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VOLUME 69
The Usefulness of Piecewise Potential Vorticity Inversion
BJØRN RØSTING
Norwegian Meteorological Institute, Oslo, Norway
JÓN EGILL KRISTJÁNSSON
Department of Geosciences, University of Oslo, Oslo, Norway
(Manuscript received 29 April 2011, in final form 10 October 2011)
ABSTRACT
It is today widely accepted that potential vorticity (PV) thinking is a highly useful approach for understanding important aspects of dynamic meteorology and for validation of output from state-of-the-art
numerical weather prediction (NWP) models. Egger recently presented a critical view on piecewise potential
vorticity inversion (PPVI). This was done by defining a PV anomaly by retaining the observed PV field in
a specific region, while changing the observed PV fields to zero elsewhere. Inversion of such a modified PV
field yields a flow vastly different from the observed. On the basis of this result it was argued that PPVI is
useless for understanding the dynamics of the flow.
The present paper argues that the results presented by Egger are incomplete in the context of PPVI, since
the complementary cases were not considered and that the results also depend on the idealized model formulations. The complementary case is defined by changing the observed PV to zero in the specific region,
while retaining the observed PV field elsewhere.
By including the complementary cases, it can be demonstrated that the streamfunction fields associated with
the PV and boundary temperature anomalies presented by Egger add up to yield the observed streamfunction
field, as expected if PPVI is to be valid. It follows that PPVI is indeed valid and useful in these cases.
1. Introduction
Hoskins et al. (1985) provided a theoretical survey
of the concept of potential vorticity (PV), along with
applications of PV and low-level boundary temperature anomalies to the interpretation of atmospheric
dynamics. Although the PV aspect of atmospheric dynamics had been well known for several decades, having been described by Rossby (1940), Ertel (1942), and
Charney (1947), the paper by Hoskins et al. introduced,
most likely for the first time, the concept of ‘‘PV thinking.’’ PV thinking means adopting PV in order to describe and understand the dynamics of the atmosphere,
such as baroclinic and barotropic instability. This is
achieved, for instance, through PV inversion, by which
geopotential heights, winds, and temperature fields are
Corresponding author address: Bjørn Røsting, Norwegian Meteorological Institute, Forecasting Division, POB 43, Blindern,
0313 Oslo, Norway.
E-mail: bjorn.rosting@met.no
DOI: 10.1175/JAS-D-11-0115.1
Ó 2012 American Meteorological Society
obtained, provided that suitable balance and boundary
conditions are specified.
The classical approach to baroclinic instability, described by Eady (1949), is essentially based on the PV
concept, in which the (quasigeostrophic) PV field confined between an upper lid (i.e., the tropopause) and the
surface is specified, along with the temperature distribution at lower and upper boundaries. Analytical solutions of this problem yield growing waves with the
largest growth rate for wavelengths on the order of
4000–6000 km.
PV can also be applied in identifying errors in numerical weather prediction (NWP) analyses and simulations
(e.g., Mansfield 1996). This is achieved by comparing the
PV fields retrieved from NWP models with features observed in water vapor (WV) images. This method is based
on the strong absorption of radiation in the 6.3–6.7-mm
water vapor bands; hence PV-rich stratospheric air
is easily identified in WV images (e.g., Santurette and
Georgiev 2005). When errors in the PV fields are identified by an observed mismatch between PV fields and
WV features, it is possible to manually correct the PV
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fields. The corrected PV fields are then inverted, yielding
a modified numerical analysis that is balanced and suitable for initializing numerical reruns (e.g., Demirtas and
Thorpe 1999; Manders et al. 2007; Røsting et al. 2003;
Røsting and Kristjánsson 2006, 2008; Verkley et al. 2005).
These procedures represent an efficient use of PV
thinking.
An important aspect of PV thinking is piecewise PV
inversion (PPVI). In PPVI, the PV field is partitioned,
and the contributions of the different parts of the PV
field to the flow are quantified by piecewise inversion
(Davis 1992; Kristjánsson et al. 1999). The feasibility of
such an approach for Ertel’s nonlinear PV was first demonstrated by Davis and Emanuel (1991).
Criticism of PPVI pertaining to its usefulness for understanding atmospheric dynamics was recently put forward by Egger (2008, hereafter E2008). Although some
of the arguments concerning PPVI in E2008 may be
pertinent, particularly the discussion of inverted tendency
fields, others call for comments. The criticism advanced in
E2008 is based on experiments to be discussed in section 3
below, in which the observed PV is replaced by PV 5 0 in
a selected region. Inversion of such a modified PV field
yields a flow different from the observed flow. In E2008 it
was claimed that such a result implies that PPVI in general is useless for understanding the dynamics of the flow.
Below, we present evidence that this drastic claim is unjustified. Methven and de Vries (2008) addressed the
limitations of the mathematical model describing the
baroclinic cases of E2008 and pointed out the importance
of dynamic evolution rather than pure diagnosis based on
PPVI in assessing the benefits of PV thinking.
This paper is structured as follows. In section 2 we give
a brief presentation of the PV inversion technique used
in many case studies. In section 3 we address some of the
cases presented in E2008. A case study based on PPVI is
presented in section 4, and discussion and conclusions
are provided in section 5.
We now recapture the PPVI technique, since knowledge
of this technique is essential for the subsequent discussion.
In general, studies requiring PV inversion adopt the
Ertel PV:
1
h $u,
r
Dq
›q
1
1
[
1 u $q 5 h $u_ 1 $ 3 Fr $u,
Dt
›t
r
r
(1)
where r is density, h is the absolute vorticity vector, and
$u is the three-dimensional gradient of potential temperature. PV is expressed in PV units (PVU; 1 PVU 5
1026 m2 s21 K kg21).
(2)
where u is the wind vector, Fr is the friction force, and u
denotes diabatic heating.
The mathematical expression for the Ertel PV can be
simplified by using isentropic coordinates (x, y, u):
›u
,
q 5 (zu 1 f ) 2g
›p
(3)
where zu is relative vorticity on an isentropic surface and
f is the Coriolis parameter (planetary vorticity), g is the
acceleration of gravity, and 2›u/›p is an expression of
static stability.
Relations (2) and (3) show that for adiabatic and inviscid flow q is conserved for an air parcel moving on
an isentropic surface.
Potential temperature fields ub are specified at the
lower and upper boundaries (e.g., at 950 and 150 hPa)
and the ub anomalies can be regarded as PV anomalies;
that is, a warm (cold) anomaly at the lower boundary can
be regarded as a positive (negative) PV anomaly. Likewise, a warm (cold) anomaly at the upper boundary can
be treated as a negative (positive) PV anomaly (Hoskins
et al. 1985). Hence such anomalies can be inverted
separately.
The total PV and boundary potential temperature
fields are decomposed into mean and perturbation terms.
Hence the total PV is expressed as
q 5 q 1 q9
(4)
and the expression for the boundary potential temperature field is analogous:
ub 5 ub 1 ub9 .
2. Piecewise PV inversion
q5
PV is changed because of diabatic effects and friction
as shown in the following equation:
(5)
The mean fields q and ub are temporal or spatial means,
with temporal means taken as an average over, for example, 48 h.
The perturbation terms in Eqs. (4) and (5) may be
decomposed into several anomalies. Usually one is interested in specific PV and ub anomalies that appear to
have a large influence on the phenomena of interest. Such
anomalies are usually clearly defined as temporally and
spatially coherent features.
PV anomalies that are far away from the region of interest or difficult to define temporally are generally defined as parts of a residual term, denoted by qres.
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Following Kristjánsson et al. (1999), the perturbation
field in Eq. (4) is expressed as
n
q9 5
å qi ,
(6)
i51
where qi denotes the anomalies.
For N selected PV anomalies the above relation becomes
N
q9 5
å qi 1 qres ,
(7)
i51
where the residual term is
n
qres 5
å
i5N11
qi .
(8)
The expressions for the ub fields are similar.
PPVI deals with obtaining uniquely the familiar fields
of geopotential height or streamfunction, winds, and temperature distribution associated with specific PV and
ub anomalies.
To close the set of equations required for PV inversion a balance condition is required, as well as lateral
and horizontal boundary conditions. We adopt Charney’s
balance condition (Charney 1955) given by
›(›C/›x, ›C/›y)
,
= 2 F 5 $ ( f $C) 1 2
›(x, y)
(9)
where F is the geopotential and C is the streamfunction.
The reason why Ertel’s PV Eq. (1) and the Charney
balance condition [Eq. (9)] are often preferred in PV
inversion studies is that the flow is properly described by
Eqs. (2) and (9), including flow with fairly large Rossby
numbers (e.g., Ro ; 1), allowing for strongly curved
flow. The disadvantage of adopting these equations directly is their nonlinearity. This means that the streamfunction fields associated with their respective PV and
ub anomalies, as well as contributions from the mean and
residual fields of PV and ub, do not add up to yield the
total streamfunction field, as they do for quasigeostrophic
PV (QPV).
Davis and Emanuel (1991) and Davis (1992) demonstrated how the nonlinearity problem can be resolved.
For completeness this will now be briefly recaptured.
Adopting the hydrostatic relation on the right-hand
side of Eq. (1) yields a relation for q containing products
of second derivatives of C and F [see, e.g., relation (1.4)
in Davis (1992) for details]. Adopting this relation,
Davis and Emanuel (1991) demonstrated how Ertel’s PV
VOLUME 69
can be inverted yielding unambiguous results, as for inversion of QPV. The method is referred to as the full
linear (FL) method because all terms are retained through
collecting the nonlinear terms in the coefficient of the
linear operator.
Davis (1992) presented two different PV inversion
methods: subtraction from the total (ST) and addition to
the mean (AM). He demonstrated that the average of
these two methods yields results close to those of the FL
method except for cases when the PV perturbations are
much larger than the mean PV field.
The above description of the FL, ST, and AM methods
applies to boundary temperature anomalies as well.
Rather than adopting the FL method we use the
combination of the ST and AM methods in the case
study presented in section 4 below. The method has also
been successfully adopted in several case studies (e.g.,
Kristjánsson et al. 1999; Thorsteinsson et al. 1999;
Røsting and Kristjánsson 2006).
Although quasigeostrophic theory is a weak anomaly
theory, the inversion of QPV is useful for demonstrating
the principle of PV inversion and in theoretical studies
of atmospheric flows (e.g., the Eady model). The results
from such studies in E2008 lead to the conclusion that
PPVI in many cases fails to contribute to our understanding of the dynamics of the flow. Since PPVI has been
adopted in many recent case studies (e.g., Bracegirdle
and Gray 2009; McInnes et al. 2009) and constitutes
a central part of PV thinking, the criticism in E2008
needs to be addressed.
We now discuss some of the results presented by
E2008. The main purpose of the following discussion is
to investigate to what extent PPVI is valid and useful in
the cases adopted in E2008.
3. Results from elementary tests of piecewise
PV inversion
In E2008, in connection with Fig. 1, it was stated that
the winds in region D2 are induced by the positive PV
anomaly Z1 in region D1. However, basing our arguments on the PPVI technique (as described in the previous section), we note that the winds induced by Z1
only contribute to the total wind field in region D2 in the
same way as one of the components qi in Eq. (6) is only
one of many contributions to the total perturbation field.
To what extent Z1 influences the winds in D2 depends
on the distance between the PV anomalies Z1 and A2
and their amplitudes.
The same argument is valid for the ‘‘complementary
case’’ (i.e., the winds associated with the negative PV
anomaly A2 contribute to the total wind field in region
D1 and may influence Z1 to a certain extent). The total
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FIG. 1. Schematic presentation of a domain D divided into subdomains D1 and D2; Z1 and A2 represent positive and negative PV
anomalies, respectively. Reproduced from Egger (2008).
streamfunction ctotal in D2 is then obtained through adding the contributions from A2 and Z1. The same procedure yields ctotal in D1. Hence performing inversions for
both regions D1 and D2 is a necessary requirement for
assessing the validity of PPVI.
FIG. 2. Observed streamfunction cob in dotted lines, forming
ridges R1 and R2 and troughs T1 and T2 in regions D1 and D2,
respectively. Solid lines show c obtained from inversion of the PV
field after replacing the observed PV with PV 5 0 in region D2.
Reproduced from Egger (2008).
regions shown in Fig. 2, where q9 5 0 in region D1 and
q95 qob in D2:
a. The barotropic case: Rossby waves in zonal
mean flow
q9 5
In the equation for PV in a barotropic flow (e.g.,
Holton 2004) we may define the planetary vorticity f0 1
by as the mean PV field. Introducing a height increment
z9 5 g21f0c, we obtain through linearization an expression for perturbation PV:
q9 5 = 2 c 2 d 2 c,
(10)
where d21 5 (gH)1/2 f021 is the Rossby radius of deformation, with H being the mean height of the fluid.
We now consider a one-dimensional barotropic Rossby
wave: here Eq. (10) becomes
q9 5
›2 c
2 d2 c.
›x2
(11)
In this example from E2008, the observed streamfunction field is described by cob 5 P sin(kx) from which
the observed QPV anomaly field is calculated by inserting cob in Eq. (11).
The case presented in E2008 now proceeds as follows:
region D1 is defined for 2L/2 # x # 0 and D2 for 0 # x #
L/2 as shown in Fig. 2, reproduced from E2008. Following E2008, the PV field is modified by letting q9 5 0
in the region D2.
Figure 2 shows, besides cob, the streamfunction c9
obtained through inversion of the modified PV (solid
contours).
To test the validity of PPVI for the flow in this example, we now define the complementary case for the
0
in D1
.
2(k2 1 d2 )P sin(kx) in D2
(12)
Carrying out the inversion of PV in this case we obtain
the streamfunction c0, assuming, as for the case of c9,
that c0 and ›c0/›x are continuous at the origin. The solution is
c0 5
2B sinh[d(x 1 L/4]
P sin(kx) 1 B sinh[d(x 1 L/4]
in D1
,
in D2
(13)
where B 5 2Pk/[2dcosh(dL/4)]; that is, B is identical to
the coefficient A in the case presented in E2008.
The distribution of c0 in this complementary case
[Eq. (13)] is antisymmetric to that displayed in Fig. 2,
with respect to the origin.
Addition of the two complementary fields yields, for
D1 and D2, c 5 c9 1 c0 5 P sin(kx) 5 cob. Hence addition of the fields associated with the PV in D1 and D2
yields the total observed PV field, as expected for piecewise PV inversion.
b. Baroclinic waves in zonal shear flow
In the baroclinic case described in E2008 the flow is
confined between a surface plane and a rigid lid at the
top, with the height of the model atmosphere being H.
The dynamical development of the flow in this case is
described by the equations of the Eady model (e.g.,
Holton 2004; Egger 2008). Figure 3, reproduced from
E2008, shows the tropopause at Hs 5 3H/4.
Following E2008, an observed field is given as a modal
streamfunction:
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FIG. 3. Idealized presentation of a two-dimensional atmosphere with depth H 5 20 km. The tropopause is at z 5 3H/4 5 Hs, whereas n 5
3p/H and k 5 2p/L are the vertical and horizontal wavenumber, respectively. Shown are (a) the observed streamfunction (positive values are
solid contours, while negative values are dashed) and (b) the streamfunction obtained through inversion of the PV field after replacing the
observed PV with PV 5 0 in the troposphere (D2). The contours, as in (a), show the streamfunction. Reproduced from Egger (2008).
^ sin(kx).
cob 5 P sin(kx) sin[n(z 2 H)] 5 c(z)
(14)
^ and ›c/›z
^
(ii) Continuity requirements for c
at the
tropopause z 5 3H/4.
The horizontal wavenumber is k 5 2p/L, and n 5 3p/H
is the vertical wavenumber. Figure 3a, retrieved from
E2008, shows the structure of the observed streamfunction cob.
Now using quasigeostrophic PV, the perturbation is
expressed by
^ is shown in Fig. 3b.
The distribution of c
In the complementary case, not considered by E2008,
q9ob is confined to the troposphere (D2), while q9ob 5 0 in
the stratosphere (D1). This PV distribution yields the
following solution:
in D1 (stratosphere)
q9ob 5 =2 cob 1 ( f02 /N02 )(›2 cob /›z2 ),
^ 5 Ee2gz 1 Fegz ,
c
(15)
where N02 [ (g/u0 )(du0 /dz) is the squared Brunt–Väisälä
frequency, expressing static stability, here assumed to be
constant.
By inserting the expression for cob into Eq. (15), an
equation for the amplitude of the streamfunction in the
x–z plane is obtained:
^
N02 q^
›2 c
2^
c
5
2
g
,
›z2
f02
(16)
where g 5 kN0/f0 and q^ is the amplitude of the PV
perturbation.
With modifications of the observed PV—that is,
changing PV to q9ob 5 0 in the troposphere [i.e., below Hs
(D2)] while retaining the original q9ob in the stratosphere
[i.e., above Hs (D1)]—Egger retrieved the solution of
^ through inversion of the modified PV field.
c
The coefficients are determined by the boundary
conditions:
^ 5c
^ at z 5 0 and z 5 H.
(i) c
ob
(17)
and in D2 (troposphere)
^5c
^ 1 Ge2gz 1 Kegz
c
ob
(18)
The coefficients in Eqs. (17) and (18) may be determined
by using the boundary conditions (i) and (ii).
Adding the streamfunctions in E2008 to the ones in
Eqs. (17) and (18), in regions D1 and D2 respectively, the
^ is obtained. Hence, as in the
observed streamfunction c
ob
barotropic case, PPVI yields the desired results.
c. Baroclinic waves in zonal shear flow: Impacts
from the lower boundary temperature anomalies
The final example from E2008 to be discussed addresses
the impact on the flow from lower boundary temperature
anomalies.
With suitable boundary conditions the atmospheric
flow (in region D1) associated with the temperature
anomalies at the lower boundary (region D2) becomes
c * 5 2nP sin(k x) sinh[g(z 2 H)] /g cosh(gH).
(19)
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The validity of PPVI in this case may be verified by
again including the complementary case, where q9 5 q9ob
in 0 , z # H.
The atmospheric flow for the complementary case, in D1
(0 , z # H), is
^5c
^ 1 Ae2gz 1 Begz .
c
ob
(20)
The coefficients in Eq. (20) are determined by the following boundary conditions:
^5
(i) At the top of the model atmosphere (z 5 H), c
^ 5 0.
c
ob
^
(ii) At the lower boundary ›c/›z
5 0 as z / 0.
Through application of the assumptions (i) and (ii) the
inverted streamfunction for the complementary case is
retrieved:
c** 5 P sin(k x) sin[n(z 2 H)]
1 nP sin(kx) sinh[g (z 2 H)]/ g cosh(g H).
(iii) c / 0 when r / ‘.
The solution yields an induced cyclonic wind field that
is strongest at the surface of the PV anomaly (i.e., at
r 5 r0). Beyond the surface the winds become evanescent with increasing distance r [see Fig. 6.10 in Holton
(2004), originally from Thorpe and Bishop (1995)].
We now assume that D1 is the region 0 # r # r0 and
D2 is the adjacent region r . r0. We note from Eq. (23)
that this problem avoids the requirement q9 5 0 in D1.
This means that PV inversion in this case yields the c
field associated with a specific positive PV anomaly.
The solution of Eq. (23) is quite different from the ones
presented in E2008, such as the solution presented in
Fig. 2 for the barotropic case where the associated winds
appear to remain strong far away from the PV anomaly.
In fact, the results from the case described by Eq. (23)
have a strong resemblance to NWP simulated flows associated with isolated PV anomalies.
(21)
The addition of expressions (19) and (21), which describe
the two complementary cases, yields for region D1
c 5 c* 1 c** 5 P sin(k x) sin[n(z 2 H)] [ cob .
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(22)
Thus, we find that the two fields obtained from piecewise
QPV inversion for the whole region, comprising D1 and
D2, again add to yield the observed field. Hence, also for
this case the criticism of PPVI by E2008 is refuted.
d. The impact of an isolated QPV anomaly
The cases presented in E2008 appear to depend
strongly on the mathematical formulation of the problems,
including the necessity of introducing the conditions c 5 0,
implying q9 5 0 in a selected domain, as seen from, for
instance, Eq. (12). We now compare the inversion of the
PV anomalies (11) and (15) with an example discussed by
Thorpe and Bishop (1995) and Holton (2004). Expressing the QPV anomaly (15) with a three-dimensional
Laplacian operator through the transformation z^ 5 (N0 /f0 )z
and introducing spherical coordinates, we get through
symmetry
q0 0 # r # r0
1 › 2 ›c
r
5
.
(23)
2
0 r . r0
›r
r ›r
In this example the QPV anomaly is assumed to consist
of a constant q0 within a ball-shaped region.
Relation (23) is solved by integration with the following continuity and boundary conditions:
(i) c is defined at the origin.
(ii) Continuity of c and ›c/›r at the boundary (r 5 r0)
of the PV anomaly.
4. Case study
We now present a case study describing a real weather
situation, in which the results from PPVI, using the
technique described in section 2 (i.e., combining the ST
and AM methods) explain properly important aspects of
cyclogenesis. This implies explaining the mutual intensification of the upper-level PV anomaly and the lowlevel (boundary) temperature anomaly, as well as the
strong frontogenesis that took place. The fields in Fig. 4
are based on a successful NWP simulation of the severe
North Atlantic winter storm of 10–11 January 2006. The
position of the surface low is indicated by L.
Two snapshots of the development are shown: the first
one at 0600 UTC 10 January shown in Figs. 4a,b and the
second 6 h later in Figs. 4c,d.
PPVI reveals the following sequence of events:
(i) A positive upper-level PV (UPV) anomaly is shown
in Fig. 4a. The winds obtained from PPVI are strong
at the edge of the UPV anomaly and weaken with
distance away from it. There is pronounced lowlevel temperature advection by the wind associated
with the UPV anomaly (Fig. 4b). Strong cold advection takes place west and south of the cyclone center,
while warm advection is pronounced in the warm
frontal region north and northwest of the surface
low. This pattern of temperature advection enhances
the lower boundary warm anomaly observed at the
cyclone center denoted by L.
(ii) Six hours later (Fig. 4c) the UPV anomaly has become
stronger because of the impacts from the winds
induced by the strengthened lower boundary warm
anomaly. Figure 4d shows that the warm front has
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FIG. 4. Fields based on a simulation by the Norwegian High-Resolution Limited-Area Model (HIRLAM12). The position of the surface
low is indicated by L. (a) A positive UPV anomaly (contours every 0.5 PVU) and the associated winds obtained through PPVI, presented
at the 400-hPa level. Fields are valid at 0600 UTC 10 Jan 2006. (b) Potential temperature (contours every 2 K) at 950 hPa and the winds
(obtained through PPVI) at 900 hPa associated with the UPV anomaly shown in (a). The fields are valid at 0600 UTC 10 Jan 2006. (c) As in
(a), but valid at 1200 UTC 10 Jan 2006. (d) As in (b), but valid at 1200 UTC 10 Jan 2006.
intensified and the cold advection west and south of
the cyclone center has become stronger over the 6-h
period through the intensification of the winds related
to a stronger positive UPV anomaly. In turn the cold
advection to the south of the cyclone center enhances
the lower boundary warm anomaly observed at L in
Fig. 4d. Hence the mutual interaction between the
warm anomaly at the lower boundary and the UPV
anomaly continues and becomes stronger.
Figure 4 only shows one selected UPV anomaly and
the low-level boundary temperature fields. Nevertheless,
the UPV and boundary temperature anomalies presented in Fig. 4 have a crucial impact on the initial stage of
the cyclogenesis.
This example illustrates that PPVI is very useful for
analyzing instantaneous flow patterns, as well as the dynamics of cyclogenesis.
5. Discussion and conclusions
By presenting some examples of PV inversion, Egger
(2008) claimed that PPVI is generally not useful for
understanding atmospheric dynamics. Here, by reanalyzing some of the cases presented in E2008 we demonstrate that there are no reasons to discard PPVI as
a valid diagnostic tool. In E2008 the observed PV field
was changed to qob 5 0 in a selected domain (D2), while
qob was retained in the remaining adjacent domain (D1).
Regarding the PV field in the adjacent domain (D1) as
the anomaly, inversion of this anomaly for the whole
integration region (D1 1 D2) yields a streamfunction
different from the one that is observed.
We have argued that this procedure is misleading, and
that in order to test the validity of PPVI it is necessary to
invert the PV field also in the complementary cases. The
complementary cases are obtained by retaining qob and
letting qob 5 0 in the selected (D2) and adjacent (D1)
domains respectively.
Addition of the field obtained through PV inversion in
the complementary case should then yield the total
(observed) field if PPVI is valid. As we have shown, by
including the complementary cases, the formalism of
piecewise PV inversion indeed remains valid in the cases
presented in E2008.
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The examples presented in E2008 do not include interaction between PV anomalies, a shortcoming also
pointed out by Methven and de Vries (2008). Such interactions constitute an important part of assessing the
dynamics through PPVI. The results from E2008 are
also model dependent, which is illustrated by the example given by Eq. (23). Inversion of the QPV anomaly
given by Eq. (23) yields a solution resembling that obtained through PPVI for a real case presented in Fig. 4.
The solution of Eq. (23) is very different from those
of the cases presented in E2008, illustrating the dependence of the results from PPVI on the mathematical and
dynamical formulation of the problem.
Based on the above examples and case study (Fig. 4),
it appears that PPVI is a highly useful tool for diagnosing
dynamical processes in the atmosphere. In particular,
PPVI provides a method for assessing which regions of
the atmospheric flow are important for specific dynamical developments.
Although the various contributions from individual PV
anomalies, obtained through PPVI, cannot be observed in
the real atmosphere, nonetheless the addition of the contributions from all the PV anomalies, including those from
the residual and mean PV, yields the observed streamfunction field. This suggests that the various contributions
from different PV anomalies are conceptually meaningful
(e.g., Thorsteinsson et al. 1999). This is also confirmed by
case studies of real situations in which frontogenesis and
low-level developments can be understood through the
impact from specific PV anomalies as shown in section 4
(e.g., Davis and Emanuel 1991; Stoelinga 1996; Røsting
et al. 2003; Bracegirdle and Gray 2009).
An interesting example of application of PPVI was
recently provided by Hinssen et al. (2011). They studied
the impact of the stratospheric PV anomaly associated
with the polar winter vortex on the tropospheric flow.
Given the large horizontal dimension of the stratospheric polar vortex, an associated large-scale PV anomaly was defined, and its effect on the tropospheric flow
turned out to be noticeable. During sudden stratospheric
warmings (SSWs) a breakup or substantial weakening of
the stratospheric PV anomaly and the associated polar
vortex takes place. PPVI shows that the westerly mean
flow in the troposphere weakens and may even become
easterly in some regions after an SSW.
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