Reabsorption of Lee-Wave Energy in Bottom-Intensified Currents

Yue Wu aWoods Hole Oceanographic Institution, Woods Hole, Massachusetts

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Eric Kunze bNorthWest Research Associates, Redmond, Washington

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Amit Tandon cUniversity of Massachusetts Dartmouth, Dartmouth, Massachusetts

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Amala Mahadevan aWoods Hole Oceanographic Institution, Woods Hole, Massachusetts

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Abstract

While lee-wave generation has been argued to be a major sink for the 1-TW wind work on the ocean’s circulation, microstructure measurements in the Antarctic Circumpolar Currents find dissipation rates as much as an order of magnitude weaker than linear lee-wave generation predictions in bottom-intensified currents. Wave action conservation suggests that a substantial fraction of lee-wave radiation can be reabsorbed into bottom-intensified flows. Numerical simulations are conducted here to investigate generation, reabsorption, and dissipation of internal lee waves in a bottom-intensified, laterally confined jet that resembles a localized abyssal current over bottom topography. For the case of monochromatic topography with |kU0| ≈ 0.9N, where k is the along-stream topographic wavenumber, |U0| is the near-bottom flow speed, and N is the buoyancy frequency; Reynolds-decomposed energy conservation is consistent with linear wave action conservation predictions that only 14% of lee-wave generation is dissipated, with the bulk of lee-wave energy flux reabsorbed by the bottom-intensified flow. Thus, water column reabsorption needs to be taken into account as a possible mechanism for reducing the lee-wave dissipative sink for balanced circulation.

© 2023 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: Yue Wu, ywu.ocean@gmail.com

Abstract

While lee-wave generation has been argued to be a major sink for the 1-TW wind work on the ocean’s circulation, microstructure measurements in the Antarctic Circumpolar Currents find dissipation rates as much as an order of magnitude weaker than linear lee-wave generation predictions in bottom-intensified currents. Wave action conservation suggests that a substantial fraction of lee-wave radiation can be reabsorbed into bottom-intensified flows. Numerical simulations are conducted here to investigate generation, reabsorption, and dissipation of internal lee waves in a bottom-intensified, laterally confined jet that resembles a localized abyssal current over bottom topography. For the case of monochromatic topography with |kU0| ≈ 0.9N, where k is the along-stream topographic wavenumber, |U0| is the near-bottom flow speed, and N is the buoyancy frequency; Reynolds-decomposed energy conservation is consistent with linear wave action conservation predictions that only 14% of lee-wave generation is dissipated, with the bulk of lee-wave energy flux reabsorbed by the bottom-intensified flow. Thus, water column reabsorption needs to be taken into account as a possible mechanism for reducing the lee-wave dissipative sink for balanced circulation.

© 2023 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: Yue Wu, ywu.ocean@gmail.com

1. Introduction

Roughly 1 TW (=1012 W) of wind work powers the balanced ocean circulation (Wunsch 1998; Scott and Xu 2009; Ferrari and Wunsch 2009). But how this power input is dissipated to maintain a steady state remains uncertain. Among the suggested energy sinks summarized in Kunze and Lien (2019), dissipation through internal lee-wave generation has been estimated to be potentially the largest, though with considerable uncertainty, accounting for anywhere from 15% to 75% of the 1-TW input (Scott et al. 2011; Nikurashin and Ferrari 2011; Melet et al. 2014; Wright et al. 2014; Nikurashin et al. 2014; Trossman et al. 2016; Yang et al. 2018). A common assumption in this recent ocean literature is that all lee-wave generation is lost to turbulent dissipation due to wave breaking in a one-way forward energy cascade (from large to small scale). However, microstructure measurements in the Drake Passage and on the flanks of the Kerguelen Plateau in the Southern Ocean (Brearley et al. 2013; Sheen et al. 2013; Waterman et al. 2013, 2014; Cusack et al. 2017, 2020) found that depth-integrated dissipation rates fall short of linear predictions of lee-wave generation (Bell 1975; Nikurashin and Ferrari 2010b), as well as inferences based on finescale parameterizations (e.g., Polzin et al. 1995), by as much as an order of magnitude. Apart from instrumental and sampling biases, which were explored and discounted by Waterman et al. (2014), this suppression of turbulence might be explained in one or more of several ways:

  1. linear theory overpredicts lee-wave generation by ignoring topographic blocking and splitting (e.g., Nikurashin et al. 2014; Trossman et al. 2013, 2015);

  2. partial reabsorption of lee-wave energy into bottom-intensified currents through wave action conservation (Eliassen and Palm 1961; Booker and Bretherton 1967; Jones 1967; Bretherton and Garrett 1968; Jones and Houghton 1971; Dunkerton 1984; Shaw and Shepherd 2008; Kunze and Lien 2019);

  3. remote dissipation in the form of free waves escaping the generating current (Kunze et al. 1995);

  4. advection of lee-wave energy downstream from localized generating topography (Zheng and Nikurashin 2019).

Among these mechanisms, reabsorption encapsulates momentum and energy exchange between lee waves and sheared mean flow but has largely been ignored in recent oceanic lee-wave studies, with the exception of Kunze and Lien (2019) and Baker and Mashayek (2021). Conservation of wave action suggests that, in a steady flow field, wave action E/ωL is conserved rather than wave energy E (Jones 1967; Bretherton and Garrett 1968). As Doppler-shifting alters lee wave’s intrinsic (Lagrangian) frequency ωL = |ωEkU| = |kU| (where lee-wave Eulerian frequency ωE = 0 to maintain stationary phase with respect to the topography, k is the along-stream topographic wavenumber, and U is the flow velocity), sheared U results in exchange of lee-wave energy E with background mean flow. For lee waves generated by bottom-intensified flows, the decrease in U with height above bottom implies a critical layer at the depth where |kU| = |f|. If there is no loss to turbulence as a lee wave propagates upward to its critical depth, linear wave action conservation dictates that the dissipative energy fraction is |f|/|kU0|, while the reabsorbed fraction |(kU0f)|/|kU0|, where U0 is the local bottom flow speed. Kunze and Lien (2019) predicted dissipative fractions of ∼0.5 for typical abyssal |N/f| ratios and topographic height h spectra.

a. Wave–mean interaction in the ocean

Wave–mean interaction is well established in atmospheric studies where the effect of mountain waves on the background wind has been parameterized as gravity wave drag (e.g., McFarlane 1987). However, in the ocean, previous efforts to parameterize the effect of internal waves on balanced flow as eddy viscosity and diffusivity (Müller 1976) were not supported observationally (e.g., Frankignoul 1976; Frankignoul and Joyce 1979; Ruddick and Joyce 1979; Brown and Owens 1981). This inconsistency is likely due to (i) treatment of the wave’s effect solely as a damping mechanism for balanced flow while momentum/energy exchange between wave and mean fields can have either sign (e.g., Nagai et al. 2015; Rocha et al. 2018; Thomas and Daniel 2021) so that waves can either decelerate or accelerate balanced flow (e.g., Muench and Kunze 1999, 2000) and (ii) neglect of wave reflection at the ocean surface, which is a fundamental difference from the atmospheric setting for internal waves (Olbers and Eden 2017; Baker and Mashayek 2021).

Unlike the atmosphere with its upper radiation boundary condition, the ocean has a free surface that reflects internal gravity waves (IGWs). Upward- and downward-propagating waves traveling across the same shear (Fig. 1a) will have opposing effects on the mean flow to cancel out in the absence of dissipation. However, when dissipation is taken into account, the wave field will not be perfectly symmetric in the vertical. For example, Muench and Kunze (1999, 2000) found that internal waves within equatorial deep jets transfer energy to the mean to sustain the jets. Nagai et al. (2015) simulated near-inertial waves spontaneously generated by frontal instabilities where wave energy is reabsorbed into the mean with no more than 15% dissipating or radiating away. Shakespeare and Hogg (2017, 2018) simulated interior amplification of internal waves through interactions with vertical mean shear. In all the above cases, dissipation is important to break the symmetry that might be created from repeated reflections so that net wave-mean interaction occurs.

Fig. 1.
Fig. 1.

Comparison of scenarios where internal waves are (a) free to reflect off the surface and bottom to create a vertically symmetric wave field and (b) critically trapped in a bottom-intensified current.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0058.1

b. Reabsorption of lee-wave energy

Lee-wave generation arises when abyssal currents interact with topography in the wavenumber band satisfying |f| < |kU| < N, where f is the Coriolis frequency and N is the buoyancy frequency. Because lee-wave generation is thought to be a major sink for the 1 TW of wind work, the focus here will be on balanced flow interactions with topography. Previous ocean modeling (Scott et al. 2011; Nikurashin and Ferrari 2011; Trossman et al. 2013, 2015, 2016; Wright et al. 2014; Melet et al. 2015) has assumed that all lee-wave generation is lost to turbulent dissipation. However, wave action E/|kU| conservation dictates reabsorption of lee-wave energy as |kU| → |f| in bottom-intensified flows, resulting in only a fraction available for dissipation, consistent with the observed suppression of turbulence (e.g., Waterman et al. 2014). Many abyssal currents are bottom intensified, for example, deep western boundary currents (Warren 1976) and overflows (Price and O’Neil Baringer 1994; Johnson et al. 1994; Girton and Sanford 2003). In particular, recent measurements have found bottom-intensified flows in the Drake Passage and on the flanks of the Kerguelen Plateau associated with Antarctic Circumpolar Current jets (Sheen et al. 2013; Brearley et al. 2013; Waterman et al. 2013, 2014). Waterman et al. (2014) reported that turbulent dissipation rates were below linear lee-wave generation predictions in these bottom-intensified currents. Brearley et al. (2013) found that near-bottom currents were uncorrelated with surface currents in the Scotia Sea and that turbulent dissipation inferences better correlated with near-bottom than surface currents. Sheen et al. (2013) reported bottom-intensified currents of ∼0.4 m s−1 in the bottom 1000 m at a site beneath the polar front (their Fig. 2d). There are also observations of bottom intensification of flows over topography under the Kuroshio Extension in the western North Pacific (Bishop et al. 2012). Blocking (Klymak et al. 2010) and splitting of flows by high–topographic Froude number conditions at lower topographic wavenumbers (|k| < |f/U|) are expected to amplify near-bottom flow on vertical scales of O[(f2k2U2)1/2/(Nk)] (Hogg 1973; McWilliams 1974; Bell 1975), which will be felt in the lee-wave generating wavenumber band |f/U| < k < |N/U|. Because of this, near-bottom flow intensification may be widespread over topography.

Lee waves are confined inside the |U| = |f/k| isotach of their generating flow. In a typical bottom-intensified abyssal current of finite width, lee waves encounter turning points on the sides of the jet where the cross-stream wavenumber passes through zero, so that the waves reflect back toward the jet’s center, and critical layers in the vertical where |kU| approaches |f| from above (|kU| ↓ |f|), vertical wavenumber m → ∞, vertical group velocity cgz → 0 and lee waves stall (Jones 1967; Lighthill 1967; Olbers 1981). Unstable shear, wave breaking, and turbulence production are expected at vertical critical layers, resulting in bottom trapping of lee waves and turbulent dissipation (Bell 1975; Kunze 1985; Kunze and Lien 2019). Wave action conservation dictates that at least |f|/|kU| of the lee-wave generation will be lost to turbulence production. In contrast, if trapping conditions by critical layers are absent, for example, in barotropic or surface-intensified flow, wave reflections will occur at the free surface or vertical turning points (|kU| ↑ N) (Fig. 1a; Baker and Mashayek 2021). It is trapping by bottom-intensified flows that results in net energy reabsorption by the mean (Fig. 1b).

In this paper, we build on previous numerical modeling of oceanic lee waves in depth-independent mean flow (e.g., Nikurashin and Ferrari 2010b,a; Nikurashin et al. 2014) by considering a bottom-intensified, laterally confined, O(0.1) Rossby-number jet mimicking a localized abyssal current to examine the relative roles of lee-wave energy reabsorption versus dissipation, and test analytic predictions based on wave action conservation (Kunze and Lien 2019) which imply that as much as |kU0f|/|kU0| of the lee-wave generation could be reabsorbed back into the mean flow. While most recent literature on oceanic lee waves has focused on the generation problem (e.g., Nikurashin and Ferrari 2010b,a; Zheng and Nikurashin 2019; Trossman et al. 2015), here the roles of wave-mean interactions and reabsorption as lee waves propagate in the overlying water column are highlighted. Energy rather than momentum budgets are the focus to test the predictions of wave action E/|kU| conservation and relate to energy dissipation. In section 2, the numerical model and its initial and boundary conditions are described. In section 3, Reynolds decomposition is applied to obtain energy conservation for the total, mean, and lee-wave fields, with emphasis on exchange terms that act to reallocate energy between the mean and lee waves. Simulation results and energy budgets for the total, mean, and wave fields are presented in section 4. Conclusions and discussion are in section 5.

2. Numerical modeling

Provided that the flow is bottom intensified, reabsorption and dissipation do not depend on the jet’s geometry, for example, width, height, and bottom speed, so details of the flow configuration do not impact our results. To examine the hypothesis that a significant fraction of lee-wave energy can be reabsorbed by bottom-intensified flow, an idealized numerical experiment with zonal channel flow is set up using the process study ocean model (PSOM; Mahadevan et al. 1996a,b) with f = −1.3 × 10−4 rad s−1 corresponding to 63.4°S latitude and N = 10−3 rad s−1 as typical abyssal stratification. We first set up a three-dimensional domain with the x, y, and z axes corresponding to zonal, meridional, and upward, respectively. A bottom-intensified, laterally confined zonal jet in thermal-wind equilibrium with sloping isopycnals is initialized with maximum speed of |U0| = 0.18 m s−1, decaying over 1900 m vertically and Ly = 7 km meridionally (Fig. 2), where the vertical mean shear resembles Fig. 14a of Waterman et al. (2014), and the meridional width is selected such that the jet’s Rossby number is ∼0.1 and Burger number ∼0.1. The jet’s zonal velocity is
U(y,z)=U0sech2(πyLy)(zztzbzt)2,
where zt = −100 m and zb = −2000 m are the top and bottom bounds of the jet, respectively.
Fig. 2.
Fig. 2.

Initial conditions for the bottom-intensified, laterally confined jet for the (a) density anomaly ρ(y, z) − ρ0, (b) zonal velocity U(y, z), (c) vertical profiles of along-axis zonal velocity U(z) and vertical shear Uz(z), and (d) across-stream sections of bottom zonal velocity U(y) and meridional shear Uy(y).

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0058.1

The simulation is zonally periodic. The south, north, and bottom boundaries are rigid and free slip (frictionless), while the top boundary is a free surface. The jet vanishes 2 km from the north and south boundaries to prevent wave–boundary interactions which would add unnecessary complexity to the problem. Bottom topography is a sinusoidal function of the zonal (along-stream) coordinate x with single wavelength λx = 1.2 km corresponding to lee-wave intrinsic frequency |kU0|≈ 9 × 10−4 rad s−1 = 0.9N, and amplitude a = 5 m (topographic Froude number Frt = amaN/U0 = 0.03 ≪ 1 for linear generation). A single critical layer is predicted at roughly 800-m depth on the jet’s axis where |kU| ≈ |f|. The minimum expected dissipative fraction |f|/|kU0| ∼ 0.15 based on linear wave action conservation for our choices of Coriolis frequency f, topographic wavenumber k, and bottom flow speed U0. For typical ocean topographic height spectra, most lee-wave generation occurs at superinertial intrinsic frequencies (|kU0| ≫ |f|). Since the primary goal here is to illustrate the partition between lee-wave reabsorption and dissipation in the water column, simulating linear generation at a single topographic wavelength is a justifiable simplification.

The horizontal grid size is 100 m, sufficient to resolve the topographic wavelength. In the vertical, a modified sigma coordinate system is composed of four sections with z = 0 being the free surface at rest and z = −2000 m the average water depth: (i) a surface layer whose thickness varies with the free-surface height, (ii) a middepth section from 5- to 1925-m depth where grid cells are rectilinear with uniform 8-m vertical resolution, (iii) a deep section from 1925- to 1995-m depth that transitions from the rectilinear grids to slopes parallel to topography, and (iv) a 5-m-thick bottom boundary layer.

Wave breaking and turbulence are not resolved. Their effects are parameterized with eddy viscosities and diffusivities. In the horizontal, biharmonic viscosity and diffusivity Ah = Kh = 125 m4 s−1 are employed to dissipate gridscale variance with a damping time scale of 0.58 days while preserving scales of interest. In the vertical, Laplacian viscosity and diffusivity Aυ = Kυ = 4 × 10−3 m2 s−1 are prescribed to remove lee-wave energy with a damping time scale of 0.05 days near the critical layer. Dissipation is dominated by vertical gridscale damping at the critical layer because Aυm2Ahk4 and Kυm2Khk4. To prevent shear instabilities at the critical layer, lee waves are dissipated at the vertical grid scale by the prescribed eddy viscosity and diffusivity to ensure that the maximum gradient Froude number Frg=S/N=(du/dz)2+(dυ/dz)2/N does not exceed 1.3 (Fig. 3) and so remains less than the critical value of 2 (Miles 1961). Other damping schemes were attempted, in particular the reduced-shear parameterization of Kunze et al. (1990), but proved to be numerically unstable.

Fig. 3.
Fig. 3.

(left) Across-stream and (right) along-stream sections of gradient Froude number Frg=S/N=(du/dz)2+(dυ/dz)2/N. Locations of cross sections are indicated by the dotted vertical lines. Bottom topography is plotted as the green dashed curve with its amplitude multiplied by a factor of 5 for visibility. Maximum Frg in this steady case is limited to 1.3 by gridscale damping to prevent shear instability.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0058.1

3. Energetics

The governing equations for a Boussinesq fluid on an f-plane are
DuiDtεijfuj+pxibδi3=viscousterms,
DbDt+N2w=diffusiveterms,
uixi=0,
where i and j run from 1 to 3, εij = 1 (−1) if (i, j) is an even (odd) permutation of (1, 2) and zero otherwise, and δij = 1 if i = j and zero otherwise; D/Dt = ∂/∂t + uj(∂/∂xj) is the material or Lagrangian time derivative, ui = (u, υ, w) 3D velocity, p reduced pressure deviation from hydrostatic equilibrium [that is, flattened isopycnals ρ*(z) to achieve a state of a minimum potential energy (Winters et al. 1995)], b=g[ρρ*(z)]/ρ0 buoyancy relative to reference density ρ0, and N2=g[dρ*(z)/dz]/ρ0 buoyancy frequency squared.

a. Total energy conservation

Taking the product of (1) with ui, and (2) with N−2b, then summing gives total energy conservation in an Eulerian reference frame,
t(KE+APE)=uixi(KE+APE)advectionxi(uip)pressure-work+totaldissipation,
where kinetic energy KE=(1/2)ui2=(1/2)(u2+υ2+w2) and available potential energy APE=(1/2)N2b2. This form of APE is an approximation to the more-complete definitions (e.g., Winters et al. 1995; MacCready and Giddings 2016) valid for small perturbations (Gill 1982).

b. Mean energy conservation

To investigate energy exchange between mean and wave fields, (1) and (2) are Reynolds-decomposed to separate mean (overbar) and lee-wave (prime) fields, u=u¯+u and b=b¯+b, following Hasselmann (1971), with the overbar denoting a zonal average and bold symbols denoting vectors. For lee waves, u¯=b¯=0. For sinusoidal topography and a periodic channel simulation, zonal averaging effectively separates mean and wave fields. Averaging over nonlinear advection terms gives rise to terms associated with nonzero lee-wave momentum- and buoyancy-fluxes uiuj¯ and buj¯ that act on the mean.

The mean momentum and buoyancy equations are
u¯it+u¯ju¯ixjεijfu¯j+p¯xib¯δi3=xj(uiuj¯)+viscousterms,
b¯t+u¯jb¯xj+N2u¯3=xj(buj¯)+diffusiveterms.
To obtain mean energy conservation, (5) and (6) are multiplied by u¯i and N2b¯, respectively, and summed,
t(MKE+MAPE)=uiuj¯u¯ixj+N2buj¯b¯xjexchangeu¯jxj(MKE+MAPE)advectionxj(uiuj¯u¯i+N2buj¯b¯)redistributionxi(u¯ip¯)pressure-work+meandissipation,
(Müller 1976), where mean kinetic energy (MKE) =(1/2)ui¯2=(1/2)(u¯2+υ¯a2+w¯a2) and mean available potential energy MAPE=(1/2)N2b¯2. Subscript a indicates that the velocity component is ageostrophic and part of the cross-stream circulation. The first term on the rhs of (7) is the mean energy exchange with waves and the second term is the advection of mean energy by mean velocities. The third term arises from redistribution of mean momentum and buoyancy by lee-wave fluxes uiuj¯ and buj¯, that is, lee-wave drag. In our case, the dominant contribution is the vertical component for j = 3, that is, (/z)(uiw¯u¯i+N2bw¯b¯), which represents vertical transmission into the water column of the loss of mean energy to lee-wave generation at the bottom. The exchange and redistribution terms can be combined as [uiuj¯(u¯i/xj)+N2buj¯(b¯/xj)](/xj)(uiuj¯u¯i+N2buj¯b¯)=u¯i(/xj)(uiuj¯)N2b¯(/xj)(buj¯) and interpreted as gradients of the lee-wave Reynolds terms acting on the means. The fourth term is mean pressure-work and the last term is the dissipation of the mean.

c. Wave energy conservation

Wave momentum and buoyancy equations are obtained by subtracting mean (5) and (6) from total (1) and (2),
uit+u¯juixj+uju¯ixjεijfuj+pxibδi3=xj(uiujuiuj¯)+viscousterms,
bt+u¯jbxj+ujb¯xj+N2u3=xj(ujbujb¯)+diffusiveterms.
Wave energy conservation is derived by multiplying (8) and (9) by ui and N2b, respectively, zonally averaging and summing,
t(IKE¯+IAPE¯)=uiuj¯u¯ixjN2buj¯b¯xjexchangeu¯jxj(IKE¯+IAPE¯)advectionxj(12uiuiuj¯+12N2bbuj¯)redistributionxi(uip¯)pressure-work+wavedissipation,
(Müller 1976; Polzin 2010), where internal-wave kinetic energy IKE=(1/2)ui2=(1/2)(u2+υ2+w2) and internal-wave available potential energy IAPE=(1/2)N2b2. The first term on the rhs of (10) is wave energy exchange with the mean, which has the same magnitude but opposite sign to the exchange term in (7). The second term is advection of wave energy by mean velocities. The third term is redistribution of wave momentum and buoyancy by wave fluxes; it is higher order and was ignored in Müller (1976) but is retained here because lee-wave fluctuations are comparable to the mean near the critical layer. It can also be written as wave advection of wave energy, (/xj)[(1/2)uiuiuj¯+(1/2)N2bbuj¯]=(/xj)[uj(IKE)¯+uj(IAPE)¯] (Scotti and White 2014). It is a nonlinear modification of the lee-wave energy-flux convergence since the energy-flux convergence (pressure-work) and wave redistribution terms can be combined as (/xi)[(cgi+ui)IE], where cgIE=cg(IKE+IAPE)=up is the linear energy-flux and u′IE is the nonlinear energy-flux (e.g., Moum et al. 2007). The fourth term is the lee-wave pressure-work, which is the convergence of the lee-wave energy-flux up¯ (i.e., generation), and the last term is lee-wave dissipation. The sum of (7) and (10) is closed.
In the numerical simulation budget calculation, wave redistribution terms are not computed explicitly from the triple correlations uiuiuj¯ and N2bbuj¯ but replaced by total advection minus mean advection plus redistribution minus wave advection,
uixi(KE+APE)+u¯jxj(MKE+MAPE)+xj(uiuj¯u¯i+N2buj¯b¯)+u¯jxj(IKE+IAPE),
that is, as a residual. Such an approach ensures overall energy conservation because the total (4) and mean budgets (7) are accurate up to the second order in the lee-wave perturbations, while numerical evaluation of triple covariances from model outputs (third order in the lee-wave perturbations) has nonnegligible errors that vary with the number of topographic wavelengths that the zonal average is defined over (four in our experiment) and the time of snapshot when the simulation is considered steady and results are taken. It raises uncertainty about attribution to wave redistribution, amounting to a truncation of wave energy conservation to the second order. We will refer to the residual in the wave energy budget as wave nonlinearity or wave redistribution but caution that this interpretation has not be confirmed with direct calculation due to aforementioned numerical error.

d. Restoration of the jet

Without external forcing, the experiment is an initial-value problem in which both the jet and lee waves spin down due to gridscale damping. To investigate and quantify the exchange of energy between mean and wave fields in a steady state, the kinetic and available potential energy of the jet are continuously restored. Restoration is of the form,
DΨDt=1τ(Ψ¯Ψ¯ini),
where Ψ represents the density and zonal velocity for the instantaneous jet and Ψini represents the initial jet. This is intended to mimic replenishment of balanced energy by large-scale forcing. Results are insensitive to the value of restoration time scale τ, which is chosen as 1 day in the simulation. Restoration imposes an extra term on the rhs of the total and mean energy conservation (4) and (7)
1τ[u¯(u¯u¯ini)+N2b¯(b¯b¯ini)],
while its contribution to wave energy conservation (10) is zero.

4. Results

Linear lee waves with dominant frequency |kU0| ≈ 0.9N are generated by the bottom topography and propagate upward while remaining trapped within the jet, achieving steady state after 5 days (Fig. 4). As they approach the critical layer at 800-m depth, their vertical wavelengths decrease while horizontal velocities u′, υ′ increase (IKE and IAPE intensify; Fig. 5b). Cross-stream asymmetry (Fig. 4c) is due to the difference in the bandwidth of the internal-wave band due to differences in stratification N and effective Coriolis frequency on the two sides of the jet. The south (left) side is more stratified (Fig. 2a) and has negative (cyclonic) relative vorticity (Fig. 2d).

Fig. 4.
Fig. 4.

(left) Across-stream and (right) along-stream sections of simulated lee-wave zonal, meridional, and vertical velocities (u′, υ′, w′) on day 8. Bottom topography is plotted as green dashed curve with its amplitude amplified by a factor of 5 for visibility.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0058.1

Fig. 5.
Fig. 5.

Vertical profiles of horizontally averaged (a) MKE and MAPE and (b) IKE and IAPE. Mean energies are O(10−3) J kg−1 at the bottom and O(10−4) J kg−1 at the critical depth. Lee-wave energies are ∼5 × 10−6 J kg−1 near the bottom and ∼1 × 10−5 J kg−1 at ∼1100-m depth below the critical layer.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0058.1

Velocity profiles (Fig. 6) at the center of the jet show that u′ and υ′ are 180° and 90° out of phase with w′, respectively. Horizontal-plane fluid trajectories are elongated ellipses near the bottom with |u′| > |υ′|, becoming more circular with |u′| ≈ |υ′| near the critical layer, consistent with irrotational near-buoyancy waves becoming near-inertial as they propagate upward into decreasing |kU|.

Fig. 6.
Fig. 6.

Vertical profiles of lee-wave velocities (u′, υ′, w′) in the left three panels, as well as horizontal velocity vector (u′, υ′) in the right panel, at the center of the jet. Red/blue stars mark depths where w′(z) has local extrema.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0058.1

At steady state, the jet’s zonal velocity u¯ (Fig. 7a) does not fully restore to its initial conditions (deviation in Fig. 7b) because of continuous competition between restoration and lee-wave modulation of the jet. Ageostrophic υ¯a and w¯a (Figs. 7c,d) indicate a cross-stream secondary circulation due to geostrophic adjustment associated with this competition acting to sharpen horizontal density gradients and maintain the jet’s structure (Fig. 7b). It is concentrated at the critical layer on the jet axis and, less dramatically, along the jet flanks.

Fig. 7.
Fig. 7.

Across-stream sections of (a) zonally averaged zonal velocity u¯(y,z) and (b) its deviation from initial conditions (u¯uini), (c) zonally averaged ageostrophic meridional velocity υ¯a, and (d) vertical velocity w¯a on day 8.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0058.1

a. Wave energy budget

The dominant balance in the horizontally averaged lee-wave energy budget (10) is between a pressure-work source {or energy-flux convergence [Fig. 8a(3)]} and a reabsorption sink {loss through exchange with the mean [Fig. 8a(1)]} both of magnitude of (2–3) × 10−10 W kg−1 below the critical depth. These signify that a considerable fraction of lee-wave energy is lost back to the jet. The exchange term is negative in the lee-wave budget, indicating reabsorption of lee-wave energy into the mean, so is also referred to as reabsorption. The vertical integral of lee-wave pressure-work [Fig. 8a(3)], that is, lee-wave vertical energy-flux or generation, is consistent with linear generation theory, wp¯(1/2)Ua2(N2k2U2)(k2U2f2)2×107m3s3 (Bell 1975). Near the critical layer, there is a dissipation sink of O(10−10) W kg−1 [Fig. 8a(4)], but its vertical integral is much smaller than either the pressure-work [Fig. 8a(3)] or reabsorption [Fig. 8a(1)]. Local dissipation rates (without horizontal averaging) are as high as O(10−8) W kg−1 (Fig. 9).

Fig. 8.
Fig. 8.

Energy budget terms for the (a) wave (10), (b) mean (7), and (c) total fields (4). The wave budget is primarily a balance between linear lee-wave energy-flux convergence, that is, (a:3) pressure-work, (a:2), nonlinearity, and (a:1) loss through reabsorption by the mean. The mean balance is primarily between (b:1) reabsorption of lee-wave energy and (b:2) redistribution into the water column of the loss to lee-wave generation at the bottom.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0058.1

Fig. 9.
Fig. 9.

(a) Cross-stream and (b) along-stream sections of numerical dissipation rates ϵ. Bottom topography is plotted as the green dashed curve with an amplification factor of 5. Dissipation hotspots up to O(10−8) W kg−1 at 800-m depth are associated with the lee-wave critical layer at the jet’s axis where |kU| ≈ |f|.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0058.1

Below the critical depth, there is a wave advection plus nonlinearity sink of roughly 10−10 W kg−1 [Fig. 8a(2)] dominated by wave nonlinearity, which was computed as a residual (section 3c). Wave advection is O(10−13), negligibly small. If wave nonlinearity can be attributed to wave redistribution (nonlinear wave pressure-work), (/z)[w(IKE+IAPE)¯] (Moum et al. 2007), it indicates a nonlinear reduction of the lee-wave energy-flux convergence [pressure-work; Fig. 8a(3)] because it has the opposite sign. Integrated over the domain, reabsorption into the mean, wave nonlinearity, and dissipation correspond to approximately 55%, 29%, and 14% of linear lee-wave generation, respectively.

b. Mean energy budget

The predominant balance in the horizontally averaged mean energy budget (7) is between a reabsorption source [exchange; Fig. 8b(1)] and an advection plus redistribution sink [Fig. 8b(2)]. Exchange is dominated by uw¯(u¯/z) (Fig. 10a), and advection plus redistribution by the redistribution term (/z)(uw¯u¯) (Fig. 10e). The advection contribution (Fig. 10d) is due to the ageostrophic w¯a, localized near the critical layer, moving wave energy. The dominant redistribution term (/z)(uw¯u¯) is the vertical derivative of uw¯u¯ and represents vertical transmission into the water column of the mean energy loss to lee-wave generation, that is, wp¯, at the bottom. That is, wp¯=[uw¯(f/N2)υb¯]u¯ at the bottom (Eliassen and Palm 1961; Shaw and Shepherd 2008; Kunze and Lien 2019), where uw¯(f/N2)υb¯ is the Eliassen–Palm flux, because the cross-stream buoyancy-flux is small in our simulations. Dissipation of the jet [Fig. 8b(4)] is negligible because the scale selectivity of vertical damping acts primarily on lee waves. Restoration [Fig. 8b(5)] replenishes the jet’s kinetic and potential energies, extending up to the critical depth because of redistribution of the bottom lee-wave sink over this depth range.

Fig. 10.
Fig. 10.

(left) Exchange and (right) advection plus redistribution terms in mean energy conservation (7). Exchange is dominated by uw¯(u¯/z) and advection plus redistribution by the redistribution term (/z)(uw¯u¯), which represents transmission into the water column of the loss of mean energy to lee-wave generation at the bottom.

Citation: Journal of Physical Oceanography 53, 2; 10.1175/JPO-D-22-0058.1

c. Total energy budget

With restoration injecting energy into the jet and dissipation extracting energy at the grid scale, total energy is balanced so that the time derivative of total KE plus APE is nearly zero. As an internal process, exchanges [Figs. 8a(1),b(1)] do not appear in the total budget. Total dissipation [Fig. 8c(4)] is identical to lee-wave dissipation [Fig. 8a(4)]. Restoration [Fig. 8c(5)] is identical to restoration of the mean [Fig. 8b(5)].

5. Summary

In this paper, lee-wave generation, propagation, interaction, and dissipation in a steady, bottom-intensified and laterally confined jet are numerically examined in a zonally periodic channel. The jet flows over monochromatic sinusoidal topography in the limit of topographic Froude number Frt ≪ 1. Lee waves are generated with dominant intrinsic frequency |kU0| ≈ 0.9N ≈ 7.23|f| and are trapped inside of the jet by the |U| = |f/k| isotach.

For these monochromatic lee waves, the wave budget is dominated by a generation source and a reabsorption sink. Dissipation is concentrated near the critical layer, but its water-column-integrated value is relatively weak, only about 14% of lee-wave generation. This dissipative fraction is consistent with the minimum wave action conservation prediction |f/kU0| of Kunze and Lien (2019), but this may be a coincidence, because they assumed linear lee-wave propagation. Wave dynamics in the water column can be nonlinear, especially when a critical layer is present, so that the linear partition of lee-wave generation into exchange with the mean (reabsorption in our case) and dissipation is modified by nonlinearity that may not have been properly taken into account because nonlinear redistribution was treated as a residual.

The mean budget is dominated by a reabsorption source [Fig. 8b(1)] with the opposite sign to the lee-wave energy-flux convergence [Fig. 8a(3)], and redistribution into the water column of the loss to lee-wave generation at the bottom. The mean budget also has contributions from restoration, as well as pressure-work associated with a cross-stream ageostrophic circulation.

This research explored energy budgets for different components (lee waves, mean flow, and total) and highlighted energy exchange between mean and lee-wave fields. A bottom-intensified, laterally confined jet that mimics a localized abyssal current leads to reabsorption of lee-wave energy into the mean, which could not be seen in previous oceanic lee-wave modeling studies using depth-independent flow (e.g., Nikurashin and Ferrari 2010b,a; Nikurashin et al. 2014; Mayer and Fringer 2020) but is well known in atmospheric studies (Eliassen and Palm 1961; Jones 1967; Bretherton and Garrett 1968).

6. Discussion

Lee waves have been advocated as a major sink for the 1-TW balanced power input (Scott et al. 2011; Nikurashin and Ferrari 2011; Melet et al. 2014; Wright et al. 2014; Nikurashin et al. 2014; Trossman et al. 2016; Yang et al. 2018). Most estimates for this sink have assumed that lee waves represent a one-way (forward) energy cascade to turbulent dissipation. However, microstructure measurements in the Southern Ocean (e.g., Waterman et al. 2014) report a shortfall of dissipation, in particular in bottom-intensified flows. Wave action conservation suggests reabsorption of lee-wave energy in bottom-intensified flows (Kunze and Lien 2019). Thus, the common assumption of equating lee-wave dissipation with bottom generation in the recent ocean modeling literature is challenged. Our numerical study has confirmed that, in bottom-intensified flow, the loss of balanced energy to lee-wave generation is partially reversible, with reabsorption being of O(1) importance in reducing turbulent dissipation relative to lee-wave generation, making lee waves a less-efficient sink for balanced energy. Thus, the lee-wave sink of balanced flow might be overestimated. The extent to which lee waves represent the dominant sink for balanced energy requires a better census of bottom-intensified flow in the ocean.

The idealized numerical experiment described here confirms a nondissipative fate for lee waves (i.e., reabsorption, mechanism two in the introduction) but is not a complete description. The topography is limited to one dimensional and monochromatic, and the topographic Froude number Frt is small to prevent nonlinear generation (mechanism one). The adopted gridscale damping strongly dissipates lee waves near the critical layers before shear can trigger instabilities so that the propagation of free waves leading to remote dissipation (mechanism three) is suppressed. Likewise, remote dissipation by lee-wave energy swept downstream (mechanism four) is excluded because the topographic amplitude is homogeneous and the channel periodic. Only bottom-intensified currents were considered, corresponding to lee-wave reabsorption and reduced dissipation. The case of surface-intensified flow has been explored by Baker and Mashayek (2021), where extraction of balanced energy by lee waves during upward propagation reverses upon reflection downward from the surface (Fig. 1a). In this case, lee waves can represent a larger sink for balanced flow than just their initial generation, but this depends on where in the water column and flow they dissipate.

Investigation of a broadband topographic height spectrum (e.g., Goff and Jordan 1988) is a next step to allow excitation of a continuous lee-wave wavenumber spectrum |f/U0| < k < |N/U0|. Critical layers will then be more broadly distributed through the water column. The reabsorptive and dissipative fractions for broadband lee waves will hence be affected. Kunze and Lien (2019) found approximate equipartition for typical abyssal topographic spectral slopes and |N/f| ratios. For subinertial wavenumbers below |f/U0|, reduction of effective topographic width by nonlinear generation allows possible lee-wave generation (Klymak 2018; Klymak et al. 2021) and hence must also be taken into account. Blocking and splitting for |k| < |f/U0| can lead to bottom modification of mean flows on vertical scales of O[(f2k2U2)1/2/(Nk)] (e.g., Klymak et al. 2010). It can either accelerate or decelerate the near-bottom flow experienced in the lee-wave-generating band |f|/|U0| < |k| < N/|U0|, which tends to be linear with topographic Froude numbers am < 1 because of both shrinking topographic amplitude a and vertical wavenumber m with increasing topographic wavenumber k.

Nevertheless, to better quantify lee-wave energy partition in mean shear, impacts of broadband topographic spectra and nonlinear generation must be studied. With multiple competing processes at play, determining lee wave’s role in dissipating versus redistributing balanced energy requires further research for better parameterizations of wave drag and mixing in ocean general circulation models (Melet et al. 2015).

Acknowledgments.

The authors thank two anonymous reviewers for their comments. This research was supported by NSF Grants OCE-1756279 (WHOI), OCE-1756093 (NWRA), and OCE-1755313 (UMass Dartmouth).

Data availability statement.

The process study ocean model is illustrated on the lab web page at https://mahadevan.whoi.edu/PSOM. Model configuration and code has been uploaded for an example experiment in the GitHub archive of the PSOM, version 1.0, at https://github.com/PSOM/V1.0/tree/master/code/leewaves. Simulation outputs and analysis on which this paper is based are too large to be retained or publicly archived with available resources but will be made available to collaborators or interested individuals upon request.

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