New ordinary differential equations (ODEs) for the evolution of spectral components are derived from the complex Ginzburg–Landau equation (CGLe) for one-dimensional spatial domains without boundaries (free evolution) and with one fixed boundary (semi-free evolution). For such evolution, a complex or imaginary diffusion term creates a tendency for waves to lengthen. This requires a novel ansatz and auxiliary condition that treat wavenumbers as time-varying. The ansatz consists of a discrete spatial Fourier transform modified with a time-dependent wavenumber for the peak spectral component. The wavenumbers of the other components are fixed relative to this wavenumber. The new auxiliary condition is the terminal condition for complex diffusion (after wavenumbers evolve to zero, they remain at zero). The derived free and semi-free ODEs are solved along characteristic lines located symmetrically about a fixed spatial point. Waves lengthen with time away from this point in both directions. Laboratory experiments on the formation of channel sandbars, theoretically described by the CGLe, show two regions whose evolutionary behaviour is qualitatively predicted by the free and semi-free evolution equations. This analysis applies to other time-dependent partial differential equations with complex or imaginary diffusion terms. New freely evolving solutions are derived for the complex heat equation and Schrödinger equation (linear and nonlinear).

Most numerical models are run and compared to data in a subjective
manner. This paper demonstrates how model performance statistics can be
used to calibrate and/or validate hydrodynamic models in a more
objective way. Statistics were also used to compare model runs that used
different amounts of field data in order to inform the debate about the
optimum mix of modelling and measurement.
The hydrodynamics around the mouth of the Teign estuary (UK) have been
simulated using two coastal area numerical modelling systems. Model
performance statistics were calculated to assess the accuracy of the
predictions of the measured currents at a number of locations around the
estuary mouth. The relative mean absolute error was used as it is
applicable to vectors as well as scalars and measures all types of
errors. An adjusted relative mean absolute error was also used to reduce
the effect of measurement error. A classification table was adopted that
categorises the results according to the size of the error. In addition,
time series and scatter plots were used to judge the performance of the
modelling systems.
Calm conditions during a spring tide were simulated, as was a relatively
large storm. The two modelling systems gave more or less equal
performances when run in engineering mode (where default values were
used for most of the system settings). In each case, one modelling
system performed better than the other at some locations and worse than
it at other locations. One model was also run using a scientific
approach, where different amounts of information were used to alter the
model settings and sensitivity tests were performed. The model
performance statistics showed that using more data does not necessarily
lead to better model predictions. New methods for incorporating data
into the operation of a model need to be evaluated thoroughly before
they can be used without site-specific calibration.

Since its introduction in 1978, the Battjes and Janssen model has proven
to be a popular framework for estimating the cross-shore
root-mean-square wave height H-rms transformation of random breaking
waves in shallow water. Previous model tests have shown that wave
heights in the bar trough of single bar systems and in the inner troughs
of multiple bar systems are overpredicted by up to 60\% when standard
settings for the free model parameter gamma (a wave height-to-depth
ratio) are used. In this paper, a new functional form for gamma is
derived empirically by an inverse modelling of gamma from a
high-resolution (in the cross-shore) 300-h H-rms data set collected at
Duck, NC, USA. We find that, in contrast to the standard setting, gamma
is not cross-shore constant, but depends systematically on the product
of the local wavenumber k and water depth It. Model verification with
other data at Duck, and data collected at Egmond and Terschelling
(Netherlands), spanning a total of about 1600 h, shows that cross-shore
H-rms profiles modelled with the locally varying gamma are indeed in
better agreement with measurements than model predictions using the
cross-shore constant gamma. In particular, model accuracy in inner bar
troughs increases by up to 80\%. Additional verifications with data
collected on planar laboratory beaches show the new functional form of
gamma to be applicable to non-barred beaches as well. Our optimum gamma
cannot. be compared directly to field and laboratory measurements of
height-to-depth ratios and we do not know of a physical mechanism why
gamma should depend positively on kh.