## Abstract

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.

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## Data availability

The data used in this investigation can be obtained at https://www.ndbc.noaa.gov.

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## 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,

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.

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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). https://doi.org/10.1007/s42286-021-00048-y

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### Keywords

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