Tropical Cyclone Motion
Tropical Cyclone Motion
Tropical cyclone tracks (1979-1988) Tropical cyclone tracks (1979 1988)
Mean direction of TC motion Mean direction of TC motion
J. Atmos. Sci.
Two-dimensional barotropic flow
Two dimensional barotropic flow
The partitioning problem p g p
The partitioning problem
The partitioning problem
Symmetric vortex in a uniform flow
Symmetric vortex in a uniform flow
Vortex motion on a beta-plane
Vortex motion on a beta-plane
Vortex motion on a beta-plane
Vortex motion on a beta-plane
Vortex motion on a beta-plane
Some calculations
Vortex track
Vo e c
More on tropical cyclone More on tropical cyclone
asymmetries
asymmetries
A line vortex
A li t h th t ti l i d fil
A line vortex has the tangential wind profile:
v
v = Γ
2πr
C
2πrCirculation = 2 rvdr
C
= 1 rv 0
rv 0 r
r r
r
Two-vortex interaction: line vortices: same sign
C
v = Γ
2πd
C
d
2πd v = Γ
C
v 2πd
Vortices rotate around each other about their common centre
Two-vortex interaction: line vortices, opposite signspp g
Γ
Γ
Γ
v = 2πd
C
v = Γ
2πd
d
C
Vortices translate in the direction normal to the line between them with speed Γ
p v =v =
2πd
Reference
Fundamentals of Geophysical Fluid Dynamics J.
McWilliams (2006) CUP( )
Chapter 3: Barotropic and Vortex Dynamics
Vorticity and velocity distribution
2 2
2 2
x y
u v
y x
y x
Can “invert” the vorticity to obtain the streamfunction when y suitable boundary conditions on the streamfunction are given.
Biot-Savart law
Biot-Savart law
L
d
r lV 3
4 L r
V
Biot-Savart law
Integral over a volume Veg ove vo u e V
3
1
4 | |
dV
ω ru
4
V | |r 3dV r
V du
+ wavenumber-one vorticity asymmetry
+
vorticity asymmetry
The partitioning problem The partitioning problem
Recall, for a moving reference frame, g
Small asymmetry
Asymmetry onlysy e y o y
Large asymmetry
Small asymmetry Large asymmetry
Small asymmetry -plane Large asymmetry
Two-vortex interaction
Track of the large vortex
f-plane -plane
Vortex interaction
Two like-signed potential vortices will circle around a common centre without getting closer (Fujiwhara effect).
Two like-signed vortices with a finite vorticity core willg y merge when their distance of separation is smaller than some critical value.
This merger process is the predominant mechanism for the evolution of two-dimensional turbulence, and has for
i i i
this reason been studied extensively.
The existence of a critical distance has been confirmed by a number of high-resolution numerical simulations of inviscid two-dimensional flows as well as by laboratory experiments on interacting barotropic vortices in a experiments on interacting barotropic vortices in a rotating fluid.
Vortices further apart
Dye visualization of the merger of two cyclonic vortices at successive times.
D i li ti f th d t t th Dye visualizations of the merger process demonstrate the formation of cusps and the existence of long filaments.
These characteristic features of vortex merging can be well These characteristic features of vortex merging can be well captured by simple point-vortex models in which each
vortex is represented by a point vortex surrounded by a contour of passive tracers. The method of contour
kinematics is used to calculate numerically the evolution of the material contours A typical calculated evolution of a the material contours. A typical calculated evolution of a two-point-vortex configuration is shown in the next figure.
C l l t d l ti f i iti ll i l t f i Calculated evolution of initially circular contours of passive
tracers which are advected by the co-rotating velocity field induced by the two point vortices (not shown). The distance induced by the two point vortices (not shown). The distance
between the point vortices was artificially decreased.
