ATMS 502
Computer problem 5, page 1
Fall 2006 
Jewett
Oct. 31, 2006
ATMS 502  CS 505  CSE 566
Jewett
Computer Problem 5
2D nonlinear quasicompressible flow
Due
: Thursday, Nov. 16
(recommended!)
/ Thursday, Nov. 30 (last date accepted)
Problem
: colliding dry outflows – nonlinear, quasicompressible flow
Methods
: directional splitting; non/monotonic piecewise linear advection, LaxW.
A. Equations
The set of equations includes four unknowns: horizontal
(u)
and vertical
(w)
wind,
potential temperature
(
θ
)
, and pressure
(p)
. There are advection, diffusion, and
pressure/buoyancy terms, with sound waves present. The continuous equation form:
umomentum:
u
t
=
"
uu
x
"
wu
z
"
1
#
$
p
x
+
K
$
u
xx
+
$
u
zz
( )
;
$
u
=
u
"
U
(
z
)
wmomentum:
w
t
=
"
uw
x
"
ww
z
"
1
$
p
z
+
g
$
%
+
K
m
w
xx
+
w
zz
( )
;
$
=
"
(
z
)
(thermodynamic):
"
t
=
#
u
( )
x
#
w
( )
z
+
u
x
+
w
z
( )
Perturbation
pressure:
"
p
t
=
#
c
s
2
$
u
x
+
z
w
( )
’
(
)
*
+
Equations of state and potential temperature:
p
=
R
d
T
and
=
T
p
0
p
$
%
’
(
)
R
d
/
C
p
Dependent variables: u(x,z,t), w(x,z,t),
θ
(x,z,t), p’(x,z,t), with units of m s
1
, m s
1
, ˚K
and Pa (pascals) for u,w,
θ
, and p’, and g kg
1
for
. The discrete equation form:
u:
2
t
u
=
#
u
x
x
u
( )
(
n
)
x
#
w
x
z
u
( )
(
n
)
z
#
1
x
%
p
(
n
#
1)
+
K
xx
u
+
zz
%
u
( )
(
n
#
1)
w:
2
t
w
=
#
u
z
x
w
( )
(
n
)
x
#
w
z
z
w
( )
(
n
)
z
#
1
( )
z
z
%
p
(
n
#
1)
+
g
#
’
(
)
*
+
,
(
n
)
z
+
K
xx
w
+
zz
w
( )
(
n
#
1)
:
t
=
$
1
%
t
F
i
+
1/2
u
,
( )
$
F
i
$
1/2
u
,
( )
[ ]
(
n
)
+
(
n
)
x
u
[ ]
(
n
)
$
1
%
t
F
k
+
1/2
w
,
( )
$
F
k
$
1/2
w
,
( )
[ ]
(
n
)
+
(
n
)
z
w
[ ]
(
n
)
’
(
(
)
*
+
+
+
K
xx
+
zz
,
( )
(
n
)
or
t
=
$
u
2
x
+
u
2
%
t
2
xx
$
w
2
z
+
w
2
%
t
2
zz

.
/
0
1
2
(
n
)
+
K
xx
+
zz
,
( )
(
n
)
"
p
:
2
t
#
p
=
$
c
s
2
x
u
(
n
+
1)
+
z
( )
z
w
(
n
+
1)
{ }
’
(
)
*
+