Cyclonic and anticyclonic contributions to atmospheric energetics

Scientific Reports
| (2021) 11:13202 | 
https://doi.org/10.1038/s41598-021-92548-7
www.nature.com/scientificreports/
where 
u
and 
v
denote local zonal and meridional wind velocities, respectively, and subscripts 
x
and 
y
partial 
derivatives in the zonal and meridional directions, respectively. The curvature, defined as
can be derived from the definition of curvature of a two-dimensional curve implicitly represented by 
ψ
x
,
y
=
c
;
The curvature or curvature vorticity enables us to circumvent the difficulties in determining areas of cyclonic and 
anticyclonic circulations, because it is free from shear vorticity and thus extracts vortex circulation with a certain 
radius. A similar quantity, curvature vorticity multiplied by a scalar wind speed (named Eulerian Centripetal 
Acceleration or ECA), was utilized to track 500-hPa mobile pressure troughs
45
,
48
. As shown in Supplementary 
Fig. S2e, ECA well captures centers of upper-tropospheric troughs along a strong westerly jet, which was the 
purpose of contriving ECA
45
. Meanwhile, ECA is less effective in representing the center of an upper-tropospheric 
cut-off low than the curvature (Supplementary Fig. S1c). Another aspect of ECA is a distinct difference in its 
amplitude between the upper and lower troposphere (Supplementary Figs. S2e and S2f.), which is due to the 
direct contribution of squared wind speed. At this point, curvature is suitable for identifying three-dimensional 
cyclonic and anticyclonic vortices and evaluating those contributions to the atmospheric energy cycle, whereas 
ECA is compatible with the identification of centers of troughs at a given mid- or upper-tropospheric level. In 
this study, curvature is weakly smoothed by applying a 9-point horizontal smoothing (weight is 0.5 next to the 
center point and 0.3 at corners) twice when used for determining the direction of local circulation (cyclonic or 
anticyclonic).
In addition to the potential application of curvature to meteorological and climatic phenomena including 
cut-off lows, the decomposition of vorticity into the curvature and shear terms can be useful for other fields of 
geoscience, because curvature is calculated purely locally with no laborious procedures required. For example, 
we can distinguish and identify ocean eddies and jets along the western boundary currents, or determine the 
boundary of a given warm/cold core eddy. In the case of the meandering Kuroshio Extension as in Supplemen-
tary Fig. S12a, relative vorticity includes mixed contributions from the vortex and shear terms (Supplementary 
Fig. S12b). The curvature term better depicts eddies (Supplementary Fig. S12c), while the shear term helps us 
identify oceanic jets (Supplementary Fig. S12d). For example, a mesoscale cyclonic eddy associated with the 
meandering Kuroshio is better resolved as an isolated vorticity maximum around [32°N, 139°E], whereas shear 
vorticity depicts more continuous bands of positive and negative values than relative vorticity, representing the 
meandering Kuroshio current and its eastward Extension. This decomposition can therefore be helpful for elu-
cidating dynamical processes involved in the maintenance and variability of the oceanic jet under the possible 
feedback forcing from eddies. The identification of ocean eddies through curvature and curvature vorticity based 
on horizontal flow fields may thus be more straightforward than, for example, the commonly used Okubo-Weiss 
(OW) parameter
49
,
50
or identification based on sea surface height
51
.
It may be informative to show the relationship between the curvature utilized in this study and the OW 
parameter, which is defined as
The last equality holds for the case of horizontally non-divergent flow. The last term is related to Gaussian 
curvature of three-dimensional surface, since for a given surface 
z
=
ψ
x
,
y
, Gaussian curvature is defined as
The numerator is clearly one fourth of the OW parameter where 
ψ
x
,
y
represents the streamfunction. 
The curvature used in this study focuses on a two-dimensional isocurve, while the OW parameter focuses on 
a curved surface.
One should be cautious in calculating Eulerian statistics based on the vorticity decomposition. In the present 
study, for example, V’V’ is calculated from a high-pass-filtered field of total (not decomposed) meridional wind 
as shown in Supplementary Fig. S3. It might be possible to calculate eddy variance and covariance from decom-
posed velocities such as 
v
=
v
C
+
v
A
+
v
S
, where subscripts “C”, “A”, and “S” denote velocities derived from 
positive and negative curvature vorticity and shear vorticity terms, respectively (e.g., 
v
C
=
∂
∂
x
−
∇
2
−
1
ζ
C
; 
ζ
C
denotes positive curvature vorticity). However, those second (or higher) order statistics can have non-negligible 
contributions from “cross terms” in such a way that 
V
′
V
′
=
V
′
C
V
′
C
+
V
′
A
V
′
A
+
V
′
S
V
′
S
+
V
′
C
V
′
A
+
V
′
A
V
′
S
+
V
′
S
V
′
C
, 
and the correlation coefficients between the decomposed velocity components may not necessarily be small. 
(3)
V
R
S
=
1
V
2
−
uvu
x
+
u
2
v
x
−
v
2
u
y
+
uvv
y
(4)
κ
2
≡
1
R
S
=
1
V
3
−
uvu
x
+
u
2
v
x
−
v
2
u
y
+
uvv
y
(5)
κ
2
≡
ψ
xx
ψ
xy
ψ
x
ψ
yx
ψ
yy
ψ
y
ψ
x
ψ
y
0
ψ
2
x
+
ψ
2
y
3
/
2
(6)
W
=
v
x
+
u
y
2
+
u
x
−
v
y
2
−
v
x
−
u
y
2
=
4
v
x
u
y
−
u
2
x
(7)
κ
3
≡
ψ
xx
ψ
yy
−
ψ
2
xy
1
+
ψ
2
x
+
ψ
2
y

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