Revisiting the Generation of Internal Waves by Resonant Interaction with Surface Waves

AUGUST 2016 O L B E R S A N D E D E N 2335

DIRK OLBERS Alfred Wegener Institute for Polar and Marine Research, Bremerhaven, Germany

CARSTEN EDEN Institut fur€ Meereskunde, Universitat€ Hamburg, Hamburg, Germany

(Manuscript received 1 April 2015, in ﬁnal form 29 December 2015)

ABSTRACT

Two surface waves can interact to produce an internal gravity wave by nonlinear resonant coupling. The process has been called spontaneous creation (SC) because it operates without internal waves being initially present. Previous studies have shown that the generated internal waves have high frequency close to the local Brunt–Väisälä frequency and wavelengths that are much larger than those of the participating surface waves, and that the spectral transfer rate of energy to the internal wave field is small compared to other generation processes. The aim of the present analysis is to provide a global map of the energy transfer into the internal 2 wave field by surface–internal wave interaction, which is found to be about 10 3 TW in total, based on a realistic wind-sea spectrum (depending on wind speed), mixed layer depths, and stratification below the mixed layer taken from a state-of-the-art numerical ocean model. Unlike previous calculations of the spectral transfer rate based on a vertical mode decomposition, the authors use an analytical framework that directly derives the energy flux of generated internal waves radiating downward from the mixed layer base. Since the radiated waves are of high frequency, they are trapped and dissipated in the upper ocean. The radiative flux thus feeds only a small portion of the water column, unlike in cases of wind-driven near-inertial waves that spread over the entire ocean depth before dissipating. The authors also give an estimate of the interior dissipation and implied vertical diffusivities due to this process. In an extended appendix, they review the modal description of the SC interaction process, completed by the corresponding counterpart, the modulation interaction process (MI), where a preexisting internal wave is modulated by a surface wave and interacts with another one. MI establishes a damping of the internal wave field, thus acting against SC. The authors show 2 that SC overcomes MI for wind speeds exceeding about 10 m s 1.

1. Introduction generation by mesoscale eddies or the mean flow (Nikurashin and Ferrari 2011) and dissipation of bal- Internal gravity waves have a major share of the en- anced flow (Molemaker et al. 2010), but there are sev- ergy contained in oceanic motions. Important sources eral other processes that can generate internal waves are waves radiating out of the mixed layer because of (e.g., Olbers 1983), among which is the forcing by non- near-inertial motions excited by wind fluctuations (see, linear coupling to wind-generated surface waves, the e.g., Alford 2001; Rimac et al. 2013) and waves generated topic of the present study. If internal waves break, their at the ocean bottom, originating from the conversion of energy is partly used to mix the mean stratification. barotropic into baroclinic tides at submarine topography Although this diapycnal mixing is weak in the interior, (see, e.g., Falahat et al. 2014). Other forcing functions such breaking internal waves generate large amounts of that have been studied in recent years are lee-wave potential energy that then drives large-scale oceanic motions. Unlike the internal wave energy generated by near-inertial motions in the mixed layer and the tides Corresponding author address: Carsten Eden, Institut für Meereskunde, Universität Hamburg, Bundesstrasse 53, Hamburg that have low frequencies and propagate through the 20146, Germany. entire water column, internal waves generated by reso- E-mail: [email protected]. nant interactions of surface waves are of high frequency

DOI: 10.1175/JPO-D-15-0064.1

Ó 2016 American Meteorological Society Unauthenticated | Downloaded 10/01/21 05:50 AM UTC 2336 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 46 and thus likely to be trapped in the upper ocean. The all, such that we focus on the spontaneous creation interaction of surface and internal waves thus could process in the main part of the present study but con- contribute to mixing just below the mixed layer. sider both processes in appendix C. The relative im- Since the stress (vertical flux of horizontal momen- portance of SC and MI interactions can be assessed tum) induced by linear surface waves is zero (see, e.g., easily, realizing that the cross section of the interactions Hasselmann 1970), nonlinear effects need to be con- is identical when formulating the scattering in terms of sidered, and the evaluation of the transfer of energy the action spectra. We show in appendix C that the ratio 5 from surface to internal waves enters the field of non- of MI to SC is (Ucrit/U) , with a critical wind speed 2 linear resonant interactions among the wave branches. around 10–15 m s 1, depending on the Brunt–Väisälä The process has been investigated before, however, with frequency and the parameters of the surface and internal disperse and partly inconclusive results. Restricting the wave spectra. discussion to studies that deal with the interactions of Although the flux by interaction with surface waves is waves in random fields (see, e.g., Hasselmann 1966, lower than other sources of internal waves in the main 1967), only a few remain: Kenyon (1968) computed the pycnocline even for strong winds, it might still be im- transfer from an observed swell spectrum to the first portant since it is likely to be dissipated just below the baroclinic wave mode in shallow water and concluded mixed layer. We thus revisit the process with the aim to that the observed level of internal wave energy cannot provide a globally integrated estimate of the flux and its be explained by the process. Olbers and Herterich (1979, interior dissipation. hereinafter OH79) analyzed the transfer to deep-water In section 2 we start by outlining the surface layer baroclinic wave modes from wind-sea spectra as a model with nonlinear forcing due to surface waves with a function of wind speed and for different stratifications. spectral and statistical treatment of the interaction. We They find that high-frequency internal waves are excited derive the coupling conditions to the ocean interior and with wavelengths that are large compared to the ones of the spectral energy flux into the internal wave field at the the surface waves, but at a rate that is small in general. mixed layer base in general form. Section 3 shows the Only for the case of a shallow mixed layer, high wind dependency of the energy flux on the parameters of a speeds, and strong stratification below the mixed layer, simple factorized wind-sea spectrum, the mixed depth, 2 internal waves are generated at the level of 0.1 mW m 2. and the Brunt–Väisälä frequency below the mixed layer This level is typically a lower limit of the energy transfer base. We present a parameterization of the energy trans- by near-inertial motions in the mixed layer (see, e.g., fer rate in terms of the local wind speed, mixed layer Rimac et al. 2016) and far below the transfer rates by the depth, and the Brunt–Väisälä frequency, resulting from a tides (see, e.g., Falahat et al. 2014). numerical evaluation of the spectral transfer integral. A Watson (1990, hereinafter W90; 1994, hereinafter global estimate of the flux is provided, and its interior W94) has analyzed surface-internal wave interactions dissipation is based on realistic parameters. The last sec- in a more general setting. W90 applies the modal in- tion contains a summary and conclusions. teraction framework for idealized conditions, and W94 presents a calculation of energy transfers for the North 2. Theoretical framework Pacific. As OH79, Watson considers what he calls spontaneous creation (SC)—two surface waves generate We abandon here the baroclinic vertical mode frame- an internal wave—and he also discusses a modulation work used by OH79, W90, W94, and others and instead interaction process (MI), where a preexisting internal directly calculate the energy flux of internal waves radi- wave is modulated by a surface wave and interacts with ating from the surface mixed layer into the interior another surface wave. MI is included in the general stratified ocean. Radiation physics rather than modes, framework of OH79 but was not evaluated. It was first similar to our approach, are also considered by Moehlis studied in detail by Dysthe and Das (1981). By this in- and Llewellyn Smith (2001) for a wind-driven case. We teraction, following W90 and W94, the internal mode believe that this is a much simpler framework than the loses energy to the surface waves. The SC process, an- one by OH79 and W90, since the flux can be interpreted alyzed in the present study, cannot be separated in as being induced by ‘‘surface wave pumping,’’ generated Watson’s calculations from the MI process, but from his by the wave-induced vertical velocity in the surface layer, results we note that the MI dominates SC up to a critical much like ‘‘Ekman pumping’’ is induced by the di- wind speed. In strong wind regimes, the SC process vergence of the wind stress and ‘‘inertial pumping’’ is overcomes the MI process. Note that this regime (high induced by the divergence of wind-generated, near- winds, low mixed layer depths) is the one where the SC inertial waves in the mixed layer (Gill 1984). In fact, our surface-internal wave interactions become relevant at formulation would allow us to treat these surface-related

Unauthenticated | Downloaded 10/01/21 05:50 AM UTC AUGUST 2016 O L B E R S A N D E D E N 2337 processes in a similar way to the surface wave forcing we where u is the horizontal velocity, p is the pressure consider here. The flux, by a radiation condition, de- (scaled by the constant density), and w is the vertical pends only on the Brunt–Väisälä frequency just below velocity. All vectors are horizontal, and the vector u : the mixed layer but is independent of specific baroclinic denotes an anticlockwise rotation of the vector u by 908. mode structure, unlike OH79. On the other hand, our The terms (X, Z) on the right-hand side are the forcing analysis ignores the (small) part of the surface wave functions (stress divergence) for horizontal and vertical forcing below the mixed layer, which we believe is un- momentum, induced by surface waves. The surface wave important, since amplitudes of surface waves rapidly fields are marked by the index s. At the surface z 5 z we decay with depth.1 consider the kinematic and the dynamic boundary con- When forcing the modal system with finite depth and dition for a free surface. Expansion of the conditions specified stratification below the mixed layer by time- around the mean sea level z 5 0 yields dependent pumping at the top, a pressure field is set up ›z ›ws and allows for resonance with the applied pumping. The 2 5 52 s =zs 1 zs w Y u jz50 and resonant pressure field will project on the vertical modes ›t ›z jz50 and the resulting energy flux (vertical velocity 3 pressure) ›ps p 2 gz 5 P 52zs , (4) will be due to the resonating modes and must therefore ›z jz50 depend on their structure, as can be seen in the results of OH79. This is not the case in our semi-infinite system where the nonlinear terms Y and P are again induced where the flux is independent on the interior stratifi- by surface waves, restricted here to the quadratic cation and depth. Therefore, we cannot expect a com- terms of the expansion about the mean sea surface (g is plete match of both approaches, but we expect close gravity acceleration). A solution is sought now in spec- correspondence. We compare radiation and modal tral space where, for example, the vertical velocity is computations in appendix C. represented as ð ð a. The mixed layer and forcing by surface gravity ^ 5 1 2 2i(kx2vt) waves w(k, v, z) 3 d x dt w(x, z, t)e , (5) (2p) We consider a mixed layer from the ocean surface to a depth z 52d that has constant density and resides on with the complex wave amplitude w^, the wavenumber k, top of a stratified layer where internal wave propagation and the frequency v. Similar representations hold for the becomes possible with a nonzero Brunt–Väisälä fre- other variables. All (positive and negative) frequencies quency N(z). The surface wave forcing will be integrated are considered, but since w^*(k, v) 5 w^(2k, 2v), the from the surface to z 52d, such that this depth range analysis may be restricted to nonnegative frequencies. should contain all (or most) of the surface wave forcing. Here and in the following, the complex conjugate is de- Since the forcing is quadratic in the vertical structure noted by an asterisk. After the spectral transformation, function of the surface waves, the forcing indeed di- we follow OH79 and Olbers (1986) to combine (1)–(4) minishes very rapidly with increasing depth. The part of to a single equation for w^(k, v): the forcing that reaches into the stratified interior will ›2w^ Q^ not be considered here. The equations of motion for the 2 q2w^ 52 , surface layer are written in the form ›z2 v2 2 f 2 ^ ^ ›X 2 ^ ›u s s › s s Q 52(vk 1 if k) 1 ivk Z, (6) 1 f u 1 =p 5 X 52= u u 2 w u , (1) : ›z ›t : ›z ›w ›p › 1 5 Z 52= usws 2 wsws, and (2) with the surface boundary condition at z 5 0, ›t ›z ›z ^ ›w ›w^ Q = u 1 5 0, (3) 2 gw^ 52 0 , ›z ›z v2 2 f 2 ^ 2 ^ ^ ^ Q 5 k (ivP 2 gY) 2 vk 1 if k X(z 5 0) (7) 0 :

1 21 Consider, for instance, a wind speed of 10 m s , for which the where q2 5 k2v2/(v2 2 f 2), g 5 gk2/(v2 2 f 2). Note that mean wavelength of a saturated wind sea is 83 m (see below) and ^ ^ 2 d 5 50 m. The forcing then decreases by a factor of about 5 3 10 4 Q derives from the wave-induced stress and Q0 from the below z 52d, such that, indeed, almost all of the surface wave wave-induced forcing at the upper boundary. Given the forcing is contained in the mixed layer above z 52d. forcing, two boundary conditions are required to solve

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(6), that is, one condition in addition to the surface over m1 56and the integral over k1; the same applies condition given by (7). This additional condition will be for index 2. For the resonance condition v 5 v1 1 v2, ^ obtained by considering the matching to the lower the coefficient in Q0 is given by stratified layer that allows for internal wave propagation qffiffiffiffiffiffiffiffiffiffi k k and thus radiation of the energy generated by the 0 5 2 2 1 2 C12 i k1k2vk 1 . (13) forcing. k1k2 b. Matching the mixed layer forcing to the interior ^ ^ The forcing term Q yields a similar form as Q0, but it To match the mixed layer field to the interior, we take turns out that the corresponding coupling coefficient ^ ^ vanishes identically.2 This property is important because continuity of w and ›w/›z at the mixed layer base ^ z 52d. In the stratified layer below the mixed layer, a Q derives entirely from interior forcing. Hence, only the perturbation with wavevector k and frequency v is an weak nonlinearities of the surface boundary condition generate the transfer between surface and internal internal wave with a vertical velocity wîw 5 wîw(k, v, z) governed by waves (as in OH79, where the volume force only con- tributes below the mixed layer). ›2w^ The ensemble of surface waves is considered as a iw 1 m2(z)w^ 5 0, (8) 5 ›z2 iw random process with a zero mean, ha1i 0, and a second moment, ha1a2i, where the brackets denote the statisti- with the vertical wavenumber m given by the local dis- cal expectation. The second moment is given by the persion relation surface wave spectrum F(k1):

2 5 2 2 2 2 = 2 2 2 m (z) k (N (z) v ) (v f ). (9) 5 5 1 1 ha1a2i ham (k1)am (k2)i F(k1)dm ,2m d(k1 k2). 1 2 2 1 2 The internal wave propagates in a variable environment (14) with the Brunt–Väisälä frequency profileÐ N(z), where ^ 6 5 6 z 0 0 the phase of wiw is exp if(z) exp i m(z ) dz , such The assumption here is that the surface wave field is a that m 5 ›f/›z. We must require a positive vertical horizontally homogeneous stochastic process. The nor- wavenumber m to have downward group velocity, which malization of the eigenfunction fm(z)in(11) ensures is the radiation condition. The matching conditions are that the total energy is given by ð ð ›w^ ›w^ 0 w^ 5 w^ and 5 iw at z 52d. (10) 5 1 s s 1 s s 1 s s 5 2 iw F0 (u u w w ) dz gz z d kF(k). ›z ›z 2 2‘ (15) c. Forcing by an ensemble of surface gravity wave The forcing terms Q^ and Q^ are now established for a The energy of surface waves is equally partitioned be- 0 5 s 2 given ensemble of surface waves. The vertical velocity of tween kinetic and potential energies; thus, F0 gh(z ) i. the surface wave field is expressed as d. The solution in terms of the pumping velocity ð ^ 5 i(kx2v t) We complete the problem [(6)] by requiring w W at s 5 å 2 m f w (x, z, t) d kam(k)e m(z), (11) 52 5 m56 z d with a yet unknown W W(k, v), which is a surface-wave-induced pumping velocity at the base of 5 1/2 kz 5 5 with the vertical eigenfunction fm(z) k e and a1(k) the mixed layer. The unknown W W(k, v) is de- 1/2 a*2(2k)andv1(k) 52v2(2k) 5 (gk) . The other sur- termined by the matching conditions from (10). The face wave fields follow from the polarization vector solution w^(z)of(6) with the appropriate boundary s s s s ; 0 2 0 2 5 (u , w , p , z ) (ifmk/k , fm, ifmvm/k , ifm(z 0)/vm) conditions in (7) and (10), and especially for of the linear wave solution for surface waves. We denote w^(z 52d) 5 W, can be found in terms of the appropri- ^ the vertical derivative by a prime. For Q0 we find ate Green’s function. The analysis is detailed in appendix A, and the result for the pumping velocity is ^ 5 å 0 2 2 2 2 Q0 a1a2C12d(k k1 k2)d(v v1 v2), (12) 12

mj where aj 5 a are the amplitudes of the surface waves, 2 kj This important result was pointed out by an anonymous re- 5 56 ^ j 1, 2, mj . In expressions like (12), the index 1 viewer. Up to then we used that Q vanishes on the resonance stands for (m1, k1), and the sum over 1 stands for the sum surface.

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^ Q 1 2 gd by the dispersion relation (9). We consider only waves W(k, v) 52 0 v2 2 f 2 ag~ 1 b~q 2 im(2d)(1 2 gd) that propagate downward so that m is positive for positive v, and thus 1 3 , (16) ð ð Ð 2 ‘ z l sinhqd coshqd 2 i kx1s m(z0) dz02vt 5 [ 2d ] wiw(x, z, t) d k dv a(k, v)e , 2‘ with the coefficients (20) lqd coshqd 2 1 2 qd sinhqd a~ 5 and l sinhqd 2 coshqd where s is the sign of v and the amplitudes satisfy the re- ality condition a(k, v) 5 a*(2k, 2v). We equate w(2d) sinhqd 1 l(1 2 coshqd) b~ 5 , (17) with w at z 52d (and thus neglect the tiny velocity l sinhqd 2 coshqd iw induced by surface waves) and obtain where l 5 g/q 5 gk/[v(v2 2 f 2)1/2] ’ gk/v2. We use now ^ a v 5 W v the expression for Q0, resulting from (12),in(16) and (k, ) (k, ), (21) obtain W as which is the relation between the wave amplitude and the 5 å 2 2 2 2 forcing fluctuations. Only frequencies in the range f , W(k, v) D12a1a2d(k k1 k2)d(v v1 v2), 12 v , Nd (and the negative mirror of the interval) are al- (18) lowed for propagation. The response at frequencies out- side this interval is evanescent (m becomes imaginary) where D12 is the interaction coefficient, characterizing and must be dissipated locally. This part of the response is the interaction of the waves with labels 1 and 2. In disregarded in the present approach. It might, however, practice we have l 1, and we arrive for this limit at be important for mixing on the mixed layer base. W 52 2mm m We define the spectrum (k, v)ofwiw(z d)by 5 1 2 D12 D2kk k 1 2 pffiffiffiffiffiffiffiffiffiffi 0 0 5 2 0 2 0 W i k k v2/g k k ha(k, v)a*(k , v )i d(k k )d(v v ) (k, v), (22) 5 1 2 1 2 1 2 , v coshqd 1 i(N2 2 v2)1/2 sinhqd k k d 1 2 with W (k, v) 5 hW(k, v)W*(k, v)i, so that the energy (19) of the vertical velocity of the propagating waves at the mixed layer base becomes where Nd 5 N(z 52d)istheBrunt–Väisälä frequency just 2 mm1m2 below the mixed layer base. The indices in D2kk k should ð ð 1 2 Nd indicate that the triad sum of the wave vectors is zero, and 1 2 5 1 2 W 1 W 2 hwiwijz52d d k dv½ (k, v) (k, v). similarly for the frequencies due to the d functions in (18). 2 2 f Writing (18) in dimensionlesspffiffiffiffi form by normalizing the ve- (23) W a k v k locities and j j by the phase speed /p,ffiffiffiffiffiffiffiffiffiffi one finds that the dimensionless coupling coefficient (D12 / k1k2)v/k is of The relation to the spectrum of the total (mechanical) 2 order v2/gk, which is typically O(10 3). It is therefore the energy of downward propagating waves at z 52d is thus large gap between the internalpffiffiffiffiffi wave frequency v and the [see, e.g., Olbers et al. 2012, their Eq. (7.22)] surface wave frequency gk that generates small co- 2 2 efficients and thus weak coupling. Note that the small ratio 2 1 N 2 f pffiffiffiffiffi E (k, v) 5 d ½W (k, v) 1 W (k, 2v). (24) v/ gk enters the coupling coefficient in a quadratic way 2 v2 2 f 2 and the transfer integral (see below) as fourth power. The pumping induced by W(k, v) at the mixed layer base The vertical flux of energy is hwiwpiwijz52d with the E 2 establishes the generation of internal waves that radiate spectral form cvert (k, v), where downward from the surface layer into the interior. The sta- ›v 1 (v2 2 f 2)3/2(N2 2 v2)1/2 tistical properties of the surface wave field imply the van- c 5 52 d (25) 5 vert 2 2 2 ishing of the mean pumping velocity, hW(k, v)i 0. The ›m kv Nd f radiation flux, however, depends on the second moment of the pumping velocity, as derived in the following section. is the vertical group velocity of downward propagating waves. We thus arrive at e. The spectral energy flux at the mixed layer base ð ð Nd 2 The vertical velocity wîw of the interior wave field F 5 5 F tot hwiwpiwijz52d d k dv (k, v) (26) follows from (8) with the vertical wavenumber m, given f

Unauthenticated | Downloaded 10/01/21 05:50 AM UTC 2340 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 46 with for the total downward energy flux Ftot and its spectrum F just below the level z 52d. Given the surface 1 (v2 2 f 2)1/2(N2 2 v2)1/2 F(k, v) 52 d wave spectrum, the energy flux radiating into the in- 2 vk ternal wave field of stratified lower layer can now be 3½W (k, v) 1 W (k, 2v) (27) calculated. The pumping velocity spectrum becomes

ð ð W 5 2 2 2 2 2 2 1 2 1 (k, v) d k1 d k2jD( k, k1, k2)j F(k1)F(k2)d(k k1 k2)d(v v1 v2) (28)

2 2 5 1/2 1/2 for a general surface wave spectrum, with D( k, k1, k2) (v2 2 f 2) (N2 2 v2) 212 F(k, u, v) 52 d W (k, u, v), D2 2 . We will evaluate W and the ﬂux F in circular kk1 k2 v coordinates (k, u) instead of the wavenumber vector k. The surface wave spectrum F(k)d2k 5 F(k, u)dkdu (29) is transformed accordingly, that is, F(k) 5 F(k, u)/k. Hence, with

ð W u 5 u 2 2 2 u u 1 2 1 (k, , v) dk1 d 1 jD( k, k1, k k1)j F(k1, 1)F(k2, 2) d(v v1 v2). (30) k2

The way to calculate the pumping velocity spectrum W growing sea state, also called the Joint North Sea from a general and a factorized surface wave spectrum is Wave Project (JONSWAP) spectrum: detailed in appendix B. ﬃﬃ 22 p 2 2 G(x) 5 (5/2)x23e25/4x 1ln c exp[2( x 21) /2s ]=n, (32)

3. The energy flux for specific wind-sea spectra where s 5 sa for x , 1ands 5 sb for x . 1, and n is a normalization constant. The form of the Pierson– In principle, it is possible to calculate the energy flux Moskowitz spectrum is obtained for c 5 1andn 5 1, [(29)] for any given surface wave spectrum globally for while typical parameters for the JONSWAP spectrum are any given time. However, the global distribution of the c 5 3.3, s 5 0.07, s 5 0.09, n 5 1.4143 (Hasselmann et al. spectrum is unknown to us and is in fact not needed in a b 1976). As directional distribution we use that detail. Instead, we approximate the surface wave spectrum following standard methods. The surface wave 1 A(u) 5 s coth(sp) sech2(su). (33) spectrum F(k, u) is factorized with respect to wave- 2 number and angular dependence: A simple parameterization for the expectation of the s 2 5 5 22 mean surface displacement is h(z ) i F0/g (a/5)km F(k, u) 5 (F /k )G(k/k )A(u, k/k ), (31) 2 0 m m m with the Phillips constant a 5 8.1 3 10 3. The wave- 1/2 number km or the equivalent frequency vm 5 (gkm) is 5 s 2 s where F0 gh(z ) i is the total energy, z is the sea parameterized in terms of the wind speed U at 10 m 2 surface elevation due to the surface wave, and km is a height as km 5 0.77g/U , which leads to fixed wavenumber characterizing the peak of the spectrum. The function G(x), describing the spectral U 4 h(zs)2i 5 2:8 3 1025 m2 . (34) shape, is dimensionless and normalized by integration ms21 over x 5 k/km, and the directional distribution A(u, x) is normalized to one by integration over u.Apro- These parameterizations can be derived from Hasselmann totype of a wind-sea spectral shape G(x)isthatof et al. (1976) and Rosenthal (1986). Pierson and Moskowitz (1964). It applies to the wind- With these parameterizations in terms of the wind sea part of the surface waves, that is, the part of the speed, we follow OH79 and also W90 and W94,whouse spectrum that is in equilibrium with the local wind. the Donelan spectrum (Donelan et al. 1985), an offspring Here, we use the modification of this spectrum by of JONSWAP. We are aware that these spectra, de- Hasselmann et al. (1973), which is appropriate for a veloped for growing seas, are rather fetch limited than

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Ð 22 21 21 21 FIG. 1. (a) An example of the energy flux spectrum F(k, u, v) du [(mW m )(m s ) ] into the internal wave field (downward 5 5 21 5 ä ä ä 5 positive) for an anisotropic JONSWAP spectrumÐ with s 3.5 and U 13.7 m s , d 100 m, and Brunt–V is l frequency Nd 2 2 2 2 0.0157 s 1. (b) The associated frequency spectrum F(k, u, v) du dk [(mW m 2)(s 1) 1]. equilibrium, and that the U4 scaling of an equilibrium relatively small rate compared to, for example, the spectrum would not apply in these cases, but rather U4 wind-stress-driven energy transfer to internal waves that 2 times dimensionless fetch x~5 x/(U2/g)(x is the actual is typically 0.1 to 1 mW m 2 (e.g., Rimac et al. 2013). fetch). The problem is that the fetch is basically unknown However, as in OH79, it turns out that the flux strongly in the open ocean (x~is of order 103–4, large values apply to depends on the environmental parameters. Figure 2 the equilibrium scaling). Since the Pierson–Moskovitz shows the dependency of the total flux Ftot on the wind form is only rarely observed (K. Hasselmann 2015, per- speed U, the mixed layer depth d and the Brunt–Väisälä sonal communication), we decided to use the JONSWAP frequency Nd. Shallow mixed layers with strong Brunt– form together with the equilibrium wind scaling. Väisälä frequency and strong winds lead to a larger ex- We evaluate the flux spectrum F(k, u, v)fromthe citement of internal waves. For instance, the total flux can 22 21 surface wave spectrum by numerical integration.Ð Figure 1a become as large as 0.13 mW m for U 5 13.7 m s , d 5 3 21 shows the resulting flux F(k, v) 5 F(k, u, v) du,in- 100 m, and Nd 5 0.026 s (15 cph). One has to keep in tegrated over the horizontal direction, for a typical ex- mind, however, that wind speed and mixed layer depth ample using the JONSWAP spectrum [(32)]withan are related to each other and that stronger winds usually anisotropic angular distribution [(33)]withs 5 3.5 for a imply deeper mixed layers. This points toward the need 21 2 2 wind speed U 5 13.7 m s where hz i 5 1m and km 5 to use a consistent dataset for U, d,andNd, which we will 2 0.04 m 1 (wavelength 156 m), mixed layer depth is d 5 take below from a numerical ocean model simulation. 100 m, and the Brunt–Väisälä frequency at the mixed layer The black lines in Fig. 2 demonstrate that 2 base is 0.0157 s 1 (9 cph). The internal wave excitement is predominantly occurring for high frequencies (v * N /2) 7 4 d F 5F u n 5 U (U, d, N ) with u 2 , and long wave lengths (k km). FigureÐ 1b shows the flux tot d 01 1 (d/d )2 10 m s 1 as density in frequency space, F(v) 5 F(k, u, v) du dk. 0 N The angular distribution is not shown. For a surface wave 5 d n 2 , (35) spectrum with narrow angular support, the internal wave 0:0158 s 1 radiates at almost right angles to the dominant direction of 2 2 where d 5 150 m and F 5 2.83 3 10 3 mW m 2 the surface waves. 0 0 provides a reasonable bulk fit to the numerically obtained The total flux Ftot integrated over all wavenumbers and 2 data. This bulk fit for the flux F is motivated by a frequencies for these parameters is 0.025 mW m 2,a tot nondimensional version of the integral (28). The flux F 5 2 3 spectrum can be written as (k, v) F0 vmNd/g G 2 G 3 In all following figures and for all explicit values for fluxes, we (kd, v/Nd, v /gk), where is a dimensionless function of 2 multiply fluxes with a constant density 1000 kg m 3 to obtain units its arguments. The dimensional factor in front of G is then 22 26 3 23 7 ; 7 4 of mW m for 10 m s . proportional to U u . The proportionality Nd may be

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22 21 FIG. 2. The total flux Ftot (mW m ) as function of U, d, and Nd: (a) Ftot(U) for d 5 100 m and Nd 5 0.0052 s (3 cph, blue), Nd 5 21 21 21 0.0157 s (9 cph, red), and Nd 5 0.026 s (15 cph, yellow); (b) Ftot(d)forU 5 13.7 m s , and the same values for Nd as in (a); and (c) Ftot(Nd) 2 for U 5 13.7 m s , and for d 5 400 (green) to d 5 30 m (blue). The circles indicate the values obtained by the numerical evaluations. ; 7 ; 1 2 2 ; 4 Each figure also includes a black line, showing a bulk fit given by U in (a), 1/(1 d /d0)in(b),and Nd in (c). derived from the interaction coefficient; the dependence on layer parameterization after Gaspar et al. (1990).Wind d is a heuristic fit4 to curves in Fig. 2.InFig. 3 all numeri- stress forcing and U is taken from the year 2010 of the cally evaluated data for Ftot from Fig. 2 are compared with reanalysis by Kalnay et al. (1996) with a resolution of 6 h the bulk fit [(35)], which shows reasonable agreement. and replaces the climatological wind stress forcing with Analogous results are also found by OH79 using the monthly frequency in a 500-yr-long spinup period of the modal framework; they analyzed the transfer to deep- model. The wind stress is also used to calculate the sur- water internal wave modes from wind-sea spectra as face boundary condition of the mixed layer model by function of wind speed and stratification using different Gaspar et al. (1990), that is, to parameterize the input of Brunt–Väisälä frequency profiles. High-frequency in- turbulent kinetic energy by the wind. More details on the ternal wave modes with wavelengths that are large com- model setup and performance can be found in Eden pared to the ones at the surface waves are excited. They et al. (2014). also find that the transfer is very small in most cases. If the Figure 4 shows the annual mean total flux Ftot and its mixed layer is shallow, the wind speed and the Brunt– maximum during the year entering the internal wave Väisälä frequency are large, internal waves are generated field diagnosed from the simulated mixed layer depth 2 at a level of 0.1 mW m 2. The transfer to modes of a and Brunt–Väisälä frequency at the mixed layer base, Brunt–Väisälä frequency profile without a strong seasonal thermocline (small N under the mixed layer) is up to three ordersofmagnitudesmaller(seeTable1inOH79). The parametric dependence ;u7n4 is also found. These find- ings are confirmed in new computations of the modal transfer for an exponential stratification with a mixed layer on top. Details are given in appendix C. The bulk fit of (35) can now be used to estimate the global energy flux to the internal wave field. To work with a mixed layer depth that is dynamically consistent with the wind speed, we use a global ocean circulation model with high vertical resolution and high-frequency wind forcing. The ocean model is identical to the energetically consistent model used in Eden et al. (2014;their experiment CONSIST-SURF), which contains a mixed

4 As pointed out by a reviewer, the interaction coefficient should lead to an exponential dependence and, in fact, replacement of 1 2 2 2 1/(1 d /d0) by exp( adkm) yields an equally good fit; however, different a are required for the range of high and low fluxes. A fit with constant a is worse than (35). Apparently, a depends on Nd and d,but FIG. 3. The bulk fit of Ftot(U, d, Nd) given by (35) compared to all the functional form remains unknown and could not be retrieved. values from numerical evaluation of Ftot (circles in Fig. 2).

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22 21 FIG. 4. (a) Mean total ﬂux Ftot leaving the mixed layer {log10[Ftot (mW m ) ]} in 2010. (b) As in (a), but maximum of Ftot.

2 and the wind speed U taken from the reanalysis product. 3.7 3 10 4 TW during July/August to its maximum of 2 2 The largest fluxes show up in the storm track regions of 8.2 3 10 4 TW in March and 1.2 3 10 3 TW in September/ the oceans; while toward the equator, the flux and its October. Even the maximal globally averaged values are maximum almost vanish. The flux varies a lot in space more than an order of magnitude smaller than the flux and time: Fig. 5a shows a time series of the flux Ftot in supported by wind fluctuations at near-inertial fre- the Pacific Ocean along 1608W, demonstrating that it is quencies. In Rimac et al. (2016) the wind input to near- the large wind speeds during fall and early winter in mid- inertial energy in the mixed layer is estimated as to high latitudes in both hemispheres that drives the 0.34 TW, of which 90% dissipate in the mixed layer and 2 flux maxima. The globally integrated annual mean flux only 10% radiate out, that is, 3.4 3 10 2 TW leave the 2 is 6.9 3 10 4 TW, and it varies from its minimum of mixed layer and are available for interior mixing.

22 21 FIG. 5. (a) The total ﬂux Ftot {log10[Ftot (mW m ) ]} along 1608W. (b) Mean interior internal wave dissipation Fdiss in 2010 of the total 2 23 21 2 23 21 ﬂux Ftot {log10[« (m s ) ]} along 1608W. (c) Vertical maximum Fdiss {log10[« (m s ) ]} along 1608W. (d) As in (c), but for vertical 2 21 21 diffusivity K {log10[K (m s ) ]}.

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The spectral shape of the flux in the frequency do- framework. It is based on matching the vertical velocity in main, shown in Fig. 1, can be well approximated by the the mixed layer generated by the nonlinear forcing of function surface waves with the vertical velocity of an internal ð wave propagating from the mixed layer base into the (v2 2 f 2)1/2(N2 2 v2)1/2 F(k, u, v) dk du 5 n d interior. The generated energy flux by this radiation c v process can be interpreted as being induced by surface v2 2 f 2 wave pumping, generated by the wave-induced vertical 3 , (36) 2 2 2 1 2 velocity in the surface layer. The physics is analogous v f 4Nd to Ekman pumping, induced by the divergence of the where Nd denotes the Brunt–Väisälä frequency at the wind stress, and inertial pumping, induced by the di- base of theÐ mixed layer and nc is a normalization factor vergence of wind-generated near-inertial motions in that sets F dv dk du 5Ftot. Motivation to use this form the mixed layer (Gill 1984). However, unlike Ekman or comes from the group velocity in (27) and the frequency inertial pumping, the surface wave pumping is due to dependency of the interaction coefficient (19). Using triad interactions between two surface waves and one this spectral shape, it is possible to calculate the depth internal wave, like modal triad interaction of internal z0 52b(v) below which internal wave propagation waves in the interior. stops. This depth is given by v 5 N(z0) where the waves Since surface waves may have an order of magnitude at the frequency v have their turning point. Assuming larger energy content than internal gravity waves, such that the flux F(v) from the mixed layer is dissipated an interaction is potentially a large source for the in- equally over the depth range b–d, it becomes possible to ternal wave field. The coupling of the wave types is, calculate the total interior wave dissipation Fdiss by in- however, rather weak, characterized by an interaction tegration in frequency domain. The turning point b de- coefficient of order v2/gk that enters the spectral trans- pends on v but is typically only 50 to 400 m below the fer integral as square. The ratio reflects the large spectral mixed layer (not shown) for the Brunt–Väisälä fre- gap between the internalpffiffiffiffiffiffi wave frequency v and the quency profiles of the model. Figure 5b shows the annual surface wave frequency gk. On the other hand, the F mean interior dissipation of internal waves Fdiss along energy flux tot depends quadratically on the surface 1608W generated by the flux Ftot during 2010, shown as a wave energy. As a result, the local wind speed U enters F time series in Fig. 5a. Along 1608W, the flux Ftot is large as the seventh power in our expression for tot, using in the northern North Pacific and the Southern Ocean standard dependency of the wind sea spectra from U, compared to other longitudes. Enhanced values of Fdiss while the Brunt–Väisälä frequency Nd at the mixed layer are restricted close to the mixed layer with values be- base enters the flux via its fourth power. We have de- 210 29 2 23 tween 10 and 10 m s , comparable to the low rived the flux Ftot radiating from the mixed layer base in range of observational estimates of mixing rates below analytical form and also fitted the parametrical de- the mixed layer (Kunze et al. 2006; Waterhouse et al. pendency in terms of U, Nd at the mixed layer base, and 2014). In case of strong storm events in combination the mixed layer depth d. The basic results of OH79 are with shallow mixed layers, for example, during fall in the confirmed, also by new calculations of the modal energy northern North Pacific or during February to April in transfer described in appendix C, which, besides the the Southern Ocean, the vertical maximum of Fdiss can spontaneous creation process (SC) treated in the main 2 2 2 locally reach values up to 10 8 to 10 7 m2 s 3 along the part of this study, also includes the modulation in- section, as seen in Fig. 5c. Using the Osborn–Cox re- teraction process (MI; W90, W94). We have shown that 2 lation K 5 d/(1 1 d)Fdiss/N (Osborn 1980) with the the importance of SC overcomes MI at wind speeds 2 mixing efficiency d ’ 0.2, it is possible to calculate the around 10 m s 1. For lower wind speeds, both processes vertical diffusivity K from the internal wave dissipation. have only minor transfer rates at all. F The diffusivity K calculated from the mean Fdiss remains Using a model-based estimate of the flux tot dem- low, but for large fluxes Ftot, the diffusivity calculated onstrates that, in the global integral, it is only about 23 from the instantaneous Fdiss locally reaches values of 0.5–1 3 10 TW, that is, two orders of magnitude 2 2 2 10 5 to 10 4 m2 s 1 , as shown in Fig. 5d. smaller than the flux due to the inertial pumping. Locally in space and time, however, it can reach similar magnitudes. Since the internal waves generated by interaction 4. Summary and discussion with surface waves are of high frequency, their turning The transfer of energy from a surface wave field to point lies close to the mixed layer, such that it is likely that internal waves below a mixed layer is revisited here, the flux Ftot is also dissipated close to the mixed layer. building on previous work by OH79, but using a simpler This is different from the fluxes due to inertial pumping

Unauthenticated | Downloaded 10/01/21 05:50 AM UTC AUGUST 2016 O L B E R S A N D E D E N 2345 and the tides that generate low-frequency waves pene- conditions. We thus deﬁne w^ 5 w~ 1 az 1 b,wherew~ trating the entire depth range of the ocean. Our estimate satisﬁes of the implied dissipation shows that it sometimes reaches ›2w~ magnitudes comparable to observational estimates close 2 q2w~ 5 q2(az 1 b). (A1) to the mixed layer, in particular during strong wind events, ›z2 but in general stays below them. Note that we use Q^ [ 0 for the surface wave forcing. Homogeneous boundary conditions for w~ are ob- ü Acknowledgments. The authors thank J rgen Willebrand, tained for Sergey Danilov, and two anonymous reviewers for helpful comments. gW 1 Q W 1 dQ a 5 0 and b 5 0 , (A2) 1 2 gd 1 2 gd

Q 52^ 2 2 2 APPENDIX A where 0 Q0/(v f ). The solution of (A1) is written as ð Solution for the Pumping Velocity 0 ~ 5 2 1 The solution from (6) with the boundary conditions w(z) q dj G(z, j)(az b), (A3) 2d of (7) and w^(2d) 5 W is written in terms of Green’s function for the problem with homogeneous boundary with Green’s function

1 sinhq(z 1 d)(coshqj 1 l sinhqj) for 2d # z # j # 0 G(z, j) 5 , (A4) q(l sinhqd 2 coshqd) (coshqz 1 l sinhqz) sinhq(j 1 d) for 2d # j # z # 0 where l 5 g/q. The matching conditions [(10)] result in which leads to expression (16) for W.

›w^ ›w~ 5 1 a 5 imW . (A5) ›z 2d ›z 2d APPENDIX B ~ We evaluate w explicitly and ﬁnd Derivation of the Spectrum of the Pumping Velocity 1 Q ~ 1 1 Q ~ 2 2 5 To calculate the energy ﬂux F radiating out of the mixed ðgW 0Þa ðW d 0Þbq im(1 gd)W 0, tot layer, we need the spectrum of the pumping velocity (A6) W (k, v). From (18) the squared modulus of W becomes

W (k, v) 5 hW(k, v)W*(k, v)i 2mm m 2mm m 5 å 1 2 3 4 2 2 2 2 2 2 2 2 D2kk k (D2kk k )*ha1a2a*3a*4id(k k1 k2)d(v v1 v2)d(k k3 k4)d(v v3 v4). (B1) 1234 1 2 3 4

The fourth-order moment can be broken into products for k 5 0 the interaction coefﬁcient vanishes. For the of second-order moments by the Gaussian assumption, second term we have 1 5 3, 2 5 4, and for the third 1 5 4, 5 1 1 5 ha1a2a*3a*4 i ha1a2 iha*3a*4 i ha1a*3 iha2a*4i ha1a*4iha2a*3i. 2 3. Since D is symmetric in the last two arguments, we The ﬁrst term does not contribute here because obtain for these terms the same result (this yields a factor 5 1 5 ha1a2i ha1a2*2i implies from (14) that k1 k2 0, and 2). Inserting the surface wave spectrum, we arrive at

1 2mm m W 5 å 1 2 2 2 2 2 2 (k, v) jD2kk k j F1F2d(k k1 k2)d(v v1 v2). (B2) 2 12 1 2

The sum over m1 and m2 mustbewrittenout.Onthereso- (19), there is no dependence of D on the frequency indices m1 nance surface where the interaction coefﬁcient is given by and m2. The expression is needed for positive and negative v.

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Consider first v . 0. The term from m1 5 m2 52does cannot be satisfied. Hence, using the convention of not contribute because the resonance condition (14), we find

ð ð 1 W (k, v) 5 d2k d2k F(k )F(k ) 2 1 2 1 2 211 2 212 2 3fjD2 j d(k 2 k 2 k )d(v 2 v 2 v )12jD2 2 j d(k 2 k 1 k )d(v 2 v 1 v )g, (B3) kk1k2 1 2 1 2 kk1 k2 1 2 1 2 where the surface wave spectrum is introduced and the we obtain a similar expression for W (k, 2v). It turns out frequencies v, v1,andv2 are now to be taken as positive. that the squared interaction coefficient does not depend on The first term in the curly brackets will not contribute the sign of v and the frequency indices m1 and m2,sothat because the condition v 5 v1 1 v2 cannot be satisfied for W (k, v) 5 W (k, 2v). The spectrum is then expressed surface waves that have frequencies vj v. For negative by (28). For the evaluation we factorize the surface wave v the combination m1 5 m2 51does not contribute, and spectrum as in (31). The integral then finally becomes

ð 2 k 21 F 1max 1 dv W u 5 0 2 2 2 2 å u u (k, , v) dk1jD( k, k1, k k1)j G(x1)G(x2) A( 1, x1)A( 2, x2). (B4) k k du u2u m k1min 2 1 sign( 1)

Further details, such as the functions k1min, k1max, u1, u2 analysis of OH79 results in the transfer rate for the and dv2/du1 and the method of integration (the in- spontaneous creation process (SC) to mode n, given by tegrand becomes singular on the boundaries of in- ð tegration), are found in OH79. We evaluate this form of SC 5 2 2 2 2 Sn (k) d k1F(k1)F(k1 k)Td(v1 v2 v), (C1) the integral numerically to calculate the energy flux spectrum and the total flux Ftot radiating out of the mixed layer in (26) and (27). with the scattering cross section ð 0 2 5 1 2 N (z) APPENDIX C T 2p(1 cosa) dz. fkn(z)f1(z)f2(z) 2H g 2 1 0 The Modal Treatment of SC and MI Mechanisms (fkn/g)f1f2jz50 In this section we briefly review the modal transfer between internal waves (IW) and surface waves (SW) of The modulation interaction process (MI) is then repre- OH79 and compare with W90 and W94. The modal sented by (see OH79 or W94)

ð T SMI(k) 5 E (k) d2k [v F(k ) 2 v F(k 2 k)] d(v 2 v 2 v) 5 l (k)E (k). (C2) n n 1 2 1 1 1 v 1 2 n n

Ð The IW mode is normalized according to N2f2 dz 5 v2 where N(z) 5 0. In all applications in this section, we take pffiffiffiffiffi kn 5 ksz 5 5 and the SW mode is given by fs kse .Variablea is d 100 m and b 1300 m. The IW eigenfunctions are the angle between k and k1 . As in the main text, F(k)isthe exponential Asinhkz in the mixed layer and Bessel 2 T 5 2 SW energy spectrum with integral ghz i (contrary to functions k(j) B[Jk(j) Yk(j)Jk(j2)/Yk(j2)] with Watson’s notation, where F denotes the SW action spec- j 5 kbN(z)/v, k 5 kb, j2 5 j(2H) 5 kN(2H)/v. trum). We have repeated an SC case from OH79 with The amplitudes A and B are related by continu- ä ä ä T 0 5 constant Brunt–V is l frequency and found reasonable ity. The dispersion relation (N0/v)tanhkd k (j1) 2T 5 2 5 agreement. k(j1)withj1 j( d) kN0/v follows by conti- The new calculations for SC and MI reported are nuity of the derivative of the mode and is solved for an exponential Brunt–Väisälä frequency N(z) 5 numerically. For this exponential model, the cross

N0exp[(z 1 d)/b] with a mixed layer 0 . z .2d on top, section becomes

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21 FIG. C1. Transfer for SC mechanism for an anisotropic JONSWAP spectrum with s 5 3.5 and U 5 13.7 m s , and a Brunt–Väisälä 21 frequency N0 5 0.0157 s (same conditions as for Fig. 1). (a) The SC transfer as function of scaled wavenumber for modes 1 (blue) and higher. (b) The modal dependence of the ﬂux, integrated over wavenumbers. (c) Dispersion relation of the ﬁrst 10 modes.

4 T 2 normalization and some reasonable approximations 2 v (j ) b 2 T 5 2p(1 1 cosa) k k B2 2 k 1 1 e qdj I , 2 2 1 2 g2 k sinhkd k2 1 12 [cosa ’ 0, v1 v2 ’ cg2 (k1 k2), where cg is the group velocity, k ’ k ] we find agreement, except that our (C3) 1 2 cross section is 2 times the one of Watson. where The program to compute the scattering integrals (C1) and (C2) is similar to the one used for the radiation flux ð 2 SC MI k form [(B4)]ofSC.WecomputeS and ln 5 S /En 2 5 /b 5 T 2 n n B with I(j) djj k(j) (C4) u F 5 I(j ) 2 I(j ) inÐ polar coordinates (k, Ð). For SC we display n(k) 1 2 u SC u F 5 F d Sn (k, )and n dk n(k). The case shown in and Fig. C1 is for the same parameters as Fig. 1. The figure also ð shows the dispersion relation vn(k) of the first 10 modes j 1 (k 1k )b11 obtained during the numerical integration. The radiation 5 1 2 T 2 I12 dj (j/j1) k(j). (C5) 2 j form yielded a rate 0.025 mW m (see section 3), the 2 2 modal form yields 0.022 mW m 2 tothefirstmodeand 2 The normalization integral I of modes is obtained ana- 0.025 mW m 2 as sum over the 10 modes calculated. lytically, whereas the integral I12 must be calculated Such a perfect agreement is fortuitous; for example, for 21 numerically. The corresponding term in the cross sec- the same conditions except that N0 5 0.0052 s ,we 2 2 tion is found to be generally very small and negligible. find 4.5 3 10 4 mW m 2 for the radiative form and 2 2 The cross section given in W94 [their Eq. (26)], though 1.83 10 4 mW m 2 for the modal form. OH79 is used, looks at first fundamentally different; Since we will consider the isotropic Garrett and Munk however, implementing Watson’s different mode (Garrett and Munk 1972; Munk 1981) spectrum for the

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FIG. C2. As in Fig. C1, but for MI mechanism. (a) The decay time for the MI transfer as a function of scaled wavenumbers for modes 1 (blue) and higher. (b) As in (a), but for the transfer rate. (c) Modal distribution of internal wave energy, as explained in the text. (d) The modal dependence of the transfer rate, integrated over wavenumbers.

u 5 IW energyÐ En(k, ) En(k)/2p, it is sufficient to look for (stars and line). As in most other cases considered for the ln(k) 5 du ln(k, u). The modal MI transfer is then present study, we find ln to be negative so that IWs are MI 5 MI completelyÐ described by Sn (k) En(k)ln(k)/2p and dampened and Sn is a transfer from IW to SW as in W90 MI 5 ä ä ä Sn dk En(k)ln(k)/2p. We use the Garrett and Munk and W94. For the high wind speed and Brunt–V is l model (GM76; Cairns and Williams 1976) form of the frequency, the MI transfer shown in Fig. C2 is much 2 IW spectrum (integrated over direction) as a function of smaller than the corresponding SC transfer: MI has 10 3 2 2 n and k, mW m 2 for the 10 modes and SC has 0.025 mW m 2.For low wind speeds the behavior reverses, as demonstrated in fN m2 5 ^ 2 0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fig. C3, showing transfer rates for SC and MI for wind E (k) EH(n) p with 21 n p 2 2 1 2 2 2 1 2 speeds from 5 to 20 m s and two values of the Brunt– (N0 k f m ) k m Väisälä frequency. H H n 5 + ( ) 2 2 , (C6) We summarize these results: n 1 n+ d the SC and MI transfer is mainly in mode 1; where we assume the WKB form m 5 np/b for the vertical d ln, and hence the MI transfer, is overall negative (not ^ 3 22 3 22 wavenumber and E 5 4m s [ 4 3 10 Jm for the so at very large N0 and very small U, not shown) and total IW energy. The modal part H(n)isnormalized IW loses energy to SW; 7 2 (H+ 5 2:13 for n+ 5 3). We show MI transfers in Fig. C2 d the SC transfer behaves as ;U and MI as ;U ,soSC 21 21 for U 5 13.7 m s and N0 5 0.0157 s .Inthethirdpanel overcomes MI at a critical wind speed (around 2 of the figure, we show the modal energy distribution in- 10 m s 1 for standard GM); tegrated over all wavenumbers (1) and integrated for the d both the SC and MI transfer strongly increase with N0, 4 wavenumber range participating in the SW–IW interaction SC roughly as N0 (not shown); and

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The relative importance of SC and MI interactions can be easily assessed from the scattering integrals (C1) and (C2). One obtains for the ratio of MI terms and SC terms

E v F 2 v F E A 2 A MI/SC ’ n 2 1 1 2 5 n 1 2 , (C7) vn F1F2 vn A1A2

where En is the IW spectral energy level and Aj 5 Fj/vj, j 5 1, 2 is the corresponding SW spectral action level at the respective interacting wavenumbers and frequen-

cies. We proceed with A1 2 A2 ’ k›A1/›k1 (because kj k) and use the steep descent of the SW spectrum 23 Fj 5 (F0/km)m(k1/km) , m 5 5/2n (see section 3)to evaluate the derivative. We arrive at (A1 2 A2)/(A1A2) ’ 5/2 (7/2)kvm/(mF0)(k1/kn) , which enters the following estimate with k1 ’ km. The IW energy level and frequency is taken in the high-frequency limit En ’ ^ 2 EH(n)mn(f /N0)/k , vn ’ N0k/mn with mn 5 np/b, which then leads to

^ 7 2 f EH(n) vm MI/SC ’ 2 . (C8) 2m p vn F0

The ﬁnal step is implementation of the wind speed

parameterizations vm 5 0.88g/U, F0 5 gbU4 (see section 3), which yields

FIG. C3. Transfer for SC (blue) and MI (red) mechanisms as 7 3 0:88 f EH^ (n) function of mode number and wind speed. The lowest and highest MI/SC U25 5 (U /U)5 . (C9) 5 ’ 2 crit curves are for mode 10 and mode 1, respectively. (top) For N0 pm bvn 23 21 21 5.2 3 10 s and (bottom) for N0 5 0.0157 s . The black dashed 2 7 lines display a U and U dependence. The standard GM parameters and b from (34) yield 21 Ucrit 5 10 m s for the ﬁrst mode. Consistent with the

22 results in Fig. C3, the critical speed depends on the d the SC rate is of order 0.1 mW m for strong winds mode number and increases with it. For the scenario and large N . 0 of W90 and W94, with a higher IW energy and smaller 21 The IW spectrum used by W90 and W94 [Eq. (19) in b, we find a larger Ucrit 5 14 m s . W94] is a high-frequency approximation of the standard 2 2 5 2 5 2 GM [(C6)], valid for k (fm/N) k‘ (fnp/bN0) REFERENCES and k2 m2. Because of the lack of modal normalization and a higher N , we found that the energy of Watson’s IW Alford, M., 2001: Internal swell generation: The spatial distribution 0 of energy flux from the wind to mixed layer near-inertial spectrum is about twice that of (C6). Moreover, problems motions. J. Phys. Oceanogr., 31, 2359–2368, doi:10.1175/ may arise because the approximated spectrum becomes 1520-0485(2001)031,2359:ISGTSD.2.0.CO;2. singular at low wavenumbers. The SW–IW interactions do Cairns, J. L., and G. O. Williams, 1976: Internal wave observations not know about the low-wavenumber limit k‘ of the spec- from a midwater float, 2. J. Geophys. Res., 81, 1943–1950, trum, and thus far too large spectral values can be picked doi:10.1029/JC081i012p01943. Donelan, M. A., J. Hamilton, and W. Hui, 1985: Directional spectra from the approximated spectrum. A further point is that of wind-generated waves. Philos. Trans. Roy. Soc. London, the Donelan form of the SW spectrum, used by Watson A315, 509–562, doi:10.1098/rsta.1985.0054. [Eq. (2.19) of W90], has less energy for the same wind Dysthe, K., and K. Das, 1981: Coupling between a surface- speed than the JONSWAP spectrum (roughly a factor 0.6). wave spectrum and an internal wave: Modulational in- We believe that these are the reasons for the much higher teraction. J. Fluid Mech., 104, 483–503, doi:10.1017/ S0022112081003017. MI rates than SC rates in Watson’s calculations: the re- Eden, C., L. Czeschel, and D. Olbers, 2014: Towards energetically versal between MI and SC takes place at wind speeds consistent ocean models. J. Phys. Oceanogr., 44, 3160–3184, 2 around 15 m s 1 (see below). doi:10.1175/JPO-D-13-0260.1.

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