Q. J. R. Meteorol. Soc.
(2006), 132 , pp. 2625–2652 doi: 10.1256/qj.05.188
By B. RØSTING
1 ∗
ANSSON
2
1 Norwegian Meteorological Institute, Oslo, Norway
2
Department of Geosciences, University of Oslo, Norway
(Received 1 September 2005; revised 25 July 2006)
S UMMARY
In this paper simulations of three selected severe winter storms over the North Atlantic are presented.
Modifications of potential vorticity (PV) fields according to features in water vapour (WV) images are combined with information from singular vectors (SVs) in an attempt to improve the initial state over data-sparse regions.
The apparent mismatch between features in the WV images and the upper-level PV anomalies in the numerical analyses is corrected, mainly at levels indicated as sensitive by the fastest-growing SVs. The procedure is based on the fact that modification of the analysis in regions where SVs show a pronounced signal may have a large impact on the simulation. Model reruns, based on the inverted corrected PV fields, were then performed. Three cases are presented: the two French Christmas storms of 1999, and the storm affecting UK, the North Sea and southern Norway on 30 October 2000. In these cases a substantial improvement of the simulations of the storms was achieved.
The PV modifications were carried out by a digital analysis system, implemented at the Norwegian
Meteorological Institute. This system allows the PV modifications to be done interactively within an operational time limit.
K EYWORDS : Singular vector analysis Water vapour images
1.
I NTRODUCTION
Despite large advances in numerical weather prediction in the last decade, serious forecast failures still occur on occasion, often due to poor data coverage in sensitive areas, e.g. over the North Atlantic or the Pacific Ocean. For this reason a large body of investigations has been presented which address the important task of improving numerical simulations of severe storms. Many of these have dealt with assimilation methods, i.e. three- and four-dimensional variational analyses (3D-Var, 4D-Var) (e.g. Rabier et al.
2000; Gustafsson et al.
2001). Particularly, 4D-Var seems to be highly successful in improving the initial state of numerical simulations, also involving assimilation of retrieved radiances from polar-orbiting satellites, such as EUMETSAT ATOVS† Retransmission
Service (EARS) data which are available for assimilation in numerical weather prediction (NWP) models.
Several investigations have described the use of singular vectors (SVs, Buizza and
Palmer 1995) in improving NWP analyses and forecasts. SVs are mainly used for constructing the initial spread of analyses, due to sparsity of observations, in mediumrange forecasting. This is done by perturbing the numerical analysis within the range of uncertainty of the initial state due to sparsity of observations, as in the ensemble prediction system (EPS), e.g. Jung et al.
(2005). These perturbations, introduced in regions indicated as sensitive by SV analysis, are expressed mathematically by the
SVs which form basis vectors in phase space. Some studies have demonstrated that
SVs are useful in short-range forecasting as well (e.g. Buizza and Montani 1999;
∗
Corresponding author: The Norwegian Meteorological Institute, PO Box 43 Blindern, N-0313 Oslo, Norway.
e-mail: bjorn.rosting@met.no
† Advanced TIROS Operational Vertical Sounder.
c Royal Meteorological Society, 2006.
2625

2626
Montani et al.
1999; Browning et al.
2000; Røsting et al.
2003), since an analysis error in the region indicated as sensitive by the fastest-growing SVs will have a large impact on the simulation. In such sensitive regions the simulation is likely to benefit from additional observations, including remote-sensing types, e.g. water vapour (WV) images.
This paper deals with the use of WV images combined with SV and potential vorticity (PV) analysis in attempting to improve numerical analyses and forecasts.
Earlier investigations (e.g. Demirtas and Thorpe 1999; Røsting et al.
2003; Hello and
Arbogast 2004) have shown that numerical analyses corrected by PV modifications according to WV signals can be beneficial for the forecast, particularly in cases with large forecast errors. The PV structure of the leading SVs is adopted as an additional tool in our investigations, reducing the subjectivity of the method by objectively highlighting the regions where modifications are likely to have a large impact on the numerical simulation. However, since SV growth is linear and based on simplified physics, SVs may fail to identify regions susceptible to severe cyclogenesis.
An advantage of the PV–WV–SV method is that it may cope with large analysis errors, larger than the typical analysis errors given by the error covariance matrix used in assimilation techniques. Corrections of large errors may be rejected by the assimilation routines since the corrected fields deviate too much from the background analysis. The new analyses obtained by PV inversion, after PV modification, are in this study used directly as an initial state, without involving the assimilation routines.
In Røsting et al.
(2003) successful experimental reruns of the Christmas storm of
1997 were described. Operational NWP models failed completely in predicting this development in the short range. It was demonstrated how PV fields can be modified according to information from WV images. Information from the leading SVs was invoked as well in order to determine the 3D structure of the modifications. Inversion of the corrected PV fields yielded a new analysis, from which a new NWP simulation was carried out. The new analysis was used directly for the simulation, without using variational techniques. That investigation was mainly a sensitivity study, the sensitivity being assessed by the impacts from various PV anomalies subject to PV surgery on the numerical simulations of the storms.
The purpose of this paper is to test further the combined use of PV, WV and
SV analysis in improving numerical simulations. Three storm cases, which caused substantial damage and were generally poorly simulated by the current operational NWP routines, have been selected. Though cases of forecast failure, as in the cases studied below, are rare with state-of-the-art NWP models and 4D-Var assimilation schemes, they still occur due to lack of observations in data-sparse areas such as the North
Atlantic. Such forecast errors are particularly serious since they tend to involve cases of rapid cyclogenesis which can cause substantial damage and threaten human life. We believe that manual intervention by the method described in this investigation has a large potential for use in an operational environment, in preparing an improved subjective analysis consistently in 3D. Following the corrections of the initial state, a short-range
NWP forecast is carried out. Output from such a model simulation should be regarded as an important supplementary tool, together with data retrieved from operational NWP models and all kinds of available observations, in diagnosing and forecasting severe weather.
In section 2 we briefly recall some basic theoretical concepts on which our investigations are based and describe the methods which are used. In section 3 the selected cases are described. The experiments (model reruns) are presented in section 4, followed by a discussion in section 5. Conclusions are presented in section 6.

IMPROVED SIMULATION OF WINTER STORMS BY PV MODIFICATION 2627
2.
T
HEORY AND METHODS
( a ) Theory
Ertel’s potential vorticity, q
, is given by q =
1
ρ
η · ∇ θ,
(1) where
ρ is density,
η is the absolute vorticity vector and
∇ θ is the 3D gradient of potential temperature. PV is expressed in PV units, defined as 1 PVU
=
10
−
6 m
2 s
−
1
K kg
−
1
.
PV is modified by diabatic effects and friction according to
D q
D t
≡
∂q
∂t
+ u
· ∇ q =
1
ρ
η · ∇ ˙ +
1
ρ
∇ ×
F r
· ∇ θ,
(2) where u is the wind vector, F r is the friction force and
θ ˙ denotes diabatic heating.
Equation (2) shows that PV is conserved for a particle in adiabatic and inviscid flow.
From Eq. (2) one can demonstrate how PV is generally modified by diabatic effects and friction. When, following Hoskins et al.
(1985), Eq. (2) is integrated over a volume
τ and Stokes theorem is invoked, one obtains
D
D t
τ qρ d
τ =
S
θ ˙ η + θ ∇ ×
F r
· n d
S.
(3)
Here
S is the surface area of the volume
τ
. From this relation it is seen that the massintegrated PV contained within a volume changes due to diabatic effects and friction on its surface only. Diabatic effects and friction within the volume will merely redistribute
PV while preserving the total mass-integrated PV within the volume.
Now considering flow with small Rossby number (geostrophic scaling) and large
Richardson number, by omitting friction, assuming hydrostatic flow and using isentropic coordinates, Eq. (2) becomes
∂q
∂t
= − v θ
· ∇
θ q + q 2
∂
∂θ
θ ˙ q
,
(4) where v θ and
∇
θ are the horizontal wind vector and horizontal gradient operator, respectively, on a
θ
(isentropic) surface. The second term on the right-hand side (RHS) of Eq. (4) shows that PV is enhanced (depleted) in the region below (above) a diabatic heating maximum. Hence a PV dipole is created with the connecting axis aligned along the absolute vorticity vector, as seen by the first term on the RHS of Eq. (2).
The theory of SVs is thoroughly documented elsewhere (e.g. Buizza and Palmer
1995) and is not repeated here. However, a noteworthy aspect of SVs is the growth of the leading SVs in terms of their PV structure (Montani 1998). The (total) PV, winds and the effect from diabatic heating may be partitioned as q = q + q u
= u
+ u
S = S + S
(5)
(6)
(7) where q and u represent the basic state flow changing slowly with time, while q and u denote rapidly changing anomalies of the flow. The total effect from diabatic heating
S
, given by the first term on the RHS of Eq. (2), is expressed by the large-scale heating

2628
Figure 1.
(a) Schematic presentation of upper-level basic state PV ( q
, thin contours) and PV anomalies ( q
, bold contours). (b) Schematic presentation of vertical structure of an initial singular vector (thin contours), and PV anomalies (bold dashed contours). The contour interval is arbitrary. See text for discussion of areas A, B, C, D.
term affecting the basic state flow,
S
, and the released latent heating term associated with the eddies, denoted by
S
.
Friction is now omitted and the following relations are valid:
D q
D t
D q
D t
≡
≡
∂q
∂t
∂q
∂t
+
+ u u
·
· ∇
∇ q q
=
=
S,
S.
By subtracting Eq. (9) from Eq. (8) and using Eqs. (5), (6) and (7), one then gets
(8)
(9)
¯ q
D t
= − u
· ∇ q − u
· ∇ q + S .
(10)
Finally, through linearization (neglecting the term u
· ∇ q
) and by assuming that the diabatic contribution affects q only, i.e.
S =
0, the following expression is obtained:
¯ q
D t
≡
∂
∂t
+ ¯ · ∇ q = − u
· ∇ q.
(11)
Hence q is modified by exchanging PV with the basic state flow.
From Eq. (11) it follows that a PV perturbation tends to change if it is located in a region of strong basic state PV gradients. Such gradients are typical at the tropopause close to the upper-level jets and at lower levels of the troposphere where basic state lowlevel temperature gradients with associated warm (cold) temperature anomalies can be regarded as low-level positive (negative) PV anomalies (Hoskins et al.
1985).
We now describe some implications of Eq. (11). In Fig. 1(a), q max ( q min) denotes the maximum (minimum) basic state PV, and the bold contours represent the (idealized) streamlines related to anomalies ( q
). In the confluent region of q
, i.e. at A in Fig. 1(a), the advection term
− u
· ∇ q is large and negative ahead of the anomaly, but small and positive behind it. Thus the PV perturbation tends to weaken according to Eq. (11).
In the region of strong uniform gradients of q
, such as at B, an anomaly will change according to the asymmetry of its shape. In the diffluent region of q
, at C in Fig. 1(a),
∇ q and u point in opposite directions in the regions of strong gradients of q behind the anomaly, while the gradient of q is much weaker downstream. Thus the perturbation

IMPROVED SIMULATION OF WINTER STORMS BY PV MODIFICATION 2629 tends to grow. This is also consistent with results derived from the Eliassen–Palm flux, which show that the amplitude of a PV anomaly will grow when it moves into a region with weaker basic state PV gradient (e.g. Andrews et al.
1987). The PV anomaly at D is located in a region with weak basic state PV gradients and will accordingly not change much.
Figure 1(b) shows schematically a typical cross-section of a leading initial SV. At initial time the SVs generally have a strong vertical upstream tilt. The PV structure inferred from cases discussed below suggests such an upstream tilt with height, highlighting the baroclinicity of the perturbations. During the rapid growth of the SV, the upstream tilt of the SV structure decreases as energy propagates upwards (e.g. Buizza and Palmer 1995; Montani et al.
1999). Hence the perturbation wind u increases at upper levels where interaction with the strong gradients of upper-level PV close to the jet takes place, according to Eq. (11).
The bold dashed contours show (schematically) the cross-section of the 2 PVU contours, showing the dynamical tropopause at the upper (250–300 hPa) levels with associated PV anomalies. Low-level positive PV anomalies are also indicated. At A, modification of the upper-level PV anomaly may have a significant effect on a numerical rerun, while any change at lower levels will have little or no impact. At C, the lower part of the troposphere is sensitive and modification of the flow may be beneficial for a model rerun; here the upward propagation of energy may form a PV anomaly at the tropopause and promote its subsequent rapid growth. In the cases studied below the dynamically active region subject to PV modification is collocated with the part of the leading SV which has a vertical structure as the one seen at B in Fig. 1(b). In these cases the SVs are located in regions of pronounced gradients of basic state PV at upper- and midtropospheric levels. Strong low-level baroclinic zones are also present and these are reflected by the basic state low-level PV pattern seen in the cases below.
Hence for the cases studied below, PV modifications throughout the troposphere are expected to have a large impact on the numerical simulation.
By rapid and linear growth, targeted SVs are designed to reach peak amplitude within a specific region (the verification area) at a specific time, referred to as the optimization time.
( b ) Methods
We now describe the methods which are used in this investigation.
In several investigations where PV is used to describe atmospheric dynamics, PV is broken up into PV components which form ‘building blocks’ of the atmospheric flow
(e.g. Fehlmann and Davies 1999; Thorsteinsson et al.
1999; Røsting et al.
2003). By inverting the PV field, the entire flow structure is obtained. The PV inversion routine used in this investigation combines two inversion methods: adding the selected PV anomalies to the mean PV (average taken over e.g. 48 hours), referred to as ‘addition to mean’, and subtracting the PV anomalies from the total PV, described as ‘subtraction from total’ (Davis 1992). This procedure yields almost the same result as the ‘full linear’ method of Davis and Emanuel (1991), according to Davis (1992). The procedure of defining PV anomalies is described in (e.g.) Kristjansson et al.
(1999). The inversion is based on Charney’s balance condition (Charney 1955) while horizontal boundary conditions are given by the hydrostatic relation. Vertical boundary conditions are given by the geopotential and stream function.
In the investigation of the Christmas storm of 1997 (Røsting et al.
2003), several
PV anomalies likely to influence the storm were identified. Grid-point values of selected
PV anomalies were modified and this was a rather cumbersome process. In the present

2630 felt3 Pot.Virvling 270K (+0) 1999 Ŧ 12 Ŧ 25 18 UTC
Figure 2.
Model area for the experimental simulations by HIRLAM with 20 km horizontal resolution and 40 model levels, shown by the fine-mesh resolution. (For clarity, only every fifth grid line is shown.) The HIRLAM boundary fields are provided for the area with the coarser resolution (50 km, 31 model levels). PV inversion is carried out in this large region.
study the DIgital ANAlysis system (DIANA) used operationally at the Norwegian
Meteorological Institute (NMI) for producing synoptic surface weather charts has been modified to include correction of PV and relative humidity (RH) fields on pressure surfaces. These modifications are introduced according to features in WV images which are superimposed on PV and RH fields from the numerical analysis. Thus the total PV
(and RH) fields are now modified interactively, without the need for defining separately the relevant PV anomalies. However, PV inversion by the method developed by Davis
(1992) and adopted in (e.g.) Kristj´ansson et al.
(1999), is based on dividing the PV field into an average and a perturbation part. This technical problem is coped with in the following way:
(i) Calculate PV in the large region with coarse grid shown in Fig. 2, at WMO mandatory levels
∗
, yielding the total PV field, q tot
.
(ii) Define a positive (proxy) PV anomaly which equals the whole positive PV perturbation field in the large region, at all levels, q
(iii) Modify the PV field q tot
.
(iv) Replace q
1 by q mod
1
.
according to WV/SV information, yielding q and let q = q tot
+ q
(v) A new PV field is then given as q
NEW mod
.
= q − q tot
= (q tot
+ q mod
) − q mod tot
= q mod
.
This procedure allows the use of PV inversion with the method developed by Davis
(1992).
In this investigation we combine two methods of identifying sensitive regions in the atmosphere, in attempting to improve the analysis and forecasts. These methods are:
(i) A method that exploits the strong relationship between PV anomalies and features in WV images in the baroclinic zones. Since WV is an efficient absorber and emitter at 6.2–6.7
μ m, a WV image clearly shows dry and moist regions in the upper
∗
1000, 900, 800, 700, 600, 400, 300, 250, 200, 150, 100 hPa.

IMPROVED SIMULATION OF WINTER STORMS BY PV MODIFICATION 2631 troposphere as dark and bright features, respectively. It is further recognized that in the baroclinic zones there is a close correspondence between such dark (bright) features in the WV images and positive (negative) upper-level PV anomalies (e.g. Appenzeller and Davies 1992; Mansfield 1996). A mismatch between features in WV images and the corresponding numerical PV fields indicates analysis errors and hence calls for an adjustment of PV anomalies, i.e. moving and changing the amplitudes of such anomalies.
(ii) The objective, adjoint methods, by which SVs are calculated. The leading SVs point to regions where analysis errors, if they exist, will grow rapidly and hence identify regions where additional observations, such as derived from satellite imagery, will have a large positive impact on the numerical resimulation.
The reason for involving SV analysis is the rather large degree of freedom in deciding on the 3D structure of the PV modification. The horizontal extent of the PV modification is fairly clear from the WV–PV relation. However, the vertical structure of such modifications is more difficult to determine. In order to cover as well as possible the most sensitive parts of the flow, including low-level PV anomalies, the location of which is difficult to infer from satellite images, information on SV structure is used as an additional tool in the PV modification.
SVs often have a wide spatial extent, as seen below, frequently taking the shape of a wave train covering a large region, such as the region between North America and western Europe. This is particularly evident at upper-tropospheric levels where SV structure may reflect possible downstream impacts to the verification region. At such levels, energy propagation along the jets is most rapid, normally at 30
◦ longitude day
−
1
.
If analysis errors exist, their effects may reach the verification region from remote regions within 48 hours or less, the time depending on the strength of the upper-level jet. Hence the SVs may cover a large spatial area, particularly when the verification area and optimizing time are large.
In real-time experiments (e.g. FASTEX), flight campaigns were designed to provide additional observations by dropsondes in sensitive regions. The need to have SVs confined to a smaller region is then obvious from planning reasons. By performing a scaling of the leading SVs relative to SV1, the sensitive regions are made smaller, however at the risk of losing the signal in true sensitive regions (relation (12) in
Buizza and Montani 1999). In the present investigation we select the sensitive region of interest by identifying the dynamically active regions, inferred from satellite imagery and synoptic reports if available. The need for spatially confined SVs in theoretical studies is then not strictly necessary. However, for clarity, there may be a need for optimally highlighting the sensitive regions of interest when combining the WV–PV and SV methods. In this study we have attempted to make the SV signals more spatially confined by choosing a smaller verification region (e.g. Montani 1998) and shorter optimization time, particularly required in cases of imminent severe cyclogenesis.
The vertical extension of the PV modifications is sought to be confined mainly to levels indicated as sensitive according to the fastest-growing SVs. The modification includes low levels of the troposphere (e.g. 700–900 hPa), for consistency reasons as seen from Eq. (3), and also supported by the pronounced SV signal observed in many cases (e.g. the Hallowe’en storm case, studied below).
After completing the PV modification, fields of winds and geopotential heights are obtained by PV inversion, and these fields constitute, together with RH fields (which may also be modified), a corrected balanced analysis for our experimental simulations.

2632
Our control simulation and experimental reruns are based on a version of HIRLAM
(High-Resolution Limited-Area Model) with 20 km horizontal resolution and 40 model levels. The regional NWP model area for these simulations covers a part of the North
Atlantic, Greenland and western Europe, corresponding to the fine mesh in Fig. 2. It is believed that a resolution of 20 km horizontally and 40 model levels is adequate for resolving the diabatic effects taking place in the storms, resulting in the mesoscale structures of the systems described below (e.g. Kristj´ansson 1990).
The boundary fields for our experiments are obtained from the European Centre for Medium-range Weather Forecasts (ECMWF), and the PV subject to modifications in our investigation is calculated at the WMO mandatory levels. These calculations are based on fields obtained by a HIRLAM simulation using 50 km horizontal resolution and 31 model levels. The area for these simulations is shown as the region with coarser grid in Fig. 2.
SV calculations are conducted with the ECMWF spectral model T42L31, using the
ECMWF PrepIfs System (ECMWF 1999), with an optimization time appropriate for the different cases, i.e. equal to the time for the cyclone to reach maximum intensity within a geographically defined verification region (e.g. Buizza and Montani 1999).
The verification areas for the cases in this study are defined as the areas confined to a 15
◦ ×
20
◦ latitude–longitude region (but 20
◦ ×
20
◦ in the case of the T1 storm) containing the cyclone at optimization time. Optimization time is rather short for the cases studied in this investigations, since severe cyclogenesis was imminent.
In these investigations we consider the five leading SVs. Buizza and Montani (1999) presented an objective method for assessing the rapidly growing analysis error, referred to as the ‘pseudo-inverse initial perturbation’, determined by projecting the forecast error onto the leading SVs. Then the analysis error is calculated by adjoint methods (i.e.
inverting the ‘forward propagator’). They showed that the analysis error can be reduced by
∼
15%, which is an upper bound achieved by this procedure. Ten SVs were used, however the method appears to be insensitive to a number of SVs beyond four. Thus, considering only the five leading SVs should be sufficient.
( c ) Limitations of the method
There are some possible adverse effects related to the PV inversion routine and SV analyses used in our investigations.
(i) PV inversion requires positive PV. In regions of low static stability, HIRLAM simulations occasionally give regions with slightly negative static stability, resulting in negative PV (by Eq. (1)). This problem is solved technically in the PV inversion routine, by a method which allows the iteration to converge (see Kristj´ansson et al.
1999 for details). However, the resulting inverted fields in such regions may deviate substantially from the observed fields, e.g. in the range 80 to 120 m geopotential height higher than observed. These effects tend to be local, however, and are in the cases presented here located geographically far from the strong baroclinic zones and thus far from the
Atlantic storm tracks.
(ii) The analysis obtained from PV inversion does not include the divergent wind component and this may be thought to have a detrimental effect on the numerical simulation (Browning et al.
2000). However, in our numerical experiments, divergence appears to be rapidly restored (after about 3 hours simulation). This is also supported by the rapid development of low-level PV during the first 12 hours of the simulations performed in this study, arising from condensational heating in the rising air (Eq. (4)), not shown.

IMPROVED SIMULATION OF WINTER STORMS BY PV MODIFICATION 2633
(iii) In some of the cases presented below, SV calculations were based on analyses which contained significant errors. The calculation of SVs is based on linearization of the model operator, referred to as the tangent linear propagator (e.g. Buizza and Palmer
1995). The latter is calculated along the nonlinear trajectory, i.e. the NWP forecast itself.
Thus the SV technique may fail to highlight properly the true sensitive regions in such cases of serious forecast failures.
In the cases studied below, despite serious forecast failures of mesoscale and smaller synoptic-scale phenomena, the leading SVs did indicate sensitive regions located in observed dynamically active areas. We believe this is due to the fairly good model simulation of the large-scale structure of the flow, on which the calculation of the SVs, with coarse T42L31 resolution, is based.
The structure of the leading SVs seems to depend much on the size of the verification area and the optimizing time. By starting the simulations for the French Christmas storms 6 hours earlier, and with a large verification area, essentially covering most of
Europe, SVs 2–5 were more successful in pointing to the true, observed sensitive regions than SV1 (not shown).
( d ) Summary
We now summarize the key steps of the WV–PV–SV method as follows:
•
Within baroclinic zones, WV images show dark and bright features which provide an indication of PV anomaly location.
•
A possible mismatch between the PV field in the NWP analysis and features in the WV image is identified.
•
The PV structure of the fastest-growing SVs (i.e. 1–5) is inspected in order to identify sensitive regions objectively.
•
In case of a mismatch between the PV field and the WV signals, the PV is modified accordingly. The spatial structure of the modification should include levels indicated as sensitive by the leading SVs. In the cases presented below, the leading
SVs all show sensitivity in the dynamically active region, for the selected verification area and optimization time. If the leading SVs fail to indicate any sensitivity (due to reasons mentioned in subsection 2(c)) in a region which is clearly dynamically active, the vertical structure of the PV modification is made according to experience from several studies of PV anomalies associated with strong cyclogenesis. Hence
PV modifications are introduced at tropopause level according to the WV signal. An additional requirement should be that the dynamical tropopause, given by PV
=
2 PVU, associated with a positive PV anomaly, is frequently located at, but rarely much below,
500 hPa. When negative PV anomalies are introduced or enhanced at upper levels, in order to match cloud heads with location inferred from satellite images, low-level positive PV anomalies associated with the cyclone should be enhanced as well. This is motivated by the principle of trying to maintain some consistency in PV modifications, according to diabatic effects described by Eqs. (3) and (4). The RH may be modified, in order to be consistent with the corrected PV field, e.g. RH should be reduced in regions where upper-level PV anomalies are enhanced or introduced, such as in tropopause foldings.
•
The modified PV anomalies are tuned to yield an inverted field (geopotential heights and winds) which is consistent with the subjective analysis, i.e. with strong weight on the available observations. The 1000 hPa inverted field is interpolated to mean sea level pressure. The tuning is performed by repeating the key steps described here until the inverted fields are as close to the observed ones as desirable. Since the

2634 method is done interactively the time required will be reasonably short, e.g. within an operational time limit.
The inverted fields, in terms of geopotential height and horizontal velocity components together with RH, constitute a new ‘analysis’ which contains balanced fields, and assimilation routines are not invoked. However, it is possible to combine PV modification as described here with variational methods, as in Verkley et al.
(2005). In that case, after carrying out a variational analysis (i.e. minimizing the cost function), the new analysis is used as background field in a new formulation of the cost function which contains a PV term replacing the observation term. The PV term is mathematically quite similar to the observation term, measuring the distance between modified PV and PV retrieved from the model output. After minimizing this new cost function, followed by normal mode initialization, an improved numerical analysis was obtained. A numerical rerun based on the new analysis was substantially improved.
3.
S YNOPTIC DESCRIPTIONS
We now describe the three cases investigated. The first two took place during
Christmas 1999 and affected western Europe, particularly France. They are referred to as the T1 and T2 storms; T1 occurred on 25–26 December, followed by T2 on 27
December. The third storm is referred to as the Hallowe’en storm, which affected Great
Britain, the North Sea and southern Norway on 30 October 2000.
Figure 3(a) shows the surface synoptic analysis of the T1 cyclone at 18 UTC on
25 December, while Fig. 3(b) shows the strong zonal 300 hPa jet, indicated by isotachs.
Observations at 300 hPa indicate a jet much stronger than the one shown in the numerical analysis, by
∼
50 kt near the exit region of the jet. The observed track and central surface pressure development of the T1 storm are also presented in this figure.
In many of the numerical analyses and forecasts at the time (not shown), the central pressure of the cyclone at 18 UTC on 25 December was about 10 hPa higher than observed, and the strong intensification of the cyclone was not caught by the operational
NWP models in the short range (Baleste et al.
2001).
The subjective analyses showed a moderate deepening of the cyclone until about
00 UTC on 26 December. At that time the cyclone was located in the region below the left exit of the strong upper-level jet, resulting in the explosive deepening of the storm over the next 6 hours (Fig. 3(b)). The central surface pressure reached
∼
960 hPa at
06 UTC on 26 December.
Figure 3(c) shows the T2 cyclone at 06 UTC on 27 December 1999, while Fig. 3(d) presents the 300 hPa winds (in knots) at the same time, indicating the upper-level jet.
The inserted wind observations suggest that the upper-level jet was underestimated in the NWP analysis by at least 25–30 kt. Six hours earlier, the sounding from Brest showed an exceptional maximum wind speed of 285 kt (147 m s
−
1
) at 333 hPa (Baleste et al.
2001), indicating that the observed winds exceeded the NWP analysed winds by about
140 kts at that time! It seems very likely that the NWP model forecast failure of the
T2 storm was influenced by the poor simulation of the upper-level jet; this issue will be discussed in section 5. The observed track and central pressure development of the T2 storm are presented in Fig. 3(d). The cyclone developed rapidly after 06
UTC
, reaching peak intensity at 18
UTC with central pressure down to
∼
964 hPa. The deepening of the cyclone was so rapid that the 4D-Var system at ECMWF failed in adjusting the analysis to new observations.

IMPROVED SIMULATION OF WINTER STORMS BY PV MODIFICATION 2635
129
8400
163
8710
180
8870
291
8400
285
8770
271
8440
279
8540
270
8690
183
8470
221
8690
262
9050
254
8950
303
9300
308
9390 T1control FF.KNOP 300hPa (+0) 1999 Ŧ 12 Ŧ 25 18 UTC
12 Ŧ 25 18:00 (17:00 Ŧ 19:00 )
T2control FF.KNOP 300hPa (+0) 1999
Ŧ
12
Ŧ
27 06 UTC
TEMP 300hPa 1999 Ŧ 12 Ŧ 27 06:00 (05:00 Ŧ 07:00 )
139
8440 125
8600
286
8450
276
8500
231
8400
169
8470
138
8420
285
8820
295
8650 290
8600
246
8550
264
8580
X
8860
261
8760
273
9160
286
9360 293
9350
278
9170
103
8840
156
8650
271
8670
285
8890
290
9030
206
8580
259
8880
285
9060
226
9150
211
9060
221
9180
236
9290
218
245
8950
TEMP 300hPa 2000 Ŧ
Ŧ 10 Ŧ 29 18 UTC
Ŧ 29 18:00 (17:00 Ŧ 19:00 )
Figure 3.
(a) Subjective analysis of the incipient T1 storm at 18 UTC on 25 December 1999. (b) NWP analysis of
300 hPa isotachs (shading, knots) for the T1 storm at the same time, with observed 300 hPa winds and geopotential heights inserted. The cyclone track is shown with central pressures (hPa) at the indicated times, starting at 12 UTC on 25 December 1999 at left. (c) is as (a), but for the T2 storm at 06 UTC on 27 December 1999. (d): As (b), but for the T2 storm case, with central pressures starting at 00 UTC on 27 December. (e) is as (a), but for the
Hallowe’en storm at 18 UTC on 29 October 2000. (f) is as (b), but for the Hallowe’en storm case, with central pressures starting at 18 UTC on 29 October.

2636
Figure 3(e) shows the synoptic analysis of the incipient Hallowe’en storm at 18
UTC on 29 October 2000, located at 50
◦
N, 20
◦
W, while Fig. 3(f) shows the 300 hPa winds at the same time, giving an indication of the jet. According to the observations, the winds are better represented in this NWP analysis than in the previous ones. The observed track and central pressure development of the storm are shown in Fig. 3(f). This storm shows a steady and very rapid deepening over 24 hours (
∼
30 hPa (12 hours)
−
1
), reaching peak intensity at 18–21 UTC on 30 October, i.e. central surface pressure at
∼
946 hPa.
A thorough investigation of the mesoscale structure of this storm, as it travelled across
UK, is provided by Browning (2005). The cyclone was located below the left exit of the upper-level jet already at 00 UTC on 30 October. Contrary to the previous cases, the storm was to some extent forecast by the NWP models at the time, but the deepening was underestimated (by 10–15 hPa) and its propagation was somewhat too slow in the operational Norwegian HIRLAM simulation (not shown). Some improvement was achieved by HIRLAM test simulations adopting 3D-Var assimilation, about to become operational at the NMI at the time.
4.
E XPERIMENTS
( a ) The T1 storm
In order to investigate to what extent the analysis of the westerly flow across the
North Atlantic was sensitive to modifications (e.g changes based on information inferred from WV images), we now study the structure of the leading SVs. Figures 4(a–c) and
4(d–f) show the PV structure of the initial SV1 and SV3 respectively, at 18 UTC 25
December 1999, presented at levels 13 (
∼
300 hPa), 18 (
∼
500 hPa) and 24 (
∼
850 hPa).
Their peak amplitude is reached within the verification region defined between 40–
60
◦
N, 05
◦
W–15
◦
E at an optimization time of 18 hours. SV2 has weaker signals than
SVs 1 and 3 at upper levels west of the cyclone, while SV4 shows much of the same structure as SV3. SV5 has a very strong upper-level signal close to the cyclone. All the five leading SVs have nearly identical signals at lower levels.
The strongest signals given by the leading SVs are located over the North Atlantic at middle and upper levels of the troposphere, upstream of the surface low whose position at 18
UTC is marked by letter L in Figs. 4(c) and (e). They clearly indicate the importance of an upper-level forcing. The SVs give a weaker, though more concentrated, signal at lower levels at the location of the T1 cyclone.
Figure 5(a) shows the WV image valid at 18 UTC on 25 December, with the control
(original) PV field superimposed on the 400 hPa and 800 hPa surfaces. The cloud head and dry stratospheric intrusion associated with the incipient T1 storm can be identified at A and B respectively. Letter D indicates the 800 hPa PV anomaly. The PV contours are at this time aligned anticyclonically across the bright region (A), with the strongest gradient along the cloud-head edge, as would be expected for a well-developed cloud head. The dark area at B appears to be well collocated with the position of the positive upper-level PV anomaly in this area, however a more pronounced ‘PV trough’ associated with the dry intrusion would be expected. We may thus conclude that any discrepancy between the WV signal and the upper-level PV field in the NWP analysis is related to the amplitude rather than the location of the upper-level PV anomalies.
Figure 5(c) shows the
+
15 hour forecast surface pressure in the control run of the
T1 storm valid at 09
UTC on 26 December, based on unmodified PV. Also shown are the cyclone track and the central pressure development from this simulation. Comparing to
Fig. 3(b), it is seen that the control run grossly underestimated the central deepening, the track is also too far south and the propagation speed of the cyclone is too slow.

IMPROVED SIMULATION OF WINTER STORMS BY PV MODIFICATION 2637
a b
50N
60N
00
20W
L
c
d
L
e f
Figure 4.
Potential vorticity structure of initial singular vectors 1 and 3 at 18 UTC on 25 December 1999 with optimization time of 18 hours: (a) SV1 at level 13 (
∼
300 hPa), (b) SV1 at level 18 (
∼
500 hPa), (c) SV1 at level 24
(
∼
850 hPa), (d) SV3 at level 13 (
∼
300 hPa), (e) SV3 at level 18 (
∼
500 hPa), and (f) SV3 at level 24 (
∼
850 hPa).
Bold solid (dashed) contours show the positive (negative) values of the SV, with contour interval 0.004 PVU. Thin contours depict basic state PV with contour interval 0.5 PVU. The position of the incipient cyclone is indicated by letter ‘L’ in (c) and (e).

2638
T1origPV pvtfn 400hPa (+0) 1999 Ŧ 12 Ŧ 25 18 UTC
METEOSAT7 Vanndamp WV_CAL 1999 Ŧ 12 Ŧ 25 18:00
T1modPV800 pvtfn 800hPa (+0) 1999
Ŧ
12
Ŧ
25 18 UTC
METEOSAT7 Vanndamp WV_CAL 1999 Ŧ 12 Ŧ 25 18:00
Figure 5.
(a) EUMETSAT WV image at 18 UTC on 25 December 1999, with contours of PV on the 400 hPa
(solid) and 800 hPa (dashed) surfaces from the control run superimposed. The letters A, B, C and D show the regions which are modified; see explanation in the text. (b) EUMETSAT WV image at the same time, with contours of modified PV as in (a). (c) Sea level pressure in CONTROL run at 09 UTC on 26 December 1999, with the track and central pressure development as in Fig. 3(b). (d) Sea level pressure from rerun at 09 UTC on
26 December 1999, based on the analysis obtained from inversion of the modified PV, with the track and central pressure development as in Fig. 3(b).
We now adjust the PV field shown in Fig. 5(a) according to information in the WV image. The arrows shown in Fig. 5(a) indicate the regions where PV modification is required, according to the discussion above.
The following PV adjustments are made:
(i) Upper-level PV is enhanced at B, with a sharper PV trough introduced in the dark area of the dry intrusion. At C, PV contours are pushed slightly north across the bright region in the image. The modified PV is shown in Fig. 5(b) for the 400 hPa surface.
The PV is left mainly unchanged in the cloud-head region (A), except for some minor adjustments in the regions adjacent to area B.
(ii) PV at lower levels, 700–900 hPa, is moved closer to the storm, maintaining its amplitude of
∼
1.5 PVU. This modification is based on previous case-studies revealing the location of such PV anomalies (e.g. Shutts 1990; Kristj´ansson et al.
1999). SVs 1 and 2 show signals at these lower levels (Figs. 4(c) and (f)), with peak amplitude close

IMPROVED SIMULATION OF WINTER STORMS BY PV MODIFICATION 2639 to the cyclone centre, also supporting the location of the PV modification. Figure 5(b) shows the modified PV fields at the 400 and 800 hPa levels.
The modified PV field is now inverted, yielding a new analysis. Figure 5(d) shows the modified 21-hour forecast of the T1 storm, as well as the track and development of central pressure produced by this model rerun. There is a substantial improvement of the simulated cyclone development, compared to the control run (Fig. 5(c)). However, the track is now slightly too far north of the observed and the propagation of the storm is still slower than observed (Fig. 3(b)). The deepening between 00 and 09
UTC is still underestimated, though greatly improved.
( b ) The T2 storm
Figures 6(a–c), 6(d–f) show the PV structure of the initial SV1 and SV2 respectively, at 06 UTC 27 December, with optimization time of 12 hours. The verification area in this case is 40
◦
–55
◦
N, 10
◦
W–10
◦
E. The singular vectors are presented at levels 13
(
∼
300 hPa), 18 (
∼
500 hPa) and 24 (
∼
850 hPa). The leading SVs show distinct signals in the upper and middle troposphere, e.g. at levels 13 and 18, close to the T2 cyclone which is located at 48
◦
N, 18
◦
W; the position is shown by the letter L in Figs. 6(c) and (f).
Consequently,the regions particularly sensitive to perturbations affecting the T2 storm are located at and below the upper-level jet at 45–55
◦
N, 00–30
◦
W, see Fig. 3(d).
The signals are weaker than at upper levels, but still pronounced and localized at the cyclone in the lower troposphere (e.g. level 24), highlighting the importance of the lowlevel part of the baroclinic zone for the deepening of the T2 cyclone. Singular vectors
3–5 are less localized than SV1 and SV2, showing very strong signals in the upper troposphere over the North Atlantic, including the position of the incipient T2 storm
(not shown).
Figure 7(a) shows the WV image valid at 06 UTC on 27 December with the unmodified PV fields on the 400 hPa and 800 hPa surfaces superimposed. The PV contours are running anticyclonically across the cloud head A. However, at this late stage of the cloud-head development, the anticyclonically oriented PV gradients should have been extended further north, as observed in the T1 case above (Figs. 5(a) and (b)).
The PV ‘trough’ at the rear of the developing wave is well positioned according to the WV signal at B. The low-level PV anomaly is indicated by C, giving the position on the 800 hPa surface.
Figure 7(c) presents the control simulation of the cyclone at
+
15 hours, valid at 21
UTC on 27 December, as well as the track and central pressure in the control simulation, all based on the unmodified PV. Comparing to the observed one (Fig. 3(d)), the simulated track is seen to be too far south and the deepening of the cyclone is seriously underestimated. This simulation was not much different from most operational runs at the time (not shown).
There is a distinct signal by SV1, SV2 in regions A and B throughout the troposphere in the dynamically active region, as seen by comparing Figs. 6(b), 6(e) and
7(a). The low-level PV anomaly C is also located in a sensitive area, see Figs. 6(c), 6(f).
Hence PV is modified in the regions indicated by the arrows in Fig. 7(a) by the following adjustments:
(i) PV is now reduced (by
∼
1–2 PVU) in the cloud-head region A between 300 and 600 hPa, pushing the high PV to the north. This results in a picture resembling the one in Fig. 5(a) for the T1 storm. The modification is also supported by the presence of convective cells north of the cloud head, suggesting that this is a region containing deep

2640
a b
00
60N
20W
50N
L
c d f
L e
Figure 6.
Potential vorticity structure of initial singular vectors 1 and 2 at 06 UTC on 27 December 1999 with optimization time of 12 hours: (a) SV1 at level 13 (
∼
300 hPa), (b) SV1 at level 18 (
∼
500 hPa), (c) SV1 at level 24
(
∼
850 hPa), (d) SV2 at level 13 (
∼
300 hPa), (e) SV2 at level 18 (
∼
500 hPa), and (f) SV2 at level 24 (
∼
850 hPa).
Contours are as in Fig. 4. The position of the incipient cyclone is indicated by letter ‘L’ in (c) and (f).

IMPROVED SIMULATION OF WINTER STORMS BY PV MODIFICATION 2641 pvinput2 pvtfn 800hPa (+0) 1999
Ŧ
12
Ŧ
27 06 UTC
METEOSAT7 Vanndamp WV_CAL 1999 Ŧ 12 Ŧ 27 06:00 pvinput pvxfn 800hPa (+0) 1999 Ŧ 12 Ŧ 27 06 UTC pvinput2 pvxfn 400hPa (+0) 1999 Ŧ 12 Ŧ 27 06 UTC
METEOSAT7 Vanndamp WV_CAL 1999 Ŧ 12 Ŧ 27 06:00
Figure 7.
(a) EUMETSAT WV image at 06 UTC on 27 December 1999, with the PV on the 400 hPa and 800 hPa surfaces from the control run, as in Fig. 5(a). The letters A, B and C show the regions where PV is modified; see explanation in the text. (b) EUMETSAT WV image at the same time, with contours of modified PV as in
(a). (c) Sea level pressure from CONTROL run at 21 UTC on 27 December 1999, with the track and central pressure development as in Fig. 3(d). (d) Sea level pressure from rerun at 21 UTC on 27 December 1999, based on the analysis obtained from inversion of the modified PV, with the track and central pressure development as in
Fig. 3(d).
cold air with associated lowered tropopause. A displacement of the high PV over the cloud head to the region north of it can then be justified.
(ii) The PV maximum (positive PV anomaly) at B is moved slightly to the east, and enhanced at levels between 300 and 700 hPa, as ‘PV troughs’ are generally collocated with the darkest part in the WV images, with PV maxima tucking in behind. Figure 7(b) shows the adjusted PV field (on the 400 hPa surface), displaying a large change in PV pattern as seen by comparison with Fig. 7(a).
(iii) A slight displacement and enhancement (
∼
0.5 PVU) of the low-level PV at C at levels 700–900 hPa to a position closer to the surface cyclone has been included, as seen in Fig. 7(b), for the same reason as mentioned in the T1 case (e.g. Shutts 1990;
Kristj´ansson et al.
1999). Based on the discussion in subsection 2(a), enhancement of low-level PV is also justified for consistency reasons, i.e. reduction of PV in the cloud head is consistent with enhanced PV in the lower troposphere, close to the low-level cyclone and associated low-level front.

2642
The rerun based on the modified PV field yields a significant improvement of the simulated cyclone development, as seen in Fig. 7(d) which shows the new forecast valid at 21
UTC on 27 December as well as the track and central surface pressure development.
The peak intensity in this model simulation was reached at
+
21 to
+
24 hrs, with central pressure dropping to
∼
971 hPa. There are also strong pressure gradients at the rear of the low centre, and these are features expected to be augmented by release of latent heat (e.g. Grøn˚as 1995). The track of the cyclone is now improved compared to the control run (Fig. 7(c)). However, the propagation speed was still slower than observed and the rapid central pressure fall between 06 and 12 UTC was still not caught properly by the simulation, as seen by comparing Figs. 3(d) and 7(d). Though being far from a perfect simulation, this was nevertheless a major improvement over the control (and operational) run, with central surface pressure dropping 15–18 hPa lower than in the control run.
( c ) The Hallowe’en storm
The leading initial SVs (1–5), in this case at 18
UTC on 29 October 2000, are calculated with an optimization time of 24 hours, and with verification area confined to 50–65
◦
N, 10
◦
W–10
◦
E. They show signals throughout the troposphere over a region stretching from southern Greenland and Newfoundland to western Europe, however with peak amplitude at all levels close to the incipient cyclone at 50
◦
N, 20
◦
W. The position is shown by the letter L in Figs. 8(a) and (d). Figures 8(a–c), 8(d–f) show the PV structure of SV1 and SV2 respectively, at 18 UTC on 29 October 2000. They are presented at levels 13 (
∼
300 hPa), 16 (
∼
400 hPa) and 24 (
∼
850 hPa). Contrary to the T1 and T2 storms, the strongest signal by the SVs 1–2 is located in the lower troposphere, as also observed in many case-studies of severe storms (e.g. Buizza and Palmer 1995).
SVs 3–5 have a less localized distribution over the North Atlantic, and a much stronger signal at upper levels, including regions at the incipient storm. These SVs have, on the other hand, a weaker signal than SVs 1–2 at lower levels near the cyclone (not shown).
Figure 9(a) shows the WV image at 18
UTC
29 October 2000 with the unmodified
PV field on the 400 hPa and 800 hPa surfaces. A possible mismatch between the WV signatures and the model PV field is seen at A, where a slightly more anticyclonic curvature of the PV field across the bright regions is expected. At B the location of the positive upper-level PV anomaly appears to be correctly positioned. C indicates a lowlevel PV anomaly (i.e. at levels 900–700 hPa), presented here on the 800 hPa surface.
D indicates a region of a very clear mismatch of the PV/WV signal. The dry intrusion and the bright comma-shaped feature are not captured at all by the model analysis.
Figure 9(c) presents the
+
24 hours control simulation, valid at 18 UTC on 30
October as well as the track and central pressure development of the storm, based on unmodified PV. The figure illustrates that the control run underestimates the intensity and propagation speed of the storm, as seen by comparing with Fig. 3(f). The operational
HIRLAM performed somewhat better at the time (not shown) than our control run, possibly due to the neglect of the divergent wind component in the control analysis.
The SV analysis and the WV signal both indicate that regions A and B are sensitive
(Fig. 8(a), (b), (e)), suggesting that PV modifications there might have a large impact on a model rerun. However, the SV signal at D, downstream of the incipient cyclone, is weak.
PV modification should be carried out in the regions indicated by the arrows. In this case the following PV adjustments are performed:

IMPROVED SIMULATION OF WINTER STORMS BY PV MODIFICATION 2643
L
a b
00
60N
50N
20W
c d
L
e f
Figure 8.
Potential vorticity structure of initial singular vectors 1 and 2 at 18 UTC on 29 October 2000 with optimization time of 24 hours: (a) SV1 at level 13 (
∼
300 hPa), (b) SV1 at level 16 (
∼
400 hPa), (c) SV1 at level 24
(
∼
850 hPa), (d) SV2 at level 13 (
∼
300 hPa), (e) SV2 at level 16 (
∼
400 hPa), and (f) SV2 at level 24 (
∼
850 hPa).
Contours are as in Fig. 4. The position of the incipient cyclone is indicated by letter ‘L’ in (a) and (d).

2644
Invertfields pvtfn 400hPa (+0) 2000
Ŧ
10
Ŧ
29 18 UTC
Invertfields_2 pvtfn 800hPa (+0) 2000 Ŧ 10 Ŧ 29 18 UTC
METEOSAT7 Vanndamp WV_CAL 2000 Ŧ 10 Ŧ 29 18:00
Invertfields_2 pvxfn 400hPa (+0) 2000 Ŧ 10 Ŧ 29 18 UTC
Invertfields pvxfn 800hPa (+0) 2000 Ŧ 10 Ŧ 29 18 UTC
METEOSAT7 Vanndamp WV_CAL 2000 Ŧ 10 Ŧ 29 18:00
Figure 9.
(a) EUMETSAT WV image at 18 UTC on 29 October 2000, with the PV on the 400 hPa and 800 hPa surfaces from the control run, as in Fig. 5(a). The letters A, B, C and D show the regions where PV is modified; see explanation in the text. (b) EUMETSAT WV image at the same time, with contours of modified PV, as in (a).
(c) Sea level pressure from CONTROL run at 18 UTC on 30 October 2000, with the track and central pressure development as in Fig. 3(f). (d) Sea level pressure from rerun at 18 UTC on 30 October 2000, based on the analysis obtained from inversion of the modified PV, with the track and central pressure development as in Fig. 3(f).
(i) The upper-level PV field is now reduced at A at levels between 300 and 700 hPa, by
∼
0.5 PVU, in order to correct the mismatch seen in the WV image. However, the modification is modest since the more grey-shade features in the image suggest that PVrich dry stratospheric air is over-running this part of the polar-frontal cloud band. Hence
PV at upper levels is likely to be only slightly affected by the diabatic effects at this early stage of the cyclone development. The positive PV anomaly in area B is enhanced at levels between 300 and 700 hPa in order to adjust (lower) the pressure according to observations, i.e. the amplitude of the PV field is enhanced. In region D, PV is adjusted between 300 and 500 hPa to match the WV features.
(ii) The position of the incipient cyclone in the control numerical analysis seems to be nearly correct according to observations (Figs. 3(e), 9(c)). Hence the low-level PV anomaly at 700–900 hPa is not moved but enhanced slightly (by
∼
0.5 PVU), since lowlevel PV anomalies are likely to attain values between 1 and 1.5 PVU after cyclogenesis

IMPROVED SIMULATION OF WINTER STORMS BY PV MODIFICATION 2645 and associated strong ascent have started. The consistency between reduction of upperlevel PV (at A) and enhanced low-level PV justifies this modification, and so does the very strong SV signal at these low levels (Figs. 8(c), (f)). Figure 9(b) shows the adjusted
PV fields on the 400 hPa and 800 hPa surfaces.
Figure 9(d) shows the
+
24 hours modified NWP forecast valid at 18 UTC on
30 October, as well as the track and central pressure development of the storm by the new simulation. The new simulation is quite satisfactory. Central pressure dropped to
958 hPa at 06 UTC on 30 October, 3 hPa higher than the observed pressure and 19 hPa deeper than in the control run (Fig. 3(f), 9(c) and 9(d)). At 18 UTC on 30 October, the central minimum pressure was
∼
950 hPa, about 4 hPa higher than observed. The cyclone moved slightly faster than in the control run (Figs. 9(c), (d)), however the track was slightly too far to the left compared to the analysed positions. After
+
30 hours, the simulated storm, as in the control run, moved somewhat slower than observed.
5.
D ISCUSSION
In this section we provide a further discussion of the method described above and address some unresolved problems with the results from PV inversion carried out in the investigation.
In particular we address the following issues:
(i) The missing divergent winds in the analyses obtained by PV inversion.
(ii) The limitations of the method in dealing with downstream developments.
(iii) Determination of the 3D structure of the PV modifications.
( a ) Balance condition
It has been remarked (subsection 2(b)) that the PV inversion provides the nondivergent winds only and that this limitation may have a detrimental effect on the modified simulations. We speculate that the pressure rise or lack of deepening seen in the first three hours of some experimental runs, based on analyses obtained by PV inversion, can be attributed to this neglect of the divergent wind components. These pressure tendencies were certainly not observed and cannot be ascribed to any real dynamical effects; examples are seen in Figs. 5(c) and 7(c) for the first 6 hours of simulation of the T1 and T2 storms, respectively. After 3 hours of simulation the divergent winds are restored.
In the simulation of the French Christmas storms, the prediction of the storms was improved, though propagation and development were slower than observed. Divergent motions were probably very important in these cases, particularly associated with the exceptionally strong upper-level jet, which reached 285 kt or 147 m s
−
1 at 300 hPa over Brest at 00 UTC on 27 December (Baleste et al.
2001). After another 3 hours of simulation the cyclones deepened, and this is presumably the time required for restoring the divergent motions associated with the cyclone. On the other hand, when the weak low-level (
∼
900–700 hPa) PV anomaly close to the cyclone centre was enhanced by
∼
1 PVU (see Figs. 5(b) and 7(b)), the initial erroneous filling of the cyclone weakened and even disappeared (Figs. 5(d) and 7(d)). By introducing low-level
PV anomalies, irrotational wind components developed faster during the initial 3 hours of the simulation as the low-level PV anomaly was advected by the ambient flow. Hence the balance of the flow may have been more rapidly restored in this case. In the case of the Hallowe’en storm, the erroneous pressure rise at the beginning of the simulation did not occur. The reason was perhaps that the upper-level PV advection was strong enough

2646 to counteract the detrimental effects from non-divergent winds in the new analysis, by rapidly restoring the divergent flow in the first hours of the simulation.
In some experiments connected to the cases studied in this research, RH fields were modified to be consistent with the corrected PV fields. The effects from such initially modified RH fields on the numerical reruns had only negligible effects on the forecast, compared to the reruns retaining the original RH fields. The RH fields seem to be consistently restored according to the new PV fields after 3 hours of simulation time.
( b ) Reasons for forecast failures for the T2 storm
In most operational NWP simulations, the T2 storm moved eastwards across the
Atlantic, over regions too far south for an interaction with the strong upper-level
PV gradients associated with the jet across the north Atlantic to take place. In test simulations, upper-level PV anomalies and low-level potential temperature anomalies off the eastern coast of Canada were modified. These modifications were introduced in regions indicated as strongly sensitive at upper levels by the leading SVs, see
Fig. 4. Nevertheless, the model reruns failed to catch the T2 storm. One reason for this somewhat disappointing result could be imperfections in the ECMWF boundary fields which were probably crucial for the forecast failure beyond
∼
24 hours, as well as affecting the usefulness of the SVs (see subsection 5(c)). Simulations based on sensitivities, combining adjoint and variational methods, have succeeded in better simulating the T2 storm (Hello and Arbogast 2004).
Another possible reason for the difficulties in simulating the development of the
French storms may be attributed to model deficiencies. Though a 20 km horizontal grid spacing has been adopted, this may still be inadequate for proper simulation of the mesoscale structures of the storms. Simulation of the T2 storm with 50 km horizontal grid spacing produced a central minimum pressure
∼
5 hPa higher than in the present simulation. More notable was the lack of the strong pressure gradients at the rear of the cyclone in that case (not shown). Another reason is possibly the limited integration domain. Analysis errors in the boundary fields over the east coast of North America and western North Atlantic are likely to influence the forecast for western Europe at
∼
48 hours or even less in the presence of the extraordinarily strong upper-level zonal jets. In more ‘normal’ conditions, downstream propagation of energy along the jets takes place at a group velocity of
∼
30
◦ longitude day
−
1
. Such an influence is also reflected by the strong upper-level SV signal located upstream of the cyclones studied in this research (Figs. 4, 6 and 8 above). The upstream SV signal is more pronounced with reruns starting 6 hours earlier, longer optimization time and larger verification area (not shown).
( c ) Sensitive regions
In this subsection the usefulness of SVs as an additional tool in correcting PV anomalies is expanded on. In some test simulations, modifications of PV in non-sensitive regions according to the leading SVs had no (or just a modest) effect on the simulations.
We now focus on the rerun of the Hallowe’en storm described in section 4(c), and refer to this simulation as EXP1. In another experiment, PV was not modified in region
D, while PV was otherwise modified exactly as done in section 4(c) (Fig. 9(b)). We refer to this experiment as EXP2.
Figure 10 shows the 24-hour simulations of EXP1 and EXP2. The difference between the two simulations is small, hence the large PV modification in region D had a very small effect on the numerical rerun. As mentioned in subsection 4(c), the SV signal was weak in region D (over UK, see Fig. 8).

IMPROVED SIMULATION OF WINTER STORMS BY PV MODIFICATION 2647
Hal7 MSLP (+24) 2000 Ŧ 10 Ŧ 30 18 UTC
Figure 10.
Surface pressure (hPa) on 30 October 2000 from numerical reruns of the Hallowe’en storm: EXP1
(solid contours), and EXP2 (dashed contours).
Conversely, the effect on the simulation is large when PV modification is introduced in sensitive areas, e.g. regions A, B and C in Fig. 9(a).
The SVs are to some extent calculated based on analyses containing large errors, hence the SV technique may turn out to be inadequate in pointing to the real sensitive region. Nevertheless, in the present cases the leading SVs strongly highlight the dynamically active regions. By choosing a small verification region containing the cyclone at optimization time, the SV signal becomes more localized geographically, particularly at lower tropospheric levels. Thus SV analysis seems to be a useful tool in the diagnostics of the PV analysis on which the PV modification is based. SV analysis may reduce the subjectivity of the method, by indicating the most sensitive regions subject to PV modification and in this way helping to determine the 3D structure of the modification.
In this way the selected corrections will have a large impact on the numerical rerun.
( d ) PV structure in modified simulations
The modified simulations of the storms were to a large extent successful. Even so, there were some difficulties in properly simulating the track and deepening rate of the storms. Beside the probable limiting effects related to the too-coarse resolution, the difficulties may be attributed to the underestimation of the upper-level jet as discussed in subsection 5(b). In this case the upper-level PV anomalies might have been a consequence of, rather than the reason for, the cyclone development.
The development of the upper-level PV fields in the simulations of the Hallowe’en storm is illustrated in Fig. 11. Figure 11(a) shows the WV image at 18
UTC on 30
October with the PV field on the 310 K isentropic surface from the control run at
+
24 hours (corresponding to the time of strongest cyclone intensity) superimposed.
The upper-level PV trough is lagging behind its expected position, i.e. the dry intrusion at A. The location of the 24-hour forecast PV field (310 K surface) in the modified run (Fig. 11(b)), while not in perfect agreement, shows a much better correspondence

2648
Hal_controlpv Pot.Virvling 310K (+24) 2000
Ŧ
10
Ŧ
30 18 UTC
METEOSAT7 Vanndamp WV_CAL 2000 Ŧ 10 Ŧ 30 18:00
Hal6pv Pot.Virvling 310K (+24) 2000
Ŧ
10
Ŧ
30 18 UTC
METEOSAT7 Vanndamp WV_CAL 2000 Ŧ 10 Ŧ 30 18:00
Figure 11.
(a) EUMETSAT WV image at 18 UTC on 30 October 2000, with PV on the 310 K (isentropic) surface from the control run superimposed. (b) is as (a), but with PV superimposed from the simulation based on modified
PV in the initial state.
with the WV signal. The PV trough is now covering the dark area A, and PV contours are curving anticyclonically along the edge of the hook-shaped bright feature marked with letters B, though there is some mismatch in the southern part of the region. Thus the simulation fairly consistently preserves the PV structure of the flow and the model results are in agreement with a conceptual model (not shown) describing the dry sinking
PV-rich stratospheric air within a developing extratropical cyclone (e.g. Young et al.
1987; Browning 1997).
We want to mention another possible limitation of the PV modification method.
When PV is modified in regions of dry intrusions (e.g. at B in Fig. 7(a)), only the part of the PV field which corresponds to the dry intrusion as inferred from the WV image is modified. The leading edge of the dry intrusion generally undercuts medium- and high-level clouds which are associated with the warm conveyor belt (e.g. Browning
2005). The PV in this leading part of the dry intrusion is not modified in our work.
PV in such a region is generally reduced to less than 2 PVU
∗ and the leading edge of the dry intrusion is also a mesoscale feature of the flow. Since PV inversion is a smoothing process (i.e. solving the inverse of a Laplacian-like operator), mesoscale PV anomalies such as the lower parts of dry intrusions may have a minor contribution to the inverted fields, i.e. geopotential heights, winds and temperature. A correct 3D structure is quickly established in the numerical reruns (i.e. after
∼
3 hours run) carried out in this investigation. However, the effect from neglecting the PV of the undercutting part of the dry intrusion in the new analysis remains uncertain.
The adjustment of PV associated with cloud heads is less straightforward than in the case of dry intrusions. At an early stage of the cyclone development, the emerging cloud head is farther down in the troposphere, and the upper-level jet and consequently the PV features (e.g. at 300–400 hPa levels) are not affected. The grey shade seen to the north of B in Fig. 7(a), and the grey shade observed at A in Fig. 9(a) may be examples of such features. During cyclogenesis, the cloud head expands, upper-level PV is reduced due to diabatic heating (Eqs. (3) and (4)), and PV contours gradually become oriented in an
∗
For adiabatic and inviscid flow, Eq. (3) describes the conservation of PV substance ( qρ
) rather than PV within the volume. Following an air parcel, an increase of
ρ leads to a decrease of q
.

IMPROVED SIMULATION OF WINTER STORMS BY PV MODIFICATION 2649 anticyclonic pattern across the cloud head, with the strongest gradients along the cloudhead edge, e.g. Fig. 5(a) and the modified PV pattern in Fig. 7(b). This pattern becomes most distinct at the mature stage of cyclogenesis, as seen in the hook-shaped feature B in Fig. 11(b). A possible way of determining the vertical depth of the cloud head would be to use NWP moisture fields. However, NWP RH fields are likely to be wrong in cases with large analysis errors. For instance in the cloud-head region A shown in Fig. 7(a),
RH fields at lower levels showed rather low values (less than 50% at 700 hPa), while the highest RH values between 80 and 90% at 700 hPa were located in the dark WV region
B of the dry intrusion. Currently, adopting the DIANA system, cloud-top temperatures can be read directly from the IR images, and the cloud-top heights can be estimated if there are soundings in the areas. Unfortunately, since soundings are scarce over the oceans, PV modifications associated with cloud heads may be uncertain.
( e ) Recent case
We have so far described comparatively old cases and one may argue that recent advances in NWP developments, particularly regarding 4D-Var, with more efficient use of remote-sensing data, has rendered the need for manual intervention of the kind described here superfluous. For this reason we have carried out similar experiments with the 7–8 January 2005 storm. This storm was quite well forecast by the state-ofthe-art NWP models, but there were even in the short range serious forecast errors in cyclone track by some NWP models. For instance, the Norwegian HIRLAM simulation initialized at 06
UTC on 7 January brought the storm too far north, along the west coast of Norway, while the 00 and 12
UTC simulations brought the cyclone more correctly farther south, across southern Scandinavia. However, the forecast propagation speed was exaggerated and the deepening was stronger than observed in the operational run starting at 12
UTC on 7 January. Initial modifications of PV based on the method described in this paper produced an improved simulation regarding the cyclone track, propagation speed and surface pressure. The leading SVs in this case had a strong signal in the dynamically active region. This experiment will be described in a forthcoming paper.
6.
S UMMARY AND CONCLUSIONS
This research has dealt with identification and correction of errors in NWP analyses by comparing the NWP PV fields with features in WV images. The added information from the five fastest-growing SVs helped to identify the sensitive areas and the 3D structure of the corrections. Three cases of major forecast failures of severe cyclogenesis affecting north-west Europe have been investigated and the method described above has been successful in improving the numerical analyses and simulations of the storms.
The present study follows up a previous case study (Røsting et al.
2003), in which the method was shown to be quite successful.
The corrections of the PV field described in the present study are based on modification and displacement of PV anomalies by interactive methods. Current NWP models rarely produce substantial forecast failures in the short range (i.e. out to 48 hours). However, analysis errors may still occur over the vast data-sparse areas such as the North
Atlantic and the regions from Greenland to northern Canada. From the latter regions, upper-level PV anomalies are frequently advected eastwards, strengthening upper-level jet streaks across the North Atlantic. The methods of PV modification described above are particularly useful for manual intervention in such cases with strong zonal jets across the North Atlantic, when satellite images occasionally provide the only early warning

2650 of analysis errors. Such information may be available before errors are detected by the
(often few) synoptic observations.
The method applied here allows correction of large analysis errors, i.e. those much larger than those contained in the error covariance matrix used in assimilation techniques
(i.e. 3D-Var and 4D-Var). Rather than invoking the assimilation procedure, the new analysis is obtained directly by inverting the corrected PV field.
In order to identify the regions of the flow which are particularly sensitive to new observations and hence analysis changes, SV analysis is applied. In our study the signals from the WV–PV relationship and some of the leading SVs are generally in agreement in highlighting the region of cyclogenesis. Correction of the PV fields according to
WV signals in regions which are also sensitive according to some of the leading SVs had a significant positive impact on the model rerun. Conversely, the correction of PV anomalies in non-sensitive regions had essentially no effects on the model rerun.
Since the SVs used for this study are based on NWP analyses containing large errors, the fastest-growing SVs may have failed to highlight properly the true sensitive regions. However, for small verification regions and short optimization times, the true sensitive regions appeared to be identified by the leading SVs. This could be due to the rather coarse resolution of the SVs, e.g. the large-scale structure of the flow was fairly well simulated.
The modifications of the PV fields are also based on experience from several case-studies and hence on conceptual models describing severe cyclogenesis. There is usually a consistency between the SV signal and the general knowledge provided by conceptual models. However, SVs constitute a more precise tool, indicating the location and structure of the modifications in complex flow patterns. One example is the case of the Hallowe’en storm where the WV–PV signal was strong over UK, but the SV signal was very weak. In this case PV modification over UK had a rather negligible effect on the NWP rerun (subsection 5(c) and Fig. 10).
The modification of the RH fields had a negligible impact on the results from the numerical reruns in this investigation. Apparently the moisture fields are rapidly adjusted after 3 to 6 hours of simulation time.
In the present study we have tested the method for three cases, i.e. the two French
Christmas storms of 1999 (the T1 and T2 storms) and the Hallowe’en storm of 2000.
These storms, particularly the French storms, were poorly simulated by the operational short-range NWP models at the time. The storms were generally well simulated by the method described in this paper, even though downstream effects may have had an impact on the developments of the French storms. The neglect of the divergent winds in the modified analyses may have had an adverse effect on the simulations, but divergent winds are rapidly restored after about 3–6 hours of the numerical simulation.
Encouragingly, the modified runs were to a large extent able to describe properly some mesoscale phenomena related to strong cyclogenesis, e.g. the dry intrusions, spatially and temporally consistent with features in the WV images (Fig. 11).
Though the results are quite promising, more than three cases are certainly needed to assess the generality of the results. However, several other North Atlantic winter storms have been subjected to related methods (Demirtas and Thorpe 1999; Røsting et al.
2003; Hello and Arbogast 2004), yielding substantial forecast improvements in all cases. As mentioned in subsection 5(e), numerical reruns of the storm on 7–8 January
2005, based on the methods described in this research, have been highly successful.
Our results are obtained with the benefit of hindsight as the cases described here have been thoroughly studied and analysed after they occurred, and test runs in real time will be necessary to fully test the potential of the method. The PV–WV–SV

IMPROVED SIMULATION OF WINTER STORMS BY PV MODIFICATION 2651 method has great potential to become operational since PV fields can now be modified interactively—a procedure which substantially reduces the time required for preparing a new analysis and a rerun of the model to less than one hour in many cases.
A CKNOWLEDGEMENTS
The basic inversion routine used in our research was originally supplied by Christopher A. Davis and the PV inversion package was supplied by Sigurdur Thorsteinsson and co-workers at the Icelandic Meteorological Office. The routine for calculating the
PV structure of singular vectors was provided by Andrea Montani. Anstein Foss modified the necessary software for doing PV inversions at the Norwegian Meteorological
Institute (NMI) and also prepared software for graphical presentation. Support was also provided by the division for research and development at NMI, assisting in running simulations on the SGI ORIGIN 3800. Anstein Foss and collaborators at NMI have developed the DIANA system, which for the purpose of this study has been modified to include editing of PV (and RH) fields interactively. We are also grateful to Wim Verkley and Jan Barkmeijer at KNMI for useful and stimulating discussion on the methods described in this paper. We thank two anonymous reviewers who helped to improve this paper, particularly regarding comments and advice on the use of singular vectors and the relation between cloud heads and PV fields.
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