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  • View in gallery

    The stability frequency N (102 s−1) (black), the mean flow U (m s−1) (green), the mean vertical group velocity c0 (cm s−1) (red), 10 × Λ (blue), and the parameter μ (102 s m−2) (cyan) as a function of depth (km).

  • View in gallery

    (a) Upward propagating energy compartment (10−2 m2 s−2) as a function of time (days) and depth (m), with αυ = 0 and N as shown in Fig. 1 and U = 0. (b) As in (a), but with αυ = 1/3 day−1. (c),(d) As in (a),(b), but for the downward propagating compartment .

  • View in gallery

    (a) Stationary energy compartments (10−3 m2 s−2) E (solid black) and ΔE (solid red) as a function of depth (km) for the case without dissipation and forcing, and with U = 0 and αυ = 1/3 day−1. Also shown are E (dashed black) and ΔE (dashed red) for a case with forcing and dissipation. (b) Stationary eastward and westward energy compartments, Ee (black) and Ew (red), and energy in vertical asymmetry ΔEe and ΔEw (green) for the case without dissipation and forcing, and with U as in Fig. 1 and αυ = 1/3 day−1. (c) Stationary energy compartments Ee (black), Ew (red), ΔEe (green), and ΔEw (blue) for a case with dissipation and forcing with μ and U as shown in Fig. 1 and αυ = 1/3 day−1.

  • View in gallery

    (a) Downward and eastward propagating energy compartment (10−3 m2 s−2) as a function of time (days) and depth (m), for a case without dissipation and forcing, with αυ = 0, and N and U as shown in Fig. 1. The contour lines indicate the shear zone ∂zU. (b) As in (a), but with αυ = 1/3 day−1. (c),(d) As in (a),(b), but for the upward and westward propagating compartment .

  • View in gallery

    (a) Downward and eastward propagating energy compartment with αυ = 1/3 day−1 for the case with dissipation and forcing at the bottom, and N and U as shown in Fig. 1. The contour lines indicate the shear zone ∂zU. (b) As in (a), but upward and eastward propagating compartment . (c),(d) As in (a),(b), but for the westward propagating compartment . Note the different color scales in the upper and lower row.

  • View in gallery

    The parameter Λ as function of N/f.

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A Closure for Internal Wave–Mean Flow Interaction. Part I: Energy Conversion

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  • 1 Alfred Wegener Institut für Polar- und Meeresforschung, Bremerhaven, and MARUM, Universität Bremen, Bremen, Germany
  • | 2 Institut für Meereskunde, Universität Hamburg, Hamburg, Germany
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Abstract

When internal (inertia-)gravity waves propagate in a vertically sheared geostrophic (eddying or mean) flow, they exchange energy with the flow. A novel concept parameterizing internal wave–mean flow interaction in ocean circulation models is demonstrated, based on the description of the entire wave field by the wave-energy density in physical and wavenumber space and its prognostic computation by the radiative transfer equation. The concept enables a simplification of the radiative transfer equation with a small number of reasonable assumptions and a derivation of simple but consistent parameterizations in terms of spectrally integrated energy compartments that are used as prognostic model variables. The effect of the waves on the mean flow in this paradigm is in accordance with the nonacceleration theorem: only in the presence of dissipation do waves globally exchange energy with the mean flow in the time mean. The exchange can have either direction. These basic features of wave–mean flow interaction are theoretically derived in a Wentzel–Kramers–Brillouin (WKB) approximation of the wave dynamics and confirmed in a suite of numerical experiments with unidirectional shear flow.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Carsten Eden, carsten.eden@uni-hamburg.de

Abstract

When internal (inertia-)gravity waves propagate in a vertically sheared geostrophic (eddying or mean) flow, they exchange energy with the flow. A novel concept parameterizing internal wave–mean flow interaction in ocean circulation models is demonstrated, based on the description of the entire wave field by the wave-energy density in physical and wavenumber space and its prognostic computation by the radiative transfer equation. The concept enables a simplification of the radiative transfer equation with a small number of reasonable assumptions and a derivation of simple but consistent parameterizations in terms of spectrally integrated energy compartments that are used as prognostic model variables. The effect of the waves on the mean flow in this paradigm is in accordance with the nonacceleration theorem: only in the presence of dissipation do waves globally exchange energy with the mean flow in the time mean. The exchange can have either direction. These basic features of wave–mean flow interaction are theoretically derived in a Wentzel–Kramers–Brillouin (WKB) approximation of the wave dynamics and confirmed in a suite of numerical experiments with unidirectional shear flow.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Carsten Eden, carsten.eden@uni-hamburg.de

1. Introduction

Internal gravity waves play an important role in the energy cycle of ocean motions. When they break, the wave field provides energy to small-scale turbulence in the interior of the ocean, which drives density mixing and increases the potential energy of the large-scale circulation (Wunsch and Ferrari 2004). This process is thought to be of importance for the global meridional overturning circulation. Prominent sources energizing the oceanic internal wave field are near-inertial waves excited by wind stress fluctuations at the surface (Alford 2001; Rimac et al. 2013) and the scattering of the barotropic tide into baroclinic waves at topography (Nycander 2005; Falahat et al. 2014). Several other generation processes over a broad range of frequencies have been discussed (see, e.g., Olbers 1983) also involving interactions with the geostrophically balanced circulation, as, for example, through the dissipation of mesoscale eddies by spontaneous wave emission or other processes (Ford et al. 2000; Molemaker et al. 2010; Tandon and Garrett 1996; Eden and Greatbatch 2008; Brüggemann and Eden 2015) and the generation of lee waves by large-scale currents or mesoscale eddies flowing over topography (e.g., Nikurashin and Ferrari 2011; Scott et al. 2011; Trossman et al. 2016).

Whatever their generation mechanism, internal gravity waves can interact with the mean flow. When waves are propagating in a vertically sheared mean flow, they exchange energy with the mean flow, provided there is dissipation and forcing of the wave amplitude. The effect is generally called wave–mean flow interaction and, specifically in the atmospheric literature, gravity wave drag. The process is of importance for the dynamics of the upper atmosphere (e.g., Miller et al. 1989; Fritts and Alexander 2003; Muraschko et al. 2015), and the direction of the energy exchange can be from the mean flow to the waves or vice versa. Furthermore, the deformation of the wave paths by the mean flow can lead to critical layers where the wave energy is transferred to small-scale turbulence. A basic overview of the wave driving of the middle atmosphere is given in Becker (2012), and an overview of the gravity wave schemes currently used in global atmospheric models is given in Alexander et al. (2010).

In the ocean, gravity wave drag is less often considered, but it is the focus of the present study. While early theoretical considerations have predicted rather large wave-induced viscosities (Müller 1976), the observational estimates in the ocean (e.g., Frankignoul and Joyce 1979; Ruddick and Joyce 1979; Brown and Owens 1981) suggest a small effect. Recent reviews by Polzin (2008, 2010) confirm this finding. An exception is the wave–mean flow interactions in the deep equatorial jets studied by Muench and Kunze (1999, 2000). In their model, the jets are driven by the divergence of wave-induced momentum flux, created by internal waves that encounter critical layers.

Müller (1976) considers the concept of a wave-induced viscosity, which appears attractive and logical when appealing to the corresponding problem of mean transport processes induced by particle propagation and collisions in a Boltzmann gas (see, e.g., Huang 1987). The interaction of a small-scale random internal gravity wave field with a large-scale mean flow leads to a perturbation of the wave spectrum that is vertically asymmetric and anisotropic. This part is proportional to the mean shear. Müller (1976) assumes that nonlinear wave–wave interactions relax the actual perturbed energy spectrum toward vertical symmetry of upward and downward propagating energy and toward horizontal isotropy. The theory by Müller (1976) predicts very large viscosities, inconsistent with the observational estimates. The reason is that the waves do not interact with the same shear for a long time but are prone to reflection at the top and bottom of the ocean. Waves that, for example, gain energy from the shear flow on the way up will lose the same amount after reflection. This is a fundamental difference from the atmospheric setting of wave drag, where reflection at the top is missing and which seems to be more aligned with Müller’s theory.1 Still, despite small viscosities on average in the ocean and thus a small effect on the large-scale circulation, gravity wave drag may become an important source of oceanic internal gravity waves, as suggested by Polzin (2010). Since energy can be transferred by gravity wave drag from the geostrophically balanced flow to waves, this process may also be of importance for the dissipation of balanced flow.

Here, we revisit the topic of internal wave–mean flow interaction in the ocean and develop a simple closure of the process for use in ocean circulation models. In section 2, we discuss single internal gravity waves propagating in a horizontal mean flow with vertical shear in the Wentzel–Kramers–Brillouin (WKB) approximation (see, e.g., Olbers et al. 2012), to point out some principal properties of the wave–mean flow interaction in terms of energy transfers. Section 3 revisits the continuous description of a wave field using the radiative transfer equation including wave–mean flow interaction, which is used in section 4 to derive integrated energy compartments for the interaction with a unidirectional flow. Section 5 details simple closures for wave–wave interactions and dissipation for these compartments, which are used in section 6 for numerical integrations. These exemplify the effects of wave–wave interactions and dissipation on wave–mean flow interaction in some important idealized cases. The last section provides a short summary and discussion. A companion paper (Eden and Olbers 2017, hereinafter Part II) will deal with arbitrary configurations of the mean flow and the wave effects in the momentum equation.

2. Single waves

Consider an internal gravity wave propagating in a horizontal2 mean flow U(z, t) with a vertical shear and a stability frequency N(z), with horizontal wavenumber vector k, vertical wavenumber m, and intrinsic (non-Doppler shifted) frequency ω. Assuming that the vertical wavelength 2π/m and the wave period 2π/ω of the wave remain small compared to the vertical and temporal scales of U and N, respectively, the WKB approximation can be applied. The intrinsic frequency then obeys the local intrinsic dispersion relation ω ≡ Ω(k, m, z) for internal gravity waves, which may be expressed in the form
e1
where k = |k| and σ = −sign(m) is positive for upward propagating waves (m < 0) and negative for downward propagating waves (m > 0). The (Doppler shifted) frequency of encounter then satisfies . The ray of a wave (x, z) is given by the lateral and vertical group velocity,
e2
where denotes the absolute value of the vertical group velocity. Vertical wavenumber and the frequency of the wave change along the ray according to
e3
where the dot denotes the rate of change taken along the ray of the wave, that is, for any property ψ(z, t) (e.g., Olbers et al. 2012). The intrinsic frequency ω and the vertical wavenumber m are changing because of a shear of the mean flow U. Since both N and U are assumed to lack any lateral dependency, the horizontal wavenumber stays constant, . Wave capture by horizontal shear (Bühler and Mcintyre 2005) thus does not occur. The vertical wavenumber can be obtained from Eq. (1), which is consistent with Eq. (3).
These wave kinematic ray tracing equations are supplemented by a dynamical one that governs the amplitude or energy E of the wave. Under WKB conditions the wave action A = E/ω is conserved (Bretherton and Garrett 1968; Whitham 1970; Lighthill 1978), which means
e4
The energy E of the wave is not conserved, but there is exchange with the energy of the mean flow. To concentrate on this interaction of the wave with the mean flow, we assume for the rest of this section that , so that . A more precise formulation of energy conservation is found in appendix A. Then
e5
with
e6
and k = k(cosϕ, sinϕ). Here ϕ is the direction of the wave vector. The time scale τC characterizes the energy exchange of the wave with the mean flow and both signs σ and γ determine its direction. For a shear representative of the interior ocean τC is of order of hours for high-frequency waves with ω/N = O(1), while it is much larger for near-inertial waves. Note that τC changes along the ray because ω, c, and C change. The direction ϕ of the horizontal wave path remains constant because of the assumption of lateral homogeneity.

From the second relation in Eq. (5), it follows that the intrinsic frequency ω is decreasing for σγ > 0. Under this condition ω may approach the Coriolis frequency f, and ultimately, the wave proceeds into a critical layer where breaking must occur because the vertical wavelength reduces dramatically. Conservation of action tells us that the initial action E0/ω0 = E/ω is conserved so the amount E0f/ω0 of the initial energy must be transferred to turbulence by breaking. In the following discussion, however, we ignore critical layer effects for simplicity.

The interaction with the mean flow depends on the sign σ, that is, on the upward or downward propagation direction of the wave, and on the sign γ, that is, on the direction of the lateral wave propagation with respect to the mean flow shear. Let denote the energy of an upward/downward wave group with frequency , respectively. The energy and frequency equations become
e7
For positive γ, the energy and frequency E+ and ω+ of an upward propagating wave are decreasing, such that the wave loses energy to the mean flow. A downward propagating wave, on the other hand, receives energy E from the mean flow for positive γ and its frequency ω also increases. If γ is negative, E+ and ω+ increase and E and ω decrease, that is, the upward propagating wave receives and the downward propagating wave loses energy to the mean flow. The process is thus completely symmetric (in absence of any dissipative effects).

In the ocean, we can expect wave reflection at the surface and the bottom, such that an upward propagating wave that has received (or lost) energy from (to) the mean flow will become after reflection a downward propagating wave, which will lose (receive) the same amount of energy to (from) the mean flow, returning to the previous level, since the horizontal wave propagation and thus γ has not changed during the reflection process. The wave–mean flow interaction is thus reversible without any net effect, as long as no other dissipative or diabatic processes, such as wave breaking due to gravitational or shear instability, become important and convert the energy to a different form before the process reverses (and as long as we do not include horizontal anisotropies, which makes the situation more complicated). This is an example of the well-known nonacceleration theorem (often called noninteraction theorem) by Boyd (1976) and Andrews and McIntyre (1976). Note that in the atmosphere, surface reflection and the related symmetrization is missing and the wave–mean flow interaction may lead to the familiar gravity wave drag with significant importance for the large-scale circulation.

Because of the reflection at the surface and the bottom, a gravity wave field with a large extent of symmetry in upward or downward propagation direction is observed in the ocean (Garrett and Munk 1975; Munk 1981). Consider a couple of waves that are initially symmetric in σ, that is, E+ = E. The total energy E = E+ + E is then initially conserved since with ΔE = E+E, which vanishes initially. Any asymmetry in the wave field given by ΔE is thus indicative of an energy exchange with the mean flow. The changes of the asymmetry ΔE are given by . For positive γ, the asymmetry ΔE will become negative and the total energy E will increase, while for negative γ, the asymmetry ΔE will become positive and the total energy E will also increase. The energy exchange is thus directed from mean flow to the waves for both signs of γ. However, after reflection of the waves the roles of E+ and E are exchanged, and for both signs of γ the asymmetry will decay and the waves again lose energy to the mean flow. Because of the reflection, the process is thus completely reversible even in the presence of asymmetries in the wave field as long as no diabatic or dissipative processes are involved. Such processes are thus essential for a net energy transfer due to wave–mean flow interaction.

3. Continuous wave field

Consider now an ensemble of weakly interacting gravity waves instead of a single wave. The wave field is described by its energy spectrum in wavenumber space (k, m) and physical space (x, z) at time t. The quantity represents the contribution to the energy of the wave field from waves in the volume dkdm centered at the wavenumber vector (k, m) in wavenumber space. In the same way, the wave action spectrum becomes now a field in wavenumber and physical space (see, e.g., Olbers et al. 2012).

Instead of using the wavenumber space, both energy and wave action can also be described as functions of m, ω, and ϕ, where ω is the intrinsic frequency and ϕ denotes the wavenumber angle with k = k(cosϕ, sinϕ). It is in fact convenient to use the space instead of (k, m) in the current discussion. A governing equation for the wave field—the radiation balance or radiative transfer equation—can be formulated for the action spectrum in this space as
e8
where and denote the lateral and vertical group velocity, respectively; , , and denote the corresponding wave refractions, as defined above ( is nonzero only if the dispersion relation varies with horizontal position so it plays no role in the following); and S a source that represents all processes affecting the wave field, except for propagation and refraction. Such processes contain wave–wave interactions and dissipation of waves by gravitational or shear instability but also the forcing.
To focus on the wave–mean flow interaction, we will ignore any lateral inhomogeneity in Ω and for simplicity, such that the second and last term on the left-hand side of Eq. (8) vanish, but note that ϕ still figures as coordinate and continues to be of importance for the wave–mean flow interaction, as we have seen in the discussion of single wave packets. A way to treat lateral inhomogeneities in is discussed in Eden and Olbers (2014). Formulated for the spectral energy instead of , the radiative transfer equation Eq. (8) then becomes
e9
assuming that ∂tΩ = 0 as before, that is, that the stability frequency N stays constant in time.

4. Energy compartments for unidirectional flow

As in Olbers and Eden (2013), we do not attempt to solve Eq. (9) directly but rather integrate it in wavenumber–frequency space, forming equations for energy compartments that are then solved. We begin by integrating over the frequency ω and separately over negative and positive vertical wavenumbers m, but not yet over the wavenumber direction ϕ, since the direction of wave propagation was shown above to be important for the sign of the interaction with the mean flow. The integration in negative and positive m and all ω of Eq. (9) yields
e10
with the energy compartments of upward (m < 0) and downward (m > 0) propagating waves given by
e11
Here ml denotes a low wavenumber cutoff. As in Olbers and Eden (2013), the integral of the second term on the left hand side of Eq. (9) was evaluated assuming a factorized form of the internal gravity wave spectrum , with functions A and B given by Cairns and Williams (1976) and Munk (1981) [which is the Garrett–Munk (GM76) spectrum], which are supposed to be identical for σ = ±1. This leads to
e12
using and where k needs to be replaced in the integral with the dispersion relation Eq. (1). The integral of the first term on the right-hand side of Eq. (9) was evaluated using the same factorized spectrum for after one partial integration, such that
e13
using and the normalizations and . Both integrals c0 and Λ can be readily evaluated using the functions A(m) and B(m) given by Cairns and Williams (1976) and Munk (1981) and become positive functions of N/f. The mean vertical group velocity c0 is given in Olbers and Eden (2013) and Λ is given in the appendix B.

The Eqs. (10) require knowledge of the processes described by S. Olbers and Eden (2013) discuss their effect in terms of energy compartments that have also been integrated in ϕ. The wave–wave interactions must conserve total energy; therefore, this nonlinear effect vanishes in the corresponding equation for , while their effect is parameterized as linear damping in the equation for the difference in Olbers and Eden (2013). Without further processes, the wave–wave interactions were assumed to damp any asymmetries in the energy of upward or downward propagating waves on a time scale τυ on the order of days. This closure, together with one for the dissipation of gravity waves, was key to construct a simple model for the propagation, interaction, and dissipation of gravity waves in the ocean, and we will use this concept here as well and extend it with the wave–mean flow interaction.

To simplify the wave–mean flow interaction, we consider an unidirectional flow and rotate the local coordinate system such that U = (U, 0) and C = ∂zU cosϕ. Suppose that ∂zU > 0. We have learned from the discussion of single wave packets and also see from Eq. (10) that decreases due to the wave–mean flow term for the angles −π/2 < ϕ < π/2 (γ > 0) and increases in the other half-space (γ < 0). The opposite is true for , and for the case ∂zU < 0 the relations for and reverse. Therefore, we define energy compartments integrated over the respective half-spaces of ϕ as
e14
where denotes the energy of the eastward and upward or downward propagating waves, while denotes the energy of the westward and upward or downward propagating waves. The Eqs. (10) are integrated over the half-spaces in ϕ in the same way, which yields
e15
and
e16
We have defined by and the operators that represent the integration over the respective eastward (e) and westward (w) directions, as given by Eq. (14), for all frequencies and the negative and positive vertical wavenumber domains, as in Eq. (11). It now becomes necessary to form equations for to express the integral in the above terms involving the mean flow. However, these equations will couple to , and so on, which forms a hierarchy of moments that needs to be truncated at some point. For simplicity, the truncation at lowest order is used here, that is, , which yields
e17
e18
As expected, the only difference between the compartments of the eastward propagating waves and the one of the westward propagating waves is the sign of the interaction with the mean flow.

Note that critical layers are not considered in the present approach. The waves are assumed to freely propagate through the water column to the surface or the bottom, except for the WKB interaction with the mean flow. However, when parts of the wave field approach critical layers, a fraction of the fluxes and will transfer energy to small-scale turbulence. An example of a calculation of this fraction is given in Muench and Kunze (1999). Here we ignore this effect for simplicity, but an implementation of the critical layer effect into the present concept will be attempted in future work.

5. Closures for wave–wave interactions and dissipation

To treat the integral of ωS in an analogous way as in Olbers and Eden (2013), it is convenient to consider, instead of , the sum and the difference in energy of the eastward and upward and downward propagating waves, and correspondingly for . Before doing this, the term S = Sww + Sdiss needs to be decomposed into wave–wave interactions Sww and dissipation Sdiss by gravitational or shear instability. Wave–wave interactions conserve total energy, and it is assumed that this is also true when integrated over a single half-space in ϕ. Note that this would hold for a horizontally isotropic spectrum such as the GM76 spectrum. Therefore, the corresponding integrals of Sww vanish in the equations for Ee and Ew. The dissipation Sdiss is assumed to be symmetric in upward or downward propagating waves and will therefore vanish in the equations for ΔEe and ΔEw. This yields
e19
Following Olbers and Eden (2013), the effect of the wave–wave interactions are parameterized as linear damping of ΔEe and ΔEw, respectively,
e20
with a time scale τυ = 1/αυ on the order of days. The parameter αυ is discussed in more detail in Olbers and Eden (2013). Its order of magnitude can be motivated from the Langevin rate in the scattering integral that results from Sww using the weak interaction assumption for the relaxation of a spectral distortion in the GM76 spectrum. As in Olbers and Eden (2013), αυ is taken constant here for simplicity. The choice to exchange energy between and and separately between and is motivated from the triad interactions of the elastic scattering mechanism where the scattered waves have almost the same horizontal wavenumber vector (see, e.g., McComas 1977). Also as in Olbers and Eden (2013), the dissipation is parameterized as a quadratic functional of the total wave energy E = Ee + Ew,
e21
with a function μ = μ(f/N). The choice of μ for Internal Wave Dissipation, Energetics and Mixing (IDEMIX) is discussed in detail in Olbers and Eden (2013). The function is found by analytical means by McComas and Müller (1981) and by Henyey et al. (1986) using an Eikonal approach, and is also used to estimate gravity wave dissipation from finescale observations (Gregg 1989; Polzin et al. 1995; Sun et al. 1999). Adding the expression for Ee and Ew yields the corresponding expression for dissipation μE2. For ∂zU = 0, that is, ignoring the wave–mean flow interaction, and adding E = Ee + Ew and ΔE = ΔEe + ΔEw, the resulting model is identical to the one in Olbers and Eden (2013).
Vertical boundary conditions for the energy compartments are reflection at the top and the bottom, that is, at the surface the energy compartment of upward propagating waves is reflected to the respective downward propagating compartment, and correspondingly for the bottom. This results in a vanishing difference of upward and downward propagating compartments at the vertical boundaries. However, a possible forcing of the wave field at the bottom or the surface—for example, by generation of gravity waves by barotropic tides or other flow over topography, or generation of waves at the surface mixed layer by inertial pumping—will manifest itself in a nonvanishing difference (Olbers and Eden 2013) according to
e22
where denotes a possible energy flux into the wave field at the boundaries.

6. Numerical integrations

Figure 1 shows N(z) and U(z) profiles that we use to integrate Eqs. (19) numerically. The Coriolis frequency is set to f = 7.3 × 10−5 s−1. The stability frequency N is an exponential function with a decay scale of 800 m, the mean flow U has amplitude 0.5 m s−1, and a shear zone is centered at 500 m with a vertical scale of 200 m. The mean vertical group velocity c0 and the parameter Λ are functions of N/f; both have maxima at depth where N becomes small. The dissipation parameter μ, which is also a function of N/f, on the other hand, decreases with depth.

Fig. 1.
Fig. 1.

The stability frequency N (102 s−1) (black), the mean flow U (m s−1) (green), the mean vertical group velocity c0 (cm s−1) (red), 10 × Λ (blue), and the parameter μ (102 s m−2) (cyan) as a function of depth (km).

Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0054.1

Consider a situation without dissipation and forcing. We also start without mean flow. It is then appropriate to use the energy compartment of upward propagating waves in all ϕ, and similarly for the energy compartment for the downward propagating waves. Figures 2a and 2c show E+ and E for the case without wave–wave interactions, that is, for αυ = 1/τυ = 0. Both compartments are initialized as a Gaussian profile centered at 1200 m depth with a decay scale of 200 m with amplitude of 10−3 m2 s−2. As expected, E+ propagates initially upward, while E propagates downward. At the surface (bottom) E+ (E) is reflected into E (E+) and propagates downward (upward) again and so forth, and only numerical artifacts will affect the amplitudes and shapes of the signal during later periods of the cycle. The increase in amplitude toward the surface in both E+ and E is related to the increase in N with height.

Fig. 2.
Fig. 2.

(a) Upward propagating energy compartment (10−2 m2 s−2) as a function of time (days) and depth (m), with αυ = 0 and N as shown in Fig. 1 and U = 0. (b) As in (a), but with αυ = 1/3 day−1. (c),(d) As in (a),(b), but for the downward propagating compartment .

Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0054.1

Switching on the wave–wave interactions by setting αυ = 1/3 day−1, the compartments E+ and E start to interact and energy is scattered between the upward and the downward waves. After 20 days the process leads to an almost symmetric distribution in upward and downward wave energies, and the upward and downward propagation of the compartments is hardly seen anymore. Using the stationary version of ΔE = ΔEe + ΔEw in the equation for total energy E = Ee + Ew yields in the absence of dissipation the simple diffusion equation (see Olbers and Eden 2013), which yields after a few days integration the stationary state for E and ΔE shown in Fig. 3a. While ΔE = 0, that is, a completely symmetric state has been reached, E features a surface maximum due to the dependency of c0 on N.

Fig. 3.
Fig. 3.

(a) Stationary energy compartments (10−3 m2 s−2) E (solid black) and ΔE (solid red) as a function of depth (km) for the case without dissipation and forcing, and with U = 0 and αυ = 1/3 day−1. Also shown are E (dashed black) and ΔE (dashed red) for a case with forcing and dissipation. (b) Stationary eastward and westward energy compartments, Ee (black) and Ew (red), and energy in vertical asymmetry ΔEe and ΔEw (green) for the case without dissipation and forcing, and with U as in Fig. 1 and αυ = 1/3 day−1. (c) Stationary energy compartments Ee (black), Ew (red), ΔEe (green), and ΔEw (blue) for a case with dissipation and forcing with μ and U as shown in Fig. 1 and αυ = 1/3 day−1.

Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0054.1

Consider now a case with the mean flow as shown in Fig. 1, but still without dissipation and forcing, and also without wave–wave interactions for the moment, that is, with αυ = 0. Figure 4a shows , the energy compartment of downward and eastward, that is, along-flow propagating waves. Initial conditions are chosen as for the case with U = 0. The perturbation in first propagates down to the bottom and reflects into , which propagates up (not shown). After surface reflection, gains energy from the mean flow when propagating downward through the shear zone indicated by contour lines, while after bottom reflection, (not shown) loses the same amount of energy to the mean flow again while propagating upward through the shear zone, and so forth. The counterflow propagating energy compartment gains energy for the upward propagating part of the wave field as shown in Fig. 4c, while it loses that energy again to the mean flow after surface reflection, and so forth. This is what we have expected from the discussion of single waves, and there is thus no net energy transfer between the waves and the mean flow.

Fig. 4.
Fig. 4.

(a) Downward and eastward propagating energy compartment (10−3 m2 s−2) as a function of time (days) and depth (m), for a case without dissipation and forcing, with αυ = 0, and N and U as shown in Fig. 1. The contour lines indicate the shear zone ∂zU. (b) As in (a), but with αυ = 1/3 day−1. (c),(d) As in (a),(b), but for the upward and westward propagating compartment .

Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0054.1

This behavior does not change for the same case but including now the wave–wave interactions, that is, setting αυ = 1/3 day−1. The scattering between and and also for and leads to a symmetrization as before, but an anisotropy with respect to lateral propagation direction develops: Fig. 3b shows the energy compartments of the eastward and westward propagating waves and , respectively, and their asymmetry and in the stationary state. The wave–wave interactions damp any asymmetry such that ΔEe = ΔEw = 0. However, while the energy of the eastward or along-flow propagating part of the waves Ee is concentrated below the shear zone, the energy of the westward or counterflow propagating part of the waves Ew is concentrated above the shear zone, that is, the wave field features a strong anisotropy in ϕ above and below the shear zone. But since there is as much wave energy propagating into as from the shear zone for both compartments Ee and Ew, since ΔEe = ΔEw = 0, there is no net energy gain by the waves or the mean flow. Since the wave–wave interactions conserve total energy and there is thus no dissipation, the net effect of the waves on the mean flow vanishes, as seen above for the consideration of single waves, and in agreement with the nonacceleration theorem by Andrews and McIntyre (1976).

Only for a case with dissipation, that is, using now the closures [Eq. (21)] with the parameter μ as shown in Fig. 1, do we expect a net wave–mean flow interaction. To obtain a stationary case in the numerical simulation, we now also implement forcing at the bottom by specifying a flux = 0.25 × 10−6 m3 s−3 into both ΔEe and ΔEw, which aims to resemble the generation of gravity waves by the barotropic tides over topography with a realistic amplitude. Figure 5 shows the spinup of the energy compartments , , , and , while Fig. 3c shows the stationary solution in terms of Ee, Ew, ΔEe, and ΔEw. Again, upward and downward propagating signals can be seen in the compartments, originating from the bottom and reflected at the surface and the bottom into the respective complementary compartments. As before for the case without dissipation, receives energy from the mean flow while propagating from below into the shear zone and is reflected into , which loses energy to the mean flow again while propagating downward. But during that upward and downward propagation and reflection, part of the energy is now dissipated and not all of the energy is transferred back again to the mean flow, such that a net energy transfer from the mean flow to the compartment is generated. Because of the dissipation also becomes a little larger than in the stationary state after vertical symmetrization, which leads to a positive ΔEw. For the Ee compartment, the opposite happens: while propagates from below into the shear zone, it loses energy and gains energy after reflection during the downward propagation, but during the propagation and reflection some energy is dissipated. Variable becomes larger than in stationary state, and thus ΔEe > 0 since energy is supplied from below.

Fig. 5.
Fig. 5.

(a) Downward and eastward propagating energy compartment with αυ = 1/3 day−1 for the case with dissipation and forcing at the bottom, and N and U as shown in Fig. 1. The contour lines indicate the shear zone ∂zU. (b) As in (a), but upward and eastward propagating compartment . (c),(d) As in (a),(b), but for the westward propagating compartment . Note the different color scales in the upper and lower row.

Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0054.1

Since the compartment gains energy during upward propagation through the shear zone and loses energy [see Eqs. (17) and (18)], the Ew compartment accumulates more energy above the shear zone and thus ΔEw > ΔEe. The net energy transfer from the mean flow to the waves is given from Eq. (19) by , and it is thus positive in this case, that is, it would correspond to a wave drag on the mean flow. For the profile shown in Fig. 1, the energy transfer from the mean flow to the waves is positive for cases with ∂zU > 0 and ∂zU < 0 and bottom and surface forcing (not shown). However, a different profile of U can generate locally a net transfer from the mean flow to the waves, that is, a negative wave drag. A detailed analysis on this issue is given in Part II, treating also the three-dimensional case and arbitrary configurations of the mean flow.

7. Summary and discussion

Internal gravity waves interacting with a mean shear flow strictly follow the nonacceleration theorem of Boyd (1976) and Andrews and McIntyre (1976) if adiabatic conditions apply. Then, the energy exchange with the mean flow of a pair of waves with the same horizontal wavenumber and opposing vertical wavenumbers (or opposing horizontal wavenumbers and same vertical wavenumbers) vanishes. The same is true for a single wave propagating upward or downward in a shear flow when reflections at the ocean’s surface and bottom (or interior turning points) are considered and averaging over up–down cycles is applied. Here the waves gain (lose) energy propagating upward or downward and lose (gain) the same amount when reversing.

The nonacceleration constraint is only broken when dissipative effects are active, for example, friction, wave breaking, or encounters of critical layers. In addition to the dissipation, the waves are then affected by a net energy exchange with the mean flow, which, depending on the shear configuration, can be of either sign. The waves then exchange energy with the mean flow depending on the vertical asymmetry (difference between upward and downward propagating energies) and horizontal anisotropy of the wave field. On the other hand, these properties are also excited by the interaction with the mean flow, that is, an initially symmetric and isotropic wave spectrum is modulated toward asymmetry and anisotropy by the interaction process.

The present study discusses these effects in detail for single waves as well as a random ensemble of waves in a horizontal shear flow, applying the WKB method for three-dimensionally propagating waves and describing the wave–mean flow interaction by wave action conservation and the appropriate radiation balance equation for the wave spectrum. The latter is integrated in wavenumber space, similar to previous IDEMIX models (Olbers and Eden 2013; Eden and Olbers 2014), to consider the interaction of integrated energy compartments among themselves and with the mean shear. A key assumption for our approach is the assumption that the wave field is close to a representative empirical spectral shape. The present analysis is restricted to a shear flow with uniform direction and the feedback of the interaction on the mean flow is not yet considered. Part II generalizes the results to arbitrary configurations and the full interaction problem. For unidirectional flow, four compartments are the minimum to adequately represent the interactions, taking care of the up–down differences and along-flow and counterflow anisotropies. The results of the approach are four compartment equations that can be integrated by numerical time stepping.

For practical use, the effect of nonlinear wave–wave interaction needs to be parameterized in the compartment equations. We have used here the simple closures that are also used by Olbers and Eden (2013). These closures rely on the parameterization for the wave dissipation as suggested by McComas and Müller (1981) and Henyey et al. (1986), and on linear damping toward symmetry in upward and downward propagating waves. The former is also used in similar form to estimate wave dissipation from finescale observations (Gregg 1989; Polzin et al. 1995; Sun et al. 1999). As in Olbers and Eden (2013), the time scale τυ for the latter is taken constant for simplicity here. Its value is on the order of days motivated by the Langevin rate in the scattering integral of weakly interaction waves with a spectral distortion. On the other hand, it is clear that τυ differs for different wavenumbers such that a constant value is only a zero-order approximation for the effect. A possible way to determine τυ for a spectral distortion in a continuous GM76-like wave field exactly from the scattering integral is demonstrated in appendix A of Eden and Olbers (2014), but it remains unclear how to generalize this for τυ of a continuous spectrum.

One possible way to obtain an improved expression for τυ and also for the dissipation are numerical calculations of the scattering integral for a given spectrum as done in Lvov et al. (2012). Another way is given by direct numerical simulations of the wave field, which would, however, be computational rather demanding. In such simulations, the impact of mean shear flow on the dissipation and symmetrization of the wave field could also be investigated. Furthermore, the changes of the mean vertical group velocity c0 and the mean growth rate by wave drag Λ for large deviations of the actual spectrum from the factorized (GM76) spectrum could be estimated in such numerical simulations.

The compartment model for unidirectional flow presented here using the simple closures for the effect of wave–wave interaction is initialized with a simple exponential stratification and a shear zone concentrated in the thermocline and otherwise identical parameterizations as in previous IDEMIX versions, in particular for wave–wave interactions and wave breaking. We have demonstrated the following behavior:

  • without wave–wave interactions and dissipation, the wave field is in a cyclic state due to reflection at top and bottom, with vanishing global energy exchange with the mean flow in the time mean;
  • including wave–wave interactions symmetrizes the energy spectrum, but conserves total energy, yielding a wave field that is symmetric and that still does not globally exchange energy with the mean flow in the time mean; and
  • only when wave dissipation (by wave breaking) is considered does a global nonzero net exchange of energy with the mean flow occur in the time mean. The exchange may have either sign.
Our model builds upon previous work on wave–mean flow interaction (Müller 1976; Polzin 2008, 2010) but extends the solution technique and corrects for deficits. The most important correction is the implementation of reflective boundary conditions instead of an unphysical damping along the wave rays in a vertically infinite ocean as in Müller (1976). The alteration presented here establishes a basic cyclic state for the wave field and avoids the large wave-induced viscosities found by Müller (1976). Second, the new model realizes that wave–wave interactions do not affect the total wave energy but rather impact the asymmetries and anisotropies of the wave state. Finally, the compartment model simplifies the radiation balance by integration in wavenumber space, assuming a spectral shape for the calculation of two integrated parameters. Contrary to Müller (1976), however, there is no expansion of the interaction problem, assuming small perturbations of the wave field by the interaction with the mean flow.

The present version of the compartment model is studied as a test bed for the general configuration, that is, interaction with a nonuniform shear including the implied acceleration or deceleration of the mean flow. In addition, in the present version critical layers have no direct manifestation, though one could argue that they contribute to overall dissipation by the implemented parameterization of wave breaking. The extra transfer of energy to turbulence in critical layers and the general interaction problem will be considered in a follow-up paper. The extension of the present study to an arbitrary configuration of the mean flow and the wave effects in the momentum equation is considered in Part II.

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.

APPENDIX A

Conservation of Energy

Time-independent N and U(z) result in ωenc = constant and . Furthermore, constant N then leads to
e23
where as before. Note that c is positive by definition while ∂c/∂ω > 0 for most of the internal wave frequency range (for ). Hence, the divergence term plays a counteracting role in relation to the wave–mean flow interaction, and it turns out that its effect is almost always dominating in the abovementioned frequency range. This example questions the loose interpretation of action conservation as and (e.g., Müller et al. 2005; Bühler and Mcintyre 2005).
However, proceeding to integrated values, the situation is different. Integrating over the total extent of a wave packet with boundaries moving with the local group velocity, we note that the integrated action is constant. We define
e24
where is a weighted mean frequency for the packet. Then the action conservation leads to
e25
The difference to Eq. (23) is that we now consider energy of the wave packet, not energy density E.

APPENDIX B

The Parameter Λ

With from the factorized frequency spectrum by Cairns and Williams (1976) and Munk (1981), and with the normalization factor nB = (2/π)/[1 − 2/πarcsin(f/N)], the parameter Λ is given by
e26
The integral can be solved in closed form, which yields
e27
Note that Λ is a positive and decreasing function of N/f. It is displayed in Fig. B1.
Fig. B1.
Fig. B1.

The parameter Λ as function of N/f.

Citation: Journal of Physical Oceanography 47, 6; 10.1175/JPO-D-16-0054.1

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1

Note that if the oceanic waves are dissipated at the surface as observed by Pinkel (2005) for sea ice covered regions, a larger effect of wave drag might become possible.

2

All vectors are two-dimensional here.

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