J.L. Hench et al. / Continental Shelf Research 22 (2002) 26152631
2629
coordinates are related to those in the x2y
systems are related by the chain rule
coordinate system by
q qs
q qn
q
qx
qs qx qn qx
Msi Mxi cos a Myi sin a;
A:3
q
q
sin a ;
A:11
cos a
qs
qn
Mni Myi cos a Mxi sin a;
A:4
q qs
q qn
q
where the index i 1 : 5 as in Eqs. (A.1) and (A.2)
qy
qs qy qn qy
above. The along- and across-stream velocities can
q
q
be related to the x2y velocity components using
sin a cos a ;
A:12
qs
qn
the same orthogonal rotation
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where qs=qx cos a; qs=qy sin a; qn=qx
Us U cos a V sin a U 2 V 2;
A:5
sin a; and qn=qy cos a from Fig. 8. The final
relationship needed for the rotation is
Un V cos a U sin a 0:
A:6
qa
1
;
A:13
qs Rs
From Eq. (A.6), a arctanV =U : Eqs. (A.5) and
(A.6) can also be rearranged to express U ; V in
where Rsx; y; t is the streamwise radius of
terms of Us and a
curvature (cf. Kalkwijk and de Vriend, 1980; Gill,
1982), with curvature to the left assumed positive.
U Us cos a;
A:7
Expanding the spatial derivatives in Eqs. (A.9) and
(A.10) with the chain rule, substituting in the
V Us sin a:
A:8
expression for the radius of curvature, and
simplifying yields the momentum equations in
Substituting the x2y momentum terms from
s2n coordinates
Eqs. (A.1) and (A.2) into Eqs. (A.3) and (A.4),
qZ Cf Us2
qUs
qUs
and replacing U and V using Eqs. (A.7) and (A.8)
Us
g
0;
A:14
s
qz} |fflfflfflfflffl{zfflqffls} |fflffl{zq} |fflfflffl{Hfflffl }
t
gives
zffl
fflffl
|{
fflffl ffl
Ms5
Ms1
Ms2
Ms4
qUs
qUs
qUs
sin a
Us cos a
|{ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflqfflyfflfflffl}
t
qx
qz}
qa Us2
qZ
ffl fflffl
fUs g
0:
A:15
Us
q} Rs |ffl{zffl} |fflffl{zqffln
Ms1
t
Ms2
|ffl {zffl |ffl{zffl} M
ffl}
Cf Us2
n3
qZ
qZ
Mn1
Mn4
Mn2
g cos a sin a
0;
A:9
qx
qy |fflfflffl {Hfflffl }
zffl
Following the above derivation, the mapping
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Ms5
between the x2y and s2n momentum equations is
Ms4
Local streamwise acceleration:
qa
qa
qa
qUs
qU
qV
Us Us2 cos a sin a
cos a
A:16
sin a:
|ffl{zfflt |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflqxfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflqfflyfflffl}
q}
qt
qt
qt
{z
ffl fflffl
Mn1
Mn2
Local rotary acceleration:
qZ
qZ
qV
qU
qa
fUs g cos a sin a
0:
A:10
|ffl{zffl}
Us
cos a
A:17
sin a:
qy
qx
Mn3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
qt
qt
qt
Mn4
Streamwise acceleration:
With this transformation Ms3 and Mn5 are zero by
qUs
qU
qU
V
U
Us
cos a
definition (i.e. bottom friction acts entirely in the
qx
qy
qs
streamwise direction, and Coriolis only in the
qV
qV
normal direction). To complete the transforma-
V
A:18
U
sin a:
qx
qy
tion, derivatives in the x2y and s2n coordinate
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