Morphodynamic relationships for ebb and flood delta volumes at Florida’s tidal entrances

Abstract

Tidal entrances constitute an important boundary condition for the coastal ocean regime. Based on data from 67 sandy entrances in Florida, morphodynamic relationships between tidal prism, entrance throat area, and ebb and flood delta volumes are revisited. The main source of sand in these deltas is the littoral zone as opposed to the river. It is found that, as a rule of thumb, the volume of a mature ebb delta is equal to one fifth of the prism at the spring range of tide. Data for the coasts of Florida taken together indicate that the volume of mature flood delta varies with one third power of the prism. The Atlantic Coast flood delta volumes by themselves do not show any correlation with the prism, presumably because entrance depths have been altered by dredging. A case study of the closure of a land barrier breach at Matanzas Inlet illustrates the application of the derived morphodynamic relationships between prism, throat area, and ebb delta volume. These relationships are useful for prediction of changes in the throat area and the ebb delta volume when morphologic changes occur over time scales consistent with the reestablishment of equilibrium.

Introduction

Tidal entrances constitute an important dynamic boundary condition with respect to the coastal ocean regime due to their role as interfaces for exchange between seawater and embayed waters. Generalized morphodynamic relationships applicable to the entrances serve as useful tools for characterizing the physical state and the dynamic behavior of entrances. At sandy entrances, the three most important morphologic features are the channel, the ebb delta, and the flood delta. Because deltas exist due to water flow through the channel, the ebb and the flood delta volumes, V _e and V _f, respectively, depend on the channel throat cross-sectional area, A _c, i.e., the minimum flow area below mean tide level. As A _c is known to vary with the spring tidal prism, P, i.e., the water volume that enters the bay at (annual mean) spring tide, V _e and V _f, vary with P. The relationships A _c(P), V _e(P), and V _f(P) play a key role in characterizing the link between the hydrodynamic and the morphologic regimes of the entrances and, in turn, the roles they play as sources or sinks of marine sediment. In this paper, we have reviewed the physical significance of these relationships based on data from sixty-seven sandy entrances in Florida (Fig. 1), where the maintenance of stable passages for navigation is vital to coastal commerce and recreational interests.

Fig. 1

Florida’s tidal entrances

Area–prism relationship

Florida’s Atlantic Coast with 28 entrances and the Gulf of Mexico Coast with 39 entrances are characterized by micro-tidal (<2 m) ranges R _o (Tables 1 and 2). The Atlantic tides are semi-diurnal, the panhandle coast tides from Pensacola Pass to East Pass at Carrabelle are mainly diurnal, and the lower Gulf Coast tides south of Anclote Pass are mixed. Representative values of the nearshore wave energy flux E _f in nominally 10-m water depth (Tables 1 and 2) indicate that the coastal wave energy in Florida is low to moderate compared to the US shorelines taken together, which also includes the high-wave-energy Pacific Coast (Walton and Adams 1976). Moreover, wave energy along the Gulf Coast is generally lower than along the Atlantic Coast. This difference is seemingly reflected in the median grain diameter of sand d ₅₀ of the ebb delta or the beach close to it (given in Tables 1 and 2). The d ₅₀ value is finer along the Gulf Coast (0.3 mm mean) compared to the Atlantic Coast (0.4 mm) presumably because smaller grains are winnowed out by the larger waves of the Atlantic. This difference is, however, too small to be of significance in differentiating the dynamics of the deltas at the two coasts and will not be invoked in the present analysis.

Table 1 Atlantic Coast entrances

Table 2 Gulf Coast entrances

The well-known empirical relationship between the throat area and the spring tidal prism

$$A_{{\text\rm{c}}} = aP^{m} $$

(1)

is applicable to entrance channels in live-bed sedimentary equilibrium. The coefficients a and m vary from entrance to entrance; however, O’Brien (1969) showed that for 28 US entrances, a=4.69×10⁻⁴ and m=0.85 as best-fit values are applicable to all entrances when P is measured in cubic meters (m³) and A _c in square meters (m²). For a sinusoidal variation of the flow discharge at the tidal frequency, P is related to mean discharge $\overline{Q}$ over flood or ebb flow duration by

$$P = \frac{{T\overline{Q} }} {2}$$

(2)

where T is the tidal period. Combining Eqs. 1 and 2 gives

$$A_{{\text\rm{c}}} = a{\left( {\frac{T} {2}} \right)}^{m} \overline{Q} ^{m} $$

(3)

For illustrative purposes, taking T=44,700 s for a semi-diurnal tide, we obtain

$$A = 2.30\overline{Q} ^{{0.85}} $$

(4)

which is similar to

$$A = 1.51Q^{{0.83}}_{{\text{r}}} $$

(5)

where A is the river cross-sectional area and Q _r is the river discharge. Equation 5, known as the regime equation, was empirically derived for several non-tidal rivers in the US by Blench (1961). The analogy between river and tidal flows was pointed out by Mason (1973).

The transition between river-dominated flow and tide-dominated flow depends on the ratio of tide-mean discharge to river discharge ${\overline{Q} } \mathord{\left/ {\vphantom {{\overline{Q} } {Q_{r} }}} \right. \kern-\nulldelimiterspace} {Q_{r} }$. For river flow, ${\overline{Q} } \mathord{\left/ {\vphantom {{\overline{Q} } {Q_{r} }}} \right. \kern-\nulldelimiterspace} {Q_{r} } = 0$, and the influence of the river on tidal flow becomes minor when ${\overline{Q} } \mathord{\left/ {\vphantom {{\overline{Q} } {Q_{r} }}} \right. \kern-\nulldelimiterspace} {Q_{r} } \geqslant 20$. Although many small entrances in Florida are inundated by river outflows during spates, such events are relatively rare and, on an annual mean basis, the ratio ${\overline{Q} } \mathord{\left/ {\vphantom {{\overline{Q} } {Q_{r} }}} \right. \kern-\nulldelimiterspace} {Q_{r} }$ at all entrances is well above 20 (Bruun 1978). This, in turn, provides the justification for dealing with delta volumes in terms of their dependence on the tidal prism, without invoking the influence of river discharge.

The basis of Eq. 4 can be examined by considering sand transport in rivers. At a given flow velocity u (=Q _r/A), when the bottom shear stress τ is greater than the critical shear stress for erosion τ _c under live-bed equilibrium, the number of sand grains eroding from the bed (by saltation) per unit bed area per unit time equals the number depositing from suspension per unit bed area per unit time. The resulting sand unit load q _s, i.e., sand volume per unit time per unit channel width, is dependent on τ. A well-known expression for this dependence is the Einstein-Brown approximation of the Einstein’s bedload equation for sand transport (see, e.g., Graf 1971)

$$ϕ = 40ψ ^{3} $$

(6)

in which the bedload function φ and the Shields’ parameter ψ, are, respectively,

$$Ï• = \frac{{q_{{\text\rm{s}}} }} {{w_{{\text\rm{s}}} d}}$$

(7)

$$ψ = \frac{τ } {{\gamma {\left( {s - 1} \right)}d}} = \frac{{ρ u^{2}_{ * } }} {{\gamma {\left( {s - 1} \right)}d}}$$

(8)

with

$$u_{ * } = u{\left( {\frac{f} {8}} \right)}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} $$

(9)

In the above equations, w _s is the settling velocity of a grain of diameter d, γ is the unit weight of water, s=γ _s/γ is the specific weight of sand grains, γ _s is their unit weight, $u_{ * } = {\sqrt {{g\tau } \mathord{\left/ {\vphantom {{g\tau } \gamma }} \right. \kern-\nulldelimiterspace} \gamma } }$ is the flow friction velocity, and f is the Darcy–Weisbach friction factor.

In sufficiently energetic rivers and tidal entrances, suspended load transport usually dominates over bedload. Christensen and Chiu (1973) argued that, as both modes of transport involve sediment exchange between the bed and the suspension, q _s can be considered to represent the total load, i.e., the sum of suspended load and bedload. They supported this assumption by showing the application of an Einstein-type bedload equation to data on total load of sand measured in flumes. As this approach obviates the need to differentiate between the two types of loads, we will consider Eq. 6 to represent the total load. From Eqs. 6 through 9, we obtain

$$\frac{{q_{{\text\rm{s}}} }} {{w_{{\text\rm{s}}} }} = \frac{{0.078f^{3} }} {{g^{3} {\left( {s - 1} \right)}^{3} d^{2} }}u^{6} $$

(10)

where q _s now denotes total unit load. This equation can be restated as

$$q_{{\text{s}}} = \frac{{0.078f^{3} w_{{\text\rm{s}}} }} {{g^{3} {\left( {s - 1} \right)}^{3} d^{2} }}{\left( {\frac{{\overline{Q} }} {{A_{{\text\rm{c}}} }}} \right)}^{6} = K_{1} {\left( {\frac{{\overline{Q} }} {{A_{{\text\rm{c}}} }}} \right)}^{6} $$

(11)

in which $K_{1} = {0.078f^{3} w_{{\text{s}}} } \mathord{\left/ {\vphantom {{0.078f^{3} w_{{\text{s}}} } {g^{3} {\left( {s - 1} \right)}^{3} d^{2} }}} \right. \kern-\nulldelimiterspace} {g^{3} {\left( {s - 1} \right)}^{3} d^{2} }.$ For a given entrance channel, we may treat K ₁ as a constant. For such a channel in live-bed equilibrium, the ratio ${\overline{Q} } \mathord{\left/ {\vphantom {{\overline{Q} } {A_{c} }}} \right. \kern-\nulldelimiterspace} {A_{c} }$ is obtained from Eq. 3 as

$$ \frac{{\overline{Q} }} {{A_{{\text{c}}} }} = K_{2} \overline{Q} ^{{1 - m}} $$

(12)

where $K_{2} = {{\left( {2 \mathord{\left/ {\vphantom {2 T}} \right. \kern-\nulldelimiterspace} T} \right)}^{2} } \mathord{\left/ {\vphantom {{{\left( {2 \mathord{\left/ {\vphantom {2 T}} \right. \kern-\nulldelimiterspace} T} \right)}^{2} } a}} \right. \kern-\nulldelimiterspace} a.$ Hence, combining Eqs. 11 and 12, we get

$$ q_{{{\text{so}}}} = K_{1} K^{6}_{2} \overline{Q} ^{{6{\left( {1 - m} \right)}}} $$

(13)

where q _s=q _so is now the unit load of sand under live-bed equilibrium. An expression obtained by Kraus (1998) using a different formula for sediment transport can be interpreted in the same way as Eq. 13. It is a restatement of Eq. 1, and as m=0.85, for a given entrance, q _so is proportional to $\overline{Q} ^{{0.09}} .$ A set of data on 162 entrances along the US shorelines indicates that m ranges between 0.84 and 1.10 (Jarrett 1976). Thus, the exponent of $\overline{Q}$ ranges from 0.96 to −0.60, implying a reversal in the dependence of q _so on $\overline{Q}$ from almost linear to approximately $1 \mathord{\left/ {\vphantom {1 {\overline{Q} }}} \right. \kern-\nulldelimiterspace} {\overline{Q} }$.

For 8 out of 28 entrances without jetties, O’Brien (1969) showed that Eq. 1 could be applied with a=2×10⁻⁵ and m=1, supporting an earlier observation of LeConte (1905) inferred from data on five California entrances. According to LeConte, “Nature requires 33 sq. ft of mean-tide section for each and every million cubic feet of tidal waters passing in and out at spring tides”. Given the area in square meters and the prism in cubic meters, this amounts to a=1.08×10⁻⁴ and m=1. For sheltered harbors, LeConte recommended a=1.41×10⁻⁴ and m=1. The value of m equal to unity in these instances means that q _so is independent of $\overline{Q}$ and K ₂=T/2a. Area–prism data in Tables 3 and 4, compiled from the literature by Powell (2003) and plotted in Fig. 2, corroborate this observation. The best-fit values of the coefficients of Eq. 1 are a=7.25×10⁻⁵ and m=0.97 (which is close to 1) with a correlation coefficient r ²=0.67. Port Canaveral Entrance is an artificial channel without a direct connection to the interior waters, and its throat area, maintained by dredging, is much larger than the equilibrium value corresponding to the entrance’s tidal prism. Excluding this entrance yields a=6.25×10⁻⁵ and m=1 with r ²=0.91.

Table 3 Atlantic Coast entrance deltas

Table 4 Gulf Coast entrance deltas

Fig. 2

Area–prism data for Florida entrances. The best-fit line excludes Port Canaveral Entrance, an artificial harbor without a bay

Equation 13 implies that, at an entrance which conforms to Eq. 1, the sand unit load is q _so. Thus, for entrances for which K ₁, K ₂, $\overline{Q}$, and m are known, q _so can be calculated. Consider this example: T=44,712 s (semi-diurnal tide), a=6.25×10⁻⁵, m=1, w _s=0.04 m s⁻¹ (corresponding to d=3×10⁻³ m), f=0.028 (from Bruun 1978), and s=2.65 (quartz sand). With these values, we obtain K ₁=1.79×10⁻⁴ and K ₂=0.716. Hence, from Eq. 13, q _so=2.41×10⁻⁵ m² s⁻¹. So, for an entrance that is, say, 100-m wide, taking 2,650 kg m⁻³ as the granular density of sand, this value of q _so amounts to the transport of 143 t of sand over one half tidal period (ebb or flood) or 2.05×10⁵ t year⁻¹.

The sand entering Florida’s entrances is predominantly littoral, i.e., from the open coasts as opposed to the interior waters. The annual rate of sand transport into six entrances based on dredging records is given in Table 5 (Jones and Mehta 1980). This net rate should be comparable to the gross rate during flood tidal flow provided that all or, at least, the majority of sand deposits in the channel. Dredging records and growth rates of flood deltas indicate at least an order of magnitude higher rate of sand deposition in the channel compared to the ebb delta. In other words, the majority of sand which enters during flood tide deposits in the channel (Bruun 1978). Therefore, we may take the sand transport rate derived from deposition in the channel to be an approximate measure of q _so.

Table 5 Annual sand loads at six entrances from dredging records

For the net unit load q _so, Table 5 indicates a range of 8.01×10⁻⁶ to 4.23×10⁻⁵ m² s⁻¹, and 1.1×10⁵ to 2.95×10⁵ t year⁻¹ for the net load, i.e., unit load multiplied by channel width. The net load is surprisingly uniform, and neither it nor the unit load correlates with the wave energy flux, E _f, the main transporting agent for littoral drift. This suggests that the morphology of the channel and the deltas plays an important role in governing sand transport in the channel.

Ebb delta volume vs prism

The volume V _e and the crescentic shape of the ebb delta (Fig. 3) are determined by the tidal current and the sea waves. At Jupiter Inlet on the Atlantic Coast (Fig. 4), the ebb delta grew rapidly in the initial years after this entrance was fully opened by capital dredging and stabilized by jetties in 1947. The mean growth rate was 38,000 m³ per year during the first decade. After about two decades, the delta attained a quasi-equilibrium state or maturity under the counterbalancing roles of tide in depositing sand and waves in removing it. Year-to-year changes in this balance caused the volume to oscillate about a long-term mean of about 5×10⁵ m³, notionally bounded by a high 7.8×10⁵ m³ and a low 2.2×10⁵ m³. Dombrowski and Mehta (1996) characterized these bounds by a dimensionless number representing the ratio of wave energy flux to tidal energy flux. As the latter can be assumed to remain invariant on the annual mean basis, the wave energy determines the bounds; the upper bound corresponds to a low wave height and the lower bound to a larger wave height. Devine and Mehta (1999) showed that the delta lobe face (Fig. 3a) recedes by erosion under high waves and recovers when wave action subsides, in a manner similar to the submerged portion of a sandy beach.

Fig. 3

a Ebb delta features b Idealized geometry

Fig. 4

Growth of ebb delta at Jupiter Inlet since 1947

The ebb delta volumes in Tables 3 and 4 were calculated by applying the graphical method of Dean and Walton (1975) to isobaths at the delta along with those at the nearby shorelines unaffected by the delta. The volumes represent deltas that are, at least, a decade old and, in many cases, older than five decades. At such mature deltas, the following argument allows us to ascertain the dependence of the delta volume on the tidal prism.

Figure 3b is a rendition of the ebb delta idealized as a box of lateral area A _e, planform area A _E and length L _e. The delta volume is

$$V_{{\text\rm{e}}} = A_{{\text\rm{e}}} L_{{\text\rm{e}}} $$

(14)

As A _e can be expected to vary with the size of the entrance, we will assume that A _e is geometrically related to the throat area A _c according to

$$ A_{{\text{e}}} = c_{0} A^{{n_{0} }}_{{\text{c}}} $$

(15)

where c ₀ and n ₀ are entrance-specific constants. Values of c ₀ are generally not available; however, De Vriend et al. (2002) selected n ₀=1. Substituting Eq. 1 into Eq. 15 gives

$$ A_{{\text{e}}} = c_{0} a^{m} P^{{n_{0} m}} $$

(16)

Based on the data of Gibeaut and Davis (1993) from 21 entrances on the Gulf Coast between Caxambas Pass and Hurricane Pass (Pinellas), Dombrowski and Mehta (1996) obtained the following approximate dependence of A _E on the prism P

$$ A_{{\text{E}}} = c_{1} P^{{n_{1} }} $$

(17)

where $c_{1} = {2.54} \mathord{\left/ {\vphantom {{2.54} {R^{{n_{1} }}_{{\text{o}}} }}} \right. \kern-\nulldelimiterspace} {R^{{n_{1} }}_{{\text{o}}} }$ and n ₁=0.81 with r ²=0.65. In Table 2, the tidal range R _o along the segment of the Gulf Coast where these entrances occur is seen to be reasonably uniform, varying narrowly between 0.7 and 0.9 m. We may accordingly and conveniently take c ₁ to be independent of R _o and further assume that Eq. 17 is applicable to all entrances.

From Fig. 3b, by way of geometric scaling $L_{{\text\rm{e}}} \sim A^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}_{{\text\rm{E}}}$ or in general

$$ L_{{\text{e}}} = c_{2} A^{{n_{2} }}_{{\text{E}}} $$

(18)

an inference supported by the analysis of Vincent and Corson (1981) on 67 entrances along the US shorelines. They reported c ₂=0.3 and n ₂=0.58 (which is reasonably close to 0.50). Combining Eqs. 17 and 18, we obtain

$$ L_{{\text{e}}} = c_{2} c^{{n_{2} }}_{1} P^{{n_{1} n_{2} }} $$

(19)

where c ₂ is another constant. Finally, combining Eqs. 14, 16, and 19 yields

$$ V_{{\text{e}}} = c_{0} c^{{n_{2} }}_{1} c_{2} a^{m} P^{{n_{0} m + n_{1} n_{2} }} $$

(20)

where $c_{{\text{e}}} = c_{0} c^{{n_{2} }}_{1} c_{2} a^{m}$ and $n_{{\text{e}}} = n_{0} m + n_{1} n_{2} .$ Selecting n ₀=1, m=0.97, n ₁=0.81, and n ₂=0.58, we obtain n _e=1.44.

The best-fit values of c _e and n _e for the Florida entrances are given in Table 6. In Fig. 5, three lines are shown, one including all 57 entrances for which ebb deltas are reported, one line for the 19 Atlantic Coast entrances, and the last line for the 38 Gulf Coast entrances. Gulf data are also given without including Willys Cut and Hurricane Pass (Pinellas). These adjacent entrances effectively contain a single ebb delta whose value is uncertain because the bathymetry used was of poor quality and uncertain date.

Table 6 Coefficients for ebb delta volume vs tidal prism relation

Fig. 5

Ebb delta volume vs tidal prism

The value n _e=1.26 for the Atlantic Coast is smaller than 1.44 by 13% but only 2% greater than n _e=1.23 obtained consistently by Walton and Adams (1976) for all three US shorelines. These investigators also showed that c _e decreases with increasing wave energy, indicating that, for a given prism, the ebb delta volume decreases with increasing wave energy. This is so because, with increasing wave energy, the sand-winnowing role of waves increases relative to the sand-depositing role of ebb flow. Unfortunately, as the wave energy flux E _f along the coasts of Florida varies over a narrow range compared with the US shorelines as a whole (Walton and Adams 1976), any effect of E _f on delta volume cannot be discerned from the “noisy” V _e values.

Dombrowski and Mehta (1996) concluded that, due to uncertainties associated with bathymetric data and subjectivity in delineating the ebb delta geometry, the error in estimating V _e according to the Dean and Walton (1975) method can be as much as ±10% and possibly higher in cases involving complicated shoreline and entrance configurations. At 16 entrances between St. Marys and Hillsboro on the Atlantic Coast, the mean depth is generally more than twice in dredged entrances (7 m) compared to the ones that remain natural or undredged (3.1 m). Thus, channel dredging has influenced delta volume, possibly quite significantly in some cases.

From Table 6, we observe that a plausible value of n _e for the Florida coast as a whole is equal to 1. With this value, we obtain c _e=0.2. In other words, as a rule of thumb, the mean ebb delta volume is one fifth of the spring tidal prism.

Flood delta volume vs prism

Flood deltas are shaped by tidal flow without the counteracting influence of sea waves, even as these waves “push” littoral sand into the entrance. At a new entrance, the delta volume increases rapidly in the initial years. At Sikes Cut on the Gulf Coast, the growth rate of the delta in Appalachicola Bay during the first three decades, since this entrance was opened in 1954, was seemingly constant at 233,000 m³ year⁻¹ (Fig. 6). In contrast, at St. Lucie Inlet on the Atlantic Coast, where the bay area is restricted by the width of the embayment, the growth rate (500,000 m³ year⁻¹, Fig. 6) decreased after the first decade from the time the entrance was newly dredged in 1892. About four decades later, the volume stabilized, with practically no further change within a ±15% error band characteristic of generally well-defined flood delta estimates. In general, where an identifiable flood delta volume is limited by bay size, sand continues to arrive from the sea and to deposit in the upstream reaches (Carr de Betts and Mehta 2001). The excess sand is often dredged to maintain navigable depths, e.g., in the Intracoastal Waterway between the barrier island and the mainland.

Fig. 6

Growth of flood deltas at Sikes Cut (since 1954) and St. Lucie Inlet (since 1892)

Regardless of bay size, the thickness h _f of mature deltas appears to vary over a narrow range. Based on the histogram of the thicknesses of 61 flood deltas in Florida, Carr de Betts and Mehta (2001) observed that the majority of values ranged between 0.7 and 1.1 m, with a mean very close to 1 m. Most of these deltas occurred in conditions defined by narrow ranges of characteristic physical parameters: micro-tidal range, water depth of 2 to 5 m, strength of flood current from 0.8 to 1.1 m s⁻¹, and grain size from 0.2 to 0.7 mm. One may surmise that the constant delta thickness of 1 m is a manifestation of live-bed sedimentary equilibrium locally over the flood delta. At a new entrance, once the thickness of the deposit reaches 1 m, subsequent incoming sand is pushed further inward, causing the delta to expand and fan out.

In analogy with Eq. 14, the flood delta volume is

$$V_{{\text\rm{f}}} = A_{{\text\rm{f}}} L_{{\text\rm{f}}} = h_{{\text\rm{f}}} w_{{\text\rm{f}}} L_{{\text\rm{f}}} $$

(21)

where A _f is the cross-sectional area, L _f the length, and w _f the width of the delta “box”. The simplest idealization of delta growth can be parameterized in terms of a constant h _f, e.g., 1 m, a constant w _f governed by the flood-flow jet and L _f increasing along the longitudinal distance ζ at a decreasing rate until the delta acquires a mature, quasi-steady volume.

The width w _f may be estimated as follows. A simple model for the increase in flow jet width w with ζ is given by

$$ \frac{{w{\left( \zeta \right)}}} {{w_{{\text{c}}} }} = e^{{\frac{f} {{16h}}\zeta }} $$

(22)

where w _c is the entrance width and h is the water depth in the bay (Bruun 1978). As the jet spreads its velocity u decreases and a delta is formed beyond a distance ζ=L _s, where u<u _cr, the critical velocity for erosion. From flow continuity

$$u{\left( {L_{{\text\rm{s}}} } \right)} = u_{{{\text\rm{cr}}}} = \frac{{\overline{Q} }} {{w{\left( {L_{{\text\rm{s}}} } \right)}h}}$$

(23)

The critical velocity is obtained from Eqs. 8 and 9 as

$$u_{{{\text\rm{cr}}}} = {\left[ {\frac{{8gψ _{{{\text\rm{cr}}}} }} {f}{\left( {s - 1} \right)}} \right]}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} d^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} $$

(24)

where ψ_cr is the critical value of the Shields’ parameter. We now combine Eqs. 3, 22, 23, and 24 to obtain

$$L_{{\text\rm{s}}} = \frac{{16h}} {f}\ln {\left( {\frac{{\frac{2} {T}{\left( {\frac{{A_{{\text\rm{c}}} }} {a}} \right)}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }} {{w_{{\text\rm{c}}} h{\left[ {\frac{{8gψ _{{{\text\rm{cr}}}} }} {f}{\left( {s - 1} \right)}} \right]}^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} d^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}} \right)}$$

(25)

By conveniently taking w _f to be the jet width at distance L _s, this width can be obtained from Eq. 22. We will consider Sikes Cut with characteristic parameters: h=3.4 m, A _c=360 m² (at ζ=0, the jet origin), w _c=65 m (at ζ=0), T=86,400 s, a=6.25×10⁻⁵, m=1, d=3×10⁻³ m, f=0.025, ψ_cr=0.06, and s=2.65. This yields L _s=170 m, and from Eq. 22 (with ζ=170 m), w _f=71 m. Figure 7 appears to corroborate this value of L _s as the approximate distance beyond which the flood delta occurs. The delta has fanned out in an “explosive” mode, consistent with a rapidly widening flow jet. Such a behavior is prominent when the depth h decreases with distance, i.e., when the bay is shallower than the entrance. An extension of Eq. 22, which enables the simulation of the delta fan, is given in Bruun (1978).

Fig. 7

Flood delta at Sikes Cut

The wider the entrance, the greater the width of the flood delta; so, for a mature delta, as h _f and L _f may be held constant, one would infer that w _f∼P ^1/3. Thus, from Eq. 21, we obtain V _f∼P ^1/3. In general,

$$V_{{\text\rm{f}}} = c_{{\text\rm{f}}} P^{{n_{f} }} $$

(26)

where c _f and n _f have qualitatively the same meaning as c _e and n _e, respectively. These coefficients are given in Table 7 based on 57 entrances in Tables 3 and 4. The prism correlates much more weakly with V _f (with low r ² values in Table 7) than with V _e (higher r ², Table 6). For all entrances combined, n _f is 0.34, i.e., nearly one third. The Atlantic Coast entrances by themselves show no correlation, possibly due to significant dredging at several entrances. A case in point is St. Marys Entrance, where the flood delta actually decreased from 5.5 to 1.6×10⁶ m³ when the channel was deepened between 1984 and 1992. On the Gulf Coast, the correlation improves (r ² increases from 0.38 to 0.47) if Clam Pass, which has a small delta that may not be mature, is excluded from the best-fit analysis (Fig. 8).

Table 7 Coefficients for flood delta volume vs tidal prism relation

Fig. 8

Flood delta volume vs tidal prism; Florida’s Gulf Coast entrances

An application

Matanzas Inlet on the Atlantic Coast offers a case where we may test the area–prism and ebb delta volume vs prism relationships. In 1964, Hurricane Dora broke through the narrow Rattlesnake Island separating Matanzas River from the Intracoastal Waterway (Fig. 9). By 1976, the breach was transporting 71% of the tidal prism, and the resulting cross-current was adversely affecting navigation in the waterway. In December 1976, the breach was closed by placement of rock and sand. The outcome is summarized in Table 8. Comparing the April 1977 post-closure prism, throat area, and ebb delta volume with the corresponding pre-closure values of October 1976, we observe that, due to a partial loss of the effective bay area, the prism decreased by 55%. This decrease concurrently caused the throat area to contract by 15%. One million cubic meters of sand representing about 33% of the ebb delta moved into the entrance channel and nearly blocked it (Mehta and Sheppard 1979). A portion of sand moved from the channel into the northern arm of the Matanzas River, blocking vessel access between the Intracoastal Waterway and the entrance.

Fig. 9

Matanzas Inlet on the Atlantic Coast of Florida

Table 8 Effect of breach closure at Matanzas Inlet on ebb delta volume

Based on the line in Fig. 2, the estimated equilibrium throat area in April 1977 would be 323 m², as opposed to 697 m² achieved. From Table 6 (Atlantic Coast entrances), the equilibrium ebb delta volume is estimated to be 1.09 vs 2.00×10⁶ m³ realized. These observations imply that the 4-month period, between breach closure in December 1976 and the next bathymetric survey in April 1977, was insufficient for the entrance morphology to adjust to the decreased tidal prism.

To recover the prism, a relief channel was dredged through the northern arm of the river. This operation was successful and resulted in a 171% increase of the prism by 1996. The throat area increased by 31% (to 910 m²), which compares favorably with a 26% increase in the estimated equilibrium area (875 m²). The ebb delta concurrently increased by 140% to 4.80×10⁶ m³. The corresponding estimated value is 3.84×10⁶ m³, amounting to a 92% increase. This is lower than 140% but can be taken to be of the same order of magnitude.

Concluding comments

The relationship between tide-mean water discharge and flood- or ebb-mean sediment load can be construed as an indicator of the dependence of ebb and flood delta volumes on the tidal prism. The morphodynamic relationship between ebb delta volume and tidal prism, originally reported by Walton and Adams (1976), results from scaling arguments for delta geometry. For the coast of Florida as a whole, the volume varies linearly with the prism and, on that basis, is one fifth of the prism. The main source of sand in the deltas is the littoral zone as opposed to the river.

The flood delta volumes for the coast of Florida indicate that the volume at mature deltas varies with prism to a power close to one third. Scaling arguments similar to those for the ebb delta show that this dependence is consistent with the morphodynamics of flood delta development. The Atlantic Coast delta volumes by themselves do not show any correlation with the prism presumably because, at many entrances, the depths have been altered by dredging.

The effect of the closure of a storm-induced breach near Matanzas Inlet illustrates the applicability of the area–prism and ebb delta volume vs prism relationships. Four months after breach closure and concomitant changes in the entrance morphology, the throat area reduction was over-predicted, as was the reduction in ebb delta volume. However, after several years, as the entrance reverted to a new state of equilibrium, the predicted changes in the throat area and the ebb delta volume were of the same order of magnitude as those experienced. We, therefore, conclude that the morphodynamic relationships are useful for the prediction of changes in the throat area and the ebb delta volume when morphologic changes occur over time scales consistent with the reestablishment of equilibrium.