7.1.1.1. Case 1: The HOS-Ocean formulation in relation to SWD¶

The formulation applied in HOS-Ocean, for long-crested waves propagating over a constant water depth \(d\), is explained in [DBTouzeF16]. From equations (6) and (8) in that publication, the velocity potential \(\phi\) and surface elevation \(\zeta\) are defined as

\[\phi(\mathbf{x},t)= \sum_{j=0}^n \mathcal{Re} \Bigl\{A_j(t)\, \frac{\cosh k_j(z+d)}{\cosh k_j d} e^{i k_j x} \Bigr\}, \qquad k_j = j \Delta k_x\]

\[\zeta(x, y, t)= \sum_{j=0}^n \mathcal{Re} \Bigl\{B_j^{\zeta}(t)\, e^{i k_j x} \Bigr\}\]

We have adjusted the symbols for wave elevation, water depth and loop index to simplify the comparison with the SWD documentation. When HOS-Ocean has calculated the temporal spectral amplitudes \(A_j(t)\) and \(B_j^{\zeta}(t)\) the kinematics can be calculated everywhere using the above equations.

We compare the formulation above to shape 2 as defined in the theory section of our documentation:

\[\phi(\mathbf{x}, t)= \sum_{j=0}^n \mathcal{Re} \Bigl\{c_j(t)\, e^{-i k_j x} \Bigr\} \frac{\cosh k_j(z+d)}{\cosh k_j d}\]

\[\zeta(x, y, t)= \sum_{j=0}^n \mathcal{Re} \Bigl\{h_j(t)\, e^{-i k_j x} \Bigr\}\]

It is noted that the sign in the exponential terms are different in theses formulations. We can apply the SWD formulation if and only if the kinematics in the physical space is identical to the HOS-Ocean formulation. To satisfy this requirement it follows from pure algebra that:

\[c_j(t) = \bar{A}_j(t), \qquad h_j(t) = \bar{B}_j^{\zeta}(t)\]

\(\bar{A}_j(t)\) and \(\bar{B}_j^{\zeta}(t)\) denote the complex conjugate of \(A_j(t)\) and \(B_j^{\zeta}(t)\) respectively.

Hence, when HOS-Ocean has established \(A_j(t)\) and \(B_j^{\zeta}(t)\) at the end of each time step, the program may in an SWD output routine calculate \(c_j(t)\) and \(h_j(t)\) from above equations and write these SWD specific amplitudes to the SWD-file.

In addition SWD requires output of \(\frac{dc_j(t)}{dt}\) and \(\frac{dh_j(t)}{dt}\). It follows from differentiation that

\[\frac{dc_j(t)}{dt} = \frac{d\bar{A}_j(t)}{dt}, \qquad \frac{dh_j(t)}{dt} = \frac{d\bar{B}_j^{\zeta}(t)}{dt}\]

It may be tempting to evaluate the right-hand side of these equations using some kind of finite-difference schemes. However, as pointed out in the SWD theory section, higher accuracy is obtained if these expressions are evaluated from the dynamic and kinematic free surface conditions. This is because these expressions can be evaluated using analytical spatial gradients as a replacement for the temporal gradient. In practice these terms can often be extracted from the right-hand side of the ODE equation system.

Noting that the physical boundary conditions explicitly define the temporal slopes of these amplitudes, this is also important information regarding temporal interpolation to be applied in the application program. This is accounted for in the SWD API assuming sound input of \(\frac{dc_j(t)}{dt}\) and \(\frac{dh_j(t)}{dt}\).

In practice, a copy of the subroutine swd_write_shape_1_or_2.f90 may be adapted for convenient SWD output. This code takes care of the low level C-stream output.

Note

It was assumed that the wave train propagates in the positive x-direction in HOS-Ocean, the same convention as assumed in SWD. If the wave moves in the negative direction in HOS-Ocean, we must flip the sign of the x-axis to comply with the SWD convention. Hence, we then must apply \(c_j(t)=A_j(t)\) and \(h_j(t)=B_j^{\zeta}(t)\) as a result of conjugating twice.

DBTouzeF16: Guillaume Ducrozet, Félicien Bonnefoy, David [Le Touzé], and Pierre Ferrant. Hos-ocean: open-source solver for nonlinear waves in open ocean based on high-order spectral method. Computer Physics Communications, 203:245 – 254, 2016. URL: http://www.sciencedirect.com/science/article/pii/S0010465516300327, doi:https://doi.org/10.1016/j.cpc.2016.02.017.