8.2 1\vo-Dimensional Shallow-Water Flow
413
lems. The difficulties are readily apparent in two relatively simple examples: twodimensional shallow-water flow and two-dimensional vertically stratified flow.
Boundary conditions for the two-dimensional shallow-water system will be discussed in this section . Stratified flow will be considered in Section 8.3.
The two-dimensional shallow-water equations, linearized about a basic state
with constant mean flow (U, V) and depth H, may be written
o
U
o
where the notation follows that used in Section 8.1.1. Suppose that a solution is
sought in the limited domain 0 :s x :s L, 0 :s y :s L. In contrast to the onedimensional case, the two-dimensional system cannot be reduced to a set of three
scalar equations because no transformation of variables will simultaneously diagonalize both coefficient matrices . Nevertheless, the coefficient matrix multiplying
the x-derivative can be diagonalized through the same change of variables used in
the one-dimensional problem . Defining d == u - TJ/c and e == u + TJ/c as before,
the two-dimensional system becomes
U -c o
o
o
U
o
o
V
-c/2
This equation is useful fOT determining the number of boundary conditions that
should be specified at the x -boundaries. In the case c > U > 0, the signal in v
and e is propagating inward through the boundary at x = O. and the signal in d
is propagating inward through the boundary at x = L. In order to obtain a wellposed problem , one might therefore attempt to specify two conditions at x = 0 of
the form
e(O, t) = a)d(O, 1) + I)(t),
and one condition at x = L of the form
v(O,1) = a2d(0, t) + h(t),
d(L, t) = a3e(L, t) + a4v(L, t) + !J(t) .
(8.25)
This approach follows the guideline that the number of conditions specified at
each boundary should be equal to the number of eigenvalues associated with outward propagation through each boundary. Even so, not all choices of a), .. . , a4
yield a well-posed problem . Oliger and Sundström (1978) and Sundström and
Elvius (1979) provide details about various allowable values for a) , .. . , a4. One
413
lems. The difficulties are readily apparent in two relatively simple examples: twodimensional shallow-water flow and two-dimensional vertically stratified flow.
Boundary conditions for the two-dimensional shallow-water system will be discussed in this section . Stratified flow will be considered in Section 8.3.
The two-dimensional shallow-water equations, linearized about a basic state
with constant mean flow (U, V) and depth H, may be written
o
U
o
where the notation follows that used in Section 8.1.1. Suppose that a solution is
sought in the limited domain 0 :s x :s L, 0 :s y :s L. In contrast to the onedimensional case, the two-dimensional system cannot be reduced to a set of three
scalar equations because no transformation of variables will simultaneously diagonalize both coefficient matrices . Nevertheless, the coefficient matrix multiplying
the x-derivative can be diagonalized through the same change of variables used in
the one-dimensional problem . Defining d == u - TJ/c and e == u + TJ/c as before,
the two-dimensional system becomes
U -c o
o
o
U
o
o
V
-c/2
This equation is useful fOT determining the number of boundary conditions that
should be specified at the x -boundaries. In the case c > U > 0, the signal in v
and e is propagating inward through the boundary at x = O. and the signal in d
is propagating inward through the boundary at x = L. In order to obtain a wellposed problem , one might therefore attempt to specify two conditions at x = 0 of
the form
e(O, t) = a)d(O, 1) + I)(t),
and one condition at x = L of the form
v(O,1) = a2d(0, t) + h(t),
d(L, t) = a3e(L, t) + a4v(L, t) + !J(t) .
(8.25)
This approach follows the guideline that the number of conditions specified at
each boundary should be equal to the number of eigenvalues associated with outward propagation through each boundary. Even so, not all choices of a), .. . , a4
yield a well-posed problem . Oliger and Sundström (1978) and Sundström and
Elvius (1979) provide details about various allowable values for a) , .. . , a4. One