Transcript Experiments with crazy paving

The Rossby wave paradigm
and its problems
Ian James
School of Mathematics, Meteorology and
Physics
University of Reading
Ian N. James
University of Reading
[email protected]
School of Mathematics Meteorology & Physics
1
Basic properties
• Transverse wave motion in a fluid with a gradient of
potential vorticity
• Dispersion relationship:
c

k
U 
k
qy
2
 l2 
2



2
q
k
2q y kl


y

cg 
i
j  U  2 2 2 i  2 2 2
k
l
k  l  k  l 


j

• Highly dispersive; anisotropic propagation
• Basic paradigm of low frequency flow
Ian N. James
University of Reading
[email protected]
School of Mathematics Meteorology & Physics
2
Physical picture
• Displaced parcel acquires vorticity relative to
environment.
• “Action at a distance”   2
<0
<0
A
B
>0
Ian N. James
University of Reading
[email protected]
School of Mathematics Meteorology & Physics
3
Ray tracing
• Packets propagate with
group velocity (WKBJ)
m
q y
U c
l   m2  k2
• “Refractive index” m
• Pacific-North American
pattern (=0 Rossby
wave)
Ian N. James
University of Reading
[email protected]
School of Mathematics Meteorology & Physics
4
Fluid instability
• Positive feedback
between trains of
Rossby waves
• Stationary wrt each
other: sheared flow,
change of sign of [q]y
• Barotropic instability
(horizontal)
• Baroclinic instability
(vertical)
Ian N. James
University of Reading
[email protected]
School of Mathematics Meteorology & Physics
5
Definition of PV gradient
• Quasi-geostrophic form:
qy    U yy 
f2
f2
 ln N 2 
U zz  2 U z
2
N
N
z
3
E
(
k
)

k
• Barotropic: if
then q y  k 1 / 2
5 .2
2
U
5 .1
UiU0
1
qyi.iy
[q]y
5 .0
0
0 16
2
0 16
4
0 16
6
0 16
8
Ian N. James
University of Reading
[email protected]
School of Mathematics Meteorology & Physics
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20.1
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Towards a Rossby wave programme
• Rationale for smoothing of PV field?
• How do Rossby waves propagate in the
presence of small scale [q]y anomalies?
• Scattering of Rossby waves in an
inhomogeneous medium?
Ian N. James
University of Reading
[email protected]
School of Mathematics Meteorology & Physics
7