Modeling Concepts - Coastal Lagoons: Ecosystem Processes and Modeling for Sustainable Use and Development - page 266

In lake or water reservoir problems very often the retention or residence time,
which is also estimated 4b by Equation (6.32) as the ratio of the water volume to the
inflow discharge, is used. This time is usually interpreted as the “lifetime” the water
from the inflow water or an incoming particle resides in the lake or reservoir. 4b
It is also possible to use Equations (6.33)-(6.35) to estimate the flushing time
from periodic and/or stochastic water exchanges with the adjacent sea resulting from
tidal or meteorological forcing at the entrance. In this case, these exchanges and the
corresponding flushing times take the form:

V
q

h
T

avg

flush

q =∑ q i , q =⋅ ⋅ 2 S

i

, τ i = (6.36)

flush i avg

i i

where is the contribution of the i th harmonic oscillation to the ventilation stream
(see Section 6.3.2.2.); and are the average surface area and water depth of
the lagoon, respectively; and and are the amplitude and period, respectively, of
the i th harmonic of the water-level oscillations. The resulting flushing time becomes

q i

S avg H avg

h i T i

H

1
Σ

avg

τ flush = ⋅ (6.37)

h
T

2

i

i

Harmonic analysis of the water-level variations caused by irregular winds often
shows that it is the higher frequency terms that contribute significantly to the flushing
marine water flux in Equation (6.36). These terms prevail due to their short period,
despite their small amplitude in comparison with the significant harmonic terms,
which constitute the main deviation of sea level from its statistical mean value. These
short period and small amplitude level variations contribute to the main part of the
marine water pumping into the lagoon area. This is why the accuracy of statistical
determinations of the amplitudes and frequencies for such short period harmonics
should be carefully established.
However, estimating the marine water flux in the lagoon from Equation (6.36)
is not straightforward. We cannot just include in this equation all harmonic terms
obtained from the harmonic analysis. This may lead to a nonconvergent series of
(1/ n ) type because of the n- fold frequencies in the Fourier analysis and because of
the possibility that the h i value will not decrease monotonously with n . In other
words, including more short period terms may not guarantee the convergence of the
flushing time value.
As an example, let us consider the statistical analysis of a 3-year time series of
water-level variations at the Vistula Lagoon entrance, sampled at a 12-h interval. A
formal Fourier analysis of this time series yielded a Fourier series with 1092 har-
monic terms. As the Nyquist frequency is 1/2 dt , which corresponds to a period of
1 day, terms with periods less than 3 Nyquist periods were excluded from the series
to obtain more reliable results. This reduced the number of harmonic terms in the
Fourier series from 1092 to 367. Furthermore, harmonic terms with amplitudes less