930

JOURNAL OF PHYSICAL OCEANOGRAPHY

VOLUME 33

lets Research Program (CIRP), ONR (N00014-97-C-

6010), and NSF (OCE-0094938).

APPENDIX

The use of a streamwise-normal coordinate system

allows a more intuitive physical interpretation for

strongly curving flow fields. Here we derive a form of

the fully nonlinear transient frictional shallow water

equations by transforming the familiar *x**y *equations.

Here *U*(*x, y, t*), *V*(*x, y, t*), and (*x, y, t*) are computed

on a fixed *x**y *grid for all time steps. The elevation and

velocity fields are used to reconstruct each term in the

FIG. A1. Coordinate system definition sketch for the *x**y *to *s**n*

transformation. At each time step, an *s**n *coordinate system is es-

Individual *x**y *momentum terms *M*xi and *M*yi are des-

tablished (aligned with the local velocity vector **U **at each compu-

ignated as

tational node. The local axes rotation angle is (*x, y, t*). Vector **M**i

U

U U

U

V

f*V*

g

tions, with *x**y *components *M*xi and *M*yi (e.g., *g * / *x *and *g * / *y*),

t

x

y

x

and is rotated onto the *s**n *axes to determine local *s**n *components

M

U2

V2

H

varies so that at all points the alongstream velocity *U*s

U

0,

(A1)

is equivalent to the speed, and the across-stream velocity

0. From Fig. A1 it should be apparent that mo-

M

mentum terms in *s**n *coordinates are related to those in

the *x**y *coordinate system by

V

U V

V

V

f*U*

g

Myi sin ,

Mxi cos

(A3)

t

x

y

y

M

Mxi sin ,

Myi cos

(A4)

where the index *i * 1:5 as in Eqs. (A1) and (A2) above.

U2

V2

H

The *s**n *velocities can be related to the *x**y *velocity

V

0.

(A2)

components using the same orthogonal rotation:

M

V sin

U2

U cos

V 2,

(A5)

U sin

0.

V cos

(A6)

For simplicity, horizontal diffusion terms are omitted

from (A1) and (A2) as they did not contribute signifi-

arctan(*V*/*U *). Equations (A5) and

From Eq. (A6),

cantly to the momentum balances of our inlet simula-

(A6) can also be rearranged to express (*U, V *) in terms

tions. At each time step, we define a two-dimensional,

of *U*s and :

orthogonal, curvilinear coordinate system such that at

Us cos ,

(A7)

each grid point, one coordinate direction points in the

streamwise direction (*s*) and the other points in the

Us sin .

(A8)

across-stream or normal direction (*n*) in the right-hand

sense (see Fig. A1). The orientation of the *s**n *coor-

Substituting the *x**y *momentum terms from (A1) and

dinate system relative to the original fixed *x**y *coordi-

(A2) into (A3) and (A4), and replacing *U *and *V *using

nate system, given by the streamline angle (*x, y, t*),

(A7) and (A8), gives

t

Us

Us

Us

Cf U 2

sin

g cos

sin

0,

Us cos

(A9)

x

y

x

y

M