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1
2
The usefulness of piecewise potential vorticity
inversion
by
3
Bjørn Røsting
4
1
2
1
and Jón Egill Kristjánsson
The Norwegian Meteorological Institute, Oslo, Norway
Department of Geosciences, University of Oslo, Norway
1
2
5
Abstract
6
It is today widely accepted that potential vorticity (PV) thinking is a highly
7
useful approach for understanding important aspects of dynamic meteorol-
8
ogy and for validation of output from state-of-the-art numerical weather pre-
9
diction (NWP) models. Egger (2008) recently presented a critical view on
10
piecewise potential vorticity inversion (PPVI). This was done by defining a
11
PV anomaly by retaining the observed PV field in a specific region, while
12
changing the observed PV fields to zero elsewhere. Inversion of such a modi-
13
fied PV field yields a flow vastly different from the observed. On the basis of
14
this result it was argued that piecewise potential vorticity inversion (PPVI)
15
is useless for understanding the dynamics of the flow.
16
In the present paper we argue that the results presented by Egger are
17
incomplete in the context of PPVI, since the complementary cases were not
18
considered and that the results also depend on the idealized model formula-
19
tions. The complementary case is defined by changing the observed PV to
20
zero in the specific region, while retaining the observed PV field elsewhere.
21
By including the complementary cases, we demonstrate that the stream-
22
function fields associated with the PV and boundary temperature anomalies
23
presented by Egger (2008), add up to yield the observed streamfunction field,
24
as expected if PPVI is to be valid. It follows that PPVI is indeed valid and
25
useful in these cases.
2
26
1
Introduction
27
Hoskins et al. (1985) provided a theoretical survey of the concept of potential
28
vorticity (PV), along with applications of PV and low level boundary temper-
29
ature anomalies to the interpretation of atmospheric dynamics. Though the
30
PV aspect of atmospheric dynamics had been well known for several decades,
31
being described by Rossby (1940), Ertel (1942) and Charney (1947), the pa-
32
per by Hoskins et al. introduced, most likely for the first time, the concept
33
of PV thinking. PV thinking means adopting PV in order to describe and
34
understand the dynamics of the atmosphere, such as e.g. baroclinic and
35
barotropic instability. This is achieved e.g. through PV inversion, by which
36
geopotential heights, winds and temperature fields are obtained, provided
37
that suitable balance and boundary conditions are specified.
38
The classical approach to baroclinic instability, described by Eady (1949),
39
is essentially based on the PV-concept, where the (quasi-geostrophic) PV
40
field confined between an upper lid, i.e. the tropopause, and the surface
41
are specified, along with the temperature distribution at lower and upper
42
boundaries. Analytical solutions of this problem yield growing waves with
43
the largest growth rate for wavelengths on the order of 4000 - 6000 km.
44
PV can also be applied in identifying errors in NWP analyses and simu-
45
lations (e.g. Mansfield 1996). This is achieved by comparing the PV fields
46
retrieved from numerical weather prediction (NWP) models with features
47
observed in water vapor (WV) images. This method is based on the strong
3
48
absorption of radiation in the 6.3-6.7 µm water vapor bands, hence PV-
49
rich stratospheric air is easily identified in WV images (e.g. Santurette and
50
Georgiev 2005). When errors in the PV fields are identified by an observed
51
mismatch between PV fields and WV features, it is possible to manually
52
correct the PV fields. The corrected PV fields are then inverted, yielding a
53
modified numerical analysis that is balanced and suitable for initializing nu-
54
merical reruns (e.g. Demirtas and Thorpe 1999, Manders et al. 2007, Røsting
55
et al 2003, Røsting and Kristjánsson 2006, 2008, Verkley et al. 2005). These
56
procedures represent an efficient use of PV thinking.
57
An important aspect of PV thinking is piecewise PV inversion (PPVI).
58
In PPVI, the PV field is partitioned, and the contributions of the different
59
parts of the PV field to the flow are quantified by piecewise inversion (Davis
60
1992, Kristjánsson et al 1999). The feasibility of such an approach for Ertel’s
61
nonlinear PV was first demonstrated by Davis and Emanuel (1991).
62
Criticism of PPVI pertaining to its usefulness for understanding atmo-
63
spheric dynamics was recently put forward by Egger (2008), referred to as
64
E2008 below. Though some of the arguments on PPVI in E2008 may be
65
pertinent, particularly the discussion of inverted tendency fields, others call
66
for comments. The criticism advanced in E2008 is based on experiments to
67
be discussed in section 3 below, in which the observed PV is replaced by
68
P V = 0 in a selected region. Inversion of such a modified PV field yields a
69
flow different from the observed. In E2008 it was claimed that such a result
70
implies that PPVI in general is useless for understanding the dynamics of
4
71
the flow. Below, we present evidence that this drastic claim is unjustified.
72
Methven and de Vries (2008) addressed the limitations of the mathematical
73
model describing the baroclinic cases of E2008 and pointed out the impor-
74
tance of dynamic evolution rather than pure diagnosis based on PPVI in
75
assessing the benefits of PV thinking.
This paper is structured as follows:
76
77
In section 2 we give a brief presentation of the PV inversion technique used
78
in many case studies. In section 3 we address some of the cases presented in
79
E2008. A case study based on PPVI is presented in section 4, and discussion
80
and conclusions are provided in section 5.
81
2
82
83
84
Piecewise PV inversion
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
q = η · ∇θ
ρ
(1)
85
where ρ is density, η is the absolute vorticity vector and ∇θ is the three
86
dimensional gradient of potential temperature. PV is expressed in PV-units,
87
defined as 1PVU=10−6 m2 s−1 Kkg −1 .
88
89
PV is changed due to diabatic effects and friction as shown in the following
equation:
5
Dq
∂q
1
1
≡
+ u · ∇q = η · ∇θ̇ + ∇ × Fr · ∇θ
Dt
∂t
ρ
ρ
(2)
90
where u is the wind vector, Fr is the friction force and θ̇ denotes diabatic
91
heating.
92
93
The mathematical expression for the Ertel PV can be simplified by using
isentropic coordinates (x,y,θ):
q = (ζθ + f )(−g
∂θ
)
∂p
(3)
94
where ζθ is relative vorticity on an isentropic surface and f is the coriolis
95
∂θ
is
parameter (planetary vorticity), g is the acceleration of gravity and − ∂p
96
an expression of static stability.
97
98
Relations (2) and (3) show that for adiabatic and inviscid flow q is conserved for an air parcel moving on an isentropic surface.
99
Potential temperature fields (θb ) are specified at the lower and upper
100
boundaries, e.g. at 950 and 150 hPa and the θb anomalies can be regarded
101
as PV anomalies, i.e. a warm (cold) anomaly at the lower boundary can
102
be regarded as a positive (negative) PV anomaly. Likewise, a warm (cold)
103
anomaly at the upper boundary can be treated as a negative (positive) PV
104
anomaly (Hoskins et al. 1985). Hence such anomalies can be inverted sepa-
105
rately.
106
The total PV (q) and boundary potential temperature (θb ) fields are de-
107
composed into mean and perturbation terms. Hence the total PV is expressed
6
108
as:
q = q + q′
109
and the expression for the boundary potential temperature field is analogous:
θb = θb + θ′ b
(5)
The mean fields, q and θb are temporal or spatial means, with temporal
110
111
(4)
means taken as an average over e.g. 48 hours.
112
The perturbation terms in equations (4) and (5) may be decomposed into
113
several anomalies. Usually one is interested in specific PV and θb anomalies
114
that appear to have a large influence on the phenomena of interest. Such
115
anomalies are usually clearly defined as temporally and spatially coherent
116
features.
117
PV anomalies that are far away from the region of interest, or difficult to
118
define temporally are generally defined as parts of a residual term, denoted
119
by qres .
120
Following Kristjánsson et al. (1999), the perturbation field in (4) is expressed
121
as:
q′ =
n
X
i=1
122
where qi denotes the anomalies.
7
qi
(6)
123
For N selected PV anomalies the above relation becomes:
′
q =
N
X
qi + qres
(7)
i=1
124
where the residual term is:
n
X
qres =
qi
(8)
i=N +1
125
The expressions for the θb fields are similar.
126
PPVI deals with obtaining uniquely the familiar fields of geopotential
127
height or streamfunction, winds and temperature distribution associated with
128
specific PV and θb anomalies.
129
To close the set of equations required for PV inversion a balance condition
130
is required, as well as lateral and horizontal boundary conditions. We adopt
131
Charney’s balance condition (Charney 1955) given by:
∇2 Φ = ∇ · (f ∇Ψ) + 2
132
∂(∂Ψ/∂x, ∂Ψ/∂y)
∂(x, y)
(9)
where Φ is the geopotential and Ψ is the streamfunction.
133
The reason why Ertel’s PV (1) and the Charney balance condition (9) are
134
often preferred in PV inversion studies is that the flow is properly described
135
by (2) and (9), including flow with fairly large Rossby numbers, i.e. Ro ∼ 1,
136
e.g. allowing for strongly curved flow. The disadvantage of adopting these
137
equations directly is their non-linearity. This means that the streamfunction
8
138
fields associated with their respective PV and θb -anomalies, as well as con-
139
tributions from the mean and residual fields of PV and θb do not add up
140
to yield the total streamfunction field, as they do for quasi-geostrophic PV
141
(QPV).
142
Davis and Emanuel (1991) and Davis (1992) demonstrated how the non-
143
linearity problem can be resolved. For completeness this will now be briefly
144
recaptured.
145
Adopting the hydrostatic relation on the right hand side of (1) yields a
146
relation for q containing products of second derivatives of Ψ and Φ. See
147
e.g. relation (1.4) in Davis (1992) for details. Adopting this relation, Davis
148
and Emanuel (1991) demonstrated how Ertel’s PV can be inverted yielding
149
unambiguous results, as for inversion of QPV. The method is referred to as
150
the full linear (FL) method, because all terms are retained through collecting
151
the non-linear terms in the coefficient of the linear operator.
152
Davis (1992) presented two different PV inversion methods, i.e. subtrac-
153
tion from the total (ST) and addition to the mean (AM). He demonstrated
154
that the average of these two methods yields results close to those of the FL
155
method except for cases when the PV perturbations are much larger than
156
the mean PV-field.
157
158
The above description of the FL, ST and AM methods applies to boundary temperature (θb ) anomalies as well.
159
Rather than adopting the FL method we use the combination of the ST
160
and AM methods in the case study presented in section 4 below. The method
9
161
has also been succsessfully adopted in several case studies (e.g. Kristjánsson
162
et. al 1999, Thorsteinsson et. al 1999, Røsting and Kristjánsson 2006).
163
Though quasi-geostrophic theory is a weak anomaly theory, the inversion
164
of QPV is useful for demonstrating the principle of PV inversion and in
165
theoretical studies of atmospheric flows (e.g. the Eady model). The results
166
from such studies in E2008 led to the conclusion that PPVI in many cases fails
167
to contribute to our understanding of the dynamics of the flow. Since PPVI
168
has been adopted in many recent case studies (e.g. Bracegirdle and Gray
169
2009, McInnes et al. 2009) and constitutes a central part of PV thinking,
170
the criticism in E2008 needs to be addressed.
171
We now discuss some of the results presented by E2008. The main purpose
172
of the following discussion is to investigate to what extent PPVI is valid and
173
useful in the cases adopted in E2008.
174
3
175
Results from elementary tests of piecewise
PV inversion
176
In E2008, in connection with Figure 1, it was stated that the winds in region
177
D2 are induced by the positive PV anomaly Z1 in region D1. However,
178
basing our arguments on the PPVI technique (as described in the previous
179
section), we note that the winds induced by Z1 only contribute to the total
180
wind field in region D2 in the same way as one of the components qi in (6)
181
is only one of many contributions to the total perturbation field q ′ . To what
10
182
extent Z1 influences the winds in D2 depends on the distance between the
183
PV anomalies Z1 and A2 and their amplitudes.
184
The same argument is valid for the ”complementary case”, i.e. the winds
185
associated with the negative PV anomaly A2 contribute to the total wind
186
field in region D1 and may influence Z1 to a certain extent. The total stream-
187
function ψtotal in D2 is then obtained through adding the contributions from
188
A2 and Z1. The same procedure yields ψtotal in D1. Hence performing in-
189
versions for both regions D1 and D2 is a necessary requirement for assessing
190
the validity of PPVI.
191
3.1
192
The barotropic case - Rossby waves in zonal mean
flow
193
In the equation for PV in a barotropic flow (e.g. Holton 2004) we may define
194
the planetary vorticity f0 + βy as the mean PV field. Introducing a height
195
increment η = g −1 f0 ψ, we obtain through linearization an expression for
196
perturbation PV:
q ′ = ∇2 ψ − δ 2 ψ
(10)
197
where δ −1 = (gH)1/2f0 −1 is the Rossby radius of deformation. H is the mean
198
height of the fluid.
199
200
We now consider a one-dimensional barotropic Rossby wave, hence equation (10) becomes
11
q′ =
∂2ψ
− δ2ψ
∂x2
(11)
201
In this example from E2008, the observed streamfunction field is described
202
by ψob = P sin(kx) from which the observed QPV anomaly field is calculated
203
by inserting ψob in equation (11).
204
The case presented in E2008 now proceeds as follows: Region D1 is defined
205
for (-L/2 ≤ x ≤ 0) and D2 for (0≤ x ≤ L/2) as shown in Figure 2, reproduced
206
from E2008. Following E2008, the PV field is modified by letting q ′ = 0 in
207
the region D2.
208
209
Figure 2 shows, besides ψob , the streamfunction ψ ′ obtained through inversion of the modified PV (solid contours).
210
To test the validity of PPVI for the flow in this example we now define
211
the complementary case for the regions shown in Figure 2, where q ′ = 0 in
212
region D1, and with q ′ = qob in D2 :
q′ =







0 in D1;
(12)
−(k 2 + δ 2 )P sin(kx) in D2.
213
Carrying out the inversion of PV in this case we obtain the streamfunction
214
ψ ′′ , assuming that, as for the case of ψ ′ , that ψ ′′ and ∂ψ ′′ /∂x are continuous
215
at the origin. The solution is:
12
ψ ′′ =







−Bsinh[δ(x + L/4] in D1;
(13)
P sin(kx) + Bsinh[δ(x − L/4] in D2.
216
where B = −P k/[2δcosh(δL/4)], i.e. B is identical to the coefficient A in
217
the case presented in E2008.
218
219
The distribution of ψ ′′ in this complementary case (13) is antisymmetric
to that displayed in Figure 2, with respect to the origin.
220
Addition of the two complementary fields yields for D1 and D2, ψ =
221
ψ ′ + ψ ′′ = P sin(kx) = ψob . Hence addition of the fields associated with
222
the PV in D1 and D2 yields the total, observed PV field, as expected for
223
piecewise PV inversion.
224
3.2
225
In the baroclinic case described in E2008 the flow is confined between a
226
surface plane and a rigid lid at the top, the height of the model atmosphere
227
being H. The dynamical development of the flow in this case is described by
228
the equations of the Eady model (e.g. Holton 2004, Egger 2008). Figure 3,
229
reproduced from E2008, shows the tropopause at Hs = 3H/4.
230
Baroclinic waves in zonal shear flow
Following E2008, an observed field is given as a modal streamfunction :
ψob = P sin(kx)sin[n(z − H)] = ψ̂(z)sin(kx)
13
(14)
231
232
233
234
235
The horizontal wavenumber is k = 2π/L, and n = 3π/H is the vertical
wavenumber.
Figure 3a, retrieved from E2008, shows the structure of the observed
streamfunction ψob .
Now using quasi-geostrophic PV, the perturbation is expressed by
′
qob
= ∇2 ψob + (f02 /N02 )∂ 2 ψob /∂z 2
(15)
236
where N02 ≡ (g/θ0 )dθ0 /dz is the Brunt-Vaisala frequency squared, expressing
237
static stability, here assumed to be constant.
238
239
By inserting the expression for ψob into (15), an equation for the amplitude
of the streamfunction in the x-z plane is obtained:
∂ 2 ψ̂
N02 q̂
2
−
γ
ψ̂
=
∂z 2
f02
240
(16)
where γ = kN0 /f0 and q̂ is the amplitude of the PV perturbation.
241
′
With modifications of the observed PV, i.e. changing PV to qob
= 0 in
242
′
the troposphere, i.e. below Hs (D2), while retaining the original qob
in the
243
stratosphere, i.e. above Hs (D1), Egger retrieved the solution of ψ̂ through
244
inversion of the modified PV field.
245
The coefficients are determined by the boundary conditions:
246
(i) ψ̂ = ψˆob at z = 0 and z = H
247
(ii) Continuity requirements for ψ̂ and ∂ ψ̂/∂z at the tropopause z = 3H/4.
248
The distribution of ψ̂ is shown in Figure 3b.
14
249
′
In the complementary case, not considered by E2008, qob
is confined to
250
′
the troposphere (D2), while qob
= 0 in the stratosphere (D1). This PV
251
distribution yields the following solution:
252
in D1, stratosphere
253
ψ̂ = Ee−γz + F eγz
(17)
ψ̂ = ψˆob + Ge−γz + Keγz
(18)
in D2, troposphere
254
The coefficients in (17) and (18) may be determined by using the boundary
255
conditions (i) and (ii).
256
Adding the streamfunctions in E2008 to the ones in (17) and (18) , in regions
257
D1 and D2 respectively, the observed streamfunction ψˆob is obtained. Hence,
258
as in the barotropic case, PPVI yields the desired results.
259
3.3
260
Baroclinic waves in zonal shear flow - impacts from
the lower boundary temperature anomalies
261
The final example from E2008 to be discussed addresses the impact on the
262
flow from lower boundary temperature anomalies.
263
With suitable boundary conditions the atmospheric flow (in region D1)
264
associated with the temperature anomalies at the lower boundary (region
15
265
D2) becomes:
ψ ∗ = −nP sin(kx)sinh[γ(z − H)]/γcosh(γH)
266
(19)
The validity of PPVI in this case may be verified by again including the
267
′
complementary case, where q ′ = qob
in 0 < z ≤ H.
268
The atmospheric flow for the complementary case is:
269
in D1 (0 < z ≤ H)
ψ̂ = ψˆob + Ae−γz + Beγz
(20)
270
The coefficients in equation (20) are determined by the following boundary
271
conditions:
272
(i) At the top of the model atmosphere (z = H), ψ̂ = ψˆob = 0
273
(ii) At the lower boundary ∂ ψ̂/∂z = 0 as z → 0
274
275
Through application of the assumptions (i) - (ii) the inverted streamfunction for the complementary case is retrieved:
ψ ∗∗ = P sin(kx)sin[n(z − H)] + nP sin(kx)sinh[γ(z − H)]/γcosh(γH) (21)
276
Addition of expressions (19) and (21), that describe the two complementary
277
cases, yields for region D1:
16
ψ = ψ ∗ + ψ ∗∗ = P sin(kx)sin[n(z − H)] ≡ ψob
(22)
278
Thus, we find that the two fields obtained from piecewise QPV inversion for
279
the whole region, comprised by D1 and D2, again add to yield the observed
280
field. Hence, also for this case the criticism of PPVI by E2008 is refuted.
281
3.4
282
The cases presented in E2008 appear to depend strongly on the mathematical
283
formulation of the problems, including the necessity of introducing the condi-
284
tions ψ = 0, implying q ′ = 0 in a selected domain, as seen from e.g. equation
285
(12). We now compare the inversion of the PV anomalies (11) and (15)
286
with an example discussed by Thorpe and Bishop (1995) and Holton (2004).
287
Expressing the QPV anomaly (15) with a three dimensional Laplacian op-
288
erator through the transformation ẑ = (N0 /f0 )z and introducing spherical
289
coordinates, we get due to symmetry:
The impact of an isolated QPV anomaly




q0
1 ∂ 2 ∂ψ
(r
)
=

r 2 ∂r
∂r

 0
0 ≤ r ≤ r0 ;
(23)
r > r0 .
290
In this example the QPV anomaly is assumed to consist of a constant q0
291
within a ball-shaped region.
292
293
Relation (23) is solved by integration with the following continuity and
boundary conditions:
17
294
(i) ψ is defined at the origin.
295
(ii) Continuity of ψ and ∂ψ/∂r at the boundary (r = r0 ) of the PV anomaly.
296
(iii) ψ → 0 when r → ∞.
297
The solution yields an induced cyclonic wind field that is strongest at the
298
surface of the PV anomaly, i.e. at r = r0 . Beyond the surface the winds
299
become evanescent with increasing distance (r), see Figure 6.10 in Holton,
300
2004 (originally from Thorpe and Bishop, 1995).
301
We now assume that D1 is the region 0 ≤ r ≤ r0 and D2 is the adjacent
302
region r > r0 . We note from (23) that this problem avoids the requirement
303
q ′ = 0 in D1. This means that PV inversion in this case yields the ψ-field
304
associated with a specific positive PV anomaly.
305
The solution of (23) is quite different from the ones presented in E2008,
306
e.g. the solution presented in Figure 2 for the barotropic case where the
307
associated winds appear to remain strong far away from the PV anomaly. In
308
fact the results from the case described by (23) have a strong resemblance to
309
NWP simulated flows associated with isolated PV anomalies.
310
4
311
We now present a case study describing a real weather situation, in which the
312
results from PPVI, using the technique described in section 2, i.e. combining
313
the ST and AM methods, explain properly important aspects of cyclogene-
Case study
18
314
sis. This implies explaining the mutual intensification of the upper level PV
315
anomaly and the low level (boundary) temperature anomaly, as well as the
316
strong frontogenesis that took place. The fields in Figure 4 are based on
317
a successful NWP simulation of the severe North Atlantic winter storm of
318
10-11 January 2006. The position of the surface low is indicated by ”L”.
319
Two snapshots of the development are shown, the first one at 06UTC 10
320
January shown in Figures 4a,b and the second 6 hours later in Figures 4c,d.
PPVI reveals the following sequence of events:
321
322
(i) An upper level positive PV (UPV) anomaly is shown in Figure 4a. The
323
winds obtained from PPVI are strong at the edge of the UPV anomaly and
324
weaken with distance away from it. There is pronounced low level temper-
325
ature advection by the wind associated with the UPV anomaly (Figure 4b).
326
Strong cold advection takes place west and south of the cyclone center, while
327
warm advection is pronounced in the warm frontal region north and north-
328
west of the surface low. This pattern of temperature advection enhances the
329
lower boundary warm anomaly observed at the cyclone center denoted by
330
′′
331
(ii) Six hours later (Figure 4c) the UPV anomaly has become stronger due
332
to the impacts from the winds induced by the strengthened lower boundary
333
warm anomaly. Figure 4d shows that the warm front has intensified and
334
the cold advection west and south of the cyclone center has become stronger
335
over the 6 hour period through the intensification of the winds related to
336
a stronger positive UPV anomaly. In turn the cold advection to the south
L′′ .
19
337
of the cyclone center enhances the lower boundary warm anomaly observed
338
at the ′′ L′′ in Figure 4d. Hence the mutual interaction between the warm
339
anomaly at the lower boundary and the UPV anomaly continues and becomes
340
stronger.
341
Figure 4 only shows one selected UPV anomaly as well as low level bound-
342
ary temperature fields. Nevertheless, the UPV and boundary temperature
343
anomalies presented in Figure 4 have a crucial impact on the initial stage of
344
the cyclogenesis.
This example illustrates that PPVI is very useful for analyzing instanta-
345
346
neous flow patterns, as well as the dynamics of cyclogenesis.
347
5
348
By presenting some examples of PV inversion, Egger (2008) claimed that
349
PPVI is generally not useful for understanding atmospheric dynamics. Here,
350
by reanalyzing some of the cases presented in E2008 we demonstrate that
351
there are no reasons to discard PPVI as a valid diagnostic tool. In E2008 the
352
observed PV field was changed to qob = 0 in a selected domain (D2), while
353
qob was retained in the remaining, adjacent domain (D1). Regarding the PV
354
field in the adjacent domain (D1) as the anomaly, inversion of this anomaly
355
for the whole integration region (D1 + D2) yields a streamfunction different
356
from the one that is observed.
357
Discussion and conclusions
We have argued that this procedure is misleading, and that in order to
20
358
test the validity of PPVI it is necessary to invert the PV field also in the
359
complementary cases. The complementary cases are obtained by retaining
360
qob and letting qob = 0 in the selected (D2) and adjacent (D1) domains
361
respectively.
362
Addition of the field obtained through PV inversion in the complementary
363
case should then yield the total (observed) field if PPVI is valid. As we have
364
shown, by including the complementary cases, the formalism of piecewise PV
365
inversion indeed remains valid in the cases presented in E2008.
366
The examples presented in E2008 do not include interaction between PV
367
anomalies, a shortcoming also pointed out by Methven and de Vries (2008).
368
Such interactions constitute an important part of assessing the dynamics
369
through PPVI. The results from E2008 are also model dependent, this is
370
illustrated by the example given by equation (23). Inversion of the QPV
371
anomaly given by equation (23) yields a solution resembling that obtained
372
through PPVI for a real case presented in Figure 4. The solution of (23)
373
is very different from those of the cases presented in E2008, illustrating the
374
dependency of the results from PPVI on the mathematical and dynamical
375
formulation of the problem.
376
Based on the above examples and case study (Figure 4), it appears that
377
PPVI is a highly useful tool for diagnosing dynamical processes in the atmo-
378
sphere. In particular PPVI provides a method for assessing which regions of
379
the atmospheric flow that are important for specific dynamical developments.
380
Though the various contributions from individual PV anomalies, obtained
21
381
through PPVI, cannot be observed in the real atmosphere, addition of the
382
contributions from all the PV anomalies, including those from the resid-
383
ual and mean PV, yields the observed streamfunction field. This suggests
384
that the various contributions from different PV anomalies are conceptually
385
meaningful (e.g. Thorsteinsson et al. 1999). This is also confirmed by case
386
studies of real situations where frontogenesis and low level developments can
387
be understood through the impact from specific PV anomalies as shown in
388
section 4 (e.g. Davis and Emanuel 1991, Stoelinga 1996, Røsting et al. 2003,
389
Bracegirdle and Gray 2009).
390
An interesting example of application of PPVI was recently provided
391
by Hinssen et al (2011). They studied the impact of the stratospheric PV
392
anomaly associated with the polar winter vortex on the tropospheric flow.
393
Given the large horizontal dimension of the stratospheric polar vortex, an
394
associated large scale PV anomaly was defined, and its effect on the tro-
395
pospheric flow turned out to be noticeable. During sudden stratospheric
396
warmings (SSW) a breakup or substantial weakening of the stratospheric
397
PV anomaly and the associated polar vortex takes place. PPVI shows that
398
the westerly flow in the troposphere weakens and may even become easterly
399
in some regions after a SSW.
22
400
6
References
401
Bracegirdle,T.J. and Gray,S.L. 2009: The dynamics of a polar low assessed
402
using potential vorticity inversion. Q.J.R. Meteorol.Soc.,135,880-893
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Charney,J.G. 1947: The dynamics of long waves in a baroclinic westerly
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current. J.Meteor., 4, 135-163
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Charney,J.G. 1955: The use of the primitive equations of motions in numer-
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ical prediction. Tellus, 7, 22-26
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Davis, C.A. 1992: Piecewise potential vorticity inversion. J.Atmos.Sci., 49,
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1397-1411
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Davis,C.A. and Emanuel,K.A. 1991: Potential vorticity diagnostics of cyclo-
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genesis. Mon. Weather Rev., 119,1929-1953
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Demirtas,M. and Thorpe,A.J. 1999: Sensitivity of short range weather fore-
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casts to local potential vorticity modifications. Mon.Weather Rev., 127, 922-
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Eady, E.T. 1949: Long waves and cyclone waves. Tellus 1(3), 33-52
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Egger.J. 2008: Piecewise Potential Vorticity Inversion: Elementary Tests. J.
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Ertel, H. 1942: Ein neuer hydrodynamischer Wirbelsatz. Meteorologische
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Zeitschrift 59, 277-281
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Hinssen, Y., van Delden, A. and Opsteegh,T. 2011: Influence of sudden
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stratospheric warmings on tropospheric winds. Meteorologische Zeitschrift,
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Holton, J.R. 2004: An Introduction to Dynamic Meteorology. Fourth Edi-
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tion. ISBN: 0-12-354015-1, Elsevier Academic Press
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Hoskins,B.J., McIntyre,M.E. and Robertson,A.W. 1985: On the use and sig-
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nificance of isentropic potential vorticity maps. Q.J.R. Meteorol.Soc., 111,
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Kristjánsson, J.E., Thorsteinsson,S. and Ulfarsson,G.F. 1999: Potential vor-
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ticity based interpretation of the ’Greenhouse Low’, 2-3 February 1991. Tel-
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lus, 51A, 233-248
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Manders, A.M.M., Verkley, W.T.M., Diepeveen, J.J. and Moene, A.R. 2007:
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Application of a potential vorticity modification method to a case of rapid
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cyclogenesis over the Atlantic Ocean. Q.J.R. Meteorol. Soc.,133, 1755-1770
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Mansfield,D. 1996: The use of potential vorticity as an operational forecast-
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ing tool. Meteorol.Appl.,3,195-210
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McInnes, H., Kristjánsson, J.E., Schyberg, H. and Røsting, B. 2009: An
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assessment of a Greenland lee cyclone during the Greenland Flow Distortion
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experiment - an observational approach. Q.J.R. Meteorol.Soc.,135,1968-1985
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Methven, J. and de Vries, H. 2008: Comments on “Piecewise Potential Vor-
440
ticity Inversion: Elementary Tests”. J. Atmos. Sci., 65, 3003-3008
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Rossby, C. G. 1940: Planetary flow patterns in the atmosphere. Q.J.R.Meteorol.Soc.,
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66, 68-87
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Røsting, B., Kristjánsson, J. E. and Sunde,J 2003: The sensitivity of numer24
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Røsting, B. and Kristjánsson, J. E. 2006: Improving simulations of severe
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winter storms by initial modification of potential vorticity in sensitive regions.
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Røsting, B. and Kristjánsson, J. E. 2008:A successful resimulation of the 7-8
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January 2005 winter storm through initial potential vorticity modification in
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465
ing a numerical weather analysis in terms of potential vorticity using three
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467
1736
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7
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Fig.1. Schematic presentation of a domain D divided into subdomains D1 and
470
D2. Z1 and A2 represent positive and negative PV anomalies respectively.
471
Reproduced from Egger (2008).
472
Fig.2. Observed streamfunction (ψob ) in dotted lines, forming ridges R1 and
473
R2 and troughs T1 and T2 in regions D1 and D2 respectively. Solid lines
474
show ψ obtained from inversion of the PV field after replacing the observed
475
PV with PV=0 in region D2. Reproduced from Egger (2008).
476
Fig.3. Idealized presentation of a two-dimensional atmosphere with depth H
477
= 20km. The tropopause is at z = 3H/4 = Hs , whereas n = 3π/H, k =
478
2π/L are vertical and horizontal wavenumber respectively. In a) the observed
479
streamfunction ψob is displayed. Positive values are in solid contours, while
480
negative values are dashed. In b) the streamfunction is obtained through
481
inversion of the PV field after replacing the observed PV with PV = 0 in the
482
troposphere (D2). The contours show the streamfunction, contours as in a).
483
Reproduced from Egger (2008).
484
Fig.4. Fields based on a simulation by the Norwegian HIRLAM12 model.
485
The position of the surface low is indicated by ”L”.
486
a) A positive UPV anomaly (contours every 0.5 PVU) and the associated
Figure captions
26
487
winds obtained through PPVI, presented at the 400 hPa level. Fields valid
488
at 06 UTC 10 January 2006.
489
b) Potential temperature (contours every 2K) at 950 hPa and the winds
490
(obtained through PPVI) at 900 hPa associated with the UPV anomaly
491
shown in a. The fields are valid at 06 UTC 10 January 2006.
492
c) Same as in a, but valid at 12 UTC 10 January 2006.
493
d) Same as in b, but valid at 12 UTC 10 January 2006.
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