Closure depth

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The concept of closure depth refers to sandy shores. It is recommended to read this article in conjunction with the article Shoreface profile.


Definition of Closure depth (DoC):
Closure depth is conceived as the offshore limit beyond which cross-shore sediment transport or its net contribution to the coastal sediment balance is negligible for the relevant wave climate and time scale.
This is the common definition for Closure depth (DoC), other definitions can be discussed in the article


Two operational proxies are often used in the scientific literature:

  1. Hallermeier distinguished inner and outer closure depths, which are estimated from wave-induced sediment-mobility criteria (Hallermeier 1981[1], 1983[2]). Hallermeier’s limits refer to wave-induced seabed activity. They do not include sediment transport generated independently by tidal, wind-driven, density-driven or ocean currents.
  2. Kraus et al. (1998[3]) estimated closure depth as the cross-shore location beyond which no significant bed-level change is detected in repeated profile surveys over the considered time interval.

Hallermeier interpreted the inner limit as the approximate boundary between the highly active littoral zone (the upper shoreface) and a less active shoal zone (the lower shoreface), and the outer limit as the approximate boundary of the shoal zone, beyond which the annual median wave condition cannot initiate sediment motion.

Kraus et al. assumed that the depth beyond which no detectable bed-level change occurs represents the boundary across which no significant net sediment transport occurs.

Both approaches are therefore indirect proxies for the conceptual closure depth, but neither is based on direct measurement of the net cross-shore sediment flux. The absence of detectable bed-level change does not necessarily imply negligible net sediment transport. Small systematic bed-level changes distributed over the broad lower shoreface can represent substantial sediment volumes, while sediment transport can also occur without local bed-level change if the transport has little spatial divergence[4].


Relevance of the closure depth concept

The closure depth is closely related to the concept of 'active coastal zone', which is based on the assumption that wave-induced sedimentary exchange processes between the inner continental shelf and the coast are mainly confined to a coastal strip - the shoreface. For wave-dominated coasts, the outer closure depth is often taken as the seaward boundary of the active coastal zone considered in long-term coastal sediment budgets. The inner closure depth is more relevant to the upper shoreface and shorter-term profile adjustment. Closure depth determines the assumed width and volume of the profile participating in coastal adjustment, see Shoreface profile. An error in closure depth therefore propagates directly into estimates of nourishment volume and shoreline displacement. Closure depth is a crucial parameter in models for estimating shoreline retreat due sea level rise (Bruun rule) or shoreline adaptation to changes in the nearshore wave climate resulting from natural causes or human interventions (one-line models). No unique, generally valid formula for predicting the closure depth has been established so far.

Fig. 1. Schematic illustration of the closure-depth concepts of Hallermeier and Kraus et al. The active coastal zone is represented here by the upper shoreface, where cross-shore sediment exchange occurs frequently over timescales of years to decades. The inner closure depth is located where the mean beach profile converges with the storm profile associated with wave conditions exceeded for approximately 12 hours per year.

Sand transport in the coastal zone is mainly driven by waves. However, this only holds within a zone of limited cross-shore width. In deep water, waves and seabed hardly interact, because wave orbital motion decreases exponentially with depth (see Shallow-water wave theory). The strip along the coast where sand transport is mainly driven by waves is called the active coastal zone. The closure depth is commonly used as an estimate of the seaward boundary of the active coastal zone for the timescale and processes considered. Wave-induced sand transport has both a longshore and a cross-shore component. The longshore component plays a major role in structural long-term shore accretion (transport convergence) or shore erosion (transport divergence). The cross-shore component is mainly responsible for large seasonal sand transfers up and down the beach, especially by alternating mild weather and storm weather conditions (Fig. 1).

The active coastal zone is a zone of intense morphodynamics: the mutual interaction between wave dynamics and seabed/beach morphology. The influence of nearshore interventions (e.g. jetties, groynes, breakwaters, dredging, dumping) that influence the wave climate will trigger a morphodynamic feedback that will extend across the shoreface, approximately to the closure depth corresponding to the timescale considered. This also holds for the impact of sea level rise. Morphodynamic feedbacks are at the core of the concept of equilibrium shoreface profile. At the annual-decadal time scale (scale of alternating storm and mild weather conditions) morphodynamic activity involves mainly the upper shoreface, the zone of wave transformation, refraction and breaking. At the decadal-centennial time scale sediment exchange between the upper and lower shoreface cannot be ignored. The concept of active coastal zone therefore has a temporal component; the closure depth at shorter timescales is smaller than the closure depth at longer time scales. The shorter timescale and smaller closure depth are most relevant for estimating the impact of storms (see Dune erosion); the longer timescale and greater closure depth are most relevant for estimating the impact of sea level rise (see the article Bruun rule).

Process-based models calculate sediment transport throughout the model domain and therefore need not prescribe closure depth as an internal transport boundary. However, their results depend on the offshore extent of the model, boundary conditions and the representation of slow lower-shoreface processes. For regional and long-term studies, simpler models using an assumed closure depth remain widely used because they are computationally more practical.

Closure depth based on wave-driven sand transport

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

  • an “inner” (closer to shore) closure depth [math]h_{in}[/math] corresponding to the seaward boundary of the littoral zone, and
  • an “outer” or “lower” (further from shore) closure depth [math]h_{out}[/math] corresponding to the seaward boundary of the shoal zone.

Hallermeier associated the littoral and shoal zones approximately with what are now commonly called the upper and lower shoreface.

Hallermeier (1981[1]) derived for the inner closure depth the formula

[math]h_{in} = 2.28 H_{12h/y}- 68.5 (\Large\frac{ H_{12h/y}^2}{g T_{12h/y}^2}\normalsize), \qquad (1)[/math]

where [math]g[/math] is the acceleration of gravity, [math]H_{12h/y}[/math] is the significant wave height exceeded for a total of approximately 12 hours in an average year, corresponding to about 0.137% of the time, [math] T_{12h/y}[/math] is the wave period associated with these conditions. The wave data should represent the non-breaking wave conditions reaching the shoreface.

The first term in the formula (1) is directly proportional to wave height and is the main contributor to the inner closure depth (Depth of Closure 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. The theoretical background of the inner DoC [math]h_{in}[/math] is based on its relation to the sediment mobility number. Grain size does not appear explicitly in Hallermeier’s inner-limit formula. This empirical result was obtained for fine- to medium-sand conditions (typical sediment diameters, ranging between 0.16 and 0.42 mm) and should not be interpreted as general proof that closure depth is independent of sediment properties[5][1][6].

The outer closure depth [math]h_{out}[/math] is the estimated maximum depth beyond which the annual median wave condition cannot initiate motion of the specified bed sediment. For the outer closure depth Hallermeier (1983[2]) derived the formula

[math]h_{out} = 0.018 \,H_s T_s \, \sqrt{\Large\frac{g}{d_{50} (s-1)}\normalsize}, \qquad (2)[/math]

where [math]H_s[/math] and [math]T_s[/math] are the significant wave height and period respectively, [math]d_{50}[/math] is the median sediment diameter and [math]s[/math] is the ratio of specific gravity of sand to that of fluid (about 2.65). The outer closure depth [math]h_{out}[/math] depends on both hydrodynamic and sedimentological parameters, in contrast with [math]h_{in}[/math]. Both closure depths refer to MLW (mean low water) conditions. Hallermeier considered the estimates (1) and (2) appropriate for exposed microtidal coastal environments with high wave energy and low tidal ranges[5]. Since the outer closure depth Eq. (2) is based on another criterion than the inner closure depth Eq. (1), it may happen that in some cases the inner closure depth is deeper than the outer closure depth[7]. Such a result indicates that the two empirical estimates do not form a physically consistent zonation for the local wave and sediment conditions; the formulas should not then be interpreted literally as ordered profile boundaries.

Valiente et al. (2019[4]) define the ‘wave base’ as the outer limit of the lower shoreface and estimate its position from the maximum depth of significant bed activity and sediment transport, termed the Depth of Transport (DoT).

Houston (1995[8]) simplified Hallermeier’s formulation using properties of a Pierson-Moskowitz wave spectrum (see Statistical description of wave parameters) and a modified exponential distribution of significant wave height over time to express the DoC in terms of mean annual significant wave height. Following Houston’s approach, the Hallermeier equation can be expressed in the form:

[math]h_{in} = 6.75 \overline {H_{s}} . \qquad (3) [/math]

Equation (3) has the advantage to incorporate only a single parameter to estimate the closure depth (DoC) without the need to determine the wave height and period exceeded 12 hours in a particular time interval. Houston’s expression is convenient where only annual mean significant wave height is available, but it relies on assumed wave-height statistics. Where a sufficiently long local wave record exists, the 12-hour exceedance condition should be determined directly.

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. Udo et al. (2020[9]) tested equations (1) and (2) using observed values for [math]H_{12h/y}[/math] and [math]T_{12h/y}[/math] at different field sites around Japan where observed profile data were available for at least 5 years. Inner closure depth (DoC) was determined by analysing the envelope of the profile data in a way similar to Kraus et al.[3]. They concluded that the coefficients in equation (1), although providing acceptable estimates, appear to be location dependent, overestimating DoC along the Pacific Ocean side and underestimating DoC along the Sea of Japan side. The empirical coefficients are not universal and should be validated against regional profile observations where available. A useful extension of calculating the closure depth (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[10]). 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. This event-related limit should not automatically be equated with the long-term closure depth, because cumulative sediment response also depends on storm duration, wave period, water level and event sequence.

Closure depth based on repeated shoreface profile measurements

The Hallermeier formulas estimate depths associated with specified wave-induced seabed-activity or mobility criteria. Direct measurements of net cross-shore sediment flux at these depths are rarely available, so field comparisons generally use indirect morphological or sedimentological indicators. The closure depth in the field is therefore often related to morphological or sedimentary features:

  1. a zone of minimum sand accretion or erosion (according to the definition of closure depth)
  2. a discontinuity of the shoreface slope or a discontinuity in the shoreface sediment composition.
Fig. 2. Mean profile, envelope and standard deviation of repeated profile elevations at Ocean City, Maryland. The depth is referenced to the National Geodetic Vertical Datum (NGVD).The profile envelope converges near 5.5 m depth, providing a profile-based closure-depth estimate for the four-year survey period. Renewed variability farther offshore occurs over inherited shoals and does not necessarily represent sediment exchange with the present beach. Adapted from Kraus et al. (1998[3]).

Kraus et al. (1998[3] proposed that the inner closure depth [math]h_{in}[/math] should correspond to the most landward location where the shoreface profiles from repeated measurements converge. The outer closure depth [math]h_{out}[/math] should correspond to the location where no significant bed level change is detected from repeated profile measurements. This is illustrated in Fig. 2 showing the result of repeated surveys of a shoreface profile on the US Atlantic coast. 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 envelope tends to converge in the depth range of 5 to 6 m for this particular profile, seaward of which the profile elevations separate over the crests of the shoals. 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 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. For the profile data shown in Fig. 2, corresponding to a 4-year time interval, [math]h_{in}[/math] is approximately 5.5 m. The outer depth [math]h_{out}[/math] is situated around the 10 m depth contour, where no significant bed level change is observed during the survey period. The Hallermeier formula (1) predicts an inner closure depth [math]h_{in} \approx[/math] 6 m, in reasonable agreement with the profile data. Formula (2) gives for the outer closure depth at Ocean City ([math]H_s[/math]=1 m, [math]T_s[/math] = 6 s, [math]d_{50}[/math]=0.3 mm)[11] a value of about 11 m, also in reasonable agreement.

Barrineau et al. (2021[12]) performed a similar study but for a much larger number of transects at different sites along the US East coast. They conclude that the method of Kraus et al. yields estimates for the inner closure depth [math]h_{in}[/math] comparable to the results of the Hallermeier formula (1) for wave dominated coastal sections. However, for coastal sections where the energy of tides and waves is of the same order (mixed-energy coasts), the estimates for the closure depth according to the method of Kraus et al. appear to be substantially smaller than the results of the Hallermeier formula (1). The authors also note the influence of coastal inlets on the DoC estimates. Near inlets and on mixed wave–tide coasts, profile change can be strongly influenced by tidal channels, deltas and alongshore redistribution. The Hallermeier wave-only formulas should not be expected to reproduce these patterns.

Birkemeier (1985[13]) compared Eq. (1) with observations over a 16-months period of coastal profiles on the Pacific Ocean and the Gulf of Mexico. He concluded that Eq. (1) overpredicts the closure depth derived from these data and proposed a modified version based on the profile-convergence proxy:

[math]h_{in} = 1.75 H_{12h/y}- 57.9 (\Large\frac{ H_{12h/y}^2}{g T_{12h/y}^2}\normalsize). \qquad (4)[/math]

It produces a smaller estimate of the depth of closure than the Hallermeier equation (1) for given wave conditions.

Other methods for closure depth estimation

Aragones et al. (2018[14]) derived depth of closure estimates by analysing trend changes in the grain size distribution of several coastal profiles of the Mediterranean coast near Valencia. They argued that the DoC is situated at a certain depth where the tendency of seaward decreasing grain size [math]d_{50}[/math] changes into an increasing trend before decreasing further offshore again. At the studied Mediterranean sites, a change in the cross-shore grain-size trend coincided with the profile-convergence depth. Whether this indicator applies elsewhere depends on local sediment sources, sorting processes and inherited stratigraphy. However, the depth was substantially smaller than the closure depth predicted by Hallermeier's formula (1) and also smaller than predicted by the formula (4) of Birkemeier.

McFall et al. (2021[7]) compiled data on the stability of dredged sediment deposits placed on the shoreface for 20 sites along with the Atlantic and Pacific US coasts. Nearshore placements landward of the inner limit Eq. (1) were generally mobile, whereas placements near or beyond the outer limit Eq. (2) were substantially more stable. The closure-depth formulas can therefore assist preliminary siting, although local currents and morphology must also be considered.

Limitation of the closure depth concept

The wave-induced seabed-activity criteria underlying the Hallermeier formulas (1) and (2) do not necessarily correspond to the criterion of (almost) zero depth profile change. Zero depth profile change may arise, for example, in the situation where depth change due to sediment transport across the DoC contour derived from profile data is compensated by accretion or erosion due to gradients in longshore sediment transport[15]. Another issue is that depth change of the lower shoreface which is below the detection limit can amount to a considerable sediment flux when integrated over the whole lower shoreface [4]. A mean change of only 1 cm over a lower shoreface 2 km wide represents 20 m3 of sediment per meter of shoreline. Insight into the local coastal dynamics is therefore required when applying the closure depth concept.

The Hallermeier formulas are based on the assumption that sediment transport on the shoreface is dominated by wave action and that the contribution of other drivers of sediment transport can be ignored. This may be a reasonable assumption for many shores, especially exposed microtidal coasts. However, it cannot be expected that the Hallermeier formulas are applicable in situations where this is not the case. Examples are: (1) macrotidal coasts, where combined wave- and tide-driven sediment transport across the DoC contour can be much larger than sediment transport driven only by waves[4]; (2) coasts where important sediment transport across the shoreface result from up- and downwelling currents induced by wind stress and density gradients[15]. In some situations the determination of wave characteristics on the lower shoreface may not be straightforward. Valiente et al. (2019[4]) observed that on the macrotidal, embayed and high-energy coastline of SW England, offshore wave conditions do not well represent wave conditions at the toe of the lower shoreface due to the presence of rocky headlands. Wave conditions representative of the local shoreface should therefore be used, not untransformed offshore buoy data where refraction, sheltering or depth limitation is important.

Hartman and Kennedy (2016[16]) argue that considering only the annual 12h extreme conditions for the inner DoC ignores the potential impact of more frequent less severe storm conditions. This can be important in tropical regions where cyclones generate extreme waves, but with such short durations that they have little opportunity to move large amounts of sediment.

Conclusion

Closure-depth estimates depend on the criterion and method used. A depth inferred from detectable profile change is not necessarily equivalent to one calculated from sediment-mobility or wave-statistical criteria. Closure depth can also vary alongshore and change over time as wave conditions, tidal currents, sediment characteristics or shoreface morphology change. It should therefore be treated as an uncertain model parameter rather than as a precisely known and fixed contour[17]. Reported values should specify the estimation method, reference water level, observation period and detection threshold. Where closure depth strongly influences a shoreline prediction, calculations should preferably be repeated for a plausible range of values.


Related articles

Shoreface profile
Active coastal zone
Bruun rule
Littoral drift and shoreline modelling


References

  1. ↑ 1.0 1.1 1.2 1.3 1.4 Hallermeier, R.J. 1981. A profile zonation for seasonal sand beaches from wave climate. Coastal Engineering 4: 253–277 Cite error: Invalid <ref> tag; name "H81" defined multiple times with different content
  2. ↑ 2.0 2.1 2.2 Hallermeier, R.J. 1983. Sand transport limits in coastal structure design. Proceedings Coastal Structures ’83, American Society of Civil Engineers, pp. 703–716 Cite error: Invalid <ref> tag; name "H83" defined multiple times with different content
  3. ↑ 3.0 3.1 3.2 3.3 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.
  4. ↑ 4.0 4.1 4.2 4.3 4.4 Valiente, N.G., Masselink, G., Scott, T., Conley, D. and McCarroll, R.J. (2019) Role of waves and tides on depth of closure and potential for headland bypassing. Mar. Geol. 407, 60–75
  5. ↑ 5.0 5.1 Hallermeier, R. J. 1978. Uses for a calculated limit depth to beach erosion. Proceedings, 16th Coastal Engineering Conference, American Society of Civil Engineers, pp. 1493 - 1512
  6. ↑ Royer, E., Wang, P. and Cheng, J. 2023. Determining depth of closure based on time-series beach profiles and empirical formulas: A case study along the Florida coast. Shore & Beach, 91(1): 3-22
  7. ↑ 7.0 7.1 McFall, B.C., Brutsche, K.E., Priestas, A.M. and Krafft, D.R. 2021. Evaluation techniques for the beneficial use of dredged sediment placed in the nearshore. J. Waterw. Port Coast. Ocean Eng. 147, 04021016
  8. ↑ 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.
  9. ↑ Udo, K., Ranasinghe, R. and Takeda, Y. 2020. An assessment of measured and computed depth of closure around Japan. Sci. Rep. 10, 2987
  10. ↑ Kraus, N. C., and Harikai, S. 1983. Numerical model of the shoreline change at Oarai Beach CoastaI Engineering 7: 1-28.
  11. ↑ Stauble, D.K., Garcia, A.W., Kraus, N.C., Grosskop, W.G. and Bass, G.P. 1993. Beach Nourishment Project Response and Design Evaluation: Ocean City, Maryland Report 1 1988-1992. USACE
  12. ↑ Barrineau, P., Janmaat, R. and Kana, T. 2021. Empirical depths of closure along the US East coast. Coastal Engineering 170, 104009
  13. ↑ Birkemeier, W. A. 1985. Field data on seaward limit of profile change. Journal of Waterway, Port, Coastal and Ocean Engineering 111(3): 598-602.
  14. ↑ Aragonés, L., Ignacio Pagán, J., López, I. and Serra, J.C. 2018. Depth of closure: New calculation method based on sediment data. Int. J. Sediment Res. 33: 198–207
  15. ↑ 15.0 15.1 Anthony, E.J. and Aagaard, T. 2020. The lower shoreface: Morphodynamics and sediment connectivity with the upper shoreface and beach. Earth-Science Reviews 210, 103334
  16. ↑ Hartman, M. and Kennedy, A.B. 2016. Depth of closure over large regions using airborne bathymetric lidar. Marine Geology 379: 52–63
  17. ↑ Durkin, C.J., Seenath, A. and Knaapen, M.A.F. 2025. A critical review of closure depth theories and uncertainties: implications for shoreline modelling and coastal management. Ocean and Coastal Management 267, 107732


The main author of this article is Grzegorz, Rozynski
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