# Closure depth

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 Definition of Closure depth: This very short entry boils down to a simple definition, supported by the well known Hallermeier-equation, see Hallermeier (1981) [1], characterizing the depth of closure by significant waves occurring 12 hours in a given year. Since a clear potential exists to expand this definition into a more detailed contribution, such an attempt is presented below. This is the common definition for Closure depth, other definitions can be discussed in the article

Hallermeier (1981, 1983) [1] [2] defined three profile zones, i.e. the littoral zone, shoal or buffer zone and offshore zone. This partition defined two closure depths, namely:

• an “inner” (closer to shore) closure depth hin at the seaward limit of the littoral zone, and
• an “outer” or “lower” (further from shore) closure depth hout at the seaward limit of the shoal/buffer zone.

From Hallermeier (1981): [1]

(1)

$h_{in} = 2.28 H_{s} - 68.5 (\frac{H_{s}^2}{gT^2})$

where Hs is the effective wave height just seaward of the breaker zone that is exceeded for 12 hours per year, i.e. the significant wave height with a probability of yearly exceedance of 0.137%, T is the wave period associated with Hs, and g is the acceleration of gravity.

From Hallermeier (1983): [2]

(2)

$h_{out} = 0.018 H_{m} T_{m} \sqrt \frac{g}{D_{50} (s-1)}$

where Hm and Tm are the median wave height and period respectively, D50 is the median sediment diameter and s is the ratio of specific gravity of sand to that of fluid (about 2.65). As we can see hin is solely related to the wave parameters, irrespective of typical sediment diameters, ranging between 0.16 and 0.42 mm, as validated and described by Hallermeier (1978, 1981) [3] [1], whereas hout requires both hydrodynamic and sedimentological parameters. They are relevant for MLW (mean low water) conditions.

The concepts of hin and hout are illustrated in Fig. 1, Krauss et al. (1998) [4]. It shows the result of surveys on a profile survey line at the foot of 56th Street, Ocean City, Maryland. The envelope of recorded elevations (above and below the mean) in any profile survey over the 4-year interval of available data and the standard deviation of depths are plotted as functions of the distance offshore. The envelopes tend to converge in the depth range of 18 to 26 ft (5.5 to 8.0 m) for this particular profile, seaward of which the profile elevations separate over the crests of the shoals. The depth in Fig. 1 is referenced to the National Geodetic Vertical Datum (NGVD), Krauss et al. (1998) [4].

Formation of the offshore shoals can be attributed to earlier coastal and geologic processes and not to offshore movement of sediment from the present beach. The convergence of the profile envelope landward of the offshore shoals indicates that movement of sediment on the shoals is not directly related to sediment exchange on the nearshore profile. Therefore, hin for the profile data shown in Fig. 1, corresponding to 4-year time interval, is approximately 18 ft or 5.5 m.

 Fig. 1 Mean, envelope and standard deviation of profile survey elevations 56th Street, Ocean City, Maryland, from Krauss et al. (1998) [4]

Krauss et al. (1998) [4] supplied the following, detailed definition of depth of closure associated with the littoral processes (hin): the depth of closure (DoC) for a given or characteristic time interval is the most landward depth seaward of which there is no significant change in bottom elevation and no significant net sediment transport between the nearshore and the offshore.
This definition is general in that it applies to the open coast where nearshore waves and wave induced currents are the dominant sediment-transporting mechanisms, as well as to non-wave dominated locations, such as the beach adjacent to a long jetty, at which sediment may be jetted offshore by a-large rip current or an ebb tidal shoal where the tidal current is a major contributor to the sediment-transporting processes. The theoretical background of DoC is based on its relation to the critical Froude number, describing the threshold of erosive seabed agitation by wave action, Hallermeier (1978, 1981) [3] [1]:

(3)

$FR = \frac {U_{b}^2}{(s-1)gh} = 0.03$

where Ub is the maximum horizontal wave-induced fluid velocity at water depth h. Using linear wave theory and the value of s-1 = 1.6 for quartz sand in seawater, Hallermeier expressed this equation in the form of equation (1). The first term in that equation is directly proportional to wave height and is the main contributor to DoC. The second term provides a small correction associated with the wave steepness. Equation (1) can be further generalized to incorporate other time scales by introducing into it a significant wave height exceeded 12 hours in a particular time interval.

Birkemeier (1985) [5] modified the Hallermeier equation (1) to:

(4)

$h_{in} = 1.75 H_{s} - 57.9 (\frac{H_{s}^2}{gT^2})$

It produces a smaller estimate of the depth of closure than the Hallermeier equation (1) for given wave conditions. Houston (1995) [6] simplified Birkemeier’s formulation using properties of a Pierson-Moskowitz wave spectrum (Rijkswaterstaat 1986) [7] and a modified exponential distribution of significant wave height over time to express the DoC in terms of mean annual significant wave height $\overline {H_{s}}$ according to $h_{in} = 6.75 \overline {H_{s}}$. Following Houston’s approach, the Hallermeier equation can similarly be expressed in the form:

(5)

$h_{in} = 8.9 \overline {H_{s}}$

Equation 5 has the advantage to incorporate only a single parameter to estimate DoC without the need to determine the wave height and period exceeded 12 hours in a particular time interval. However, it is not applicable for estimating DoC for a particular storm event as discussed below.

In the original formulation of Hallermeier (1981) [1] and subsequent modifications, the DoC was defined based on the largest wave height exceeded 12 hours per year. This definition incorporates a time element, but the exact event associated with the value of wave height is ambiguous. Depending on changes in storm activity and wave conditions from year to year, the predicted DoC can vary substantially. In order to determine a representative value of the DoC based on this definition, wave conditions averaged over a period of several years must be employed.

A useful extension of calculating the DoC based on average annual wave conditions is to relate the DoC to a particular time period of interest over which specific storm events or seasonal wave conditions occur. A similar “wave-by-wave” interpretation of the DoC was introduced by Kraus and Harikai (1983) [8]. In a wave-by-wave or event approach, the DoC can be associated with a recurrence frequency or return period for a particular storm. The return period should be associated with the wave height, not with the storm surge.

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## References

1. Hallermeier, R. J. (1981). A Profile Zonation for Seasonal Sand Beaches from Wave Climate. Coastal Engineering, Vol. 4, 253-277.
2. Hallermeier, R. J. (1983). Sand Transport Limits in Coastal Structure Design, Proceedings, Coastal Structures ’83, American Society of Civil Engineers, pp. 703-716.
3. Hallermeier, R. J. (1978). Uses for a calculated limit depth to beach erosion. Proceedings, 16th Coastal Engineering Conference, American Society of Civil Engineers, 1493- 15 12.
4. Kraus, N. C., Larson, M. and Wise, R. A. (1998) Depth of Closure in Beach-fill Design, Coastal Engineering Technical Note CETN II-40, 3/98, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS.
5. Birkemeier, W. A. (1985). Field data on seaward limit of profile change. Journal of Waterway, Port, Coastal and Ocean Engineering 111(3), 598-602.
6. Houston, J. R. (1995). Beach-fill volume required to produce specified dry beach width, Coastal Engineering Technical Note 11-32, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS.
7. Rijkswaterstaat (1986). Manual on artificial beach nourishment, Delft Hydraulics Laboratory, The Netherlands.
8. Kraus, N. C., and Harikai, S. (1983). Numerical model of the shoreline change at Oarai Beach, CoastaI Engineering 7(l), l-28.