Douglas C. Speirs and William S. C. Gurney. 2001. Population persistence in rivers and estuaries. Ecology 82:1219-1237.
Appendix C. Flow in a tidal river.
We seek a solution of the equations
describing the dynamics of sea-surface elevation () and x-velocity ()
(C.1) |
Our chosen solution must satisfy
the following flow boundary conditions at the landward () and seaward () ends of the system
(C.2) |
and the conditions
of zero wind-stress at the surface and zero slip at the bottom,
i.e.
(C.3) |
The problem defined by Eqs. (C.1)
- (C.3) is completely linear, so we may expect the solution to be a superposition
of the flow generated by the river () and the flow generated by the tide
(). Since the river input is constant,
we expect the river generated flow to be steady (independent of ) and uniform (independent of ), so we look for a solution of the form
(C.4) |
If we associate a surface elevation
with the river flow
and with the tidal flow,
then the assumption that the river generated flow is steady and uniform implies
that
and
, where
is a constant. This is consistent with Eqs. (C.1) if
(C.5) |
and
(C.6) |
The general solution of Eq. (C.5)
is
, where and
are arbitrary constants. To obtain a zero gradient at (required by equation C.3) we set B=0. To ensure that in partial fulfilment of the
requirements of Eq. (C.2) we set . Finally we set
, thus matching the requirements of the second
element of Eq. (C.3). Our final solution is
(C.7) |
To solve Eq. (C.6)
we follow Chen et al. (1997) and assume that the solution
takes the general form
(C.8) |
Substituting this
form into Eqs. (C.6) yields
(C.9) |
where represents the average value of over the water column. Differentiating the first element
of Eq. (C.9) with respect to time and back substituting for
yields
(C.10) |
Now, as a trial,
suppose that
(C.11) |
so that (C.10) can
be rewritten
(C.12) |
This has solutions
of the form
(C.13) |
so our generic solution for becomes a linear combination of terms
which, without loss of generality, we can write as
(C.14) |
To ensure that our
complete solution matches the boundary conditions (C.2) and (C.3)
we select the following combination
(C.15) |
Thus our final solution
for the x-component of scaled velocity becomes
(C.16) |
where
(C.17) |
To obtain the z-component of velocity
we note that and water
is incompressible, so
(C.18) |
Back substituting
from equation (C.16) then gives
(C.19) |