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W4: Gravity Wave Parameterisation for the Ocean

Principal investigators: Dr. Christian Mertens (University of Bremen), Prof. Dirk Olbers (MARUM, University of Bremen), Dr. Friederike Pollmann (University of Hamburg)

The recently proposed parameterization module "Internal wave Dissipation Energy and MIXing" (IDEMIX) describes the generation, propagation, interaction, and dissipation of the internal gravity wave field and can be used in ocean general circulation models to account for vertical mixing (and friction) in the interior of the ocean (Olbers & Eden, 2013). It is based on the radiative transfer equation of a weakly interacting internal wave field, for which spectrally integrated energy compartments are used as prognostic model variables. IDEMIX is central to the concept of an energetically consistent ocean model, since it enables to link all sources and sinks of internal wave energy and furthermore all parameterized forms of energy in an ocean model without spurious sources and sinks of energy. 

Space-time scales of important oceanic processes (pink areas) and scales explicitly resolved by ocean models (grey rectangular areas). The lower left rectangle represents modern global ocean climate models and the upper right rectangle eddy resolving basin-scale models. Also shown are dispersion curves (solid lines) for linear gravity waves (upper set) and planetary waves (lower set). Vertical dotted lines indicate the external (Ro) and the first internal (Ri) Rossby radii and the Ozmidov length scale Lo.

The objectives of W4 are to better understand internal gravity wave generation, interaction, and dissipation processes in order to extend and improve the IDEMIX model. This includes the implementation of new and improved forcing functions, such as the energy exchanges with mesoscale eddies or the anisotropic internal tide generation (Pollmann & Nycander, 2023). Moreover, the IDEMIX energy equation was coupled to an evolution equation for the spectral shape parameter bandwidth (Olbers et al., 2023). The IDEMIX versions of increasing complexity are evaluated against available fine- and microstructure observations of ocean mixing, internal wave energy, and spectral shape. The basic IDEMIX version was implemented in the state-of-the-art ocean models ICON and FESOM, where it reproduced the observed mixing work variability substantially better than the reference scenario with a constant background diffusivity (Brüggemann et al., 2024). Future plans are to investigate the spatio-temporal variability of internal wave spectral shape from the full-depth WOCE/CLIVAR hydrographic profiles and to derive and evaluate a simplified IDEMIX model (EASYMIX), allowing for easier applicability and reduced computational expenses. In collaboration with subproject S3, the different IDEMIX model variants will be applied in climate change simulations.

In phase I (2016-2020) we worked on improving IDEMIX by identifying strengths and weaknesses in a global evaluation, by adding new or extending existing forcing functions, and by broadening our understanding of the wave processes.

Evaluation of the basic IDEMIX version: The internal gravity wave energy estimated from Argo CTD-profiles by extending the finestructure method (a), which estimates turbulent kinetic energy dissipation and vertical diffusivity from finescale (\(O\) (10-100) m) shear and/or strain variability, is generally well reproduced by IDEMIX (b), but some deviations remain e.g. in regions of strong mesoscale (e.g. Drake Passage), tidal (e.g. around Hawaii), or wind (e.g. subtropical gyres of Pacific Ocean) forcing (Pollmann et al., 2017).

Adding lee waves to IDEMIX: In the North Atlantic and the Southern Ocean, the lee wave stress on the mean flow (a) is locally comparable to that induced by the surface wind forcing (b). Coupling the new lee wave closure to a realistic, eddy-permitting ocean model generates 0.27 TW of lee wave energy globally, which is a substantial fraction of the total internal wave field’s energy (Eden et al., submitted). The investigation of how this lee wave forcing affects wave and ocean dynamics is part of the ongoing PhD work of Thomas Eriksen (see also Reports/Implementation of Lee Waves in IDEMIX here).

Improved understanding of wave processes: The numerical evaluation of the kinetic equation, describing wave energy changes due to nonlinear wave-wave interactions, shows energy dissipation mainly at frequencies between 2\(f\) and 3\(f\) (\(f\) is the local Coriolis frequency; white lines show (2, 3, 4, 5, 10, 20)\(f\) for a wide range of horizontal (\(k\)) and vertical (\(m\)) wavenumbers (left). In this frequency range, the energy dissipation (in \(10^{-9} m^2s^{-3}\)) depends on the spectral slope \(s\) (middle), a relation not yet accounted for the theoretical parameterizations that form the basis of the finestructure method and of mixing parameterizations like IDEMIX (Eden et al., 2019). Fitting the Garrett-Munk (GM) spectral model to vertical wavenumber spectra obtained from Argo floats demonstrates that the spectral slope \(s\) varies geographically (right) around the GM-value of 2 (Pollmann, 2020).

Improved understanding of wave processes: The most energetic, principal, lunar, semi-diurnal internal tide (\(M_2\)) loses its energy to the higher-mode continuum via nonlinear wave-wave interactions in the vicinity of prominent generation regions (Hawaii, mid-ocean ridges) with a strong dependence on latitude (Olbers et al., 2020). The maximum zonally averaged energy transfers occur near 29°, where the \(M_2\)-tide frequency \(\omega_{M_2}\) equals twice the local Coriolis frequency and the only possible resonant sum interaction with \(\omega_{M_2} = \omega_1 + \omega_2 \) is parametric subharmonic instability, for which \(\omega_1 \approx \omega_2 \approx \omega_{M_2} / 2\) .

Improving the tidal forcing of IDEMIX: A new method to calculate not only the magnitude but also the direction of the barotropic-to-baroclinic tidal energy transfer while resolving the modal structure of the generated internal tides was derived and evaluated (Pollmann et al., 2019). Using this anisotropic tidal forcing (superscript \(\Phi\)) in IDEMIX instead of assuming the same average forcing in all directions changes the vertically integrated \(M_2\) tide energy \(\hat{E}_{M_2}\) by up to a factor of two, the continuum energy levels \(E_{IW}\) in the upper ocean by up to a factor of three, and the upper ocean turbulent kinetic energy dissipation rates \(\epsilon_{TKE}\) by up to a factor of four. The comparison to in-situ measurements, satellite observations, and numerical simulations using Stormtide is underway in collaboration with subproject W2.

Fig. 1: (Bottom) Turbulent kinetic energy (TKE) dissipation rate estimated from hydrographic profiles collected at 47°N in the Atlantic Ocean, using the finescale parameterization, and (top) a comparison of the vertically integrated TKE dissipation rates from two IDEMIX simulations and these observations (PhD work of D. Garcia Santacruz)

In phase 2, one aim of subproject W4 was to derive wave-driven mixing estimates from full-depth hydrographic and velocity profiles at 47°N and 16°N in the Atlantic Ocean. These were then compared to  the corresponding output of the internal wave model IDEMIX (see Objectives for details), testing different tuning parameter, configuration and forcing settings. The IDEMIX 2020 configuration, implemented in the state-of-the-art ocean models ICON and FESOM (Brüggemann et al., 2024) agrees well with the observations (Fig. 1). Among all the configurations tested, the setup with lee wave forcing and a modal bandwidth of j* = 9 gives the closest agreement with the observations (not shown).

Another aim of phase 2 was to incorporate in IDEMIX the variability of the spectral shape (vertical wavenumber slope and bandwidth) seen in Argo float-based observations (fig. 2; Pollmann, 2020). These feature notable deviations from the canonical Garrett-Munk model (Garrett and Munk, 1975, Cairns and Williams, 1976) parameters (a spectral slope of -2 and a wavenumber scale m* of 0.01 rad/m), which are currently assumed in IDEMIX. Building on the parametric approach of Hasselmann et al. (1973, 1976), the true energy spectrum is approximated by a parametric form (the empirical Garrett-Munk energy spectrum), which is characterized by a number of free, space- and time-dependent parameters. Each parameter is linked to the true energy spectrum through a certain algorithm, such that the radiation balance (the evolution equation for energy) can be projected onto these parameters and evolution equations for them can be derived. We did so for the wavenumber scale m*, showing that there is a  power law relation between energy and wavenumber scale m*, which is supported by the observed spectral parameters and energy levels (Olbers et al., 2023). Fig. 3 illustrates the performance in comparison to the original IDEMIX formulation (IDEMIX 2020, as shown also in Fig. 1) in an idealized, single-column setup, highlighting differences in particular for weak stratification.

Fig. 2: The first global-scale maps of internal wave energy, bandwidth, spectral slope and wavenumber scale in the upper ocean (300-500 m) derived from Argo float profiles (Pollmann, 2020).
Fig. 3: Differences between the original and parametric IDEMIX model in an idealized, single-column application, highlighting also the influence of adding turning point physics (PhD work of A. Patel)

Phase 2 also saw the global application of the new methodology to compute the anisotropic internal tide generation, derived in phase 1 (Pollmann et al., 2019): Fig. 4 illustrates the notable directional dependence of the barotropic-to-baroclinic energy conversion of the first mode of the principal lunar semi-diurnal constituent (M2-tide). Such directionally dependent internal tide generation estimates are valuable forcing functions for sophisticated parameterizations of tidally driven mixing (e.g. IDEMIX) and will be applied in different TRR 181 subprojects in phase 3.

Fig. 4: Barotropic-to-baroclinic energy conversion in northeastward and northwestward direction (approx. 4°-angleincrements) of the mode-1 M2-tide estimated by applying the modal, direction-resolving linear theory of Pollmann et al., 2019, to the global ocean (after Pollmann & Nycander, 2023).

References:
Garrett & Munk, 1975: Space-time scales of internal waves: A progress report. J. Geophys. Res., 80, 291–297, doi.org/10. 1029/JC080i003p00291.
Cairns, J. L., and G. O. Williams, 1976: Internal wave observations from a midwater float, 2. J. Geophys. Res., 81, 1943–1950, https://doi.org/10.1029/JC081i012p01943.

Hasselmann et al., 1973: Measurements of wind-wave growth and swell decay during the joint North Sea wave project (JONSWAP). Dtsch. Hydrogr. Z., 12, 1–95.

Hasselmann et al, 1976: W. Sell, D. B. Ross, and P. M¨uller, 1976: A parametric wave prediction model. J. Phys. Oceanogr., 6, 200–228, https://doi.org10.1175/1520-0485(1976)006,0200:APWPM.2.0.CO;2.

 

The central goals of subproject W4 are to better understand the internal wave life cycle and its link to ocean mixing through observations, theory, and numerical modeling, and, building on these results, to improve the energetically consistent IDEMIX framework for the representation of internal wave effects in state-of-the-art ocean models.
In phase 3, we will

  • develop a simplified IDEMIX framework called EASYMIX. The motivation is to reduce the computational costs and moreover provide a low-level, easy-to-use variant that will increase the IDEMIX user base. We will investigate the effect both in terms of accuracy and computational costs of different simplification steps, both in present-day and future-climate simulations.
  • characterize the internal wave spectral variability and their link to mixing from WOCE/CLIVAR transects. This complements our earlier analysis based on Argo-float observations, which are limited to the ocean’s upper 2000 m. In combination, these data sets will provide valuable references to constrain, among others, the parametric IDEMIX model or to refine parameterizations involving the spectral shape as tuning parameters. They will moreover allow us to identify patterns and dependencies on environmental conditions.
  • investigate how different incarnations of the finestructure method, which derives wave-driven mixing from hydrographic information, differ. Eden et al., 2019, found a slope-dependence in their numerical evaluation of the scattering integral and its influence will be tested in contrast with high-resolution microstructure observations.

Implementation of Lee Waves in IDEMIX

I’m investigating what and how big of a role lee waves play in transferring energy between large scale geostrophic motions and scmall scale turbulent mixing.

Thomas Eriksen, PhD W4

The purpose of my project is to investigate what and how big of a role lee waves play in transferring energy between large scale geostrophic motions and small scale turbulent mixing. Lee waves are formed when geostrophic motions interact with bottom topography. They radiate away from the topography and eventually break. When they break, the kinetic energy that they contain is used for dissipation, which, ultimately, raises potential energy. The issue of their role in the general circulation has been raised due to observed increased mixing rates near the ocean bottom in the Drake Passage and the Scotia Sea.

Previous estimates of the energy transfer from geostrophic motions into lee waves are around 1/3 of the energy input into gravity waves from winds. However, they are very few and differ roughly by a factor of 4. Furthermore, this energy transfer estimate has so far only been diagnosed and not used as in integral part of an ocean model. The contribution of lee waves in driving the large scale motions themselves – the overturning circulation, for example – are therefore largely unknown. The proper way of including lee waves in an energetically consistent ocean model would thus be to diagnose the energy c o n t a i n e d in lee waves e v e r y w h e r e in the ocean, let this energy travel and eventually be used for dissipation – in my case using an internal wave model – and then subtract it from its source.

This is exactly what is done in my model. The objective of my study is therefore to extend the IDEMIX model with an inclusion of lee wave energetics. This means that the energy being transferred into lee waves will be able to affect the rest of the ocean through diapycnal diffusivity – similarly to other types of gravity waves.

So far in my study, I have diagnosed the global energy transfer into lee waves to around 0.3TW. This is in accordance with previous estimates. The implementation of lee wave energetics into IDEMIX is underway. The lee wave energy flux is split into four directional compartments (N, S, E, W) will enter the gravity wave field as a bottom boundary flux, and the wave energy will thus be able to travel in the same manner as energy from other gravity waves. This is a fundamentally different way of treating lee waves compared to previous studies.

The next step is to study the differences in diapycnal diffusivity in model runs with and without lee waves. To what degree lee waves are able to account for the observed increased mixing rates in the deep Southern Ocean is till an open question, which I would like to answer. After this, I would like to address the question of what role lee waves play in setting the overturning circulation.

How the background mean flow effects internal gravity waves

From my work, hopefully general rules may be seen that can be included in parameterisations for internal gravity waves.

Rachael Ewins, PhD in W4

I am investigating the effect background mean flow has on the propagation of internal gravity waves. From this hopefully general rules may be seen that can be included in parameterisations for internal gravity waves. For this ray tracing is used to follow the positions and properties of wave packets that interact with an idealised current.

The test wave packets are populated randomly over a range of physical positions and also phase space, which allows exploration of the importance to various properties to how the test wave packets interact with the background current. The key property that is being tracked is the energy of the packets and from this the transfer of energy to and from the current can be seen.

Ray tracing simply propagates the position and wave numbers of the wave packets over a series of time steps given that background properties of background flow velocity, the local buoyancy frequency. The energy of the wave packets can be followed due to the conservation of Action. The results means that individual wave packets can be followed to different end conditions namely critical layer absorption, wave capture or refraction away from the current flow. The net energy transfer from the waves to the background flow (or from) can be seen by the end energy of the waves that enter critical layers or are captured by the current.

By varying the properties of the background current the effects of various shears in the current can be seen which will lead to more information about the key properties of both internal wave and background flow that lead to wave captures and critical layer absorption. In addition the background flow can be changed into configuration to simulate eddies, using the same processes.