7.1 Tbe Projection Method
337
density from tbe momentum equations and tbereby express the Euler equations as
a closed system of five prognostic equations in five unknowns . When sound waves
are filtered from tbe goveming equations using tbe ineompressible, Boussinesq,
anelastic, or pseudo-ineompressible approximations, the approximate mass eonservation equations for these filtered systems do not depend on the loeal time
derivative of the true density, and as a eonsequenee, they eannot be used to obtain
a prognostie equation for pressure. Sinee a prognostie equation is not available
for the ealculation of pressure, the filtered systems are not strietly byperbolie, and
their numerieal solution eannot be obtained entirely through the use of explicit
finite-differenee sehernes.
As an example, eonsider the Boussinesq equations (1.51), (1.56), and (1.60),
which may be written in the form
V·v=o,
dp'
df5
-+w-=O,
dt
ov
1
dz
I
I
- + - V p =F(v,p) ,
ot Po
(7.1)
(7.2)
(7.3)
wbere
I
p'
F(v, p ) = -v· Vv - g-k
Po
and p' and p' are the deviations of the pressure and density from their values in
a hydrostatically balaneed referenee state, fi(z) and f5(z). The unknown variables
are the three velocity eomponents, the perturbation density, and the perturbation
pressure. There is no prognostic equation available to determine the time tendeney
of p', The perturbation pressure field at a given instant ean, however, be diagnosed from the instantaneous velocity and perturbation density fields by solving
the Poisson equation
(7.4)
which ean be derived by taking the divergenee of (7.3) and then using (7.1). The
perturbation pressure satisfying (7.4) is the instantaneous pressure distribution
that will keep the evolving veloeity field nondivergent.
7.1.1 Forward-in-Time Implementation
The projection method (Chorin 1968) is a classical teebnique that may be used
to obtain numerieal solutions to the Boussinesq system. Suppose the momentum
equation is integrated over a time interval /lt to yield
l
l n
In
+
J ov
l
l n +
+
(7.5)
-dt=ot
In
-V pl dt + l
Po
In
J
F(V,p')dt,
, 1
I
n
337
density from tbe momentum equations and tbereby express the Euler equations as
a closed system of five prognostic equations in five unknowns . When sound waves
are filtered from tbe goveming equations using tbe ineompressible, Boussinesq,
anelastic, or pseudo-ineompressible approximations, the approximate mass eonservation equations for these filtered systems do not depend on the loeal time
derivative of the true density, and as a eonsequenee, they eannot be used to obtain
a prognostie equation for pressure. Sinee a prognostie equation is not available
for the ealculation of pressure, the filtered systems are not strietly byperbolie, and
their numerieal solution eannot be obtained entirely through the use of explicit
finite-differenee sehernes.
As an example, eonsider the Boussinesq equations (1.51), (1.56), and (1.60),
which may be written in the form
V·v=o,
dp'
df5
-+w-=O,
dt
ov
1
dz
I
I
- + - V p =F(v,p) ,
ot Po
(7.1)
(7.2)
(7.3)
wbere
I
p'
F(v, p ) = -v· Vv - g-k
Po
and p' and p' are the deviations of the pressure and density from their values in
a hydrostatically balaneed referenee state, fi(z) and f5(z). The unknown variables
are the three velocity eomponents, the perturbation density, and the perturbation
pressure. There is no prognostic equation available to determine the time tendeney
of p', The perturbation pressure field at a given instant ean, however, be diagnosed from the instantaneous velocity and perturbation density fields by solving
the Poisson equation
(7.4)
which ean be derived by taking the divergenee of (7.3) and then using (7.1). The
perturbation pressure satisfying (7.4) is the instantaneous pressure distribution
that will keep the evolving veloeity field nondivergent.
7.1.1 Forward-in-Time Implementation
The projection method (Chorin 1968) is a classical teebnique that may be used
to obtain numerieal solutions to the Boussinesq system. Suppose the momentum
equation is integrated over a time interval /lt to yield
l
l n
In
+
J ov
l
l n +
+
(7.5)
-dt=ot
In
-V pl dt + l
Po
In
J
F(V,p')dt,
, 1
I
n