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  • View in gallery

    The ratio of the vertical eddy diffusivity to the vertical eddy viscosity, vs the layer Richardson number, calculated from the parameterizations (3.3) and (3.9), with two values of the parameter A3 (solid and dashed lines). The same quantity calculated from the Hudson River observations of Peters (1999) is plotted as stars. The parameterization (3.9) with A3 = 0 is designed to fit the observations, except for one instance, which is highlighted as a filled star.

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    Model quasi-steady salt intrusion vs QR, for three estuaries, as labeled. The salt intrusion is defined by a different salinity in each case: e.g., X13.5 means the salt intrusion defined by s = 13.5. Observed curves are also plotted, as is a curve proportional to Q−1/3R for reference.

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    Quasi-steady model solution for the Hudson River, under moderate-flow conditions and average tides. (a) The dimensionless salinity Σ vs x is plotted as a thick line, with the stratification shown as a gray band around it. The surface velocity and ν (the diffusive fraction of up-estuary salt flux) are also shown. (b) The salinity vs x and z. (c) Distributions of the parameterized diffusivities. (d) The overturning streamfunction normalized by QR, showing the exchange flow (the sense of the circulation is generally clockwise on the page).

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    (a) Adjustment times and (b) sensitivities for 35 numerical simulations. Theoretical values (from Table 1) are plotted against numerical model results. Both exchange-dominated and tidal-stirring-dominated cases are considered. In either case, the forcing is due to variation of either river flow or tidal mixing, with a period of 50 days. Cases with the longest adjustment times correspond to those with the smallest sensitivities.

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    Time-dependent numerical simulation of the Hudson River estuary. (a) The forcing variation, which consists of a spring–neap cycle of tidal amplitude and a switch from low river flow to high and back again. (b) The response of the model stratification with time, where the parameter Φ is the normalized stratification averaged over the length of the salt intrusion. (c) The length of the salt intrusion vs time. In (b) and (c), a quasi-steady model solution is also plotted to indicate how important unsteady effects might be. (right) Salinity sections at three times. These are from low-flow neap, low-flow spring, and high-flow periods. During low-flow periods, the stratification is relatively sensitive to the spring–neap cycle and the salt intrusion is not, in comparison with the quasi-steady solution. This result is because the salt intrusion does not have time to adjust.

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Estuarine Adjustment

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Abstract

Subtidal adjustment of estuarine salinity and circulation to changing river flow or tidal mixing is explored using a simplified numerical model. The model employs tidally averaged, width-averaged physics, following Hansen and Rattray, extended to include 1) time dependence, 2) tidally averaged mixing parameterizations, and 3) arbitrary variation of channel depth and width. By linearizing the volume-integrated salt budget, the time-dependent system may be distilled to a first-order, forced, damped, ordinary differential equation. From this equation, analytical expressions for the adjustment time and sensitivity of the length of the salt intrusion are developed. For estuaries in which the up-estuary salt flux is dominated by vertically segregated gravitational circulation, this adjustment time is predicted to be TADJ = (1/6)L/u, where L is the length of the salt intrusion and u is the section-averaged velocity (i.e., that due to the river flow). The importance of the adjustment time becomes apparent when considering forcing time scales. Seasonal river-flow variation is much slower than typical adjustment times in systems such as the Hudson River estuary, and thus the response may be quasi steady. Spring–neap mixing variation, in contrast, has a period comparable to typical adjustment times, and so unsteady effects are more important. In this case, the stratification may change greatly while the salt intrusion is relatively unperturbed.

Corresponding author address: Parker MacCready, University of Washington, Box 355351, Seattle, WA 98915-5351. Email: parker@ocean.washington.edu

Abstract

Subtidal adjustment of estuarine salinity and circulation to changing river flow or tidal mixing is explored using a simplified numerical model. The model employs tidally averaged, width-averaged physics, following Hansen and Rattray, extended to include 1) time dependence, 2) tidally averaged mixing parameterizations, and 3) arbitrary variation of channel depth and width. By linearizing the volume-integrated salt budget, the time-dependent system may be distilled to a first-order, forced, damped, ordinary differential equation. From this equation, analytical expressions for the adjustment time and sensitivity of the length of the salt intrusion are developed. For estuaries in which the up-estuary salt flux is dominated by vertically segregated gravitational circulation, this adjustment time is predicted to be TADJ = (1/6)L/u, where L is the length of the salt intrusion and u is the section-averaged velocity (i.e., that due to the river flow). The importance of the adjustment time becomes apparent when considering forcing time scales. Seasonal river-flow variation is much slower than typical adjustment times in systems such as the Hudson River estuary, and thus the response may be quasi steady. Spring–neap mixing variation, in contrast, has a period comparable to typical adjustment times, and so unsteady effects are more important. In this case, the stratification may change greatly while the salt intrusion is relatively unperturbed.

Corresponding author address: Parker MacCready, University of Washington, Box 355351, Seattle, WA 98915-5351. Email: parker@ocean.washington.edu

1. Introduction

The response of the estuarine salt intrusion to changes in river flow is of fundamental interest to both shellfish and water-quality managers, among others. A classical approach has been to develop empirical curves that relate the length of the salt intrusion L to the river volume flux QR (Abood 1974; Monismith et al. 2002). These curves typically result in a power law of the form LQnR. This approach appears to be justified, because the fits to data are reasonably good. Further, the specific value of n (e.g., −⅓) could sometimes be explained theoretically (MacCready 1999; Monismith et al. 2002; Hetland and Geyer 2004). This approach assumes a quasi-steady system, consistent with the theoretical models of Hansen and Rattray (1965) and Chatwin (1976). Kranenburg (1986) was the first to challenge the quasi-steady paradigm. He developed a theory that predicted the adjustment time TADJ for straight or widening channels and that spanned from well-mixed to partially mixed systems. His results are mostly supported in this paper. This contribution attempts to give a clear explanation of the TADJ predictions and to explore their implications using a simple numerical model that may be readily applied to real estuaries. MacCready (1999) reviews some of the intervening developments in the understanding of estuarine adjustment, but in general little progress has been made since Kranenburg (1986). The exception is that we have a better understanding of how variations in tidal mixing interact with L (Park and Kuo 1996; MacCready 1999) and with up-estuary salt flux (Bowen 2000). Hetland and Geyer (2004) used a 3D numerical model of tidally averaged flow to test quasi-steady sensitivity of the salt intrusion to various parameters. They also explored time-dependent response and revived Kranenburg’s notion that the adjustment time should vary with the freshwater filling time—a very appealing simplification that is made use of here.

2. Model development

Here a mathematical model of tidally averaged, width-averaged estuarine salinity structure in a rectangular channel of varying cross section is developed. The numerical solution to the equations is developed in this section, and approximate analytical solutions are developed in section 4. The model is tested against observations in section 3. The model is basically a time-dependent version of the Hansen and Rattray (1965) equations, and I attempt to make it a better predictive tool by developing parameterizations of the tidally averaged mixing coefficients. A steady version of the model was described in MacCready (2004). The approach here does not attempt to represent adequately the behavior of systems with strong lateral segregation of flow.

The state variables are u(x, z, t) and s(x, z, t), the tidally averaged, width-averaged velocity and salinity, where positive u and x are toward the mouth. Dividing variables into depth-mean (overbar) and deviation from depth-mean (prime) parts gives u = u(x, t) + u′(x, z, t) and s = s(x, t) + s′(x, z, t). Of these four variables, one is assumed to be known: uQR/A, where A(x) is the cross-sectional area. Then u′ and s′ are assumed to be in quasi-steady balance with the local along-channel salinity gradient sx, the mixing coefficients, and the river flow (subscripts x and t will denote partial differentiation). The quantities u′ and s′ have the standard cubic and quintic vertical profiles as described in MacCready (2004):
i1520-0485-37-8-2133-e21
where
i1520-0485-37-8-2133-e22
The magnitude of the exchange flow is uE = gβsxH3/(48KM), ζz/H, and H is the channel depth. Also, KM and KS are the vertical eddy viscosity and diffusivity. The term β ≅ 7.7 × 10−4 psu−1 appears in the linearized equation of state ρ = ρ0(1 + βs), where ρ is the density and ρ0 is the density of freshwater. The system thus is entirely described by the time and space evolution of the remaining state variable s(x, t), which is governed by [MacCready 2004, his Eq. (7)]:
i1520-0485-37-8-2133-e23
Here KH is an along-channel eddy diffusivity due to tidal correlations of velocity, salinity, and area. Nondimensionalizing the salinity as Σ ≡ s/socn, where socn is the oceanic salinity, (2.3) may be rewritten as
i1520-0485-37-8-2133-e24
where the terms on the right are associated with the three main physical mechanisms for along-channel salt transport. Using the polynomial forms for u′ and s′, the “exchange” term may be expressed in terms of Σx, allowing (2.4) to be rewritten as
i1520-0485-37-8-2133-e25
I have introduced the length scales:
i1520-0485-37-8-2133-e26
Of the four length scales, LE3 is typically the most important for partially mixed systems, whereas LH dominates in well-mixed cases. The parameter c ≡ (gΔρH/ρ0)1/2 is like a maximum internal wave speed, where Δρ is the density difference between river and ocean waters. These length scales are identical to those given in MacCready (2004) except that here KS and KM are allowed to vary separately.
Equation (2.5) is then solved numerically using forward time integration, with centered spatial derivatives (FTCS; Press et al. 1992). At each time step the rhs of (2.5) is evaluated and is used to advance Σ forward in time at each spatial point. At the mouth, the physical boundary condition is that the deepest salinity is equal to socn. This may be written, using expression (2.1) for s′, as
i1520-0485-37-8-2133-e27
This is a quadratic that may be solved analytically for Σ0. Here Σ0 is Σ at the mouth (x = 0), and Σ1 is Σ one grid point up-estuary (x = −Δx). Also TE2 = H2/(KSKM)1/2 and TE1 = H2/KS are vertical mixing time scales that appear in (2.6). The numerical domain is made long enough that the salt intrusion never extends beyond the up-estuary limit of the grid, where Σ = 0. The grid spacing Δx is chosen so that the length of the salt intrusion is well resolved. In a typical case, Δx = 1 km is used.
The time step Δt in the FTCS scheme is changed throughout the integration to optimize the model speed while ensuring numerical stability. To do this it is noted that, neglecting the river-flow term, (2.5) may be written approximately as
i1520-0485-37-8-2133-e28
where I have written the whole term in brackets in (2.5) as an effective diffusivity Keff, thereby casting (2.5) as a heat equation. Then numerical stability requires (Press et al. 1992, p. 847) that
i1520-0485-37-8-2133-e29
where Kmaxeff is the maximum value of Keff along the channel at the previous time step. Stable simulations are reliably achieved using Δt from (2.9) with a constant of 0.4 instead of 0.5.

3. Mixing parameterizations

The model specification is only complete when one has values for the mixing terms KM, KS, and KH; KM and KS are the vertical eddy viscosity and diffusivity, which relate tidally averaged turbulent fluxes to tidally averaged shear and stratification. It is assumed that they are constant over depth, but they are allowed to vary with time and along-channel distance. Parameterizations for these are sought in terms of known channel parameters (H, B, UT) and state variables—in particular, the tidally averaged stratification Φ; B is the channel width, UT is the amplitude of the depth-averaged tidal velocity, and Φ ≡ (sbottomstop)/socn. To allow comparison with observations, H is defined as the channel thalweg depth and B is defined as A/H so that the cross-sectional area is correct.

A parameterization for KM may be derived from the results of Geyer et al. (2000). They evaluated the momentum balance in the Hudson River estuary using a two-layer analytical model with no interfacial friction and found that the tidally averaged shear was approximately governed by
i1520-0485-37-8-2133-e31
where R = 2CD3UT3rms. Here Δu is the difference between outflowing and inflowing velocities averaged over upper- and lower-layer thicknesses h1 and h2, CD3 = 3.1 × 10−3 is a drag coefficient referenced to 3 m above the bottom, and UT3rms is the root-mean-square tidal velocity at that depth. Geyer et al. (2000) discovered that the momentum balance in the lower layer was dominated by the along-channel pressure gradient and bottom friction. This sets the velocity of the inflowing deep water. They found essentially zero interfacial friction, and the velocity of the outflowing upper layer was thus set by mass conservation. Our model, which has KM constant with depth, cannot reproduce the stress profile found by Geyer et al. (2000), because our KMu/∂z will always have nonnegligible stress at middepth. However, I can attempt to ensure that the model gives the correct relation between sx and the exchange flow. The cubic profile (2.1) of u has Δu ≅ 1.1uE (for uuE). This may be combined with (3.1) from Geyer et al. (2000) (assuming h1 = h2) and solved for KM to find
i1520-0485-37-8-2133-e32
Then, accounting for the differences in velocity and drag coefficient of depth-averaged tidal current amplitude UT versus the 3-m rms tidal velocity UT3rms, the final expression for KM is found to be
i1520-0485-37-8-2133-e33
where A0 = 0.065 and CD = 2.6 × 10−3. However, in the comparisons with observations (shown below) it is found that a reasonable fit required A0 = 0.0325, one-half of the value derived above. The reason for the discrepancy may lie in the lateral averaging, which would require consideration of nonrectangular cross sections. This smaller value is used in all of the simulations below in which the mixing parameterizations are employed. A parameterization of the same form was used by MacCready (1999), with larger A0. An expression of this kind would also result from a mixing-length argument. Increasing stratification should have little effect on (3.3) because the stress is mostly in the relatively unstratified bottom boundary layer. However, note that some authors, for example Uncles and Stephens (1990), have found that the decrease of KM by stratification is significant.
The vertical eddy diffusivity, in contrast, is focused on middepth fluxes between lower and upper layers and thus will likely be more strongly affected by stratification. Neglecting lateral variation and tidal changes in surface height, one may define the eddy diffusivity appropriate for the tidally averaged salt balance by
i1520-0485-37-8-2133-e34
where a tilde denotes a quantity with tidal and depth variation, an overbar is depth averaging, and angle brackets are for tidal averaging. Note that 〈z〉 = sz in the notation of section 2. In physical terms, KS is the eddy diffusivity one would use with the tidally averaged stratification to achieve the tidally averaged vertical turbulent salt flux. It is assumed that the depth averaged, tidally averaged rate of loss of barotropic tidal energy goes, in part, into averaged potential energy gain due to mixing. The ratio of these is the flux Richardson number (Dyer 1997, p. 54)
i1520-0485-37-8-2133-e35
where = UT sin(ωTt). It is further assumed that this expression scales with tidally averaged properties, giving
i1520-0485-37-8-2133-e36
In solving for KS, I find that
i1520-0485-37-8-2133-e37
where RiL is analogous to the layer Richardson number (Lewis 1996; Dyer 1997, p. 42) but is defined with the tidally averaged stratification and by assuming that the shear scales with UT:
i1520-0485-37-8-2133-e38
Note that c and UT are known environmental parameters, whereas Φ is a state variable. Expression (3.7) decreases with RiL, an expected result of increasing stratification; however, Rf is an unknown function of RiL. It is proposed that KS may be parameterized using a function of the following form:
i1520-0485-37-8-2133-e39
with A1, A2, and A3 being empirically derived constants. The maximum value of KS is A1CDUTH, much like KM in (3.3), and so it is expected that A1A0. The rate of decrease of KS with increasing RiL is governed by A2, analogous to the parameterization in Munk and Anderson (1948). The minimum value of KS as RiL → ∞ is given by A3 times its maximum value (0 ≤ A3 ≤ 1); A3 is intended to represent the potential effects of a channel with shallow sides. On the sides, the most stratified water (which is at middepth in the center of the channel) will intersect with the sloping sidewalls and will be subject to turbulence generated by the bottom boundary layer. Nonzero A3 implies that, even with strong midchannel stratification, turbulent buoyancy flux cannot be entirely shut down on the edges.

To determine A1 and A2, I will fit to inferred values of turbulent salt flux from observations made in the Hudson River estuary (Peters 1999). Because these data were generally taken in 15 m of water and not on the sidewalls, I will perform the fit by assuming A3 = 0 and will then use other criteria later to estimate A3. Peters (1999, his Table 4) gives turbulent salt flux (inferred from dissipation) at the depth of the greatest stratification, integrated over the full tidal cycle, for 12 cases. For these he also gives UT (his Table 2), and one may take the tidally averaged bottom-to-top salinity difference graphically from his Fig. 7. For our polynomial form of s, one may show that the peak value of ∂s/∂z is equal to 1.7(sbottomsstop)/H, allowing us to express Peters’s observations in terms of the variables in (3.9), with A3 = 0. The results of this comparison are plotted in Fig. 1. A reasonable fit between prediction and observations is found using A1 = 0.022 and A2 = 3.33. There is one observation, highlighted in Fig. 1, that my parameterization does not predict at all well. This occurred during spring tide, with the lowest observed tidally averaged stratification (1.5 psu). There is no obvious explanation for this discrepancy. In finishing the parameterization of KS, it is assumed that A3 = 0.3, the reasons for which are discussed at the end of this section.

The final parameterization that is required is for KH, the along-channel mixing term. Banas et al. (2004) found a good fit for observations in Willapa Bay using KH = 0.035UTB, with the mixing being due to lateral stirring by tidal eddies in that complex, multichannel system. This parameterization is adopted here but is generalized to cases in which the tidal excursion LTUTTT/π (where TT is the tidal period) is smaller than B and thus is the limiting factor for the stirring. Near the mouth of the estuary, along-channel flux is enhanced by efficient exchange between the mouth and the coastal ocean (Stommel and Farmer 1952). MacCready (2004) expressed this idea in terms of a KH, which decreased from a theoretically derived maximum value near the mouth KHM, to a smaller value landward. Based on 3D numerical experiments (not shown here) it is assumed that this portion of KH near the mouth decays linearly from KHM to zero over LT, and that KHM is decreased from the value derived in MacCready (2004) by a factor ε (0 ≤ ε ≤ 1) because the presence of the river plume will decrease the efficiency of estuary–ocean exchange. To summarize, in the current model I will use
i1520-0485-37-8-2133-e310
where x is the along-channel coordinate, with x = 0 at the mouth and negative in the up-estuary direction. The value of the diffusivity at the mouth is given by
i1520-0485-37-8-2133-e311
A reasonable fit to observations (below) is found with ε = 0.1, and thus KHM is a factor-of-10 smaller than that proposed in MacCready (2004). This parameter is very poorly constrained, however, and would be a good topic for future studies.

One may test this model and its mixing parameterizations, using observations of the length of the salt intrusion L versus river flow. Doing this for Delaware Bay (Garvine et al. 1992), northern San Francisco Bay (Monismith et al. 2002) and the Hudson River (Abood 1974) gives the results plotted in Fig. 2. The model compares fairly well to San Francisco Bay and the Hudson, notably capturing the observed insensitivity of L to QR in the San Francisco Bay case. On the other hand, the model seriously overpredicts the length of the salt intrusion in the Delaware Bay. This is potentially because the Delaware bathymetry has a deep middle channel cut into a very wide, shallow bay, whereas our model treats the estuarine cross section as rectangular. The salt flux in the Delaware may also be more laterally segregated, whereas this model assumes it to be vertically segregated. The fit to Delaware observations was actually better in MacCready (2004); however, in that case the diffusivities were chosen to achieve the best fit instead of being given by the parameterization derived in this section. There is no obvious way to remedy this short of reworking the model formulation to account for different cross-sectional shapes or salt flux mechanisms. Examples of equilibrium fields for the Hudson River solution are plotted in Fig. 3. The Hudson is modeled as a straight channel, 14 m deep, and 1.14 km wide with QR = 300 m3 s−1 and UT = 0.9 m s−1. Northern San Francisco Bay has variable depth and width, as described in MacCready (2004), and UT = 0.85 m s−1. Delaware Bay has H = 14 m, and UT = 0.8 m s−1. The area A is 0.32 km2 over the 20 km nearest the mouth and decays exponentially up-estuary from that, with a length scale of 25 km, to a minimum of A = 104 m2. Extensive experimentation with different mixing parameterization coefficients gave rise to the values given in the text above, but one cannot be certain that these are optimum choices.

These systems span a wide range of coastal plain estuaries, from wide to shallow and from well mixed to highly stratified, depending on QR. The model captures the basic response of L to QR in two of the three cases. The model differs from that derived in MacCready (2004) in that it incorporates time dependence and the above mixing parameterizations. I proceed, assuming that the model gives a reasonable approximation of estuarine dynamics over a wide range of forcing. The purpose of this model is to explore estuarine adjustment over a wide range of parameter space, and so its simplicity and rapid numerical integration are real benefits. Specific estuaries are best left to detailed 3D models, as epitomized by the recent work in the Hudson of Warner et al. (2005).

4. Linear adjustment time theory

Before using the model to explore responses to unsteady forcing, it is useful to develop a theoretical expectation. The derivation here is in some ways similar to that in MacCready (1999) but is substantially simplified by expressing the adjustment time in terms of the hydraulic replacement time L/u, as suggested by Kranenburg (1986), Bowen (2000), and Hetland and Geyer (2004). It differs from Kranenburg (1986) in that the separate contributions of the exchange flow versus tidal stirring are made explicit, as are the physics of the response to different types of forcing. In addition, the consequences of the adjustment time for estuarine sensitivity are explored.

The along-channel integral of the salt balance (2.5) gives
i1520-0485-37-8-2133-e41
where x = R is some position up-estuary at which s = 0, and x = 0 at the mouth. In deriving (4.1) it has been assumed that A is equal to a constant and that the exchange flow is dominated by the LE3 term (Monismith et al. 2002; MacCready 2004). Next, by assuming that the salinity structure may be described very approximately as decreasing linearly from socn to zero over the distance L, one may scale terms in (4.1) as
i1520-0485-37-8-2133-e42
Substituting these into (4.1) gives
i1520-0485-37-8-2133-e43
thus the problem is reduced from a PDE in Σ(x, t) to a nonlinear ODE in L(t). It is also implicit in (4.3) that the values of LE3 and LH at x = 0 are representative of their values over L. One may express L as the sum of a mean state and some variation about that state: L(t) = L0 + L′(t). Then, for small variations |L′| ≪ L0, one may linearize (4.3) using the leading terms in the power series expansions of L−3 and L−1 to find an equation that is linear in L′:
i1520-0485-37-8-2133-e44
This may be forced in a variety of ways, allowing river flow, tidal mixing, or even the oceanic salinity to change. These may also be written in terms of small variation about a mean value. To first order in the perturbation terms, these are
i1520-0485-37-8-2133-e45
where KV = KV0 + KV, [KV is an approximation of the vertical mixing coefficients, given by KV = (KSK2M)1/3] and KH = KH0 + KH; also
i1520-0485-37-8-2133-e46
Substituting these into (4.4) and again retaining terms only to first order, it is found (dividing by L0) that
i1520-0485-37-8-2133-e47
In deriving (4.7) I have made use of the identity 1 = (LE30/L0)3 + LH0/L0, which may be shown from (4.4) by setting all primed quantities to zero. A dimensionless form of (4.7) may be derived by defining dimensionless quantities
i1520-0485-37-8-2133-e48
The quantity T0 is approximately the freshwater replacement time over the length of the salt intrusion. For the sake of obtaining a simple theoretical prediction it is assumed that tidal mixing is solely proportional to the amplitude of the tidal velocity, with UT = UT0 + U ′T, so that KV/KV0 = KH/KH0 = U ′T/UT0. Thus for the moment I am suspending the more complicated mixing parameterizations from the previous section. Under this assumption, and applying (4.8), one may write the equation for λ concisely. To allow comparison with numerical simulations (shown below) this is done in two limits. First, for exchange dominated flows (L3E30/L30 → 1; LH0/L0 → 0) the governing equation is
i1520-0485-37-8-2133-e49
In the tidal-stirring dominated limit (L3E30/L30 → 0; LH0/L0 → 1) the governing equation is
i1520-0485-37-8-2133-e410
These both have the form of a forced, damped, linear, first-order ODE, which may be written generally as
i1520-0485-37-8-2133-e411
where α and γ are numerical constants (±1 and 3), and F is the dimensionless forcing: u′/u0, Σ′, or U ′T/UT0. An equation of this form was used in Hetland and Geyer (2004) to analyze their numerical results. What has been done here is to demonstrate how such a form arises naturally from the estuarine system of equations. Monismith et al. (2002) also used an equation similar in form to (4.11), but with a nonunity exponent on the damping term.
If (4.11) is an adequate representation of the linearized estuarine adjustment problem, it gives a number of useful results. For step changes in forcing, (4.11) predicts exponential adjustment toward a new steady state in a time
i1520-0485-37-8-2133-e412
where both the dimensionless (τADJ) and dimensional (TADJ) forms of the adjustment time are given. MacCready (1999) argues that a full characterization of the adjustment problem also requires that one know the sensitivity, defined as
i1520-0485-37-8-2133-e413
MacCready (1999) gave theoretical expressions for Sens while considering changes in a system that had fully equilibrated to a step change in forcing. However, a more important aspect of the sensitivity arises in systems with continuously varying forcing. In mathematical terms, consider the case of periodic forcing F = F0 sin(Ωτ), where Ω = ωT0 is the dimensionless forcing frequency. Then the solution to (4.11) is
i1520-0485-37-8-2133-e414
A solution of similar form was given in Hetland and Geyer (2004). The term Ω/γ may also be written as ωTADJ so that it expresses the ratio of the adjustment time to the forcing period (times 2π). For very gradual forcing variation ωTADJ → 0, and so the response is quasi steady, given by λ = αF/γ. However, as the forcing variation becomes more rapid the response of λ lags the quasi-steady response and decreases in amplitude. For (4.14), the sensitivity is
i1520-0485-37-8-2133-e415
The theoretical results are summarized in Table 1.

There is also a more intuitive way to understand the theoretical adjustment times given in Table 1. MacCready (1999) noted that the adjustment process could be thought of in terms of an adjustment to small step changes in forcing. Then the adjustment time is given by the net change in salt of the system over the course of the adjustment, divided by the rate of addition/removal of salt at the start of the adjustment. In looking at Table 1, it is seen that the adjustment time in the exchange-dominated cases is given by (1/6)L0/u0, both for changes in river flow and changes in tidal mixing. This seems curious, given that the two forcing processes are so different. For either process a factor of ½ in the adjustment time results from the fact that the net salt in the system is proportional to one-half of the length of the salt intrusion. This leaves us to explain the remaining factor of ⅓. For the case of changing river flow, the salt intrusion length varies as u−1/3; therefore, in our linearized approach the net salt will vary as −(⅓)u. The initial rate of change of net salt is due to increased section-averaged advection at the mouth and so varies as −u. The ratio of these is ⅓, which is the desired result. In the case of varying tidal mixing, on the other hand, the salt intrusion length varies as K−1V, or −KV in the series expansion. The initial rate of change of net salt (after the stratification and exchange flow have adjusted to the new mixing levels, but before the length of the salt intrusion has adjusted) varies as K−3V, or −3KV in the series expansion. The ratio of these is again ⅓, but it is clear that the physics leading to this result is very different in the two cases.

The theoretical predictions may be tested using the numerical model developed in section 2. The model is run in two limits; in both cases I do not use the mixing parameterization of section 3 and instead set the mixing coefficients by hand (all constant in x but potentially functions of time). In the exchange-dominated limit, I set KH = 0, KM = 14 cm2 s−1, and KS = KM/3. For the tidal-stirring-dominated limit, I set KM = KS = 100 cm−1 s−1, effectively making the system well mixed in the vertical direction, and set KH = 200 m2 s−1. For both cases, the estuary is 10 m deep, 1000 m wide, and 400 km long (longer than the greatest salt intrusion in all cases). Eighteen simulations in each limit were performed, using combinations with average values of river flow given by QR = (50, 100, 200) m3 s−1 and mixing coefficients multiplied by factors of 0.5, 1, and 2. For each of these nine combinations, either QR or UT was varied periodically in time by ±10%, with a period of 50 days. Simulations lasted four forcing periods, by which time a nearly periodic solution was obtained. The salt intrusion length L was calculated for the final forcing period as 2∫Σ dx. The time averages of L and u were used to calculate T0, τ, Ω, L0, and λ. A least squares fit of λ to a function of the form λ = C1 sin(Ωτ) + C2 cos(Ωτ)gives estimates for C1 and C2. From these, one may make two direct comparisons between the model results and the theory, using the relations
i1520-0485-37-8-2133-e416
These are plotted in Fig. 4 for all simulations (35 out of 36) for which the calculated TADJ was less than 50 days. These spanned a range of L0 from 7 to 158 km. Overall, there is reasonable agreement between the analytical theory and numerical model despite the many simplifying assumptions used to derive the linear equation in (4.11). Most of the model adjustment times are within a factor of 2 of the theory, with the theory always underpredicting exchange-dominated cases, as was seen also in Hetland and Geyer (2004). The magnitude of the model sensitivity is generally less than that of the theory. The model adjustment times span from 1.6 to 45 days, and the factor (1 + Ω2/γ2)−1/2 from Sens goes from 0.98 to 0.17. In physical terms, estuaries in which the adjustment time is comparable to the forcing period will exhibit much-decreased sensitivity. This is because the length of the salt intrusion does not have time to equilibrate with the forcing. The simulations done here range from nearly equilibrated to very nonequilibrated.

5. Consequences of unsteadiness

In running the complete model (with the full mixing parameterizations from section 3) for San Francisco Bay, Delaware Bay, and the Hudson River estuary, the adjustment times given in Table 2 are found. Adjustment times listed there are calculated using the model configurations from Fig. 2, and varying QR by ±10% about either 100 or 1000 m3 s−1, with forcing periods of 50 or 100 days. The longer forcing periods are required to obtain reliable estimates for cases with longer adjustment times. Results for varying UT were similar, although in some cases (especially Delaware Bay) no TADJ could be calculated, perhaps because of the competing tendencies of the exchange flow and tidal stirring. The adjustment times vary from T0 to T0/6, whereas the theory from section 4 implies a range from T0 to T0/3. Another trend is that all of the high-flow cases are a greater fraction of T0 than are the low-flow cases, contrary to what one might expect given that the high-flow cases tend to be more exchange dominated. Thus, although the theory may be a reasonable guide to adjustment times, it can also be off by as much as a factor of 3, and detailed observations and modeling are clearly required for specific estuaries. The San Francisco Bay adjustment times in Table 2 bracket the observational estimate of 12 days given in Monismith et al. (2002), although they found no significant trend of the adjustment time relative to QR. The long adjustment times in Delaware Bay suggest that it may be significantly out of equilibrium much of the time, similar to the results of Banas et al. (2004) for Willapa Bay.

Vertical stratification and the exchange flow are more strongly affected by changes in mixing when the length of the estuary is not in equilibrium (Park and Kuo 1996; MacCready 1999; Hetland and Geyer 2004). This may be shown from the linear theory of the previous section. For uuE, the dimensionless stratification is
i1520-0485-37-8-2133-e51
and uE = c2ΣxH2/(48KM); therefore,
i1520-0485-37-8-2133-e52
Expanding KS, KM, and L in terms of small variations about a mean value, one may express Φ to leading order as
i1520-0485-37-8-2133-e53
For the case of an exchange-dominated estuary forced by varying UT, (4.9) becomes λτ = −3U ′T/UT0 − 3λ, which may be substituted into (5.3) to give
i1520-0485-37-8-2133-e54
This predicts that Φ can grow much larger or smaller than its equilibrium value when |λτ| is not small relative to 1. From (4.14) one may show that the dependence of |λτ| on ωTADJ is (ωTADJ)/[1 + (ωTADJ)2]1/2, which varies between 0 and 1, and so one would expect greater Φ variation when λ is out of equilibrium.

The interplay among adjustment time, salt intrusion length, and stratification in a more realistic example may be illustrated using the numerical model. The Hudson River case is forced with changing river flow and a typical spring–neap variation of UT (Fig. 5). This has the full mixing parameterizations derived in section 3. Also plotted is the quasi-steady solution, which is calculated by allowing the estuary to equilibrate with the instantaneous values of QR and UT at each time step of the integration. For low-flow conditions, ωTADJ = 2π(31 days)/(14 days) = 14, implying that the sensitivity of L is reduced to 7% of its quasi-steady value. During high flow, on the other hand, it is only reduced to 60%. In the converse situation, the relative sensitivity of stratification is greater during low flow. The adjustment time is also apparent in the lag of L relative to the steps in QR, with the step from high to low flow being slower (Kranenburg 1986). For this system, it is apparent that the quasi-steady solution may stray significantly from the time-dependent solution.

6. Summary and conclusions

Starting from the same classical equations used by Hansen and Rattray (1965), Chatwin (1976), and others, I have developed a model of tidally averaged, width-averaged estuarine circulation and salinity. The model improves on its predecessors by allowing variation of channel depth and width and allowing time-dependent forcing. I have also made an attempt to develop plausible parameterizations of the effective eddy viscosity, diffusivity, and tidal stirring coefficients (KM, KS, and KH), building on work by Geyer et al. (2000), MacCready (1999, 2004), Peters (1999), and Banas et al. (2004). An important step in these parameterizations is that they depend only on the channel geometry, the tidal current amplitude (UT, a known forcing parameter), and the tidally averaged stratification (a state variable). By doing this, spring–neap variation in UT can sensibly affect the mixing coefficients. Also, the mixing coefficients are no longer treated as fitting parameters, as was done in Hansen and Rattray (1965) (although I have certainly engaged in some fitting in choosing the unknown constants in the three mixing coefficients, as well as in choosing their functional forms!). The resulting model did a reasonable job of reproducing the curves for salt intrusion versus QR for the Hudson River estuary and north San Francisco Bay. It was less successful in the Delaware Bay, predicting a salt intrusion that is too long, especially during low flow.

The model appears to be able to work for systems ranging from well mixed to strongly stratified (see Fig. 5), but it cannot create a salt wedge. This is because the x momentum equation has been linearized and sx is assumed to be constant in z. The vertical structure functions (cubic and quintic profiles for velocity and salinity) rely on the assumptions that 1) channel geometry varies slowly along the channel, 2) sx is constant in z, and 3) the mixing coefficients are constant in z. As a consequence, the model is completely inappropriate for fjordlike systems such as the Puget Sound. A final important caveat about model limitations is that I have not addressed channel cross-sectional shape in any sensible way, and this has been suggested as an important consideration (Fischer 1972; Smith 1980).

The time-dependent model equations were linearized and scaled down to the point at which the evolution of the salt intrusion could be represented by a dimensionless, first-order, linear ODE with forcing and damping. This equation offers, it is hoped, a much clearer view of the system response characteristics than those given earlier by Kranenburg (1986) and MacCready (1999). An equation of the same form was used by Monismith et al. (2002) and Hetland and Geyer (2004). The contribution of this paper is to justify rigorously why an equation of this form is pertinent and to predict what its coefficients should be. Results that follow from this equation have, to some extent, appeared in the previous literature. In this paper, I am able to offer explanations for some of Kranenburg’s (1986) adjustment time results and give theoretical support for Kranenburg’s assertion that the adjustment time should be the same for changes in both QR and UT, despite the strongly different sensitivity to these forcing terms.

Last, the potential utility of the numerical model is demonstrated in section 5, in which it is used to simulate the Hudson River response to order-of-magnitude changes in QR superimposed on the spring–neap cycle of UT. Although the model has significant limitations in comparison with the remarkable 3D simulations of Warner et al. (2005), it is fast to run (about 2 min for the simulation in Fig. 5) and is very easy to reconfigure.

Acknowledgments

This work was generously supported by NSF Grant OCE-0136760 and NOAA Grant NA04NOS4730156. I am indebted to Rocky Geyer, James Lerczak, Rob Hetland, Neil Banas, and Stephen Monismith for many discussions of these ideas.

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Fig. 1.
Fig. 1.

The ratio of the vertical eddy diffusivity to the vertical eddy viscosity, vs the layer Richardson number, calculated from the parameterizations (3.3) and (3.9), with two values of the parameter A3 (solid and dashed lines). The same quantity calculated from the Hudson River observations of Peters (1999) is plotted as stars. The parameterization (3.9) with A3 = 0 is designed to fit the observations, except for one instance, which is highlighted as a filled star.

Citation: Journal of Physical Oceanography 37, 8; 10.1175/JPO3082.1

Fig. 2.
Fig. 2.

Model quasi-steady salt intrusion vs QR, for three estuaries, as labeled. The salt intrusion is defined by a different salinity in each case: e.g., X13.5 means the salt intrusion defined by s = 13.5. Observed curves are also plotted, as is a curve proportional to Q−1/3R for reference.

Citation: Journal of Physical Oceanography 37, 8; 10.1175/JPO3082.1

Fig. 3.
Fig. 3.

Quasi-steady model solution for the Hudson River, under moderate-flow conditions and average tides. (a) The dimensionless salinity Σ vs x is plotted as a thick line, with the stratification shown as a gray band around it. The surface velocity and ν (the diffusive fraction of up-estuary salt flux) are also shown. (b) The salinity vs x and z. (c) Distributions of the parameterized diffusivities. (d) The overturning streamfunction normalized by QR, showing the exchange flow (the sense of the circulation is generally clockwise on the page).

Citation: Journal of Physical Oceanography 37, 8; 10.1175/JPO3082.1

Fig. 4.
Fig. 4.

(a) Adjustment times and (b) sensitivities for 35 numerical simulations. Theoretical values (from Table 1) are plotted against numerical model results. Both exchange-dominated and tidal-stirring-dominated cases are considered. In either case, the forcing is due to variation of either river flow or tidal mixing, with a period of 50 days. Cases with the longest adjustment times correspond to those with the smallest sensitivities.

Citation: Journal of Physical Oceanography 37, 8; 10.1175/JPO3082.1

Fig. 5.
Fig. 5.

Time-dependent numerical simulation of the Hudson River estuary. (a) The forcing variation, which consists of a spring–neap cycle of tidal amplitude and a switch from low river flow to high and back again. (b) The response of the model stratification with time, where the parameter Φ is the normalized stratification averaged over the length of the salt intrusion. (c) The length of the salt intrusion vs time. In (b) and (c), a quasi-steady model solution is also plotted to indicate how important unsteady effects might be. (right) Salinity sections at three times. These are from low-flow neap, low-flow spring, and high-flow periods. During low-flow periods, the stratification is relatively sensitive to the spring–neap cycle and the salt intrusion is not, in comparison with the quasi-steady solution. This result is because the salt intrusion does not have time to adjust.

Citation: Journal of Physical Oceanography 37, 8; 10.1175/JPO3082.1

Table 1.

Theoretical predictions for the adjustment time and sensitivity, given in dimensional format.

Table 1.
Table 2.

Adjustment times and freshwater replacement times calculated for three realistic cases using the numerical model, with varying river flow. For each case, both high- and low-flow situations are considered. The analytical theory predicts, for exchange-dominated cases, that TADJ = (⅓)T0, whereas these results have a range of TADJ = (1/6–1)T0.

Table 2.
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