Geoscience Reference
In-Depth Information
The helicity is calculated from (
18.8
), (
18.5
)to(
18.6
)as
H D
.
r
˛
Sr
ˇ/
.1=
S
/
rS
.
Sr
ˇ/:
(18.16)
The helicity consists of two parts representing the irrotational and rotational
components of
v
in (
18.13
),
H D H
˛
C H
ˇ
;
(18.17)
where the irrotational part is
H
˛
WD
.
r
˛/
.1=
S
/
rS
.
Sr
ˇ/;
(18.18)
and the rotational part is
H
ˇ
WD
.
Sr
ˇ/
.1=
S
/
rS
.
Sr
ˇ/;
(18.19)
where H
˛
and H
ˇ
are used for simplicity instead of H
irrot
and H
rot
which were used
in the earlier publications (
Sasaki 2009
,
2010
).
Because
.
Sr
ˇ/
.
Sr
ˇ/
D
0
,(
18.19
) becomes
H
ˇ
D
0
(18.20)
and
H D H
˛
:
(18.21)
Therefore, the helicity is given, using (
18.6
);
!
D
.1=
S
/
rS
.
Sr
ˇ/
,as
H
˛
D
.
r
˛/
!:
(18.22)
Comparing the old form of helicity expressed by (
18.8
), the new form (
18.22
)
shows an important difference, because the rotational term denoted by
.
Sr
ˇ/
in
the above vector product vanishes. Since
, two independent
thermodynamical parameters for baroclinicity in general, (
18.22
) becomes
!
includes rSandr
ˇ
H
˛
D H
˛;
BC
WD u
D
C v
D
C w
D
:
(18.23)
where the subscript BC sands for baroclinic,
u
D
;
v
D
and
w
D
(or u
˛
;
v
˛
and w
˛
respectively) represent the irrotational velocity components (r
˛/
on Cartesian x,y,
and z coordinates, and
are the three-dimensional components of vorticity.
This supports Beltrami relation and the tilting of a horizontal vortex tube into the
vertical, and a high value of helicity (relative helicity ! 1.0) in the supercell stage.
However, the vortex tube at the mature tornadic stage is vertical and hits the
ground perpendicularly, so we expect a drastic change from the supercell stage to
the mature tornado stage to satisfy the boundary condition of the vortex tube at
the ground surface. Therefore, (
18.23
) becomes drastically different from the tilting
process, expressed by,
,
,and
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