounced in the large vertical wavelengths O(100–1000) m, suggesting that the near-inertial internal waves generated by Katrina were dominated by low-order vertical modes. 2) In the wake of Katrina, the mooring record shows a decrease in the vertical scale of near-inertial motion as time progresses. The near-inertial waves in the early part of the record were characterized by large vertical scales and thus carried weak shear variance. In contrast, the near-inertial waves in the later part of the record showed smaller vertical scales, producing stronger shear variance and thus contributing more dominantly to the elevated diapycnal mixing. 3) Both diapycnal diffusivity and turbulent kinetic dissipation rate within 150–470 m were significantly elevated in the wake of Katrina. The mean diffusivity in the wake of Katrina increased to 11.7 3 1025 m2 s21, which is a factor of 6–7 greater than the mean value of 1.8 3 1025 m2 s21 in the hurricanefree period. 4) Both the elevated near-inertial and superinertial shear variances contributed to the strong diapycnal mixing in the wake of Katrina, but the former played a more dominant role. The energetic nearinertial shear was mainly due to the pronounced wind work on near-inertial motions during the passage of Katrina. The elevated superinertial shear variance, however, was largely derived from energy transfer from the near-inertial internal waves through nonlinear wave-wave interactions. 5) The diapycnal mixing below the thermocline in the northern Gulf of Mexico also exhibited a significant subseasonal variation with diffusivity in January 2006 about 3 times the value in November and December 2005. This strong month-to-month variation was mainly attributed to the change of near-inertial wind work that generated more energetic near-inertial shear in January, although stratification change also contributed to the subseasonal variation of diapycnal mixing. There is debate about whether the substantial nearinertial energy within the surface mixed layer generated

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by surface winds can radiate downward to a great depth and influence diapycnal mixing below thermocline (Wunsch and Ferrari 2004). Our results provide some evidence that at least in the northern Gulf of Mexico diapycnal mixing below the thermocline can be affected by near-inertial internal waves generated either by an extreme weather event—that is, hurricane—or monthto-month weather variability. Furthermore, our analyses show that the energy transfer rate from near-inertial to superinertial internal waves is comparable to the turbulent dissipation rate (Fig. 8a; Table 2). For instance, the mean shear production rate within 150–470 m was estimated at 12.3 3 1029 m2 s23 in the wake of Katrina, just below the turbulent dissipation rate of 13.8 3 1029 m2 s23 in the same depth range. In the hurricane-free period, both the production rate and turbulent dissipation rate decreased to ;2.0 3 1029 m2 s23. As the wind-generated near-inertial internal waves are characterized by vertical scales much larger than those of superinertial internal waves, the similarity between energy transfer and dissipation rate appears to be consistent with a scenario that near-inertial wave energy generated by the winds cascades toward small scales to furnish the mixing in the upper ocean. However, to draw a definitive conclusion is still premature as both the estimates for energy transfer rate and turbulent dissipation rate contain large uncertainties, making rigorous quantitative comparisons difficult. Clearly, direct measurements of vertical velocity and turbulent dissipation rate are required to further validate our results and quantify the role of near-inertial wind work in subthermocline diapycnal mixing. This will be critically important for improving our understanding of the energy source for diapycnal mixing. At present internal waves are not well resolved in climate models as result of insufficient resolution. Diapycnal mixing induced by internal waves is typically specified as a constant eddy diffusivity of 1.0 3 1025 m2 s21 (Large et al. 1994) or parameterized based on internal tide dissipation (Montenegro et al. 2007; Jayne 2009). The results of this study suggest that these parameterizations may not be sufficient to represent the spatial and temporal structure of the diapycnal mixing in the upper ocean because of influences of near-inertial internal waves. The spatial and temporal inhomogeneity of diapycnal mixing is important to improve model representation and prediction of large-scale ocean circulations and climate (Richards et al. 2009; Saenko and Merrifield 2005; Wunsch and Ferrari 2004). A comprehensive parameterization of diapycnal mixing induced by wind-generated near-inertial waves requires both a reliable estimate of near-inertial wind work in the mixed layer and thorough knowledge of downward radiation of near-inertial energy into the deep ocean. The latter has

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turned out to be challenging as it is affected by interactions with mesoscale eddies (Kunze 1985; Balmforth et al. 1998; Lee and Niiler 1998; Zhai et al. 2005, 2007). Nevertheless, there is some light at the end of the tunnel for overcoming this difficulty, as climate models are improving in their representation of large-scale near-inertial internal waves (Jochum et al. 2013), even though highfrequency small-scale internal waves are far away from being fully resolved in the climate models in the coming decade. The Gulf of Mexico is subject to high risk of oil spills, as exemplified by the Deepwater Horizon oil spill on 20 April 2010. Oil spills can potentially cause serious environmental and economic consequences resulting in resource losses along coastal zones. Accurate prediction of oil spills is of great importance to reduce their impact on the marine environment and provide critical information for cleanup operations (Liu et al. 2013). Diapycnal mixing is an effective mechanism for driving the across-isopycnal flux of pollutants from oil spills. Furthermore, change of physical and chemical properties of oil spills may occur during their mixing with seawater or other chemical materials like surfactants, which in turn can have a substantial effect on trapping oil plumes (McLaughlin 2005; Adalsteinsson et al. 2013). As demonstrated in this study, near-inertial internal waves may contribute significantly to diapycnal mixing. The Gulf of Mexico is a region of energetic near-inertial oscillations. In addition to the effect of hurricanes and winter storms highlighted in this study, near-inertial internal waves can also be generated by land–sea-breeze circulations as a result of the strong projection of diurnal winds onto the near-inertial band (Zhang et al. 2009) and parametric subharmonic instability around the critical latitude 28.98N (Zhang et al. 2010). The ability of numerical models in simulating and predicting the vertical transport of oil spills should be improved with improvement in simulation of near-inertial internal waves. Acknowledgments. We thank the Earth System Research Laboratory for providing NCEP/NCAR 6-h wind stress. This research was made possible in part by a grant from BP/The Gulf of Mexico Research Initiative through the Gulf Integrated Spill Response Consortium. Z. Jing is partially supported by the Chinese Scholarship Council. P. Chang acknowledges support from the National Program on Key Basic Research Project (973 Program) Grant 2014CB745000 and the China National Global Change Major Research Project Grant 2013CB956204. S. DiMarco acknowledges BOEM as the source of observations, Contract 1435-01-04-CT34239, ‘‘Survey of deepwater currents in the Eastern Gulf of Mexico’’ U.S. Dept. of the Interior, Bureau of

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Offshore Energy Management. We thank Dr. Z. Wang (TAMU) for providing the comparisons between the finescale parameterization and microstructure measurements.

APPENDIX A Method of Computing Dissipation Rates The method in computing the dissipation rate in (1) mainly follows Kunze et al. (2006). A linear fit is first removed from the velocity profiles within 150–470 m. Then buoyancy-frequency-normalized velocity spectra uV/N(k) are formed by Fourier transforming the normalized velocity (u/N, y/N), where N is the vertical mean buoyancy frequency. In these calculations, a multitaper technique was used to minimize the spectral leakage. The time-bandwidth product is chosen as 2, leading to the lowest resolved vertical wavenumber kmin to be 2p/160 rad m21. As the shear is the derivative of velocity with respect to depth, the buoyancy-frequencynormalized shear spectra are computed as uS/N (k) 5 k2 uV /N (k)Scorr (k) ,

(A1)

where Scorr is the spectral correction accounting for the smoothing of ADCP-measured velocity resulting from the range averaging. Here Scorr 5 sinc24 [(kDz)/2p] with sinc(x) 5 sin(px)/(px) and Dz 5 8 m as the bin length (Polzin et al. 2002). Following Gargett’s (1990) concerns about underestimating the shear variance if the spectrum becomes saturated at wavenumbers smaller than 2p/10 rad m21, the buoyancy-frequency-normalized shear variance 2 hS2 i/N is determined by integrating uS/N from kmin to a maximum wavenumber kmax so that hS2 i N

2

5

ð kmax kmin

uS/N (k) dk 5 0:66:

N

2

5

3pE0 bj* 2

ð kmax kmin

k2 (k 1 k*)

2

of thermocline, j* 5 3 is the reference mode number, and k* 5 (pj*N)/(bN0 ) is the reference wavenumber with N0 5 5.2 3 1023 rad s21.

APPENDIX B Influence of Shear-Strain Variance Ratio The shear–strain variance ratio is usually estimated as Rv 5

(A2)

It should be noted that the shear spectra tend to be contaminated by measurement noise at k . 2p/ 60 rad m21 (Fig. A1). When kmax determined from (A2) is larger than 2p/60 rad m21, kmax is set to be 2p/ 2 60 rad m21 and hS2 i/N is recomputed by integrating uS/N from kmin to kmax 5 2p/60 rad m21. The GM shear variance is computed over the same wavenumber band, so that 2 SGM

FIG. A1. Vertical wavenumber spectra for buoyancy-frequencynormalized shear. Spectra are binned with respect to the GMnormalized shear variance with the legend representing the ratio of the number of profiles with GM-normalized shear variance within a certain range to the number of total profiles. Numbers in the parentheses are those in the wake of Katrina with those outside the parentheses for the entire time period. Horizontal gray dashed line represents the GM model spectrum with the dotted line representing the saturated model spectrum (Gargett 1990). Vertical dashed lines denote 2p/160 and 2p/60 rad m21. Black dashed– dotted line denotes the noise model spectrum.

dk ,

(A3)

(Gregg and Kunze 1991) where E0 5 6.3 3 1025 is the dimensionless energy level, b 5 1300 m is the scale depth

hS2 i 2

N hj2z i

,

(B1)

2

where hS2 i/N and hj2z i are the buoyancy normalized shear variance and strain variance, respectively, computed from the wavenumber band 2p/160 ; 2p/ 60 rad m21. Shear and strain variance can be further decomposed into hS2 i 5 hS2i i 1 hS2s i

(B2)

hj2z i 5 hj2z,i i 1 hj2z,s i

(B3)

respectively, where the subscripts i and s denote nearinertial and superinertial internal waves, respectively. In the following analysis, three assumptions are made. First, the near-inertial shear dominates the superinertial shear at the wavenumber band 2p/160 ; 2p/60 rad m21

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FIG. B1. Scatterplot between the near-inertial shear variance and diffusivity computed from the parameterized Rv in the wake of Katrina. Variable R represents the correlation coefficient.

(Fig. 4) so that hS2 i ’ hS2i i. Second, strain associated with the near-inertial internal waves is negligible so that hj2z i ’ hj2z,s i. Third, the superinertial internal waves in the wake of Katrina do not change significantly so hj2z,s i can be treated as a constant. Under these three assumptions, Rv increases linearly with hS2i i and can be parameterized as Rv 5 7hS2i i/hS2i iref , where hS2i iref is a constant reference value. Here hS2i iref is chosen as 3.3 3 1026 s22, which is the mean value during November 2005– January 2006. It can be seen from Fig. B1 that the diffusivity computed from the parameterized Rv is still highly correlated to the near-inertial shear variance with a correlation coefficient r of 0.67 in the wake of Katrina (Fig. B1). Note that the chosen value 3.3 3 1026 s22 for hS2i iref is somewhat arbitrary. However, r ranges from 0.64 to 0.69 for hS2i iref from 1.2 3 1026 s22 (the smallest near-inertial shear variance in the wake of Katrina) to 5.7 3 1026 s22 (the mean near-inertial shear variance in the wake of Katrina). Also note that all three assumptions do not hold exactly. But all three assumptions tend to overestimate the effect of Rv on the diffusivity. Therefore, the variation of Rv does not have a substantial impact on our conclusions. REFERENCES Adalsteinsson, D. D., and Coauthors, 2013: Subsurface trapping of oil plumes in stratification: Laboratory investigations. Monitoring and Modeling the Deepwater Horizon Oil Spill: A Record-Breaking Enterprise, Geophys. Monogr., Vol. 195, Amer. Geophys. Union, 257–262. Alford, M. H., 2003: Improved global maps and 54-years history of wind-work on ocean inertial motions. Geophys. Res. Lett., 30, 1424, doi:10.1029/2002GL016614.

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VOLUME 45

Zhang, X., S. F. DiMarco, D. C. Smith IV, M. K. Howard, A. E. Jochens, and R. D. Hetland, 2009: Near-resonant ocean response to sea breeze on a stratified continental shelf. J. Phys. Oceanogr., 39, 2137–2155, doi:10.1175/ 2009JPO4054.1. ——, D. C. Smith IV, S. F. DiMarco, and R. D. Hetland, 2010: A numerical study of sea-breeze-driven ocean Poincare wave propagation and mixing near the critical latitude. J. Phys. Oceanogr., 40, 48–66, doi:10.1175/ 2009JPO4216.1.