Skip to main content

Numerical Considerations for Quantifying Air–Water Turbulence with Moment Field Equations


We investigate energy transfer of air–water interactions and develop a numerical method that captures its temporal variability and generates and tracks the short waves that form in the water surface as a result of the air–water turbulence. We solve a novel system of balance equations derived from the Navier–Stokes equations known as moment field equations. The main advantage of our approach is that we do not assume a priori that the stochastic random variables that quantify the turbulent energy transfer between air and water are Gaussian. We generate non-conservative multifractal measures of turbulent energy transfer using a recursive integration process and a self-affine velocity kernel. The kernel exactly satisfies the (duration limited) kinetic equation for waves as well as invariant scaling properties of the Navier–Stokes equations. This allows us to derive source terms for the moment field equations using a turbulent diffusion operator. The operator quantifies energy transfer along a space time path associated with pressure instabilities in the air–sea interface and transfers the statistical shape (or fractal dimension) of the atmosphere to the wind-sea. Because we use observational data to begin the recursive integration process, the ocean–atmosphere interaction is inherently built into the model. Numerical results from application of our methods to air–sea turbulence off the coast of New Jersey and New York indicate that our methods produce measures of turbulent energy transfer that match theory and observation, and, correspondingly, significant wave heights and average wave periods predicted by our model qualitatively match buoy data.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Data availability

The data used in this investigation can be obtained at


  1. 1.

    Baker, G.R., Overman, E.A.: The art of scientific computing, Draft XVII, (2011)

  2. 2.

    Blake, E.S., Kimberlain, T.B., Berg, R.J., Cangialosi, J.P., Beven II, J.L.: Tropical Cyclone Report Hurricane Sandy (AL182012) 22-29 October 2012, National Hurricane Center, 12 February (2013)

  3. 3.

    Benzi, R., Paladin, G., Parisi, G., Vulpiani, A.: On the multifractal nature of fully developed turbulence and chaotic systems. J. Phys. A: Math. Gen. 17, 3521–3531 (1984)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Benzi, R., Ciliberto, S., Tripiccione, R., Baudet, C., Massaioli, F., Succi, S.: Extended self-similarity in turbulent flows. Phys. Rev. E 48, R29 (1993)

    Article  Google Scholar 

  5. 5.

    Cavaleri, L.: Wave modeling-missing the peaks. J. Phys. Oceanogr. 39, 2757–2778 (2009)

    Article  Google Scholar 

  6. 6.

    Cockburn, B., Shu, C.-W.: Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16(3), 173–261 (2001)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Conroy, C.J., Kubatko, E.J., Nappi, A., Sebian, R., West, D., Mandli, K.T.: hp discontinuous Galerkin methods for parametric, wind-driven water wave models. Adv. Water Resour. 119, 70–83 (2018)

    Article  Google Scholar 

  8. 8.

    Conroy, C.J., Mandli, K.T., Kubatko, E.J.: Quantifying air-water turbulence with moment field equations. J. Fluid Mech. 1–35 (2021).

  9. 9.

    Dietrich, J.C., Zijlema, M., Westerink, J.J., Holthuijsen, L.H., Dawson, C., Luettich Jr., R.A., Jensen, R.E., Smith, J.M., Stelling, G.S., Stone, G.W.: Modeling hurricane waves and storm surge using integrally-coupled, scalable computations. Coast. Eng. 58, 45–65 (2011)

    Article  Google Scholar 

  10. 10.

    Dietrich, J.C., Tanaka, S., Westerink, J.J., Dawson, C.N., Luettich Jr., R.A., Zijlema, M., Holthuijsen, L.H., Smith, J.M., Westerink, L.G., Westerink, H.J.: Performance of the Unstructured-Mesh, SWAN+ADCIRC Model in Computing Hurricane Waves and Surge. J. Sci. Comput. 52(2), 468–497 (2012)

    Article  Google Scholar 

  11. 11.

    Dubuc, B., Quiniou, J.F., Roques-Carmes, C., Tricot, C., Zucker, S.W.: Evaluating the fractal dimension of profiles. Phys. Rev. A 39(3), 1500–1512 (1989)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Frisch, U.: Turbulence: the legacy of A.N. Kolmogorov, Cambridge University Press, (1996)

  13. 13.

    Hasselmann, K., Ross, D.B., Müller, P., Sell, W.: A Parametric Wave Prediction Model. J. Phys. Oceanogr. 6, 200–228 (1975)

    Article  Google Scholar 

  14. 14.

    Hentschel, H.G.E., Procaccia, I.: Fractal nature of turbulence as manifested in turbulent diffusion. Phys. Rev. A 27(2), 1266–1269 (1983)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Holthuijsen, L.H., Booij, N., Herbers, T.H.C.: A prediction model for stationary, short-crested waves in shallow water and ambient currents. Coast. Eng. 13, 23–54 (1989)

    Article  Google Scholar 

  16. 16.

    Holthuijsen, L.H.: Waves In Oceanic and Coastal Waters. Cambridge University Press, Cambridge (2007)

    Book  Google Scholar 

  17. 17.

    Janssen, P.: The Interaction of Ocean Waves and Wind. Cambridge University Press, Cambridge (2004)

    Book  Google Scholar 

  18. 18.

    Jensen, R.E., Cialone, A., Smith, J.M., Bryant, M.A., Hesser, T.J.: Regional wave modeling and evaluation for the North Atlantic coast comprehensive study, J. Waterway Port Coastal Ocean Eng. (2016),

  19. 19.

    Kolmogorov, A.N.: The local structure of turbulence in incompressible viscous fluid for very large Reynolds’ numbers. Doklady Akademiia Nauk SSSR 30, 301–305 (1941)

  20. 20.

    Kolmogorov, A.N.: On analytical methods in probability theory, Selected Works of A.N. Kolmogorov Volume II: Probability Theory and Mathematical Statistics, Springer Science+Business Media Dordrecht, pp.62-108, (1992)

  21. 21.

    Kubatko, E.J., Westerink, J.J., Dawson, C.: \(hp\) discontinuous Galerkin methods for advection dominated problems in shallow water flow. Comput. Methods Appl. Mech. Eng. 196, 437–451 (2006)

    Article  Google Scholar 

  22. 22.

    Kubatko, E.J., Yeager, B.A., Ketcheson, D.I.: Optimal Strong-Stability-Preserving Runge-Kutta Time Discretizations for Discontinuous Galerkin Methods. J. Sci. Comput. 60, 313–344 (2014)

    MathSciNet  Article  Google Scholar 

  23. 23.

    LeVeque, R.J.: Finite volume methods for hyperbolic problems, Cambridge University Press, New York, (2004)

  24. 24.

    Liberto, T.D., Colle, B.A., Georgas, N., Blumberg, A.F., Taylor, A.A.: Verification of a multimodel storm surge ensemble around New York City and long island for the cool season. Wea. Forecast. 26, 922–939 (2011)

    Article  Google Scholar 

  25. 25.

    Mandelbrot, B.B.: Intermittent turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier. J. Fluid Mech. 62, 331–358 (1974)

    Article  Google Scholar 

  26. 26.

    Mandelbrot, B.B.: Fractals: form, chance and dimension, (1977)

  27. 27.

    Mellor, G.L., Donelan, M.A., Oey, L.Y.: A surface wave model for coupling with numerical ocean circulation models. J. Atmos. Ocean. Technol. 25, 1785–1807 (2008)

    Article  Google Scholar 

  28. 28.

    Meneveau, C., Sreenivasan, K.: Simple multifractal cascade model for fully developed turbulence. Phys. Rev. Lett. 59(13), 1424–1427 (1987)

    Article  Google Scholar 

  29. 29.

    Miles, J.W.: On the generation of surface waves by shear flows, J. Fluid Mech., pp. 185–204

  30. 30.

    Ning, L., Emanuel, K., Vanmarcke, E.: Hurricane Risk Analysis, Safety, Reliability, Risk and Life-Cycle Performance of Structures and Infrastructures (January 9): 1291-1297 (2014)

  31. 31.

    National Oceanic and Atmospheric Administration, National Data Buoy Center, Buoy 44065,

  32. 32.

    National Oceanic and Atmospheric Administration, National Data Buoy Center, Buoy 45005,

  33. 33.

    Phillips, O.M.: On the generation of waves by turbulent wind. J. Fluid Mech. 2(5), 417–445 (1957)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Resio, D.T., Vincent, L., Ardag, D.: Characteristics of directional wave spectra and implications for detailed-balance wave modeling. Ocean Model. (2015).

    Article  Google Scholar 

  35. 35.

    Richardson, L.F.: Atmospheric diffusion shown on a distance-neighbour graph. Proc. R. Soc. A Lond. 110(756), 709–737, (1926)

  36. 36.

    Salmon, R.: Lectures on Geophysical Fluid Dynamics. Oxford University Press, Oxford (1998)

    Book  Google Scholar 

  37. 37.

    Syu, C.Y., Kirchoff, R.H.: The fractal dimension of the wind. J. Solar Energy 115, 151–154 (1993)

    Article  Google Scholar 

  38. 38.

    Taleb, N.N.: Statistical consequences of fat tails: real world preasymptotics, epistemology, and applications, STEM Academic press, (2020)

  39. 39.

    Tanaka, M.: Verification of Hasselmann’s energy transfer among surface gravity waves by direct numerical simulations of primitive equations. J. Fluid Mech. 444, 199–221 (2001)

  40. 40.

    Zakharov, V., Resio, D., Pushkarev, A.: Balanced source terms for wave generation within the Hasselmann equation. Nonlinear Process. Geophys. 24, 581–597 (2017)

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Colton J. Conroy.

Ethics declarations

Conflict of interest

The authors report no conflict of interest.

Computer code availability

The computer code used in this investigation can be obtained at

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Steady Energy Matching Condition for Duration- and Fetch-Limited Waves

Appendix A: Steady Energy Matching Condition for Duration- and Fetch-Limited Waves

In the case of a purely steady wind (\(q \equiv 0\)) blowing over a fetch-limited body of water, we ensure the duration-limited source terms supply a consistent amount of energy to the moment field by setting Eq. (23) equal to (48), and solve for \(t_l\), which yields,

$$\begin{aligned} \nu (t_l)= & {} \nu (\chi ), \nonumber \\ a\left[ \frac{g}{\left| \left| \mathbf {u}_0\right| \right| }\left( t_{l}\right) \right] ^{-3/7}= & {} A\left[ \frac{g}{\left| \left| \mathbf {u}_{\chi }\right| \right| ^2}\chi \right] ^{3/7\left( p-1\right) }, \nonumber \\ t_l= & {} \left( \frac{\left| \left| \mathbf {u}_0\right| \right| }{g}\right) \left( \frac{A}{a}\right) ^{-7/3} \left[ \frac{g}{\left| \left| \mathbf {u}_{\chi }\right| \right| ^2}\chi \right] ^{\left( 1-p\right) }. \end{aligned}$$

We then set \(t = t_l\) in the source terms in Eqs. (7) and (8). There are two important points to make note of i. \(t_l\) is a function of the fetch coordinate, \(\chi \in [0,L_{\chi }]\), where \(L_{\chi }\) is the total fetch length, and ii. in Hasselmann’s parameterization, the energy of the water waves is a function of the wave frequency, and this is why we choose to match the frequency of the duration- and fetch-limited solutions (this also conveniently ensures that the first-order moments will match as well). It can be noted that in simulations of physical wave records, q is rarely (if ever) exactly equal to 0. In fact, in our companion paper [8] we model surface wave heights over the fetch-limited domain of Lake Erie and q never equals a true zero and model results match observations quantitatively well.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Conroy, C.J., Mandli, K.T. & Kubatko, E.J. Numerical Considerations for Quantifying Air–Water Turbulence with Moment Field Equations. Water Waves 3, 319–354 (2021).

Download citation


  • Wind-wave interaction
  • Turbulent energy transfer
  • Moment field equations
  • Recursive integration