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1
The QG Vorticity equation
The complete derivation of the QG vorticity equation can be
found in Chapter 6.2.1. Please read this section and keep in mind
that extra approximations have been made besides those that lead to
the vorticity equation (4.22). Those approximations include
i. Va ! Vg or Va / Vg "O(Ro)
ii. The advection velocity is geostrophic
iii. f ! f0 + "y , i.e., midlatitude beta-plane
approximation
The resultant QG vorticity equation is
Dg !g + f( )Dt
= " f0#$#p
or
!"g!t
= #Vg $% "g + f( ) + f0 !&!p (6.19)
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In Cartesian coordinates the geostrophic wind (with constant-f)
is defined as
Vg ! f0"1k # $%
Thus, the geostrophic vorticity, !g = k "# $ Vg , can be
expressed as
!g ="vg"x
#"ug"y
=1f0
"2$"x2
+"2$"y2
%&'
()*=1f0+2$ (6.15)
For a pure 2-D motion, the vorticity equation can be written
as
!!t"2# = $ $
1f0
!#!y
!!x
+1f0
!#!x
!!y
%&'
()*("2# + f ) (4.28’)
For a 3-D QG flow, we have to deal with the stretching term as
well, we will demonstrate how to express the conservation of QG PV
in terms of geopotential height in the QG PV equation (from page
5).
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2
Example of vorticity advection for a 2-D case
(4.28’) can be rewritten as
!Vg "# $g + f( ) = !Vg "#$g ! %vg The two term on the rhs
represent the geostrophic advections of relative vorticity and
planetary vorticity, respectively. For disturbances in the
westerlies, these two effects tend to have opposite signs.
Advection of relative vorticity tends to move the vorticity
pattern and hence the troughs and ridges eastward (downstream).
However, the advection of planetary vorticity tends to move the
troughs and ridges westward against the advecting wind field. The
latter motion is called retrograde motion or retrogression.
The net effect of advection on the evolution of the wave pattern
depends on the scale of the wave perturbations. To demonstrate the
point, we consider an idealized geopotential distribution on a
beta-plane, consisting of the sum of a zonally averaged part, which
depends linearly on y, and a zonally varying part (wave part,
representing a synoptic wave disturbance) that has a sinusoidal
dependence in x and y:
!(x, y) = !0 " f0Uy + f0Asin kx cos ly (6.20)
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Here y = a(! " !0 ) .
Then the corresponding geosstrophic wind are given by
ug = !
1f0
"#"y
=U + lAsin kx sin ly
vg =1f0
"#"x
= kAcoskx cos ly
The geostrophic vorticity is then simply
!g = f0"1#2$ = "(k2 + l2 )Asin kx cos ly
With the aid of these relations it can be shown that in this
simple case the advection of relative vorticity by the wave
component of the geostrophic wind vanishes and so that the
advection of the relative vorticity is
!ug"#g"x
! vg"#g"y
= !U"#g"x
= +kU(k2 + l2 )Acoskx cos ly
While the advection of the planetary vorticity is
!"vg = !"kAcoskx cos ly
Therefore, if k2 + l2( ) < ! /U , the synoptic wave should
move westward (retrogression) and if k2 + l2( ) > ! /U , the
wave should move eastward. And waves of intermediate wavelength,
i.e., k2 + l2( ) = ! /U can be stationary.
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The 3-D dynamics governed by the conservation of QG PV is far
more complex, a topic to be discussed next.
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The QG Potential Vorticity equation (3D)
The QG PV equation is derived from utilizing both the QG
vorticity equation and the thermodynamic equation.
From the homework of Chapter 2
!T!t
+ u!T!x
+ v!T!y
"#$
%&'+ w((d ) () =
JCp
(2.55’)
Recall that ! " #$gw and that %d # %( ) / $g = Sp & #
T'('(p
,
So,
!T!t
+ Vg "#hT$%&
'()*
+ pR
$%&
'(), = J
Cp (6.13a)
where ! " #RT0 p#1d ln$0 / dp and !0 is the potential
temperature corresponding to the
basic state temperature T0 . ! " 2.5 #10$6m2Pa$2s$2 in the
mid-troposphere.
From (6.2), that is
!"!p
= #$ = # RTp
(6.13a) can be expressed as
!!t
+ Vg "#h$%&
'()
*!+!p
$%&
'()* ,- = . J
p (6.13b)
Multiplying (6.13b) through by f0!
and differentiating with respect to p yields:
!!p
f0"
!#!p
$%&
'()+
!!p
f0"Vg *+
!,!p
$%&
'()
-
./
0
12 + f0
!3!p
= 4 f0!!p
5 J" p
$%&
'()
(6.22)
where we use ! to denote !"!t
and ! " R /Cp .
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While the QG vorticity equation can be written in terms of
geopotential tendency as follows.
1f0!2" + Vg #!
1f0!2$ + f
%&'
()*= f0
+,+p
Substituting it in (6.22) through f0!"!p
, we have
1f0!2 +
""p
f0#
""p
$%&
'()
*
+,
-
./ 0 + Vg 1!
1f0!22 + f
$%&
'()+
""p
f0#Vg 1!
"2"p
$%&
'()
*
+,
-
./ = 0 (6.23)
for the adiabatic motion.
Noting that ! = ! (p) , the equation above can be reorganized
as
!!t
1f0"2 +
!!p
f0#
!!p
$%&
'()
*
+,
-
./0 + Vg 1"
1f0"20 + f + !
!pf0#
!0!p
$%&
'()
*
+,
-
./ +
f0#
!Vg!p
1"!0!p
$%&
'()= 0
However, for the thermal wind relation, f0!Vg / !p = k " # !$ /
!p( ) , so the last term drops off the equation and we have
finally:
!!t
+ Vg "#$%&
'()q =
DgqDt
= 0 (6.24)
where q is the quasi-geostrophic potential equation defined
by
q ! 1f0"2# + f + $
$pf0%
$#$p
&'(
)*+
(6.25)
For diabatic flow, the QG PV equation should be
DgqDt
= ! f0""p
# J$ p
%&'
()*
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Geopotential tendency
(6.23) is often referred to the geopotential tendency equation
as it can be used for the purpose of diagnosing the tendency of the
geopotential.
!2 +""p
f02
#""p
$%&
'()
*
+,
-
./ 0
A! "### $###
= 1 f0Vg 2!1f0!23 + f
$%&
'()
B! "#### $####
1""p
1f02
#Vg 2! 1
"3"p
$%&
'()
*
+,
-
./
C! "#### $#### (6.23)
A: local geopotential tendency
B: vorticity advection
C: vertical differential thickness advection
Note the !2" # $" . So equation (6.23) provides an immediate
qualitative implication for the tendency of geopotential with known
distribution of ! . Term B is usually the dominant forcing term in
the upper troposphere, wherein a rising (falling) geopotential is
associated with negative (positive) advection of absolute
vorticity. Note that the advection term, as its name implies, does
not change the strength of the disturbance at the levels where the
advection is occurring, but only acts to propagate the disturbance
horizontally and to spread it vertically as will be shown later in
the lecture.
Term C represents a major mechanism for amplification or decay
of midlatitude synoptic system. Term C is also proportional to the
minus the rate of change of temperature advection with respect to
pressure (i.e., plus the rate of change wrt height). It is also
referred to as differential temperature advection.
Differential temperature advection enhances upper level height
anomalies in developing disturbances. Below the 500-hPa ridge there
is strong warm advection associated with the warm front, whereas
below the 500-hPa trough there is strong cold advection associated
with the cold front. The former increases thickness, thus builds
the upper level ridge; the later decreases the thickness thus
deepens the upper level trough. Above the 500-hPa level the
temperature gradient is usually flatter, advection tends to be
small. Thus, in contrast to Term B in (6.23), the effect of forcing
term C is concentrated in the lower troposphere.
In the region of warm advection !Vg "#(!$% / $p) > 0 , as Vg
gas a component down the temperature gradient. But the advection
also decrease with height, therefore
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! "Vg #$("!% / !p)&' () / !p > 0 . Conversely, beneath
the 500-hPa trough there is cold advection decreasing with height,
that gives rise to deepening effect on geopotential.
! = "#"t!
""p
$Vg %& $"#"p
'()
*+,
-
./
0
12
> 0 at the ridge< 0 at the trough
345
65
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The temperature advection pattern described above indirectly
implies conversion of potential energy to kinetic energy.
