1. Introduction
Extratropical cyclones over the North Atlantic and Europe are notorious for causing large damage to property and loss of life. For example, these storms accounted for over 10% of the total insured losses of the top 40 global natural and man-made catastrophes over the period from 1970 to 2008 (Enz et al. 2009; Haylock 2011). In the past decade or two, a number of high-impact cyclones such as Lothar (Ulbrich et al. 2001; Wernli et al. 2002) and Martin (Ulbrich et al. 2001; Walser et al. 2006) in 1999, Kyrill in 2007 (Fink et al. 2009), and Klaus in 2009 all had catastrophic impacts on Europe. The focus of this study is on a storm referred to as Xynthia, which crossed western Europe between 27 February and 1 March 2010, and caused more fatalities than any of the aforementioned cyclones. High-wind gusts and flooding on 27–28 February 2010 resulted in more than 60 fatalities and caused extreme damage to coastal regions, transportation systems, and extensive electrical power outages. The majority of the fatalities occurred in France associated with a storm surge that arrived at high tide along with waves up to 7.5 m high, and caused a sea wall to fail in the coastal town of L'Aiguillon-sur-Mer, France. The total economic loss associated with Xynthia is estimated to be in excess of $4.5 billion (U.S. dollars) (3.5 billion Euro) (Liberato et al. 2013). An overview of Xynthia and the impacts of the storm are provided by Liberato et al. (2013).
Given the socioeconomic impact of these storms, accurate prediction of severe extratropical cyclones is of growing importance and interest. In spite of the great improvements in numerical weather prediction (NWP) achieved over the past several decades (e.g., Simmons and Hollingsworth 2002; Jung et al. 2006, 2010), these severe cyclones are often a challenge for the operational NWP models to predict (e.g., Shutts 1990; Ulbrich et al. 2001; Jung et al. 2004; Walser et al. 2006), and questions remain regarding their predictability (e.g., Leutbecher et al. 2002; Hoskins and Coutinho 2005). These intense cyclones are often especially sensitive to the initial state, such as the October 1987 storm that resulted in widespread wind damage in southern England (Shutts 1990), and the high-impact extratropical cyclones Lothar and Martin that caused extensive damage across Europe (Walser et al. 2006). Winter snowstorms that have a large impact on the United States also often exhibit a high degree of sensitivity to the initial state (e.g., Langland et al. 2002; Kleist and Morgan 2005a,b). The severe cyclone Xynthia offers an excellent opportunity to further explore predictability and initial condition sensitivity issues.
The mechanisms for the development of these severe extratropical cyclones point to a myriad of dynamical aspects and contributing factors. For example, the October 1987 storm attained a very low central pressure (951 hPa) and the key factor in the rapid intensification was related to the interaction of positive low-level and upper-level potential vorticity (PV) anomalies (Hoskins and Berrisford 1988). The rapid development of Kyrill was attributed to the superposition of three polar jets, and related jet streak dynamics and secondary circulations, as well as a dry-air intrusion that occurred during a complex two-stage intensification (Fink et al. 2009). Lothar developed to the south of a very intense upper-level jet, initially as a shallow cyclone with intense condensational heating that supported a well-defined positive low-level PV anomaly (Wernli et al. 2002) similar to that of a diabatic Rossby wave (e.g., Snyder and Lindzen 1991; Parker and Thorpe 1995). Lothar was not associated with any PV anomalies at the tropopause level during the early phase of development (Ulbrich et al. 2001; Wernli et al. 2002), which is in contrast to Kyrill and the October 1987 storm.
Adjoint-based systems and tools have been applied to extratropical cyclones for applications that includes initial condition sensitivity and predictability studies (Gelaro et al. 1998; Langland et al. 2002; Coutinho et al. 2004; Hoskins and Coutinho 2005), synoptic-scale dynamics (Reynolds and Gelaro 2001; Reynolds et al. 2001; Kleist and Morgan 2005a), and targeted observing strategies (Gelaro et al. 1999; Langland et al. 1999; Szunyogh et al. 2000; Leutbecher et al. 2002). A relatively high-resolution nested adjoint modeling system is used in this study to quantify the initial condition sensitivity and predictability of processes that influence the rapid development of Xynthia. We make use of the gradient fields derived from the adjoint model to interpret the initial condition sensitivity. Furthermore, perturbations are constructed from these gradient calculations with initial magnitudes comparable to analysis errors to investigate the growth of structures that are relevant for the predictability of extreme events such as Xynthia.
A number of studies have highlighted the importance of initial condition sensitivity in limiting the predictability of extratropical cyclones. The “surprise” snowstorm of 24–25 January 2000 in which the operational numerical weather prediction guidance over the eastern United States was particularly poor, has received considerable attention (Langland et al. 2002; Zhang et al. 2002, 2003). Forecast errors in regions of high sensitivity over the eastern Pacific were found to propagate faster than the phase speed of the trough and ridge associated with downstream development dynamics, which has important implications for predictability (Langland et al. 2002). The mesoscale ensemble results of Zhang et al. (2002) demonstrate that the short-range forecasts of precipitation for this case are very sensitive to the fidelity of the initial state. The ensemble results also reveal that forecast changes arise from the rapid growth of errors at scales below 500 km in association with moist processes. Kleist and Morgan (2005a) used an adjoint model to identify initial condition sensitivities, which were maximized in a region of enhanced low-level baroclinicity with an upshear vertical tilt in the vicinity of a short-wave trough in this case. Likewise, in a study of North Pacific cyclones, Reynolds et al. (2001) found that singular vectors (SVs), which represent the fastest-growing perturbations (Molteni and Palmer 1993; Buizza and Palmer 1995), have an upshear tilt in the midtroposphere and are typically positioned below prominent PV features. Badger and Hoskins (2001) found that optimal growth occurs in the presence of a mid or low-level PV anomaly with small vertical and horizontal scale. An initial period of rapid growth occurs associated with the “unshielding” of midtropospheric small-scale perturbations (Orr 1907; Farrell 1982) and is proportional to the vertical shear (e.g., Errico and Vukicevic 1992; Langland et al. 1995; Reynolds et al. 2001). Ancell and Mass (2006) noted that the perturbation growth rates are dependent on horizontal resolution, with the adjoint sensitivities predicting larger changes in the response function with increased horizontal resolution.
Other studies of extratropical cyclone predictability have emphasized the increased perturbation and error growth that occurs in the presence of moisture (Tan et al. 2004; Hoskins and Coutinho 2005), which is consistent with the overall importance of moist processes for rapid development of severe extratropical cyclones (e.g., Wernli et al. 2002). The ensemble results of Zhang et al. (2007) for an idealized baroclinic wave suggest a three-stage error-growth progression. The initial stage features error growth due to small-scale convective instability that quickly saturates followed by a progression from unbalanced convective scales to large-scale balanced motions, and in the final stage, error growth occurs as a result of baroclinic instability. In contrast to this three-stage model, Durran et al. (2013) used a large ensemble to study the predictability of two Pacific Northwest cyclones and found the growth of large-scale perturbations to be more rapid than the growth of perturbations at the smallest scales, with no evidence of upscale-growth of small-scale perturbations.
The overall objective of this study is to quantify the sensitivity of the intensification of Xynthia to the initial state and to explore the predictability characteristics of this high-impact cyclone. The sensitivity of numerical predictions to the initial moisture state is of particular importance in this event because Xynthia developed rapidly along a filament of anomalously high moisture content in the lower- and midtroposphere. The focus on moisture sensitivity is also of relevance from a climate change perspective, since the strength of extratropical storms is sensitive to projected future moisture changes (e.g., Booth et al. 2013). Section 2 contains a description of the models including the adjoint and tangent linear model formulations. A synoptic-scale overview is presented in section 3. Section 4 contains an interpretation of the adjoint sensitivity results and the summary and conclusions can be found in section 5.
2. Nonlinear and adjoint numerical model description
a. Nonlinear numerical model
The nonlinear numerical simulations of the evolution of Xynthia are performed using the atmospheric module of the Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS;1 Hodur 1997), which is based on a finite-difference approximation to the fully compressible, nonhydrostatic equations and makes use of a terrain-following vertical coordinate transformation. The vertical acoustic modes are solved using a semi-implicit formulation to efficiently integrate the compressible equations (Klemp and Wilhelmson 1978). The finite-difference schemes are of second-order accuracy in this study, although higher-order options are available. Fourth-order accurate horizontal diffusion is used for all variables with the exception of perturbation pressure, to mitigate nonlinear instability through the damping of short-wavelength horizontal scales.
The nonlinear model prognostic variables include the u, υ, and w components of the wind, the perturbation Exner function (related to the atmospheric pressure), potential temperature, water vapor, microphysical species, and turbulent kinetic energy (TKE). A terrain-following height coordinate and a horizontally staggered C grid are used. Cloud microphysical processes are represented using explicit moist physics based on a modified version of Rutledge and Hobbs (1983). The subgrid-scale deep convection is parameterized following a modified Kuo convective parameterization (Molinari 1985). The planetary boundary layer and free-atmospheric turbulent mixing and diffusion are parameterized using a prognostic equation for the TKE budget (Hodur 1997). A surface-layer parameterization based on Louis (1979) is used to represent the surface fluxes and a force-restore method is used for the surface energy budget. The nonlinear, adjoint, and tangent linear models use identical physical parameterizations. The physical parameterizations in the adjoint and tangent linear models are formulated with no additional simplifying assumptions relative to the nonlinear model. To avoid significant challenges associated with the more nonlinear aspects of the physics, ice processes and radiative processes are neglected.
The nonlinear, adjoint, and tangent linear models are applied in a nested grid mesh mode. The horizontal grid increment for the coarse and fine meshes are 45 and 15 km, respectively, each with 45 vertical levels. The coarse mesh contains 201 × 161 grid points and the fine mesh comprises 121 × 121 points. A sponge upper-boundary condition is applied to mitigate the reflection of vertically propagating gravity waves over the top 10 km of the model, with the model top located at 30 km. Topography is based on a 1-km resolution digital elevation model [i.e., the Global Land One-km Base Elevation (GLOBE), http://www.ngdc.noaa.gov/mgg/topo/globe.html].
The initial conditions are created from multivariate optimum interpolation (MVOI) analyses (Barker 1992) of upper-air sounding, surface, commercial aircraft, and satellite data that are quality controlled. The analysis background fields and the lateral boundary conditions for the outer most grid mesh are based on the Navy Operational Global Atmospheric Prediction System (NOGAPS) forecast fields (Hogan and Rosmond 1991; Peng et al. 2004).
b. Adjoint and tangent linear models
The tangent linear and adjoint COAMPS (Amerault et al. 2008) models include the nonhydrostatic dynamical core, as well as the TKE, cumulus, and explicit moist physics parameterizations. In this study, only warm-rain processes are included to minimize the inherent nonlinearities associated with the microphysics, although the adjoint and tangent linear models include ice, snow, and graupel microphysical species (Doyle et al. 2012). The decision points or switches that may result in discontinuities are identical in the nonlinear, tangent linear, and adjoint models (Zou et al. 1993; Vukicevic and Errico 1993). The nonlinear model's trajectory is saved every time step to provide sufficient accuracy for the adjoint and tangent linear models. Gradients and perturbations associated with the vertical diffusion are neglected in the adjoint and tangent linear models (Mahfouf 1999). These adjoint simulations are considerably higher resolution (45- and 15-km grid meshes) than applied in previous adjoint- and singular vector-based studies of extratropical cyclones [e.g., 150 km in Langland et al. (2002), 175 km in Coutinho et al. (2004), 60 km in Kleist and Morgan (2005a), 24–216 km in Ancell and Mass (2006), and 45 km in Ancell and Mass (2008)].
c. Adjoint optimal perturbations
3. Synoptic-scale overview
The extratropical cyclone Xynthia developed over the subtropical ocean to the south of the Azores Islands on 26 February 2010 associated with a short wave embedded in a broad trough over the central and eastern North Atlantic. An overview of the development of Xynthia from a synoptic-scale and mesoscale perspective is provided in Liberato et al. (2013) and we provide a brief synopsis here. At 1200 UTC 26 February 2010, a shallow cyclone was located to the south of the Azores positioned near a comma-shaped cloud shield (Fig. 1). Approximately 12 h later, the cyclone began to intensify very rapidly at a rate of 19 hPa (24 h)−1, as it moved northeastward toward Portugal and the Bay of Biscay; eventually crossing the coastline of France at 0000 UTC 28 February. During the following several days it weakened and continued to move northeastward along the coastline of northern France and the North Sea, and then it crossed the southern Baltic Sea to southern Finland, where it had a central pressure of 990 hPa. From a climatological perspective, the track of Xynthia was unusual; most severe European storms develop farther northward over the Atlantic and then move eastward over western and central Europe (e.g., Hoskins and Coutinho 2005). Near the time of landfall, gusts at hurricane force 30–45 m s−1 were recorded at numerous stations along the coast of France. These strong winds over the Bay of Biscay provided the forcing for locally generated ocean waves in excess of 6 m and a storm surge of just under 2 m that contributed to the extensive flooding and loss of life along the coast of France. The highest reported gust in western Europe was 66 m s−1 on the Pic du Midi in the French Pyrenees (2877 m above sea level) on 27 February.
Meteosat-9 infrared imagery valid at 1345 UTC 26 Feb 2010 (courtesy of University of Wisconsin–CIMSS). The Xynthia minimum central pressure and position based on Deutscher Wetterdienst analyses are shown every 6 h beginning at 1200 UTC 26 Feb 2010.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
Meteosat-9 infrared imagery valid at 1345 UTC 26 Feb 2010 (courtesy of University of Wisconsin–CIMSS). The Xynthia minimum central pressure and position based on Deutscher Wetterdienst analyses are shown every 6 h beginning at 1200 UTC 26 Feb 2010.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
Meteosat-9 infrared imagery valid at 1345 UTC 26 Feb 2010 (courtesy of University of Wisconsin–CIMSS). The Xynthia minimum central pressure and position based on Deutscher Wetterdienst analyses are shown every 6 h beginning at 1200 UTC 26 Feb 2010.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
The extratropical cyclone developed along a filament of enhanced moisture, which emanated from the subtropics and was oriented along a southwest to northeast corridor, as apparent in the water vapor composite valid at 1200 UTC 26 February shown in Fig. 2a. These relatively narrow filaments of water vapor have been referred to as “atmospheric rivers” since they are responsible for more than 90% of the horizontal water vapor transport in midlatitudes (Zhu and Newell 1998) and have been major contributing factors for heavy precipitation and flooding events, particularly in coastal regions (Ralph et al. 2004, 2011; Neiman et al. 2008). The vertically integrated precipitable water anomaly derived the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis and valid at the same time (Fig. 2b) highlights the anomalously high content of moisture in the atmospheric river that extended from the south-central Atlantic to the southern portion of the Iberian Peninsula. The development of Xynthia took place along the northern edge of a region of anomalously warm sea surface temperatures (SSTs), as shown in Fig. 3. Anomalously warm SSTs may have also contributed to the rapid intensification of Lothar (Wernli et al. 2002). At the southeast flank of the Xynthia, very warm subtropical air advected northward into Spain and later to central Europe.
(a) The morphed composite based on Special Sensor Microwave Imager (SSM/I)/Advanced Microwave Scanning Radiometer for Earth Observing System (AMSR-E) total precipitable water (mm) valid at 1200 UTC 26 Feb 2010 (courtesy of University of Wisconsin–CIMSS). (b) The vertically integrated precipitable water anomaly based on the NCEP–NCAR reanalysis (courtesy of Physical Science Division, NOAA/ESRL).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
(a) The morphed composite based on Special Sensor Microwave Imager (SSM/I)/Advanced Microwave Scanning Radiometer for Earth Observing System (AMSR-E) total precipitable water (mm) valid at 1200 UTC 26 Feb 2010 (courtesy of University of Wisconsin–CIMSS). (b) The vertically integrated precipitable water anomaly based on the NCEP–NCAR reanalysis (courtesy of Physical Science Division, NOAA/ESRL).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
(a) The morphed composite based on Special Sensor Microwave Imager (SSM/I)/Advanced Microwave Scanning Radiometer for Earth Observing System (AMSR-E) total precipitable water (mm) valid at 1200 UTC 26 Feb 2010 (courtesy of University of Wisconsin–CIMSS). (b) The vertically integrated precipitable water anomaly based on the NCEP–NCAR reanalysis (courtesy of Physical Science Division, NOAA/ESRL).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
The sea surface temperature composite anomaly valid on 26 Feb 2010 based on the NCEP–NCAR reanalysis (courtesy of Physical Science Division, NOAA/ESRL).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
The sea surface temperature composite anomaly valid on 26 Feb 2010 based on the NCEP–NCAR reanalysis (courtesy of Physical Science Division, NOAA/ESRL).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
The sea surface temperature composite anomaly valid on 26 Feb 2010 based on the NCEP–NCAR reanalysis (courtesy of Physical Science Division, NOAA/ESRL).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
4. Adjoint sensitivity results
An analysis of the nonlinear, adjoint, and tangent linear model simulations of Xynthia is presented in this section in order to explore the sensitivity to the initial state and related predictability issues. A number of different nonlinear, adjoint, and tangent linear numerical simulations are performed to elucidate various aspects of the sensitivity and predictability of the storm.
a. Nonlinear model simulation
The nonlinear model 36-h integration begins with a cold start initialization at 1200 UTC 26 February and extends to the final time at 0000 UTC 28 February. The initial sea level pressure and 10-m winds for the coarse mesh (Fig. 4a), highlight the early stages of Xynthia prior to intensification with a central pressure of 996 hPa and location to the south of the Azores. The PV and winds on the 320-K surface at the initial time (Fig. 4c) feature a PV anomaly and tropopause depression associated with the Xynthia surface cyclone, and a strong polar jet to the north with a wind speed maximum greater than 70 m s−1. The nonlinear model captures the rapid development of Xynthia with simulated central pressures of 984 hPa (analyzed 986 hPa) at 12 h, 964 hPa (analyzed 969 hPa) at 24 h, and 960 hPa (analyzed 967 hPa) at 36 h. The sea level pressure and 10-m winds at 36 h (Fig. 4b) show a very intense and fully developed system located near the coast of France with near-surface wind speeds in excess of 25 m s−1. The PV on the 320-K surface (Fig. 4d) shows a tropopause fold to the west of the surface cyclone and a polar jet streak located to the southwest with the jet exit region located near the Iberian Peninsula.
The sea level pressure (every 4 hPa), 10-m wind speed according to the color scale (isotach interval of 2.5 m s−1 above 5 m s−1), and wind vectors (every eight grid points) for (a) 1200 UTC 26 Feb and (b) 0000 UTC 28 Feb 2010 for the outer grid mesh (Δx = 45 km). The location of the fine mesh is shown by the black box in (a) and (b). The potential vorticity and wind speed (red istotachs every 10 m s−1 above 40 m s−1) and wind vectors (every eight grid points) on the 320-K surface for (c) 1200 UTC 26 Feb and (d) 0000 UTC 28 Feb 2010. The potential vorticity is shown by the color scale [every 0.1 potential vorticity units (PVU, 1 PVU = 10−6 K m2 kg−1 s−1)] and is shaded in gray above 2 PVU.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
The sea level pressure (every 4 hPa), 10-m wind speed according to the color scale (isotach interval of 2.5 m s−1 above 5 m s−1), and wind vectors (every eight grid points) for (a) 1200 UTC 26 Feb and (b) 0000 UTC 28 Feb 2010 for the outer grid mesh (Δx = 45 km). The location of the fine mesh is shown by the black box in (a) and (b). The potential vorticity and wind speed (red istotachs every 10 m s−1 above 40 m s−1) and wind vectors (every eight grid points) on the 320-K surface for (c) 1200 UTC 26 Feb and (d) 0000 UTC 28 Feb 2010. The potential vorticity is shown by the color scale [every 0.1 potential vorticity units (PVU, 1 PVU = 10−6 K m2 kg−1 s−1)] and is shaded in gray above 2 PVU.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
The sea level pressure (every 4 hPa), 10-m wind speed according to the color scale (isotach interval of 2.5 m s−1 above 5 m s−1), and wind vectors (every eight grid points) for (a) 1200 UTC 26 Feb and (b) 0000 UTC 28 Feb 2010 for the outer grid mesh (Δx = 45 km). The location of the fine mesh is shown by the black box in (a) and (b). The potential vorticity and wind speed (red istotachs every 10 m s−1 above 40 m s−1) and wind vectors (every eight grid points) on the 320-K surface for (c) 1200 UTC 26 Feb and (d) 0000 UTC 28 Feb 2010. The potential vorticity is shown by the color scale [every 0.1 potential vorticity units (PVU, 1 PVU = 10−6 K m2 kg−1 s−1)] and is shaded in gray above 2 PVU.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
To further evaluate the skill and fidelity of the nonlinear model simulation, the 10-m wind speed valid at 2200 UTC 27 February is compared with the Advanced Scatterometer (ASCAT) winds valid near the same time. As shown in Figs. 5a,b, the model simulation captures the low-level wind structure, including the wind speed maximum along the southern portion of Xynthia where the sea level pressure gradient is especially strong (Fig. 5c). The low-level wind maximum is positioned on the cold side of the bent-back warm front (Shapiro and Keyser 1990), which is well defined in the 925-hPa potential temperature (Fig. 5c), and is consistent with the cyclonically turning cold conveyor belt, as shown in Browning (1990), Schultz (2001), and D. M. Schultz and J. M. Sienkiewicz (2013, personal communication).
(a) The 10-m wind speed (m s−1 color) and wind barbs from the Advanced Scatterometer (ASCAT) valid at 2200 UTC 27 Feb 2010. The fine grid mesh (Δx = 15 km) nonlinear model simulated (b) 10-m wind speed (color scale with interval every 2.5 m s−1) and wind vectors (every five grid points), and (c) sea level pressure (every 4 hPa in black contours) and 925-hPa potential temperature (color scale every 1-K dashed contours) valid at 2200 UTC 27 Feb 2010 (34 h).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
(a) The 10-m wind speed (m s−1 color) and wind barbs from the Advanced Scatterometer (ASCAT) valid at 2200 UTC 27 Feb 2010. The fine grid mesh (Δx = 15 km) nonlinear model simulated (b) 10-m wind speed (color scale with interval every 2.5 m s−1) and wind vectors (every five grid points), and (c) sea level pressure (every 4 hPa in black contours) and 925-hPa potential temperature (color scale every 1-K dashed contours) valid at 2200 UTC 27 Feb 2010 (34 h).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
(a) The 10-m wind speed (m s−1 color) and wind barbs from the Advanced Scatterometer (ASCAT) valid at 2200 UTC 27 Feb 2010. The fine grid mesh (Δx = 15 km) nonlinear model simulated (b) 10-m wind speed (color scale with interval every 2.5 m s−1) and wind vectors (every five grid points), and (c) sea level pressure (every 4 hPa in black contours) and 925-hPa potential temperature (color scale every 1-K dashed contours) valid at 2200 UTC 27 Feb 2010 (34 h).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
b. Accuracy of adjoint and tangent linear models, and tangent linear approximation
The COAMPS tangent linear and adjoint models have undergone a suite of tests for correctness and accuracy. A gradient check and perturbation test have been conducted, the results of which indicate that the adjoint has been accurately coded. The results of these tests for the codes are discussed in Amerault et al. (2008). The gradient fields are used to construct optimal perturbations in order to test the validity of the tangent linear approximation.
The optimal perturbations derived from the adjoint gradients and evolved in the tangent linear and nonlinear models are used to evaluate the validity of the tangent linear approximation for the integration length (36 h) and resolutions considered in this study [see similar evaluation for this model in Doyle et al. (2012) for a tropical application]. The nonlinear perturbation is defined as the difference between the nonlinear forecast from the control state and the nonlinear forecast from the perturbed state. The tangent linear model is considered useful when the nonlinear and tangent-linear-evolved perturbations at the final time are similar in magnitude and pattern. We examine both the coarse mesh (45-km resolution grid) and nested grid (15-km resolution grid) simulations. Both of these simulations include microphysical and convective parameterizations. The evolved zonal velocity perturbations u′ at 850 hPa in the nonlinear and tangent linear models are shown in Figs. 6a and 6b, respectively, for the coarse mesh at the 36-h integration time. The evolved zonal velocity perturbation patterns are quite similar for the nonlinear and tangent linear simulations. The domain-wide correlations (all vertical levels) between the nonlinear and tangent linear model simulations at the final time (36 h) are 0.83 for both the u-wind component and potential temperature, for the coarse mesh simulation.
Evolved adjoint optimal perturbations after 36 h of integration at 850 hPa for the zonal wind component valid at 0000 UTC 28 Feb 2010 for the outer grid mesh (Δx = 45 km) corresponding to the (a) nonlinear and (b) tangent linear models. Evolved optimal perturbations after 36 h of for the fine mesh (Δx = 15 km) are shown for the (c) nonlinear and (d) tangent linear models. The color scale interval is 2 m s−1 in (a) and (b), and 4 m s−1 in (c) and (d). The box in (c) and (d) shows the horizontal extent of the adjoint response function.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
Evolved adjoint optimal perturbations after 36 h of integration at 850 hPa for the zonal wind component valid at 0000 UTC 28 Feb 2010 for the outer grid mesh (Δx = 45 km) corresponding to the (a) nonlinear and (b) tangent linear models. Evolved optimal perturbations after 36 h of for the fine mesh (Δx = 15 km) are shown for the (c) nonlinear and (d) tangent linear models. The color scale interval is 2 m s−1 in (a) and (b), and 4 m s−1 in (c) and (d). The box in (c) and (d) shows the horizontal extent of the adjoint response function.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
Evolved adjoint optimal perturbations after 36 h of integration at 850 hPa for the zonal wind component valid at 0000 UTC 28 Feb 2010 for the outer grid mesh (Δx = 45 km) corresponding to the (a) nonlinear and (b) tangent linear models. Evolved optimal perturbations after 36 h of for the fine mesh (Δx = 15 km) are shown for the (c) nonlinear and (d) tangent linear models. The color scale interval is 2 m s−1 in (a) and (b), and 4 m s−1 in (c) and (d). The box in (c) and (d) shows the horizontal extent of the adjoint response function.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
At higher resolution, nonlinearities become more important and saturation processes involving moist physical parameterizations, which often include discrete branches and on/off switches, are more prominent. Ancell and Mass (2006) found that for perturbations made in sensitive regions, the tangent linear approximation degrades at finer grid spacing. In spite of these challenges associated with an increase in resolution, the evolved u′ perturbations (36 h) for the moist nonlinear and tangent linear simulations that use a 15-km grid increment in the nested grid mesh, shown in Figs. 6c and 6d, respectively, are overall very similar. However, the magnitude of the perturbations are considerably larger in the tangent linear than nonlinear model, in part because the perturbation growth saturates because of nonlinearities in the nonlinear simulation. The correlations between the tangent linear and nonlinear simulations at the final time on the fine mesh are 0.63 and 0.58 for the 850-hPa u-wind component and potential temperature, respectively. The correlations for the fine mesh grid are less than that of the coarse mesh; however, this is to be expected because of the greater importance of nonlinearities at higher resolution, as well as the increasing influence of moisture processes on these scales, which introduce additional nonlinearities. Overall, the agreement between evolved perturbations in the nonlinear and tangent linear models is reasonable, which lends confidence that the tangent linear model is useful for both the coarse and fine mesh resolutions over the 36-h integration period.
c. Adjoint sensitivity
The adjoint and tangent linear models are applied to investigate the initial condition sensitivity for Xynthia using an initialization time of 1200 UTC 26 February 2010. The sensitivity of the 36-h kinetic energy in the box (in the lowest 860 m) on the fine grid mesh (location of the box shown in Figs. 6c,d) to the initial state at 700 hPa is presented in Fig. 7. The sensitivity of winds, temperature, and moisture are a maximum in the 850- to 500-hPa layer, and the 700-hPa layer sensitivity is representative. The υ-wind component sensitivity, shown in Fig. 7a, indicates a positive sensitivity maximum (e.g., stronger southerly winds will increase the strength of the storm) at the leading edge of the short-wave trough associated with the incipient cyclone, with an extension to the north and northwest near a secondary short-wave trough. The u-wind component sensitivity (not shown) indicates positive and negative regions positioned along the short-wave trough. The two short waves phase-lock during the rapid development of Xynthia and the trough becomes meridionally aligned and deep by the 36-h time. The sensitivity to the initial 700-hPa PV (Fig. 7d), computed as a proxy using the adjoint optimal perturbations, indicates a sensitivity maximum near the northern portion of the PV anomaly and the short wave associated with the developing cyclone. Enhancements to this PV anomaly and extension to the north will lead to an intensification of the low-level winds, and increase in aerial extent of strong winds, at the 36-h time.
The adjoint sensitivity valid at the initial time of 1200 UTC 26 Feb 2010 at 700 hPa for the (a) υ-wind component (color scale with interval every 0.005 m s−1), (b) water vapor [color interval every 0.01 m2 s−2 (g kg−1)−1], and (c) potential temperature (color scale with interval every 0.01 m2 s−2 K−1) for the coarse mesh (Δx = 45 km). The 700-hPa geopotential height analysis is shown in (a) with an interval of 30 m. Water vapor greater than 4 g kg−1 is shown by the hatching in (b). The sensitivities are scaled by 105 km−3. The 700-hPa potential vorticity derived from the adjoint optimal perturbations is shown in (d) (color scale with interval every 0.005 PVU). The hatched region in (d) represents potential vorticity from the analysis greater than 0.7 PVU. The location of a vertical cross section shown in Fig. 8 is indicated by the solid black line in (a).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
The adjoint sensitivity valid at the initial time of 1200 UTC 26 Feb 2010 at 700 hPa for the (a) υ-wind component (color scale with interval every 0.005 m s−1), (b) water vapor [color interval every 0.01 m2 s−2 (g kg−1)−1], and (c) potential temperature (color scale with interval every 0.01 m2 s−2 K−1) for the coarse mesh (Δx = 45 km). The 700-hPa geopotential height analysis is shown in (a) with an interval of 30 m. Water vapor greater than 4 g kg−1 is shown by the hatching in (b). The sensitivities are scaled by 105 km−3. The 700-hPa potential vorticity derived from the adjoint optimal perturbations is shown in (d) (color scale with interval every 0.005 PVU). The hatched region in (d) represents potential vorticity from the analysis greater than 0.7 PVU. The location of a vertical cross section shown in Fig. 8 is indicated by the solid black line in (a).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
The adjoint sensitivity valid at the initial time of 1200 UTC 26 Feb 2010 at 700 hPa for the (a) υ-wind component (color scale with interval every 0.005 m s−1), (b) water vapor [color interval every 0.01 m2 s−2 (g kg−1)−1], and (c) potential temperature (color scale with interval every 0.01 m2 s−2 K−1) for the coarse mesh (Δx = 45 km). The 700-hPa geopotential height analysis is shown in (a) with an interval of 30 m. Water vapor greater than 4 g kg−1 is shown by the hatching in (b). The sensitivities are scaled by 105 km−3. The 700-hPa potential vorticity derived from the adjoint optimal perturbations is shown in (d) (color scale with interval every 0.005 PVU). The hatched region in (d) represents potential vorticity from the analysis greater than 0.7 PVU. The location of a vertical cross section shown in Fig. 8 is indicated by the solid black line in (a).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
The sensitivity of the 36-h kinetic energy to the initial 700-hPa water vapor, displayed in Fig. 7b, is characterized by a narrow maximum elongated in the southwest–northeast direction and oriented along the 700-hPa mean wind direction at the leading edge of the short-wave trough. The water vapor sensitivity is located near the northern edge of the enhanced region of precipitable water (Figs. 2a,b). Likewise, the potential temperature sensitivity (Fig. 7c) shows a similar elongated structure. The results suggest that moistening and warming along the narrow filament of sensitivity at the initial time leads to a strengthening of the storm. Although a large region of enhanced moisture is indicated in Fig. 2b, only a relatively small portion of this atmospheric river at the initial time was critically important for the development of Xynthia.
The moisture sensitivity [(m2 s−2) (g kg−1)−1] has maximum numerical values that are approximately 5–10 times greater than the horizontal wind sensitivity (m s−1) and 1.5 times greater than the potential temperature sensitivity [(m2 s−2) K−1]. These sensitivity fields for each variable have different units making their direct comparison somewhat less clear. However, when these sensitivities are scaled by typical values of observational uncertainty, approximately, 1–1.5 g kg−1, 1.8 m s−1, and 1 K, the moisture sensitivity clearly dominates with temperature sensitivity second largest. The dominance of the moisture sensitivity is overall consistent with numerous other studies that underscore the importance of moisture and diabatic heating for extratropical cyclogenesis (e.g., Kuo et al. 1991).
The vertical structure of the water vapor adjoint sensitivity and PV optimal perturbation fields is shown in Figs. 8a and 8b, respectively. The vertical cross section has a northwest–southeast orientation approximately normal to the banded moisture and temperature sensitivity region at 700 hPa (Fig. 7a) and the low- and midlevel front. The water vapor sensitivity is a maximum along the sloping warm frontal zone, with a secondary maximum just above the boundary layer near the 1-km altitude. The sensitivity suggests moistening along the narrow sloping band of positive sensitivity will lead to further intensification of the winds at the 36-h time near the coast of France. The regions of negative water vapor sensitivity coincide with near saturated conditions at 700 hPa (~3 km). The adjoint results suggest that reducing the moisture in these regions will lead to an intensification of the winds at the final time through an increase of the moisture gradient along the front, which in turn influences the diabatic processes.
(a) Vertical cross section of the adjoint water vapor sensitivity [color scale with interval every 0.025 m2 s−2 (g kg−1)−1] and analysis potential temperature (contours every 4 K) valid at the initial time of 1200 UTC 26 Feb 2010. The 5 and 10 g kg−1 contours from the water vapor analysis are shown in (a). (b) The potential vorticity derived from the adjoint optimal perturbations (color scale with interval every 0.005 PVU) and the analysis wind speed normal to the cross-sectional plane (location shown in Fig. 7a) (gray shading at an interval of 10 m s−1 with the shading beginning at 10 m s−1). The analysis potential vorticity of 0.5 PVU is shown by the bold white dashed contour in (b), with the bold solid purple contour corresponding to the PV in the perturbed analysis. The analysis potential temperature is shown every 8 K with the dashed contours in (b). The cross sections are shown for the coarse mesh (Δx = 45 km). The sensitivity is scaled by 105 km−3.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
(a) Vertical cross section of the adjoint water vapor sensitivity [color scale with interval every 0.025 m2 s−2 (g kg−1)−1] and analysis potential temperature (contours every 4 K) valid at the initial time of 1200 UTC 26 Feb 2010. The 5 and 10 g kg−1 contours from the water vapor analysis are shown in (a). (b) The potential vorticity derived from the adjoint optimal perturbations (color scale with interval every 0.005 PVU) and the analysis wind speed normal to the cross-sectional plane (location shown in Fig. 7a) (gray shading at an interval of 10 m s−1 with the shading beginning at 10 m s−1). The analysis potential vorticity of 0.5 PVU is shown by the bold white dashed contour in (b), with the bold solid purple contour corresponding to the PV in the perturbed analysis. The analysis potential temperature is shown every 8 K with the dashed contours in (b). The cross sections are shown for the coarse mesh (Δx = 45 km). The sensitivity is scaled by 105 km−3.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
(a) Vertical cross section of the adjoint water vapor sensitivity [color scale with interval every 0.025 m2 s−2 (g kg−1)−1] and analysis potential temperature (contours every 4 K) valid at the initial time of 1200 UTC 26 Feb 2010. The 5 and 10 g kg−1 contours from the water vapor analysis are shown in (a). (b) The potential vorticity derived from the adjoint optimal perturbations (color scale with interval every 0.005 PVU) and the analysis wind speed normal to the cross-sectional plane (location shown in Fig. 7a) (gray shading at an interval of 10 m s−1 with the shading beginning at 10 m s−1). The analysis potential vorticity of 0.5 PVU is shown by the bold white dashed contour in (b), with the bold solid purple contour corresponding to the PV in the perturbed analysis. The analysis potential temperature is shown every 8 K with the dashed contours in (b). The cross sections are shown for the coarse mesh (Δx = 45 km). The sensitivity is scaled by 105 km−3.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
A proxy for the PV sensitivity, computed using the adjoint optimal perturbations (Fig. 8b), also exhibits a narrow maximum along the sloping frontal zone, although the slope of the PV sensitivity bands is shallower than that of the water vapor sensitivity. The bands of maximum PV sensitivity are positioned beneath the upper-level jet in a region of strong vertical wind shear, qualitatively similar to the coarser-resolution singular vectors of Reynolds et al. (2001). When the optimal perturbation is added to the control analysis, the PV sensitivity maximum connects the low-level PV anomaly with a narrow PV region in the upper portion of the front (as shown by the bold purple contour corresponding to 0.5 PVU in Fig. 8b), which may be an indication of a tropopause fold. Increasing and sharpening the PV concentrated in this sloping maximum along the front will promote a strengthening of the winds at the 36-h time.
The 36-h forecast of Xynthia's intensity is most sensitive to regions in the lower and midtroposphere along the warm frontal zone. This sensitivity is further illustrated through examination of the water vapor sensitivity interpolated to the 300-K surface, shown in Fig. 9. The moisture sensitivity is a maximum in a narrow region along the warm conveyor belt (e.g., Harrold 1973; Browning et al. 1973; Carlson 1980; Eckhardt et al. 2004) and embedded within the low-level jet and in the region of strongest ascent along the isentropic surface. It is noteworthy that the positive moisture sensitivity maximum in the sloped ascent region is flanked by negative sensitivity minima to the east and west, which underscores the necessity of accurately representing the moisture field in the warm conveyor belt in the initial state, in particular the position and gradients associated with the moist filament (Figs. 2a,b). This implies that moisture gradients within atmospheric rivers are potentially important.
The adjoint water vapor sensitivity [color scale with interval every 0.02 m2 s−2 (g kg−1)−1], and wind vectors (plotted every five grid points) and pressure (contour interval every 50 hPa) from the analysis, valid at the initial time of 1200 UTC 26 Feb 2010 and displayed on the 300-K isentropic surface for the coarse mesh (Δx = 45 km). The sensitivity is scaled by 105 km−3.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
The adjoint water vapor sensitivity [color scale with interval every 0.02 m2 s−2 (g kg−1)−1], and wind vectors (plotted every five grid points) and pressure (contour interval every 50 hPa) from the analysis, valid at the initial time of 1200 UTC 26 Feb 2010 and displayed on the 300-K isentropic surface for the coarse mesh (Δx = 45 km). The sensitivity is scaled by 105 km−3.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
The adjoint water vapor sensitivity [color scale with interval every 0.02 m2 s−2 (g kg−1)−1], and wind vectors (plotted every five grid points) and pressure (contour interval every 50 hPa) from the analysis, valid at the initial time of 1200 UTC 26 Feb 2010 and displayed on the 300-K isentropic surface for the coarse mesh (Δx = 45 km). The sensitivity is scaled by 105 km−3.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
The development and intensification of Xynthia occurred over the relatively warm subtropical waters west of Africa, where the SST anomalies were 1°–2°C warmer than the climatological average (Fig. 3). One question is whether the relatively warm SSTs, either locally over the Bay of Biscay or more remotely along the coast of Africa, played a role in the intensification of Xynthia. The sensitivity of the strength of the winds associated with the cyclone at the 36-h time to the initial SSTs is shown in Fig. 10. The SST sensitivity is almost exclusively positive with the maximum sensitivity located to the right of the track of the surface cyclone, which is generally in the warm sector of the storm. The strength of the storm is most sensitive to the SST during the period when Xynthia undergoes rapid intensification. In this region, the near-surface wind speeds were strong (>20 m s−1) and the SSTs warm (14°–19°C), which resulted in enhanced sensible and latent heat fluxes at the air–sea interface. It is noteworthy that the sensitivity to the ground surface temperature is much smaller than the SST sensitivity, and only a confined region of sensitivity over northern Spain is apparent, which provides further support for the notion that the sea surface fluxes were important for the intensification of Xynthia.
The adjoint sea surface temperature sensitivity (color scale with interval every 0.02 m2 s−2 K−1), and sea level pressure (contour interval 2 hPa) from the analysis, valid at the initial time of 1200 UTC 26 Feb 2010 and displayed for a subdomain on the coarse mesh (Δx = 45 km). Over land regions, the sensitivity to the ground surface temperature is displayed. The sensitivity is scaled by 108 km−2.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
The adjoint sea surface temperature sensitivity (color scale with interval every 0.02 m2 s−2 K−1), and sea level pressure (contour interval 2 hPa) from the analysis, valid at the initial time of 1200 UTC 26 Feb 2010 and displayed for a subdomain on the coarse mesh (Δx = 45 km). Over land regions, the sensitivity to the ground surface temperature is displayed. The sensitivity is scaled by 108 km−2.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
The adjoint sea surface temperature sensitivity (color scale with interval every 0.02 m2 s−2 K−1), and sea level pressure (contour interval 2 hPa) from the analysis, valid at the initial time of 1200 UTC 26 Feb 2010 and displayed for a subdomain on the coarse mesh (Δx = 45 km). Over land regions, the sensitivity to the ground surface temperature is displayed. The sensitivity is scaled by 108 km−2.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
While quantifying the analysis errors is inherently difficult, assessing the differences between two cycling analysis systems, in this case COAMPS and NOGAPS, can provide some insight into the nature of the analysis uncertainty and error growth. The analysis difference between the two modeling systems (COAMPS − NOGAPS) at 850 hPa for the water vapor, potential temperature, and potential vorticity is shown in Figs. 11a–c, respectively, along with the hatched regions of large sensitivity (and adjoint perturbations for PV). The differences between the moisture analyses (Fig. 11a) are particularly large in the region of the moist plume where the moisture sensitivity is generally large (hatching in Fig. 11a). The differences in the water vapor between the two analyses are greater than 1 g kg−1 over large regions, particularly along the coast of Africa. Potential temperature differences (Fig. 11b) of 2–4 K between the two analyses are found in regions of large temperature sensitivity (hatching in Fig. 11b). The potential vorticity analysis differences (Fig. 11c) are a maximum near the short-wave trough associated with the development of Xynthia, with a stronger low-level PV maximum in the vicinity of the incipient cyclone analyzed by the higher-resolution COAMPS. The PV analysis differences project on to the regions of greatest PV adjoint perturbations (hatching in Fig. 11c). These differences can be viewed as a proxy for typical analysis errors, which will project on to the sensitive regions to some degree, as identified by the adjoint. The differences are in line with the maximum magnitude of the adjoint optimal perturbations for the winds, temperature, and moisture, namely, 1 m s−1, 1 K, and 1 g kg−1, respectively.
Differences between the COAMPS and NOGAPS analyses at 850-hPa at the initial time 1200 UTC 26 Feb 2010 for the (a) water vapor (color scale with an interval of 0.5 g kg−1), (b) potential temperature (color scale with an interval of 1 K), and (c) potential vorticity (color scale with an interval of 0.2 PVU). The hatched regions correspond to areas of sensitivity as shown in (a) greater (red) and less (blue) than 0.01 m2 s−2 (g kg−1)−1 and (b) greater (red) and less (blue) than 0.01 m2 s−2 K−1. Regions of PV optimal perturbations greater (red) and less (blue) than 0.001 PVU are shown in (c).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
Differences between the COAMPS and NOGAPS analyses at 850-hPa at the initial time 1200 UTC 26 Feb 2010 for the (a) water vapor (color scale with an interval of 0.5 g kg−1), (b) potential temperature (color scale with an interval of 1 K), and (c) potential vorticity (color scale with an interval of 0.2 PVU). The hatched regions correspond to areas of sensitivity as shown in (a) greater (red) and less (blue) than 0.01 m2 s−2 (g kg−1)−1 and (b) greater (red) and less (blue) than 0.01 m2 s−2 K−1. Regions of PV optimal perturbations greater (red) and less (blue) than 0.001 PVU are shown in (c).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
Differences between the COAMPS and NOGAPS analyses at 850-hPa at the initial time 1200 UTC 26 Feb 2010 for the (a) water vapor (color scale with an interval of 0.5 g kg−1), (b) potential temperature (color scale with an interval of 1 K), and (c) potential vorticity (color scale with an interval of 0.2 PVU). The hatched regions correspond to areas of sensitivity as shown in (a) greater (red) and less (blue) than 0.01 m2 s−2 (g kg−1)−1 and (b) greater (red) and less (blue) than 0.01 m2 s−2 K−1. Regions of PV optimal perturbations greater (red) and less (blue) than 0.001 PVU are shown in (c).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
d. Adjoint optimal perturbation characteristics
The results of several numerical experiments provide further insight into the characteristics of the adjoint optimal perturbations, as well as the predictability of intense cyclones such as Xynthia. A pair of simulations is conducted with positive and negative signs on the adjoint optimal perturbations that are introduced into the nonlinear COAMPS model at the initial time (1200 UTC 26 February 2010) and evolved for 36 h. The 10-m wind speed and sea level pressure for the control, and the positive and negative perturbation experiments are displayed in Fig. 12. Kinetic energy diagnostics, horizontally and vertically integrated over the response function box (over a 600 × 600 km2 area or 40 × 40 grid cells and in the lowest 860-m or 11 model levels) and the entire fine mesh domain (over a 1800 × 1800 km2 area or 120 × 120 grid cells and in the lowest 860-m or 11 model levels), as well as the wind speed perturbation maxima and minima at 10 m in the low-level jet are summarized in Table 1. The adjoint optimal perturbation maxima for the wind speed grow rapidly from ~1 m s−1 at the initial time to 20.6 m s−1 and 36.2 m s−1 at 36 h in the nonlinear and tangent linear models, respectively. The wind speed maximum in the perturbed nonlinear forecast increases only slightly over the control forecast (from 33.8 to 34.8 m s−1), however, the aerial extent of the low-level jet increases substantially, as evident from comparison of Figs. 12a,c, and apparent in the evolved wind speed (and surface pressure) adjoint perturbations (36 h) in Fig. 12e. The evolved adjoint wind speed and surface pressure perturbations for the full perturbation experiment (Fig. 12e) show a marked increase in the perturbation wind speed to the north of the control low-level jet, which increases the spatial extent of the high winds. In contrast, the simulation that used the opposite sign of the perturbation, shown in Fig. 12d, exhibits a surface wind field that has a low-level jet that is spatially confined along the coast with a sea level pressure minimum that is elongated in the southwest–northeast direction. The evolved adjoint wind speed and surface pressure perturbations (Fig. 12f) show the marked decrease in the low-level winds and the reduction in the pressure gradient in the center of the Bay of Biscay. The kinetic energy integrated over the response function box and for the fine mesh domain (Table 1) quantifies the changes that occur with the introduction of the optimal perturbations. Introduction of the positive perturbation into the nonlinear model leads to a modest increase in the state kinetic energy by 7% relative to the control at the 36 h. In the tangent linear model, the optimal perturbation grows more rapidly such that the kinetic energy integrated over the response function box nearly doubles over the 36-h period. The opposite signed perturbations introduced into the nonlinear model result in a ~25% reduction in the state kinetic energy relative to the control. The fact that the opposite sign perturbation experiment yields a much greater impact relative to the optimal perturbation experiment in the nonlinear model suggests that the nonlinearities are preventing the storm from intensifying significantly beyond the control simulation. Introduction of a water vapor–only adjoint perturbation into the nonlinear model (with moist parameterizations active) results in a low-level wind maximum (Fig. 12b) and integrated kinetic energy (Table 1) that is quite similar to that of the full perturbation case. The low-level wind perturbations from the water vapor–only experiment have a similar spatial pattern and slightly reduced magnitude relative to the full perturbation experiment. Using just the water vapor perturbation, the growth rates are sufficiently fast to reach approximately 75% of the full perturbation wind speed. This result once again underscores the importance of the initial moisture field, particularly the small-scale features within the moist filament, for the development of the intense pressure gradient and the strong low-level winds.
Nonlinear forecasts after 36 h of integration valid at 0000 UTC 28 Feb 2010 for the fine mesh (Δx = 15 km). The total 10-m speed (color scale on right with an interval of 2.5 m s−1) and sea level pressure (contour interval of 4 hPa) for (a) control simulation (and no adjoint perturbations), (b) simulation initialized with water vapor adjoint perturbations only, (c) simulation initialized with the adjoint perturbations, (d) simulation initialized with the adjoint perturbations introduced at the initial time with the opposite sign, (e) evolved adjoint wind speed perturbations (color shading with interval of 2 m s−1) after 36 h corresponding to (c), and (f) evolved adjoint wind speed perturbations (color shading with interval of 2 m s−1) after 36 h corresponding to (d). The evolved surface pressure perturbations are shown in (e) and (f) with a contour interval of 2 hPa (dashed is negative).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
Nonlinear forecasts after 36 h of integration valid at 0000 UTC 28 Feb 2010 for the fine mesh (Δx = 15 km). The total 10-m speed (color scale on right with an interval of 2.5 m s−1) and sea level pressure (contour interval of 4 hPa) for (a) control simulation (and no adjoint perturbations), (b) simulation initialized with water vapor adjoint perturbations only, (c) simulation initialized with the adjoint perturbations, (d) simulation initialized with the adjoint perturbations introduced at the initial time with the opposite sign, (e) evolved adjoint wind speed perturbations (color shading with interval of 2 m s−1) after 36 h corresponding to (c), and (f) evolved adjoint wind speed perturbations (color shading with interval of 2 m s−1) after 36 h corresponding to (d). The evolved surface pressure perturbations are shown in (e) and (f) with a contour interval of 2 hPa (dashed is negative).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
Nonlinear forecasts after 36 h of integration valid at 0000 UTC 28 Feb 2010 for the fine mesh (Δx = 15 km). The total 10-m speed (color scale on right with an interval of 2.5 m s−1) and sea level pressure (contour interval of 4 hPa) for (a) control simulation (and no adjoint perturbations), (b) simulation initialized with water vapor adjoint perturbations only, (c) simulation initialized with the adjoint perturbations, (d) simulation initialized with the adjoint perturbations introduced at the initial time with the opposite sign, (e) evolved adjoint wind speed perturbations (color shading with interval of 2 m s−1) after 36 h corresponding to (c), and (f) evolved adjoint wind speed perturbations (color shading with interval of 2 m s−1) after 36 h corresponding to (d). The evolved surface pressure perturbations are shown in (e) and (f) with a contour interval of 2 hPa (dashed is negative).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
Diagnostics based on the optimal perturbations evolved in the nonlinear model at the 36-h time period (0000 UTC 28 Feb). The quantities shown are the vertically integrated perturbation kinetic energy (KE) in the optimization box, perturbation kinetic energy over the second grid mesh (same depth as the optimization box), and maximum wind speed along with the maximum and minimum perturbation wind speed at 10 m in the low-level jet (LLJ). The experiments shown are the control, positive perturbation, negative perturbation, and moisture perturbation only.
The evolution of the vertical structure of the adjoint perturbation is complex in this case, although many similarities are apparent to previous adjoint and singular vector studies (e.g., Badger and Hoskins 2001; Reynolds et al. 2001; Ancell and Mass 2006). Initially, the sensitivity fields and adjoint optimal perturbations exhibit a marked up-shear tilt in this study along the sloping warm conveyor belt and frontal system. The sloped perturbations extract energy from the mean flow as they are untilted by it (Farrell 1988, 1989; Lacarra and Talagrand 1988; Borges and Hartmann 1992). Buizza and Palmer (1995) describe an amplifying Rossby wave packet that is characterized by a phase tilt against the vertical shear that leads to the group velocity being focused toward the jet core, which results in an increase in the intrinsic frequency and energy growth, while the propagation and refraction of the wave packet into the jet leads to a decrease in the tilt with time. In the case of Xynthia, the sloping PV optimal perturbation maximum implies that a sharpening and redistribution of the PV, essentially providing a more continuous PV maximum between the low-level and upper-level anomalies (Fig. 8), will lead to intensification of the low-level winds. These results are broadly consistent with the simple-model results of Badger and Hoskins (2001) and Morgan (2001) that suggest that the interaction between internal PV anomalies and surface thermal anomalies is an important part of SV perturbation growth and evolution. However, this interaction discussed in these studies occurs after the initial fast-growth phase dominated by the PV unshielding.
The perturbation growth is quite sensitive to the location of the adjoint optimal perturbations. When the initial adjoint optimal perturbations are shifted by 10 grid cells to the west (or 450 km), the resultant growth in the nonlinear model is reduced considerably, as apparent in Fig. 13, which compares the 925-hPa u-wind component perturbation at 36 h on the fine mesh for the simulation with the unshifted initial-time perturbation field, (Fig. 13a) and the simulation with the shifted initial-time perturbation field (Fig. 13b). The maximum perturbation 10-m wind speed is 3 m s−1 after 36 h, in contrast to 20.6 m s−1 in the positive (unshifted) perturbation simulation.
Adjoint optimal perturbations evolved in the nonlinear model after 36 h of integration valid at 0000 UTC 28 Feb 2010 for the fine mesh (Δx = 15 km). The perturbation u-wind component (color scale on right with an interval of 2 m s−1) at 925 hPa is shown for the (a) simulation with the adjoint perturbations and (b) simulation initialized with the adjoint perturbations introduced at the initial time and shifted spatially by 10 grid cells to the west.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
Adjoint optimal perturbations evolved in the nonlinear model after 36 h of integration valid at 0000 UTC 28 Feb 2010 for the fine mesh (Δx = 15 km). The perturbation u-wind component (color scale on right with an interval of 2 m s−1) at 925 hPa is shown for the (a) simulation with the adjoint perturbations and (b) simulation initialized with the adjoint perturbations introduced at the initial time and shifted spatially by 10 grid cells to the west.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
Adjoint optimal perturbations evolved in the nonlinear model after 36 h of integration valid at 0000 UTC 28 Feb 2010 for the fine mesh (Δx = 15 km). The perturbation u-wind component (color scale on right with an interval of 2 m s−1) at 925 hPa is shown for the (a) simulation with the adjoint perturbations and (b) simulation initialized with the adjoint perturbations introduced at the initial time and shifted spatially by 10 grid cells to the west.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
Vertical profiles of the domain-averaged (coarse mesh) total energy (J kg−1) of the adjoint optimal perturbations at the (a) initial time and (b) after 36 h. The kinetic energy (red), available potential energy (green), and total energy (black) are shown. The total energy from an experiment with no moist processes (purple) is shown in (b).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
Vertical profiles of the domain-averaged (coarse mesh) total energy (J kg−1) of the adjoint optimal perturbations at the (a) initial time and (b) after 36 h. The kinetic energy (red), available potential energy (green), and total energy (black) are shown. The total energy from an experiment with no moist processes (purple) is shown in (b).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
Vertical profiles of the domain-averaged (coarse mesh) total energy (J kg−1) of the adjoint optimal perturbations at the (a) initial time and (b) after 36 h. The kinetic energy (red), available potential energy (green), and total energy (black) are shown. The total energy from an experiment with no moist processes (purple) is shown in (b).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
The domain-averaged total dry energy and individual terms in Eq. (8) for the initial perturbations (Fig. 14a) indicate a well-defined peak in the middle troposphere with a maximum in the 4–6-km layer and a secondary maximum near 1.5 km. The total energy is dominated by the available potential energy at the initial time. This is to be expected because the temperature sensitivity is approximately 5 times greater than the wind sensitivity (e.g., Fig. 6). Rapid total energy perturbation growth (by a factor of over 104) ensues during the 36-h adjoint integration period with the kinetic energy dominant by the final time (Fig. 14b). The growth is large throughout the troposphere, with the strongest growth near the jet stream level (~8 km) and a low-level secondary maximum (~2 km). The upper-level maximum at final time occurs at the same altitude as the upper-tropospheric jet.
The evolution of the perturbation u-wind component power spectrum for the coarse mesh domain at 1500 m, shown in Fig. 15, provides further insight into the characteristics of the adjoint optimal perturbations. The spectra are an average of every other two-dimensional zonal slice over the domain for the coarse mesh. The spectra are shown every 2 h during the integration of the tangent linear model. The initial-time adjoint optimal perturbations have an energy peak at a ~900-km wavelength with secondary maxima at 1300 and 700 km. During the 6–24-h period, a primary or secondary maximum in the spectrum occurs at relatively short wavelengths of ~600–800 km. Perturbation growth overall occurs rapidly and generally shifts upscale to a ~1200 km wavelength after 36 h. This rapid perturbation growth is consistent with unshielding (Orr 1907; Farrell 1982) of the midtropospheric optimal perturbations and the subsequent vertical superposition of PV anomalies that occurs in the region of sloped ascent and latent heating along the warm conveyor belt, which in turn reinforces the baroclinic growth.
Power spectrum for the u-wind component adjoint optimal perturbations for the coarse mesh at 1500 m shown every 2 h over the 36-h integration. The darker blue colors correspond to the earlier times in the integration, and lighter blue denotes the later times. The red dashed line represents the spectrum at the initial time, scaled by 103.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
Power spectrum for the u-wind component adjoint optimal perturbations for the coarse mesh at 1500 m shown every 2 h over the 36-h integration. The darker blue colors correspond to the earlier times in the integration, and lighter blue denotes the later times. The red dashed line represents the spectrum at the initial time, scaled by 103.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
Power spectrum for the u-wind component adjoint optimal perturbations for the coarse mesh at 1500 m shown every 2 h over the 36-h integration. The darker blue colors correspond to the earlier times in the integration, and lighter blue denotes the later times. The red dashed line represents the spectrum at the initial time, scaled by 103.
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
e. Dry simulation
The adjoint sensitivity results highlight the importance of moisture within the ascending warm conveyor belt region. To examine the role of moisture further, a simulation was conducted without moist processes including latent heating in the nonlinear, adjoint, and tangent linear models. The dry nonlinear forecast has a weaker and slower-moving cyclone by the 36-h time with a central pressure of 980 hPa and maximum 10-m wind speed of 12.5 m s−1 over the Bay of Biscay in contrast to the 960-hPa central pressure and 32 m s−1 wind speed maximum over the bay in the control. The sensitivity of the 36-h forecast kinetic energy, within the response function box surrounding the strongest winds (as previously discussed), to the initial potential temperature at 700 hPa for the dry simulation is shown in Fig. 16a. The sensitivity is a maximum in the sloped region along the warm front, similar to the control sensitivity (e.g., Fig. 7c), however, the maximum sensitivity in the dry simulation is spatially more expansive and attains a smaller sensitivity maximum (0.08 m2 s−2 K−1), only one-third that of the control (0.25 m2 s−2 K−1). The growth rate of the adjoint optimal perturbations is considerably slower in the dry simulation relative to the control. For example, the optimal u-wind component perturbation at 925 hPa evolved in the nonlinear model at 36 h, shown in Fig. 16b, indicates a maximum of 19 m s−1, which is ~40% weaker than the control maximum of 31 m s−1 (Fig. 13a). The vertically integrated total energy of the evolved adjoint optimal perturbations at the 36-h time is reduced by a factor of 20 in the dry simulation relative to the control (Fig. 14b). It should be noted that a simulation using the moist trajectory with an adjoint that excludes moist processes yields relatively similar sensitivity and perturbation growth results to the simulation that excludes moist processes in both the nonlinear trajectory and adjoint. Overall these results are consistent with the findings of Ancell and Mass (2008) in that the inclusion of moisture in both the forward and adjoint models produces quite different sensitivity fields relative to a fully dry nonlinear and adjoint simulation. Additionally, they found that a simulation, which excludes moisture from the adjoint model but retains moisture in the nonlinear trajectory, to be very similar to that from a fully dry nonlinear and adjoint integration.
(a) The adjoint potential temperature sensitivity (color scale with interval every 0.01 m2 s−2 K−1) valid at the initial time of 1200 UTC 26 Feb 2010 at 700 hPa for the coarse mesh (Δx = 45 km) for the dry simulation. The 700-hPa geopotential height analysis with an interval of 30 m. The sensitivity is scaled by 105 km−3. (b) Adjoint optimal u-wind component perturbation field (color scale with an interval of 2 m s−1) at 925 hPa evolved in the nonlinear model at 36 h valid at 0000 UTC 28 Feb 2010 for the fine mesh (Δx = 15 km).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
(a) The adjoint potential temperature sensitivity (color scale with interval every 0.01 m2 s−2 K−1) valid at the initial time of 1200 UTC 26 Feb 2010 at 700 hPa for the coarse mesh (Δx = 45 km) for the dry simulation. The 700-hPa geopotential height analysis with an interval of 30 m. The sensitivity is scaled by 105 km−3. (b) Adjoint optimal u-wind component perturbation field (color scale with an interval of 2 m s−1) at 925 hPa evolved in the nonlinear model at 36 h valid at 0000 UTC 28 Feb 2010 for the fine mesh (Δx = 15 km).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
(a) The adjoint potential temperature sensitivity (color scale with interval every 0.01 m2 s−2 K−1) valid at the initial time of 1200 UTC 26 Feb 2010 at 700 hPa for the coarse mesh (Δx = 45 km) for the dry simulation. The 700-hPa geopotential height analysis with an interval of 30 m. The sensitivity is scaled by 105 km−3. (b) Adjoint optimal u-wind component perturbation field (color scale with an interval of 2 m s−1) at 925 hPa evolved in the nonlinear model at 36 h valid at 0000 UTC 28 Feb 2010 for the fine mesh (Δx = 15 km).
Citation: Monthly Weather Review 142, 1; 10.1175/MWR-D-13-00201.1
5. Summary and conclusions
To quantify the initial condition sensitivity and predictability of the high-impact extratropical cyclone Xynthia, we have applied a high-resolution nonhydrostatic nested adjoint modeling system with microphysics to simulate the life cycle of this storm over the east Atlantic. The high resolution and moist capabilities of the adjoint modeling system allow us to address sensitivity and predictability of severe extratropical cyclones such as Xynthia more completely and with greater fidelity than possible in previous studies. The extratropical cyclone Xynthia had a large socioeconomic impact on Europe and resulted in more than 60 fatalities, many of which were associated with flooding and a storm surge induced by a very strong low-level wind maximum along the bent-back warm front. Xynthia developed beneath a tropopause PV anomaly along a narrow filament of lower- and midtropospheric water vapor, sometimes referred to as an atmospheric river, which played a key role in the intensification.
The adjoint diagnostics indicate that the intensity and aerial extent of severe winds associated with the front just prior to landfall were particularly sensitive to perturbations in the moisture and temperature fields and to a lesser degree the wind fields. The sensitivity maxima are generally found in the low- and midlevels and oriented in a sloped region along the warm front and maximized within the warm conveyor belt. The moisture sensitivity indicates that only a relatively small filament of moisture within the atmospheric river at the initial time was critically important for the development of Xynthia. The moisture sensitivity has maximum values that are approximately 5–10 times greater than the horizontal wind sensitivity and ~1.5 times greater than the temperature sensitivity when the fields are scaled by typical values of observational uncertainty. The PV optimal perturbations at the initial time are positioned beneath the upper-level jet in a region of strong vertical wind shear with a sloping maximum that connects the low-level PV anomaly with a narrow PV region in the upper portion of the front.
The development and intensification of Xynthia occurred along the northern portion of a region where the SST anomalies were 1°–2°C warmer than the climatological average. The inclusion of surface flux and boundary layer parameterizations in the adjoint provides a method to quantify the sensitivity of the strong winds at the final time (36 h) to the initial SST. The SST sensitivity maximum is positioned in a swath to the right of the track of the surface cyclone within the warm sector of the storm. The strength of the near surface winds at 36 h is most sensitive to the SST during the period when Xynthia undergoes rapid intensification, with the strongest sensitivity located well to the southwest of the final position of the cyclone.
The nested application of the nonhydrostatic moist adjoint modeling system provides an opportunity to examine perturbation growth on finer scales than previously addressed. Adjoint-based optimal perturbations introduced into the tangent linear and nonlinear models, with initial magnitudes comparable to analysis errors (~1 m s−1, 1 K, and 1 g kg−1), exhibit rapid growth with a perturbation 10-m wind speed maximum in excess of 20 m s−1 at the 36-h time. In contrast, initial perturbations of the opposite sign lead to substantial weakening of the low-level jet (perturbation wind speed minimum of less than −20 m s−1 opposing the basic state flow) and a marked reduction in the spatial extent of the strong low-level winds, which would produce weaker waves and a smaller surge. When the initial adjoint optimal perturbations are shifted by 10 grid cells to the west (or 450 km), the resultant growth is reduced substantially, such that the maximum perturbation 10-m wind speed is 3 m s−1 after 36 h, in contrast to more than 20 m s−1 in the control perturbation simulation. This result suggests that the overall growth is quite sensitive to the location and structure of the initial perturbations, which is particularly important for predictability given the nature of the finescale gradients associated with the front. Analysis differences between the COAMPS and NOGAPS cycling data assimilation systems project onto these mesoscale sensitivity regions, and to the degree that these reflect potential analysis errors, point to the potential for rapid forecast error growth.
The adjoint optimal perturbations show a total energy maximum in the midtroposphere (near 500 hPa) that grows rapidly over the 36-h integration period and the growth proceeds throughout the depth of the troposphere. At the initial time, the total energy is dominated by the available potential energy. Rapid growth of the perturbation total energy ensues during the 36-h integration, with the kinetic energy dominant by the final time. The sensitivity fields and adjoint optimal perturbations exhibit a marked up-shear tilt along the sloping warm conveyor belt and frontal system and the perturbations extract energy from the mean flow as they are untilted by the shear, which is consistent with the PV unshielding mechanism. The evolution of the perturbation u-wind component power spectrum indicates that perturbation growth occurs rapidly and shifts upscale to a ~1200-km wavelength after 36 h.
This study demonstrates the utility of a high-resolution nonhydrostatic adjoint modeling system to provide an efficient and accurate method to diagnose sensitivity and quantify predictability of high-impact mesoscale phenomena, such as Xynthia. The adjoint results suggest that in spite of the severe socioeconomic damage attributed to the cyclone, the situation could have been considerably worse. The adjoint sensitivity and optimal perturbation experiments point to scenarios with even stronger low-level jets that are more spatially expansive as reasonable possibilities, which presumably would have posed a greater flooding and wind damage risk. The results underscore the need for more accurate moisture observations and data assimilation systems that can adequately assimilate these observations in order to reduce the forecast uncertainties as much as possible in severe extratropical cyclones. However, given the nature of the sensitivities and the potential for rapid perturbation and error growth, the intrinsic predictability of these severe cyclones is likely limited. The results motivate the need for high-resolution ensembles to quantify the forecast uncertainty and provide probabilistic guidance to mitigate the impact of severe extratropical cyclones.
Acknowledgments
This research is supported by the Chief of Naval Research through the NRL Base Program, PE 0601153N. We appreciate the comments and helpful suggestions of the editor, David M. Schultz, and two anonymous reviewers. We thank Larry O'Neil of Oregon State University for providing Fig. 5a. Computational resources were supported in part by a grant of HPC time from the Department of Defense Major Shared Resource Centers, Stennis Space Center, Mississippi.
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