Rossby wave propagation
Propagation…
Three basic concepts:
• Propagation in the vertical
• Propagation in the y-z plane
• Propagation in the x-y plane
1. Vertical propagation
•Reference back to Charney & Drazin (1961)
•Recall the QG equations from MET 205A
•QGVE


 Vg   g  f   f o
t
p
•QGTE
T
p
 Vg T 

t
Rd
Vertical propagation
•These were combined as follows:
•We defined a tendency  and developed the tendency
equation (and used it to diagnose height tendencies)
•We also developed the omega equation.
•A further equation – not much emphasized – was the quasigeostrophic potential vorticity equation (QGPVE), found by
elimination of vertical velocity.
Vertical propagation
•If we use log-pressure vertical coordinates, the result is:



V


g

q  0
 t

•Where
2
f
   
2
o
q   f 
 o

2
o N z  z 
Vertical propagation
•Here,  is streamfunction.
•We now linearize this in the usual way, assuming a constant
basic state wind U (see Holton pp 421-422).
•We next assume the usual wave-like solution to the linearized
equation:
 '( x, y, z, t )  ( z)exp i(kx  ly  kct )  z / 2H 
•Where z is log-pressure height, H is scale height, and
everything else is as usual.
Vertical propagation
•Upon substitution, we get the vertical structure equation for
(z):
d 
2

m
0
2
dz
2
•Where:
2
N
m  2
fo
2
1
 
2
2 
U  c  (k  l )   4 H 2
Vertical propagation
•Obviously, the quantity m2 is crucial – as it was in the vertical
propagation (or not!) of gravity waves.
•When m2 > 0, the wave can propagate in the vertical, and
(z) is wave-like.
•When m2 < 0, the wave does not propagate, and the
solution decays expenentially with height (given the normal
upper BCs).
Vertical propagation
•Specializing to stationary waves, where c=0, we have:
•Stationary waves WILL propagate in the vertical if the mean
wind U satisifies:
1
 2 2

f
0  U   (k  l ) 
 Uc
2
4N H 

2
o
2
•i.e., we require 0 < U < Uc … U must be westerly but not too
strong.
Vertical propagation
•If the mean wind is easterly (U < 0), stationary waves cannot
propagate in the vertical.
•Likewise, if mean winds are westerly but too strong, there is no
propagation.
•What does this tell us about the the observed atmosphere?
Vertical propagation
•Observations? See ppt slide…
•Obs show the presence of stationary planetary-scale (Rossby)
waves in winter (U>0) – but not in summer (U<0).
•The theory above helps us to understand this.
•Further – the results are wavenumber-dependent.
•Consider winter and assume 0<U<Uc.
Vertical propagation
•Note that “m” depends on zonal scale (Lx) thru k:
Uc 

 2 2
f o2 
(k  l )  4 N 2 H 2 


•Consider zonal waves N=1, 2, and 3.
•Lx decreases as N increases, which means that k increases as
N increases, which means that Uc decreases as N increases.
•So propagation becomes more difficult as N increases…the
“window of opportunity” [0<U<Uc] shrinks as N increases.
Vertical propagation
•This means that we are MOST LIKELY to see wave 1 in the
stratosphere, less likely to see wave 2, and even less likely to
see waves 3 etc. – precisely as observed!
Uc 

 2 2
f o2 
(k  l )  4 N 2 H 2 


•Thus, stratospheric dynamics (at least for stationary waves) is
dominated by large-scale waves.
Vertical propagation - summary
•For stationary waves, theory verifies observations (or vice
versa) that the largest waves can propagate vertically when
flow is westerly, but not easterly.
•Thus we expect large-scale waves, but not transient eddyscale waves to propagate upward (smaller waves are trapped
in the troposphere).
•Theory gets more complicated if we let U=U(z) – see Charney
& Drazin.
2. Propagation in the y-z plane
•Reference back to Matsuno (1970)
•Matsuno extended these ideas to 2D (y-z)
•These ideas were also developed in part II of the EP paper.
Propagation in the y-z plane
•Matsuno again considered a QG atmosphere, this time in
spherical coordinates (Charney & Drazin – beta plane).
•He also considered the linearized QGPVE, and this time
assumed a more general solution of the form:
 '( x, y, z, t )  ( y, z)exp ik ( x  ct )
•He allowed U=U(y,z) now, and thus the amplitude of the eddy
{(y,z)} is also a function of y and z – this is to be solved for.
•Overall this is more realistic (than U=constant).
Propagation in the y-z plane
•Matsuno thus obtained a PDE for the amplitude:
f2
  cos    f 2  2 
2
 ns   0.
 2
 2
2
2
a cos    f
  N z
•A second order PDE for amplitude, which was solved
numerically.
•The only thing to be prescribed was the mean wind, U, which
was taken from an analytical expression to be representative of
the observed atmosphere.
Propagation in the y-z plane
•In the equation, we have
2
2
s
f
2
ns 
 2

2
aU a cos  4 N 2 H 2
q
•Here, an important term is s = zonal wavenumber (integer).
•The quantity ns2 acts as a “refractive index”, as we will see,
and note here that it depends on the mean wind (U) and on
wavenumber (s).
Propagation in the y-z plane
•The results? Matsuno computed structures (amplitude and
phase… is assumed complex) for waves 1 and 2.
•Matsuno found qualitatively good agreement between his
results and observations, in both phase and amplitude.
•In particular, in regions where ns2 is negative, wave amplitudes
are small, indicating that Rossby waves propagate away from
these regions.
•Conversely, in regions where ns2 is positive and large, wave
amplitudes are also large.
Propagation in the y-z plane
Propagation in the y-z plane
Propagation in the y-z plane
Propagation in the y-z plane
•In fact, we can develop – based on the EP paper – a quantity
called the Eliassen-Palm Flux vector (F) and use it to show
wave propagation.
•Without going deep into details, we can write for the QG case:
o f o Rd
Fy  o u ' v ',Fz 
v 'T '
2
N H
•It can be shown that the direction of F is the same as the
direction of wave propagation (F // cg), and also that div(F)
indicates the wave forcing on the mean flow.
•See Holton Cht 10, 12 for more.
Propagation in the y-z plane
Propagation in the y-z plane
Summary
•Planetary-scale (stationary Rossby) waves can propagate both
vertically and meridionally through a background flow varying
with latitude and height.
•The ability to propagate can be measured in terms of both a
refractive index, and the EP flux vector.
•Both will be used in the next section on propagation in the x-y
plane.