## 1. Introduction

Despite being a small-scale phenomenon, turbulent mixing shapes regional- to global-scale processes, ranging from the distribution of passive tracers to the driving of the meridional overturning circulation (Munk and Wunsch 1998; Wunsch and Ferrari 2004; Kunze and Llewellyn Smith 2004). It is therefore crucial to find consistent parameterizations for this mixing in ocean models, where it remains unresolved and will presumably continue to be so even in the face of future computer power increases (Eden et al. 2014). Typically, small-scale turbulence is represented by a vertical mixing of fluid properties, often in terms of a vertical diffusivity with a fixed value or varying as a function of depth or stability frequency (Munk 1966; Bryan and Lewis 1979; Cummins et al. 1990). None of these approaches, however, accounts for the energy source for mixing. To overcome this issue, the model Internal Wave Dissipation, Energy and Mixing (IDEMIX), an energetically consistent parameterization for the diapycnal diffusivity induced by breaking internal gravity waves (Olbers and Eden 2013), was developed. We here present a first assessment of its performance.

Internal gravity waves are a ubiquitous feature of the global ocean and can be excited by a fluctuating wind stress at the surface (near-inertial waves), the scattering of the barotropic tide at rough topography (internal tides), the dissipation of mesoscale eddies or the geostrophic adjustment of large-scale disturbances (for a brief review, cf. Müller and Olbers 1975; Garrett and Munk 1979; St. Laurent et al. 2012). A prominent characteristic of internal waves is their continuous energy spectrum both in terms of wavenumber and frequency; with the exception of the near-inertial and the tidal frequency bands, the energy spectrum can be described as a continuum of almost universal shape known as the Garrett–Munk (GM) spectrum (Garrett and Munk 1972). The continuousness of the spectrum suggests an energy transfer in wavenumber space, generally thought to be induced by nonlinear wave–wave interactions (Olbers 1976; McComas 1977). At the high-wavenumber end of the spectrum, the energy is converted to turbulent kinetic energy, which in turn cascades to ever smaller scales until it is dissipated into internal energy or creates potential energy by density mixing.

Not only because of the intricate properties of internal gravity waves, but also for reasons of physical consistency and the fact that observations of small-scale turbulence show strong temporal and spatial variations (Polzin and Lvov 2011; Whalen et al. 2012), a mixing parameterization based on internal wave energetics is desirable. One such attempt was made by Müller and Natarov (2003) with the Internal Wave Action Model (IWAM); its application, however, is severely hindered by the fact that it involves six spatial dimensions. IDEMIX, while also based on the spectral radiation balance for internal waves, avoids this problem by way of integrating in wavenumber space. Assumptions about the effect of the integrated terms as well as a parameterization for internal wave energy dissipation then lead to a single partial differential equation for the internal wave energy. There are, to date, several IDEMIX versions of increasing complexity; in this study, we will mainly focus on the first version as described in Olbers and Eden (2013), in which all internal waves are treated as part of a laterally isotropic continuum.

The aim of this study is to evaluate the model IDEMIX through a comparison with observations. Measurements that resolve turbulence, however, are sparse, especially on longer spatial and temporal scales. We therefore estimate the turbulent kinetic energy (TKE) dissipation rate and diapycnal diffusivity from finestructure data. The underlying concept is that turbulent mixing is the consequence of a nonlinear energy transfer through the internal wave spectrum and that it can thus be described in terms of internal gravity wave energetics (Olbers 1976; McComas and Müller 1981; Henyey et al. 1986; Gregg 1989; Polzin et al. 1995). Assuming furthermore that the observed variance in shear or strain on the 10–100-m scale can be attributed solely to internal gravity waves, more commonly available datasets such as CTD or ADCP casts can be used to infer TKE dissipation rates and diapycnal diffusivities (Kunze et al. 2006; Whalen et al. 2015). A validation of the finestructure method was for example given by Whalen et al. (2015), who compared strain-based finescale estimates to microstructure observations and found that the mean dissipation rates agreed within a factor of 3 for 96% of the comparisons. We here show that this method also allows the estimation of internal gravity wave energy from finescale strain or density information.

This paper is structured as follows: In section 2, we briefly present the model IDEMIX, referring the reader to Olbers and Eden (2013) and Eden and Olbers (2014) for a detailed description. Section 3 deals with the finestructure method and details how to derive TKE dissipation rates and internal wave energy from finescale strain information. These fields are compared to those modeled by IDEMIX in section 4, where we also analyze the sensitivity of this model–data comparison to tuning parameter and forcing settings in IDEMIX. A discussion of these results, involving a sensitivity analysis of the finestructure method, is given in section 5; a summary and concluding remarks are presented in section 6.

## 2. The model IDEMIX

*E*together with a parameterization for its dissipation

*ε*

_{IW}and the definition of bottom and surface forcing terms (Olbers and Eden 2013; Eden and Olbers 2014). The internal wave energy varies according to

*c*

_{0}is a weighted average group velocity,

*z*is the vertical coordinate, and

*τ*

_{υ}is a time scale on the order of days, describing a relaxation toward a symmetric state due to the parameterized effect of nonlinear wave–wave interactions. This equation is obtained from the radiative transfer balance for weakly interacting oceanic internal gravity waves (Hasselmann 1967), making the following assumptions and approximations:

The radiation balance is expressed in terms of the spectral energy and divided into upward- and downward-propagating parts, which are integrated over all horizontal wavenumbers

**k**= (*k*_{1},*k*_{2}) and the appropriate half space of vertical wavenumbers*m*, assuming lateral homogeneity [i.e.,*ω*= Ω(**k**,*m*,*z*)] and that all generation processes are confined to the top and bottom boundaries.Equations for the sum

*E*and the difference Δ*E*of the integrated upward- and downward-propagating energy are expressed and simplified assuming that the dissipation of internal wave energy is symmetric with respect to*m.*The integrated effect of nonlinear wave–wave interactions is assumed to eliminate these differences Δ*E*and is hence parameterized by a relaxation toward a symmetric state with a decay time scale*τ*_{υ}*.*The emerging vertical energy flux terms are expressed in terms of

*E*and Δ*E.*The associated average velocities are set to be equal for upward- and downward-propagating waves,*c*^{+}≈*c*^{−}≈*c*_{0}, and calculated analytically assuming that the energy spectrum can be described by the GM model.Considering time scales much longer than

*τ*_{υ}, the equations for the total and asymmetric energy can be combined to form Eq. (1).

*υ*

_{0}and

*τ*

_{h}denote the lateral group velocity and a time scale on which lateral anisotropies in the wave field are eliminated by nonlinear wave–wave interactions set to 15 days.

*ε*

_{IW}is equated with the energy flux at the high-wavenumber end of the spectrum, where the vertical shear is large and shear instability and convective overturns are most likely to occur. Assuming furthermore that this flux is induced by nonlinear wave–wave interactions, an expression for

*ε*

_{IW}can be found based on a variety of already existing scaling laws (e.g., Olbers 1976) and parameterizations (e.g., McComas and Müller 1981; Henyey et al. 1986; Sun and Kunze 1999; Polzin 2004). For the sake of simplicity, the parameterization implemented in IDEMIX is a combination of the formulations by McComas and Müller (1981), who calculated the energy flux across the high vertical wavenumber cutoff of the GM spectrum induced by parametric subharmonic instability (PSI) and induced diffusion, and by Henyey et al. (1986), who used an Eikonal technique to estimate the energy flux toward high vertical wavenumbers:

*N*

_{0}and

*f*

_{0}and the parameters

*μ*

_{0}=

*μ*/arccosh(

*N*

_{0}/

*f*

_{0}) and

*f*

_{e}=

*f*arccosh(

*N*/

*f*). The quantity

*μ*≈ 2 is a constant from the parameterization by McComas and Müller (1981), so that

*μ*

_{0}= ⅔ for

*N*

_{0}/

*f*

_{0}≈ 10 [note the factor of 2 error in the value for

*μ*used in Olbers and Eden (2013)].

**u**= (

**u**

_{}_{h},

*w*) is the three-dimensional velocity vector, ε

_{TKE}is the dissipation of TKE, and κ is the vertical diffusivity, with overbars denoting the mean quantities and primes the turbulent fluctuations. Assuming that the so-called mixing efficiency is constant (

*δ*≈ 0.2; Osborn 1980) leads to the balance

*κN*

^{2}=

*δε*

_{TKE}, so that the vertical diffusivity

*κ*can be expressed in terms of the dissipation of internal wave energy

*ε*

_{IW}:

*F*

_{surf}represents internal waves radiating out of the mixed layer that is forced by a fluctuating wind stress; following Jochum et al. (2013), it is computed as 20% of the wind input into the near-inertial band in the mixed layer. The bottom energy flux

*F*

_{bot}is estimated as the conversion of barotropic tidal energy into internal wave energy using the parameterization by Jayne (2009), which is based on the barotropic tidal energy and the bottom roughness [cf. Fig. 2 in Olbers and Eden (2013) for global maps of

*F*

_{surf}and

*F*

_{bot}]. Another important source of internal gravity waves is related to the wave field’s interaction with mesoscale features, for example, through the generation of lee waves by mesoscale eddies (Nikurashin and Ferrari 2011). In the ocean model used in this study, their effect is parameterized by along-isopycnal mixing and an additional eddy-driven velocity for tracers, where the corresponding diffusivities are obtained from the closure by Eden and Greatbatch (2008). These lateral diffusivities are the same as those obtained in the parameterization by Gent and McWilliams (1990) and are estimated via a mixing length assumption as well as an assumption for the form of the eddy dissipation based on the dissipation rates in small-scale turbulence; compare Eden et al. (2014) and Eden (2016) for details on the eddy closure and the link between the various parameterizations included in the ocean model. Little is known about the details of the energy sinks of mesoscale eddies; suggested pathways include lee-wave generation by eddy–topography interaction (Nikurashin and Ferrari 2011); generation of ageostrophic instabilities, mainly in the surface mixed layer (Molemaker et al. 2005); or a direct kinetic energy cascade to smaller scales (Capet et al. 2008; Brüggemann and Eden 2015). We here follow the approach by Eden et al. (2014), using the setup for which the best agreement with observations was observed, that is, their scenario CONSIST-SURF. In this experiment, the dissipated eddy energy

*ε*

_{eddy}is partly injected into the internal wave field at the bottom (representing lee-wave generation) and partly into small-scale turbulence at the surface (representing dissipation via ageostrophic instability). The internal wave energy equation thus reads

Newer versions of the model IDEMIX explicitly treat near-inertial waves and internal tides as well as their interaction with the wave continuum (IDEMIX2; Eden and Olbers 2014). For these low-mode compartments, the assumption of lateral isotropy no longer holds. IDEMIX2 thus resolves lateral propagation and refraction by integrating the energy over the wavenumbers 0 ≤ *m* ≤ *m*_{l}, where *m*_{l} is the wavenumber separating the continuum from the low modes, as well as over the local near-inertial frequency band for the near-inertial waves. The interaction of near-inertial waves and the continuum is sufficiently small to be disregarded (cf. appendix 1 in Eden and Olbers 2014); for the M_{2} tidal constituent, wave–wave and topography interaction terms are, for example, derived from Müller and Xu (1992) and Olbers et al. (2012). Since only the M_{2} tide is considered in this model version, 50% of the tidal forcing *F*_{bot} are injected at the bottom to account for the effect of the other tidal constituents.

Note that several other assumptions made in the derivation of IDEMIX1 (e.g., the absence of interior sources) can easily be relaxed and different or additional forcing functions can readily be implemented. In this study, however, we focus on the versions described above.

## 3. Method

### a. Finestructure estimates of turbulent mixing

We closely follow the approach by Whalen et al. (2012) and estimate the TKE dissipation rate from Argo data. The Argo program maintains an array of almost 4000 freely drifting floats that are equipped with CTD sensors, which profile conductivity, temperature, and pressure down to 2000 m every 10 days (https://doi.org/10.17882/42182). We use all profiles from the years 2006–15 that have a quality flag “A” (all real-time quality tests passed) for all three CTD sensors and a vertical resolution of at most 10 m, taking the thoroughly tested and corrected delayed-mode data whenever possible. Starting at the bottom of each profile, we divide these into half-overlapping segments of 200-m length and calculate the buoyancy frequency *N*^{2} based on the adiabatic leveling method as in IOC et al. (2010). Both for CTD profiles as well as these 200-m segments, we apply the same quality control measures as detailed in Whalen et al. (2012).

^{2}10% taper, and spatially Fourier transformed to obtain the power spectra

*m*is again the vertical wavenumber. To correct for the loss of variance due to first differencing (the gradient operator inherent in the computation of

*N*

^{2}), the Fourier amplitudes are divided by the transfer function

*z*is the vertical resolution of the segment and

*λ*

_{z}= 2

*π*/

*m*is the vertical wavelength. They are also divided by

*h*

_{t}is the window function and

*L*is the length of the data window, to correct for the application of the taper (von Storch and Zwiers 2001).

*m*

_{1}= 2

*π*/100 m

^{−1}and

*m*

_{2}= 2

*π*/10 m

^{−1}. As an additional constraint,

*m*

_{2}is adjusted such that

*E*

_{GM}, a wavenumber-dependent function

*A*

_{GM}, and a frequency-dependent function

*B*

_{GM}(Cairns and Williams 1976). Details on these functions and on how to solve Eq. (8) are given in the following section. The GM model parameters used are the idealized profile

*N*=

*N*

_{0}

*e*

^{−z/b}with the scale depth

*b*= 1300 m and the reference buoyancy

*N*

_{0}= 5.24 × 10

^{−3}s

^{−1}; the dimensionless energy level

*E*

_{0}= 6.3 × 10

^{−5}, obtained by scaling the GM energy density

*E*

_{GM}= 3 × 10

^{−3}m

^{2}s

^{−2}by (

*bN*

_{0})

^{2}; the modal bandwidth

*ε*

_{IW}[i.a., those by McComas and Müller (1981) and Henyey et al. (1986), combined in Eq. (3)] to observations. To this end, he replaced the GM energy density

*E*

_{GM}in these laws by the observed energy

*E*

_{IW}, which cannot be measured directly and was hence inferred from the relation

*S*

_{10}is the observed shear measured over 10-m depth intervals, and

*S*

_{GM}is the corresponding value for the GM model. The resultant expression for

*ε*

_{IW}was found to agree within a factor of 2 with microstructure measurements of TKE dissipation rates

*ε*

_{TKE}in the midlatitude thermocline. This motivated the evaluation against and subsequent adjustment to other observations of

*ε*

_{TKE}(cf. Wijesekera et al. 1993; Polzin et al. 1995; Gregg et al. 2003; Kunze et al. 2006) and the application of the refined expression as a parameterization of TKE dissipation rates. The formulation used in this study is given by

*ε*

_{0}= 6.73 × 10

^{−10}W kg

^{−1}(Whalen et al. 2012). The function

*L*

_{f}(

*f*,

*N*) is a latitudinal correction for the dependence of internal wave characteristics on the Coriolis frequency

*f*:

*f*

_{30}is the Coriolis frequency at 30° latitude. The function

*h*(

*R*

_{ω}) accounts for the frequency content of the internal gravity waves:

*R*

_{ω}is the shear-to-strain ratio, which is the ratio of horizontal kinetic energy and available potential energy; because of the lack of shear information, it has to be set constant. Following Whalen et al. (2012), we use the GM model value of three, which is a reasonable estimate in the upper ocean (Kunze et al. 2006) and reduces the function

*h*(

*R*

_{ω}) to unity. If

*R*

_{ω}underestimates the actual shear-to-strain ratio,

*h*(

*R*

_{ω}) is also too small and vice versa.

*δ*= 0.2, the diffusivity is finally estimated as

### b. Internal wave energy

*E*(

*z*) is computed from vertical spectra of strain and potential density assuming that the total energy spectrum

*S*

_{E}features the same wavenumber and frequency dependence as the GM energy spectrum:

*m*

_{h}and

*m*

_{l}are high- and low-wavenumber cutoffs, respectively.

^{1}The terms

*n*

_{A}and

*n*

_{B}are normalization factors defined such that

*A*

_{GM}and

*B*

_{GM}integrate to unity and

*ξ*

_{z}or potential density

*ρ*′ can be found (e.g., Willebrand et al. 1977; Munk 1981; Olbers et al. 2012):

*ρ*′ =

*ρ*−

*ρ*

_{fit}is the potential density perturbation,

*ρ*

_{0}= 1027 kg m

^{−3}is the constant background density, and

*ρ*

_{fit}is a vertical fit to the data within each 200-m segment (equivalent to

*N*

_{fit}). To calculate the total internal wave energy

*E*(

*z*) from Argo data,

*S*

_{E}(

*m*,

*ω*) is expressed using Eq. (13), and the above equations are integrated over all frequencies and a suitable wavenumber range (the same as considered for the dissipation rate estimates, i.e., between

*m*

_{1}= 2

*π*/100 m

^{−1}and

*m*

_{2}= 2

*π*/10 m

^{−1}):

*G*and

*C*

_{1}describe the integrated frequency and wavenumber part of Eq. (16), respectively, and are given in the appendix. When considering density variance, the procedure is the same but for the function

*C*

_{1}, which changes due to the different wavenumber dependence in Eq. (17) (cf. appendix for details).

Note that the frequency-dependent part of the GM model [cf. Eq. (14)] is zero at the equator, while it yields nonzero results upon integration over all frequencies [cf. Eq. (A1)]. Because of this discrepancy between the integrated and frequency-dependent GM model formulations, the dissipation rate and energy estimates close to the equator need to be treated carefully. For the comparison with IDEMIX we therefore only consider latitudes higher than three degrees.

## 4. Results

In this section, we present global maps of dissipation rates and energy levels obtained from Argo data and the evaluation of the parameterization IDEMIX based on these finestructure estimates. A discussion of these results is provided in section 5.

### a. Argo-based estimates of dissipation rates and internal wave energy

Global maps of average TKE dissipation rates estimated from finestructure strain data [analogous to Fig. 1 in Whalen et al. (2012)] are depicted in Fig. 1. We consider the depth ranges 250–500, 500–1000, and 1000–2000 m, where especially in the Atlantic and the Southern Ocean data coverage strongly decreases with depth. In each depth range, the spatial variations in dissipation rates span at least three orders of magnitude. Strong mixing is observed in the western boundary currents and regions of high mesoscale activity, particularly in the subtropical gyres and the western boundary of the Pacific Ocean and in the Antarctic Circumpolar Current (ACC). The former signal is most prominent close to the surface, while mixing hot spots like Drake Passage or the Kuroshio and Gulf Stream, as well as their extensions, can be identified as such at all depths. In general, a decrease in dissipation rates of about an order of magnitude between the upper and lower depth range can be observed, although locally sometimes the inverse is the case.

Internal gravity wave energy is shown in Fig. 2 for the depth range 250–500 m. Highly energetic regions with values of more than 10^{−2} m^{2} s^{−2} (i.e., 3 times higher than the GM model value) are the subtropical northwest Pacific, the wind-driven gyres of the Pacific and the tropics. In general, energy levels decrease by about two orders of magnitude from the equator to the poles. Longitudinal gradients are smaller and exist only outside of the tropics, where energy levels decrease slightly from west to east in all ocean basins. These signals are the same at all depths. On average, energy levels *E*_{ρ}) are 2.0 (3.5) times higher in the upper depth range than in the lower one, but energy levels increasing with depth can also be observed at some locations. Depending on the depth range, energy levels computed from potential density spectra are up to a factor of 2 higher than those calculated from strain spectra.

### b. Evaluation of IDEMIX

IDEMIX is coupled to the primitive equation model pyOM, in which energy is exchanged consistently between the mean flow, parameterized mesoscale eddies, TKE, and, in the form of IDEMIX, internal gravity waves [cf. Eden et al. (2014), Eden (2016), section 3 for details on the model, and https://wiki.zmaw.de/ifm/TO/pyOM2 for the documentation and code]. We use a horizontal resolution of 1° with 115 vertical levels and run the full model until it reaches a dynamic equilibrium (about 200 years). The standard tuning parameter settings are *μ*_{0} = ⅓, *τ*_{υ} = 2 days, and

#### 1) Global maps

The TKE dissipation rates modeled by IDEMIX are shown in Figs. 3a–c. They vary horizontally by three orders of magnitude, with maximum values in the western boundary currents of all three ocean basins. Elevated dissipation rates can be observed near the equator and in parts of the ACC as well as in the central Atlantic and near the continental margins where the tidal forcing signal becomes apparent. The dissipation rates are modeled to be weakest over wide areas of the central and eastern ocean basins and in parts of the Southern Ocean. They decrease on average by a factor of 3 between the upper and lower depth range. A qualitative comparison to Fig. 1 demonstrates that IDEMIX reproduces the mixing patterns obtained from Argo data; overall, the spatial variations and their magnitude agree well. However, the different mixing hot spots are more spatially confined (e.g., the western boundary currents and their extensions), and the large areas of high dissipation rates observed in the subtropical gyres of the western Pacific are but partially mirrored in IDEMIX. Moreover, IDEMIX simulates higher dissipation rates at the continental margins and lower dissipation rates in regions of weak mixing, for example, south of Australia or in the central to eastern North Atlantic, particularly in the upper ocean. As a consequence, horizontal correlation coefficients are rather low, varying between 0.1 and 0.2 below and above 1000 m, respectively (cf. Table 1). In a regional comparison, values of up to 0.4 are found in the Southern Ocean and the northwest Pacific (120°–180°E, 3°–39°N), while correlation coefficients remain below 0.1 at all depths in the North Atlantic (3°–78°N, 75°W–30°E). In a global comparison, 60%–75% of the data agree within a factor of 3 with the Argo based estimates (cf. Table 2). Comparable (and higher) values are determined for the Southern Ocean and the northwest Pacific, while they remain below 60% at any depth in the North Atlantic. Differentiating between regions of high and low dissipation rates (taking a threshold value of *ε*_{TKE,crit} = 3 × 10^{−9} W kg^{−1}), IDEMIX is seen to perform better in regions of weak dissipation: In a global comparison, about 70% of the data agree within a factor of 3 above 1000 m (where there are a significant number of estimates available for both scenarios), compared to about 50% of the data in regions of strong dissipation. On regional scales, the behavior is similar, with the noteworthy exception of the Southern Ocean, where above 500 m about 70% of the data agree within a factor of 3, both in the weak and strong dissipation scenarios.

Coefficients of horizontal correlation between IDEMIX- and Argo-based estimates. The upper depth range is 250–500 m, the middle 500–1000 m, and the lower 1000–2000 m. The scenario *F*_{eddy} = 0 corresponds to setting 0.2*ε*_{eddy} = 0 in Eq. (6). The area between 3°S and 3°N is excluded from this otherwise global comparison because of a singularity of the GM model at the equator.

As in Table 1, but for percentage of IDEMIX data agreeing with Argo-based estimates within a factor of 2 (in parentheses) and 3.

The performance of IDEMIX is even better with respect to internal wave energy levels (cf. Figs. 3d–f; note that we consider *E*_{ρ} in order to base this model–data comparison on the same spectra as for dissipation rates): the magnitudes as well as the spatial variation characterized by a high equator to pole and, at higher latitudes, a weaker west-to-east gradient are well reproduced. In detail, however, there are some shortcomings: similar to the case of TKE dissipation rates, regions of high energy levels—particularly the strong signal in the central and eastern subtropical Pacific—are often smaller than in the Argo-based global maps and at the continental margins, energy levels are higher than estimated from the finestructure data. Both the qualitative and the quantitative agreement with Argo data is better than that of dissipation rates: Horizontal correlation coefficients amount to around 0.7 in the upper and middle and 0.3 in the lower depth range, and in all depth ranges, more than 90% of the energy estimates agree within a factor of 3 with the Argo-derived values (cf. Tables 1 and 2). This also holds true in a regional comparison, where the best quantitative agreement is again observed in the northwestern Pacific Ocean; the best qualitative agreement is achieved in the global comparison as well as in the upper North Atlantic. The performance of IDEMIX is equally good for regions of high and low energy (setting ^{2} s^{−2} as the threshold value), with more than 90% of the Argo and IDEMIX data agreeing within a factor of 3 in all depth ranges in a global and a regional comparison.

The quantitative differences between Argo- and IDEMIX-derived TKE dissipation rates and energy levels discussed above are supported and elucidated further by a comparison of the respective distributions of the estimates (cf. Fig. 4 for the upper depth range); the two curves overlap most strongly in the northwest Pacific, and the overlap is generally larger for energy levels than for dissipation rates. Typically, the distribution of Argo-based estimates is shifted toward lower values compared to that of the IDEMIX-derived estimates and features a less sharp peak, which also holds true in the center and lower depth range (not shown). The distribution of IDEMIX values is in some cases characterized by a second, albeit much smaller, maximum, which is not observed in the more smoothly varying Argo distribution.

Figure 5 illustrates how dissipation rates derived from Argo data (Fig. 5a) and computed by IDEMIX (Fig. 5b), respectively, vary with depth in the subtropical Pacific Ocean, where the number of available Argo estimates is highest. The spatial variation computed by IDEMIX can be seen to reproduce the high dissipation rates estimated from Argo data at the surface with the maximum at around 180°–200° longitude and with elevated mixing rates reaching farther down at around 150°–160° and 180°–200° longitude. The low dissipation rates east of 200° longitude in the Argo-based estimates, reaching closer to the surface farther eastward, are also simulated by IDEMIX. Additional biases arise in terms of magnitudes and spatial gradients, which are weaker in IDEMIX than in the finestructure data, and in terms of the detailed spatial pattern: for example, the two streaks of high dissipation rates reaching down from near the surface at 150°–160° and 180°–200° longitude are more pronounced and extend deeper down in IDEMIX than in the Argo-derived map. Moreover, the strong signal in the Argo-derived dissipation rates near 210° longitude around 1000-m depth, which can be linked to the vigorous tidal forcing at the Hawaiian Ridge, is not reproduced by IDEMIX. At other latitudes and in other ocean basins, for example, in the North Atlantic, IDEMIX performs much worse with respect to vertical variations of dissipation rates and energy levels (not shown).

#### 2) Sensitivity analysis

In this section, we describe the sensitivity of how well IDEMIX reproduces Argo-based estimates of *ε*_{TKE} to the model’s tuning parameters and forcing settings. The analysis was carried out using the same model setup as described in the section above, but in order to save computational time, the resolution was set to 2.8° × 2.8° in the horizontal with 45 vertical levels. Note that we do not explore the three-dimensional parameter space but keep two parameters at their reference value—as before, *μ*_{0} = ⅓, *τ*_{υ} = 2 days—while varying the third. Representative values for the global ocean range between *τ*_{υ} ≤ 10 days (Olbers 1974). The values of *μ*_{0} = *μ*/[arccosh(*N*_{0}/*f*_{0})], with *μ* ≈ 2, tested here correspond to ratios *N*_{0}/*f*_{0} ranging between 1.4 for *μ*_{0} = 2 and 10^{3} for *μ*_{0} = ⅙. This is an adequate representation of the Argo-based buoyancy frequency estimates (not shown), which reach values of *N*_{0}/*f*_{0} ≈ 10^{2} in the tropics and subtropics above 500 m, while they amount to *N*_{0}/*f*_{0} ≈ 10 in the middepth range (500–1000 m) and remain around unity below.

We investigate both the qualitative and quantitative agreement between Argo- and IDEMIX-based estimates in terms of horizontal correlation coefficients and the percentage of data agreeing within a factor of 3 (cf. Fig. 6 for internal wave energy). The quantitative agreement is best for intermediate values of *μ*_{0} and *τ*_{υ}. For the standard settings *μ*_{0} = ⅓, the agreement is not only high or maximized, but also almost the same in all depth ranges. Increasing *μ*_{0} above their standard values affects the agreement most strongly in the upper and least so in the lower depth range. Correlation coefficients, on the other hand, are barely affected by the investigated variations of *μ*_{0}, or *τ*_{υ}. A clear distinction with depth is observed, with values of around 0.3 in the lower and around 0.6–0.7 in the upper and middepth range. Intermediate values of *μ*_{0} as well as low values of *τ*_{υ} yield the strongest horizontal correlation. In terms of TKE dissipation rates (not shown), the quantitative agreement is less sensitive, especially with respect to variations of *μ*_{0}. Correlation coefficients exhibit a stronger sensitivity than for energy, particularly in the upper depth range. The best agreement is not always achieved for the same parameter settings as for energy and not always for the same settings at all depths.

The sensitivity of the quantitative and qualitative agreement between Argo- and IDEMIX-based dissipation rate and energy level estimates to different (forcing) scenarios is enlisted in Tables 1 and 2. Taking 10% or 30% instead of 20% as the amount of the near-inertial energy that leaves the mixed layer and acts as *F*_{surf} on the internal wave field is still well within the range given by Furuichi et al. (2008) or Alford et al. (2012); the other scenarios, however, are arbitrarily chosen to illustrate the general influence of the respective forcing term. Most of the variations, such as using IDEMIX2 instead of IDEMIX1 or halving the surface or bottom forcing, barely affect the agreement with the Argo-derived estimates (albeit slightly more on regional and seasonal scales; not shown). The strongest impact is observed when removing either the eddy or the boundary forcing completely or, in the case of dissipation rates, removing the lateral diffusion term in Eq. (6). In those cases, 10%–50% fewer data points than in the reference scenario agree with the Argo results within a factor of 3—except for energy levels in the upper (middle) depth range, where the qualitative (quantitative) agreement is actually slightly improved. Note that these results depend on the reference parameters used: taking the standard settings of Olbers and Eden (2013), that is, *μ*_{0} = *τ*_{υ} = 1 day, as the reference, the removal of the eddy forcing term in Eq. (6) leads to 4–5 times fewer energy data points agreeing with the finestructure estimates within a factor of 3 than in the reference case (not shown). The horizontal correlation coefficients (cf. Table 2) are less sensitive to variations in the forcing settings, especially with respect to energy levels. Again, the absence of the eddy or boundary forcing or of the lateral diffusion term most strongly affects the agreement with the Argo-based estimates.

The effect of the eddy and the boundary forcing is illustrated in Fig. 7 for the upper depth range of 250–500 m. Without the eddy forcing (Fig. 7c), the high dissipation rates observed in the western boundary currents are not fully reproduced. This holds especially true for the Atlantic Ocean, where TKE dissipation in the Gulf Stream and its extension is modeled to be quite weak when using bottom and surface forcing only. Also for the mixing hot spots in Drake Passage and the Agulhas Current, the eddy forcing is of central importance (Fig. 7b). A comparison to Fig. 7c also underlines that dissipation rates are too low in the ACC and the eastern Pacific when the transfer of mesoscale eddy energy to the internal wave field is not accounted for. In the middle and lower depth range, the influence of the mesoscale eddy forcing is less pronounced (not shown). The high dissipation rates observed at the continental margins, for example, in the North Pacific or around Antarctica, and in the central and western Pacific are almost exclusively related to the bottom and surface forcing (cf. Fig. 7c).

## 5. Discussion

### a. Uncertainty of the finestructure estimates

To assess the uncertainties of the evaluation of IDEMIX, the uncertainties inherent in the finestructure method need to be examined. Whalen et al. (2015) compared dissipation rates and diffusivities from microstructure profiles from six different campaigns, representing diverse environments and open-ocean conditions, to Argo-based finestructure estimates. They found a factor of 2 agreement for 81% and a factor of 3 agreement for 96% of the data. In a locally more confined comparison, Sheen et al. (2013) report a systematic overprediction of dissipation rates estimated from CTD and LADCP finestructure data collected in the ACC during the Diapycnal and Isopycnal Mixing Experiment in the Southern Ocean (DIMES) compared to microstructure observations from the same campaign, ranging from a factor of 2 to 4 for transect mean values (cf. their Tables 1 and 2). Factor of 4 differences, however, are mainly found in the bottom kilometer, where Argo floats do not reach; farther up in the water column, the uncertainty estimated by Sheen et al. (2013) agrees with the factor of 2–3 identified by Whalen et al. (2015). The sensitivity of microstructure measurements to sensor calibration, response functions, and, most importantly, the assumption of isotropic turbulence is specified as no more than factor of 2 for values greater than the noise level by Toole et al. (1994) and Moum et al. (1995) but can under some conditions become as large as a factor of 3–5 (Mashayek et al. 2013).

The uncertainty found by Whalen et al. (2012) and Sheen et al. (2013) gives an idea of how well the local internal wave field adheres to the characteristics it is assumed to have in the finestructure method, and if the measures taken to correct for those cases when it does not are sufficient. Such a case was, for example, observed by MacKinnon and Gregg (2003) on the New England shelf, where internal wave characteristics were significantly different from the well-defined, slowly evolving shape typically assumed in spectral models. Polzin et al. (2014) in detail discuss the biases due to the assumptions inherent in the finestructure method (such as spatial homogeneity or a vertical scale separation between a time-mean and a finescale variability in the buoyancy gradients) and practical issues (such as instrument noise and lack of resolution), concluding that the total bias of the finestructure method should be “substantially less than an order of magnitude over much of the ocean” (Polzin et al. 2014, p. 1414) if it was implemented with care. To elucidate this further, we analyze the method’s sensitivity to the parameter settings involved by investigating how dissipation rate and energy estimates vary in the Atlantic Ocean when our standard settings are modified. These are a combination of the settings by Whalen et al. (2012) and Kunze et al. (2006): we set *N*^{2}, the GM model version as described by Cairns and Williams (1976) with a wavenumber dependence proportional to *R*_{ω} = 3, *λ*_{min} = 10 m, and *λ*_{max} = 100 m. We change one of these parameters at a time and perform a Welch’s *t* test to determine if dissipation rates and energy levels significantly deviate from those obtained in the reference scenario for a significance level of *α* = 0.05. Note that we do not seek to suggest different settings for the finestructure strain method than the standards developed and tested during the last decades but to evaluate the significance of our model–data comparison.

Figure 8 shows the average dissipation rate and energy level calculated from Argo data from the year 2011 in the depth ranges 250–500, 500–1000, and 1000–2000 m in the Atlantic Ocean, when one of the parameters enlisted above is changed while the rest are kept at their reference value. Independent of depth, significant deviations from the reference dissipation rate are obtained when taking an earlier GM model version with a wavenumber dependence proportional to *R*_{ω} = 2 or *R*_{ω} = 5 [in the upper 2000 m, Kunze et al. (2006) observe values between 2 and 7 and even higher values in the Southern Ocean; cf. their Figs. 13–16]. Most of the other parameter variations cause significant deviations from the reference dissipation rate only in the uppermost depth range. The strongest deviation is a factor of 2 difference observed in the 1000–2000-m depth range for the scenario with

Energy estimates obtained from finescale strain spectra are less sensitive to the parameter variations investigated. The only exception is the factor 3–4 decrease in energy in the upper and lower depth range when taking GM75m instead of GM76. In all other cases, the observed differences are well below a factor of 2 and only few of them are significant according to the Welch’s *t* test, excluding the GM75m scenario in the upper depth range. The only scenario inducing significant changes independent of depth is the one with

This uncertainty due to parameter choices and different modeling approaches becomes apparent in a comparison of our dissipation rate estimates (Fig. 1) to the global maps published by Whalen et al. (2012, 2015). We observe the same order of magnitude variations and the same spatial pattern, but our dissipation rates (and diffusivities, not shown) are higher almost everywhere. These differences are smaller than the combined uncertainties described in this section and generally lie within the 90% bootstrapped confidence intervals given by Whalen et al. (2012, cf. their Fig. S4) for their averages of the years 2006–11. Inferring from their Fig. S3 that we keep significantly more data points even for the same parameter settings and input data (not shown), we link these differences not only to the disparate parameter choices (e.g., resolution or GM model version) but also to potential differences in data quality requirements and details of the data processing (e.g., despiking) or averaging routines used.

We assess the statistical uncertainty by computing 90% bootstrap confidence intervals derived for 1000 samples for all bins with at least 10 individual dissipation rate or energy estimates. For both variables, the variation decreases with depth, ranging between 60% and 80% of the mean for dissipation rates, 20% and 30% for energy levels obtained from strain and 20% and 40% for those obtained from potential density spectra. For the evaluation of IDEMIX, we focus on strain-derived energy levels in order to use the same spectra as for the comparison of dissipation rates. These energy levels are on average 1.5–2 times lower than those derived from potential density spectra (cf. Fig. 2 for the 250–500-m depth range). This difference gives a rough idea of the uncertainty of our approach to compute energy levels from finescale strain information, which involves much less testing and refinement than what the finestructure method for turbulent dissipation rates has undergone in the last decades. The decrease of the energy levels with depth mirrors the proportionality of wave energy *E*(*z*) to a decreasing buoyancy frequency *N*(*z*), as found for internal waves with the WKB approximation.^{2} Dissipation perturbs this linear wave behavior, but the Argo-derived energy levels still pick up this dependence [the same holds true for wave energy in IDEMIX, where the decrease with depth was shown to be independent of the location (top or bottom) of the forcing; see Olbers and Eden (2013)]. The resemblance of the horizontal variations of our finestructure energy estimates to global patterns of wind power input into near-inertial motions and corresponding horizontal energy fluxes (Alford 2003; Furuichi et al. 2008; Alford et al. 2016) and of energy fluxes from geostrophic motions to internal lee waves (Nikurashin and Ferrari 2011) as well as the general agreement in terms of magnitude with the observations based on CTD and LADCP data in the Southern Ocean by Waterman et al. (2013) renders us confident that our finestructure energy estimates are sufficiently reliable for the purpose of this study. The combined uncertainty resulting from procedural (strain or density based) and statistical issues thus amounts to a factor of 1.7 in the middle and up to a factor of 2.5 in the upper and lower depth range.

We estimate the overall uncertainty of our finestructure estimates as the sum of the different uncertainties described in this section, arguing that the factor 2–3 uncertainty identified by Whalen et al. (2015) in a comparison with microstructure estimates should to a large extent represent the uncertainty related to the method’s sensitivity to parameter settings discussed above (these could have been adjusted to improve the agreement with the microstructure estimates). For dissipation rate estimates, we therefore consider 1) the uncertainty of microstructure measurements, 2) the difference between finestructure and microstructure estimates, and 3) the statistical uncertainty. In the case of energy level estimates, we consider 1) the parameter sensitivity, 2) the difference between the two methods proposed (*E*_{ρ} vs

### b. Uncertainty of IDEMIX

The assumptions made during the derivation of IDEMIX do not necessarily hold everywhere in reality. Without detailed knowledge of the energy spectra in the ocean, these uncertainties cannot be quantified, but they should nevertheless be noted. One important aspect is that IDEMIX—just like the finestructure method—relies on oceanic conditions being close to those assumed in the GM model (e.g., in the computation of the representative group velocity *c*_{0}). Another source of uncertainty is the parameterization for the dissipation of internal wave energy *ε*_{IW} based on the scalings by Olbers (1976), Henyey et al. (1986), and McComas and Müller (1981), which might neglect processes that are important for the internal wave energy cascade in the real ocean. Moreover, the assumption that the nonlinear wave–wave interactions render the wave field symmetric with respect to *m*, made in the derivation of IDEMIX, might not be justified under all conditions. The same holds true for the assumption of vertical symmetry, allowing for the approximation *c*^{+} ≈ *c*^{−} ≈ *c*_{0} (cf. section 2), or, in IDEMIX2, that properties of the first baroclinic mode are representative of the entire wave field, but these are minor issues in comparison. In addition, the Osborn–Cox relation used to link internal wave energy dissipation to TKE dissipation is a reasonable approximation in the stratified interior of the ocean but less so near the boundaries.

The type and characteristics of the forcing functions included in IDEMIX could also lead to significant errors. The models and parameterizations used to compute the surface and bottom forcing (Jayne 2009; Jochum et al. 2013) suffer themselves from biases and might be too simplified under certain conditions. For example, the surface energy input in IDEMIX is set to zero near the equator because the approach of Jochum et al. (2013) to estimate the inertial velocity components is only valid outside the deep tropics (cf. Fig. 2 in Olbers and Eden 2013). The global maps of *F*_{surf} and *F*_{bot} are obtained from model simulations with the associated numerical biases and uncertainties introduced by the various parameterizations involved, and additional biases arise when extrapolating these results to the numerical grids used in the ocean model IDEMIX is coupled to. The settings chosen in the simulations presented here are also biased: the energy content of the M_{2} tidal constituent, set to 50% of the total tidal forcing in IDEMIX2, is probably an overestimate (Falahat et al. 2014). Moreover, the fraction of the wind power input into near-inertial motions that leaves the mixed layer is not a global constant (here, 20% are used) but varies in both space and time (Furuichi et al. 2008; Alford et al. 2012; G. Voelker et al. 2016, unpublished manuscript). The missing implementation of physical processes other than near-inertial wind forcing (*F*_{surf}), internal tide generation (*F*_{bot}), and the formation of lee waves by mesoscale eddies (*F*_{eddy}) also contributes to the model’s biases. We will address these issues in detail in the conclusions.

### c. Evaluation of IDEMIX

Comparing the global maps of TKE dissipation rates and internal wave energy levels shows that IDEMIX reproduces both the spatial pattern and the magnitude of the Argo-derived finestructure estimates. This comparison is most reliable in the upper depth range, where biases due to missing Argo data are smallest, as well as away from the continental margins or the equator. At these locations, the limited applicability of the finestructure method and of assumptions made in the derivation of IDEMIX or the discrepancy between the frequency-dependent and the integrated GM model increase the uncertainty of the Argo- and IDEMIX-based estimates.

The model–data agreement is better for energy levels than for dissipation rates, both in a qualitative and in a quantitative sense. The sensitivity of the quantitative and qualitative agreement to the model’s tuning parameters also differs. This is somewhat surprising, considering that in IDEMIX dissipation rates are computed from energy levels and both variables are related via the same formula that forms the basis of the finestructure method [cf. Eq. (3)]. We surmise that two aspects contribute to this issue: First, dissipation rates are described by the amount of energy leaving the internal wave spectrum and are thus—in an energetically consistent framework—dependent on the amount of energy entering the internal wave field, that is, the external forcing functions. Energy levels, on the other hand, are directly influenced by the local characteristics of the internal wave field and hence the model’s tuning parameters. Second, the much higher amount of testing and refinement inherent in the finestructure method for TKE dissipation rates compared to our method for the calculation of internal wave energy could also explain to some degree why IDEMIX performs differently with respect to energy.

Particularly the quantitative agreement between Argo- and IDEMIX-based estimates of energy levels and TKE dissipation rates is most sensitive to changes in the tuning parameter *τ*_{υ} or *μ*_{0} (nor for any realistic setting with respect to TKE dissipation rates because of their reduced sensitivity to tuning parameter variations). The improvement of the current reference settings (*τ*_{υ} = 2 days, and *μ*_{0} = 1/3) over the ones used in Olbers and Eden (2013) is at most a factor of 4 (for energy levels in the upper depth range) and hence not significant either. The same holds true for the changes induced by varying the forcing functions when their effect on the global averages is considered (cf. Tables 1 and 2); locally, however, these changes can be significant. Considering a box encompassing the Agulhas Current (30°–60°S, 10°–40°E), TKE dissipation rates are found to be decreased by a factor of 6–7 (depending on the depth range considered) compared to the reference scenario when the eddy forcing is removed. The absence of bottom and surface forcing barely affects the modeled TKE dissipation rates in this region, identifying the eddy forcing as the main contributor to the high observed dissipation rates. Energy levels are only decreased by a factor of 2 in the area around the Agulhas Current for *F*_{eddy} = 0, which supports our interpretation that TKE dissipation rates are mainly affected by the forcing functions, while energy levels are mainly influenced by the local shape of the internal gravity wave field and thus the model’s tuning parameters. In Drake Passage (50°–70°S, 50°–70°W), on the other hand, both the eddy and the surface and bottom forcing are significant above 1000-m depth, with a reduction of TKE dissipation rates by a factor of 10–13 for *F*_{eddy} = 0 and a factor of 7–8 for *F*_{bot} = *F*_{surf} = 0 compared to the reference scenario.

Note that the sensitivity analysis was carried out using a coarser resolution to save computation time. This does not affect the quantitative agreement with the Argo-based estimates, but the qualitative agreement is modified, especially with respect to dissipation rates (cf. Table 2). This could be related to the higher amount of small-scale structures resolved in the forcing functions of the 1° simulation; nevertheless, we expect a sensitivity analysis to yield comparable results to that using a 2.8° resolution since the impact of the tuning parameters or the forcing functions on the internal wave field is not resolved in any of the two and always parameterized in the same way.

For most modeling purposes, the parameter of interest is typically the diapycnal or vertical diffusivity rather than the turbulent dissipation rate. In general, IDEMIX can reproduce diffusivities equally well as dissipation rates with two major exceptions: in the Atlantic Ocean at high northern latitudes (≥60°) and in the Southern Ocean, diffusivities are much larger than suggested by finestructure estimates (not shown). This can be linked to a much too low buoyancy frequency in the model at these locations and is thus not a shortcoming of IDEMIX but of the ocean model it is coupled to. Note that when the diapycnal diffusivity is concerned, an additional bias arises due to the mixing efficiency *δ*, which is treated as constant both in IDEMIX and in the finestructure method, but has been found to be variable (cf. Gargett and Moum 1995; Mashayek et al. 2013). Another point worth noting is the well-known issue of resolving characteristic ocean features in numerical models. The weak agreement between IDEMIX- and Argo-based estimates in the North Atlantic could not only be related to the wave field’s local divergence from the global reference assumed in IDEMIX but potentially also to an inadequate representation of the Gulf Stream path to which the ocean model IDEMIX is coupled (cf. Chassignet and Marshall 2008).

## 6. Summary and conclusions

We here present a first evaluation of the mixing parameterization IDEMIX, which describes the propagation and dissipation of internal gravity wave energy in the ocean and computes the induced diapycnal diffusivities in an energetically consistent framework. The evaluation is based on a comparison with TKE dissipation rate and energy-level estimates obtained from Argo–CTD profiles; to our knowledge, ours is the first attempt to calculate energy levels from finescale strain information alone. The Argo program maintains a nearly global array of a few thousand freely drifting floats profiling the ocean’s upper 2000 m several times a month, producing a database that is well suited for the evaluation of IDEMIX in its typical application, that is, coupled to a global ocean model. The drawback of this approach is the high uncertainty associated with these finestructure estimates, which we estimate as a factor of 5. In our evaluation, we therefore only consider the large-scale variations of the TKE dissipation rate and internal wave energy fields, which cover two to three orders of magnitude.

These large-scale signals can be seen to be well reproduced by IDEMIX: regions of particularly high or low dissipation rates or energy levels are identified as such and the corresponding magnitude is usually well simulated. Discrepancies with the Argo-based estimates are mainly related to the spatial extent of these hot spots. The agreement with Argo-derived estimates differs regionally, particularly with respect to the simulation of vertical variations or the detailed data distribution. In light of the high uncertainty of the finestructure method, tuning IDEMIX to the Argo-derived estimates in order to overcome these discrepancies is not an option. Instead, we draw the following conclusions from the results presented in section 4:

The internal wave field is spatially inhomogeneous and hence not represented equally well everywhere by a global set of parameters. This could be improved by defining regionally variable parameters such as the modal bandwidth

(Polzin and Lvov 2011), but that kind of detail is so far precluded by the lack of a unifying theory. The difference between the model versions IDEMIX1 and IDEMIX2 in terms of reproducing the Argo-based dissipation rates or energy levels is very small. The additional computational power required by adding low-mode compartments to IDEMIX1 is therefore not necessary if only a general reproduction of the Argo-derived estimates is desired. Locally and also seasonally, the differences between IDEMIX1 and IDEMIX2 are more pronounced, especially in the upper ocean. None of these improvements is significant within the uncertainty of the method, but they illustrate that a realistic simulation of the detailed structure of internal wave energy and its dissipation requires the simulation of more processes than are currently considered in IDEMIX1 (and presumably IDEMIX2). In this context, the role of parametric subharmonic instability, which is modeled in IDEMIX2 (cf. appendix A of Eden and Olbers 2014), but not in IDEMIX1, can also be evaluated. This triad interaction transfers energy from a low-wavenumber component to two high-wavenumber components of half the frequency and has been suggested to substantially shape the internal wave energy budget and potentially also turbulent mixing in several numerical studies (cf. Hibiya et al. 2002; Furuichi et al. 2005; MacKinnon and Winters 2005). Observational studies, however, reach diverging conclusions, with some supporting the importance of PSI (e.g., Nagasawa et al. 2002), while others stress the minor effect of PSI on internal wave energy levels (MacKinnon et al. 2013; Zhao and Alford 2009). Additional uncertainties arise because of potentially misleading results produced by bispectrum and bicoherence estimators, typically applied to infer the presence of PSI (Chou et al. 2014), as well as resolution or dimensionality limitations of the above named numerical studies. The comparison of IDEMIX1 and IDEMIX2 indicates that with respect to the reproduction of Argo-derived dissipation rate or energy-level estimates, the role of PSI could be important locally but is negligible on the large scales analyzed here.

The different forcing functions included in IDEMIX are of varying importance in different parts of the global ocean. In most areas, mixing hot spots are induced by a combination of strong eddy and boundary (wind and tidal) forcing. In the central subtropical Pacific or the northern Indian Ocean, however, it is the boundary forcing alone that causes elevated dissipation rates and energy levels, while for example in the Gulf Stream or the Agulhas Current, the high values are brought about mainly by the mesoscale eddy forcing. In the vicinity of these currents, the absence of the eddy forcing term in IDEMIX causes significant deviations of TKE dissipation rates from the reference scenario, underlining that in these areas, the finestructure estimates cannot be reproduced without taking the energy transfer from mesoscale eddies to the internal gravity wave field into account. The different effect of removing this eddy forcing term in the three depth ranges considered suggests that the interaction between internal gravity waves and the mesoscale as well as its role for mixing differs depending on where in the water column it takes place. This is supported by the findings of Eden et al. (2014), who observed differences in Southern Ocean diffusivities depending on whether mesoscale eddy energy was transferred to internal gravity waves mainly in the mixed layer, at the bottom, or in the ocean’s interior. It is hence crucial to better understand where and how mesoscale eddies dissipate and which processes shape their interaction with the internal gravity wave field in order to realistically implement this forcing term in IDEMIX and to help reduce the bias between the model and Argo-based estimates.

The relative probability distributions for Argo- and IDEMIX-based estimates of dissipation rates and energy levels overlap to different degrees in different parts of the ocean. Moreover, the distribution for Argo-derived estimates varies more smoothly and peaks at lower values than the one describing IDEMIX-based estimates. The forcing mechanisms incorporated in IDEMIX might hence be to different degrees representative of the real forcing processes in different parts of the ocean, both with respect to their magnitude as well as their regional structure. For example, IDEMIX seems to capture much of the forcing processes at work in the real ocean in the northwest Pacific but falls short in the North Atlantic. The results presented in Tables 1 and 2 underline that decreasing the magnitude of the forcing—an apparent possibility to reduce the shift between the maxima of the distributions for Argo- and IDEMIX-based estimates—barely affects the model–data agreement. Rather, as the generally smoother distribution of the finestructure estimates suggests an interplay of several forcing mechanisms, more detail in the simulated forcing appears to be required. This is also suggested by the incomplete simulation of vertical variations in the Argo-derived TKE dissipation rates, especially near locations characterized by strong tidal forcing (Fig. 5).

Together with the observation that regions of strong dissipation are often significantly too small in IDEMIX, these conclusions point toward the need to improve the forcing functions and modeled physical processes in IDEMIX. This holds especially true for the generation of lee waves by the flow of geostrophic eddies over rough topography, which has been shown to be an important energy source for internal gravity waves, particularly in the Southern Ocean (Nikurashin and Ferrari 2011). In the current model version, this process is only crudely represented by injecting 20% of the dissipated eddy energy into the internal wave field at the ocean bottom. One possibility to add physical detail to IDEMIX is to compute, following Nikurashin and Ferrari (2011), the energy conversion from geostrophic motions to lee waves based on linear theory (Bell 1975), which requires knowledge of the bottom velocity, bottom stratification, and topographic spectra. Additionally, mesoscale eddies have been shown to shape the internal wave field not only at the ocean bottom but also near the surface, where their presence can affect the near-inertial wave field and hence potentially also turbulent mixing (Kunze 1985; Young and Jelloul 1997; Kawaguchi et al. 2016). Especially for this type of eddy–wave interaction, however, more research is required to adequately represent it in global-scale models.

This should not imply that the wind and the tidal forcings are well constrained. On the contrary, crucial aspects such as the exact amount of near-inertial energy entering the ocean interior or the directional dependence of the barotropic energy flux are currently not well understood and hence lead to additional biases in IDEMIX. The combined effort of observational, numerical, and analytical investigations will be necessary to shed light on the details of these processes and to reduce the associated uncertainties in IDEMIX.

Other processes that are still missing in IDEMIX include the interaction between surface and internal gravity waves (Olbers and Herterich 1979; Olbers and Eden 2016), additional tidal constituents (currently, IDEMIX2 describes only the M2 tide), or the interaction of gravity waves with the balanced flow (Polzin 2010). Last, note that IDEMIX (similar to the finestructure method) only computes internal wave–induced turbulence. Processes such as double-diffusive convection are also known to lead to turbulent motions in the ocean and will also need to be considered in an all-embracing turbulence model.

Although it is reasonable to assume that a more realistic description of the forcing functions in IDEMIX will improve the spatial pattern of the modeled TKE dissipation rates and hence the agreement with the Argo-based estimates, it is by no means certain that these improvements will be significant within the high uncertainty of the finestructure method. Moreover, the fact that Argo floats currently do not reach farther down than 2000 m prevents a comprehensive assessment of how well IDEMIX describes the topographically induced energy conversion, independent of the amount of detail that goes into that description. The latter issue could be solved at least to some extent in the next years with the implementation of Deep Argo, consisting of floats that profile down to 6000 m (Riser et al. 2016). This would also add much information to our maps of the strain-derived internal wave energy content, which currently only reflect the total tidal forcing in the few locations where the ocean is shallower than 2000 m and hence mainly account for the wind energy input. Locally, the solution to both problems is to evaluate IDEMIX against measurements that actually resolve turbulence. Especially in regions where a strong discrepancy between Argo- and IDEMIX-based estimates is observed, such as the subtropical Pacific Ocean or in the vicinity of island chains, an important next step in the assessment of IDEMIX is the local comparison with microstructure measurements. These have lower uncertainties than finestructure estimates and thus allow identification of the detailed shortcomings of IDEMIX and to fine-tune the model. In addition, it would also be insightful to compare other IDEMIX variables to observations, such as diapycnal diffusivities to those obtained from tracer release experiments (e.g., Ledwell et al. 1993) or internal wave energy fluxes to those derived from high-resolution glider measurements (e.g., Johnston et al. 2013).

Considering that IDEMIX was shown to improve the modeled oceanic northward heat transport—which in turn affects many climate variables—compared to other, energetically inconsistent parameterizations (Eden et al. 2014), the improvements of IDEMIX discussed in this section are of more than just theoretical interest.

## Acknowledgments

We thank Caitlin Whalen and Eric Kunze for insightful discussions and helpful comments on the finestructure method as well as the three reviewers, whose constructive criticism helped to improve the manuscript. F. P. was supported through the Cluster of Excellence cliSAP (EXC177), funded by the German Research Foundation (DFG). This work is a contribution to the Collaborative Research Centre TRR 181 on Energy Transfer in Atmosphere and Ocean funded by the DFG.

## APPENDIX

### Internal Gravity Wave Energy

*E*

_{ρ}, Eq. (18) is adjusted to account for the slightly different wavenumber dependence [cf. Eq. (17)] by replacing

*C*

_{1}by

*C*

_{2}:

*h*is the water depth. The

## REFERENCES

Alford, M. H., 2003: Redistribution of energy available for ocean mixing by long-range propagation of internal waves.

,*Nature***423**, 159–162, doi:10.1038/nature01628.Alford, M. H., M. F. Cronin, and J. M. Klymak, 2012: Annual cycle and depth penetration of wind-generated near-inertial internal waves at Ocean Station Papa in the northeast Pacific.

,*J. Phys. Oceanogr.***42**, 889–909, doi:10.1175/JPO-D-11-092.1.Alford, M. H., J. A. MacKinnon, H. L. Simmons, and J. D. Nash, 2016: Near-inertial internal gravity waves in the ocean.

,*Annu. Rev. Mar. Sci.***8**, 95–123, doi:10.1146/annurev-marine-010814-015746.Bell, T., 1975: Topographically generated internal waves in the open ocean.

,*J. Geophys. Res.***80**, 320–327, doi:10.1029/JC080i003p00320.Brüggemann, N., and C. Eden, 2015: Routes to dissipation under different dynamical conditions.

,*J. Phys. Oceanogr.***45**, 2149–2168, doi:10.1175/JPO-D-14-0205.1.Bryan, K., and L. Lewis, 1979: A water mass model of the World Ocean.

,*J. Geophys. Res.***84**, 2503–2517, doi:10.1029/JC084iC05p02503.Cairns, J. L., and G. O. Williams, 1976: Internal wave observations from a midwater float, 2.

,*J. Geophys. Res.***81**, 1943–1950, doi:10.1029/JC081i012p01943.Capet, X., J. C. McWilliams, M. J. Molemaker, and A. Shchepetkin, 2008: Mesoscale to submesoscale transition in the California Current System. Part II: Frontal processes.

,*J. Phys. Oceanogr.***38**, 44–64, doi:10.1175/2007JPO3672.1.Chassignet, E. P., and D. P. Marshall, 2008: Gulf Stream separation in numerical ocean models.

*Ocean Modeling in an Eddying Regime*,*Geophys. Monogr.*, Vol. 177, Amer. Geophys. Union, 39–61.Chou, S. H., D. S. Luther, M. D. Guiles, G. S. Carter, and T. Decloedt, 2014: An empirical investigation of nonlinear energy transfer from the M2 internal tide to diurnal wave motions in the Kauai Channel, Hawaii.

,*Geophys. Res. Lett.***41**, 505–512, doi:10.1002/2013GL058320.Cummins, P. F., G. Holloway, and E. Gargett, 1990: Sensitivity of the GFDL ocean general circulation model to a parameterization of vertical diffusion.

,*J. Phys. Oceanogr.***20**, 817–830, doi:10.1175/1520-0485(1990)020<0817:SOTGOG>2.0.CO;2.Desaubies, Y., and W. K. Smith, 1982: Statistics of Richardson number and instability in oceanic internal waves.

,*J. Phys. Oceanogr.***12**, 1245–1259, doi:10.1175/1520-0485(1982)012<1245:SORNAI>2.0.CO;2.Eden, C., 2016: Closing the energy cycle in an ocean model.

,*Ocean Modell.***101**, 30–42, doi:10.1016/j.ocemod.2016.02.005.Eden, C., and R. J. Greatbatch, 2008: Towards a mesoscale eddy closure.

,*Ocean Modell.***20**, 223–239, doi:10.1016/j.ocemod.2007.09.002.Eden, C., and D. Olbers, 2014: An energy compartment model for propagation, nonlinear interaction, and dissipation of internal gravity waves.

,*J. Phys. Oceanogr.***44**, 2093–2106, doi:10.1175/JPO-D-13-0224.1.Eden, C., L. Czeschel, and D. Olbers, 2014: Toward energetically consistent ocean models.

,*J. Phys. Oceanogr.***44**, 3160–3184, doi:10.1175/JPO-D-13-0260.1.Falahat, S., J. Nycander, F. Roquet, and M. Zarroug, 2014: Global calculation of tidal energy conversion into vertical normal modes.

,*J. Phys. Oceanogr.***44**, 3225–3244, doi:10.1175/JPO-D-14-0002.1.Furuichi, N., T. Hibiya, and Y. Niwa, 2005: Bispectral analysis of energy transfer within the two-dimensional oceanic internal wave field.

,*J. Phys. Oceanogr.***35**, 2104–2109, doi:10.1175/JPO2816.1.Furuichi, N., T. Hibiya, and Y. Niwa, 2008: Model-predicted distribution of wind-induced internal wave energy in the world’s oceans.

,*J. Geophys. Res.***113**, C09034, doi:10.1029/2008JC004768.Gargett, A. E., 1990: Do we really know how to scale the turbulent kinetic energy dissipation rate

*ε*due to breaking of oceanic internal waves?,*J. Geophys. Res.***95**, 15 971–15 974, doi:10.1029/JC095iC09p15971.Gargett, A. E., and J. Moum, 1995: Mixing efficiencies in turbulent tidal fronts: Results from direct and indirect measurements of density flux.

,*J. Phys. Oceanogr.***25**, 2583–2608, doi:10.1175/1520-0485(1995)025<2583:MEITTF>2.0.CO;2.Garrett, C., and W. Munk, 1972: Space-time scales of internal waves.

,*Geophys. Fluid Dyn.***3**, 225–264, doi:10.1080/03091927208236082.Garrett, C., and W. Munk, 1979: Internal waves in the ocean.

,*Annu. Rev. Fluid Mech.***11**, 339–369, doi:10.1146/annurev.fl.11.010179.002011.Gent, P. R., and J. C. McWilliams, 1990: Isopycnal mixing in ocean circulation models.

,*J. Phys. Oceanogr.***20**, 150–155, doi:10.1175/1520-0485(1990)020<0150:IMIOCM>2.0.CO;2.Gouretski, V. V., and K. P. Koltermann, 2004: WOCE global hydrographic climatology. Berichte des Bundesamtes fur Seeschifffahrt und Hydrographie Rep. 35/2004, 52 pp.

Gregg, M. C., 1989: Scaling turbulent dissipation in the thermocline.

,*J. Geophys. Res.***94**, 9686–9698, doi:10.1029/JC094iC07p09686.Gregg, M. C., and E. Kunze, 1991: Shear and strain in Santa Monica Basin.

,*J. Geophys. Res.***96**, 16 709–16 719, doi:10.1029/91JC01385.Gregg, M. C., T. B. Sanford, and D. P. Winkel, 2003: Reduced mixing from the breaking of internal waves in equatorial waters.

,*Nature***422**, 513–515, doi:10.1038/nature01507.Hasselmann, K., 1967: Weak-interaction theory of ocean waves.

*Basic Developments in Fluid Dynamics*, M. Holt, Ed., Academic Press, 117–182.Henyey, F. S., J. Wright, and S. M. Flatté, 1986: Energy and action flow through the internal wave field: An eikonal approach.

,*J. Geophys. Res.***91**, 8487–8495, doi:10.1029/JC091iC07p08487.Hibiya, T., M. Nagasawa, and Y. Niwa, 2002: Nonlinear energy transfer within the oceanic internal wave spectrum at mid and high latitudes.

,*J. Geophys. Res.***107**, 3207, doi:10.1029/2001JC001210.IOC, SCOR, and IAPSO, 2010: The International Thermodynamic Equation of Seawater—2010: Calculation and use of thermodynamic properties. Intergovernmental Oceanographic Commission Manuals and Guides 56, 220 pp. [Available online at http://www.teos-10.org/pubs/TEOS-10_Manual.pdf.]

Jayne, S. R., 2009: The impact of abyssal mixing parameterizations in an ocean general circulation model.

,*J. Phys. Oceanogr.***39**, 1756–1775, doi:10.1175/2009JPO4085.1.Jochum, M., B. P. Briegleb, G. Danabasoglu, W. G. Large, N. J. Norton, S. R. Jayne, M. H. Alford, and F. O. Bryan, 2013: The impact of oceanic near-inertial waves on climate.

,*J. Climate***26**, 2833–2844, doi:10.1175/JCLI-D-12-00181.1.Johnston, T., D. L. Rudnick, M. H. Alford, A. Pickering, and H. L. Simmons, 2013: Internal tidal energy fluxes in the South China Sea from density and velocity measurements by gliders.

,*J. Geophys. Res. Oceans***118**, 3939–3949, doi:10.1002/jgrc.20311.Kawaguchi, Y., S. Nishino, J. Inoue, K. Maeno, H. Takeda, and K. Oshima, 2016: Enhanced diapycnal mixing due to near-inertial internal waves propagating through an anticyclonic eddy in the ice-free Chukchi Plateau.

,*J. Phys. Oceanogr.***46**, 2457–2481, doi:10.1175/JPO-D-15-0150.1.Kunze, E., 1985: Near-inertial wave propagation in geostrophic shear.

,*J. Phys. Oceanogr.***15**, 544–565, doi:10.1175/1520-0485(1985)015<0544:NIWPIG>2.0.CO;2.Kunze, E., and S. Llewellyn Smith, 2004: The role of small-scale topography in turbulent mixing of the global ocean.

,*Oceanography***17**, 55–64, doi:10.5670/oceanog.2004.67.Kunze, E., E. Firing, J. M. Hummon, T. K. Chereskin, and A. M. Thurnherr, 2006: Global abyssal mixing inferred from lowered ADCP shear and CTD strain profiles.

,*J. Phys. Oceanogr.***36**, 1553–1576, doi:10.1175/JPO2926.1.Ledwell, J. R., A. J. Watson, and C. S. Law, 1993: Evidence for slow mixing across the pycnocline from an open-ocean tracer-release experiment.

,*Nature***364**, 701–703, doi:10.1038/364701a0.MacKinnon, J., and M. Gregg, 2003: Mixing on the late-summer New England shelf—Solibores, shear, and stratification.

,*J. Phys. Oceanogr.***33**, 1476–1492, doi:10.1175/1520-0485(2003)033<1476:MOTLNE>2.0.CO;2.MacKinnon, J., and K. Winters, 2005: Subtropical catastrophe: Significant loss of low-mode tidal energy at 28.9°.

,*Geophys. Res. Lett.***32**, L15605, doi:10.1029/2005GL023376.MacKinnon, J., M. Alford, R. Pinkel, J. Klymak, and Z. Zhao, 2013: The latitudinal dependence of shear and mixing in the Pacific transiting the critical latitude for PSI.

,*J. Phys. Oceanogr.***43**, 3–16, doi:10.1175/JPO-D-11-0107.1.Mashayek, A., C. Caulfield, and W. Peltier, 2013: Time-dependent, non-monotonic mixing in stratified turbulent shear flows: Implications for oceanographic estimates of buoyancy flux.

,*J. Fluid Mech.***736**, 570–593, doi:10.1017/jfm.2013.551.McComas, C. H., 1977: Equilibrium mechanisms within the oceanic internal wave field.

,*J. Phys. Oceanogr.***7**, 836–845, doi:10.1175/1520-0485(1977)007<0836:EMWTOI>2.0.CO;2.McComas, C. H., and P. Müller, 1981: Time scales of resonant interactions among oceanic internal waves.

,*J. Phys. Oceanogr.***11**, 139–147, doi:10.1175/1520-0485(1981)011<0139:TSORIA>2.0.CO;2.Molemaker, M. J., J. C. McWilliams, and I. Yavneh, 2005: Baroclinic instability and loss of balance.

,*J. Phys. Oceanogr.***35**, 1505–1517, doi:10.1175/JPO2770.1.Moum, J., M. Gregg, R. Lien, and M. Carr, 1995: Comparison of turbulence kinetic energy dissipation rate estimates from two ocean microstructure profilers.

,*J. Atmos. Oceanic Technol.***12**, 346–366, doi:10.1175/1520-0426(1995)012<0346:COTKED>2.0.CO;2.Müller, P., and D. J. Olbers, 1975: On the dynamics of internal waves in the deep ocean.

,*J. Geophys. Res.***80**, 3848–3860, doi:10.1029/JC080i027p03848.Müller, P., and N. Xu, 1992: Scattering of oceanic internal gravity waves off random bottom topography.

,*J. Phys. Oceanogr.***22**, 474–488, doi:10.1175/1520-0485(1992)022<0474:SOOIGW>2.0.CO;2.Müller, P., and A. Natarov, 2003: The Internal Wave Action Model (IWAM).

*Near-Boundary Processes and Their Parameterization: Proc. ‘Aha Huliko‘a Hawaiian Winter Workshop*, Honolulu, HI, University of Hawai‘i at Mānoa, 95–105.Munk, W., 1966: Abyssal recipes.

,*Deep-Sea Res. Oceanogr. Abstr.***13**, 707–730, doi:10.1016/0011-7471(66)90602-4.Munk, W., 1981: Internal waves and small-scale processes.

*Evolution of Physical Oceanography*, MIT Press, 264–291.Munk, W., and C. Wunsch, 1998: Abyssal recipes II: Energetics of tidal and wind mixing.

,*Deep-Sea Res. I***45**, 1977–2010, doi:10.1016/S0967-0637(98)00070-3.Nagasawa, M., T. Hibiya, Y. Niwa, M. Watanabe, Y. Isoda, S. Takagi, and Y. Kamei, 2002: Distribution of fine-scale shear in the deep waters of the North Pacific obtained using expendable current profilers.

,*J. Geophys. Res.***107**, 3221, doi:10.1029/2002JC001376.Nikurashin, M., and R. Ferrari, 2011: Global energy conversion rate from geostrophic flows into internal lee waves in the deep ocean.

,*Geophys. Res. Lett.***38**, L08610, doi:10.1029/2011GL046576.Olbers, D., 1974: On the energy balance of small-scale internal waves in the deep-sea. Hamburger Geophysikalische Einzelschriften 24, 91 pp.

Olbers, D., 1976: Nonlinear energy transfer and the energy balance of the internal wave field in the deep ocean.

,*J. Fluid Mech.***74**, 375–399, doi:10.1017/S0022112076001857.Olbers, D., and K. Herterich, 1979: The spectral energy transfer from surface waves to internal waves.

,*J. Fluid Mech.***92**, 349–379, doi:10.1017/S0022112079000653.Olbers, D., and C. Eden, 2013: A global model for the diapycnal diffusivity induced by internal gravity waves.

,*J. Phys. Oceanogr.***43**, 1759–1779, doi:10.1175/JPO-D-12-0207.1.Olbers, D., and C. Eden, 2016: Revisiting the generation of internal waves by resonant interaction with surface waves.

,*J. Phys. Oceanogr.***46**, 2335–2350, doi:10.1175/JPO-D-15-0064.1.Olbers, D., J. Willebrand, and C. Eden, 2012:

*Ocean Dynamics*. Springer-Verlag, 703 pp.Osborn, T., 1980: Estimates of the local rate of vertical diffusion from dissipation measurements.

,*J. Phys. Oceanogr.***10**, 83–89, doi:10.1175/1520-0485(1980)010<0083:EOTLRO>2.0.CO;2.Polzin, K., 2004: A heuristic description of internal wave dynamics.

,*J. Phys. Oceanogr.***34**, 214–230, doi:10.1175/1520-0485(2004)034<0214:AHDOIW>2.0.CO;2.Polzin, K., 2010: Mesoscale eddy-internal wave coupling. Part II: Energetics and results from POLYMODE.

,*J. Phys. Oceanogr.***40**, 789–801, doi:10.1175/2009JPO4039.1.Polzin, K., and Y. V. Lvov, 2011: Toward regional characterizations of the oceanic internal wavefield.

,*Rev. Geophys.***49**, RG4003, doi:10.1029/2010RG000329.Polzin, K., J. M. Toole, and R. W. Schmitt, 1995: Finescale parameterizations of turbulent dissipation.

,*J. Phys. Oceanogr.***25**, 306–328, doi:10.1175/1520-0485(1995)025<0306:FPOTD>2.0.CO;2.Polzin, K., A. C. Naveira Garabato, T. N. Huussen, B. M. Sloyan, and S. Waterman, 2014: Finescale parameterizations of turbulent dissipation.

,*J. Geophys. Res. Oceans***119**, 1383–1419, doi:10.1002/2013JC008979.Riser, S. C., and Coauthors, 2016: Fifteen years of ocean observations with the global Argo array.

,*Nat. Climate Change***6**, 145–153, doi:10.1038/nclimate2872.Sheen, K. L., and Coauthors, 2013: Rates and mechanisms of turbulent dissipation and mixing in the Southern Ocean: Results from the Diapycnal and Isopycnal Mixing Experiment in the Southern Ocean (DIMES).

,*J. Geophys. Res. Oceans***118**, 2774–2792, doi:10.1002/jgrc.20217.St. Laurent, L., M. H. Alford, and T. Paluszkiewicz, 2012: An introduction to the special issue on internal waves.

,*Oceanography***25**, 15–19, doi:10.5670/oceanog.2012.37.Sun, H., and E. Kunze, 1999: Internal wave–wave interactions. Part II: Spectral energy transfer and turbulence production.

,*J. Phys. Oceanogr.***29**, 2905–2919, doi:10.1175/1520-0485(1999)029<2905:IWWIPI>2.0.CO;2.Toole, J. M., R. W. Schmitt, and K. L. Polzin, 1994: Estimates of diapycnal mixing in the abyssal ocean.

,*Science***264**, 1120–1123, doi:10.1126/science.264.5162.1120.von Storch, H., and F. W. Zwiers, 2001:

*Statistical Analysis in Climate Research.*Cambridge University Press, 484 pp.Waterman, S., A. C. Naveira Garabato, and K. L. Polzin, 2013: Internal waves and turbulence in the Antarctic Circumpolar Current.

,*J. Phys. Oceanogr.***43**, 259–282, doi:10.1175/JPO-D-11-0194.1.Whalen, C. B., L. D. Talley, and J. A. MacKinnon, 2012: Spatial and temporal variability of global ocean mixing inferred from Argo profiles.

,*Geophys. Res. Lett.***39**, L18612, doi:10.1029/2012GL053196.Whalen, C. B., J. A. MacKinnon, L. D. Talley, and A. F. Waterhouse, 2015: Estimating the mean diapycnal mixing using a finescale strain parameterization.

,*J. Phys. Oceanogr.***45**, 1174–1188, doi:10.1175/JPO-D-14-0167.1.Wijesekera, H., L. Padman, T. Dillon, M. Levine, C. Paulson, and R. Pinkel, 1993: The application of internal-wave dissipation models to a region of strong mixing.

,*J. Phys. Oceanogr.***23**, 269–286, doi:10.1175/1520-0485(1993)023<0269:TAOIWD>2.0.CO;2.Willebrand, J., P. Müller, and D. Olbers, 1977: Inverse analysis of the Trimoored Internal Wave Experiment (IWEX), part 1. Berichte aus dem Institut für Meereskunde an der Christian-Albrechts-Universität Kiel 201, 117 pp.

Wunsch, C., and R. Ferrari, 2004: Vertical mixing, energy, and the general circulation of the oceans.

,*Annu. Rev. Fluid Mech.***36**, 281–314, doi:10.1146/annurev.fluid.36.050802.122121.Young, W., and M. B. Jelloul, 1997: Propagation of near-inertial oscillations through a geostrophic flow.

,*J. Mar. Res.***55**, 735–766, doi:10.1357/0022240973224283.Zhao, Z., and M. H. Alford, 2009: New altimetric estimates of mode-1 M2 internal tides in the central North Pacific Ocean.

,*J. Phys. Oceanogr.***39**, 1669–1684, doi:10.1175/2009JPO3922.1.

^{1}

Note that we refer to this version of the GM model as GM76 [due to its first appearance in Cairns and Williams (1976)] in contrast to the modified GM75 model, here denoted as GM75m, which is characterized by a wavenumber dependence *A*_{GM}(*m*) proportional to

^{2}

For linear waves satisfying the WKB conditions the vertical energy flux (group velocity times energy) remains constant, leading for *N*^{2} ≫ *ω*^{2} to *E*(*z*) ~ *N*(*z*).