## 1. Introduction

The present study extends the model of Olbers and Eden (2017, hereinafter Part I) for the interaction of an internal gravity wave field with a larger-scale mean flow to an arbitrary two-dimensional flow with vertical shear and discusses the exchange of energy and momentum between the wave field and the mean flow. We derive the equations of the interaction of the waves with the mean flow using a Wentzel–Kramers–Brillouin (WKB) approximation (see, e.g., Müller 1976; Olbers et al. 2012), establish a reduced compartment model along the framework of Internal Wave Dissipation, Energetics and Mixing (IDEMIX; Olbers and Eden 2013; Eden and Olbers 2014), and examine the effects of the wave drag in an idealized and a realistic ocean circulation model.

A general introduction into the subject of wave–mean flow interactions has been given in Part I, pointing out the particular attention paid to effects of wave drag and critical layers in the research on the atmospheric circulation, where internal-wave-induced momentum flux is known to be an important mechanism for the deposition of momentum in atmospheric mean winds and is thus implemented in parameterized form in most circulation models. In the ocean, wave–mean flow interactions have been investigated to a lesser degree and closures for gravity wave drag have only rarely been discussed for numerical ocean models.^{1}

The difference in the attention paid to wave-induced effects on the general circulation in the atmosphere and the ocean arises from the different nature of internal waves in these media. Atmospheric internal waves are predominantly excited by flow over orography or by convection. The generated wave field is therefore vertically asymmetric (mostly upward propagating waves) and horizontally anisotropic, both properties being prerequisites for a nonvanishing wave-induced momentum flux, which follows from the dispersion relation and polarization of internal gravity waves. The oceanic wave field, on the other hand, is mainly excited by barotropic tides flowing over submarine topography, and by atmospheric storms, but waves emanating from these boundary regions are thought to become quickly vertically symmetric by reflections at the surface and bottom and wave–wave interactions, the latter also leading to a nearly horizontally isotropic state.

The prevailing concept for an empirical description of the ocean internal wave state is in fact the vertically symmetric and horizontally isotropic spectral model of Garrett and Munk (1972) and Munk (1981). Such a wave state generates no wave-induced momentum flux, as can be seen from the dispersion relation and polarization of internal gravity waves. There are numerous processes that may lead to an asymmetric and anisotropic wave field, and some of which are associated with wave–mean flow interaction. However, under adiabatic conditions, the net effect of wave–mean flow interaction by an asymmetric or anisotropic wave field still yields a vanishing stress, as indicated by the nonacceleration theorem of Boyd (1976) and Andrews and McIntyre (1976). Only when wave dissipation (by wave breaking) is considered is this constraint broken and a global nonzero net exchange of energy and momentum with the mean flow shows up in the time mean. This exchange may have either sign.

These are also the ingredients of the model of wave-induced stress by Müller (1976), which is a first attempt to account for wave drag in the ocean. However, the proposed large viscosities of this model have not been found in ocean data (e.g., Polzin 2010). The reason is twofold: as shown in Part I, the model by Müller (1976) misses a proper treatment of wave reflections at the surface and bottom boundaries and interior turning points. Second, shown in the present work, the wave-induced stress has nonlocal components, that is, contributions that are not related to the local shear of the mean flow and can be upgradient or downgradient. Another noteworthy example of a wave–mean flow interaction scenario in the ocean is the studies by Muench and Kunze (1999) and Muench and Kunze (2000) suggesting that the alternating jet structure in the deep equatorial ocean is maintained by internal waves encountering critical layers and creating a momentum–flux divergence that drive the jets.

The paper is organized as follows. In section 2 we break down the radiative transfer balance of internal gravity waves to a few equations in physical space governing energy compartments, organized accordingly in up–down and azimuthal (horizontal) wave propagation. We follow here the approach developed in the IDEMIX model (Olbers and Eden 2013). In these integrated equations the closure for the wave–flow interaction is central, as is the treatment of wave–wave interactions and wave breaking. The coupling to the mean flow momentum by wave-induced stresses is derived in section 3. The formulation of a momentum balance in terms of residual momentum is also worked out. Section 4 finally describes the implementation of the coupled wave–mean flow system in a numerical ocean general circulation model, first in an idealized one-dimensional simulation and then in a realistic eddying ocean model of the North Atlantic. The paper closes with a summary and discussion of the results.

## 2. Energy compartments for general flow direction

*t*of the energy spectrum

**k**,

*m*) and physical space (

**x**,

*z*). Because using the six-dimensional space (

**k**,

*m*,

**x**,

*z*) is computationally expensive, integrated energy compartments are thus considered following the concept of IDEMIX (Olbers and Eden 2013). It was shown in Part I that integrating the radiative transfer equations in negative and positive

*m*and all

*ω*yieldswith the energy compartments of upward (

*m*< 0,

*m*> 0,

*m*

_{l}denotes a low wavenumber cutoff. The mean vertical group velocity of the wave field

*c*

_{0}and the interaction coefficient Λ related to the wave–mean flow interaction can be calculated analytically, assuming a gravity wave spectrum of known shape but unknown amplitude

*A*(

*m*) and

*B*(

*ω*) are normalized to one when integrated over

*m*and

*ω*, respectively. The mean flow is assumed horizontally homogeneous,

**U**(

*z*) = [

*U*(

*z*),

*V*(

*z*)], and appears in Eq. (1) by

*C*= ∂

_{z}

*U*cos

*ϕ*+ ∂

_{z}

*V*sin

*ϕ*with the lateral wavenumber vector and

**k**=

*k*(cos

*ϕ*, sin

*ϕ*). We note that the horizontal homogeneity excludes lateral wave refraction from the model, thus neglecting wave capture by horizontal shear (Bühler and McIntyre 2005). However, it is obvious that an extension of the model to include such effects is possible and will be discussed in later studies. The wave action source

*S*in the last term on the right hand side of Eq. (1) represents all other processes affecting the wave field, except for propagation and refraction. Such processes contain wave–wave interactions and dissipation of gravity waves by gravitational or shear instability and also the forcing is included.

*ϕ*as

^{2}Gravity wave closures for atmospheric models often consider 20 or more azimuthal (horizontal wave direction) directions, but we restrict our model to the simplest possible configuration of four directions. Integrating Eq. (1) in the four azimuthal quarter spaces yieldswith the truncation

*ϕ*in Eqs. (3) and (4). To evaluate the terms

*S*=

*S*

_{ww}+

*S*

_{diss}into effects by wave–wave interactions (

*S*

_{ww}) and dissipation (

*S*

_{diss}) by shear and gravitational instability. We then use the approximations for

*S*

_{ww}and

*S*

_{diss}as discussed in Olbers and Eden (2013) and Part I, that is,and similarly for

_{z}

*U*is replaced with ∂

_{z}

*V*. Boundary conditions for the fluxes

*c*

_{0}Δ

*E*

_{e}, etc., at the surface and bottom of the ocean are specified as in Olbers and Eden (2013). Taking the Δ

*E*

_{e/w}equation stationary for time scales exceeding

*τ*

_{υ}and insertinginto the

*E*

_{e/w}equation, and similarly for

*E*

_{n/s}, yieldswith

*c*

_{0}is akin to a diffusive flux and is also present without vertical sheared mean flow. The flux appearing in the next term is akin to an advective flux with a velocity

We use this set of equations in an ocean general circulation model. For ∂_{z}*U* = ∂_{z}*V* = 0 and adding all equations together, the original IDEMIX version of Olbers and Eden (2013) is recovered (except for a small change in the dissipation term). Note that the parameterizations for wave–wave interactions and the wave dissipation, as stated in Eq. (10), leads to a separation of the east–west compartments from the north–south energy compartments. There is, however, coupling via the momentum balance, as shown in the next section.

## 3. Consistent wave–mean flow interaction

^{3}introducing the bolus velocity

**by 90°. It was shown by several authors (e.g., Plumb and Ferrari 2005) that for quasigeostrophic scaling, the difference between the Eulerian mean velocity**

*α***U**and the bolus velocity

**U**with

**U**is identical to

^{4}

**U**

^{2}/2, can be formulated by scalar multiplication with

**U**. While the effect of the Coriolis term vanishes in that energy equation, the scalar product of

**U**with the wave-induced vertical flux divergence on the right hand side of Eq. (20) can be split into a divergence of vertical flux—which drops out in an integral over a closed volume—and an energy exchange term given bywhere

*B*is the mean buoyancy that is assumed to be in thermal wind balance with the mean flow, that is,

**U**

^{2}/2 that are discussed, for example, in Olbers et al. (2012), but that are not essential for the present discussion. The energy transfer in wavenumber space, averaged over a wave period, can be written aswhere

*m*= −

*σ*|

*m*| and the vertical group velocity

*σc*(

*ω*,

*k*), whereNote that the wave-induced momentum flux and the buoyancy flux act in opposite directions. Equation (22) simplifies then tousing

*c*

_{0}and Λ in Eq. (1). For unidirectional flow

**U**= (

*U*, 0), as considered in Part I, only the first vector component of Eq. (28) matters, and the vertical residual momentum flux for this case is given bywhere the energy compartments Δ

*E*

_{e}and Δ

*E*

_{w}are defined slightly differently from here. It is worth mentioning that waves with intermediate frequencies contribute most strongly to the wave-induced stress because near-inertial and near-buoyancy energy is suppressed by the weighting with

*g*(

*ω*). Note further that only the asymmetric and anisotropic parts of the energy spectrum support the stress.

*ϕ*as in Part I, the integrals in Eq. (28) can be evaluated and the vertical residual momentum flux becomesThe vertical divergence of the flux given by Eq. (30) shows up in the residual momentum equation Eq. (20) and is the only effect of the wave drag.

The wave-induced residual momentum flux, given by Eq. (31), has a downgradient and a nonlocal part. The first defines a positive definite viscosity, *ν*_{e,w} is the wave-induced viscosity of Müller (1976). Taking Λ ~ 2 × 10^{−2} (see Fig. 6 from Part I), *τ*_{υ} = 3 days and the energy levels from the experiment shown in Fig. 1, a viscosity of *ν*_{e,w} ≈ 0.02 m^{2} s^{−1} is found, a value that appears too large in comparison to the observational estimates (e.g., Polzin 2010) that found, in general, one order of magnitude smaller viscosities. Furthermore, larger energy levels, as, for example, assumed for the GM76 spectra, and larger relaxation times can increase the large values of *ν*_{e,w} even further by an order of magnitude.

To evaluate the nonlocal part from the energy balances Eqs. (16)–(19), consider the balance of *E*_{e} − *E*_{w} assuming steady state (and similarly for *E*_{n} − *E*_{s}). A ratio for gradient to nonlocal parts in Eq. (31) of roughly *c*_{0} = 0.01 m s^{−1} and *U* = 0.2 m s^{−1}, this becomes for the ratio ~±0.2. The nonlocal part can thus be much larger than the gradient part. The sign of the nonlocal part is not fixed; it may be upgradient or downgradient, that is, acting as a wave drag on the mean flow or an inverse drag.

## 4. Numerical integrations

Equations (16)–(19), together with Eq. (30), have been implemented in an ocean general circulation model (https://wiki.cen.uni-hamburg.de/ifm/TO/pyOM2), extending the IDEMIX version of Olbers and Eden (2013) and Eden and Olbers (2014). The model is described in Eden (2016). Its discretization closely follows the MITgcm (Marshall et al. 1997). Equations (16)–(19) and the corresponding equations for the energy compartments *E*_{n} and *E*_{s} are discretized as vertically implicit in time, while the divergence of the vertical flux Eq. (30) is explicitly added to the momentum equation of the model. For three-dimensional applications, an additional term is added to Eqs. (16)–(19) that acts in a similar manner as lateral diffusion and resembles a zero-order account for the lateral propagation of the wave energy as discussed and used in Olbers and Eden (2013). A more elaborate way to incorporate lateral wave propagation is discussed in Eden and Olbers (2014).

### a. Idealized example

Figure 1 shows results from an idealized one-dimensional simulation initialized with a mean horizontal flow **U** with either a double-exponential, jet-like vertical profile or a profile with a shape of a tanh, akin to a surface-intensified baroclinic flow. The maximum shear of the latter and the maximal amplitude of the former are located at 500 m depth, and both have a vertical decay scale of 200 m. Temperature and salinity in the model are initialized such that the stability frequency *N*(*z*) becomes an exponential function with decay scale of 800 m and an amplitude of 8 × 10^{−3} s^{−1}, as is shown in Fig. 1 of Part I. The Coriolis frequency is set to *f* = 7.3 × 10^{−5} s^{−1}. To avoid a rapid initial dynamical adjustment of the flow profile given by inertial oscillations, a forcing term **U**^{0} represents the initial condition of **U**. Otherwise, the flow is free to evolve. Shown are results after 30 days of integration, after which the energy compartments have equilibrated. Forcing of the internal wave field is specified at the bottom as a total flux of 10^{−6} m^{3} s^{−3}, which is split into the four energy compartments *E*_{e}, *E*_{w}, *E*_{n}, and *E*_{s}, while the dissipation parameter *μ* is specified as shown in Fig. 1 of Part I. Vertical symmetrization is specified by setting *τ*_{υ} = 3 days. The effect of *μ* and *τ*_{υ} are discussed in Part I; here we concentrate on the wave–mean flow interaction.

Figure 1a shows the shear in terms of the inverse time scale *E*_{w} exceeds *E*_{e} in both cases of the mean flow shear (Fig. 1b), but is much larger in the case with tanh profile, and the same holds for the differences in upward and downward propagating waves (Fig. 1c).

Figure 1d shows the vertical momentum flux *E*_{w} is much larger than *E*_{e} for the tanh profile, while both are of the same magnitude for the double-exponential mean flow profile. In any case, the dissipation acts to reduce the wave energy during the upward propagation through the shear zone in both cases, and also during the downward propagation from the surface into the shear zone after surface reflection of the waves. Thus, Δ*E*_{w} and Δ*E*_{e} are both positive, but Δ*E*_{w} exceeds Δ*E*_{e} in magnitude, leading to a positive vertical momentum flux *E*_{w} and Δ*E*_{e} are still positive but both are of the same magnitude because of the symmetry in the wave–mean flow interaction in the shear zone. Therefore, the vertical momentum flux vanishes.

Figure 1e shows the energy transfer

Dividing the flux by the gradient of the mean flow yields an equivalent vertical viscosity, shown in Fig. 1f. It is positive for the case of a tanh mean profile with maximal values exceeding 0.01 m^{2} s^{−1}. For the double-exponential case, the viscosity is negative above the maximum of the mean flow and positive below.

### b. Realistic ocean model

Figure 2 shows the energy transfer from the mean flow to the waves due to wave drag at 300 m depth in a simulation of a realistic, eddying North Atlantic ocean model for a snapshot in September. The model has ½° × ½° horizontal resolution and 45 vertical levels with thickness ranging from 10 m at the surface to 250 m at the bottom. The configuration of the model is similar to the model used in, for example, Eden and Böning (2002) and identical to the one in Thomsen et al. (2014), except that the model code is different in these studies, which may lead to some differences in the simulations. The wave field is forced with realistic fields of energy input from barotropic tides at the bottom and by wind-generated inertial pumping at the surface using the same dataset as in Olbers and Eden (2013).

The energy transfer due to the wave drag shown in Fig. 2 is significant for the kinetic energy balance of the mean flow. At 300 m depth, the horizontally integrated energy transfer from the mean flow to the waves is 19.9 × 10^{6} W m^{−1}, while it is 149.8 × 10^{6} W m^{−1} for the dissipation due to lateral biharmonic friction. This ratio of 10%–20% can also be found at other depths. The energy transfer from mean flow to small-scale turbulence by the parameterized vertical friction at 300 m depth amounts to 56.9 × 10^{6} W m^{−1} but is only significant within the mixed layer, and very low below, that is, in the interior of the model only lateral friction and the wave drag act as dissipation of mean kinetic energy. The depth integrated values of the transfer due to lateral friction, wave drag, and vertical friction are 0.224 × 10^{12}, 0.018 × 10^{12}, and 0.115 × 10^{12} W, respectively. For the balance of internal wave energy itself, the energy transfer due to the wave drag is less important, that is, on the order of 5%, since the tidal forcing amounts to more than 0. 4 × 10^{12} W in the North Atlantic.

Figure 3 exemplifies the wave–mean flow interaction at a section along 72°W across the Gulf Stream after separation from the coast for the same snapshot shown in Fig. 2. The strong eastward current (Fig. 3a) generates strong vertical shear shown in Fig. 3b in terms of the time scale *E*_{e} and *E*_{w}, respectively. As before in the idealized example of the tanh-shaped mean flow profile from above, *E*_{e} is decreased above the shear zone while *E*_{w} is increased, and below the shear zone the magnitudes of both become similar. Correspondingly, an upward momentum flux is generated by the difference in the upward and downward asymmetries of the alongflow and counterflow propagating waves, as shown in Fig. 3e, and an energy transfer from the Gulf Stream to the waves as shown in Fig. 3f. However, the baroclinic structure of the Gulf Stream is of course also more complex than the idealized example. For example, at the southward flank of the current negative,

## 5. Summary and conclusions

We have further developed the wave–mean flow interaction model of Part I, allowing for arbitrary shear direction as a function of depth. We have also examined—as the main aim of the present paper—the feedback of the waves on the mean flow, namely, the effect of wave drag. While the model for unidirectional mean flow of Part I considers only alongflow and counterflow wave direction, the extended model described here uses energy compartments for four azimuths (horizontal wave propagation direction) as the simplest possible configuration. Otherwise, the new model is of identical structure to the one by Part I and converges to the simplest version of the IDEMIX model without consideration of mean flow effects by Olbers and Eden (2013) if the mean flow vanishes.

We note that the model is based on only a few assumptions and that its simple structure comprises wave propagation, wave–wave interactions, wave dissipation, and interaction with the mean flow. There are only four parameters in the compartment model: an average vertical group velocity *c*_{0}, a relaxation time *τ*_{υ} describing annihilation of vertical asymmetries in the wave spectrum by wave–wave interactions, a dissipation parameter *μ* controlling the strength of dissipation by wave breaking, and the interaction coefficient Λ controlling the interaction of the wave field with the mean flow. These parameters are not set by tuning, but derive from wave dynamics and the GM76 spectral shape (*c*_{0} and Λ), from wave–wave interaction studies (*τ*_{υ}, for example, Olbers 1974; McComas 1977), and from parameterizations of fine-structure dissipation schemes (McComas and Müller 1981; Henyey et al. 1986; Gregg 1989).

The new wave–mean flow interaction model has been coupled to a general circulation model. We discuss in this study the effects of the wave drag in a one-dimensional setup similar to the unidirectional study of Part I. We have derived analytically that there are parts of the stress that are not locally related to the mean shear. These may be larger than a viscous, downgradient part that is also present. Taking the viscous part alone allows a viscosity to be defined that turns out to be of order 0.01 m^{2} s^{−1}. Such values are found only as maximal values of the actual diagnosed viscosities in the one-dimensional setup with large vertical shear, but in general the viscosities are much smaller. The one-dimensional setup also demonstrates, however, that the wave-induced stress is generally not downgradient, which inhibits a meaningful parameterization in terms of a wave-induced viscosity. Polzin (2010) reviews the previous estimates of viscous stress in the ocean and also provides new estimates from the POLYMODE array. Polzin postulates flux-gradient relations for the stress and the buoyancy flux. From the data, dividing the wave-induced stress and the mean shear, his estimate is *ν*_{υ} = 2.5 ± 0.3 × 10^{−3} m^{2} s^{−1}, about two orders of magnitude smaller than the value of Müller (1976). Note that dividing stress by shear, our model locally produces viscosities of either sign—that is, wave drag or inverse drag—depending on the mean flow conditions.

The main result of the present study is the implementation of the new compartment model in a realistic eddying circulation model of the North Atlantic. As expected, wave–mean flow interactions are predominantly of importance in the region of the Gulf Stream and the North Atlantic current, including their recirculation gyres. Notable results are as follows:

- Energy transfers occur both ways, from the mean flow to the waves (predominantly in the Gulf Stream and the North Atlantic Current) or reversed (predominantly on the southward flanks of the Gulf Stream). The net energy transfer is, however, from mean flow to the wave field.
- The rate of energy gain of the wave field by wave drag is small, about 5% of the energy supplied by tides.
- For the mean kinetic energy, the energy transfer due to wave drag is on average about 10% of the transfer due to lateral friction in the model, but can be the dominant dissipation mechanism for mean flow in conditions of locally strongly sheared flow.
- The magnitude of the simulated energy transfer due to wave drag in strongly sheared flow is consistent with Polzin (2010), who reports values of about 0.4 × 10
^{−6}W m^{−3}due to wave drag estimated from moorings in the vicinity of the Gulf Stream separation. The larger values of the model by Muench and Kunze (1999, 2000), namely, 4 × 10^{−6}W m^{−3}at the equator, are not supported by our results, but note that our model does not contain sufficiently strong equatorial deep jets (Eden and Dengler 2008).

## Acknowledgments

This paper is a contribution to the Collaborative Research Centre TRR 181 “Energy Transfers in Atmosphere and Ocean” funded by the German Research Foundation.

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^{1}

Notable exceptions are the wave-induced bottom form drag discussed by, for example, Naveira Garabato et al. (2013) and recently by Trossman et al. (2016).

^{2}

Note that the notation is similar to Part I but the integration limits and the compartments are different.

^{3}

The lateral advection of the mean flow **U** is written here for completeness since it is also contained in the numerical model. However, our model for the wave drag is horizontally homogeneous (see above).

^{4}

Except near the equator where the concept breaks down. This region needs special treatment and will be studied individually later.