HF radar investigation of source terms in the Hasselmann equation

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HF radar investigation of source terms in the Hasselmann equation

Transcript Of HF radar investigation of source terms in the Hasselmann equation

HF radar investigation of source terms in the Hasselmann equation


Stuart Anderson
University of Adelaide

The spark that ignited my interest
“ … the values of different wind input terms scatter by a factor of 300 – 500 % ([1], [2])”
Zakharov, V., Resio, D., and Pushkarev, A.: Balanced source terms for wave generation within the Hasselmann equation, Nonlin. Processes Geophys., 24, 581–597, https://doi.org/10.5194/npg-24-5812017, 2017.
which referenced
[1] Badulin, S. I., Pushkarev, A. N., Resio, D., and Zakharov, V. E.: Self-similarity of wind-driven seas, Nonlin. Proc. Geoph., 12, 891–945, https://doi.org/10.5194/npg-12-891-2005, 2005.
[2] Pushkarev, A. and Zakharov, V.: Limited fetch revisited: comparison of wind input terms, in surface wave modeling, Ocean Model., 103, 18–37, https://doi.org/10.1016/j.ocemod.2016.03.005, 2016.

The spread of models for wind-wave growth :
S. I. Badulin, A. N. Pushkarev, D. Resio and V. E. Zakharov, ‘Self-similarity of wind-driven seas’, Nonlinear Processes in Geophysics, 12, 891–945, 2005

The Hasselmann equation I

The evolution of the wave field is often described by the action balance equation

𝜕𝑁 𝜕𝑡 + ∇𝑥 ⋅ ∇𝜅Ω𝑁 − ∇𝜅 ⋅ ∇𝑥Ω𝑁 where the wave action density 𝑁 𝜅Ԧ is

= 𝑆𝑖𝑛 + 𝑆𝑛𝑙 + 𝑆𝑑𝑖𝑠

under steady state conditions

related to the wave displacement spectrum

𝑆 𝜅Ԧ by 𝑁 𝜅 = 𝜌𝑤𝑔𝑆 𝜅Ԧ 𝜎𝜅

Steady state requires directional bimodality (Komen et al, 1984)

with  the intrinsic frequency,

Ω = 𝜅Ԧ ⋅ 𝑈 + 𝜎

input from the wind

spectral flux due to nonlinear interactions

loss via dissipative processes

The Hasselmann equation II

For a wide range of conditions, the action balance equation reduces to the familiar form in terms of the energy spectral density - the Hasselmann equation :

𝜕𝑆 𝜅Ԧ 𝜕𝑡 + ∇𝑥 ⋅ ∇𝜅Ω𝑆 𝜅Ԧ

= 𝑆𝑖𝑛 + 𝑆𝑛𝑙 + 𝑆𝑑𝑖𝑠

and it is this that forms the basis of the main wave modelling codes. There it is convenient to use frequency-angle coordinates,


𝑆 𝜅Ԧ 𝑑𝜅Ԧ 𝑔3

𝜌𝑤𝑔 𝑁 𝜅Ԧ 𝑑𝜅Ԧ = 𝜎 𝜅 = 2𝜎4 𝑆 𝜎, 𝜑 𝑑𝜎𝑑𝜑

The challenge is to find mathematical models for the source terms 𝑆𝑖𝑛, 𝑆𝑛𝑙 and 𝑆𝑑𝑖𝑠

Dissipation mechanisms for wind-generated surface gravity waves

Nt + x  ( N) −   (xN) = Sin + Snl + Sdis

Sscat + Sfrict + Sflex + Svisc

Sdis = S wc + Smv + Stv + Sma + Swci + Siw + Sbf + Sbiwb + Swiw + Sice + SBF

white-capping molecular viscosity turbulent viscosity Marangoni damping wave-current interactions internal wave coupling bottom friction bottom-induced wave breaking wave-induced winds sea ice coupling Benjamin Feir instability / Fermi-Pastra-Ulam recurrence

Wavenumber-direction space or frequency-direction space ?
Grid design and other considerations favour frequencydirection space for numerical wave modelling (eg SWAN, WAVEWATCH III )
From a radar perspective, wavenumber-direction space is better as the scattering integrals have an elegant physical interpretation in terms of mulltiple Bragg (resoant) scattering
The Jacobian is not ill-behaved, so both forms have been used by the HF radar community
There is a nice paper on the transformation by Hsu et al, in China Ocean Eng., Vol. 25, No. 1, pp. 133 – 138, 2011
See also F. Leckler, F. Ardhuin, C. Peureux, A. Benetazzo, F. Bergamasco, and V. Dulov, ‘Analysis and Interpretation of Frequency–Wavenumber Spectra of Young Wind Waves’, JPO, vol.45, pp. 2484-2496 from which the figure is taken

The SPM2 mapping from directional wave spectrum to radar Doppler spectrum

S =  dk R (k ',k ) (k '− k + 2k.nˆ nˆ)(.) +

+ d1F (1) (k ', k ,1)S(1)(.) +

+  d1d2F (2) (k ', k ,1,2 )S(1)S(2 )(.) +


NB : incomplete 4th order contributions for non-Gaussian surfaces

Currents, wind and wave information from the ACORN SA Gulf radar system

current velocity

significant waveheight inferred surface wind

Directional wave spectrum sampling levels

• Any HF radar can deliver (a)



• Multi-frequency radars can also do (b)

• If two radars illuminate the same patch of
ocean, they can deliver (c)

• If bistatic mode is enabled, they can do (d)

• If the signal is uncorrupted and inversion
is enabled, a single radar can do (e)



ZakharovSource TermsTermsModelsInterpretation