A 4D variational assimilation scheme with partition method for nearshore wave models

Abstract

This paper summarizes the development steps of a 4D variational assimilation scheme for nearshore wave models. A partition method is applied for adjusting both wave boundary conditions and wind fields. Nonstationary conditions are assimilated by providing defined correlations of model inputs in time. The scheme is implemented into the SWAN model. Twin experiments covering both stationary and nonstationary wave conditions are carried out to assess the adequacy of the proposed scheme. Stationary experiments are carried out considering separately windsea, swells, and mixed sea. Cost functions decline to less than 5% and RMS spectrum errors are reduced to less than 10%. The nonstationary experiment covers 1 day simulation under mixed wave conditions with assimilation windows of 3 h. RMS spectrum errors are reduced to less than 10% after 30 iterations in most assimilation windows. The results show that for spacially uniform model inputs, model accuracy is improved notably by the assimilation scheme throughout the computational domain. It is found that under wave conditions in which observed spectra can be well classified, the assimilation scheme is able to improve model results significantly.

Appendices

Appendix A: Source terms in the adjoint and the gradient

S _w including three terms: wind growth, whitecapping dissipation, and bottom friction dissipation. It reads:

$$ S_{w}=S_{wg}+S_{dw}+S_{df} $$

(26)

where S _{w g} is the wind growth, S _{d w} is the whitecapping dissipation and S _{d f} is the bottom friction dissipation. The wind growth term S _{w g} applied in this study is based on Snyder et al. (1981) and Komen et al. (1984). It reads:

$$ S_{wg}=0.25\rho_{aw}(\frac{28U_{*}\cos(\theta-\theta_{w})}{2\pi f/k}-1)2\pi fE(f,\theta) $$

(27)

where ρ _{a w} is a ratio between the density of air and the density of water, 𝜃 _w is the direction of wind vector, k is the wave number and U _∗ is a friction velocity. U _∗ reads:

$$\begin{array}{@{}rcl@{}} U_{*}^{2} &=& C_{D} U_{10}^{2} \\ C_{D} &=& \left\{\begin{array}{ll} 1.2875\times 10^{-3} & U_{10} < 7.5~\mathrm{m/s} \\ (0.8+0.065U_{10}) 10^{-3} & U_{10}\geq 7.5~\mathrm{m/s} \end{array}\right. \end{array} $$

(28)

where U ₁₀ is the wind speed at 10 m elevation and C _D is the drag coefficient from Wu (1982). The expression of white capping dissipation is Janssen (1991) and Günther et al. (1992):

$$\begin{array}{@{}rcl@{}} S_{dw} &=& F_{w}(\bar{f},\bar{k},E_{t}) E(f,\theta) \\ F_{w}&=&-\frac{C_{ds}}{\bar{s}_{pm}^{p}} 2\pi \bar{f} \bar{k}^{p}E_{t}^{p/2} ((1-\delta)\frac{k}{\bar{k}}+\delta(\frac{k}{\bar{k}})^{2}) \end{array} $$

(29)

where $\bar {f}$ is a mean frequency, $\bar {k}$ is a mean wave number, E _t is the total wave energy and $\bar {s}_{pm}$ is the wave steepness in Pierson-Moskowitz spectrum (Pierson and Moskowitz 1964). C _{d s} , δ and p are tunable coefficients. Following Komen et al. (1984), the default number of C _{d s} in the SWAN model equals to 2.36 × 10⁻5, p equals to 2 and δ equals to 1. The bottom friction dissipation term S _{d f} reads:

$$ S_{df}= -C_{b}\frac{(2\pi f)^{2}}{g^{2}{\sinh}^{2} (kd)}E(f,\theta) $$

(30)

where C _b is the bottom friction coefficient set to be 0.067 m²/s³. Based on Eqs. 26 to 30, derivatives of S _w with respect to E in Eq. 16 are:

$$ \frac{\partial S_{wg}}{\partial E}=0.25\rho_{aw}(\frac{28U_{*}\cos(\theta-\theta_{w})}{2\pi f/k}-1)2\pi f $$

(31)

$$ \frac{\partial S_{df}}{\partial E}= -C_{b}\frac{(2\pi f)^{2}}{g^{2}{\sinh}^{2} (kd)} $$

(32)

$$\begin{array}{@{}rcl@{}} \frac{\partial S_{dw}}{\partial E}&=&\frac{\partial F_{w}}{\partial E}E +F_{w}\\ \frac{\partial F_{w}}{\partial E}&=& \frac{\partial F_{w}}{\partial E_{t}}\frac{\partial E_{t}}{\partial E}+\frac{\partial F_{w}}{\partial \bar{f}}\frac{\partial \bar{f}}{\partial E}+\frac{\partial F_{w}}{\partial \bar{k}}\frac{\partial \bar{k}}{\partial E} \end{array} $$

(33)

$\frac {\partial E_{t}}{\partial E}$, $\frac {\partial \bar {f}}{\partial E}$ and $\frac {\partial \bar {k}}{\partial E}$ in Eq. 33 can be calculated as De Las Heras et al. (1994):

$$\begin{array}{@{}rcl@{}} \frac{\partial E_{t}}{\partial E}&=& 1\\ \frac{\partial \bar{f}}{\partial E}&=& \frac{\bar{f}}{E_{t}}(1-\frac{\bar{f}}{f})\\ \frac{\partial \bar{k}}{\partial E}&=& \frac{2\bar{k}}{E_{t}}[1-(\frac{\bar{k}}{k})] \end{array} $$

(34)

Hence, the source terms $\frac {\partial S_{w}}{\partial E}$ in Eq. 16 can be calculated explicitly with Eqs. 26 to 34. The source term $\frac {\partial S_{s}}{\partial E}$ in Eq. 17 can also be obtained by Eq. 32 since the friction bottom dissipation is the only term in S _s .

The wind components u and v only explicitly relate to S _{w g} . Therefore, $\frac {\partial S_{w}}{\partial u}$ and $\frac {\partial S_{w}}{\partial u}$ in Eq. 18 can be obtained as

$$\begin{array}{@{}rcl@{}} \frac{\partial S_{w}}{\partial u}&=&\frac{\partial S_{wg}}{\partial u}=\frac{\partial S_{wg}}{\partial U_{*}}\frac{\partial U_{*}}{\partial u} +\frac{\partial S_{wg}}{\partial \theta_{w}}\frac{\partial \theta_{w}}{\partial u}\\ \frac{\partial S_{w}}{\partial v}&=&\frac{\partial S_{wg}}{\partial v}=\frac{\partial S_{wg}}{\partial U_{*}}\frac{\partial U_{*}}{\partial v} +\frac{\partial S_{wg}}{\partial \theta_{w}}\frac{\partial \theta_{w}}{\partial v} \end{array} $$

(35)

where $\frac {\partial S_{wg}}{\partial U_{*}}$ and $\frac {\partial S_{wg}}{\partial \theta _{w}}$ can be expressed as

$$\begin{array}{@{}rcl@{}} \frac{\partial S_{wg}}{\partial U_{*}}&=&0.25\rho_{aw}(28k\cos(\theta-\theta_{w})) E(f,\theta)\\ \frac{\partial S_{wg}}{\partial \theta_{w}}&=&0.25\rho_{aw}(28kU_{*}{\sin}(\theta-\theta_{w}))E(f,\theta) \end{array} $$

(36)

From Eq. 28, obviously, U _∗ is related to the wind speed. So we have

$$\begin{array}{@{}rcl@{}} \frac{\partial U_{*}}{\partial u} &=& \frac{\partial U_{*}}{\partial U_{10}} \frac{\partial U_{10}}{\partial u}\\ \frac{\partial U_{*}}{\partial v} &=& \frac{\partial U_{*}}{\partial U_{10}} \frac{\partial U_{10}}{\partial v} \end{array} $$

(37)

That means the only required information for Eq. 35 is the conversion from the wind speed U ₁₀ and the wind direction 𝜃 _w to the wind components u and v which can be easily obtained as:

$$\begin{array}{@{}rcl@{}} \frac{\partial U_{10}}{\partial u} &=& \frac{u }{ U_{10}} , \frac{\partial U_{10}}{\partial v} = \frac{v }{ U_{10}} \\ \frac{\partial \theta_{w}}{\partial u} &=& \left\{\begin{array}{ll} \frac{u^{2}-U_{10}^{2} }{ U_{10}^{3}{\sin}(\theta_{w})} & {\sin}(\theta_{w}) \neq 0\\ \frac{-uv}{ U_{10}^{3}\cos(\theta_{w})} &{\sin}(\theta_{w}) = 0 \end{array}\right.\\ \frac{\partial \theta_{w}}{\partial v} &=& \left\{\begin{array}{ll} \frac{U_{10}^{2}-v^{2} }{ U_{10}^{3}\cos(\theta_{w})} & \cos(\theta_{w}) \neq 0\\ \frac{uv}{ U_{10}^{3}{\sin}(\theta_{w})} &\cos(\theta_{w}) = 0 \end{array}\right. \end{array} $$

(38)

Hence, based on Eqs. 35 to 38, the gradient calculation for the wind components in Eq. 18 can be processed.

The wind growth expression used in the Cycle 4 of the WAM model is also optional (Komen et al. 1994). In that case, the calculation of the deviation is much more complicated. The details can be found in De Las Heras et al. (1994).

Appendix B: Discrete forms of the adjoints

Generally, the discrete schemes applied in the adjoints of Eqs. 16 and 17 are based on the SWAN forward model which is a implicit scheme but being solved explicitly (Team et al. 2010). The discrete forms of Eqs. 16 and 17 are:

$$\begin{array}{@{}rcl@{}} &&\frac{1}{\Delta t}((\lambda_{w})_{i,j,l,m}^{n-1}-(\lambda_{w})_{i,j,l,m}^{n})+\frac{1}{\Delta x}C_{x}((\lambda_{w})_{i-1/2,j,l,m}^{n-1}\\ &&-(\lambda_{w})_{i+1/2,j,l,m}^{n-1}) +C_{y}((\lambda_{w})_{i,j-1/2,l,m}^{n-1} -(\lambda_{w})_{i,j+1/2,l,m}^{n-1})\\ &&+\frac{1}{2\pi{\Delta} f}C_{f}((\lambda_{w})_{i,j,l-1/2,m}^{n-1}-(\lambda_{w})_{i,j,l+1/2,m}^{n-1})\\ &&+\frac{1}{\Delta \theta}C_{\theta}((\lambda_{w})_{i,j,l,m-1/2}^{n-1}- (\lambda_{w})_{i,j+1/2,l,m+1/2}^{n-1})\\ &=& (\lambda_{w})_{i,j,l,m}^{n}(S^{E}_{wg})^{n}_{i,j,l,m}+ (\lambda_{w})_{i,j,l,m}^{n}(S^{E}_{dw})^{n}_{i,j,l,m}\\ &&+(\lambda_{w})_{i,j,l,m}^{n}(S^{E}_{df})^{n}_{i,j,l,m} -(S^{N}_{ob1})^{n}_{i,j,l,m} \end{array} $$

(39)

$$\begin{array}{@{}rcl@{}} &&\frac{1}{\Delta t}((\lambda_{s})_{i,j,l,m}^{n-1}-(\lambda_{s})_{i,j,l,m}^{n})+\frac{1}{\Delta x}C_{x}((\lambda_{s})_{i-1/2,j,l,m}^{n-1}\\ &&-(\lambda_{s})_{i+1/2,j,l,m}^{n-1}) \!+C_{y}((\lambda_{s})_{i,j-1/2,l,m}^{n-1}\!- (\lambda_{s})_{i,j+1/2,l,m}^{n-1}) \\ &&+\frac{1}{2\pi{\Delta} f}C_{f}((\lambda_{s})_{i,j,l-1/2,m}^{n-1}-(\lambda_{s})_{i,j,l+1/2,m}^{n-1})\\ &&+\frac{1}{\Delta \theta}C_{\theta}((\lambda_{s})_{i,j,l,m-1/2}^{n-1} -(\lambda_{s})_{i,j+1/2,l,m+1/2}^{n-1}) \\ &=& (\lambda_{s})_{i,j,l,m}^{n}(S^{E}_{df})^{n}_{i,j,l,m}-(S^{N}_{ob2})^{n}_{i,j,l,m} \end{array} $$

(40)

Where $S^{E}_{wg}$, $S^{E}_{dw}$, $S^{E}_{df}$, $S^{N}_{ob1}$ and $S^{N}_{ob2}$ represent the discrete forms of $\frac {\partial S_{wg}}{\partial E}$, $\frac {\partial S_{dw}}{\partial E}$, $\frac {\partial S_{df}}{\partial E}$, $\frac {\partial S_{ob1}}{\partial N}$ and $\frac {\partial S_{ob1}}{\partial N}$ respectively. Their expressions read:

$$\begin{array}{@{}rcl@{}} (S^{E}_{wg})^{n}_{i,j,l,m}&=&(\frac{\partial S_{wg}}{\partial E})^{n}_{i,j,l,m}\\ (S^{E}_{dw})^{n}_{i,j,l,m}&=& (\frac{\partial S_{dw}}{\partial E})^{n}_{i,j,l,m}\\ (S^{E}_{df})^{n}_{i,j,l,m}&=& (\frac{\partial S_{df}}{\partial E})^{n}_{i,j,l,m}\\ (S^{N}_{ob1})^{n}_{i,j,l,m}&=&\frac{4\pi f W_{1}}{M_{obs}} \sum\limits_{a=1}^{M_{obs}}(hE-\hat{h}\hat{E_{a}})\\ &&\delta^{i_{a}}_{i}\delta^{j_{a}}_{j}\delta^{l_{a}}_{l}\delta^{m_{a}}_{m}\delta^{n_{a}}_{n}|^{n}_{i,j,k,m}\\ (S^{N}_{ob2})^{n}_{i,j,l,m}&=&\frac{4\pi f W_{1}}{M_{obs}} \sum\limits_{a=1}^{M_{obs}}((1-h)E-(1-\hat{h})\hat{E_{a}})\\ &&\delta^{i_{a}}_{i}\delta^{j_{a}}_{j}\delta^{l_{a}}_{l}\delta^{m_{a}}_{m}\delta^{n_{a}}_{n}|^{n}_{i,j,k,m} \end{array} $$

(41)

First order upwind schemes are employed for the discretization in both geographical and spectral space for the forward model (Team et al. 2010). But the adjoint model is calculated backward in time so that the upwind scheme must turn to the “downwind” scheme. Therefore, the schemes become:

$$\begin{array}{@{}rcl@{}} \text{If} \hspace{0.1cm} (C_{f})_{i,j,l,m} &>& 0 \hspace{0.1cm} \text{then}\\ \lambda_{i,j,l+1/2,m}&=&(1-0.5\mu) \lambda_{i,j,l+1,m}+ 0.5\mu\lambda_{i,j,l,m} \\ \lambda_{i,j,l-1/2,m}&=&(1-0.5\mu) \lambda_{i,j,l,m}+ 0.5\mu\lambda_{i,j,l-1,m} \\ \text{If} \hspace{0.1cm} (C_{f})_{i,j,l,m} &<& 0 \hspace{0.1cm} \text{then}\\ \lambda_{i,j,l+1/2,m}&=&(1-0.5\mu) \lambda_{i,j,l,m}+ 0.5\mu\lambda_{i,j,l+1,m} \\ \lambda_{i,j,l-1/2,m}&=&(1-0.5\mu) \lambda_{i,j,l-1,m}+ 0.5\mu\lambda_{i,j,l,m} \\ \text{If} \hspace{0.1cm} (C_{\theta})_{i,j,l,m} &>& 0 \hspace{0.1cm} \text{then}\\ \lambda_{i,j,l,m+1/2}&=&(1-0.5\nu) \lambda_{i,j,l,m+1}+ 0.5\nu\lambda_{i,j,l,m} \\ \lambda_{i,j,l,m-1/2}&=&(1-0.5\nu) \lambda_{i,j,l,m}+ 0.5\nu\lambda_{i,j,l,m-1} \\ \text{If} \hspace{0.1cm} (C_{\theta})_{i,j,l,m} &<& 0 \hspace{0.1cm} \text{then}\\ \lambda_{i,j,l,m+1/2}&=&(1-0.5\nu) \lambda_{i,j,l,m}+ 0.5\nu\lambda_{i,j,l,m+1} \\ \lambda_{i,j,l,m-1/2}&=&(1-0.5\nu) \lambda_{i,j,l,m-1}+ 0.5\nu\lambda_{i,j,l,m} \end{array} $$

(42)

$$\begin{array}{@{}rcl@{}} \text{If} \hspace{0.1cm} (C_{x})_{i,j,l,m} > 0 \hspace{0.1cm} \text{then} \hspace{0.1cm} \lambda_{i+1/2,j,l,m}&=&\lambda_{i+1,j,l,m} \\ \lambda_{i-1/2,j,l,m}&=&\lambda_{i,j,l,m} \\ \text{If} \hspace{0.1cm} (C_{x})_{i,j,l,m} < 0 \hspace{0.1cm} \text{then} \hspace{0.1cm} \lambda_{i+1/2,j,l,m}&=&\lambda_{i,j,l,m} \\ \lambda_{i-1/2,j,l,m}&=&\lambda_{i-1,j,l,m} \\ \text{If} \hspace{0.1cm} (C_{y})_{i,j,l,m} > 0 \hspace{0.1cm} \text{then} \hspace{0.1cm} \lambda_{i,j+1/2,l,m}&=&\lambda_{i,j+1,l,m} \\ \lambda_{i,j-1/2,l,m}&=&\lambda_{i,j,l,m} \\ \text{If} \hspace{0.1cm} (C_{y})_{i,j,l,m} < 0 \hspace{0.1cm} \text{then}\hspace{0.1cm} \lambda_{i,j+1/2,l,m}&=&\lambda_{i,j,l,m} \\ \lambda_{i,j-1/2,l,m}&=&\lambda_{i,j-1,l,m} \\ \end{array} $$

(43)

where μ = 0 and ν = 0 are set for the downwind scheme.