Wave damping due to sea ice

SWAN has an option to include wave dissipation by sea ice. Sea ice has two effects, direct and indirect. A direct dissipation of wave energy occurs due to the presence of sea ice. This dissipation is represented in SWAN using empirical formula. The temporal exponential decay rate of energy is

  $$D_{\rm ice} = S_{\rm ice}/E = -2\,c_g\,k_i$$ (2.152)

where $S_{\rm ice}$ is the sea ice sink term, and $E$ is the wave energy spectrum. Here, $k_i$ (in 1/m) is the linear exponential attenuation rate of wave amplitude in space, $a(x)=H_0\exp(-k_i\,x)$. The factor 2 above provides a conversion from amplitude decay to energy decay. The group velocity $c_g$ provides conversion from spatial decay to temporal decay. $S_{\rm ice}$ and $E$ vary with frequency and direction.


R19


In SWAN version 41.31, one option for this dissipation was available. This is a parameterization described in Collins and Rogers (2017) and is similar to the `IC4M2' method implemented in the WAVEWATCH III model. That method, in turn, is a generalization of the formula proposed by Meylan et al. (2014). This implementation in SWAN is described in Rogers (2019), and so in the present version of SWAN, it is denoted as the `R19'. With it, the model $k_i$ may vary with frequency according to

  $$k_i \left ( f \right ) = c_0 + c_1\,f + c_2\,f^2 + c_3\,f^3 + c_4\,f^4 + c_5\,f^5 + c_6\,f^6$$ (2.153)

with $c_0$ to $c_6$ the user-defined polynomial coefficients. These coefficients are dimensional; e.g. $c_2$ has units of s$^2$/m.


The default R19 setting is the case of $c_2$ =1.06$\times$10$^{-3}$ s$^2$/m and $c_4$ =2.3$\times$10$^{-2}$ s$^4$/m. This recovers the polynomial of Meylan et al. (2014), calibrated for a case of ice floes, mostly 10 to 25 m in diameter, in the marginal ice zone near Antarctica. Another calibration, for a case that is similar except with relatively thinner ice, from Rogers et al. (2021a) is $c_2$ =0.208$\times$10$^{-3}$ s$^2$/m and $c_4$ =5.18$\times$10$^{-2}$ s$^4$/m. Other polynomials are provided in Rogers et al. (2018). An example is for a case of pancake and frazil ice: $c_2$ =0.284$\times$10$^{-3}$ s$^2$/m and $c_4$ =1.53$\times$10$^{-2}$ s$^4$/m.


This `R19' method does not depend on ice thickness. In version 41.41, three new methods are introduced which depend on ice thickness, denoted as `D15', `M18', and `R21B'.


D15


The `D15' method is a purely empirical formula from Doble et al. (2015), $k_i = C_{hf,D} f^{2.13} h_{ice}$, with default $C_{hf,D}$ = 0.1 based on the same study, for pancake ice in the marginal ice zone (MIZ) of the Weddell Sea (Antarctica).


M18


The `M18' method is the “Model with Order 3 Power Law" proposed by Meylan et al. (2018). This is a simple viscous model of the form $k_i = C_{hf,M} h^1_{ice} f^3$, where $C_{hf,M}$ includes a viscosity parameter. Our implementation here has a default $C_{hf,M} =0.059$ based on calibration to the Rogers et al. (2021a) dataset (broken floes in the Antarctic MIZ) by Rogers et al. (2021b). Two earlier calibrations were performed by Liu et al. (2020): $C_{hf,M} =0.00751$ for a case of broken floes in the Antarctic MIZ and $C_{hf,M} =0.0351$ for a case of pancake and frazil ice near the Beaufort Sea.


R21B


The `R21B' method combines the Reynolds number non-dimensionalization proposed by Yu et al. (2019) with a simple monomial power fitting. This is documented in Rogers et al. (2021b). The resulting dimensional formula is $k_i = C_{hf} h^{n/2-1}_{ice} f^n$. Those authors calibrate it to the dataset of Rogers et al. (2021a), giving $n$ = 4.5 and $C_{hf}$ = 2.9.


Source term scaling


An indirect effect of sea ice is a reduction of wind input (scaling). The areal fraction of sea ice is given as $0 \leq a_{\rm ice} \leq 1$. The effect on wind input is a scaling of the wind input source functions by open water fraction $1-a_{\rm ice}$,

  $$S_{\rm in} \, \leftarrow \, \left ( 1 - a_{\rm ice} \right ) \,S_{\rm in}$$ (2.154)

with $S_{\rm in}$ the wind input term; see Eq. (2.33). This effect can be reduced or disabled (see command SET [icewind].)


Also, the ice source function is scaled with areal ice fraction, as follows

  $$S_{\rm ice} \, \leftarrow \, a_{\rm ice}\,S_{\rm ice}$$ (2.155)

The impact of sea ice on source terms in the real ocean is not known with any certainty, and instead is primarily based on intuition and guesswork; see discussion in Rogers et al. (2016). Nonlinear interactions are not scaled.


The sea ice treatment here has the following limitations:

• reflection and scattering by sea ice is not represented, and
• floe size distribution is not represented.