8.3 Two-Dirnensional Stratified Flow
429
The preceding local boundary condition is generalized to three-dimensional
problems by approximating the factor Jk 2 + e 2 in (8.59) with
al + a2(k 2 + e 2 )
I + a3(k 2 + e 2 ) .
The approximate radiation upper boundary condition that results is
level as
A vertically discretized approximation to (7.83) at level N - !
[al - a:
+ aa;2 ) ] P = N [I -a3( + :;2)]w.
8.3.3 Numericallmplementation ofthe Radiation Upper
Boundary Condition
First consider the nonlocal formulation (8.55) and suppose that the numerical
solution of the Boussinesq system is to be obtained using the projection method
on the staggered mesh shown in Fig. 3.6. An upper boundary condition can be
obtained for the Poisson equation (7.9) as folIows. Fourier transform w n along
the top computational level of the domain; compute the Fourier coefficients for
pressure from the relation ßn = NPow n /Ikl; and inverse transform to obtain pn
at a level !i.z/2 above the uppermost row of p-points. 4 These values of p" are used
as approximations to ßn+I along the boundary and provide a Dirichlet boundary
condition for (7.9). Ideally, one would formulate the Dirichlet boundary condition
using an exact expression for the boundary values of ßn+l, but recall that ßn+1
does not represent the actual pressure at any given time. Approximating ßn+1
with v" has yielded satisfactory results in tests conducted by this author. After
solving the Poisson equation, the interior velocities are updated from (7.8). As
a final step, the w n +I along the upper boundary can be obtained from the nondivergence condition, or by linear extrapolation from below. In most applications,
the mean vertical velocity is zero, and no other upper boundary conditions are
required to obtain numerical solutions to the linearized equations.
As noted by Bougeault (1983), stability considerations require that (8.55) be
implemented in a compressible model using implicit time-differencing, As an
example, suppose that solutions are to be obtained using the partial time-split
approximation to the "compressible" Boussinesq equations (7.82)-(7.85) on the
staggered grid shown in Fig. 3.6 and that N is the vertical index of the topmost
row of pressure and buoyancy points. The boundary condition can be cast into an
implicit expression for PN _! , which represents the pressure at the same vertical
2
4Here P is replaced by pi Po to match the terminology in Seetion 7.3.2.
429
The preceding local boundary condition is generalized to three-dimensional
problems by approximating the factor Jk 2 + e 2 in (8.59) with
al + a2(k 2 + e 2 )
I + a3(k 2 + e 2 ) .
The approximate radiation upper boundary condition that results is
level as
A vertically discretized approximation to (7.83) at level N - !
[al - a:
+ aa;2 ) ] P = N [I -a3( + :;2)]w.
8.3.3 Numericallmplementation ofthe Radiation Upper
Boundary Condition
First consider the nonlocal formulation (8.55) and suppose that the numerical
solution of the Boussinesq system is to be obtained using the projection method
on the staggered mesh shown in Fig. 3.6. An upper boundary condition can be
obtained for the Poisson equation (7.9) as folIows. Fourier transform w n along
the top computational level of the domain; compute the Fourier coefficients for
pressure from the relation ßn = NPow n /Ikl; and inverse transform to obtain pn
at a level !i.z/2 above the uppermost row of p-points. 4 These values of p" are used
as approximations to ßn+I along the boundary and provide a Dirichlet boundary
condition for (7.9). Ideally, one would formulate the Dirichlet boundary condition
using an exact expression for the boundary values of ßn+l, but recall that ßn+1
does not represent the actual pressure at any given time. Approximating ßn+1
with v" has yielded satisfactory results in tests conducted by this author. After
solving the Poisson equation, the interior velocities are updated from (7.8). As
a final step, the w n +I along the upper boundary can be obtained from the nondivergence condition, or by linear extrapolation from below. In most applications,
the mean vertical velocity is zero, and no other upper boundary conditions are
required to obtain numerical solutions to the linearized equations.
As noted by Bougeault (1983), stability considerations require that (8.55) be
implemented in a compressible model using implicit time-differencing, As an
example, suppose that solutions are to be obtained using the partial time-split
approximation to the "compressible" Boussinesq equations (7.82)-(7.85) on the
staggered grid shown in Fig. 3.6 and that N is the vertical index of the topmost
row of pressure and buoyancy points. The boundary condition can be cast into an
implicit expression for PN _! , which represents the pressure at the same vertical
2
4Here P is replaced by pi Po to match the terminology in Seetion 7.3.2.