Energetics and mixing of stratified, rotating flow over abyssal hills

One of the proposed mechanisms for energy loss in the ocean is through dissipation of internal waves, in particular above rough topography where internal lee waves are generated. Rates of dissipation and diapycnal mixing are often estimated using linear theory and a constant value for mixing efficiency. However, previous oceanographicmeasurements found that non-linear dynamics may be important close to topography. In order to investigate the role of non-linear interactions, we conduct idealized 3D numerical simulations of steady flow over 1D topography and vary the topographic height, which correlates to the degree of flow non-linearity. We analyze spatial distribution of energy transfer rates between internal waves and the non-geostrophic portion of time-mean flow, and of dissipation and diapycnal mixing rates. In our simulations with taller, more non-linear topographies, energy transfer rates are similar to previously unexplained oceanographic observations near topography: internal waves gain energy from time-mean flow through horizontal straining and lose energy through vertical shearing. In the tall topography simulations, buoyancy fluxes also play a significant role, consistent with observations but contrary to linear wave theory, suggesting that quasigeostrophy-based approximations and linear theory may not hold in some regions above rough topography. Both dissipation and mixing rates increase with topographic height, but their vertical distributions differ between topographic regimes. As such, vertical profile of mixing efficiency is different for linear and non-linear topographic regimes, which may need to be incorporated into parameterizations of small-scale processes in models and estimates of ocean energy loss. 5


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Energy input into the ocean from wind work, differential surface buoyancy forcing, and tides 25 is eventually lost through small-scale turbulent processes (Munk and Wunsch 1998;Wunsch and 26 Ferrari 2004;Hughes et al. 2009;Zemskova et al. 2015). One of the important pathways to 27 dissipation is the breaking of internal waves generated as a result of interactions between a steady where ℎ 0 is the maximum topographic height. The domain size is = = 2 / and = 2 . nents, i.e., What we refer to as ageostrophic time-mean (hereafter shortened into time-mean) fields (u, ) here 154 include all zero-frequency flows and most notably the lee waves, which have zero-frequency in the 155 topographic reference frame, but exclude the geostrophic flow. 156 We find a noticeable time-scale separation between the near-zero frequency ( ≈ 0) motions 157 and motions at frequencies greater than as shown by KE and APE (defined below) spectra in 158 Fig. 4, which allows us to consider the zero-frequency and internal wave reservoirs separately. which are also reported in the Eulerian reference frame. In the Eulerian framework, stationary 168 or quasi-stationary lee waves have zero-frequency, and thus may be included in the time-mean 169 component. We specifically separate the time-mean component, which includes the lee waves in 170 our simulations, from the geostrophic flow to test the assumptions of the QG limit.

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In this study, we focus on the time-mean (i.e., zero-frequency minus geostrophic flow) and 172 internal wave (i.e., ∈ [0.75 , ] frequency band) motions. Although internal waves also include 173 lee waves, throughout the text we will refer to motions with frequencies ∈ [0.75 , ] as simply 174 internal waves for brevity, as lee waves are a part of the time-mean flow. The total energy in each kinetic energy , divided by 0 , for the time-mean and internal wave components, respectively, 177 and , as = 1 2 ¯ 2 +¯ 2 +¯ 2 and = 1 2 + + .
The APE, , is the difference between the total gravitational potential energy and the background 179 potential energy, = − * , which is the minimum potential energy of the system if all water parcels 180 were resorted adiabatically according to their densities (Winters et al. 1995). We find the reference Here, * is the buoyancy of the resorted buoyancy field at a given height such that * ( * ) = . All 188 potential energy from the geostrophic flow is in the background potential energy reservoir because 189 = 2 , with 2 a positive constant. Because * is non-linear such that where = ( , , , ). In this study, we specifically consider inertial motions, i.e.,˜ . Initial time 204 0 is large enough for the dynamics to have become reasonably stationary, and which we take as 205 0 = 4 for = 1, 2 and as 0 = 8 for = 0.6, and choose = 4 , multiple of 2 / .

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We also compute the non-linear terms that force motions at = , namely: Λ represents the sum of the triadic non-linear interactions between the inertial signal ( = ) and 208 all other frequency pairs ( 1 , 2 ) such that 1 + 2 = . If Λ > 0 (< 0), then nonlinear interactions 209 transfer energy to (away from) the inertial motions. Here, we specifically emulate the energy term calculations presented in Cusack et al. (2020). 212 We focus on the energy sinks (KE dissipation and irreversible mixing) and exchange between the 213 time-mean field and internal waves, and the analysis of the full KE and APE budgets are beyond the 214 scope of this study. The energy budget terms considered in this paper are summarized in a diagram 215 in Figure 3. The internal wave components can be computed by integrating the cospectrum (real 216 part of the cross spectrum), ( ), for example, over [ , ] frequencies as in Cusack et al. (2020).

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Namely, for any two internal wave quantities and , the temporally-averaged cross correlation 218 terms are computed as We carry out the frequency analysis (i.e., computing spectra and cospectra) at each ( , , ) over the 220 last 4 of each simulation prior to any horizontal averaging in order to capture all waves. Analogous 221 to Eqn. (11), we compute the energy budget terms at near-inertial frequencies by integrating over 222 ∈ [0.75 , 1.25 ] and denote these energy budget terms with subscript or superscript . We also 223 compute the energy budget terms for super-inertial internal waves by integrating over ∈ [1. 5 , ] 224 and denote these energy budget terms with subscript or superscript (for super-inertial waves).

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To be precise, we define energy transfers to-and-from near-inertial waves and to-and-from super- Following a mean-eddy decomposition, where an "eddy" field is taken to be deviation from Time-averaged irreversible mixing rates, which are the APE dissipation rates due to diabatic fluxes 232 (Winters and D'Asaro 1996) for the time-mean component and internal waves are defined as where we recall that in our simulations, = . In Eqns. (13) and (14), we bound the integrals with 234 1 = 0.75 , 2 = 1.25 for near-inertial waves, and 1 = 1.5 , 2 = for super-inertial waves, 235 as discussed above.

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The energy exchange between the time-mean and internal wave reservoirs comprises of conver-237 sion rates of KE, ( ), and of APE, ( ), defined as and As convention, ( ), ( ) > 0 (< 0) indicate energy gain ( In this study, we will compute and discuss all of the energy conversion and dissipation rates 249 defined in §3c. However, in this subsection, we will focus specifically on the vertical KE transfer 250 term (i.e., ( )) and horizontal APE transfer term (i.e., ℎ ( )), which are often combined into field to be in thermal wind balance, such that the EP flux is defined as also assuming per the QG limit that local buoyancy perturbations from the reference density profile  (17) can be reduced to an "effective" transfer rate This approximation has two important implications: (1) as → , ( ) ≈ − ℎ ( ), such that 259 there is no energy transfer from vertical shear at the inertial frequency, and (2) at , ℎ ( ) ≈ 0, 260 such that buoyancy fluxes do not contribute to internal wave energy at higher frequencies.

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In the following section, we analyze the spatial patterns of ℎ ( ), ( ), and ℎ ( ) in our 267 simulations. We assess whether QG theory holds for these energy transfer rates and investigate 268 whether the dynamics of bottom topography-driven flows can explain the above observations. We

Conversion between mean and internal wave fields
272 a. KE transfer to internal waves 273 We first investigate the KE exchange between zero-frequency flow and internal waves, which are 274 separated into the near-inertial and super-inertial frequencies as defined in §3. Specifically, we are 275 interested in (1) identifying the regions where the near-inertial waves gain and lose energy, and

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(2) comparing the near-inertial waves with internal waves at other frequencies. We focus on the 277 near-inertial waves in particular because we previously found in Zemskova and Grisouard (2021) 278 strong non-linear wave-wave interactions between near-inertial waves and lee waves, and that these 279 interactions play an important role in KE dissipation. 280 We find that the near-inertial waves primarily gain energy through ℎ ( ) predominantly via the  (2021),¯ approximately corresponds to lee waves 293 and˜ to freely-propagating near-inertial waves, identified from their slopes and wavelengths.

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The asymmetry between KE gain by near-inertial waves downstream of topography and their 295 KE loss upstream of topography through horizontal shearing is related to how the flow has to 296 accelerate and slow down as it goes over the bump. While the flow immediately above the 297 topography accelerates as it goes over the obstacle, further above it there is a layer of slower-298 moving fluid (coined "stagnant" by Winters and Armi (2012)). This layer grows downstream of 299 the topography as the flow dissipates and slows down, such that ¯ < 0. Because is positive 300 definite − ¯ > 0 and the near-inertial waves are generated because of the horizontal gradient 301 of the zero-frequency flow velocity. Conversely, the near-inertial waves lose KE upstream, where 302 the flow accelerates to go over the obstacle, because ¯ > 0.

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Similar considerations explain how near-inertial waves lose KE to the mean flow slightly aloft of 304 the downstream generation site described above. There, the zero-frequency flow decelerates away 305 from the topography, such that ¯ < 0 going upward into the stagnant layer from the accelerated 306 near-topography layer. We recall that a freely propagating inertial wave in our mean current can where ( 1 , 2 ) are the two horizontal wavenumbers corresponding to roots of the hydrostatic 309 dispersion relation, and 2 is equal to the topographic wavenumber (see Appendix B in Zemskova 310 and Grisouard (2021)). As such, a freely propagating inertial wave with travelling along the steep 311 characteristic (i.e., 2 ) will have a non-zero vertical velocity, unlike an inertial wave travelling i.e., energy transfer from zero-frequency to near-inertial frequency motions) is − ¯ < 0 (not 316 shown), that is, the near-inertial waves lose energy to the zero-frequency flow.

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Our results suggest that as non-linearity increases, the hydraulically-controlled leeward side of 318 the topography may host a net transfer of KE from near-inertial waves back to the time-mean flow.

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Indeed, while both | ℎ ( ) | and | ( ) | increase with increasing , the increase of the vertical 320 transfer term is more substantial. In particular, averaging at a given (cf. right panels of Fig Grisouard (2021)). However, overall KE transfers to or from the higher-frequency internal waves 348 are considerably less significant compared with the near-inertial waves. In this section, we investigate whether the linear theory assumptions regarding buoyancy fluxes 363 hold for bottom-driven flows, namely that (1) they balance the vertical shear term at = and (2) 364 they are small at .

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To explain why near-inertial waves can have a direct impact on APE budgets, we recall that 366 freely propagating inertial waves travelling along the steep characteristic (cf. Eqn. (19)) can induce 367 isopycnal displacements in our set-up. The characteristics of the real part of˜ align with 2 368 (cf. dashed lines in the middle column of Fig. 7), which is not purely horizontal, whereas the 369 characteristics of the real part of˜ are essential horizontal. It suggests that while KE may 370 be preferentially transferred to near-inertial waves with flat characteristics (i.e., = 0), APE is 371 preferentially transferred to the ones propagating along slanted characteristics (i.e., ≠ 0) and 372 near-inertial waves may indeed have a footprint on the APE budget. 373 We find that the QG approximation for the EP flux does not hold, especially for our simulations 374 with taller topography. The left column of Figure 9 shows the APE transfer rates from zero-375 frequency motions to near-inertial waves due to horizontal buoyancy fluxes (i.e., ℎ ( ) ). The 376 middle column shows ( ) + ℎ ( ) , which according to the expression for the "effective" 377 energy transfer rate shown in Eqn. (18), should be zero. However, we find that ( ) and 378 ℎ ( ) generally do not cancel each other out. This is true even when we integrate the energy 379 transfer rates along planes (cf. right panels of Fig. 9).

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In particular, APE is transferred to the near-inertial waves through buoyancy fluxes both upstream 381 and downstream of the region where the near-inertial waves lose KE through vertical shear.

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Downstream of the topographic obstacle, ¯ < 0 (right panels in Fig. 6) as denser water is 383 entrained from below, especially in the larger -regime, in which vigorous overturning occurs (cf.  At super-inertial frequencies (1.5 ≤ ≤ ), APE transfer due to buoyancy fluxes is indeed 398 smaller than the KE transfer due to vertical shear for small topographies (cf. Fig. 10(a-d)). However, 406 In this section, we described a qualitative agreement between our non-linear simulations and the 407 near-bottom ocean observations with respect to the KE and APE transfer rates. In the next section, 408 we will investigate the effect of such non-linear dynamics on KE and APE dissipation rates, and 409 subsequently, mixing efficiency.
Specifically, in order to investigate the role of topography-driven dynamics, we compute over 414 four volumes bounded in terms of heights above the bottom: We can also define mixing efficiency locally to analyze 416 its spatial distribution, such that: It is important to note that 1 ∫ ≠ . In broad strokes, the distribution of mixing efficiency 418 reflects the competition between KE and APE dissipation rates, which we describe in detail below.

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For all simulations, enhanced KE dissipation is sustained by energy exchanges with near-inertial Grisouard (2021)), and mixing rate is low ( Fig. 12(a)). However, resonant non-linear wave-wave eroded by overturns, so mixing rate (i.e., ( ) in Fig. 11(b)) starts small near the topography but 446 decreases less sharply with than ( ) and, as a result, increases with .

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In contrast, the region close to topography for tall topographic regimes (e.g., = 2) is character- 2) (cf. Fig. 11(a)), and local , in particular downstream of topography where 456 hydraulics are important (cf. Fig. 12(c)). Over this vertical extent, internal waves continue to 457 gain APE from the KE reservoir in simulation with = 2, whereas the magnitude of drops 458 close to zero for simulations with smaller topographies (Fig. 11(c)). The relative importance of the 459 exchange between internal wave KE and APE also appears in the KE and APE spectra in Fig. 4:

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KE decreases and APE increases with for = 2 (and to lesser extent = 0.1), whereas for 461 = 0.6, changes in KE and APE spectra with are less prominent.

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It is noteworthy that super-inertial internal waves are responsible for a large portion of APE 463 dissipation, especially compared with the near-inertial waves (cf. Fig. 13(a-b)). We can estimate

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It is important to note that mixing rates and mixing efficiency directly above the topography are very small for this simulation with = 2. It happens both because of topographic blocking of the flow and because we do not restore the stratification in these simulations, such that there is a bottom layer with reduced buoyancy frequency, similar to other previous studies of flows over periodic hills (e.g. Klymak 2018; Mayer and Fringer 2020).
the rate of APE transfer from near-inertial to super-inertial internal waves as which is analogous to the APE transfer term from zero-frequency flow to internal waves in Eqn. (16) topographies (cf. Fig. 6(b,d) and Fig. 7(b,e)). As such, for = 2, mixing rates, which are functions 475 of buoyancy gradients, sharply decrease with height above = 0.2 by two orders of magnitude 476 ( Fig. 11(b)). Yet, higher kinetic energy dissipation rates are sustained away from the topography 477 ( Fig. 2(c)) owing to the near-inertial motions that result from resonant wave-wave interactions.

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As a result, mixing efficiency decreases away from the bottom in the simulations with non-linear 479 topography ( = 2).

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Mixing efficiency for the transitional topographic regime = 1 at intermediate depths ∈

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However, is higher both near the topography ( < 0.1) and remarkably far away from the 483 topography ( ∈ [0.3, 0.4)) for = 1 compared with = 0.6 and = 2 simulations. In the 484 = 1 simulation near the topography, there is some overturning and turbulent mixing, and energy 485 transfer from KE to APE for the internal waves (Fig. 11(c)). However, because overturning is 486 not as vigorous as in the higher topography simulation ( = 2), buoyancy gradients are not eroded 487 as much, and there is APE transfer to super-inertial internal waves both from zero-frequency 488 motions ( Fig. 10(c)) and from near-inertial motions (Fig. 11(d)). As such, close to topography, 489 irreversible mixing rates for the = 1 and = 2 simulations are similar ( Fig. 13(a-b)), whereas KE 490 dissipation rates for = 1 simulation are 1 − 2 orders of magnitude smaller than the = 2 simulation 491 ( Fig. 13(c-d)).

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In our simulations, we observe that KE is dissipated more by the zero-frequency flow (e.g., lee 493 waves) than internal waves (Fig. 13(c)), whereas APE dissipation rates from the zero-frequency 494 flow and internal waves is approximately equal, especially for = 1, 2 ( Fig. 13(a)). KE dissipation 495 rates are driven by the non-linear wave-wave interactions, primarily between the near-inertial waves 496 and zero-frequency flow, which are small above = 0.2 for the = 1 simulation (Fig. 2(b)).
However, APE dissipation rates, in particular of super-inertial internal waves, remain high even 498 at > 0.3 ( Fig. 13(b)), as buoyancy gradients and buoyancy fluxes remain large enough that 499 APE is transferred to higher frequencies ( Fig. 11(d)). Because of such balance (low KE dissipation 500 and high APE dissipation rates), over ∈ [0.3, 0.4) is larger in the simulation with = 1 501 compared with other simulations. Interestingly, for = 0.6 and = 1, the region of increased 502 mixing efficiency lies above the region of increased energy transfer to super-inertial internal 503 waves (cf. Fig. 11(a,d)). It is possible that because of larger group velocities at higher frequencies 504 (all else being equal, per the internal wave dispersion relation), super-inertial internal waves can 505 propagate vertically faster and displace isopycnals farther away from their generation site. This or 506 some other mechanism for non-local transport and dissipation of APE could be further explored in 507 a follow-up study.

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Our results highlight that although both KE and APE dissipation rates increase with topographic 509 height, their spatial distributions are significantly different, such that the effects of topography on 510 mixing efficiency are non-trivial. In previous sections, we showed that energetics of the flow over the interaction between lee waves and near-inertial waves to be important for bottom-driven flows.

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In the energetic analysis of our simulations, we indeed find that the wave-wave interactions between 553 the non-geostrophic time-mean flow (primarily lee waves) and internal waves (with frequencies 554 between and ) produce energy transfer rates that qualitatively agree with the observations.

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Our findings have two main implications. First, the "effective" transfer rate approximation has 556 been often applied to parameterize the effective vertical viscosity by combining the contribution 557 of vertical stresses and buoyancy fluxes, and then used to estimate dissipation rates (e.g., Ferrari Second, it has been previously suggested that flow over two-dimensional topography may prefer-571 entially go around a tall topographic obstacle rather than over it, such that the energy flux into the 572 internal waves is reduced for sufficiently non-linear topographies (e.g., > 0.7 in Nikurashin et al.

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(2014)). However, in this study, we find a qualitative agreement in energy transfer rates between the 574 ocean observations and our simulation with non-linear topography ( = 2), but not our simulation  APE are defined in Eqns. (7) and (8) Eqns. (13) and (14), respectively. When a term on an arrow is positive (negative), energy Overlayed is a snapshot of the flow for experiment ( , ) = (0.16, 2) at = 6.75 to highlight a breaking event downstream of the topography.. Color: normalized perturbation velocity / ; black contours: isopycnals.
Topography is homogeneous in and the domain is periodic in and .