Leeside Troughs and Fixed Rossby Waves: Quasigeostrophic Explanations
1. Observed Planetary Wave Patterns
After upper air observations became routine, it became easy to produce
contour plots of upper air patterns. Long term mean contour plots showed
that in the Northern Hemisphere winter, three major long wave
troughs/ridges appeared on average in the middle troposphere (Fig. 1),
whereas the same patterns were not observed in the Southern Hemisphere
(Fig. 2). Evidently, there are factors at work in the Northern Hemisphere
not active in the Southern Hemisphere.
Figure 1: Average January 500 mb Heights, 1975-2006, Northern
Hemisphere
Figure 2: Average January 500 mb Heights, 1975-2006, Southern
Hemisphere
In addition, it was noted that the three mean long waves observed in the
Northern Hemisphere weaken with height (Fig. 3). In other words, their
amplitudes dampen with height until the flow is nearly zonal above 200 mb
or so.
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Figure 3: Average January 700 mb Heights, 1975-2006, Northern
Hemisphere
Our understanding of these phenomena came out of the work of Carl
Rossby. We_ve already looked at some of the background on this in
previous handouts.
2. Implications of Conservation of Absolute Circulation: Ertl_s Potential
Vorticity
Last semester, we derived the simplified vorticity equation from the
Principle of Conservation of Absolute Circulation. Recall that vorticity is
circulation per unit area, or
 a  Ca A
(1)
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
If there are no solenoids, and no friction, assume that absolute circulation is
conserved. Thus, (1) becomes
d( a A)
dt 
dCa
dt  0
(2)
or
 a A  Constant

(3)
Equation (3) can be used to derive the Simplified Vorticity Equation. Then
by assumption of non-divergence, the equation reduces to Conservation of
Absolute Vorticity, which can then be used to understand Rossby Waves.

We now use Equation (3) to develop another important _conservation_
principle. Conservation of Mass states that the volume of an air column is
constant, or
Volume = Area of Air Column X Depth = Constant
(4a)
Or
Area of Air Column = (Constant)/Depth
(4b)
Put Equation (4b) into (3) to obtain a relationship between the absolute
vorticity and the depth of the air column by embedding both constants on
the right side of the equation.
a
Depth  Constant =  Pot
(5)
Equation (5) states that the ratio of the absolute vorticity to the depth of the
air column (in a barotropic system) is constant. This ratio is known as
Potential Vorticity.

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Since this derivation was predicated on Conservation of Absolute
Circulation, it is important to note that Equation (5) will help you to
understand characteristics of only the features in the large scale flow that
are barotropic or equivalent barotropic (such as Rossby Waves), and not
baroclinic waves.
With that in mind, consider a zonal jet stream in which there is no
northward variation in u (no horizontal speed shear) approaching a
mountain range (see Fig. 4).
Figure 4: Plan and Cross-section Views showing differential development
of leeside trough due to conservation of potential vorticity
As the depth of the air column decreases, Equation (5) states that it_s
absolute vorticity must also decrease. Since we are at the core of the jet
stream in which there is no horizontal shear, this decrease in absolute
vorticity must show up as either anticyclonic curvature relative vorticity
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
and/or a decrease in latitude (f). In either case, a southward turn will
develop (a ridge). Downwind of the mountain range the opposite occurs,
leading to troughs down wind of major mountain ranges.
In order for the topography to have this effect, the mountain range must
have a width and depth of synoptic-scale dimensions and be oriented at
right angles to the flow. That is to say, 1000 km or so in diameter and a
good fraction of the troposphere in depth. The complex of the highlands of
western North America and eastern Asia fulfill these criteria. Studies have
shown that the level above which the underlying topography has no effect
on flow patterns (the so-called _nodal_ surface) is around the 200 mb level,
or the top of the troposphere.
Note that, in this model, the air stream approaches its original latitude
downwind of the mountains at a 45 degree angle. This means that it will
overshoot its original latitude and produce a train of Rossby
Conversion of (5) into isentropic coordinates yields


 a  p Constant =  PotErtel's


(6)
the factor _theta /_p is related to the depth of the air column as shown in
the figure below. Recall that since potential temperature increases with
height (except in the unusual circumstance of absolute instability), the term,
_theta/_p is negative. Ertl_s Potential vorticity is conserved for frictionless,
adiabatic (isentropic) flow and is generally positive (since absolute vorticity
is almost always positive).
In the figure, consider the bounds of the air column at 200 mb and at the
surface. Since the motion is adiabatic, the two streamlines shown are also
isentropes. Note that on the windward side of the mountain the spacing
between the isentropes gets smaller_hence _theta/_p gets more largely
negative. In order for the product in (6) to remain constant, the absolute
vorticity must decrease. In the core of the jet (in which there is no shear
relative vorticity), this can only be accomplished if the Coriolis parameter
experienced by the air column is decreased; i.e, the air column turns
southward. This produces a ridge on the upstream side of the mountain.
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On the downstream side of the mountain, the vertical gradient of potential
temperature is decreased, and the the absolute vorticity of the air column
must increase, to keep the product in (6) constant. Thus, a trough that
weakens with height is found on the eastern side of major mountain ranges,
in these circumstances.
The factor -_theta /_p is also directly related to the static stability parameter.
In the diagram above vertical shrinking and horizontal divergence will lead
to an increase in -_theta /_p. This is consistent with what we learned last
semester: that the more stable the atmosphere, the more closely spaced the
isentropes in vertical cross-section.
3. Implications of the QG-omega Equation
The simplified Equation of Continuity (synoptic-scaling) is

  V  DIVh
p
(1)
which says that layer horizontal divergence is related to the vertical
motion field.

Under weak synoptic forcing (no or weak differential vorticity advection
and no or weak temperature advection, and with no frictional and/or
diurnal heating effects), there are no forcing terms on the right hand side
of the equation.
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f 02  2 2
 
0
2
 p
or
2
2
2 2
f


2
0
  
 p 2
(2)
The QG omega equation can be rewritten as given in (3) [with the
substitution of equation (1) into the right hand side of (2)].



f 02  2 2
DIVh 
  



p
 p 2
2
(3)
Inverting the Laplacian on the left gives the approximate equation
DIV h 

 
p
(4)
which says that in the layer in which omega is positive (subsidence), then
the derivative on the right returns a positive value.
In the case of leeside sinking, there is no omega at the nodal surface, and
there is great sinking at the surface. If the nodal surface is at 200 mb, a
finite difference version of the far right hand term says that divergence must
become more positive with height (more negative with decreasing height) if
downward motion is occurring. Since subsidence associated with
topography is zero at the nodal surface and maximum at the ground, this
implies that there is no divergence at the nodal surface. For divergence to
increase with height from the ground, this implies that convergence occurs
at the ground on the lee side of mountain ranges.
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This effect is often seen east of major mountain ranges as the development
of a so-called "leeside low" at the surface that weakens with height. This
low is not baroclinic, is not associated with pre-existing temperature or
vorticity advection. However, once the low is in place (often in the spring
and early summer east of the Rockies, for example) significant moisture
advection can occur east of it, and, actually, a synoptic scale "warm front"
like feature can develop as well.
At the larger scale, this implies that long wave troughs should occur east of
major mountain ranges. Again, "major" is defined as a mountain range
having a depth through at least the middle troposphere, and a width at least
as large as the width of the jet. Thus, stationary waves as shown in Figs. 1
and 3 can be explained on the basis of (4).
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