Chemistry Reference
In-Depth Information
3 Relaxation rates and total source term
In the relaxation model discussed in the introduction ('element 3'), the
wave spectrum is calculated along a transect through the current field by
tracing the waves through (
x,k
)-space, and integrating the wave action
N
along these rays. (In the present application, the current field is absent, im-
plying that action
N
and energy
E
are actually equivalent.) In general for
such a relaxation model, the action source term need not be specified in
detail, but can be written as a deviation from an equilibrium:
d
N
S
(
N
)
|
P
(
N
N
)
.
(19)
eq
d
t
This is the start of a Taylor series expansion around
S
(
N
eq
) = 0, with
d
S
P
(
)
.
N
(20)
eq
d
N
When, however,
S
(
N
) can be written as
S
(
N
)
f
N
f
N
2
f
N
3
,
(21)
1
2
3
which is in fact the case here (Eq. 18), we can actually write exactly
f
N
N
2
S
(
N
)
f
(
3
N
)
(
N
N
)
,
(22)
1
eq
eq
N
f
N
eq
1
eq
which leads to the very good approximation
N
S
(
N
)
| P
(
N
N
)
(23)
eq
N
eq
with
P
f
f
N
2
.
(24)
1
3
eq
For a real (stable) equilibrium, one needs
P .
We are now in the situation that we can use for
0
N
eq
the equilib-
rium spectrum calculated in the previous section, but we also know the
form of the total source term
S
(
N
) (Eq. 23), and we can calculate the re-
laxation rates
(
k
,
u
)
*
k
P since we have available
f
1
and
f
3
. This approach of
calculating the relaxation rate was also used by Trokhimovski (1993).
The results of calculating the relaxation rate are shown in Fig. 4. They
are somewhat surprising, in that a negative value for the relaxation rate is
found in the region where the curvature spectrum rises. This would indi-
(
,
u
)
*
