Surface wave spectral properties of centimeter to decameter wavelengths: variable spectral slope and non-equilibrium spectrum

Abstract

Understanding the wave properties in centimeter to decameter (cmDm) wavelength range is of great interest to ocean remote sensing and air-sea interaction. For more than six decades, cmDm waves are generally considered to be in the equilibrium range, and its spectral function has a constant slope: − 5 or − 4 in the 1D frequency spectrum and − 3 or − 2.5 in the 1D wavenumber spectrum. Some variations of the equilibrium spectrum model include varying the frequency spectral slope from − 4 to − 5 at some multiple of the spectral peak frequency, or incorporating a threshold velocity in the reference wind speed. Extensive efforts are then devoted to quantifying the spectral coefficient of the equilibrium spectrum function. This paper emphasizes that the observed wave spectral slope in the ocean is mostly non-constant. Therefore, the wave properties in field observations are significantly different from those inferred from assuming a constant spectral slope. The variable spectral slope is indicative of a non-equilibrium nature of ocean surface waves. Furthermore, from signal-to-noise consideration, surface slope is much more suited than elevation for evaluating the spectral slope and for quantifying the cmDm wave properties. Microwave sensors are excellent instruments for providing the ocean surface slope measurements. Several large datasets of lowpass-filtered mean square slope (LPMSS) have been reported recently. Combining the LPMSS observations with a spectrum function that accommodates a variable spectral slope, several quantitative results on the variable spectral slope and cmDm wave properties are presented.

Appendices

Appendix 1. The G (general) wind wave spectrum model

The G model is given as:

$$ S\left(\omega \right)=\alpha {g}^2{\omega}_p^{-5}{\varsigma}^{-{s}_f}\exp \left[-\frac{s_f}{4}{\varsigma}^{-4}\right]{\gamma}^{\Gamma};\Gamma =\exp \left[-\frac{{\left(1-\varsigma \right)}^2}{2{\sigma}^2}\right];\varsigma =\frac{\omega }{\omega_p}. $$

(A1)

The associated spectral parameters vary with s_f and ω_#:

$$ {\displaystyle \begin{array}{l}\alpha ={\alpha}_1\left[1-0.3\tanh \left(0.1{\omega}_{\#}\right)\right]\\ {}\gamma ={\gamma}_1\left[1-0.5\tanh \left(0.1{\omega}_{\#}\right)\right]\\ {}\sigma ={A}_{\sigma }+{a}_{\sigma}\log \left({\omega}_{\#}\right)\end{array}}, $$

(A2)

where

$$ {\displaystyle \begin{array}{l}{\alpha}_1={A}_{\alpha }{\omega}_{\#}^{a_{\alpha }}\\ {}{\gamma}_1={A}_{\gamma }+{a}_{\gamma}\log \left({\omega}_{\#}\right)\end{array}} $$

(A3)

The coefficients A_α, a_α, A_γ, a_γ, A_σ, and a_σ are functions of s_f:

$$ {\displaystyle \begin{array}{l}{A}_{\alpha }=1.30\times {10}^{-3}{s}_f+1.64\times {10}^{-3},\\ {}{a}_{\alpha }=4.83\times {10}^{-1}{s}_f-1.49,\\ {}{A}_{\gamma }=4.42\times {10}^{-1}{s}_f+3.93\times {10}^{-1},\\ {}{a}_{\gamma }=-3.63{s}_f+19.74,\\ {}{A}_{\sigma }=-5.39\times {10}^{-2}{s}_f+3.44\times {10}^{-1},\\ {}{a}_{\sigma }=2.05\times {10}^{-9}{s}_f+5.5\times {10}^{-2}.\end{array}} $$

(A4)

Appendix 2. Parametric models of TC wind and wave fields

The wind speed parametric model is based on the modified Rankine vortex model (Holland 1980; Holland et al. 2010). To account for the wind field asymmetry, the vortex wind speed is normalized by U_10mϕ, which is the maximum wind speed along a radial transect along a given azimuth angle ϕ,

$$ \kern0em \frac{U_{10}\left(r,\phi \right)}{U_{10\mathrm{m}\phi }}=\left\{\begin{array}{cc}{r}_{\ast },& {r}_{\ast}\le 1\\ {}{r}_{\ast}^{-0.5},& {r}_{\ast }>1\end{array}\right., $$

(B1)

where r is the radial distance from the TC center; the normalized radial distance is r_*=r/r_m; the azimuth angle ϕ is measured from the TC heading, increasing counterclockwise (CCW). Based on examination of archived NOAA HWIND 2D wind fields in several historical TCs, the following formula is used for U_10mϕ:

$$ \kern0em \frac{U_{10\mathrm{m}\phi }}{U_{10\mathrm{m}}}=1-{a}_{1U}\left[1-\cos \left(\phi -{\phi}_m\right)\right], $$

(B2)

where ϕ_m is the azimuth angle of the location of the overall maximum wind speed U_10m; the U_10m location is mostly in the right-hand side relative to the TC heading, with higher probability in the right-front quadrant (ϕ between 270 and 360°) than the right-back quadrant (ϕ between 180 and 270°). The wind field asymmetry factor a_1U varies mostly between about 0.1 and 0.2.

For the wave spectrum computation, the most useful characteristic wave parameter is ω_#. In the TC wave parametric model, it is given in the normalized form: ω_* = ω_#/U_10m; the prominent feature of ω_*(ϕ) is the sinusoidal variation:

$$ {\omega}_{\ast }={a}_0\left({r}_{\ast}\right)+{a}_1\left({r}_{\ast}\right)\cos \left[\phi +\delta \left({r}_{\ast}\right)\right]. $$

(B3)

The coefficients a₀, a₁, and δ vary with r_*, and they are empirically determined from the simultaneous wind and wave measurements collected in hurricane reconnaissance and research missions

$$ \kern0em {a}_0=\left\{\begin{array}{cc}0.0056{r}_{\ast },& {r}_{\ast}\le 1\\ {}0.0056{r}_{\ast}^{-0.5},& 1<{r}_{\ast}\le 3.3\\ {}0.03,& 3.3<{r}_{\ast}\end{array}\right.. $$

(B4)

$$ \kern0em \frac{a_1}{a_0}=\left\{\begin{array}{cc}-0.1,& {r}_{\ast}\le 1\\ {}-0.1{r}_{\ast}^{-1/3},& 1<{r}_{\ast}\le 2\\ {}-0.27,& 2<{r}_{\ast}\end{array}\right.. $$

(B5)

$$ \kern0em \delta =\left\{\begin{array}{cc}-60+240\left(1-{r}_{\ast}\right),& {r}_{\ast}\le 1.1\\ {}-80{r}_{\ast}^{-1/3},& 1.1<{r}_{\ast}\end{array}\right.. $$

(B6)

More details of the parametric functions of U₁₀ and ω_# inside TCs are given in Hwang and Fan (2018).