# Interaction of Langmuir Turbulence and Inertial Currents in

## Transcript Of Interaction of Langmuir Turbulence and Inertial Currents in

VOLUME 48

JOURNAL OF PHYSICAL OCEANOGRAPHY

SEPTEMBER 2018

Interaction of Langmuir Turbulence and Inertial Currents in the Ocean Surface Boundary Layer under Tropical Cyclones

DONG WANG AND TOBIAS KUKULKA

University of Delaware, Newark, Delaware

BRANDON G. REICHL,a TETSU HARA, AND ISAAC GINIS

University of Rhode Island, Narragansett, Rhode Island

PETER P. SULLIVAN

National Center for Atmospheric Research, Boulder, Colorado

(Manuscript received 15 December 2017, in ﬁnal form 27 June 2018)

ABSTRACT

Based on a large-eddy simulation approach, this study investigates the response of the ocean surface boundary layer (OSBL) and Langmuir turbulence (LT) to extreme wind and complex wave forcing under tropical cyclones (TCs). The Stokes drift vector that drives LT is determined from spectral wave simulations. During maximum TC winds, LT substantially enhances the entrainment of cool water, causing rapid OSBL deepening. This coincides with relatively strong wave forcing, weak inertial currents, and shallow OSBL depth HB, measured by smaller ratios of HB/ds, where ds denotes a Stokes drift decay length scale. LT directly affects a near-surface layer whose depth HLT is estimated from enhanced anisotropy ratios of velocity variances. During rapid OSBL deepening, HLT is proportional to HB, and LT efﬁciently transports momentum in coherent structures, locally enhancing shear instabilities in a deeper shear-driven layer, which is controlled by LT. After the TC passes, inertial currents are stronger and HB is greater while HLT is shallower and proportional to ds. During this time, the LT-affected surface layer is too shallow to directly inﬂuence the deeper shear-driven layer, so that both layers are weakly coupled. At the same time, LT reduces surface currents that play a key role in the surface energy input at a later stage. These two factors contribute to relatively small TKE levels and entrainment rates after TC passage. Therefore, our study illustrates that inertial currents need to be taken into account for a complete understanding of LT and its effects on OSBL dynamics in TC conditions.

1. Introduction The development of tropical cyclones (TCs) strongly

depends on the air–sea interactions that include heat ﬂuxes and momentum transfer (Emanuel 1991, 1999). The TC’s strong wind and associated wave forcing drives upper-ocean currents that generate vigorous turbulence.

Denotes content that is immediately available upon publication as open access.

a Current afﬁliation: Program in Atmospheric and Oceanic Sciences, Princeton University, Princeton, New Jersey.

Corresponding author: Dong Wang, [email protected]

Turbulent eddies erode the thermocline by entraining deep cool water into the warmer upper layer, resulting in ocean surface boundary layer (OSBL) deepening and sea surface cooling (Price 1981). In turn, sea surface cooling reduces air–sea heat ﬂuxes that drive the TC, resulting in a negative feedback between TC winds and sea surface temperature (Bender and Ginis 2000; Ginis 2002). The inertial resonance between the turning wind stress and surface currents is also a critical dynamical process under TCs because it increases the shear at the mixed layer base, leading to stronger mixing on the right-hand side of TCs (Price 1981; Skyllingstad et al. 2000; Sanford et al. 2011; Sullivan et al. 2012; Reichl et al. 2016b).

Recent studies indicate that wave-driven Langmuir turbulence (LT) plays an important role in upper-ocean

DOI: 10.1175/JPO-D-17-0258.1

Ó 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

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turbulence under TCs (Sullivan et al. 2012; Rabe et al. 2015; Reichl et al. 2016b,a). LT was originally observed in moderate wind conditions as parallel bands of ﬂoating material on the sea surface, which are due to strong surface current convergences of horizontal roll vortices in the OSBL (Langmuir 1938). Over the last decades, comprehensive ﬁeld observations, mostly conducted in moderate wind conditions, have revealed characteristic features of LT, such as strong surface convergence regions, downwelling jets, and the spacing of roll vortices from several meters to kilometers (Thorpe 2004; Weller and Price 1988; Farmer and Li 1995; Plueddemann et al. 1996; Smith 1992; Gargett et al. 2004; Gargett and Grosch 2014). A systematic mathematical theory of LT is based on the wave-averaged Navier–Stokes equation, the so-called Craik–Leibovich (CL) equation, and suggests that LT is driven by the CL vortex force, which is the cross-product of Stokes drift and vorticity vectors (Craik and Leibovich 1976). Physically, the Stokes drift shear tilts vertical vorticity into the direction of wave propagation, generating LT. Today LT is recognized as a fundamental upper-ocean turbulent process (McWilliams et al. 1997; Thorpe 2004; Li et al. 2005; Sullivan and McWilliams 2010; Belcher et al. 2012; D’Asaro 2014) that contributes signiﬁcantly to turbulent transport and windand wave-driven mixed layer deepening (Kukulka et al. 2009, 2010; Grant and Belcher 2011).

Previous investigations of LT involve turbulenceresolving large-eddy simulation (LES) that is based on the ﬁltered CL equations with explicit wave effects, which resolves LT and associated relatively large vortical structures (Skyllingstad and Denbo 1995; McWilliams et al. 1997). LES studies show that LT enhances the vertical ﬂuxes of momentum and heat, inducing stronger vertical velocity variance and mixed layer deepening (McWilliams et al. 1997; Li et al. 2005; Polton and Belcher 2007; Grant and Belcher 2009; Kukulka et al. 2009; Noh et al. 2009). Direct comparisons of LES results with ocean observations reveal that LES captures in detail many of the observed LT characteristics (Skyllingstad et al. 1999; Gargett et al. 2004; Li et al. 2009; Kukulka et al. 2009, 2013; D’Asaro et al. 2014). However, most LES studies are conducted in moderate wind conditions with monochromatic waves, and only a few of them examine LT in extreme TC conditions.

Sullivan et al. (2012) explored LT dynamics under a TC by forcing an LES model with realistic TC winds and waves, which were simulated by a spectral wave model. They contrasted time series of OSBL turbulence statistics at two stations: one on the right-hand side (rhs) with strong inertial resonance and the other one on the lefthand side (lhs) with weak inertial resonance. The intensity of LT strongly depends on location and time because of

the TC’s complex wind and wave forcing, yielding more energetic LT on the rhs of the TC. Furthermore, their results indicate that the direction of roll vortices due to LT is aligned with the wind direction and tracks the Lagrangian shear direction. As a result of the TC’s transient forcing conditions, wind vector and wave propagation directions are misaligned, reducing LT intensity.

Motivated by OSBL observations of depth-averaged vertical velocity variance (VVV) obtained from Lagrangian ﬂoats under Hurricane Gustav (2008), Rabe et al. (2015) investigated OSBL turbulence with LES experiments forced by Gustav (2008)’s winds and waves. Simulated VVV is only consistent with the observed VVV with LT, that is, for simulations with CL vortex force, indicating LT’s signiﬁcant role in OSBL dynamics. LES results demonstrate that LT enhances VVV and varies with complex sea states found under TCs. Misaligned wind and wave ﬁelds near the TC eye are associated with an observed suppression of VVV, which is also predicted by the LES. Thus, wind-wave misalignment can reduce VVV and suppress LT to the levels close to shear turbulence (ST).

Building on this previous work (Sullivan et al. 2012; Rabe et al. 2015), we recently designed a series of LES experiments for the full spatial TC extent to develop a turbulence closure scheme with explicit sea-statedependent LT effects (Reichl et al. 2016b) and to investigate the role of sea-state-dependent LT in the OSBL response to TCs (Reichl et al. 2016a). In regional ocean models under TCs, which are commonly based on the Reynolds-averaged Navier–Stokes (RANS) equations, LT cannot be resolved, so that smaller-scale turbulent transport processes in the OSBL have to be parameterized. We modiﬁed the K-proﬁle parameterization (KPP) model (Large et al. 1994) to match mean current and temperature proﬁles obtained from the LES model. In the KPP model, we replaced the Eulerian current with the Lagrangian current (Eulerian current plus Stokes drift) to compute the turbulent momentum ﬂux following McWilliams et al. (2012) and also introduced turbulence enhancement factors following McWilliams and Sullivan (2000). Our new KPP model with explicit sea-state-dependent LT signiﬁcantly improves estimations of LES temperature and currents compared to results of the standard (unmodiﬁed) KPP model (Reichl et al. 2016b). We next introduced this new KPP model in a regional three-dimensional coupled wave–ocean RANS model to demonstrate that sea-state-dependent LT substantially modiﬁes the threedimensional OSBL response (Reichl et al. 2016a). Results indicate that LT reduces upwelling and horizontal advection as a result of enhanced near-surface mixing and that simulations without sea-state-dependent LT cannot accurately reproduce the sea surface cooling and

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horizontal transport. More recently, Blair et al. (2017) investigated the upper-ocean response under Hurricane Edouard (2014) with our new KPP model and satellite observations, suggesting the importance of sea-statedependent LT on the mixed layer depth evolution.

In this study, we use the same LES approach as in Reichl et al. (2016b) to comprehensively investigate LT dynamics, OSBL energetics, and the inﬂuence of inertial currents on the OSBL evolution for a wide range of realistic TC conditions. We ﬁrst review our basic numerical approach and provide an overview of complex wind and wave conditions under TCs (section 2) and then illustrate that the LT-driven OSBL is not only seastate dependent but also inﬂuenced by inertial currents in TC conditions (section 3).

2. OSBL turbulence model

To analyze LT in TC conditions, we use the same wind, wave, and turbulence modeling approaches and datasets as in our previous study (Reichl et al. 2016b), which are brieﬂy reviewed in the following subsections.

a. Wind model

The TC wind ﬁeld is constructed based on a Holland wind model (Holland 1980, 2008) with the radius of maximum wind (RMW) of 50 km, maximum wind speed at 10-m height of 65 m s21, and a translation speed of 5 m s21, which represent typical TC parameters (Reichl et al. 2016b). The output of the wind model is the wind velocity at 10-m height with speed U10 over a domain with a length from 2648 to 648 km in the TC’s propagation direction (along X) and a width from 2500 to 500 km across the TC’s propagation direction (along Y) (Fig. 1, top left). In addition to the horizontally averaged turbulent statistics investigated by Reichl et al. (2016b), in this study we also output and investigate the fourdimensional spatiotemporal high-resolution LES temperature, pressure, and velocity ﬁelds.

The wind stress vector at the sea surface has the same direction as the wind velocity, and its magnitude is parameterized by t 5 raCdU120, where ra is the air density and Cd is the drag coefﬁcient that depends on U10 as follows (Sullivan et al. 2012):

8 >< 0:0012 ,

U10 , 11 m s21 ,

C 5 (0:49 1 0:065U ) 3 1023 , 11 m s21 , U , 25 m s21 ,

(1)

d >: 1:8 3 1023 , 10

10

25 m s21 , U .

10

Recent studies show the complexity of drag coefﬁcient under TCs, which varies with different storm quadrants and even wind directions (Holthuijsen et al. 2012; Hsu et al. 2017). Since there is substantial uncertainty of the drag coefﬁcient, we adopt Eq. (1) with simpler assumptions than what has been done in previous studies (Sullivan et al. 2012; Rabe et al. 2015; Reichl et al. 2016b).

b. LES model and numerical experiments setups

Following previous approaches (McWilliams et al. 1997; Skyllingstad and Denbo 1995; Kukulka et al. 2010), we use an LES model to simulate the upper-ocean response to TC’s wind and wave ﬁelds. The LES model solves the grid-ﬁltered CL equations (Craik and Leibovich 1976). LT is generated by the Craik–Leibovich vortex force, which is the cross-product of the Stokes drift us and vorticity. When the wave effect is not considered (us 5 0), the LES model simply solves gridﬁltered Navier–Stokes equations that only generate ST. The Stokes drift is obtained through the wave model, introduced in the following section (section 2c). We only solve the density equation in the LES model and assume that temperature is linearly related to density by

g 5 2 1 ›r , (2) r0 ›T

where r is the water density, r0 5 1024 kg m23 is the reference density, T is water temperature, and g 5 2 3 1024 K21 is the thermal expansion coefﬁcient.

LES experiments are performed for 18 stations across the TC translation direction along Y from 2200 to 200 km, with a minimum and maximum spacing between stations of 20 and 50 km, respectively (Table 1). The coordinate Y is zero at the TC center, and Y . 0 and Y , 0 are on the rhs and lhs of the TC eye, respectively. The LES domain size spans horizontally 750 m 3 750 m and is 240 m deep with 256 grid points in each direction for experiments in Table 1. Inside the LES domain we use xi with index i 5 1, 2, 3 to denote the spatial coordinate (x, y, z) in east, north, and vertical directions, respectively. For all experiments the initial potential temperature proﬁle features a 10-m homogeneous mixed layer with T 5 302:4 K and a constant stratiﬁcation with dT/dz 5 0:04 K m21 below the mixed layer, where z denotes the depth. Small constant surface cooling of 25 W m22 is imposed (McWilliams et al. 1997) for all simulations, which facilitates the initial

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FIG. 1. TC’s (top left) wind direction and magnitude U10, (top center) signiﬁcant wave height Hs, (top right) mean wavelength lm, (bottom left) the wind and peak wave spectrum misalignments, (bottom center) the projected surface layer Langmuir number LaSLu, and (bottom right) the depth scale of Stokes drift shear ds. The black dashed line denotes the RMW, and the black solid line indicates the path of the TC moving from right to left. The dashed magenta lines denote two LES transects which will be examined in the main text. The two black circles on one transect, marked by capital letters A and B, are two locations that will be discussed in detail. Note that the x axis represents the translating distance of the TC, and the y axis indicates the distance of each station from the TC’s path. Both distances are normalized by the RMW. The TC travels from right to left, and (X 2 X0) 5 0 is where the TC’s eye passes.

spinup of turbulence but is otherwise insigniﬁcant for the OSBL dynamics presented here. The LES model is forced at the surface with the modeled wind stress vector (section 2a), and the Stokes drift vector us is imposed based on simulated 2D wave height spectra (section 2c).

c. Wave simulation

1) WAVE MODEL

The third-generation wave model WAVEWATCH III (Tolman 2009) is used to simulate the directional

frequency spectra of surface gravity waves in TC conditions following Reichl et al. (2016b). Its computational domain is 3000 km long in the TC’s translation direction and 1800 km wide across the TC’s translation direction. The wave model has a horizontal grid spacing of 8.33 km, and the wave spectrum is discretized into 48 evenly spaced directions and 40 logarithmically spaced frequencies. The simulated wave ﬁeld is stationary in a coordinate system translating with the TC. The skill of WAVEWATCH III physics parameterizations to simulate complex features of the wave ﬁeld on the left of the translation direction remains a topic of research (see

TABLE 1. Locations of 18 stations in LES experiments.

Station No.

1

2

3

4

5

6

7

8

9

Y 2 Y0 (km)

200

150

130

110

90

70

50

30

10

Station No. Y 2 Y0 (km)

10

11

12

13

14

15

16

17

18

0

220

240

260

280

2100

2120

2140

2200

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Hsu et al. 2018). However, comparison of the version of WAVEWATCH III employed for this study with wave observations shows good skill on average to predict the sea state under extreme hurricane conditions (Fan et al. 2009).

The most energetic wave ﬁelds with large Hs and long lm are found on the rhs of the TC near the RMW at (Y 2 Y0)/RMW 5 1 (Fig. 1, top center and top right). This is because waves travel with the TC and are thus exposed to greater wind forcing, creating favorable conditions for wave development. Note that the TC translates from right [(X 2 X0) . 0] to left [(X 2 X0) , 0], and the X axis is equivalent to time t 5 (X 2 X0)/(5 m s–1). In contrast, waves on the lhs are much weaker with smaller lm and Hs because the wind direction opposes the TC direction. Longer waves far ahead of the TC near (X 2 X0)/RMW 5 –2 indicate the presence of fasttraveling swell waves, but their Hs value is small so these waves do not signiﬁcantly contribute to wave forcing (Fig. 1, top right). As waves are more developed on the rhs of the TC, the misalignment between the wind direction uw and the peak wave spectrum uP is smaller on the rhs (Fig. 1, bottom left). In other regions under the TC, however, the propagation direction of energetic waves differs signiﬁcantly from the wind direction, in particular on the lhs.

2) WAVE PARAMETERS CONTROLLING LT

The time- and space-dependent Stokes drift in the LES model is obtained by integrating the simulated wave spectra (Kenyon 1969):

ð‘ ðp us(z) 5 2 kvF(v, u)e2kz du dv , (3)

0 2p

where F(v, u) is the two-dimensional wave height spectrum, k is the horizontal wavenumber vector, k 5 jkj denotes its magnitude, v is the radian frequency, and u is the wave propagation direction. It is assumed that surface graviptyﬃﬃwﬃﬃﬃaves satisfy the deep-water dispersion relation v 5 gk, where g is the acceleration of gravity. Note that the largest parameterized wavenumber in the model is 400 m21, which speciﬁes the upper bound of radian frequency in Eq. (3) (Reichl et al. 2016b).

Strong wind-wave misalignment in TC conditions reduces LT’s intensity (Sullivan et al. 2012; Rabe et al. 2015; Reichl et al. 2016b). To scale LT for conditions with windwave misalignment, Van Roekel et al. (2012) modiﬁed the surface layer Langmuir number from Harcourt and D’Asaro (2008) that scales LT forced by equilibrium wind-wave spectra. The so-called projected surface layer Langmuir number projects the wind and wave forcing onto the Langmuir cell direction and is given by

uuvﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃuﬃﬃﬃﬃﬃﬃcﬃﬃoﬃﬃﬃsﬃﬃ(ﬃuﬃﬃﬃﬃﬃﬃ2ﬃﬃﬃﬃﬃaﬃﬃﬃ)ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

LaSLu

5tuu cos(u

* 2

a)

w

ð0

,

(4)

s

0:2H

B

20:2HB us dz

where uw is the wind direction, us is the direction of depth-averaged Stokes drift within 0:2HB, and a is the direction of the depth-averaged Lagrangian shear, which is an estimate for the direction of Langmuir cells and deﬁned by

ð0 ›(hyi 1 y )

s dz

20:2H

›z

tan(a) 5 ð0

B

›(hui 1 u )

.

(5)

›z s dz

20:2HB

The Eulerian current and the Stokes drift in the (x, y) direction are denoted as (u, y) and (us, ys), respectively, and hi denotes a horizontal spatial average inside the LES numerical domain. The spatial distribution of LaSLu with relatively small LaSLu values (Fig. 1, bottom center) indicates that LT plays an important role in OSBL

dynamics and that LT is stronger on the rhs (Reichl et al.

2016b). Interestingly, small values of LaSLu do not always coincide with large values of Hs and lm, suggesting that LT is not necessarily more pronounced for more

energetic or more developed wave ﬁelds.

Recent studies show that the Stokes drift decay length

scale inﬂuences the dynamics and structure of LT

(Sullivan et al. 2012; Gargett and Grosch 2014; Kukulka

and Harcourt 2017). To account for the effects of Stokes

drift shear for wave spectra, we compute the length scale

ds for the depth-averaged Stokes drift shear as

ð0

us(z) dz

d 5 2‘

.

(6)

s jus(0) 2 us(2‘)j

In TC conditions, we ﬁnd that ds is closely related to the penetration depth scale of the Stokes drift introduced by Sullivan et al. (2012), for which the misalignment between surface Stokes drift and Stokes drift at greater depth is considered. The differences between ds and lm in TC conditions further illustrate that lm is a poor estimator of Stokes drift decay length scale that drives LT (Fig. 1, top right and bottom right). In summary, traditional parameters that describe the sea state are not sufﬁcient to characterize LT dynamics in TC conditions.

3. Results

We ﬁrst investigate the response of the OSBL to tropical cyclones and identify regions of relatively large

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FIG. 2. (top left) The mixed layer depth HM for the LT case, (top right) the difference of HM between the LT case and the ST case, (bottom left) turbulent boundary layer depth HB, and (bottom right) the difference between HM and HB for the LT case. The regions enclosed by the solid blue contours with (X 2 X0)/RMW . 2 are where HE is shallower than HB in the bottom left panel. The lhs regions enclosed by the solid magenta contours with (X 2 X0)/RMW . 0 are where the mixed layer depth is more than 10 m deeper than the boundary layer depth in the bottom right. Other line styles have the same meaning as in Fig. 1.

entrainment of deep cold water into the OSBL for the LT and ST cases (section 3a). Then we investigate different mechanisms that induce greater entrainment in the LT case (section 3b) and the ST case (section 3c).

a. The response of OSBL to tropical cyclones

1) OSBL DEPTH

The cold wake under the TC is caused by upper-ocean mixing accompanied by mixed layer deepening (Price 1981; Sullivan et al. 2012). The mixed layer depth HM is deﬁned as the depth of largest temperature gradient where the stratiﬁcation is strongest. The spatial and

temporal distribution of HM shows an unsymmetrical pattern with greater HM on the rhs of the TC due to resonant wind forcing that drives strong inertial currents (Price 1981; Skyllingstad et al. 2000; Sanford et al. 2011) (Fig. 2, top left). LT enhances mixed layer deepening, and the contribution of LT is spatially variant (Fig. 2, top right). Enhanced mixed layer deepening due to LT is most pronounced near the RMW, especially on the rhs of the TC where strong, wind-aligned waves create favorable conditions for LT (cf. the forcing shown in the Fig. 1). However, after the storm passes, HM in both LT and ST cases approaches a similar ﬁnal value on the rhs of the TC (Fig. 2, top right). This suggests greater

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FIG. 3. LES transect of normalized turbulent stress proﬁles at (top) (Y 2 Y0)/RMW 5 1 and (bottom) (Y 2 Y0)/RMW 5 0:6 for (left) the LT cases and (right) the ST cases. Locations A and B are denoted by the black dashed lines. The solid magenta line denotes the mixed layer depth HM, and the solid black line indicates the boundary layer depth HB. The turbulent Ekman layer depth is denoted by the solid blue line, and note that we do not show HE that is below HM.

entrainment of cool water into the OSBL for the ST case when the wind subsides, which compensates the smaller mixed layer deepening near the RMW, ﬁnally resulting in a similar HM.

Because of the highly transient forcing conditions under TCs, it is useful to introduce an OSBL depth characterized by active turbulent mixing. Following previous approaches for atmospheric boundary layer turbulence (Zilitinkevich et al. 2007), we specify a turbulent boundary layer depth HB where the turbulent stress decays to 5% of its surface value. Figure 3 shows the normalized turbulent stress proﬁles at two example transects with two speciﬁc locations (transects are denoted by a dashed magenta line and locations are marked with A and B in the bottom left of Fig. 2). Location A is on the rhs of the TC near the RMW where LT rapidly changes HM. Location B is also on the rhs but about 2.5 times the RMW behind the TC’s eye in a region where HM is similar for the LT and ST case.

As expected, HB is close to HM during rapid mixed layer deepening and strong wind and wave forcing (regions outside the solid magenta contour in the bottom right of Fig. 2). However, under the eye, where winds and turbulence levels are relatively weak, HM does not accurately characterize the boundary layer depth because HB is much smaller than HM (regions with positive values in the bottom right of Fig. 2). Across the TC eye,

the winds decrease for 21 , (X 2 X0)/RMW , 0, and the normalized turbulent stress is relatively large owing to relatively strong residual turbulence (Fig. 3, bottom). After the eye passes for 0 , (X 2 X0)/RMW , 1, the normalized turbulent stress is relatively small as a result of relatively weak residual turbulence. These history effects are more pronounced for the ST case, which indicates that the turbulence in the LT case is more responsive to the transient TC forcing.

Behind the TC eye, we observe active turbulence (jttj/u2* . 0:2) at great depth (z , 20:5HB), although winds are weak (Fig. 3, top). The turbulent stress proﬁle in the ST case shows a more homogenized structure in the upper OSBL (z . 20:5HB) (Fig. 3, top right), while the turbulent stress in the LT case decays faster with depth after the TC passes (Fig. 3, top left). This difference in stress proﬁles is related to more energetic inertial currents in the ST case (discussed in section 3c).

To further examine the effect of TC’s transient forcing on the OSBL depth over the extent of the whole TC domain, we compare HB to an estimate of Ekman layer depth: HE 5 0:25u*/f , following McWilliams et al. (1997) (Fig. 2, bottom left). For typical wind-driven Ekman layers, current shear is small below HE, and, thus, turbulence levels are negligible. However, our results reveal regions for which HB exceeds HE, illustrating increased turbulence levels below HE [regions mostly on

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FIG. 4. LES transect of buoyancy ﬂuxes at (top) (Y 2 Y0)/RMW 5 1 and (bottom) (Y 2 Y0)/RMW 5 0. 6 for (left) the LT cases and (right) the ST cases. Note that the buoyancy ﬂuxes in the plot are in the logarithmic form with a base of 10. The dashed magenta line indicates 0:6HM, and other line styles are as in Fig. 3.

the rhs with (X 2 X0)/RMW . 2 enclosed by the solid blue contour in the bottom left of Fig. 2]. For (X 2 X0)/RMW . 1:8, HB exceeds HE, suggesting the generation of turbulence that is due to inertial currents instead of local winds (Fig. 3). Such relatively large HB . HE and the persistent mixed layer deepening behind the TC eye suggest the importance of inertial currents in inﬂuencing OSBL dynamics.

2) ENTRAINMENT

To understand how turbulence drives OSBL deepening, we ﬁrst examine the evolution of proﬁles of the resolved turbulent buoyancy ﬂuxes 2hw0Bi at two transects (Fig. 4). Here B 5 r0g/r0 is the buoyancy, w is the vertical velocity, and the prime denotes the deviation from the horizontal average denoted by symbol h i.

For both the LT and ST cases, the greatest buoyant ﬂuxes occur below 0:6HM, which is in the region of the shear layer. The shear layer is deﬁned as the layer with pronounced stratiﬁcation and mean current shear and approximately locates below 0:6HM (temperature and current proﬁles will be shown in sections 3b and 3c) (Skyllingstad et al. 2000; Grant and Belcher 2011).

For the transect that is tangential to the RMW, the buoyancy ﬂuxes in the LT case are larger than in the ST case near the maximum wind period (location A), suggesting that LT enhances buoyancy ﬂuxes that contribute to enhanced mixed layer deepening (Fig. 4, top left).

However, buoyancy ﬂuxes in the ST case increase behind the TC eye and are about one order larger than in the LT case at (X 2 X0)/RMW 5 2.5 (location B) (Fig. 4, top right). This is consistent with the relatively strong mixed layer deepening in the ST case behind the TC eye, which is due to the turbulence induced by strong inertial currents.

For the transect crossing the TC’s eye region, we observe two periods with strong buoyancy ﬂuxes for both the LT and ST case, which corresponds to the timing of two maximum winds as the TC eye passes (Fig. 4, bottom). The peaks of the buoyancy ﬂuxes in the LT case are greater than in the ST case, indicating LT’s signiﬁcant role in enhancing the turbulent mixing even under transient winds. Greater buoyancy ﬂuxes in the ST case are not found until (X 2 X0)/RMW . 2, where HB is deeper than HE, suggesting the presence of shear-driven turbulence due to energetic inertial currents (Fig. 2, bottom left).

To identify the inﬂuence of LT and inertial currents on mixed layer deepening over the full spatial extent of the TC, we examine the depth-integrated buoyancy ﬂuxes, which drive changes in total (depth integrated) potential energy as

d ð0 zg hri dz 5 ð0 2hw0Bi dz 1 SGS , (7)

dt 2H r0

2H

where SGS symbolizes subgridscale density ﬂuxes (second term on the rhs), and H is the depth of LES domain.

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FIG. 5. (left) The depth-integrated turbulent buoyancy ﬂuxes Ð 0 2hw0Bi dz within the computation depth in the

LT case and (right) the difference of Ð 0

2H

2hw0Bi dz between the LT case and the ST case. The line styles have the

2H

same meaning as in Fig. 1.

The greatest depth-integrated buoyancy ﬂux occurs

around the maximum wind radius where the maximum

mixed layer deepening rates are also found (Fig. 5, left). The

stronger turbulent buoyancy transport on the rhs agrees

with the larger mixed layer deepenÐing on the rhs (Fig. 2, top

left). Pronounced differences of

0 2

H

2hw

0

B

i

dz

between

the LT and the ST cases show that the enhanced entrain-

ment by LT mainly occurs near the RMW (Fig. 5, right).

In the ST case, entrainment is greater inside the TC eye as

the result of enhanced leftover turbulence under transient

winds (Fig. 3, bottom right). The greater entrainment on

the rhs behind the TC eye is caused by the stronger tur-

bulence generated by inertial currents (section 3c).

To explore the relation of wind forcing to entrain-

mÐ ent, we scale the depth-integrated buoyancy ﬂuxes

0 2

H

2hw

0

B

i

dz

by

u3*,

which

is

similar

to

the

scaling

proposed by Grant and Belcher (2009) (Fig. 6). In both

the LT and the ST cases, signiﬁcant deviatÐions from this

scaling are observed. Smaller values of

0 2H

2hw

0

B

i

dz

are mainlyÐ found on the lhs of the storm, while greater

values of

0 2

H

2h

w

0

B

i

dz

are

located

mostly

on

its

rhs

(Fig. 6, left and center). In the ST case differences are

caused by energetic inertial currents on the rhs, inducing

shear-driven turbulent mixing. This is consistent with

the stronger buoyancy ﬂuxes and more homogenized

turbulence stress proﬁles on the rhs, behind the TC eye.

In the LT case, differences are due to LT that is more

vigorous on the rhs due to stronger wave forcing and

weaker on the lhs due to wind-wave misalignment (refer

to Fig. 1). To examine the inﬂuence of LT on entrainment

under the TC’s complicated wind and wave forcing,

we consider a similar scaling approach as Grant and

Belcher (2009) but replace Lat with LaSLu, which is more

aÐ dequate in complex wind and wave conditions, so that

0 2

H

2h

w0

B

i

dz

is

expected

to

scale

with

La2SL2uu3*.

This

scaling signiﬁcantly reduces the scatter, illustrating the

importance of sea-state-dependent Langmuir turbulence

in buoyancy entrainment (Fig. 6, right).

b. Enhanced entrainment due to LT

For aligned wind-wave conditions and relatively shallow mixed layers, LT enhances turbulent entrainment through a three-step process: ﬁrst the large coherent structures in LT facilitate the transport of momentum through the boundary layer; then the shear-instability near the mixed layer base is locally enhanced, inducing greater erosion of thermocline; ﬁnally, the eroded colder water is transported upward and mixed by LT near the surface (Kukulka et al. 2010). In this section, we evaluate this process step by step in tropical cyclone conditions by examining the ﬂow ﬁeld at a location where enhanced mixed layer deepening and greater buoyancy ﬂuxes due to LT are found (marked with A in the bottom of Fig. 2).

1) CURRENT AND TEMPERATURE PROFILES

To address the enhanced shear instability by LT, we ﬁrst examine the horizontally averaged Lagrangian current and temperature proﬁles at a location A (Fig. 7, top). The Lagrangian current is the sum of the Eulerian current and Stokes drift. To better illustrate the relation between wind and current directions, we project the horizontal Eulerian currents and Stokes drift into the along-wind and crosswind direction denoted by subscripts // and ?, respectively (Fig. 7, top left).

At location A, both the LT and the ST cases show predominant along-wind Eulerian currents compared to the

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FIG. 6. The scaling of depth-integrated buoyancy ﬂuxes Ð 0 2hw0Bi dz by u3 for both (left) the ST case and (center) the LT case. (right) The scaling of Ð 02H2hw0Bi dz by La2SL2uu3* for the LT case. C2rHosses denote loca*tions inside the RMW, and circles denote locations outside the RMW. Black colors indicate locations on the rhs of the TC, and red colors indicate locations on the lhs of the TC. The black dashed line is the linear regression of the scatter.

crosswind currents. This is because the onset of the TC’s maximum wind mainly accelerates currents and the inertial currents have not fully spun up yet. With LT, the vertical proﬁle of hui// is more uniform in the upper OSBL with z . 20:5HB compared to the ST case (Fig. 7, top left). Large coherent structures in the presence of LT efﬁciently transport momentum downward, which homogenizes the upper OSBL and decreases the shear in the mean Eulerian currents. However, the near-surface Lagrangian current shear is more alike in both cases, illustrating the predominant Stokes drift shear in the LT case. In addition, the currents in the LT and the ST cases are also similar in the shear layer (z , 20:6HB), although the buoyancy entrainment at the mixed layer base substantially differs in both cases.

To assess the inﬂuence of stratiﬁcation on entrainment, we also investigate the temperature proﬁles at location A. The OSBL temperature with LT is about 0.5 K lower than without LT because of the enhanced entrainment (Fig. 7, top center). Within the shear layer, the temperature gradient in the LT case is slightly greater than the ST case, implying stronger stratiﬁcation. To better understand the competition between destabilizing current shear and stabilizing stratiﬁcation, we examine the gradient Richardson number:

2 g dhri Ri 5 dhuir20 dzdhyi2 . (8)

dz 1 dz

In the ST case, Ri gradually increases with depth and approaches a value close to the critical value (Ri 5 0:25) near the mixed layer base (solid red line in the top right of Fig. 7). In the LT case, Ri signiﬁcantly exceeds 0.25 in the upper half of the OSBL because currents

are homogenized, illustrating the importance of LT in generating turbulence (solid black line in the top right of Fig. 7). In the lower half of the OSBL, Ri approaches a local minimum near HB, indicating the importance of locally generated shear instability that effectively erodes the thermocline and enhances entrainment of cool water (Kukulka et al. 2010).

2) VELOCITY VARIANCE AND TURBULENT

ANISOTROPY

To further address the locally enhanced shear instability by LT, we examine the velocity variance proﬁle whose anisotropy and magnitude characterize the turbulence’s type and intensity, respectively.

At location A with LT, the normalized hw02i has a peak value that is larger than both hu02i and hy02i at the depth of 0:2HB, which is induced by strong downwelling ﬂows due to LT (Fig. 8, top left). Relatively large hy02i near the surface (z . 20:4HB) is associated with strong LT crosswind convergence regions above downwelling ﬂows (note that x is approximately aligned with the wind) (Thorpe 2004). Below 0.4HB, horizontal velocity variances are larger than hw02i, suggesting that shear-driven turbulence is dominant at greater depth (z , 20:4HB). Enhanced hu02i near the bottom of the OSBL indicates enhanced shear-driven turbulence at the OSBL base due to a greater momentum transport, which is expected for rapid mixed layer deepening with LT (Kukulka et al. 2010). In the ST case, normalized hw02i in the upper half of the OSBL is much smaller than in the LT case and hu02i is always larger than hy02i throughout the OSBL as expected for shear-driven turbulence (Fig. 8, top right). In addition, the peak in horizontal velocity variances near the OSBL base is absent.

The proﬁles of velocity variances in Fig. 8 demonstrate that LT changes the anisotropy of velocity

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Interaction of Langmuir Turbulence and Inertial Currents in the Ocean Surface Boundary Layer under Tropical Cyclones

DONG WANG AND TOBIAS KUKULKA

University of Delaware, Newark, Delaware

BRANDON G. REICHL,a TETSU HARA, AND ISAAC GINIS

University of Rhode Island, Narragansett, Rhode Island

PETER P. SULLIVAN

National Center for Atmospheric Research, Boulder, Colorado

(Manuscript received 15 December 2017, in ﬁnal form 27 June 2018)

ABSTRACT

Based on a large-eddy simulation approach, this study investigates the response of the ocean surface boundary layer (OSBL) and Langmuir turbulence (LT) to extreme wind and complex wave forcing under tropical cyclones (TCs). The Stokes drift vector that drives LT is determined from spectral wave simulations. During maximum TC winds, LT substantially enhances the entrainment of cool water, causing rapid OSBL deepening. This coincides with relatively strong wave forcing, weak inertial currents, and shallow OSBL depth HB, measured by smaller ratios of HB/ds, where ds denotes a Stokes drift decay length scale. LT directly affects a near-surface layer whose depth HLT is estimated from enhanced anisotropy ratios of velocity variances. During rapid OSBL deepening, HLT is proportional to HB, and LT efﬁciently transports momentum in coherent structures, locally enhancing shear instabilities in a deeper shear-driven layer, which is controlled by LT. After the TC passes, inertial currents are stronger and HB is greater while HLT is shallower and proportional to ds. During this time, the LT-affected surface layer is too shallow to directly inﬂuence the deeper shear-driven layer, so that both layers are weakly coupled. At the same time, LT reduces surface currents that play a key role in the surface energy input at a later stage. These two factors contribute to relatively small TKE levels and entrainment rates after TC passage. Therefore, our study illustrates that inertial currents need to be taken into account for a complete understanding of LT and its effects on OSBL dynamics in TC conditions.

1. Introduction The development of tropical cyclones (TCs) strongly

depends on the air–sea interactions that include heat ﬂuxes and momentum transfer (Emanuel 1991, 1999). The TC’s strong wind and associated wave forcing drives upper-ocean currents that generate vigorous turbulence.

Denotes content that is immediately available upon publication as open access.

a Current afﬁliation: Program in Atmospheric and Oceanic Sciences, Princeton University, Princeton, New Jersey.

Corresponding author: Dong Wang, [email protected]

Turbulent eddies erode the thermocline by entraining deep cool water into the warmer upper layer, resulting in ocean surface boundary layer (OSBL) deepening and sea surface cooling (Price 1981). In turn, sea surface cooling reduces air–sea heat ﬂuxes that drive the TC, resulting in a negative feedback between TC winds and sea surface temperature (Bender and Ginis 2000; Ginis 2002). The inertial resonance between the turning wind stress and surface currents is also a critical dynamical process under TCs because it increases the shear at the mixed layer base, leading to stronger mixing on the right-hand side of TCs (Price 1981; Skyllingstad et al. 2000; Sanford et al. 2011; Sullivan et al. 2012; Reichl et al. 2016b).

Recent studies indicate that wave-driven Langmuir turbulence (LT) plays an important role in upper-ocean

DOI: 10.1175/JPO-D-17-0258.1

Ó 2018 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

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turbulence under TCs (Sullivan et al. 2012; Rabe et al. 2015; Reichl et al. 2016b,a). LT was originally observed in moderate wind conditions as parallel bands of ﬂoating material on the sea surface, which are due to strong surface current convergences of horizontal roll vortices in the OSBL (Langmuir 1938). Over the last decades, comprehensive ﬁeld observations, mostly conducted in moderate wind conditions, have revealed characteristic features of LT, such as strong surface convergence regions, downwelling jets, and the spacing of roll vortices from several meters to kilometers (Thorpe 2004; Weller and Price 1988; Farmer and Li 1995; Plueddemann et al. 1996; Smith 1992; Gargett et al. 2004; Gargett and Grosch 2014). A systematic mathematical theory of LT is based on the wave-averaged Navier–Stokes equation, the so-called Craik–Leibovich (CL) equation, and suggests that LT is driven by the CL vortex force, which is the cross-product of Stokes drift and vorticity vectors (Craik and Leibovich 1976). Physically, the Stokes drift shear tilts vertical vorticity into the direction of wave propagation, generating LT. Today LT is recognized as a fundamental upper-ocean turbulent process (McWilliams et al. 1997; Thorpe 2004; Li et al. 2005; Sullivan and McWilliams 2010; Belcher et al. 2012; D’Asaro 2014) that contributes signiﬁcantly to turbulent transport and windand wave-driven mixed layer deepening (Kukulka et al. 2009, 2010; Grant and Belcher 2011).

Previous investigations of LT involve turbulenceresolving large-eddy simulation (LES) that is based on the ﬁltered CL equations with explicit wave effects, which resolves LT and associated relatively large vortical structures (Skyllingstad and Denbo 1995; McWilliams et al. 1997). LES studies show that LT enhances the vertical ﬂuxes of momentum and heat, inducing stronger vertical velocity variance and mixed layer deepening (McWilliams et al. 1997; Li et al. 2005; Polton and Belcher 2007; Grant and Belcher 2009; Kukulka et al. 2009; Noh et al. 2009). Direct comparisons of LES results with ocean observations reveal that LES captures in detail many of the observed LT characteristics (Skyllingstad et al. 1999; Gargett et al. 2004; Li et al. 2009; Kukulka et al. 2009, 2013; D’Asaro et al. 2014). However, most LES studies are conducted in moderate wind conditions with monochromatic waves, and only a few of them examine LT in extreme TC conditions.

Sullivan et al. (2012) explored LT dynamics under a TC by forcing an LES model with realistic TC winds and waves, which were simulated by a spectral wave model. They contrasted time series of OSBL turbulence statistics at two stations: one on the right-hand side (rhs) with strong inertial resonance and the other one on the lefthand side (lhs) with weak inertial resonance. The intensity of LT strongly depends on location and time because of

the TC’s complex wind and wave forcing, yielding more energetic LT on the rhs of the TC. Furthermore, their results indicate that the direction of roll vortices due to LT is aligned with the wind direction and tracks the Lagrangian shear direction. As a result of the TC’s transient forcing conditions, wind vector and wave propagation directions are misaligned, reducing LT intensity.

Motivated by OSBL observations of depth-averaged vertical velocity variance (VVV) obtained from Lagrangian ﬂoats under Hurricane Gustav (2008), Rabe et al. (2015) investigated OSBL turbulence with LES experiments forced by Gustav (2008)’s winds and waves. Simulated VVV is only consistent with the observed VVV with LT, that is, for simulations with CL vortex force, indicating LT’s signiﬁcant role in OSBL dynamics. LES results demonstrate that LT enhances VVV and varies with complex sea states found under TCs. Misaligned wind and wave ﬁelds near the TC eye are associated with an observed suppression of VVV, which is also predicted by the LES. Thus, wind-wave misalignment can reduce VVV and suppress LT to the levels close to shear turbulence (ST).

Building on this previous work (Sullivan et al. 2012; Rabe et al. 2015), we recently designed a series of LES experiments for the full spatial TC extent to develop a turbulence closure scheme with explicit sea-statedependent LT effects (Reichl et al. 2016b) and to investigate the role of sea-state-dependent LT in the OSBL response to TCs (Reichl et al. 2016a). In regional ocean models under TCs, which are commonly based on the Reynolds-averaged Navier–Stokes (RANS) equations, LT cannot be resolved, so that smaller-scale turbulent transport processes in the OSBL have to be parameterized. We modiﬁed the K-proﬁle parameterization (KPP) model (Large et al. 1994) to match mean current and temperature proﬁles obtained from the LES model. In the KPP model, we replaced the Eulerian current with the Lagrangian current (Eulerian current plus Stokes drift) to compute the turbulent momentum ﬂux following McWilliams et al. (2012) and also introduced turbulence enhancement factors following McWilliams and Sullivan (2000). Our new KPP model with explicit sea-state-dependent LT signiﬁcantly improves estimations of LES temperature and currents compared to results of the standard (unmodiﬁed) KPP model (Reichl et al. 2016b). We next introduced this new KPP model in a regional three-dimensional coupled wave–ocean RANS model to demonstrate that sea-state-dependent LT substantially modiﬁes the threedimensional OSBL response (Reichl et al. 2016a). Results indicate that LT reduces upwelling and horizontal advection as a result of enhanced near-surface mixing and that simulations without sea-state-dependent LT cannot accurately reproduce the sea surface cooling and

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horizontal transport. More recently, Blair et al. (2017) investigated the upper-ocean response under Hurricane Edouard (2014) with our new KPP model and satellite observations, suggesting the importance of sea-statedependent LT on the mixed layer depth evolution.

In this study, we use the same LES approach as in Reichl et al. (2016b) to comprehensively investigate LT dynamics, OSBL energetics, and the inﬂuence of inertial currents on the OSBL evolution for a wide range of realistic TC conditions. We ﬁrst review our basic numerical approach and provide an overview of complex wind and wave conditions under TCs (section 2) and then illustrate that the LT-driven OSBL is not only seastate dependent but also inﬂuenced by inertial currents in TC conditions (section 3).

2. OSBL turbulence model

To analyze LT in TC conditions, we use the same wind, wave, and turbulence modeling approaches and datasets as in our previous study (Reichl et al. 2016b), which are brieﬂy reviewed in the following subsections.

a. Wind model

The TC wind ﬁeld is constructed based on a Holland wind model (Holland 1980, 2008) with the radius of maximum wind (RMW) of 50 km, maximum wind speed at 10-m height of 65 m s21, and a translation speed of 5 m s21, which represent typical TC parameters (Reichl et al. 2016b). The output of the wind model is the wind velocity at 10-m height with speed U10 over a domain with a length from 2648 to 648 km in the TC’s propagation direction (along X) and a width from 2500 to 500 km across the TC’s propagation direction (along Y) (Fig. 1, top left). In addition to the horizontally averaged turbulent statistics investigated by Reichl et al. (2016b), in this study we also output and investigate the fourdimensional spatiotemporal high-resolution LES temperature, pressure, and velocity ﬁelds.

The wind stress vector at the sea surface has the same direction as the wind velocity, and its magnitude is parameterized by t 5 raCdU120, where ra is the air density and Cd is the drag coefﬁcient that depends on U10 as follows (Sullivan et al. 2012):

8 >< 0:0012 ,

U10 , 11 m s21 ,

C 5 (0:49 1 0:065U ) 3 1023 , 11 m s21 , U , 25 m s21 ,

(1)

d >: 1:8 3 1023 , 10

10

25 m s21 , U .

10

Recent studies show the complexity of drag coefﬁcient under TCs, which varies with different storm quadrants and even wind directions (Holthuijsen et al. 2012; Hsu et al. 2017). Since there is substantial uncertainty of the drag coefﬁcient, we adopt Eq. (1) with simpler assumptions than what has been done in previous studies (Sullivan et al. 2012; Rabe et al. 2015; Reichl et al. 2016b).

b. LES model and numerical experiments setups

Following previous approaches (McWilliams et al. 1997; Skyllingstad and Denbo 1995; Kukulka et al. 2010), we use an LES model to simulate the upper-ocean response to TC’s wind and wave ﬁelds. The LES model solves the grid-ﬁltered CL equations (Craik and Leibovich 1976). LT is generated by the Craik–Leibovich vortex force, which is the cross-product of the Stokes drift us and vorticity. When the wave effect is not considered (us 5 0), the LES model simply solves gridﬁltered Navier–Stokes equations that only generate ST. The Stokes drift is obtained through the wave model, introduced in the following section (section 2c). We only solve the density equation in the LES model and assume that temperature is linearly related to density by

g 5 2 1 ›r , (2) r0 ›T

where r is the water density, r0 5 1024 kg m23 is the reference density, T is water temperature, and g 5 2 3 1024 K21 is the thermal expansion coefﬁcient.

LES experiments are performed for 18 stations across the TC translation direction along Y from 2200 to 200 km, with a minimum and maximum spacing between stations of 20 and 50 km, respectively (Table 1). The coordinate Y is zero at the TC center, and Y . 0 and Y , 0 are on the rhs and lhs of the TC eye, respectively. The LES domain size spans horizontally 750 m 3 750 m and is 240 m deep with 256 grid points in each direction for experiments in Table 1. Inside the LES domain we use xi with index i 5 1, 2, 3 to denote the spatial coordinate (x, y, z) in east, north, and vertical directions, respectively. For all experiments the initial potential temperature proﬁle features a 10-m homogeneous mixed layer with T 5 302:4 K and a constant stratiﬁcation with dT/dz 5 0:04 K m21 below the mixed layer, where z denotes the depth. Small constant surface cooling of 25 W m22 is imposed (McWilliams et al. 1997) for all simulations, which facilitates the initial

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FIG. 1. TC’s (top left) wind direction and magnitude U10, (top center) signiﬁcant wave height Hs, (top right) mean wavelength lm, (bottom left) the wind and peak wave spectrum misalignments, (bottom center) the projected surface layer Langmuir number LaSLu, and (bottom right) the depth scale of Stokes drift shear ds. The black dashed line denotes the RMW, and the black solid line indicates the path of the TC moving from right to left. The dashed magenta lines denote two LES transects which will be examined in the main text. The two black circles on one transect, marked by capital letters A and B, are two locations that will be discussed in detail. Note that the x axis represents the translating distance of the TC, and the y axis indicates the distance of each station from the TC’s path. Both distances are normalized by the RMW. The TC travels from right to left, and (X 2 X0) 5 0 is where the TC’s eye passes.

spinup of turbulence but is otherwise insigniﬁcant for the OSBL dynamics presented here. The LES model is forced at the surface with the modeled wind stress vector (section 2a), and the Stokes drift vector us is imposed based on simulated 2D wave height spectra (section 2c).

c. Wave simulation

1) WAVE MODEL

The third-generation wave model WAVEWATCH III (Tolman 2009) is used to simulate the directional

frequency spectra of surface gravity waves in TC conditions following Reichl et al. (2016b). Its computational domain is 3000 km long in the TC’s translation direction and 1800 km wide across the TC’s translation direction. The wave model has a horizontal grid spacing of 8.33 km, and the wave spectrum is discretized into 48 evenly spaced directions and 40 logarithmically spaced frequencies. The simulated wave ﬁeld is stationary in a coordinate system translating with the TC. The skill of WAVEWATCH III physics parameterizations to simulate complex features of the wave ﬁeld on the left of the translation direction remains a topic of research (see

TABLE 1. Locations of 18 stations in LES experiments.

Station No.

1

2

3

4

5

6

7

8

9

Y 2 Y0 (km)

200

150

130

110

90

70

50

30

10

Station No. Y 2 Y0 (km)

10

11

12

13

14

15

16

17

18

0

220

240

260

280

2100

2120

2140

2200

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Hsu et al. 2018). However, comparison of the version of WAVEWATCH III employed for this study with wave observations shows good skill on average to predict the sea state under extreme hurricane conditions (Fan et al. 2009).

The most energetic wave ﬁelds with large Hs and long lm are found on the rhs of the TC near the RMW at (Y 2 Y0)/RMW 5 1 (Fig. 1, top center and top right). This is because waves travel with the TC and are thus exposed to greater wind forcing, creating favorable conditions for wave development. Note that the TC translates from right [(X 2 X0) . 0] to left [(X 2 X0) , 0], and the X axis is equivalent to time t 5 (X 2 X0)/(5 m s–1). In contrast, waves on the lhs are much weaker with smaller lm and Hs because the wind direction opposes the TC direction. Longer waves far ahead of the TC near (X 2 X0)/RMW 5 –2 indicate the presence of fasttraveling swell waves, but their Hs value is small so these waves do not signiﬁcantly contribute to wave forcing (Fig. 1, top right). As waves are more developed on the rhs of the TC, the misalignment between the wind direction uw and the peak wave spectrum uP is smaller on the rhs (Fig. 1, bottom left). In other regions under the TC, however, the propagation direction of energetic waves differs signiﬁcantly from the wind direction, in particular on the lhs.

2) WAVE PARAMETERS CONTROLLING LT

The time- and space-dependent Stokes drift in the LES model is obtained by integrating the simulated wave spectra (Kenyon 1969):

ð‘ ðp us(z) 5 2 kvF(v, u)e2kz du dv , (3)

0 2p

where F(v, u) is the two-dimensional wave height spectrum, k is the horizontal wavenumber vector, k 5 jkj denotes its magnitude, v is the radian frequency, and u is the wave propagation direction. It is assumed that surface graviptyﬃﬃwﬃﬃﬃaves satisfy the deep-water dispersion relation v 5 gk, where g is the acceleration of gravity. Note that the largest parameterized wavenumber in the model is 400 m21, which speciﬁes the upper bound of radian frequency in Eq. (3) (Reichl et al. 2016b).

Strong wind-wave misalignment in TC conditions reduces LT’s intensity (Sullivan et al. 2012; Rabe et al. 2015; Reichl et al. 2016b). To scale LT for conditions with windwave misalignment, Van Roekel et al. (2012) modiﬁed the surface layer Langmuir number from Harcourt and D’Asaro (2008) that scales LT forced by equilibrium wind-wave spectra. The so-called projected surface layer Langmuir number projects the wind and wave forcing onto the Langmuir cell direction and is given by

uuvﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃuﬃﬃﬃﬃﬃﬃcﬃﬃoﬃﬃﬃsﬃﬃ(ﬃuﬃﬃﬃﬃﬃﬃ2ﬃﬃﬃﬃﬃaﬃﬃﬃ)ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

LaSLu

5tuu cos(u

* 2

a)

w

ð0

,

(4)

s

0:2H

B

20:2HB us dz

where uw is the wind direction, us is the direction of depth-averaged Stokes drift within 0:2HB, and a is the direction of the depth-averaged Lagrangian shear, which is an estimate for the direction of Langmuir cells and deﬁned by

ð0 ›(hyi 1 y )

s dz

20:2H

›z

tan(a) 5 ð0

B

›(hui 1 u )

.

(5)

›z s dz

20:2HB

The Eulerian current and the Stokes drift in the (x, y) direction are denoted as (u, y) and (us, ys), respectively, and hi denotes a horizontal spatial average inside the LES numerical domain. The spatial distribution of LaSLu with relatively small LaSLu values (Fig. 1, bottom center) indicates that LT plays an important role in OSBL

dynamics and that LT is stronger on the rhs (Reichl et al.

2016b). Interestingly, small values of LaSLu do not always coincide with large values of Hs and lm, suggesting that LT is not necessarily more pronounced for more

energetic or more developed wave ﬁelds.

Recent studies show that the Stokes drift decay length

scale inﬂuences the dynamics and structure of LT

(Sullivan et al. 2012; Gargett and Grosch 2014; Kukulka

and Harcourt 2017). To account for the effects of Stokes

drift shear for wave spectra, we compute the length scale

ds for the depth-averaged Stokes drift shear as

ð0

us(z) dz

d 5 2‘

.

(6)

s jus(0) 2 us(2‘)j

In TC conditions, we ﬁnd that ds is closely related to the penetration depth scale of the Stokes drift introduced by Sullivan et al. (2012), for which the misalignment between surface Stokes drift and Stokes drift at greater depth is considered. The differences between ds and lm in TC conditions further illustrate that lm is a poor estimator of Stokes drift decay length scale that drives LT (Fig. 1, top right and bottom right). In summary, traditional parameters that describe the sea state are not sufﬁcient to characterize LT dynamics in TC conditions.

3. Results

We ﬁrst investigate the response of the OSBL to tropical cyclones and identify regions of relatively large

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FIG. 2. (top left) The mixed layer depth HM for the LT case, (top right) the difference of HM between the LT case and the ST case, (bottom left) turbulent boundary layer depth HB, and (bottom right) the difference between HM and HB for the LT case. The regions enclosed by the solid blue contours with (X 2 X0)/RMW . 2 are where HE is shallower than HB in the bottom left panel. The lhs regions enclosed by the solid magenta contours with (X 2 X0)/RMW . 0 are where the mixed layer depth is more than 10 m deeper than the boundary layer depth in the bottom right. Other line styles have the same meaning as in Fig. 1.

entrainment of deep cold water into the OSBL for the LT and ST cases (section 3a). Then we investigate different mechanisms that induce greater entrainment in the LT case (section 3b) and the ST case (section 3c).

a. The response of OSBL to tropical cyclones

1) OSBL DEPTH

The cold wake under the TC is caused by upper-ocean mixing accompanied by mixed layer deepening (Price 1981; Sullivan et al. 2012). The mixed layer depth HM is deﬁned as the depth of largest temperature gradient where the stratiﬁcation is strongest. The spatial and

temporal distribution of HM shows an unsymmetrical pattern with greater HM on the rhs of the TC due to resonant wind forcing that drives strong inertial currents (Price 1981; Skyllingstad et al. 2000; Sanford et al. 2011) (Fig. 2, top left). LT enhances mixed layer deepening, and the contribution of LT is spatially variant (Fig. 2, top right). Enhanced mixed layer deepening due to LT is most pronounced near the RMW, especially on the rhs of the TC where strong, wind-aligned waves create favorable conditions for LT (cf. the forcing shown in the Fig. 1). However, after the storm passes, HM in both LT and ST cases approaches a similar ﬁnal value on the rhs of the TC (Fig. 2, top right). This suggests greater

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FIG. 3. LES transect of normalized turbulent stress proﬁles at (top) (Y 2 Y0)/RMW 5 1 and (bottom) (Y 2 Y0)/RMW 5 0:6 for (left) the LT cases and (right) the ST cases. Locations A and B are denoted by the black dashed lines. The solid magenta line denotes the mixed layer depth HM, and the solid black line indicates the boundary layer depth HB. The turbulent Ekman layer depth is denoted by the solid blue line, and note that we do not show HE that is below HM.

entrainment of cool water into the OSBL for the ST case when the wind subsides, which compensates the smaller mixed layer deepening near the RMW, ﬁnally resulting in a similar HM.

Because of the highly transient forcing conditions under TCs, it is useful to introduce an OSBL depth characterized by active turbulent mixing. Following previous approaches for atmospheric boundary layer turbulence (Zilitinkevich et al. 2007), we specify a turbulent boundary layer depth HB where the turbulent stress decays to 5% of its surface value. Figure 3 shows the normalized turbulent stress proﬁles at two example transects with two speciﬁc locations (transects are denoted by a dashed magenta line and locations are marked with A and B in the bottom left of Fig. 2). Location A is on the rhs of the TC near the RMW where LT rapidly changes HM. Location B is also on the rhs but about 2.5 times the RMW behind the TC’s eye in a region where HM is similar for the LT and ST case.

As expected, HB is close to HM during rapid mixed layer deepening and strong wind and wave forcing (regions outside the solid magenta contour in the bottom right of Fig. 2). However, under the eye, where winds and turbulence levels are relatively weak, HM does not accurately characterize the boundary layer depth because HB is much smaller than HM (regions with positive values in the bottom right of Fig. 2). Across the TC eye,

the winds decrease for 21 , (X 2 X0)/RMW , 0, and the normalized turbulent stress is relatively large owing to relatively strong residual turbulence (Fig. 3, bottom). After the eye passes for 0 , (X 2 X0)/RMW , 1, the normalized turbulent stress is relatively small as a result of relatively weak residual turbulence. These history effects are more pronounced for the ST case, which indicates that the turbulence in the LT case is more responsive to the transient TC forcing.

Behind the TC eye, we observe active turbulence (jttj/u2* . 0:2) at great depth (z , 20:5HB), although winds are weak (Fig. 3, top). The turbulent stress proﬁle in the ST case shows a more homogenized structure in the upper OSBL (z . 20:5HB) (Fig. 3, top right), while the turbulent stress in the LT case decays faster with depth after the TC passes (Fig. 3, top left). This difference in stress proﬁles is related to more energetic inertial currents in the ST case (discussed in section 3c).

To further examine the effect of TC’s transient forcing on the OSBL depth over the extent of the whole TC domain, we compare HB to an estimate of Ekman layer depth: HE 5 0:25u*/f , following McWilliams et al. (1997) (Fig. 2, bottom left). For typical wind-driven Ekman layers, current shear is small below HE, and, thus, turbulence levels are negligible. However, our results reveal regions for which HB exceeds HE, illustrating increased turbulence levels below HE [regions mostly on

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FIG. 4. LES transect of buoyancy ﬂuxes at (top) (Y 2 Y0)/RMW 5 1 and (bottom) (Y 2 Y0)/RMW 5 0. 6 for (left) the LT cases and (right) the ST cases. Note that the buoyancy ﬂuxes in the plot are in the logarithmic form with a base of 10. The dashed magenta line indicates 0:6HM, and other line styles are as in Fig. 3.

the rhs with (X 2 X0)/RMW . 2 enclosed by the solid blue contour in the bottom left of Fig. 2]. For (X 2 X0)/RMW . 1:8, HB exceeds HE, suggesting the generation of turbulence that is due to inertial currents instead of local winds (Fig. 3). Such relatively large HB . HE and the persistent mixed layer deepening behind the TC eye suggest the importance of inertial currents in inﬂuencing OSBL dynamics.

2) ENTRAINMENT

To understand how turbulence drives OSBL deepening, we ﬁrst examine the evolution of proﬁles of the resolved turbulent buoyancy ﬂuxes 2hw0Bi at two transects (Fig. 4). Here B 5 r0g/r0 is the buoyancy, w is the vertical velocity, and the prime denotes the deviation from the horizontal average denoted by symbol h i.

For both the LT and ST cases, the greatest buoyant ﬂuxes occur below 0:6HM, which is in the region of the shear layer. The shear layer is deﬁned as the layer with pronounced stratiﬁcation and mean current shear and approximately locates below 0:6HM (temperature and current proﬁles will be shown in sections 3b and 3c) (Skyllingstad et al. 2000; Grant and Belcher 2011).

For the transect that is tangential to the RMW, the buoyancy ﬂuxes in the LT case are larger than in the ST case near the maximum wind period (location A), suggesting that LT enhances buoyancy ﬂuxes that contribute to enhanced mixed layer deepening (Fig. 4, top left).

However, buoyancy ﬂuxes in the ST case increase behind the TC eye and are about one order larger than in the LT case at (X 2 X0)/RMW 5 2.5 (location B) (Fig. 4, top right). This is consistent with the relatively strong mixed layer deepening in the ST case behind the TC eye, which is due to the turbulence induced by strong inertial currents.

For the transect crossing the TC’s eye region, we observe two periods with strong buoyancy ﬂuxes for both the LT and ST case, which corresponds to the timing of two maximum winds as the TC eye passes (Fig. 4, bottom). The peaks of the buoyancy ﬂuxes in the LT case are greater than in the ST case, indicating LT’s signiﬁcant role in enhancing the turbulent mixing even under transient winds. Greater buoyancy ﬂuxes in the ST case are not found until (X 2 X0)/RMW . 2, where HB is deeper than HE, suggesting the presence of shear-driven turbulence due to energetic inertial currents (Fig. 2, bottom left).

To identify the inﬂuence of LT and inertial currents on mixed layer deepening over the full spatial extent of the TC, we examine the depth-integrated buoyancy ﬂuxes, which drive changes in total (depth integrated) potential energy as

d ð0 zg hri dz 5 ð0 2hw0Bi dz 1 SGS , (7)

dt 2H r0

2H

where SGS symbolizes subgridscale density ﬂuxes (second term on the rhs), and H is the depth of LES domain.

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WANG ET AL.

1929

FIG. 5. (left) The depth-integrated turbulent buoyancy ﬂuxes Ð 0 2hw0Bi dz within the computation depth in the

LT case and (right) the difference of Ð 0

2H

2hw0Bi dz between the LT case and the ST case. The line styles have the

2H

same meaning as in Fig. 1.

The greatest depth-integrated buoyancy ﬂux occurs

around the maximum wind radius where the maximum

mixed layer deepening rates are also found (Fig. 5, left). The

stronger turbulent buoyancy transport on the rhs agrees

with the larger mixed layer deepenÐing on the rhs (Fig. 2, top

left). Pronounced differences of

0 2

H

2hw

0

B

i

dz

between

the LT and the ST cases show that the enhanced entrain-

ment by LT mainly occurs near the RMW (Fig. 5, right).

In the ST case, entrainment is greater inside the TC eye as

the result of enhanced leftover turbulence under transient

winds (Fig. 3, bottom right). The greater entrainment on

the rhs behind the TC eye is caused by the stronger tur-

bulence generated by inertial currents (section 3c).

To explore the relation of wind forcing to entrain-

mÐ ent, we scale the depth-integrated buoyancy ﬂuxes

0 2

H

2hw

0

B

i

dz

by

u3*,

which

is

similar

to

the

scaling

proposed by Grant and Belcher (2009) (Fig. 6). In both

the LT and the ST cases, signiﬁcant deviatÐions from this

scaling are observed. Smaller values of

0 2H

2hw

0

B

i

dz

are mainlyÐ found on the lhs of the storm, while greater

values of

0 2

H

2h

w

0

B

i

dz

are

located

mostly

on

its

rhs

(Fig. 6, left and center). In the ST case differences are

caused by energetic inertial currents on the rhs, inducing

shear-driven turbulent mixing. This is consistent with

the stronger buoyancy ﬂuxes and more homogenized

turbulence stress proﬁles on the rhs, behind the TC eye.

In the LT case, differences are due to LT that is more

vigorous on the rhs due to stronger wave forcing and

weaker on the lhs due to wind-wave misalignment (refer

to Fig. 1). To examine the inﬂuence of LT on entrainment

under the TC’s complicated wind and wave forcing,

we consider a similar scaling approach as Grant and

Belcher (2009) but replace Lat with LaSLu, which is more

aÐ dequate in complex wind and wave conditions, so that

0 2

H

2h

w0

B

i

dz

is

expected

to

scale

with

La2SL2uu3*.

This

scaling signiﬁcantly reduces the scatter, illustrating the

importance of sea-state-dependent Langmuir turbulence

in buoyancy entrainment (Fig. 6, right).

b. Enhanced entrainment due to LT

For aligned wind-wave conditions and relatively shallow mixed layers, LT enhances turbulent entrainment through a three-step process: ﬁrst the large coherent structures in LT facilitate the transport of momentum through the boundary layer; then the shear-instability near the mixed layer base is locally enhanced, inducing greater erosion of thermocline; ﬁnally, the eroded colder water is transported upward and mixed by LT near the surface (Kukulka et al. 2010). In this section, we evaluate this process step by step in tropical cyclone conditions by examining the ﬂow ﬁeld at a location where enhanced mixed layer deepening and greater buoyancy ﬂuxes due to LT are found (marked with A in the bottom of Fig. 2).

1) CURRENT AND TEMPERATURE PROFILES

To address the enhanced shear instability by LT, we ﬁrst examine the horizontally averaged Lagrangian current and temperature proﬁles at a location A (Fig. 7, top). The Lagrangian current is the sum of the Eulerian current and Stokes drift. To better illustrate the relation between wind and current directions, we project the horizontal Eulerian currents and Stokes drift into the along-wind and crosswind direction denoted by subscripts // and ?, respectively (Fig. 7, top left).

At location A, both the LT and the ST cases show predominant along-wind Eulerian currents compared to the

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FIG. 6. The scaling of depth-integrated buoyancy ﬂuxes Ð 0 2hw0Bi dz by u3 for both (left) the ST case and (center) the LT case. (right) The scaling of Ð 02H2hw0Bi dz by La2SL2uu3* for the LT case. C2rHosses denote loca*tions inside the RMW, and circles denote locations outside the RMW. Black colors indicate locations on the rhs of the TC, and red colors indicate locations on the lhs of the TC. The black dashed line is the linear regression of the scatter.

crosswind currents. This is because the onset of the TC’s maximum wind mainly accelerates currents and the inertial currents have not fully spun up yet. With LT, the vertical proﬁle of hui// is more uniform in the upper OSBL with z . 20:5HB compared to the ST case (Fig. 7, top left). Large coherent structures in the presence of LT efﬁciently transport momentum downward, which homogenizes the upper OSBL and decreases the shear in the mean Eulerian currents. However, the near-surface Lagrangian current shear is more alike in both cases, illustrating the predominant Stokes drift shear in the LT case. In addition, the currents in the LT and the ST cases are also similar in the shear layer (z , 20:6HB), although the buoyancy entrainment at the mixed layer base substantially differs in both cases.

To assess the inﬂuence of stratiﬁcation on entrainment, we also investigate the temperature proﬁles at location A. The OSBL temperature with LT is about 0.5 K lower than without LT because of the enhanced entrainment (Fig. 7, top center). Within the shear layer, the temperature gradient in the LT case is slightly greater than the ST case, implying stronger stratiﬁcation. To better understand the competition between destabilizing current shear and stabilizing stratiﬁcation, we examine the gradient Richardson number:

2 g dhri Ri 5 dhuir20 dzdhyi2 . (8)

dz 1 dz

In the ST case, Ri gradually increases with depth and approaches a value close to the critical value (Ri 5 0:25) near the mixed layer base (solid red line in the top right of Fig. 7). In the LT case, Ri signiﬁcantly exceeds 0.25 in the upper half of the OSBL because currents

are homogenized, illustrating the importance of LT in generating turbulence (solid black line in the top right of Fig. 7). In the lower half of the OSBL, Ri approaches a local minimum near HB, indicating the importance of locally generated shear instability that effectively erodes the thermocline and enhances entrainment of cool water (Kukulka et al. 2010).

2) VELOCITY VARIANCE AND TURBULENT

ANISOTROPY

To further address the locally enhanced shear instability by LT, we examine the velocity variance proﬁle whose anisotropy and magnitude characterize the turbulence’s type and intensity, respectively.

At location A with LT, the normalized hw02i has a peak value that is larger than both hu02i and hy02i at the depth of 0:2HB, which is induced by strong downwelling ﬂows due to LT (Fig. 8, top left). Relatively large hy02i near the surface (z . 20:4HB) is associated with strong LT crosswind convergence regions above downwelling ﬂows (note that x is approximately aligned with the wind) (Thorpe 2004). Below 0.4HB, horizontal velocity variances are larger than hw02i, suggesting that shear-driven turbulence is dominant at greater depth (z , 20:4HB). Enhanced hu02i near the bottom of the OSBL indicates enhanced shear-driven turbulence at the OSBL base due to a greater momentum transport, which is expected for rapid mixed layer deepening with LT (Kukulka et al. 2010). In the ST case, normalized hw02i in the upper half of the OSBL is much smaller than in the LT case and hu02i is always larger than hy02i throughout the OSBL as expected for shear-driven turbulence (Fig. 8, top right). In addition, the peak in horizontal velocity variances near the OSBL base is absent.

The proﬁles of velocity variances in Fig. 8 demonstrate that LT changes the anisotropy of velocity

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