Another phenomenon that could occur, if a current opposing the waves is
present, is that energy is transported offshore although the waves propagate
onshore. The limit for this situation to occur is given by:
cos α
U
=-
(27)
cos δ
Cgr
This condition corresponds to the solution of the dispersion relation for the
higher wave number (see Figure 2). Although it is not likely that such
conditions would occur in model applications for the nearshore, a check is
still included in NMLong-CW because the user is free to specify an arbitrary
external current.
Cross-Shore Momentum Equation
In NMLong-CW, the cross-shore momentum equation is employed to
determine changes in the mean water level (setup/setdown) induced by wave
and wind. The model includes the possibility of specifying an external
(large-scale) current and determining the effect of this current on the wave
transformation, as well as the interaction with currents calculated by
NMLong-CW (e.g., currents generated by waves and wind). In the wave
action equation (Equation 1), the inclusion of the current is straightforward
and independent of the mechanisms that are generating the current.
However, for the momentum equations, it less obvious as to how to account
for the external current, especially for the cross-shore momentum equation.
For example, one type of external flow that might be of interest to
incorporate in applying NMLong-CW is the ebb jet from a tidal inlet.
Effectively, this type of flow is generated by a momentum source (the inlet)
generating a jet that is discharged offshore under the influence of turbulent
mixing, inertia, and bottom friction. Measurements of the jet flow or simple
models employing jet theory might be accessed to estimate the velocity field,
which, in turn, could be the input for the external current to NMLong-CW.
However, in applying the cross-shore momentum equation to determine the
waves, questions arise as to how to treat the external current in such
calculations. In this context, it should be pointed out that NMLong-CW is
based upon alongshore uniformity, which could be in contradiction to the
complex and often highly 2-D flow field at an inlet. However, applied with
care, there are many situations at an inlet where applications of the model are
theoretically justified, and satisfactory results will be obtained. Because of
the alongshore uniformity assumption, considerations should always be made
with regard to the possible variation in quantities alongshore, including the
current. Such considerations involve the relationship between the spatial
scale of the current and wave motion.
In NMLong and NMLong-CW, the mean water level η is determined
using the following cross-shore momentum equation,
dη
dS
= - xx - CDρa W W cos ϕ
ρgd
(28)
dx
dx
20
Chapter 3 Wave Model
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