A Simple Method for Estimating V_x from Mixing Equations in a 1-Dimensional, Steady-State System for LOICZ Biogeochemical Modeling

Tetsuo Yanagi

For LOICZ biogeochemical modeling, it is important to estimate the mixing volume (V_X, in m³ d^-1) across the open boundary of the system (e.g., Gordon et al., 1996). Clearly this open-boundary transport is required to balance salt transport in estuaries. Less obviously, it can be an important source or sink for nutrient transport between estuaries and the coastal ocean. In the procedure recommended by the LOICZ guidelines, V_X is estimated from the water and salt budgets, where V_R is the residual volume transport associated with freshwater discharge, S_syst and S_ocn are the system and adjacent ocean salinity values, respectively; and S_R is the average of the ocean and system salinity:

(1)

Positive flux is into the system. Obviously this equation can be solved only if there is a quantifiable salinity difference between the system and the ocean, yet some systems lack a salinity gradient. V_X is still an important variable.

We offer an alternative way to estimate the mixing volume V_X without relying on a salinity difference between the system and the ocean. The water mixing across the open boundary of the system is governed by the dispersion process (Yanagi, 2000), and the magnitude of the horizontal dispersion coefficient D_H (m² s^-1) is estimated from the current shear and the diffusivity normal to the current shear by the following equations (from Taylor, 1953):

a) In the case of dominant vertical shear (narrow and deep estuarine system):

(2)

b) In the case of dominant horizontal shear (wide and shallow estuarine system):

(3)

where H (in m) is the average depth of the open boundary of the system; W (in m) is the length of the open boundary, that is the width of the system mouth, in m; U ( in m d^-1) is the residual flow velocity at the surface layer of the open boundary (if this value is not independently known, it typically has a numerical value of about 0.1 m s^-1 or 8,640 m d^-1 (Yanagi, 2000b); K_V is the vertical diffusivity (typically ~ 10^-4 m² s^-1 or 8.64 m² d^-1 in the case of stratification and 10^-3 m² s^-1 or 86.4 m² d^-1 in the case of vertically well-mixed systems; we assume no stratification for the calculations here); and K_h is the horizontal diffusivity (also in m² s^-1 or m² d^-1; varies as a function of mixing scale [see below] ).

The following criteria can be used to decide if a system should be treated as "narrow and deep" or "wide and shallow." The distance from the center of the system to its mouth is denoted L (m). A system is considered to be "narrow and deep" if L/W > 2 and W/H <500. A system is considered "wide and shallow" if L/W <2 and W/H >500.

The LOICZ notation is more readily followed if U is rescaled to m d^-1 (if not known it can be approximated as 10⁵); and K_v is re-scaled to m² d^-1 (~ 10 in a vertically well-mixed system). D_H (in m² d^-1) according to equation (2) is then approximated in a narrow, deep estuarine system:

(2a)

To retain proper dimensionality, the coefficient 1,000 has the units m² d^-1, because this coefficient includes the estimated value for K_v. Note that, if K_v is explicitly known, equation (2a) should be modified accordingly.

Okubo (1971) gives data demonstrating the validity of a well-recognized relationship between the horizontal diffusion coefficient (K_h, in cm² s^-1) the horizontal scale of the diffusion (l, in cm). The equation he offers to summarize that relationship is K_h = 0.0103 l^1.15. It is assumed here that the diffusion scale for K_h in an estuary or embayment is given by the length of the open boundary (i.e., width of the system mouth, W). Expressing K_h in m² d^-1 and W (l) in m, Okubo’s equation becomes:

(4)

Equations (3) and (4) can be combined for use with the LOICZ notation, expressing D_H in m² d^-1 in a wide, shallow estuarine system:

(3a)

D_H has the typical dimensions of diffusion (area/time), whereas the LOICZ notation is expressed as a volume exchange rate (volume/time). D_H derived from either equation (2a) or (3a) can then be used to approximate the mixing volume (V_X, in m³ d^-1) as used in the LOICZ notation:

(5)

where A denotes the cross sectional area of the open boundary of the system (m²) and F is the distance (m) between the geographic center of the system and the observation point for oceanic salinity (typically near the mouth of the system).

As detailed examples of the calculations, including explicit definition of U, D_H, and the K’s, we apply these calculations to two systems in Japan: a narrow and deep bay (Dokai Bay) and a wide, shallow bay (Hakata Bay) (Table 1). In the case of Dokai Bay, the salt balance method and the mixing equation method of estimating V_X agree within about 5%. In the case of Hakata Bay, the agreement is about a factor of two. Further examples should be explored to evaluate the general agreement between these two methods.

Table 1. Sample calculations. U, the K’s, and D_H are reported both in notation commonly given in the oceanographic literature and in the notation used here, in order to facilitate both comparison with that literature and calculations reported here.

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Last Updated 12 May 2009 by DPS