1
Atms 502
Numerical Fluid Dynamics
Tue., Oct. 24, 2006
Wave equations
in Geophysical Fluid Dynamics
References
:
•
Durran chaps. 1,7
•
Haltiner & Williams chap. 1,2
•
Ferziger and Peric
(2002, 3rd edition)
10/25/06
Atms 502  Fall 2006  Jewett
2
Continuity:
Equation of state:
Note definition of
total derivative!
Fundamental Eqns
•
Momentum
•
1
st
law  thermodynamics
No heat exchange
Q
du
dt
=
"
1
#
dp
dx
+
f
+
u
tan
$
a
%
&
’
(
)
*
v
+
F
+
dv
dt
=
"
1
#
dp
dy
"
f
+
u
tan
$
a
%
&
’
(
)
*
u
+
F
$
dw
dt
=
"
1
#
dp
dz
"
g
"
F
z
"#
dt
=
$
r
%
•
#
r
V
P
=
"
RT
d
"
dt
=
0;
"
=
T
p
0
p
#
$
%
&
’
(
R
/
c
p
da
dt
=
"
a
"
t
+
V
•
#
a
10/25/06
Atms 502  Fall 2006  Jewett
3
Fundamental Eqns
•
Exner form
Eliminates p and
ρ
from the motion
and continuity equations
π
is the nondimensional Exner function.
If use ideal gas equation + def’n of
π
,
θ
+
thermodynamics, continuity equations:
"
=
p
p
0
#
$
%
&
’
(
R
/
C
p
)
1
*
+
p
=
c
p
,
+
"
d
"
dt
=
#
R
"
c
v
$ %
r
v
10/25/06
Atms 502  Fall 2006  Jewett
4
•
Euler equations, minus Coriolis:
•
Inviscid flow: far from solid
surfaces; neglect all viscous
effects
Euler Eqns
d
r
V
dt
=
"
c
p
#
$
%
"
g
ˆ
k
d
#
dt
=
0
d
%
dt
=
"
R
%
c
v
$ &
r
v
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10/25/06
Atms 502  Fall 2006  Jewett
5
•
Euler equations in Durran:
Part of full compressible equations
Includes sound waves; restricts time step
•
Klemp and Wilhelmson (1978):
The term f
π
is often neglected
.
Pressure field: prognostic
d
"
dt
=
#
R
"
c
v
$ %
r
v
"
#
$
dt
+
c
2
%
C
p
&
v
2
"
"
x
%
&
v
u
(
)
+
"
"
y
%
&
v
v
(
)
+
"
"
z
%&
v
w
(
)
’
(
)
*
+
,
=
f
$
, where
f
$
=

u
"
#
$
"
x

v
"
#
$
"
y

w
"
#
$
"
z

R
d
#
$
C
v
"
u
"
x
+
"
v
"
y
+
"
w
"
z
.
/
0
1
2
3
+
c
2
C
p
&
v
2
d
&
v
dt
10/25/06
Atms 502  Fall 2006  Jewett
6
•
In, e.g., a Boussinesq system, you solve a
diagnostic equation for pressure at every step.
Take divergence of momentum equations
p’ is that which keeps evolving velocity field nondivergent.
Pressure field: diagnostic
"
r
v
"
t
+
1
#
0
$ %
p
=
F
(
r
v
,
%
#
)
where
F
(
r
v
,
%
#
)
=
&
r
v
’
r
$
v
&
g
%
#
#
0
r
k
d
%
#
dt
+
w
d
#
dz
=
0
r
$
’
r
v
=
0
(
)
*
*
*
*
+
*
*
*
*
,
$
2
%
p
=
#
0
r
$
’
r
F
Durran chap. 7
10/25/06
Atms 502  Fall 2006  Jewett
7
"
u
dt
=
#
r
V
$
r
%
u
#
1
&
"
p
"
x
+
’
%
2
u
"
v
dt
=
#
r
V
$
r
%
v
#
1
&
"
p
"
y
+
’
%
2
v
"
w
dt
=
#
r
V
$
r
%
w
#
1
&
"
p
"
z
+
g
(
(
+
’
%
2
w
"(
"
t
=
#
r
V
$
r
%
(
+
Q
(
x
,
y
,
z
,
t
)
+
’
%
2
(
"
p
"
t
=
#
c
s
2
&
"
u
"
x
+
"
v
"
y
)
*
+
,

.
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 Spring '09
 JEWETT
 Fluid Dynamics, dt, ATMs, Jewett

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