296
5. Finite-Vo1ume Methods
behavior differs from that of the true solution, in which the time tendency of the
tracer concentration is determined only by the velocity field and the derivatives
of the tracer-concentration field. Most of the other previously discussed methods
for representing discontinuities and steep gradients avoid this dependence on the
mean background concentration by using a different formulation of the nonlinear flux corrector. For example, the nonlinear correction used in the flux-limited
scheme described in Section 5.5 is computed as a function of the ratio of the
slopes of the numerical solution on each side of an individual grid point, and this
ratio is independent of the magnitude of any horizontally uniform background
concentration.
5.9 Curvilinear Coordinates
If the physical boundary constraining a fluid is nonrectangular, it can be advantageous to solve the goveming equations in a curvilinear coordinate system that follows the boundary. In other circumstances, it is possible to simplify the problem
by using cylindrical or spherical coordinates to exploit certain symmetries in the
fluid system. When the goveming equations are expre ssed in non-Cartesian coordinates, additional "metric" terms arise. These terms should be approximated in
a way that preserves the conservation properties of the numerical scheme and the
ability of the scheme to represent discontinuities and poorly resolved gradients.
One elegant way to treat the metric terms is to begin with the equation formulated
for an arbitrary curvilinear coordinate system and to apply one of the preceding
methods directly to the transformed system (e.g., Smolarkiewicz and Margolin
1993) . As an example, LeVeque 's algorithm for two -dimensional tracer transport
(described in Section 5.7.3) can be modified for use with curvilinear coordinates
as folIows.
Suppose that (Xl, ... • X n ) is a position vector in Cartesian coordinates, that
(Xl , . . . , X n ) is the corresponding vector in curvilinear coordinates, and that there
is a smooth mapping between the two systems for which the Jacobian of the transformation J = Det(8xi/8xj) is nonsingular. Then the velocities in the curvilinear
coordinates are related to the Cartesian velocities such that
where repeated subscripts are summed. The divergence of the velocity vector
transforms as
8Vi
-
1 8
_
= --_- (Jvü·
8Xi
(See Gal -Chen and Somerville 1975.)
J 8Xk
(5.61)
The equations goveming the transport of a passive tracer in two-dimensional
nondivergent flow may be expressed in curvilinear coordinates in either advective
5. Finite-Vo1ume Methods
behavior differs from that of the true solution, in which the time tendency of the
tracer concentration is determined only by the velocity field and the derivatives
of the tracer-concentration field. Most of the other previously discussed methods
for representing discontinuities and steep gradients avoid this dependence on the
mean background concentration by using a different formulation of the nonlinear flux corrector. For example, the nonlinear correction used in the flux-limited
scheme described in Section 5.5 is computed as a function of the ratio of the
slopes of the numerical solution on each side of an individual grid point, and this
ratio is independent of the magnitude of any horizontally uniform background
concentration.
5.9 Curvilinear Coordinates
If the physical boundary constraining a fluid is nonrectangular, it can be advantageous to solve the goveming equations in a curvilinear coordinate system that follows the boundary. In other circumstances, it is possible to simplify the problem
by using cylindrical or spherical coordinates to exploit certain symmetries in the
fluid system. When the goveming equations are expre ssed in non-Cartesian coordinates, additional "metric" terms arise. These terms should be approximated in
a way that preserves the conservation properties of the numerical scheme and the
ability of the scheme to represent discontinuities and poorly resolved gradients.
One elegant way to treat the metric terms is to begin with the equation formulated
for an arbitrary curvilinear coordinate system and to apply one of the preceding
methods directly to the transformed system (e.g., Smolarkiewicz and Margolin
1993) . As an example, LeVeque 's algorithm for two -dimensional tracer transport
(described in Section 5.7.3) can be modified for use with curvilinear coordinates
as folIows.
Suppose that (Xl, ... • X n ) is a position vector in Cartesian coordinates, that
(Xl , . . . , X n ) is the corresponding vector in curvilinear coordinates, and that there
is a smooth mapping between the two systems for which the Jacobian of the transformation J = Det(8xi/8xj) is nonsingular. Then the velocities in the curvilinear
coordinates are related to the Cartesian velocities such that
where repeated subscripts are summed. The divergence of the velocity vector
transforms as
8Vi
-
1 8
_
= --_- (Jvü·
8Xi
(See Gal -Chen and Somerville 1975.)
J 8Xk
(5.61)
The equations goveming the transport of a passive tracer in two-dimensional
nondivergent flow may be expressed in curvilinear coordinates in either advective