Skip to main content
SpringerLink
Log in

The Interior Energy Pathway: Inertia-Gravity Wave Emission by Oceanic Flows

  • Chapter
  • First Online:
Energy Transfers in Atmosphere and Ocean

Part of the book series: Mathematics of Planet Earth ((MPE,volume 1))

Abstract

We review the possible role of spontaneous emission and subsequent capture of internal gravity waves (IGWs) for dissipation in oceanic flows under conditions characteristic for the ocean circulation. Dissipation is necessary for the transfer of energy from the essentially balanced large-scale ocean circulation and mesoscale eddy fields down to smaller scales where instabilities and subsequent small-scale turbulence complete the route to dissipation. Spontaneous wave emission by flows is a viable route to dissipation. For quasi-balanced flows, characterized by a small Rossby number, the amplitudes of emitted waves are expected to be small. However, once being emitted into a three-dimensional eddying flow field, waves can undergo refraction and may be “captured.” During wave capture, the wavenumber grows exponentially, ultimately leading to breakup and dissipation. For flows with not too small Rossby number, e.g., for flows in the vicinity of strong fronts, dissipation occurs in a more complex manner. It can occur via spontaneous wave emission and subsequent wave capture, with the amplitudes of waves emitted in frontal systems being expected to be larger than amplitudes of waves emitted by quasi-balanced flows. It can also occur through turbulence and filamentation emerging from frontogenesis. So far, quantitative importance of this energy pathway—crucial for determining correct eddy viscosities in general circulation models—is not known. Toward an answer to this question, we discuss IGWs diagnostics, review spontaneous emission of both quasi-balanced and less-balanced frontal flows, and discuss recent numerical results based on a high-resolution ocean general circulation model.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 79.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Note that a global estimate of the ocean kinetic energy, which requires three-dimensional information about flow velocity, is not available from observation and has to be derived using eddy-resolving numerical simulations.

  2. 2.

    By writing the dispersion relation in the form \(\omega ({\varvec{k}},t) = \omega _{{\mathrm {i}}} ({\varvec{x}},t) + {\varvec{u}}({\varvec{x}},t) \cdot {\varvec{k}}= \varOmega ({\varvec{k}};{\varvec{x}},t)\) and assuming that the intrinsic dispersion relation and hence the intrinsic frequency \(\omega _{{\mathrm {i}}}\) is independent of \({\varvec{x}}\), their equation (6.46) reduces to our equation (2.10b).

References

  • Allen, J.S., Holm, D.D., Newberger, P.A.: Toward an extended-geostrophic Euler-Poincaré model for mesoscale oceanographic flow. In: Norbury, J., Roulstone, I. (eds.) Large-Scale Atmosphere-Ocean Dynamics, vol. 1, pp. 101–125. Cambridge University Press (2002)

    Google Scholar 

  • Badin, G.: Surface semi-geostrophic dynamics in the ocean. Geophys. Astrophys. Fluid Dyn. 107(5), 526–540 (2013)

    MathSciNet  Google Scholar 

  • Barth, J., Allen, J., Newberger, P.: On intermediate models for barotropic continental shelf and slope flow fields. Part II: Comparison of numerical model solutions in doubly periodic domains. J. Phys. Oceanogr. 20(7), 1044–1076 (1990)

    Google Scholar 

  • Blumen, W.: Geostrophic adjustment. Rev. Geophys. 10(2), 485–528 (1972)

    Google Scholar 

  • Bouchut, F., Sommer, J.L., Zeitlin, V.: Frontal geostrophic adjustment and nonlinear wave phenomena in one-dimensional rotating shallow water. Part 2. High-resolution numerical simulations. J. Fluid Mech. 514, 35–63 (2004)

    MathSciNet  MATH  Google Scholar 

  • Broutman, D., Rottman, J.W., Eckermann, S.D.: Ray methods for internal waves in the atmosphere and ocean. Annu. Rev. Fluid Mech. 36, 233–253 (2004)

    MathSciNet  MATH  Google Scholar 

  • Bühler, O., McIntyre, M.E.: Wave capture and wave-vortex duality. J. Fluid Mech. 534, 67–95 (2005)

    MathSciNet  MATH  Google Scholar 

  • Charney, J.G.: Geostrophic turbulence. J. Atmos. Sci. 28(6), 1087–1095 (1971)

    Google Scholar 

  • Cho, H.R., Shepherd, T.G., Vladimirov, V.A.: Application of the direct liapunov method to the problem of symmetric stability in the atmosphere. J. Atmos. Sci. 50(6), 822–836 (1993)

    MathSciNet  Google Scholar 

  • Constantin, P., Majda, A., Tabak, E.: Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity 7, 1495–1533 (1994)

    MathSciNet  MATH  Google Scholar 

  • Cotter, C.: Data assimilation on the exponentially accurate slow manifold. Philos. Trans. R. Soc. A 371(1991), 20120300 (2013)

    MathSciNet  MATH  Google Scholar 

  • Cotter, C.J., Reich, S.: Semigeostrophic particle motion and exponentially accurate normal forms. Multiscale Model. Simul. 5(2), 476–496 (2006)

    MathSciNet  MATH  Google Scholar 

  • Danioux, E., Vanneste, J., Klein, P., Sasaki, H.: Spontaneous inertia-gravity-wave generation by surface-intensified turbulence. J. Fluid Mech. 699, 153–173 (2012)

    MATH  Google Scholar 

  • Dritschel, D.G., Polvani, L.M., Mohebalhojeh, A.R.: The contour-advective semi-lagrangian algorithm for the shallow water equations. Mon. Weather Rev. 127(7), 1551–1565 (1999)

    Google Scholar 

  • Ford, R.: Gravity wave radiation from vortex trains in rotating shallow water. J. Fluid Mech. 281, 81–118 (1994)

    MathSciNet  MATH  Google Scholar 

  • Ford, R., McIntyre, M.E., Norton, W.A.: Balance and the slow quasimanifold: some explicit results. J. Atmos. Sci. 57(9), 1236–1254 (2000)

    MathSciNet  Google Scholar 

  • Ford, R., McIntyre, M.E., Norton, W.A.: Reply. J. Atmos. Sci. 59(19), 2878–2882 (2002)

    Google Scholar 

  • Franzke, C.L.E., Oliver, M., Rademacher, J.D.M., Badin, G.: Multi-scale methods for geophysical flows. This volume, Chapter 1 (2019)

    Google Scholar 

  • Gall, R.L., Williams, R.T., Clark, T.L.: Gravity waves generated during frontogenesis. J. Atmos. Sci. 45(15), 2204–2219 (1988)

    Google Scholar 

  • Gottwald, G.A., Mohamad, H., Oliver, M.: Optimal balance via adiabatic invariance of approximate slow manifolds. Multiscale Model. Simul. 15(4), 1404–1422 (2017)

    MathSciNet  MATH  Google Scholar 

  • Griffies, S.M., Winton, M., Anderson, W.G., Benson, R., Delworth, T.L., Dufour, C.O., Dunne, J.P., Goddard, P., Morrison, A.K., Rosati, A., Wittenberg, A.T., Yin, J., Zhang, R.: Impacts on ocean heat from transient mesoscale eddies in a hierarchy of climate models. J. Clim. 28(3), 952–977 (2015)

    Google Scholar 

  • Hoskins, B.J.: The geostrophic momentum approximation and the semi-geostrophic equations. J. Atmos. Sci. 32(2), 233–242 (1975)

    Google Scholar 

  • Hoskins, B.J., Bretherton, F.P.: Atmospheric frontogenesis models: mathematical formulation and solution. J. Atmos. Sci. 29(1), 11–37 (1972)

    Google Scholar 

  • Jones, W.L.: Ray tracing for internal gravity waves. J. Geophys. Res. 74(8), 2028–2033 (1969)

    Google Scholar 

  • Klein, P., Treguier, A.M.: Dispersion of wind-induced inertial waves by a barotropic jet. J. Mar. Res. 53(1), 1–22 (1995)

    Google Scholar 

  • Kunze, E.: Near-inertial wave propagation in geostrophic shear. J. Phys. Oceanogr. 15(5), 544–565 (1985)

    Google Scholar 

  • Kunze, E., Boss, E.: A model for vortex-trapped internal waves. J. Phys. Oceanogr. 28(10), 2104–2115 (1998)

    Google Scholar 

  • Le Sommer, J., Medvedev, S., Plougonven, R., Zeitlin, V.: Singularity formation during relaxation of jets and fronts toward the state of geostrophic equilibrium. J. Fluid Mech. 758, 287–311 (2014)

    MATH  Google Scholar 

  • Le Sommer, J., Reznik, G.M., Zeitlin, V.: Nonlinear geostrophic adjustment of long-wave disturbances in the shallow-water model on the equatorial beta-plane. J. Fluid Mech. 515, 135–170 (2004)

    MathSciNet  MATH  Google Scholar 

  • Lee, M.-M., Nurser, A.J.G., Coward, A.C., de Cuevas, B.A.: Eddy advective and diffusive transports of heat and salt in the southern ocean. J. Phys. Oceanogr. 37(5), 1376–1393 (2007)

    Google Scholar 

  • Li, H., von Storch, J.-S.: On the fluctuating buoyancy fluxes simulated in a OGCM. J. Phys. Oceanogr. 43(7), 1270–1287 (2013)

    Google Scholar 

  • Li, Z., von Storch, J.-S., Müller, M.: The \(\text{ M }_2\) internal tide simulated by a \(1/10^\circ \) OGCM. J. Phys. Oceanogr. 45(12), 3119–3135 (2015)

    Google Scholar 

  • Lighthill, M.J.: On sound generated aerodynamically. I. General theory. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 211(1107), 564–587 (1952)

    Google Scholar 

  • Llewellyn Smith, S.G.: Near-inertial oscillations of a barotropic vortex: trapped modes and time evolution. J. Phys. Oceanogr. 29(4), 747–761 (1999)

    MathSciNet  Google Scholar 

  • Lott, F., Plougonven, R., Vanneste, J.: Gravity waves generated by sheared potential vorticity anomalies. J. Atmos. Sci. 67(1), 157–170 (2010)

    Google Scholar 

  • Lott, F., Plougonven, R., Vanneste, J.: Gravity waves generated by sheared three-dimensional potential vorticity anomalies. J. Atmos. Sci. 69(7), 2134–2151 (2012)

    Google Scholar 

  • Maltrud, M., Bryan, F., Peacock, S.: Boundary impulse response functions in a century-long eddying global ocean simulation. Environ. Fluid Mech. 10(1–2), 275–295 (2010)

    Google Scholar 

  • Maltrud, M.E., McClean, J.L.: An eddy resolving global 1/10\(^{\circ }\) ocean simulation. Ocean Model. 8(1–2), 31–54 (2005)

    Google Scholar 

  • Masumoto, Y., Sasaki, H., Kagimoto, T., Komori, N., Ishida, A., Sasai, Y., Miyama, T., Motoi, T., Mitsudera, H., Takahashi, K., Sakuma, H., Yamagata, T.: A fifty-year eddy-resolving simulation of the world ocean—preliminary outcomes of OFES (OGCM for the Earth Simulator). J. Earth Simulator 1, 35–56 (2004)

    Google Scholar 

  • McIntyre, M.:. Dynamical meteorology—balanced flow. In: Pyle, J., Zhang, F. (eds.) Encyclopedia of Atmospheric Sciences, pp. 298–303, 2nd edn. Academic Press, Oxford (2015)

    Google Scholar 

  • McIntyre, M.E.: Spontaneous imbalance and hybrid vortex-gravity structures. J. Atmos. Sci. 66, 1315–1326 (2009)

    Google Scholar 

  • McIntyre, M.E., Norton, W.A.: Potential vorticity inversion on a hemisphere. J. Atmos. Sci. 57(9), 1214–1235 (2000)

    MathSciNet  Google Scholar 

  • McIntyre, M.E., Roulstone, I.: Are there higher-accuracy analogues of semi-geostrophic theory? In: Norbury, J., Roulstone, I. (eds.) Large-Scale Atmosphere-Ocean Dynamics, vol. 2, pp. 301–364. Cambridge University Press (2002)

    Google Scholar 

  • Medvedev, S.B., Zeitlin, V.: Weak turbulence of short equatorial waves. Phys. Lett. A 342(3), 217–227 (2005)

    MATH  Google Scholar 

  • Mohebalhojeh, A.R., Dritschel, D.G.: Hierarchies of balance conditions for the \(f\)-plane shallow-water equations. J. Atmos. Sci. 58(16), 2411–2426 (2001)

    MathSciNet  Google Scholar 

  • Müller, P., McWilliams, J.C., Molemaker, M.J.: Routes to dissipation in the ocean: the 2D/3D turbulence conundrum. In: Baumert, H.Z., Simpson, J., Sündermann, J. (eds.) Marine Turbulence, pp. 397–405. Cambridge University Press (2005)

    Google Scholar 

  • Muraki, D.J., Snyder, C., Rotunno, R.: The next-order corrections to quasigeostrophic theory. J. Atmos. Sci. 56(11), 1547–1560 (1999)

    MathSciNet  Google Scholar 

  • Nadiga, B.T.: Nonlinear evolution of a baroclinic wave and imbalanced dissipation. J. Fluid Mech. 756, 965–1006 (2014)

    MathSciNet  Google Scholar 

  • Nagai, T., Tandon, A., Kunze, E., Mahadevan, A.: Spontaneous generation of near-inertial waves by the Kuroshio front. J. Phys. Oceanogr. 45(9), 2381–2406 (2015)

    Google Scholar 

  • Nikurashin, M., Ferrari, R.: Global energy conversion rate from geostrophic flows into internal lee waves in the deep ocean. Geophys. Res. Lett. 38(8), L08610 (2011)

    Google Scholar 

  • Nikurashin, M., Vallis, G.K., Adcroft, A.: Routes to energy dissipation for geostrophic flows in the southern ocean. Nat. Geosci. 6(1), 48–51 (2013)

    Google Scholar 

  • Obukhov, A.: On the problem of geostrophic wind. Izv. AN SSSR Geogr. Geophys. Ser. 13, 281–286 (1949)

    Google Scholar 

  • Olbers, D., Willebrand, J., Eden, C.: Ocean Dynamics. Springer (2012)

    Google Scholar 

  • Oliver, M., Vasylkevych, S.: Generalized large-scale semigeostrophic approximations for the \(f\)-plane primitive equations. J. Phys. A: Math. Theor. 49, 184001 (2016)

    MathSciNet  MATH  Google Scholar 

  • O’Sullivan, D., Dunkerton, T.J.: Generation of inertia-gravity waves in a simulated life cycle of baroclinic instability. J. Atmos. Sci. 52, 3695–3716 (1995)

    Google Scholar 

  • Plougonven, R., Snyder, C.: Gravity waves excited by jets: propagation versus generation. Geophys. Res. Lett. 32(18), L18802 (2005)

    Google Scholar 

  • Plougonven, R., Snyder, C.: Inertia-gravity waves spontaneously generated by jets and fronts. Part I: different baroclinic life cycles. J. Atmos. Sci. 64(7), 2502–2520 (2007)

    Google Scholar 

  • Plougonven, R., Zeitlin, V.: Lagrangian approach to geostrophic adjustment of frontal anomalies in a stratified fluid. Geophys. Astrophys. Fluid Dyn. 99(2), 101–135 (2005)

    MathSciNet  MATH  Google Scholar 

  • Plougonven, R., Zhang, F.: Internal gravity waves from atmospheric jets and fronts. Rev. Geophys. 52(1), 33–76 (2014)

    Google Scholar 

  • Ragone, F., Badin, G.: A study of surface semi-geostrophic turbulence: freely decaying dynamics. J. Fluid Mech. 792, 740–774 (2016)

    MathSciNet  MATH  Google Scholar 

  • Reznik, G.M., Zeitlin, V., Jelloul, M.B.: Nonlinear theory of geostrophic adjustment. Part 1. Rotating shallow-water model. J. Fluid Mech. 445, 93–120 (2001)

    MathSciNet  MATH  Google Scholar 

  • Rossby, C.-G.: On the mutual adjustment of pressure and velocity distributions in certain simple current systems, II. J. Mar. Res. 1(3), 239–263 (1938)

    Google Scholar 

  • Salmon, R.: New equations for nearly geostrophic flow. J. Fluid Mech. 153, 461–477 (1985)

    MATH  Google Scholar 

  • Salmon, R.: Large-scale semigeostrophic equations for use in ocean circulation models. J. Fluid Mech. 318, 85–105 (1996)

    MathSciNet  MATH  Google Scholar 

  • Saujani, S., Shepherd, T.G.: Comments on “balance and the slow quasimanifold: some explicit results”. J. Atmos. Sci. 59(19), 2874–2877 (2002)

    Google Scholar 

  • Snyder, C., Muraki, D.J., Plougonven, R., Zhang, F.: Inertia-gravity waves generated within a dipole vortex. J. Atmos. Sci. 64(12), 4417–4431 (2007)

    Google Scholar 

  • Snyder, C., Plougonven, R., Muraki, D.J.: Mechanisms for spontaneous gravity wave generation within a dipole vortex. J. Atmos. Sci. 66(11), 3464–3478 (2009)

    Google Scholar 

  • Thomas, L.N.: On the effects of frontogenetic strain on symmetric instability and inertia-gravity waves. J. Fluid Mech. 711, 620–640 (2012)

    MathSciNet  MATH  Google Scholar 

  • van Haren, H.: On the polarization of oscillatory currents in the Bay of Biscay. J. Geophys. Res. Oceans 108(C9), 3290 (2003)

    Google Scholar 

  • Vanneste, J.: Exponential smallness of inertia-gravity wave generation at small Rossby number. J. Atmos. Sci. 65(5), 1622–1637 (2008)

    Google Scholar 

  • Vanneste, J.: Balance and spontaneous wave generation in geophysical flows. Ann. Rev. Fluid Mech. 45(1), 147–172 (2013)

    MathSciNet  MATH  Google Scholar 

  • Vanneste, J., Yavneh, I.: Exponentially small inertia-gravity waves and the breakdown of quasigeostrophic balance. J. Atmos. Sci. 61(2), 211–223 (2004)

    MathSciNet  Google Scholar 

  • Vanneste, J., Yavneh, I.: Unbalanced instabilities of rapidly rotating stratified shear flows. J. Fluid Mech. 584, 373–396 (2007)

    MathSciNet  MATH  Google Scholar 

  • Viúdez, A.: Spiral patterns of inertia-gravity waves in geophysical flows. J. Fluid Mech. 562, 73–82 (2006)

    MathSciNet  MATH  Google Scholar 

  • Viúdez, Á.: The origin of the stationary frontal wave packet spontaneously generated in rotating stratified vortex dipoles. J. Fluid Mech. 593, 359–383 (2007)

    MathSciNet  MATH  Google Scholar 

  • Viúdez, A., Dritschel, D.G.: Optimal potential vorticity balance of geophysical flows. J. Fluid Mech. 521, 343–352 (2004)

    MathSciNet  MATH  Google Scholar 

  • Viúdez, A., Dritschel, D.G.: Spontaneous generation of inertia-gravity wave packets by balanced geophysical flows. J. Fluid Mech. 553, 107–117 (2006)

    MathSciNet  MATH  Google Scholar 

  • von Storch, H., Zwiers, F.W.: Statistical Analysis in Climate Research, p. 484. Cambridge University Press (1999)

    Google Scholar 

  • von Storch, J.-S., Eden, C., Fast, I., Haak, H., Hernández-Deckers, D., Maier-Reimer, E., Marotzke, J., Stammer, D.: An estimate of the Lorenz energy cycle for the world ocean based on the STORM/NCEP simulation. J. Phys. Oceanogr. 42(12), 2185–2205 (2012)

    Google Scholar 

  • Wang, S., Zhang, F.: Source of gravity waves within a vortex-dipole jet revealed by a linear model. J. Atmos. Sci. 67(5), 1438–1455 (2010)

    Google Scholar 

  • Wang, S., Zhang, F., Snyder, C.: Generation and propagation of inertia-gravity waves from vortex dipoles and jets. J. Atmos. Sci. 66(5), 1294–1314 (2009)

    Google Scholar 

  • Xing, J., Davies, A.M.: On the influence of a surface coastal front on near-inertial wind-induced internal wave generation. J. Geophys. Res. Oceans 109(C1), C01023 (2004)

    Google Scholar 

  • Zeitlin, V.: Lagrangian dynamics of fronts, vortices and waves: understanding the (semi-) geostrophic adjustment. In: Fronts, Waves and Vortices in Geophysical Flows, pp. 109–137. Springer (2010)

    Google Scholar 

  • Zeitlin, V., Medvedev, S.B., Plougonven, R.: Frontal geostrophic adjustment, slow manifold and nonlinear wave phenomena in one-dimensional rotating shallow water. Part 1. Theory. J. Fluid Mech. 481, 269–290 (2003)

    MathSciNet  MATH  Google Scholar 

  • Zhang, F.: Generation of mesoscale gravity waves in upper-tropospheric jet-front systems. J. Atmos. Sci. 61(4), 440–457 (2004)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank Tuba Masur for careful proof-reading of the manuscript. GB was partially supported by DFG grants FOR1740, BA-5068/8-1, and BA-5068/9-1. MO was partially supported by DFG grant OL-155/6-1.

Author information

Authors and Affiliations

  1. Max-Planck-Institut für Meteorologie (MPI-M), Hamburg, Germany

    Jin-Song von Storch

  2. Center for Earth System Research and Sustainability (CEN), Universität Hamburg, Hamburg, Germany

    Gualtiero Badin

  3. Jacobs University, Bremen, Germany

    Marcel Oliver

Authors
  1. Jin-Song von Storch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gualtiero Badin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marcel Oliver
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marcel Oliver .

Editor information

Editors and Affiliations

  1. Center for Earth System Research and Sustainability (CEN), University of Hamburg, Hamburg, Germany

    Carsten Eden

  2. Department of Mathematics, University of Hamburg, Hamburg, Germany

    Armin Iske

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

von Storch, JS., Badin, G., Oliver, M. (2019). The Interior Energy Pathway: Inertia-Gravity Wave Emission by Oceanic Flows. In: Eden, C., Iske, A. (eds) Energy Transfers in Atmosphere and Ocean. Mathematics of Planet Earth, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-030-05704-6_2

Download citation

Publish with us

Policies and ethics

Search

Navigation