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Ecological and Hydrologic Targets for Western Biscayne National Park 25
APPENDIX A: ADVECTION VERSUS
DIFFUSION
arrived at by a conservative evaluation of these factors and
assuming an average diffusivity in the Western Bay Zone of 1
m2/s.
A one-dimensional flow of water and salt in the x direction
can be expressed in a steady-state, vertically-mixed form as:
where S is the salt content, U is the horizontal velocity, and
A is the horizontal turbulent diffusion coefficient. If one assumes that U is independent of the distance x from the coast
(which is a very reasonable assumption for a flow distributed
all along a coastline, and an unreasonable assumption for a
point source flow), and that A is likewise independent of x (a
poor but pragmatic choice) then:
Given the assumptions, the analytical solution is exponential. The importance of this solution is that, in the absence of
other transient forcing, a steady flow offshore gives a persistent exponential gradient located near the coast. As the speed
of the flow increases, this gradient will move farther offshore
and will become sharper (larger magnitude). As the mixing
becomes more intense or efficient (i.e., the magnitude of A increases) the gradient will move closer to shore and the gradient’s magnitude will decrease. The ratio of A/U is the length,
or e-folding scale, and, as such, is a good estimate of the width
of the offshore gradient region. While the velocity U along a
coastline can be determined by metering out a known volume
of water at a known rate along a length of shoreline, the horizontal turbulent diffusion coefficient A is not as simple and
is often several orders of magnitude greater than equivalent
molecular diffusivities. It is a function of the flow and resulting friction in the area and, as such, will be dependent on the
tides, winds, and topography, and can vary by several orders
of magnitude.
The advection dispersion estimate provided on page 21 is
derived from a horizontal diffusivity of A = 1 m2/s and steady
offshore velocity U= 0.001 m/s for 26 km of coastline with an
average depth of 1 m. The value of the diffusivity A has been
shown by Wang et al. (1978) to vary from 0.5 m2/s to 5 m2/s
along the western shoreline, producing a theoretical range of