Skip to main content
SpringerLink
Log in

Diffuse Interface Approaches in Atmosphere and Ocean—Modeling and Numerical Implementation

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

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

  • 1134 Accesses

Abstract

We propose to model physical effects at the sharp density interface between atmosphere and ocean with the help of diffuse interface approaches for multiphase flows with variable densities. We use the thermodynamical consistent variable density model proposed in Abels et al. (Mathematical Models and Methods in Applied Sciences 22:1150013, 2012). This results in a Cahn–Hilliard-/Navier–Stokes-type system which we complement with tangential Dirichlet boundary conditions to incorporate the effect of wind in the atmosphere. Wind is responsible for waves at the surface of the ocean, whose dynamics have an important impact on the \(\textit{CO}_{2}\)—exchange between ocean and atmosphere. We tackle this mathematical model numerically with fully adaptive and integrated numerical schemes tailored to the simulation of variable density multiphase flows governed by diffuse interface models. Here, fully adaptive, integrated, efficient, and reliable means that the mesh resolution is chosen by the numerical algorithm according to a prescribed error tolerance in the a posteriori error control on the basis of residual-based error indicators, which allow to estimate the true error from below (efficient) and from above (reliable). Our approach is based on the work of Hintermüller et al. (Journal of Computational Physics 235:810–827, 2013), Garcke et al. (Applied Numerical Mathematics 99:151–171, 2016), where a fully adaptive efficient and reliable numerical method for the simulation of two-dimensional multiphase flows with variable densities is developed. In a first step, we incorporate the stimulation of surface waves via appropriate volume forcing.

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

References

  • Abels, H., Depner, D., Garcke, H.: Existence of weak solutions for a diffuse interface model for two-phase flows of incompressible fluids with different densities. J. Math. Fluid Mech. 15(3), 453–480 (2013a)

    Article  MathSciNet  MATH  Google Scholar 

  • Abels, H., Depner, D., Garcke, H.: On an incompressible Navier-Stokes/Cahn-Hilliard system with degenerate mobility. Annales de l’Institut Henri Poincaré (C) Non Linear. Analysis 30(6), 1175–1190 (2013b)

    Google Scholar 

  • Abels, H., Garcke, H., Grün, G.: Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Model. Methods Appl. Sci. 22(3), 1150013(40) (March 2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Adams, R.A., Fournier, J.H.F.: Sobolev spaces. In: Pure and Applied Mathematics, 2nd edn, vol. 140. Elsevier (2003)

    Google Scholar 

  • Aki, G.L., Dreyer, W., Giesselmann, J., Kraus, C.: A quasi-incompressible diffuse interface model with phase transition. Math. Model. Methods Appl. Sci. 24(5), 827–861 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Aland, S., Voigt, A.: Benchmark computations of diffuse interface models for two-dimensional bubble dynamics. Int. J. Numer. Methods Fluids 69, 747–761 (2012)

    Article  MathSciNet  Google Scholar 

  • Aland, S., Lowengrub, J., Voigt, A.: Two-phase flow in complex geometries: a diffuse domain approach. CMES 57, 57(1), 77–108 (2010)

    MathSciNet  MATH  Google Scholar 

  • Anderson, D.M., McFadden, G.B., Wheeler, A.A.: Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139–165 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Baňas, L., Nürnberg, R.: A posteriori estimates for the Cahn-Hilliard equation. Math. Model. Numer. Anal. 3(5), 1003–1026 (2009)

    Article  MATH  Google Scholar 

  • Barrett, J.W., Garcke, H., Nürnberg, R.: A stable parametric finite element discretization of two-phase Navier-Stokes flow. J. Sci. Comput. 63(1), 78–117 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numerica 14, 1–137 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Blowey, J.F., Elliott, C.M.: The Cahn-Hilliard gradient theory for phase separation with non-smooth free energy. Part I: mathematical analysis. Eur. J. Appl. Math. 2, 233–280 (1991)

    Article  MATH  Google Scholar 

  • Bosch, J., Stoll, M., Benner, P.: Fast solution of Cahn-Hilliard variational inequalities using implicit time discretization and finite elements. J. Comput. Phys. 262, 38–57 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Boyer, F.: Mathematical study of multiphase flow under shear through order parameter formulation. Asymptot. Anal. 20(2), 175–212 (1999)

    MathSciNet  MATH  Google Scholar 

  • Boyer, F.: A theoretical and numerical model for the study of incompressible mixture flows. Comput. Fluids 31(1), 41–68 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Boyer, F., Chupin, L., Fabrie, P.: Numerical study of viscoelastic mixtures through a Cahn-Hilliard flow model. Eur. J. Mech.-B/Fluids 23(5), 759–780 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Boyer, F., Lapuerta, C., Minjeaud, S., Piar, B., Quintard, M.: Cahn-Hilliard/Navier-Stokes model for the simulation of three-phase flows. Transp. Porous Media 82(3), 463–483 (2010)

    Article  MathSciNet  Google Scholar 

  • Chen, L.: i FEM: an innovative finite element method package in Matlab. https://ifem.wordpress.com (2008)

  • Clément, P.: Approximation by finite element functions using local regularization. RAIRO Analyse numérique 9(2), 77–84 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  • Constantin, P., Foias, C.: Navier-Stokes-Equations. University of Chicago Press (1988)

    Google Scholar 

  • Ding, H., Spelt, P.D.M., Shu, C.: Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comput. Phys. 226(2), 2078–2095 (2007)

    Article  MATH  Google Scholar 

  • Drennan, W., Donelan, M., Terray, E., Katsaros, K.: Oceanic turbulence dissipation measurements in SWADE. J. Phys. Oceanogr. 26, 808–815 (1996)

    Article  Google Scholar 

  • Druzhinin, O.A., Elghobashi, S.: Direct numerical simulations of bubble-laden turbulent flows using the two-fluid formulation. Phys. Fluids 10, 685–697 (1998)

    Article  Google Scholar 

  • Eck, C.: Homogenization of phase field models for binary mixtures. SIAM Multicale Model. Simul. 3(1), 1–27 (2004)

    MathSciNet  MATH  Google Scholar 

  • Feng, X.: Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal. 44(3), 1049–1072 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Ganesan, S., Tobiska, L.: A coupled arbitrary lagrangian-eulerian and lagrangian method for computation of free surface flows with insoluble surfactants. J. Comput. Phys. 228, 2859–2873 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Garcke, H., Hinze, M., Kahle, C.: A stable and linear time discretization for a thermodynamically consistent model for two-phase incompressible flow. Appl. Numer. Math. 99, 151–171 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Gross, S., Reusken, A.: Numerical methods for two-phase incompressible flows. In: Springer Series in Computational Mathematics, vol. 40. Springer (2011)

    Google Scholar 

  • Grün, G.: On convergent schemes for diffuse interface models for two-phase flow of incompressible fluids with general mass densities. SIAM J. Numer. Anal. 51(6), 3036–3061 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Grün, G., Guillén-Gonzáles, F., Metzger, S.: On fully decoupled convergent schemes for diffuse interface models for two-phase flow with general mass densities. Commun. Comput. Phys. 19(5), 1473–1502 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Grün, G., Klingbeil, F.: Two-phase flow with mass density contrast: stable schemes for a thermodynamic consistent and frame indifferent diffuse interface model. J. Comput. Phys. 257(A), 708–725 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Guo, Z., Lin, P., Lowengrub, J.S.: A numerical method for the quasi-incompressible Cahn-Hilliard-Navier-Stokes equations for variable density flows with a discrete energy law. J. Comput. Phys. 276, 486–507 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Hintermüller, M., Hinze, M., Kahle, C.: An adaptive finite element Moreau-Yosida-based solver for a coupled Cahn-Hilliard/Navier-Stokes system. J. Comput. Phys. 235, 810–827 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Hintermüller, M., Hinze, M., Tber, M.H.: An adaptive finite element Moreau-Yosida-based solver for a non-smooth Cahn-Hilliard problem. Optim. Methods Softw. 25(4–5), 777–811 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Hysing, S., Turek, S., Kuzmin, D., Parolini, N., Burman, E., Ganesan, S., Tobiska, L.: Quantitative benchmark computations of two-dimensional bubble dynamics. Int. J. Numer. Methods Fluids 60(11), 1259–1288 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • James, A.J., Lowengrub, J.: A surfactant-conserving volume-of-fluid method for interfacial flows with insoluble surfactant. J. Comput. Phys. 201(2), 685–722 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Kahle, C.: Simulation and control of two-phase flow using diffuse-interface models. PhD thesis, University of Hamburg (2014)

    Google Scholar 

  • Kay, D., Loghin, D., Wathen, A.: A preconditioner for the steady state Navier-Stokes equations. SIAM J. Sci. Comput. 24(1), 237–256 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Kay, D., Styles, V., Welford, R.: Finite element approximation of a Cahn-Hilliard-Navier-Stokes system. Interfaces Free. Boundar. 10(1), 15–43 (2008)

    Article  MathSciNet  Google Scholar 

  • Kay, D., Welford, R.: A multigrid finite element solver for the Cahn-Hilliard equation. J. Comput. Phys. 212, 288–304 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  • Kay, D., Welford, R.: Efficient numerical solution of Cahn-Hilliard-Navier-Stokes fluids in 2d. SIAM J. Sci. Comput. 29(6), 2241–2257 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Kihara, N., Hanazaki, H., Mizuya, T., Ueda, H.: Relationship between airflow at the critical height and momentum transfer to the traveling waves. Phys. Fluids 19(015102) (2007)

    Article  MATH  Google Scholar 

  • Kim, J.: Phase-field models for multi-component fluid flows. Commun. Comput. Physics 12, 613–661 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Lowengrub, J., Truskinovsky, L.: Quasi-incompressible Cahn-Hilliard fluids and topological transitions. Proc. R. Soc. A 454(1978), 2617–2654 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Lubin, P., Glockner, S.: Numerical simulations of three-dimensional plunging breaking waves: generation and evolution of aerated vortex filaments. J. Fluid Mech. 767, 364–393 (2015)

    Article  MathSciNet  Google Scholar 

  • McWilliams, J., Sullivan, P., Moeng, C.: Langmuir turbulence in the ocean. J. Fluid Mech. 334, 1–30 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  • Mellado, J.P., Stevens, B., Schmidt, H., Peters, N.: Two-fluid formulation of the cloud-top mixing layer for direct numerical simulation. Theor. Comput. Fluid Dyn. 24, 511–536 (2010)

    Article  MATH  Google Scholar 

  • Polton, J., Smith, J., MacKinnon, J., Tejada-Martinez, A.: Rapid generation of high frequency internal waves beneath a wind wave forced oceanic surface mixed layer. Geophys. Res. Lett. (L13602) (2008)

    Google Scholar 

  • Shen, L., Zhang, X., Yue, D., Triantafyllou, M.: Turbulent flow over a flexible wall undergoing a streamwise travelling wave motion. J. Fluid Mech. 484, 197–221 (2003)

    Article  MATH  Google Scholar 

  • Sullivan, P., McWilliams, J.: Turbulent flow over water waves in the presence of stratification. Phys. Fluids 14, 1182–95 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Sullivan, P., McWilliams, J.: Dynamics of winds and currents coupled to surface waves. Ann. Rev. Fluid Mech. 42, 19–42 (2010)

    Article  MATH  Google Scholar 

  • Sullivan, P., McWilliams, J., Melville, W.: Surface gravity wave effects in the oceanic boundary layer: large-eddy simulation with vortex force and stochastic breakers. J. Fluid Mech. 593, 405–452 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Sullivan, P., McWilliams, J., Moeng, C.: Simulation of turbulent flow over idealized water waves. J. Fluid Mech. 404, 47–85 (2000)

    Article  MATH  Google Scholar 

  • Sutherland, P., Melville, W.: Field measurements of surface and near-surface turbulence in the presence of breaking waves. J. Phys. Oceanogr. 45, 943–965 (2015)

    Article  Google Scholar 

  • Tsai, W., Chen, S., Lu, G.: Numerical evidence of turbulence generated by nonbreaking surface waves. J. Phys. Oceanogr. 45, 174–180 (2015)

    Article  Google Scholar 

  • Tsai, W., Hung, L.: Three-dimensional modeling of small-scale processes in the upper boundary layer bounded by a dynamic ocean surface. J. Geophys. Res. 112, (C02019) (2007)

    Article  Google Scholar 

  • Wan, D., Turek, S.: An efficient multigrid-FEM method for the simulation of solid-liquid two phase flows. J. Comp. Appl. Math. 203(2), 561–580 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Verfürth, R.: A posteriori error analysis of space-time finite element discretizations of the time-dependent Stokes equations. Calcolo 47, 149–167 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Wunsch, C., Ferrari, R.: Vertical mixing, energy, and the general circulation of the oceans. Ann. Rev. Fluid Mech. 36, 281–314 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful for many discussions with Jeff Carpenter from the Helmholtz Center in Geesthacht on practical issues related to the wind-wave coupling at the interface of atmosphere and ocean. The second author acknowledges support of the TRR 181 funded by the German Research Foundation (DFG). The third author acknowledges support by the DFG through the International Research Training Group IGDK 1754 ‘Optimization and Numerical Analysis for Partial Differential Equations with Nonsmooth Structure.’

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany

    Harald Garcke

  2. Fachbereich Mathematik, Universität Hamburg, Hamburg, Germany

    Michael Hinze

  3. Zentrum Mathematik, Technische Universität München, Munich, Germany

    Christian Kahle

Authors
  1. Harald Garcke
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Hinze
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christian Kahle
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael Hinze .

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

Garcke, H., Hinze, M., Kahle, C. (2019). Diffuse Interface Approaches in Atmosphere and Ocean—Modeling and Numerical Implementation. 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_9

Download citation

Publish with us

Policies and ethics

Search

Navigation