Adjoint free four-dimensional variational data assimilation for a storm surge model of the German North Sea

Abstract

In this paper, a data assimilation scheme based on the adjoint free Four-Dimensional Variational(4DVar) method is applied to an existing storm surge model of the German North Sea. To avoid the need of an adjoint model, an ensemble-like method to explicitly represent the linear tangent equation is adopted. Results of twin experiments have shown that the method is able to recover the contaminated low dimension model parameters to their true values. The data assimilation scheme was applied to a severe storm surge event which occurred in the North Sea in December 5, 2013. By adjusting wind drag coefficient, the predictive ability of the model increased significantly. Preliminary experiments have shown that an increase in the predictive ability is attained by narrowing the data assimilation time window.

Appendix

The wind profile follows approximately a logarithm curve with height in the atmospheric boundary layer which can be expressed as:

$$ U(z)=\frac{u_{*}}{\kappa}\ln(\frac{z}{z_{0}}) $$

(A.1)

where U(z) is the wind speed at the height of z; z ₀ is the aerodynamic surface roughness; κ = 0.4 is the von Karman constant; u _∗ is the wind friction velocity defined by:

$$ u_{*}=\sqrt{\frac{\tau}{\rho_{a}}} $$

(A.2)

Combining Eqs. 1 and A.2, a relation between u _∗ and C _d can be obtained as follows:

$$ u_{*}^{2}=C_{d}U_{10}^{2} $$

(A.3)

Combining (A.1) and Eq. A.3 yields the relation between z ₀ and C _d ,

$$ z_{0}=z_{10}\exp(-\kappa / C_{d}^{1/2}) $$

(A.4)

where z ₁₀=10m. Charnock (1955) proposed a relation between z ₀ and u _∗ , i.e., $gz_{0}/u_{*}^{2}=\alpha$, where g is the gravitational acceleration and α is the Charnock coefficient. Charnock (1955) took it as a constant. Combining all the above equations, a relation between C _d and U ₁₀ is obtained as below,

$$ \alpha^{1/2}U_{10}/(gz_{10})^{1/2}=C_{d}^{-1/2}\exp(-\frac{\kappa}{2}C_{d}^{-1/2}) $$

(A.5)

Equation A.5 is almost a linear function for C _d values in the range of 1.0 × 10⁻³ to 4.0 × 10⁻³,

$$ C_{d}=(a+bU_{10}) $$

(A.6)

where b = 0.475α ^1/2. Several values of α have been proposed; α = 0.012 (Charnock 1955); α = 0.0144 (Garratt 1977) ; α = 0.035 (Kitaigorodskii and Volkov 1965) . Stewart (1974) suggested a dependency of α with the wave state. Hsu (1974) related α to the wave steepness and Donelan et al. (1993) to the wave age.