APPENDIX A
DERIVATION OF THE DEPTH AVERAGED EQUATIONS
In this appendix a derivation of the governing equations for the circulation model is
presented. This appendix closely follows the work of Ebersole and Dalrymple (1979).
Before proceeding with the derivations of the depth integrated time averaged equations
of motion and the continuity equation the Leibnitz Rule for taking the derivative of an
integral is stated since it is invoked throughout the derivation.
Ty C/(I'V)= C dJ%Tdx + my)'y)dJ£-
The Leibnitz Rule can also be used in reverse to take derivatives outside of an integral.
The kinematic boundary conditions are also reproduced since the use of the Leibnitz Rule
will yield bottom and surface terms that will be eliminated through use of these boundary
conditions. On the free surface the boundary condition is
dr) dr)
dr)
w=a + u^ + v-¡,
(A.2)
and at the bottom the boundary condition
(A.3)
In the circulation model it is assumed that the still water depth does not vary with time so
the first term of Eq. (A.3) is dropped (actually for time varying depth essentially the same
equations are obtained).
The continuity equation (or the conservation of mass equation as it is abo called) in
three dimensions is
, d(/>u) d(pv) d(pw)
dt dx dy dz
(A.4)
107