Fixed Rossby Waves: Quasigeostrophic Explanations and
Conservation of Potential Vorticity
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 3a: Average January 200 mb Heights, 1975-2006, Northern
Hemisphere
Our understanding of these phenomena came out of the work of Carl
Rossby. We have already looked at some of the background on this in
previous handouts.
2. Implications of Conservation of Absolute Circulation: Conservation of
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
z a = Ca A
(1)
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Figure 3b: Average January 700 mb Heights, 1975-2006, Northern
Hemisphere
If there are no solenoids, and no friction, assume that absolute circulation is
conserved. Thus, (1) becomes
d(z a A)
dt
=
dCa
dt
=0
(2)
or
z a A = Constant
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(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) (which assumes barotropy—no variation in
density) to develop another important conservation principle. The principle
of conservation of mass states that the volume of an air column (in which
density does not vary) is constant, or
Volume = Area of Air Column X Depth = Constant
V = A ∆z = k
(4a)
Or
A= k/∆z (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.
za
∆z
=z Pot = k
(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 and the fact that, given the constraints used in obtaining
the equation, the ratio does not change for a given air parcel is known as
Conservation of Potential Vorticity.
Since the derivation of Equation (5) 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. Since absolute circulation tends to be conserved, so too
does Potential Vorticity tend to be conserved.
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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 its
absolute vorticity must also decrease. Since the case considered here is 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 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
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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 Rossby Wave trough.
Downstream from the trough axis, the wind will again be approaching its
original latitude, again at a 45 degree angle, and will again overshoot. Thus
this process should, in theory, produce a train of Rossby Waves.
In reality, such an infinite train of waves will not occur because there are
torques, including friction, that will act to dampen the waves fairly quickly.
But it is possible that the Rossby Wave trough that appears, on average,
over eastern Europe is really part of a train of waves stimulated by the
western highlands of North America.
3. Conservation of Isentropic Potential Vorticity (IPV)
Conversion of (5) into isentropic coordinates yields
( ¶p) = -g(z + f ) (¶q ¶p) =z
-gz aq ¶q
q
ErtlsPotVort
=k
(6)
The quantity to the left of the equals sign is known as Ertl’s Potential
Vorticity or Isentropic Potential Vorticity (IPV).
The factor ∂theta /∂p is related to the depth of the air column as shown in
Fig. 5 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).
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In Fig. 5, 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.
Figure 5: Cross-section s showing effect of topography on depth of air
columns and vertical gradient of potential temperature for westerly crossmountain flow.
On the downstream side of the mountain, the vertical gradient of potential
temperature is decreased, and, therefore, 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.
From the definition of PV above, one can see that the units are Kkg-1m2s-1.
For the purposes of contouring on maps, it is convenient to define one PV
unit as 1 PVU= 10-6 Kkg-1m2s-1. Since IPV is conserved in adiabatic
frictionless flow, we will see that it can be used to trace large scale motions
in the atmosphere.
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4. Implications of the QG-omega Equation
The simplified Equation of Continuity (synoptic-scaling) is
(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.
(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)].
(3)
Inverting the Laplacian on the left gives the approximate equation
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w Ȧ(
DIVh )
¶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.
5. Leeside Low Development and Anchored Rossby Waves
The effects of the phenomena described in sections 2, 3 and 4 above
manifest themselves east of major mountain ranges as the development of
so-called "leeside lows" or troughs at the surface that weaken with height
(Fig. 6).
Figure 6. GFS forecast for 12 UTC 16 March 2009, (left to right) 300 mb,
500 mb, and Mean Sealevel. At the latitude and longitude of the Rocky
Mountains near zonal flow is occurring at 300 mb, a deformation to the
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streamlines evident at 500 mb, and a profoundly deep leeside low is found
at the surface.
These have warm core and have, essentially, the structure of a thermal low.
These lows are not baroclinic and are not associated with pre-existing
temperature or vorticity advection. However, once such a 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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