Ill
TZXq and rMX_ko are the time averaged surface and bottom shear stresses. Assuming that
the free surface pressure pn is zero (the dynamic free surface boundary condition) the right
hand side is
___ 1 d 7 1 dh~0 11 ,
RHS = / pdz + -p.h + -T--T¡i (A.25)
P dxj-hc p dx p p
The mean pressure at the bottom can be expressed as the sum of the wave induced or
dynamic pressure and the hydrostatic pressure
P-h0 = PdVn-ko + pg[K + t])
(A.26)
so that the product P-h0fa can be manipulated to read
dh0 1 dh0 13.,. _.2, .dfj
7 = 7 + 2~~+ ?) > + s
(A.27)
Terms from both the right and left hand sides are combined as the radiation stresses.
Defining
s = +<*') iz !(*. + )! piizY
~r> i
S,vSJh^iz dz ]_hPidz
s = /_7 + -¡MK + tf- ^7){/l ^
(A.28)
(A.29)
(A. 30)
and making the assumptions that the product of the dynamic pressure at the bottom and
the bottom slope is negligible, that the j¡f\0[u')2dz term can be ignored and that the
lateral friction tj caused by momentum fluxes due to turbulent fluctuations is independent
of depth and that rj is defined by tj = pu'v', the x direction equation of momentum is
written as
3 r2/l ^ d rnr/L . ldSxx 1 dS,
+ --
XV
5^ + + a-rU '<* + * + TyUV^ ++ ; 8x ,
+9(A + ,)|5+(*L.Ir + in< = o
OX p dy p p
(A.31)
By subtracting the depth integrated continuity equation and dividing by the total depth,
the final form of the x momentum equation is
8U dU_ dU_ 1
dt + 5x dy + p{h0 + fj)
dST
dS:
xv
dx
dy .
(A.32)
~~