Geoscience Reference
In-Depth Information
experiment the initial asymmetry is placed at the radius of 0.25 (
R
p
=0
.
25,
hereafter denoted as T25); the third one at the radius of 0.3 (
R
p
=0
.
3,
denoted as T30); the fourth at the radius of 0.1 (
R
p
=0
.
1, denoted as
T10); and the fifth at the radius of 0.15 (
15, denoted as T15). The
similar five sensitivity experiments with the wavenumber 3 disturbances
are denoted as H20 (
R
p
=0
.
R
p
=0
.
2), H25 (
R
p
=0
.
25), H30 (
R
p
=0
.
3),
H10 (
R
p
=0
.
1), and H15 (
R
p
=0
.
15) respectively (Fig. 3(b)).
2.3
. Diagnosis method
The diagnosis of the model output is carried out in a cylindrical coordinate
system centered at the vortex center. Each model variable is decomposed
into a symmetric and an asymmetric component, e.g.
v
,
with a bar denoting the symmetric component and a prime the departure
from the symmetric field.
The diagnostics for the energy budget is made with the following
symmetric kinetic energy [KE
u
,
u
=¯
u
+
v
=¯
v
+
¯
=
2
,
K
u
2
+¯
v
2
)] equation:
(¯
¯
¯
¯
∂
K
∂t
−
∂
(
r
u
¯
K
)
u
∂
(
ru
2
)
r∂r
v
∂
(
u
v
)
∂r
u
v
2
2¯
v
r
u
∂
φ
∂r
,
=
−
¯
−
¯
+¯
r
−
u
v
−
¯
(2.4)
r∂r
where the first term on the right-hand side of (2.4) is the flux divergence
of
¯
by the symmetric radial flow, the sum of the second, third, fourth,
and fifth terms represents the time change rate of symmetric KE due
to wave-wave interactions, and the sixth term is the energy conversion
from symmetric potential energy to symmetric kinetic energy. Note that
the second-to-fifth terms on the right-hand side involve the interaction
among the asymmetric perturbations and they are directly related to energy
transfer between the asymmetry and the symmetry.
A Fourier analysis for vorticity is based on the following formula:
K
)+
N
ζ
(
r, λ, t
)=
ζ
0
(
r, t
[
ζ
kc
(
r, t
)cos(
kλ
)+
ζ
ks
(
r, t
)sin(
kλ
)]
,
(2.5)
k
=1
where
)are
Fourier spectrum coecients. The wavenumber spectrum of the first eight
components are used to calculate the asymmetric component, and the
Fourier asymmetric vorticity amplitude is defined by the following formula:
k
is the azimuthal wavenumber,
ζ
0
(
r, t
),
ζ
kc
(
r, t
), and
ζ
ks
(
r, t
)=
)
2
+
)
2
.
A
k
(
r, t
ζ
kc
(
r, t
ζ
ks
(
r, t
