## 1. Introduction

Since the introduction of the stratification–circulation diagram by Hansen and Rattray (1966), numerous estuarine classification schemes have been proposed. The reader might ask—why revisit this topic? Our motivation for pursuing a new classification scheme stems from notable recent advances in estuarine physics, many of which are reviewed in MacCready and Geyer (2010). These advances led us to hypothesize that there might be a simple means to determine the conditions under which a sufficiently well behaved estuary will be well mixed, partially mixed, or highly stratified. We start by outlining the classical tidally averaged model as presented by MacCready and Geyer (2010). We then rewrite the equations of this model in nondimensional form. Using this new set of equations we develop our classification scheme, and then compare its predictions with field observations.

## 2. Classical tidally averaged model

The physics of estuarine circulation is governed by the competing influences of river and oceanic flows. While the former adds freshwater, the latter adds denser saltwater, which moves landward because of the combined effect of tides and gravitational circulation (or exchange flow). The complicated balance between the river, the exchange flow, and the tides determines the estuarine velocity and salinity structure.

*H*and width

*B*. The origin of the coordinate system is at the free surface at the mouth of the estuary with the horizontal (i.e.,

*x*) axis pointing seawards and the vertical (i.e.,

*z*) axis pointing upward. Therefore, both the horizontal and vertical distances within the estuary are negative quantities. To obtain the width- and tidally averaged horizontal velocity

*u*, and salinity

*s*distribution in the estuary, these quantities are first decomposed into depth-averaged (overbar) and depth-varying (prime) components:

*A*=

*BH*. The solution for both partially and well-mixed estuaries was given by Hansen and Rattray (1965) [for a recent review, see MacCready and Geyer (2010)]:

*ξ*=

*z*/

*H*∈ [−1, 0] is the normalized vertical coordinate. The subscript

*x*implies ∂/∂

*x*, where

*x*is dimensional. The vertical eddy diffusivity is

*β*≅ 7.7 × 10

^{−4}psu

^{−1}. The nondimensional salinity is defined as Σ =

*s*/

*s*

_{ocn}, where

*s*

_{ocn}is the ocean salinity. Equations (1) and (2) were derived under the assumption that the density field is governed by the linear equation of state:

*ρ*=

*ρ*

_{0}(1 +

*βs*), where

*ρ*

_{0}is the density of freshwater. The details of the derivation are well documented in MacCready (1999, 2004).

*R*is the river term;

*T*is the tidal term; and

*E*

_{1},

*E*

_{2}, and

*E*

_{3}are the different components of the exchange term. Hansen and Rattray (1965) presented Eq. (6) in a slightly different form, and MacCready (2004, 2007) introduced the length scales in Eq. (7).

*a*

_{1}= 0.035, and

*u*is the amplitude of the depth-averaged tidal velocity. Based on field studies and modeling of the Hudson River estuary, Ralston et al. (2008) obtained

_{T}*a*

_{0}= 0.028,

*C*= 0.0026, and Sc = 2.2 is a Schmidt number. We will use Eqs. (8) and (9) in the development of a nondimensional set of equations.

_{D}*T*→ 0) and the tidally dominated case (

*E*

_{3}→ 0). While these approximations have been widely used, there does not appear to have been any serious attempt to determine the conditions under which they are applicable.

## 3. Nondimensional tidally averaged model

*X*=

*u*have been nondimensionalized by

_{T}*c*to obtain the densimetric estuarine Froude number

*u*/

_{T}*c*. Substituting Eq. (12) into (11) yields

*C*

_{1}= 2.17,

*C*

_{2}= 1.34,

*C*

_{3}= 8.16 × 10

^{−5}, and the modified tidal Froude number

*C*

_{4}= 0.667,

*C*

_{5}= 47.0, and

*C*

_{6}= 70.5. In Eq. (14), the quantity

*U*=

*u*/

*c*is the nondimensional horizontal velocity (not to be confused with

*F*, which is

_{R}*C*

_{7}= 5.31 and

*C*

_{8}= 6.04. Equation (16) is actually the nondimensional version of Eq. (19) of MacCready (2004). Being a cubic equation, it can be solved analytically to evaluate the salinity gradient at the estuary mouth

## 4. Estuary classification

*C*

_{9}= 7.06 and

*C*

_{10}= 8.82. If

Estuary classification diagram with lines representing isocontours of

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0129.1

Estuary classification diagram with lines representing isocontours of

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0129.1

Estuary classification diagram with lines representing isocontours of

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0129.1

We follow Hansen and Rattray (1966) and use the condition

## 5. Discussion

Together, Eqs. (16) and (18) provide new insight into estuarine physics. Apart from broadly classifying estuaries into three categories, namely highly stratified, partially mixed, and well mixed, these equations reveal that under the given assumptions just two parameters, *F _{T}*, but the latter combined with the square root of the estuarine aspect ratio

*B*/

*H*. Moreover the equation set predicts

To test the applicability of our classification scheme we made use of the field data presented in Prandle (1985). Using these data we have computed *B*/*H*, and

Estimates of estuarine parameters calculated using the data of Prandle (1985) and values of *B* obtained from maps. Equation (18) is used to obtain

Comparison of stratification at the estuary mouth obtained from theory or from field data.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0129.1

Comparison of stratification at the estuary mouth obtained from theory or from field data.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0129.1

Comparison of stratification at the estuary mouth obtained from theory or from field data.

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0129.1

Comparison of the estuary classification scheme with the approximation

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0129.1

Comparison of the estuary classification scheme with the approximation

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0129.1

Comparison of the estuary classification scheme with the approximation

Citation: Journal of Physical Oceanography 43, 8; 10.1175/JPO-D-12-0129.1

We also compare the theoretical results with the field data of Prandle (1985) in Table 1 and Fig. 2.6 of Geyer (2010) in Fig. 4. We have chosen to plot Eq. (18) for

A plot similar to Fig. 4 is presented in Geyer (2010, Fig. 2.6). In this figure, a line [labeled Eq. (2.22)] corresponding to

Finally, we refer to the assumptions behind our theoretical analyses and their consequences. We have simplified the problem by assuming a tidally averaged estuary with rectangular geometry. In real estuaries, bathymetry can play a crucial role in determining estuarine circulation. Moreover, the appearance of just two parameters (*C*_{1}, *C*_{2}, … , *C*_{10}, given in Table 2. All these coefficients depend upon the Schmidt number (i.e., Sc), which is an empirical quantity that could conceivably vary from estuary to estuary, within a given estuary, or with time. Although Eqs. (8) and (9) are simple and elegant, they may not be very realistic. In real estuaries, both

List of coefficients used in different equations.

## 6. Conclusions

The equations governing the physics of estuarine circulation have been presented in nondimensional form. The two resulting nondimensional parameters are the estuarine Froude number

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