## 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 *Q _{R}* (Abood 1974; Monismith et al. 2002). These curves typically result in a power law of the form

*L*∝

*Q*

^{n}

_{R}. 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

*T*

_{ADJ}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

*T*

_{ADJ}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.

*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:

*≡*u

*Q*/

_{R}*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

s

_{x}, 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):whereThe magnitude of the exchange flow is

*u*

_{E}=

*g*

*β*

s

_{x}

*H*

^{3}/(48

*K*

_{M}),

*ζ*≡

*z*/

*H*, and

*H*is the channel depth. Also,

*K*and

_{M}*K*are the vertical eddy viscosity and diffusivity. The term

_{S}*β*≅ 7.7 × 10

^{−4}psu

^{−1}appears in the linearized equation of state

*ρ*=

*ρ*

_{0}(1 +

*βs*), where

*ρ*is the density and

*ρ*

_{0}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)]:Here

*K*is an along-channel eddy diffusivity due to tidal correlations of velocity, salinity, and area. Nondimensionalizing the salinity as Σ ≡

_{H}*/*s

*s*

_{ocn}, where

*s*

_{ocn}is the oceanic salinity, (2.3) may be rewritten aswhere 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 Σ

*, allowing (2.4) to be rewritten asI have introduced the length scales:Of the four length scales,*

_{x}*L*

_{E}_{3}is typically the most important for partially mixed systems, whereas

*L*dominates in well-mixed cases. The parameter

_{H}*c*≡ (

*g*Δ

*ρH*/

*ρ*

_{0})

^{1/2}is like a maximum internal wave speed, where Δ

*ρ*is the density difference between river and ocean waters. These length scales are identical to those given in MacCready (2004) except that here

*K*and

_{S}*K*are allowed to vary separately.

_{M}*s*

_{ocn}. This may be written, using expression (2.1) for

*s*′, asThis is a quadratic that may be solved analytically for Σ

_{0}. Here Σ

_{0}is Σ at the mouth (

*x*= 0), and Σ

_{1}is Σ one grid point up-estuary (

*x*= −Δ

*x*). Also

*T*

_{E}_{2}=

*H*

^{2}/(

*K*)

_{S}K_{M}^{1/2}and

*T*

_{E}_{1}=

*H*

^{2}/

*K*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 Δ

_{S}*x*is chosen so that the length of the salt intrusion is well resolved. In a typical case, Δ

*x*= 1 km is used.

*t*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 aswhere I have written the whole term in brackets in (2.5) as an effective diffusivity

*K*

_{eff}, thereby casting (2.5) as a heat equation. Then numerical stability requires (Press et al. 1992, p. 847) thatwhere

*K*

^{max}

_{eff}is the maximum value of

*K*

_{eff}along the channel at the previous time step. Stable simulations are reliably achieved using Δ

*t*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 *K _{M}*,

*K*, and

_{S}*K*;

_{H}*K*and

_{M}*K*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 (

_{S}*H*,

*B*,

*U*) and state variables—in particular, the tidally averaged stratification Φ;

_{T}*B*is the channel width,

*U*is the amplitude of the depth-averaged tidal velocity, and Φ ≡ (

_{T}*s*

_{bottom}−

*s*

_{top})/

*s*

_{ocn}. 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.

*K*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 bywhere

_{M}*R*= 2

*C*

_{D}_{3}

*U*

_{T}_{3rms}. Here Δ

*u*is the difference between outflowing and inflowing velocities averaged over upper- and lower-layer thicknesses

*h*

_{1}and

*h*

_{2},

*C*

_{D}_{3}= 3.1 × 10

^{−3}is a drag coefficient referenced to 3 m above the bottom, and

*U*

_{T}_{3rms}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

*K*constant with depth, cannot reproduce the stress profile found by Geyer et al. (2000), because our

_{M}*K*∂

_{M}*u*/∂

*z*will always have nonnegligible stress at middepth. However, I can attempt to ensure that the model gives the correct relation between

s

_{x}and the exchange flow. The cubic profile (2.1) of

*u*has Δ

*u*≅ 1.1

*u*(for

_{E}*≪*u

*u*). This may be combined with (3.1) from Geyer et al. (2000) (assuming

_{E}*h*

_{1}=

*h*

_{2}) and solved for

*K*to findThen, accounting for the differences in velocity and drag coefficient of depth-averaged tidal current amplitude

_{M}*U*versus the 3-m rms tidal velocity

_{T}*U*

_{T}_{3rms}, the final expression for

*K*is found to bewhere

_{M}*A*

_{0}= 0.065 and

*C*= 2.6 × 10

_{D}^{−3}. However, in the comparisons with observations (shown below) it is found that a reasonable fit required

*A*

_{0}= 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

*A*

_{0}. 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

*K*by stratification is significant.

_{M}*s̃*

_{z}〉 =

*s*′

_{z}in the notation of section 2. In physical terms,

*K*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)where

_{S}*U*

_{T}sin(

*ω*

_{T}

*t*). It is further assumed that this expression scales with tidally averaged properties, givingIn solving for

*K*, I find thatwhere Ri

_{S}*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*

_{L}*U*:Note that

_{T}*c*and

*U*are known environmental parameters, whereas Φ is a state variable. Expression (3.7) decreases with Ri

_{T}*, an expected result of increasing stratification; however,*

_{L}*R*is an unknown function of Ri

_{f}*. It is proposed that*

_{L}*K*may be parameterized using a function of the following form:with

_{S}*A*

_{1},

*A*

_{2}, and

*A*

_{3}being empirically derived constants. The maximum value of

*K*is

_{S}*A*

_{1}

*C*, much like

_{D}U_{T}H*K*in (3.3), and so it is expected that

_{M}*A*

_{1}≤

*A*

_{0}. The rate of decrease of

*K*with increasing Ri

_{S}*is governed by*

_{L}*A*

_{2}, analogous to the parameterization in Munk and Anderson (1948). The minimum value of

*K*as Ri

_{S}*→ ∞ is given by*

_{L}*A*

_{3}times its maximum value (0 ≤

*A*

_{3}≤ 1);

*A*

_{3}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

*A*

_{3}implies that, even with strong midchannel stratification, turbulent buoyancy flux cannot be entirely shut down on the edges.

To determine *A*_{1} and *A*_{2}, 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 *A*_{3} = 0 and will then use other criteria later to estimate *A*_{3}. 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 *U _{T}* (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(

*s*

_{bottom}−

*s*

_{stop})/

*H*, allowing us to express Peters’s observations in terms of the variables in (3.9), with

*A*

_{3}= 0. The results of this comparison are plotted in Fig. 1. A reasonable fit between prediction and observations is found using

*A*

_{1}= 0.022 and

*A*

_{2}= 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

*K*, it is assumed that

_{S}*A*

_{3}= 0.3, the reasons for which are discussed at the end of this section.

*K*, the along-channel mixing term. Banas et al. (2004) found a good fit for observations in Willapa Bay using

_{H}*K*= 0.035

_{H}*U*, 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

_{T}B*L*≡

_{T}*U*/

_{T}T_{T}*π*(where

*T*is the tidal period) is smaller than

_{T}*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

*K*, which decreased from a theoretically derived maximum value near the mouth

_{H}*K*, to a smaller value landward. Based on 3D numerical experiments (not shown here) it is assumed that this portion of

_{HM}*K*near the mouth decays linearly from

_{H}*K*to zero over

_{HM}*L*, and that

_{T}*K*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 usewhere

_{HM}*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 byA reasonable fit to observations (below) is found with ε = 0.1, and thus

*K*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.

_{HM}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 *Q _{R}* 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

*Q*= 300 m

_{R}^{3}s

^{−1}and

*U*= 0.9 m s

_{T}^{−1}. Northern San Francisco Bay has variable depth and width, as described in MacCready (2004), and

*U*= 0.85 m s

_{T}^{−1}. Delaware Bay has

*H*= 14 m, and

*U*= 0.8 m s

_{T}^{−1}. The area

*A*is 0.32 km

^{2}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*= 10

^{4}m

^{2}. 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 *Q _{R}*. The model captures the basic response of

*L*to

*Q*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).

_{R}## 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.

*x*=

*R*is some position up-estuary at which

*= 0, and*s

*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

*L*

_{E}_{3}term (Monismith et al. 2002; MacCready 2004). Next, by assuming that the salinity structure may be described very approximately as decreasing linearly from

*s*

_{ocn}to zero over the distance

*L*, one may scale terms in (4.1) asSubstituting these into (4.1) givesthus 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

*L*

_{E}_{3}and

*L*at

_{H}*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*) =

*L*

_{0}+

*L*′(

*t*). Then, for small variations |

*L*′| ≪

*L*

_{0}, 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*′: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 arewhere

*K*

_{V}=

*K*

_{V0}+

*K*′

_{V}, [

*K*is an approximation of the vertical mixing coefficients, given by

_{V}*K*

_{V}= (

*K*

_{S}

*K*

^{2}

_{M})

^{1/3}] and

*K*

_{H}=

*K*

_{H0}+

*K*′

_{H}; alsoSubstituting these into (4.4) and again retaining terms only to first order, it is found (dividing by

*L*

_{0}) thatIn deriving (4.7) I have made use of the identity 1 = (

*L*

_{E}_{30}/

*L*

_{0})

^{3}+

*L*

_{H}_{0}/

*L*

_{0}, 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 quantitiesThe quantity

*T*

_{0}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

*U*

_{T}=

*U*

_{T0}+

*U*′

_{T}, so that

*K*′

_{V}/

*K*

_{V0}=

*K*′

_{H}/

*K*

_{H0}=

*U*′

_{T}/

*U*

_{T0}. 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 (

*L*

^{3}

_{E30}/

*L*

^{3}

_{0}→ 1;

*L*

_{H}_{0}/

*L*

_{0}→ 0) the governing equation isIn the tidal-stirring dominated limit (

*L*

^{3}

_{E30}/

*L*

^{3}

_{0}→ 0;

*L*

_{H}_{0}/

*L*

_{0}→ 1) the governing equation isThese both have the form of a forced, damped, linear, first-order ODE, which may be written generally aswhere

*α*and

*γ*are numerical constants (±1 and 3), and

*F*is the dimensionless forcing:

*′/*u

u

_{0}, Σ′, or

*U*′

_{T}/

*U*

_{T0}. 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.

*τ*

_{ADJ}) and dimensional (

*T*

_{ADJ}) 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 asMacCready (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*=

*F*

_{0}sin(Ω

*τ*), where Ω =

*ωT*

_{0}is the dimensionless forcing frequency. Then the solution to (4.11) isA solution of similar form was given in Hetland and Geyer (2004). The term Ω/

*γ*may also be written as

*ωT*

_{ADJ}so that it expresses the ratio of the adjustment time to the forcing period (times 2

*π*). For very gradual forcing variation

*ωT*

_{ADJ}→ 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 isThe 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)*L*_{0}/_{0}, 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 ^{−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 −

*. 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*u

*K*

^{−1}

_{V}, or −

*K*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

_{V}*K*

^{−3}

_{V}, or −3

*K*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.

_{V}*x*but potentially functions of time). In the exchange-dominated limit, I set

*K*= 0,

_{H}*K*= 14 cm

_{M}^{2}s

^{−1}, and

*K*=

_{S}*K*/3. For the tidal-stirring-dominated limit, I set

_{M}*K*=

_{M}*K*= 100 cm

_{S}^{−1}s

^{−1}, effectively making the system well mixed in the vertical direction, and set

*K*= 200 m

_{H}^{2}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

*Q*= (50, 100, 200) m

_{R}^{3}s

^{−1}and mixing coefficients multiplied by factors of 0.5, 1, and 2. For each of these nine combinations, either

*Q*or

_{R}*U*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

_{T}*L*was calculated for the final forcing period as 2∫Σ

*dx*. The time averages of

*L*and

*were used to calculate*u

*T*

_{0},

*τ*, Ω,

*L*

_{0}, and

*λ*. A least squares fit of

*λ*to a function of the form

*λ*=

*C*

_{1}sin(Ω

*τ*) +

*C*

_{2}cos(Ω

*τ*)gives estimates for

*C*

_{1}and

*C*

_{2}. From these, one may make two direct comparisons between the model results and the theory, using the relationsThese are plotted in Fig. 4 for all simulations (35 out of 36) for which the calculated

*T*

_{ADJ}was less than 50 days. These spanned a range of

*L*

_{0}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 + Ω

^{2}/

*γ*

^{2})

^{−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 *Q _{R}* by ±10% about either 100 or 1000 m

^{3}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

*U*were similar, although in some cases (especially Delaware Bay) no

_{T}*T*

_{ADJ}could be calculated, perhaps because of the competing tendencies of the exchange flow and tidal stirring. The adjustment times vary from

*T*

_{0}to

*T*

_{0}/6, whereas the theory from section 4 implies a range from

*T*

_{0}to

*T*

_{0}/3. Another trend is that all of the high-flow cases are a greater fraction of

*T*

_{0}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

*Q*. 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.

_{R}*≪*u

*u*

_{E}, the dimensionless stratification isand

*u*=

_{E}*c*

^{2}Σ

_{x}H^{2}/(48

*K*); therefore,Expanding

_{M}*K*,

_{S}*K*, and

_{M}*L*in terms of small variations about a mean value, one may express Φ to leading order asFor the case of an exchange-dominated estuary forced by varying

*U*, (4.9) becomes

_{T}*λ*

_{τ}= −3

*U*′

_{T}/

*U*

_{T0}− 3

*λ*, which may be substituted into (5.3) to giveThis predicts that Φ can grow much larger or smaller than its equilibrium value when |

*λ*| is not small relative to 1. From (4.14) one may show that the dependence of |

_{τ}*λ*| on

_{τ}*ωT*

_{ADJ}is (

*ωT*

_{ADJ})/[1 + (

*ωT*

_{ADJ})

^{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 *U _{T}* (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

*Q*and

_{R}*U*at each time step of the integration. For low-flow conditions,

_{T}*ωT*

_{ADJ}= 2

*π*(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

*Q*, 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.

_{R}## 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 (*K _{M}*,

*K*, and

_{S}*K*), 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 (

_{H}*U*, a known forcing parameter), and the tidally averaged stratification (a state variable). By doing this, spring–neap variation in

_{T}*U*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

_{T}*Q*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.

_{R}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 _{x} 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) _{x} 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 *Q _{R}* and

*U*, despite the strongly different sensitivity to these forcing terms.

_{T}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 *Q _{R}* superimposed on the spring–neap cycle of

*U*. 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.

_{T}## 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.

## REFERENCES

Abood, K. A., 1974: Circulation in the Hudson River estuary.

*Hudson River Colloquium,*O. A. Roels, Ed., Annals of the New York Academy of Sciences, Vol. 250, New York Academy of Sciences, 38–111.Banas, N. S., , B. M. Hickey, , P. MacCready, , and J. A. Newton, 2004: Dynamics of Willapa Bay, Washington: A highly unsteady, partially mixed estuary.

,*J. Phys. Oceanogr.***34****,**2413–2427.Bowen, M. M., 2000: Mechanisms and variability of salt transport in partially-stratified estuaries. Ph.D. thesis, Massachusetts Institute of Technology/Woods Hole Oceanographic Institution, MIT/WHOI 00-04, 171 pp.

Chatwin, P. C., 1976: Some remarks on the maintenance of the salinity distribution in estuaries.

,*Estuarine Coastal Mar. Sci.***4****,**555–566.Dyer, K. R., 1997:

*Estuaries: A Physical Introduction*. 2d ed. John Wiley and Sons, 195 pp.Fischer, H. B., 1972: Mass transport mechanisms in partially stratified estuaries.

,*J. Fluid Mech.***53****,**671–687.Garvine, R. W., , R. K. McCarthy, , and K-C. Wong, 1992: The axial salinity distribution in the Delaware estuary and its weak response to river discharge.

,*Estuarine Coastal Shelf Sci.***35****,**157–165.Geyer, W. R., , J. H. Trowbridge, , and M. M. Bowen, 2000: The dynamics of a partially mixed estuary.

,*J. Phys. Oceanogr.***30****,**2035–2048.Hansen, D. V., , and M. Rattray, 1965: Gravitational circulation in straits and estuaries.

,*J. Mar. Res.***23****,**104–122.Hetland, R. D., , and W. R. Geyer, 2004: An idealized study of the structure of long, partially mixed estuaries.

,*J. Phys. Oceanogr.***34****,**2677–2691.Kranenburg, C., 1986: A time scale for long-term salt intrusion in well-mixed estuaries.

,*J. Phys. Oceanogr.***16****,**1329–1331.Lewis, R. E., 1996: Relative contributions of interfacial and bed generated mixing to the estuarine energy balance.

*Mixing in Estuaries and Coastal Seas,*C. Pattiaratchi, Ed., Coastal and Estuarine Studies, Vol. 50, Amer. Geophys. Union, 250–266.MacCready, P., 1999: Estuarine adjustment to changes in river flow and tidal mixing.

,*J. Phys. Oceanogr.***29****,**708–726.MacCready, P., 2004: Toward a unified theory of tidally-averaged estuarine salinity structure.

,*Estuaries***27****,**561–570.Monismith, S. G., , W. Kimmerer, , J. R. Burau, , and M. T. Stacey, 2002: Structure and flow-induced variability of the subtidal salinity field in northern San Francisco Bay.

,*J. Phys. Oceanogr.***32****,**3003–3018.Munk, W. H., , and F. R. Anderson, 1948: Notes on a theory of the thermocline.

,*J. Mar. Res.***7****,**276–295.Park, K., , and A. Y. Kuo, 1996: Effect of variation in vertical mixing on residual circulation on narrow, weakly nonlinear estuaries.

*Buoyancy Effects on Coastal and Estuarine Dynamics,*D. G. Aubrey and C. T. Friedrichs, Eds., Amer. Geophys. Union, 301–317.Peters, H., 1999: Spatial and temporal variability of turbulent mixing in an estuary.

,*J. Mar. Res.***57****,**805–845.Press, W. H., , S. A. Teukolsky, , W. T. Vetterling, , and B. P. Flannery, 1992:

*Numerical Recipes in C: The Art of Scientific Computing*. 2d ed. Cambridge University Press, 994 pp.Smith, R., 1980: Buoyancy effects upon longitudinal dispersion in wide well-mixed estuaries.

,*Philos. Trans. Roy. Soc. London***296A****,**467–496.Stommel, H., , and H. G. Farmer, 1952: On the nature of estuarine circulation, Part I. Woods Hole Oceanographic Institute Tech. Rep. 52-88, 131 pp.

Uncles, R. J., , and J. A. Stephens, 1990: The structure of vertical current profiles in a macrotidal, partially-mixed estuary.

,*Estuaries***13****,**349–361.Warner, J. C., , W. R. Geyer, , and J. A. Lerczak, 2005: Numerical modeling of an estuary: A comprehensive skill assessment.

,*J. Geophys. Res.***110****.**C05001, doi:10.1029/2004JC002691.

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

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 *T*_{ADJ} = (⅓)*T*_{0}, whereas these results have a range of *T*_{ADJ} = (1/6–1)*T*_{0}.