## 1. Introduction

Operational forecast busts of significant weather events still occur occasionally despite improvements in numerical weather prediction (NWP) models, data assimilation, and observational platforms. These errors are attributable to either or both (possibly flow dependent) NWP model error and imperfect specification of the model initial conditions due to a combination of insufficient observational coverage, observational error, and errors in the assimilation of the observations. While it is reasonable to expect forecast busts to occur regularly downstream of data-void regions (Langland et al. 1999), there are occasions for which short-range (less than 72 h) NWP fails in regions in which upstream data are of relatively high density, and presumably of high quality, such as over the eastern United States. For these occasions, even if analysis errors are relatively small, flow-dependent error growth must in general be quite large.

An event for which operational model guidance was poor over the eastern United States, and for which measures of the susceptibility of forecast error to grow were quite large (Langland et al. 2002, hereafter LSG), occurred on 24–25 January 2000, when a major winter storm deposited as much as 50 cm of snow throughout portions of the Carolinas, the mid-Atlantic, and New England. This event was particularly noteworthy, as operational NWP guidance remained poor for lead times as short as 24 h. In general, forecasts of the surface cyclone intensity and position by operational NWP models improved with decreasing lead times, while guidance for the distribution of an intense precipitation band remained poor. With lead times as short as 24 h, forecasts of the precipitation band remained too far to the east, and as a result, most of the precipitation that fell over the mid-Atlantic and southeastern states was underforecast. Even sophisticated operational forecasting systems such as the European Centre for Medium-Range Weather Forecasts’ Ensemble Prediction System were not particularly helpful, as very few members of the ensemble alerted forecasters to the possibility of this high-impact weather event (Buizza and Chessa 2002).

The use of unconventional data as well as four-dimensional variational data assimilation has shown some utility in improving the forecast of this event (Zupanski et al. 2002; Jang et al. 2003). However, Zhang et al. (2002, 2003) point out the inherent difficulty in forecasting the mesoscale aspects such as the precipitation, due to the extreme error growth rates associated with moist processes. Because all operational NWP guidance from models with differing model discretizations, resolutions, and physics parameterizations failed for this case, and because operational and research model forecasts and simulations initialized with different, yet plausible, initial states produced differences in the synoptic-scale aspects of the event (including cyclone intensity and strength and precipitation distribution) comparable to the sizes of the estimated forecast errors for these aspects, it is likely that a significant fraction of the forecast error can be attributed to initial condition error in particularly sensitive regions of the flow (LSG; Zhang et al. 2002).

The objective of this study is to better understand the structure and growth of errors arising from initial condition uncertainty that are associated with the poor short-term numerical forecasts of this event. Adjoint-based sensitivity studies for this event are performed to investigate the reasons why improvements were not observed in the forecast for precipitation with decreasing lead time, despite improvements in the surface cyclone intensity and position forecasts, as well as to investigate the *possible* sources for the forecast error. In addition, an evaluation of *possible* analysis errors that contributed to an estimate of the final time forecast error is made.

*R*) of an NWP forecast defined at a specific forecast time (

*t*) can be modified by changing the model initial (

_{f}**x**

_{0}) or forecast state (

**x**

*), prior to that final forecast time. This estimate,*

_{τ}*δR*, is obtained by evaluating the inner product of a sensitivity gradient (∂

*R*/∂

**x**

*, the gradient of*

_{τ}*R*with respect to the model state at some forecast time,

*τ,*0 ≤

*τ*<

*t*) with a contemporaneous perturbation to the model state (

_{f}*δ*

**x**

*):*

_{τ}^{1}is the most efficient means of calculating the required sensitivity gradient (Errico 1997).

In this study, 36- and 48-h adjoint-derived forecast sensitivity studies are conducted to investigate why improvements in the forecasted cyclone position and intensity were not necessarily associated with improvements in the forecast of vertical motion. Response functions related to both the forecasted cyclone and associated vertical motion are constructed to perform this investigation. The sensitivity gradients for each of the response functions, including for the forecast error, are interpreted, as has been done in other adjoint-based studies (Errico and Vukićević 1992; Rabier et al. 1992; Langland et al. 1995). This investigation considers the sensitivity with respect to the model forecast trajectory for model variables as well as derived variables.

*Assuming no model error*, the adjoint model can be used to identify “key analysis errors” (Rabier et al. 1996; Klinker et al. 1998; Reynolds and Gelaro 2001). While the vertically integrated initial condition sensitivity of the 72-h forecast error for this case has already been calculated for the Navy Operational Global Prediction System (NOGAPS) model (LSG), in this work, forecast sensitivities are calculated for shorter lead times and are described in greater detail. Additionally, this work emphasizes the importance of the mesoscale aspects of the 24–25 January 2000 event, in particular, the forecast of the intense precipitation. Finally, in this study, the sensitivity gradients with respect to the initial model state calculated for the forecast error are used to derive an analysis perturbation that improves the 36- and 48-h forecasts, similar to a procedure utilized in LSG for their 72-h forecast of the event.

The synoptic setting for the development of the 24–25 January 2000 storm will be described in section 2. This is followed by a description of the modeling system and data used in section 3. Section 4 contains a description of 36- and 48-h fifth-generation Pennsylvania State University–National Center for Atmospheric Research Mesoscale Model (MM5) forecasts both valid at 1200 UTC 25 January 2000. In section 5, the MM5 Adjoint Modeling System is used to compute forecast sensitivities for a variety of response functions. A description of the corrected forecasts, in which the sensitivity of forecast error to initial conditions is used to derive an analysis increment, are found in section 6. A summary of our findings and further discussion is contained in section 7.

## 2. Synoptic overview

The precursor to the surface cyclone associated with the significant weather throughout the eastern seaboard on 24–25 January 2000 was an upper-tropospheric short-wave trough, evident by the low potential temperatures on the dynamic tropopause (defined as the 1.5-PVU surface), over the north-central United States and southern Canada at 1200 UTC 23 January 2000 (Fig. 1a). During the subsequent 12 h, this short-wave trough propagated southeastward (Fig. 1b). By 1200 UTC 24 January, the short-wave trough was advancing through the southeastern United States (Fig. 1c), at which point rapid intensification of the trough at upper and middle levels commenced. At this time, the upper trough began to propagate more zonally, before heading to the northeast. This upper wave and the significant upstream ridge that accompanied it (evident by the higher upstream potential temperatures on the dynamic tropopause) were part of a larger-scale Rossby wave packet (LSG).

Surface cyclogenesis began in the Gulf of Mexico near the Florida panhandle at 0600 UTC 24 January as an expansive region of precipitation developed to the north and northwest of the nascent surface cyclone. By 1200 UTC 24 January, the surface cyclone was located over the Florida panhandle, with composite radar reflectivities indicating a region of a precipitation extending from central South Carolina to south of the Florida panhandle (Fig. 2a). To the south and east of the cyclone, a separate region of stratiform and convective precipitation was observed. By 1800 UTC 24 January (Fig. 2b), the surface cyclone had crossed the Florida peninsula and was located to the east of Jacksonville, Florida. At that time, the primary precipitation shield was located to the north of the surface cyclone from southern North Carolina and across much of South Carolina and Georgia. A second area of precipitation extended to the south of the cyclone along the developing cold front. The most rapid intensification of the surface cyclone occurred between 1800 UTC 24 January and 1200 UTC 25 January, as the cyclone propagated northeastward just offshore of the southeast coast of the United States. During this 18-h period, the cyclone’s sea level pressure minimum fell by 20 hPa, while precipitation over the Florida peninsula ended as the cold front accompanying the developing cyclone swept eastward into the Bahamas. Composite radar reflectivities indicated an extensive region of moderate to heavy precipitation located to the north and northwest of the intensifying cyclone (e.g., Fig. 2c for 0000 UTC 25 January). Surface reports confirm moderate to heavy precipitation rates occurred during this period over the Carolinas.

While the surface cyclone was deepening rapidly, latent heat release associated with the intense precipitation became increasingly important for the development. The latent heating redistributed the potential vorticity (PV) by eroding PV associated with the upper trough, as evident by the warming along the dynamic tropopause downstream of the upper trough (Figs. 1c and 1d), while increasing the PV in the middle and lower troposphere. This process was likely responsible for the rapid intensification of the lower-tropospheric cyclone and the concomitant destruction of the upper trough that had earlier triggered the surface cyclogenesis.

By 1200 UTC 25 January (nearly the time of maximum intensity of the cyclone), the surface cyclone had deepened to 982 hPa and was located east of Cape Hatteras, North Carolina (Fig. 3a). This was also the time heavy precipitation was falling throughout the mid-Atlantic United States. A composite radar image valid at 1200 UTC 25 January shows the high reflectivity associated with the moderate to intense precipitation that extended from coastal South Carolina, through the mid-Atlantic, and into areas of southern New England (Fig. 4a). In the preceding 24 h, the heaviest accumulated precipitation had occurred in the Carolinas, with as much as 50 to 70 mm of liquid equivalent precipitation having fallen (Fig. 4b).

The primary forcing for the vertical motion responsible for the precipitation over land appears to have been lower- to midtropospheric frontogenesis. The most intense vertical velocities at 1200 UTC 25 January were found near and above 700 hPa, north and west of the surface cyclone (Figs. 3b and 3c). As the midlevel trough intensified and became cutoff, the flow became more diffluent over the mid-Atlantic along a baroclinic zone (Fig. 3b). Such a configuration of wind and temperature was strongly frontogenetical. A cross section through the maximum vertical velocities (Fig. 3c) indicates that both the frontogenesis and vertical motion extended from the surface to the middle troposphere, tilting westward with height over the surface and midtropospheric frontal zones. The most intense vertical velocities were elevated, and appear to have been associated with a secondary maximum in the horizontal frontogenesis at about 750 hPa, as opposed to surface frontogenesis in the vicinity of the surface cyclone itself. The coincidence of the sloping ascent along the warm side of a frontogenetically active baroclinic zone suggests a link between the frontogenesis and vertical motion. A calculation of the vertical motion attributed to frontogenesis (not shown) confirms this interpretation.

## 3. Numerical modeling system

### a. Forecast model

For this study, the MM5 Adjoint Modeling System (Zou et al. 1997) is used. This modeling system, which is based upon version 1 of the MM5, includes the nonlinear model, its tangent-linear model (TLM), and corresponding adjoint. The MM5 is a nonhydrostatic, limited-area, sigma-coordinate, primitive equation model.

The model domain for this study is a 60 km, 96 × 65 horizontal grid, with 16 evenly spaced sigma levels in the vertical (top pressure level in the model is 100 hPa). The model is initialized from the National Centers for Environmental Prediction (NCEP) final analysis (1° × 1° global grid) interpolated to the MM5 grid, and the lateral boundaries are updated using NCEP final analyses (i.e., “perfect” boundary conditions). Physical parameterizations used in the simulation include the Grell convective scheme, a bulk aerodynamic formulation of the planetary boundary layer, horizontal and vertical diffusion, dry convective adjustment, and explicit treatment of cloud water, rain, snow, and ice. Simulations of 36 and 48 h in length, verifying at 1200 UTC 25 January 2000, were performed.

### b. TLM and adjoint

The TLM and corresponding adjoint are based on a linearization of the nonlinear model about a specific basic state created from saving the output of the nonlinear model at each time step. The TLM and adjoint integrations for this study use the same domain, data, and set of parameterizations (either the linear versions, or their adjoints) as the nonlinear model, with the exception of the moist dynamics. As a consequence, the TLM and adjoint integrations use only dry dynamics, but are integrated about a moist basic state computed by the nonlinear model.^{2} The adjoint code has been modified by the first author in a manner as described by Zou et al. (2001) in order to eliminate the nonphysical oscillation in the sensitivity gradients associated with the adjoint of the leapfrog time-stepping scheme. This modification allows for an evaluation of the forecast sensitivities with respect to the model forecast trajectory.

## 4. Description of MM5 simulations

### a. Nonlinear model control simulations

With improvements noted for shorter forecast lead times, the deficiencies of the MM5 simulations for this event were comparable to the deficiencies described for operational and research simulations conducted in other studies of this case (e.g., LSG; Kleist et al. 2001; Zhang et al. 2002). Of the two control simulations conducted (48- and 36-h simulations), the 48-h MM5 simulation valid at 1200 UTC 25 January was poorer as the cyclone was 530 km to the east of the analyzed center and the mean sea level pressure (MSLP) was 11 hPa too high (Fig. 5a). The 36-h MM5 simulation valid at the same time had smaller position (100 km too far east) and MSLP (6 hPa too high) errors (Fig. 6a).

Compared with the analysis and supported by the radar summary, both the 36- and 48-h simulations had poor forecasts of the vertical motion at 1200 UTC 25 January. The simulations had a band of upward vertical motion with approximately the correct orientation, but the band was too weak and primarily offshore (Figs. 5b and 6b). A westward tilt with height of the vertical motion forecast is evident in cross section taken through the maximum vertical motion in both the 36- and 48-h simulations (Figs. 5c and 6c), similar to the analysis valid at the same time (Fig. 3c).

A closer inspection of the 48-h forecast shows that the vertical motion is primarily being forced by surface frontogenesis (Fig. 5c), as opposed to midtropospheric frontogenesis. In fact, the frontogenesis forecasted in the 48-h simulation shows almost no westward tilt in the vertical, unlike the verifying analysis (Fig. 3c). This, in combination with the surface cyclone being forecast to be too far to the east, is consistent with the vertical motion in this simulation being too far east. The 36-h simulation places the vertical motion at 600 hPa over the same region as in the 48-h simulation (Figs. 5b and 6b), despite the fact that the 36-h simulation appears to have a better forecast of the midtropospheric wind field (Figs. 5b and 6b), with a significant region of forecasted midtropospheric frontogenesis being evident in the cross section in Fig. 6c. The forecasted vertical motion in the 36-h simulation is stronger than that in the 48-h simulation, yet no improvement in the location of vertical motion occurred, despite the fact that there was an improvement in the forecast of the surface cyclone as well as the vertical structure of the vertical motion.

To assess the potential impact of a higher resolution on the subsequent forecast, an additional 48-h simulation was performed with two additional grids (at resolutions of 20 and 6.67 km, with 25 vertical levels) nested within the simulation described above using version 3 of the MM5. This higher-resolution model had a cyclone of similar intensity and location as the lower-resolution 48-h control simulation.

### b. Linearity test

In addition to evaluating the performance of the control simulations, it is necessary to assess the validity of the tangent-linear assumption prior to discussing any adjoint-based results (Errico and Vukićević 1992; Gilmour et al. 2001). To assess the validity of this assumption, perturbations that are evolved linearly using the TLM are compared with difference fields calculated from two nonlinear forecasts. Figure 7 shows examples of the initial perturbations used in performing both 36- and 48-h linearity tests, with the largest perturbations no larger than 1°C for temperature or 1 m s^{−1} for horizontal wind. A comparison of the linearly evolved perturbation and the nonlinear difference field reveals qualitative and quantitative agreement (Fig. 8). As expected, the largest discrepancies occur off of the Carolina coast for the nonlinear model integrations, which included moist physics (Figs. 8a and 8b), whereas very good agreement can be found when the linearity test is performed with moist physics turned off in the nonlinear model (Figs. 8c and 8d). This serves as evidence that moist dynamic processes did in fact affect the development. However, despite this possible weakness, results from the adjoint model integrations can still be quite useful as an estimate to the actual forecast sensitivity so long as the limitations of the methodology are kept in mind.

## 5. Forecast sensitivity gradients

### a. Definition of response functions

*R*

_{1}) considered is an energy-weighted forecast error defined as

**e**

*is the MM5 forecast error at 1200 UTC 25 January 2000 [i.e., the deviation of the 36- or 48-h MM5 forecast state (*

_{f}**x**

*) from the NCEP final analysis (*

_{f}**x**

^{fnl}

_{f}) valid at 1200 UTC 25 January 2000 interpolated to the MM5 grid],

**P**is a local projection operator that restricts the calculation of the energy-weighted forecast error into a specific geographic region, and

**C**is a matrix of weights that converts the forecast error into units of total dry energy (as in Rabier et al. 1996; Palmer et al. 1998).

^{3}We only consider forecast error in the vicinity of the cyclone off of the east coast at verification time (Fig. 9a). The input to the adjoint model for

*R*

_{1}is then

**x**

*is the MM5 forecast state vector at the verification time.*

_{f}*R*

_{2}) considered, a measure of the strength of the lower-tropospheric cyclonic flow, is the circulation about the box defined in Fig. 9b on the lowest sigma level (

*σ*= 0.97):

**V**is the horizontal wind vector, and

*d*

**l**is a differential element along the path enclosing that box. The only nonzero gradients for the adjoint input are with respect to the horizontal wind components tangent to the sides of the box used to define

*R*

_{2}[similar to a response function used in Errico (1997)].

*R*

_{3}, is defined to be the 750-hPa horizontal frontogenesis (Petterssen 1936) within the box denoted on Fig. 9c. The expression defining

*R*

_{3}is

*D*is the total deformation and

*β*the angle between the axis of dilatation and the isentropes,

*θ*.

A fourth response function, *R*_{4}, is the forecasted vertical motion (i.e., *R*_{4} = *w*_{σ=0.66}) in the box shown in Fig. 9d. The location of the box coincides with the region of ascent observed in the analysis valid at 1200 UTC 25 January 2000. If the vertical motion is primarily being forced by horizontal frontogenesis, forecast sensitivity gradients calculated for both *R*_{3} and *R*_{4} should be correlated, as an increase or decrease in one should correspond with an increase or decrease in the other.

### b. Description of forecast sensitivity gradients for 36-h forecast

#### 1) Comparison of sensitivity for different response functions

For the 36-h forecast, the largest sensitivity for all four response functions with respect to the initial distribution of temperature at 700 hPa is geographically isolated in a region of enhanced baroclinicity, extending from eastern Texas to southwestern Virginia (Fig. 10). A positive temperature perturbation at the initial forecast time in northern Alabama would have the greatest impact in reducing the 36-h forecast error, relative to any other initial temperature perturbations of comparable magnitude at 700 hPa (Fig. 10a).^{4} Similarly, a positive temperature perturbation in extreme northwestern Alabama would result in an increase in the circulation and vertical motion forecast at the final time (Figs. 10b and 10d), but would have relatively little impact on the forecasted frontogenesis. Negative temperature perturbations in central Arkansas would decrease *R*_{1} and increase *R*_{2}, but would have little or no impact on *R*_{3} and *R*_{4}. Adding temperature perturbations in such a fashion would create a thermal ridge in Mississippi and Alabama, while enhancing the thermal trough over Arkansas, Texas, and Louisiana. The wavelike distribution of sensitivity to temperature for all four response functions implies that all four response functions are sensitive to the addition of a thermal wave at 700 hPa at the initial time.

Cross section taken along the direction of the lower-tropospheric vertical wind shear indicate that the sensitivity gradients for all four response functions with respect to temperature exhibit an upshear vertical tilt, and are maximized in the lower and midtroposphere (Fig. 11). This distribution is characteristic of sensitivity gradients and has been observed for other cases (Rabier et al. 1992; Langland et al. 1995). The sensitivity with respect to the initial distribution of other variables such as the horizontal wind components exhibits similar characteristics, being maximized in the middle and lower troposphere, and also can be characterized as having an upshear vertical tilt (not shown).

#### 2) Sensitivity of *R*_{2} with respect to vorticity

The development of the surface cyclone was triggered once the upper trough reached the southeastern United States. Evaluating the evolution of the sensitivity of *R*_{2}, a measure of the cyclone’s final time intensity, with respect to relative vorticity^{5} together with the MM5 forecast of relative vorticity, is one means of investigating the sensitivity of the surface cyclone development to the intensity and location of the precursor upper trough. During the evolution of the 36-h forecast, the maximum in sensitivity with respect to the 500-hPa relative vorticity remains in close proximity to the 500-hPa relative vorticity maximum associated with the upper trough (Fig. 12). This simply means that increasing the amplitude of the upper trough by increasing its vorticity in the regions of maximum sensitivity at any time will lead to an increase in the final time forecast of the circulation associated with the surface cyclone.

From 12 through 24 h there is a maximum in sensitivity with respect to relative vorticity, which extends well upstream of the upper-tropospheric precursor to the surface cyclogenesis (Fig. 12). This secondary maximum propagates faster than and eventually merges with the primary maximum in the vicinity of the upper trough (Fig. 12d). Additionally at these forecast times, there is negative sensitivity to relative vorticity to the north, which also propagates at the same speed of the upper trough, while decreasing in magnitude with time. This region of negative sensitivity to relative vorticity at 12 h is associated with the upstream ridge (Fig. 12a). Thus, increases in the strength of upstream ridge would be associated with increases in *R*_{2}. This observation, in addition to the sensitivity gradient “catching up” to the upper trough, points to the importance of downstream development for the evolution of this cyclone and associated forecast errors, as was suggested in LSG.

The diabatic redistribution of PV played a key role in the degrading and reshaping of the upper trough. The importance of this process is also suggested in the evolution of the sensitivity with respect to relative vorticity. By 24 h, the forecast time at which the effects of latent heating on the PV distribution first become evident, a southeastward extension in the maximum sensitivity associated with the upper trough develops just east of the Florida coast, while simultaneously, a region of negative sensitivity appears over the Mid-Atlantic states (Fig. 12c). Increasing the vorticity to the east of the upper trough while decreasing the vorticity (building a ridge) downstream of the upper trough at 24 h will lead to an increase in *R*_{2}, which is dynamically consistent with less of an erosion of the upper trough (increase in the amplitude of the downstream ridge) through diabatic processes. Indeed, the increase in the along-stream gradient of the vorticity as implied by the sensitivity gradients at 24 h implies a much more vigorous “self-development” mechanism at work.

#### 3) Description of sensitivities for *R*_{3}

The sensitivities calculated for *R*_{3} appear to be different than those calculated for the other three response functions (Figs. 10 and 11). For 18 h and beyond, *R*_{3} is particularly sensitive to the orientation and magnitude of the forecasted lower- and midtropospheric baroclinic zone (Fig. 13). Figures 13a and 13b show that *R*_{3} can be increased by adding temperature perturbations that increase the north–south 700-hPa baroclinicity east of the Virginia and Maryland coasts at 18 and 24 h, respectively. If temperature perturbations are added in such a fashion, the newly created, zonally oriented baroclinic zone can be subsequently rotated to have a southwest-to-northeast orientation, at which point confluent winds can frontogenetically interact with the baroclinic zone. Another way to increase *R*_{3} would be to increase a meridionally oriented baroclinic zone at 700 hPa over Virginia and North Carolina at 18 and 24 h (Fig. 13). The wavelike pattern in the temperature sensitivity distribution means that *R*_{3} can be most effectively increased through the addition of a thermal wave just off the mid-Atlantic coast at 700 hPa at the forecast times just described. Sensitivity vectors,^{6} which show the direction in which the wind field needs to be perturbed to increase the response function, are also consistent with this interpretation. For example, the wind perturbations implied at 24 h off of the New Jersey coast would in effect rotate the baroclinic zone in the forecast to obtain a more zonal orientation (Fig. 13b), consistent with the dipole in sensitivity with respect to temperature.

### c. Description of forecast sensitivity gradients for 48-h forecast

Sensitivity gradients calculated for the four response functions for the 48-h forecast exhibit similar characteristics to those calculated for the 36-h forecast. The largest initial condition sensitivities for the 48-h forecast are confined to the lower and midtroposphere, are of small scale, and possess an upshear vertical tilt (not shown). The sensitivities with respect to the initial distribution of 500-hPa relative vorticity for all four response functions again point to the importance of the precursor upper trough as all sensitivities at this level are along the 500-hPa trough axis that extends from the southwestern United States to Ontario (Fig. 14). There is a secondary sensitivity maximum associated with the upstream ridge over southwestern Canada for *R*_{1}, *R*_{3}, and *R*_{4} (Fig. 14). The 48-h forecast error can be reduced by intensifying the upper trough just east of the Rocky Mountains, as well as in the northern plains, or by decreasing the magnitude of the upstream ridge located over southwestern Canada (Fig. 14a).

Unlike the initial condition sensitivities calculated for the 36-h forecast, the initial condition sensitivities for the 48-h forecast exhibit significant differences between the four response functions. There is negative sensitivity with respect to relative vorticity over the Rocky Mountains for *R*_{1} and *R*_{3}, whereas the sensitivity is largely positive for the other two response functions over the nearly the same geographical areas. Furthermore, sensitivities for *R*_{2} and *R*_{4} appear to be very similar in their structure throughout the entire southwestern United States. It seems counterintuitive that adding a negative relative vorticity perturbation in eastern Wyoming would lead to a decrease in the 48-h forecast error *and* decrease in the forecasted circulation, but would also lead to an increase in the forecasted frontogenesis and a decrease in the forecasted vertical motion (Fig. 14).

The fact that there are differences in the sensitivity gradients is consistent with the notion that improvements in one aspect of the forecast are not necessarily associated with improvements in other aspects. These differences can be exploited to investigate the reasons why improvements in the forecasted surface cyclone and position for decreasing lead times were not associated with much improvement in the forecast vertical motion. In principle, the sensitivity gradients for each of the response functions can be used to construct perturbations that increase *R*_{2}, without necessarily improving *R*_{3}. It is important to note, however, that there will be key regions and levels for which these sensitivity gradients are consistent, and it is for these regions that initial condition errors, if corrected, would lead to forecast improvements as measured by all four response functions.

## 6. Correction of forecast using *R*_{1} sensitivity gradients

*R*

_{1}with respect to the model initial state may be used to identify those analysis errors that specifically contribute to

*R*

_{1}. Subtracting these “key” analysis errors from the control analysis leads to a reduction in the forecast error as measured by

*R*

_{1}. An initial condition perturbation can be constructed from the initial condition sensitivity gradient by using the inverse of the energy-weighting matrix and a scaling factor (

*α*):

**x**

^{(1)}

_{f}), from which forecast error for this new forecast can be calculated:

*α*, ||

**e**

^{1}

_{f}|| < ||

**e**

^{0}

_{f}||. The newly calculated error can be used to recompute

*R*

_{1}, and the sensitivity of the forecast error with respect to

**x**

^{1}

_{0}can be calculated. This sensitivity gradient can then be used to derive a new analysis perturbation, which should, in theory, further reduce the forecast error. This methodology using the nonlinear and adjoint models can be performed in an iterative manner,

*α*= 0.0075. This scaling factor was chosen to be small enough so as not to overcorrect the forecast on a given iteration. Applying the perturbed initial condition perturbation leads to a 46% reduction of the forecast error in the box used to define

*R*

_{1}.

An along-shear cross section shows that a component of the key analysis perturbation is associated with an initial PV perturbation (difference between the PV calculated from the perturbed and control initial conditions) that is generally confined to the middle and lower troposphere with small differences also appearing near the model top at 100 hPa (Fig. 15). The lower- and middle-tropospheric PV perturbation has a baroclinic upshear tilt characteristic of key analysis perturbations calculated in other studies. These key analysis perturbations show evidence of a thermal wind-type balance: the positive PV perturbation is associated with both positive stratification and positive vorticity perturbations as evidenced from the distributions of temperature (Fig. 15a) and wind (Fig. 15b) relative to the PV. The effects of the baroclinic shear on the perturbation can be evaluated by considering the differences between the perturbed forecast and the control forecast at various forecast times. Within 6 h (Fig. 16a), baroclinic shear has begun to render the initially upshear-tilted PV perturbations vertical. Concomitantly, significant thermal perturbations begin to develop in the lower and upper troposphere. Additionally, larger-scale, equivalent barotropic perturbations appear in the cross section as growing perturbations in the lower stratosphere are advected into the plane of the cross section. These larger-scale perturbations are initially associated with slight fluctuations in the dynamic tropopause height upstream of the lower-tropospheric, upshear-tilted perturbations.

At 12 h, the largest PV and temperature perturbations are found in the upper troposphere (Fig. 16b), associated with changes in the height of the dynamic tropopause. The perturbations in the interior (between 400 and 700 hPa) are relatively small by comparison at this time (Fig. 16b), unlike at the initial time, when the interior perturbations were of largest magnitude (Fig. 15a). Also, the interior perturbations lose some of their tilt, becoming more equivalent barotropic, which helps to enhance the near-surface and tropopause-level thermal advections, further amplifying the thermal perturbations at the surface and tropopause. These developments resemble the growth of singular vector perturbations in baroclinically sheared flows (Morgan 2001).

Despite the fact the analysis perturbations were relatively small in magnitude and scale, their effect was significant for the 48-h MM5 forecast. Although only 46% of the full forecast error in the box used to define *R*_{1} was reduced, the quality of the perturbed 48-h forecast is quite good. Figure 17a shows the improvement in the 48-h forecast of the surface cyclone, with a major improvement in cyclone position and intensity, including the error in sea level pressure minimum being reduced by 8 hPa. The forecasted vertical motion is much improved as well, with a majority of the vertical motion to the north and west of the surface cyclone being forecast well inland (Fig. 17b), much closer to the verifying analysis (Fig. 3b). The most dramatic improvement occurred in the distribution of lower- to midtropospheric frontogenesis and associated vertical motion (Figs. 17b and 17c). The control forecast had no secondary maximum in frontogenesis in the midtroposphere (Fig. 5c), which the perturbed forecast corrects. We note however, that relative to the NCEP final analysis, the corrected 48-h simulation overforecasts the strength of the frontogenesis and concomitant vertical motion.

In addition to improvements to the 48-h final time forecast, improvements to the forecast at intermediate times are also noted. Figure 18 shows 950–700-hPa-layer-averaged vertical velocity, relative humidity, and mean sea level pressure forecasts for the control and perturbed 48-h forecasts at the 24-, 30-, and 36-h forecast times, corresponding to the analysis times shown in Fig. 2. By 24 h into the forecast, it is apparent that the perturbed forecast (Fig. 18b) has a significant precipitation band, similar to what was observed on radar, north of the surface cyclone at 1200 UTC 24 January (Fig. 2a), while the control forecast (Fig. 18a) had neither the meridionally oriented precipitation band nor a distinct cyclone center. By the time the rapid deepening commences (1800 UTC 24 January), the forecast fields of the perturbed simulation (Fig. 18d) more closely resemble the contemporaneous analysis and radar observations (Fig. 2b) than do the corresponding fields from the control forecast (Fig. 18c). In particular, the regions of ascent and higher relative humidity in the perturbed forecast extend farther inland and to a better degree match the radar reflectivity. In addition, the perturbed forecast is characterized by a deeper cyclone. At 36 h into the forecast (Figs. 18e and 18f), the differences between the control and perturbed forecasts are much more apparent. The precipitation distribution that may be inferred from the relative humidity and vertical motion fields is now much farther west as is the surface cyclone position in the perturbed simulation. Convective precipitation remains over Florida in the control forecast, while it is forecast farther to the east in the perturbed forecast.

Comparisons between the observed 24-h accumulated precipitation (Fig. 4b) with the forecasted accumulation precipitation in the last 24 h of the control and perturbed 48-h forecasts (Figs. 19a and 19b, respectively) further suggest that the perturbed forecast is an improvement over the control as the accumulated precipitation in the perturbed forecast is broader in spatial coverage with heavier amounts extending well inland into Virginia and North Carolina. The perturbed simulation, however, overforecasts precipitation east of the Chesapeake Bay over Delaware, eastern Maryland, and eastern Virginia.

Zupanski et al. (2002) suggest that the NCEP Eta Model precipitation forecast errors over the mid-Atlantic and southeastern United States were attributable to the failure of the operational three-dimensional variational analysis scheme to create sufficient precipitable water and lower-tropospheric convergence in the analyses over the southeastern United States at 1200 UTC 24 January 2000. Brennan and Lackmann (2005) suggest a crucial feature missing from many of the operational and research forecast model forecasts and simulations was a diabatically generated, lower-tropospheric PV anomaly over the southeastern United States at 1200 UTC 24 January. This PV anomaly developed over the lower Mississippi Valley between 0600 and 1200 UTC 24 January and was associated with the band of precipitation described earlier in this section over Georgia and Alabama at 1200 UTC 24 January. Brennan and Lackmann demonstrate that this PV anomaly subsequently aided in the transport and concomitant convergence of moisture over the southeastern United States.

Examination of the differences between the perturbed and control simulations of precipitable water (Fig. 20b) and lower-tropospheric (900 hPa) divergence (Fig. 20a) over the southeastern United States for 1200 UTC 24 January are consistent with the results of Zupanski et al. (2002) and Brennan and Lackmann (2005). Specifically, the precipitable water values and near-surface convergence are greater in the perturbed run compared with the control run at 24 h into the model simulation. Moreover, the 800–900-hPa perturbation PV (Fig. 20c) is found immediately upstream of the region of lower-tropospheric convergence. This configuration is consistent with ascent being forced by the anomalous PV. The perturbation wind field in Fig. 20a indicates an increased southeasterly flow from the Atlantic, consistent with enhanced moisture transport into the southeastern United States.

Because of the inherent difficulties in partitioning forecast error into those components attributed to model error and those attributable to initial condition error, there is no way to determine whether the forecast improvement obtained from the adjoint sensitivities for response function *R*_{1} represent the compensation by the adjoint model of inadequacies of the forward model. Given the consistency between perturbed forecast fields and analyses in addition to actual observations (in particular radar) prior to the final time forecast, we find that the perturbed forecast is more consistent with what happened in the real atmosphere than the control simulation. This does not necessarily allow one to conclude whether initial condition errors or model deficiencies were a larger contributor to the forecast error, but exemplifies the fact that a good forecast of the event was possible despite model deficiencies.

## 7. Summary and conclusions

In terms of vertical motion and precipitation, the 24–25 January 2000 storm was poorly forecast by operational and research NWP models, even with lead times as short as 36 h. With decreasing lead times, operational models exhibited improvements in the forecasted surface cyclone position and intensity at 1200 UTC 25 January 2000, but failed to show improvements in the forecast of precipitation. MM5 simulations of 36 and 48 h for the event exhibit similar characteristics to operational NWP forecasts. While initial condition deficiencies and model error were both likely contributors to the forecast error, possible initial condition deficiencies were the primary focus of this study.

Using the MM5 Adjoint Modeling System, forecast sensitivities with respect to the forecast trajectories are calculated for various response functions, for both the 36- and 48-h forecasts verifying at 1200 UTC 25 January 2000. The response functions chosen for this study were an energy-weighted forecast error (*R*_{1}), lower-tropospheric circulation (*R*_{2}), 750-hPa frontogenesis (*R*_{3}), and vertical motion (*R*_{4}). The sensitivities for all four response functions with respect to the initial conditions were relatively isolated, and shared similar characteristics. The largest initial condition sensitivities were found to be in the middle and lower troposphere, and had a characteristic upshear baroclinic tilt.

The evolution of the sensitivity of *R*_{2} with respect to 500-hPa relative vorticity shows the importance of the upper trough for the development of the surface cyclone. In addition to large sensitivities to the vorticity associated with the upper trough, *R*_{2} was also sensitive to vorticity upstream of the upper trough, including a large negative sensitivity to the upstream ridge. The evolution of this sensitivity also suggests the importance of the destruction of the upper trough with forecast time by diabatic processes (i.e., diabatic PV redistribution). If less destruction of the upper trough had occurred at earlier forecast times, the forecasted intensity of the surface cyclone at 1200 UTC 25 January would have been greater.

Although the sensitivity gradients calculated for the four response functions were in general very similar, subtle differences in the phase and distribution of the maximum sensitivity regions imply that perturbations that will lead to changes in one aspect of the forecast do not necessarily have to be associated with changes in other aspects.

An iterative procedure using the sensitivity gradients of *R*_{1} with respect to the initial conditions was used to construct perturbations to add to the control initial conditions to improve the 36- and 48-h forecasts. The initial perturbations with respect to temperature and wind appeared to satisfy a thermal wind-type balance, as they were consistent with the initial PV perturbations.

The analysis perturbations used to improve the 36- and 48-h forecast were small both spatially and in magnitude, well within the bounds of analysis uncertainty. These initial perturbations were not associated with any appreciable changes to the tropopause, but their subsequent effect was large because of their rapid growth. Because of the baroclinic shear, the initially upshear-tilted interior PV perturbations became more barotropic with time. Potential vorticity superposition resulted in an enhancement of the perturbations winds in the lower troposphere and a subsequent enhancement of lower-tropospheric thermal perturbations 24 h into the forecast. The evolved perturbations appear to not only reduce the final time forecast error, but also have a positive impact on the entire forecast trajectory, as evident from comparisons of the vertical motion forecasts with radar reflectivities.

Despite the diagnosed improvements to the forecast trajectory associated with the adjoint-derived initial condition perturbation, it would be incorrect to infer that only analysis error has been corrected because consideration of model error has been neglected in this study. Indeed, the influence and apparent rapid growth of the lower-stratospheric perturbations that appear in the cross section shown in Fig. 16 must be further considered. It is possible that this perturbation is spurious in the sense that it may represent a correction to the model initial state and subsequent forecast trajectory that compensates for deficiencies in the model physics and resolution. Additionally, the apparent “overcorrection” of the forecasted frontogenesis and vertical motion relative to the NCEP final analysis may reflect that certain aspects of the key analysis perturbations are incorrect. It is interesting to note that when the perturbed analysis derived for the 48-h forecast was interpolated and used as the initial condition for a higher-resolution simulation with nests of 20 and 6.67 km, the resulting forecast was an improvement relative to the high-resolution control simulation. The resulting cyclone was deeper than the control forecast at this higher resolution (though still insufficiently deep relative to observations and analyses) and had a position error that was a few hundred kilometers too far to the northeast of the analyzed position. Additionally, the vertical motion and associated precipitation forecast was better in the perturbed simulation. A possible focus of a future study would be to better understand how forecast errors in NWP models associated with both physics and resolution can be compensated for by initial condition changes derived from adjoint-based results. Such a study would be useful, as it would provide insight into how operational four-dimensional variational assimilation behaves in the presence of poorly estimated model error.

## Acknowledgments

This work was completed as part of the first author’s Master’s thesis at the University of Wisconsin—Madison (Kleist 2003), supported in part by National Science Foundation Grant ATM-0121186. Both authors wish to thank Linda Keller, Amanda Adams, Drs. Jonathan Martin and John Young, and two anonymous reviewers for helpful comments on earlier versions of this manuscript. The authors also wish to thank Dr. Yong-Run Guo of NCAR for assistance with the MM5 adjoint code.

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MSLP (contours, interval 4 hPa) for the NCEP final analysis and composite radar base reflectivity (db*Z*) valid at (a) 1200 UTC 24 Jan, (b) 1800 UTC 24 Jan, and (c) 0000 UTC 25 Jan 2000.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

MSLP (contours, interval 4 hPa) for the NCEP final analysis and composite radar base reflectivity (db*Z*) valid at (a) 1200 UTC 24 Jan, (b) 1800 UTC 24 Jan, and (c) 0000 UTC 25 Jan 2000.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

MSLP (contours, interval 4 hPa) for the NCEP final analysis and composite radar base reflectivity (db*Z*) valid at (a) 1200 UTC 24 Jan, (b) 1800 UTC 24 Jan, and (c) 0000 UTC 25 Jan 2000.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

1200 UTC 25 Jan 2000 analysis of (a) sea level pressure (solid, interval 4 hPa) and surface observations of wind (barbs, kt), temperature (°C), and MSLP (hPa); (b) 700-hPa geopotential height (solid, interval 30 m), temperature (dashed, interval 3°C), vertical velocity omega (filled, interval 3 × 10^{−1} Pa s^{−1}), as well as observations of 700-hPa wind (barbs, m s^{−1}) and temperature (°C); and cross-section analysis of (c) vertical velocity omega (filled, interval 3 × 10^{−1} Pa s^{−1}), frontogenesis [solid, interval 1°C (100 km 3 h)^{−1}, positive values only], and potential temperature (dashed, interval 3 K). Line A–B in (b) denotes orientation of cross section for (c).

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

1200 UTC 25 Jan 2000 analysis of (a) sea level pressure (solid, interval 4 hPa) and surface observations of wind (barbs, kt), temperature (°C), and MSLP (hPa); (b) 700-hPa geopotential height (solid, interval 30 m), temperature (dashed, interval 3°C), vertical velocity omega (filled, interval 3 × 10^{−1} Pa s^{−1}), as well as observations of 700-hPa wind (barbs, m s^{−1}) and temperature (°C); and cross-section analysis of (c) vertical velocity omega (filled, interval 3 × 10^{−1} Pa s^{−1}), frontogenesis [solid, interval 1°C (100 km 3 h)^{−1}, positive values only], and potential temperature (dashed, interval 3 K). Line A–B in (b) denotes orientation of cross section for (c).

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

1200 UTC 25 Jan 2000 analysis of (a) sea level pressure (solid, interval 4 hPa) and surface observations of wind (barbs, kt), temperature (°C), and MSLP (hPa); (b) 700-hPa geopotential height (solid, interval 30 m), temperature (dashed, interval 3°C), vertical velocity omega (filled, interval 3 × 10^{−1} Pa s^{−1}), as well as observations of 700-hPa wind (barbs, m s^{−1}) and temperature (°C); and cross-section analysis of (c) vertical velocity omega (filled, interval 3 × 10^{−1} Pa s^{−1}), frontogenesis [solid, interval 1°C (100 km 3 h)^{−1}, positive values only], and potential temperature (dashed, interval 3 K). Line A–B in (b) denotes orientation of cross section for (c).

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

(a) Composite radar base reflectivity (db*Z*) and (b) 24-h accumulated liquid equivalent precipitation (mm) valid at 1200 UTC 25 Jan 2000.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

(a) Composite radar base reflectivity (db*Z*) and (b) 24-h accumulated liquid equivalent precipitation (mm) valid at 1200 UTC 25 Jan 2000.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

(a) Composite radar base reflectivity (db*Z*) and (b) 24-h accumulated liquid equivalent precipitation (mm) valid at 1200 UTC 25 Jan 2000.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Forty-eight-hour forecast valid at 1200 UTC 25 Jan 2000 of (a) MSLP (solid, interval 4 hPa) and 10-m winds (barbs, m s^{−1}), (b) 700-hPa geopotential height (solid, interval 30 m), wind (barbs, m s^{−1}), temperature (dashed, interval 3°C), and vertical velocity omega (filled, interval 3 × 10^{−1} Pa s^{−1}), as well as (c) cross section of vertical velocity omega (filled, interval 3 × 10^{−1} Pa s^{−1}), frontogenesis [solid, 1°C (100 km 3 h)^{−1}, positive values only], and potential temperature (dashed, interval 3 K). Line C–D in (b) denotes orientation of cross section for (c).

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Forty-eight-hour forecast valid at 1200 UTC 25 Jan 2000 of (a) MSLP (solid, interval 4 hPa) and 10-m winds (barbs, m s^{−1}), (b) 700-hPa geopotential height (solid, interval 30 m), wind (barbs, m s^{−1}), temperature (dashed, interval 3°C), and vertical velocity omega (filled, interval 3 × 10^{−1} Pa s^{−1}), as well as (c) cross section of vertical velocity omega (filled, interval 3 × 10^{−1} Pa s^{−1}), frontogenesis [solid, 1°C (100 km 3 h)^{−1}, positive values only], and potential temperature (dashed, interval 3 K). Line C–D in (b) denotes orientation of cross section for (c).

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Forty-eight-hour forecast valid at 1200 UTC 25 Jan 2000 of (a) MSLP (solid, interval 4 hPa) and 10-m winds (barbs, m s^{−1}), (b) 700-hPa geopotential height (solid, interval 30 m), wind (barbs, m s^{−1}), temperature (dashed, interval 3°C), and vertical velocity omega (filled, interval 3 × 10^{−1} Pa s^{−1}), as well as (c) cross section of vertical velocity omega (filled, interval 3 × 10^{−1} Pa s^{−1}), frontogenesis [solid, 1°C (100 km 3 h)^{−1}, positive values only], and potential temperature (dashed, interval 3 K). Line C–D in (b) denotes orientation of cross section for (c).

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Same as in Fig. 5, except for 36-h forecast valid at 1200 UTC 25 Jan 2000. Line E–F in (b) denotes orientation of cross section for (c).

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Same as in Fig. 5, except for 36-h forecast valid at 1200 UTC 25 Jan 2000. Line E–F in (b) denotes orientation of cross section for (c).

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Same as in Fig. 5, except for 36-h forecast valid at 1200 UTC 25 Jan 2000. Line E–F in (b) denotes orientation of cross section for (c).

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Initial temperature perturbation (interval 0.25 K, negative values dashed, zero contour omitted) at *σ* = 0.72 valid at (a) 0000 UTC 24 Jan 2000 and (b) 1200 UTC 23 Jan 2000. Line G–H in (b) denotes orientation of cross section for Figs. 15 and 16.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Initial temperature perturbation (interval 0.25 K, negative values dashed, zero contour omitted) at *σ* = 0.72 valid at (a) 0000 UTC 24 Jan 2000 and (b) 1200 UTC 23 Jan 2000. Line G–H in (b) denotes orientation of cross section for Figs. 15 and 16.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Initial temperature perturbation (interval 0.25 K, negative values dashed, zero contour omitted) at *σ* = 0.72 valid at (a) 0000 UTC 24 Jan 2000 and (b) 1200 UTC 23 Jan 2000. Line G–H in (b) denotes orientation of cross section for Figs. 15 and 16.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Thirty-six-hour temperature perturbation (interval 1 K) on *σ* = 0.72 valid at 1200 UTC 25 Jan 2000 calculated from the (a) difference between two nonlinear forecasts that included moist physics, (b) TLM integrated about a moist basic state, (c) difference between two nonlinear forecasts that excluded moist physics, and (d) TLM integrated about a dry basic state. For (a)–(d) the negative values are dashed, and the zero line has been omitted.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Thirty-six-hour temperature perturbation (interval 1 K) on *σ* = 0.72 valid at 1200 UTC 25 Jan 2000 calculated from the (a) difference between two nonlinear forecasts that included moist physics, (b) TLM integrated about a moist basic state, (c) difference between two nonlinear forecasts that excluded moist physics, and (d) TLM integrated about a dry basic state. For (a)–(d) the negative values are dashed, and the zero line has been omitted.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Thirty-six-hour temperature perturbation (interval 1 K) on *σ* = 0.72 valid at 1200 UTC 25 Jan 2000 calculated from the (a) difference between two nonlinear forecasts that included moist physics, (b) TLM integrated about a moist basic state, (c) difference between two nonlinear forecasts that excluded moist physics, and (d) TLM integrated about a dry basic state. For (a)–(d) the negative values are dashed, and the zero line has been omitted.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Boxes denote geographic regions isolated for defining (a) *R*_{1}, (b) *R*_{2}, (c) *R*_{3}, and (d) *R*_{4}. Also plotted are (a) 36-h MSLP forecast error (interval 2 hPa, negative values dashed, zero contour omitted), (b) 36-h forecast of winds at *σ* = 0.97 (barbs, m s^{−1}), (c) 36-h forecast of 750-hPa frontogenesis [1°C (1000 km 3 h)^{−1}, negative values dashed, zero contour omitted], and (d) 36-h forecast of vertical velocity at *σ* = 0.66 (interval 5 cm s^{−1}, negative values dashed, zero contour omitted).

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Boxes denote geographic regions isolated for defining (a) *R*_{1}, (b) *R*_{2}, (c) *R*_{3}, and (d) *R*_{4}. Also plotted are (a) 36-h MSLP forecast error (interval 2 hPa, negative values dashed, zero contour omitted), (b) 36-h forecast of winds at *σ* = 0.97 (barbs, m s^{−1}), (c) 36-h forecast of 750-hPa frontogenesis [1°C (1000 km 3 h)^{−1}, negative values dashed, zero contour omitted], and (d) 36-h forecast of vertical velocity at *σ* = 0.66 (interval 5 cm s^{−1}, negative values dashed, zero contour omitted).

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Boxes denote geographic regions isolated for defining (a) *R*_{1}, (b) *R*_{2}, (c) *R*_{3}, and (d) *R*_{4}. Also plotted are (a) 36-h MSLP forecast error (interval 2 hPa, negative values dashed, zero contour omitted), (b) 36-h forecast of winds at *σ* = 0.97 (barbs, m s^{−1}), (c) 36-h forecast of 750-hPa frontogenesis [1°C (1000 km 3 h)^{−1}, negative values dashed, zero contour omitted], and (d) 36-h forecast of vertical velocity at *σ* = 0.66 (interval 5 cm s^{−1}, negative values dashed, zero contour omitted).

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Forecast sensitivity with respect to the 00-h forecast of 700-hPa temperature valid at 0000 UTC 24 Jan 2000 for (a) −*R*_{1} (interval 3 J kg^{−1} K^{−1}), (b) *R*_{2} (interval 3 × 10^{3} s^{−1} K^{−1}), (c) *R*_{3} (interval 1 × 10^{−15} m^{−1} s^{−1}), and (d) *R*_{4} (interval 7 × 10^{−6} m s^{−1} K^{−1}). All sensitivity gradients are contoured with the darker line type, with negative values dashed and zero contour omitted. Also plotted in (a)–(d) is the analysis of temperature valid at 0000 UTC 24 Jan 2000 (light dashed, interval 3°C); line I–J denotes orientation of cross section for Fig. 11.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Forecast sensitivity with respect to the 00-h forecast of 700-hPa temperature valid at 0000 UTC 24 Jan 2000 for (a) −*R*_{1} (interval 3 J kg^{−1} K^{−1}), (b) *R*_{2} (interval 3 × 10^{3} s^{−1} K^{−1}), (c) *R*_{3} (interval 1 × 10^{−15} m^{−1} s^{−1}), and (d) *R*_{4} (interval 7 × 10^{−6} m s^{−1} K^{−1}). All sensitivity gradients are contoured with the darker line type, with negative values dashed and zero contour omitted. Also plotted in (a)–(d) is the analysis of temperature valid at 0000 UTC 24 Jan 2000 (light dashed, interval 3°C); line I–J denotes orientation of cross section for Fig. 11.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Forecast sensitivity with respect to the 00-h forecast of 700-hPa temperature valid at 0000 UTC 24 Jan 2000 for (a) −*R*_{1} (interval 3 J kg^{−1} K^{−1}), (b) *R*_{2} (interval 3 × 10^{3} s^{−1} K^{−1}), (c) *R*_{3} (interval 1 × 10^{−15} m^{−1} s^{−1}), and (d) *R*_{4} (interval 7 × 10^{−6} m s^{−1} K^{−1}). All sensitivity gradients are contoured with the darker line type, with negative values dashed and zero contour omitted. Also plotted in (a)–(d) is the analysis of temperature valid at 0000 UTC 24 Jan 2000 (light dashed, interval 3°C); line I–J denotes orientation of cross section for Fig. 11.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Vertical cross section of the sensitivity with respect to the 00-h forecast of temperature valid at 0000 UTC 24 Jan 2000 for (a) −*R*_{1} (interval 2 J kg^{−1} K^{−1}), (b) *R*_{2} (interval 1.5 × 10^{3} s^{−1} K^{−1}), (c) *R*_{3} (interval 8 × 10^{−16} m^{−1} s^{−1}), and (d) *R*_{4} (interval 7 × 10^{−6} m s^{−1} K^{−1}). All sensitivity gradients are contoured with the darker line type, with negative values dashed and zero contour omitted. Also plotted in (a)–(d) is the analysis of potential temperature valid at 0000 UTC 24 Jan 2000 (light dashed, interval 3 K); cross-section orientation is denoted by the line I–J in Fig. 10.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Vertical cross section of the sensitivity with respect to the 00-h forecast of temperature valid at 0000 UTC 24 Jan 2000 for (a) −*R*_{1} (interval 2 J kg^{−1} K^{−1}), (b) *R*_{2} (interval 1.5 × 10^{3} s^{−1} K^{−1}), (c) *R*_{3} (interval 8 × 10^{−16} m^{−1} s^{−1}), and (d) *R*_{4} (interval 7 × 10^{−6} m s^{−1} K^{−1}). All sensitivity gradients are contoured with the darker line type, with negative values dashed and zero contour omitted. Also plotted in (a)–(d) is the analysis of potential temperature valid at 0000 UTC 24 Jan 2000 (light dashed, interval 3 K); cross-section orientation is denoted by the line I–J in Fig. 10.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Vertical cross section of the sensitivity with respect to the 00-h forecast of temperature valid at 0000 UTC 24 Jan 2000 for (a) −*R*_{1} (interval 2 J kg^{−1} K^{−1}), (b) *R*_{2} (interval 1.5 × 10^{3} s^{−1} K^{−1}), (c) *R*_{3} (interval 8 × 10^{−16} m^{−1} s^{−1}), and (d) *R*_{4} (interval 7 × 10^{−6} m s^{−1} K^{−1}). All sensitivity gradients are contoured with the darker line type, with negative values dashed and zero contour omitted. Also plotted in (a)–(d) is the analysis of potential temperature valid at 0000 UTC 24 Jan 2000 (light dashed, interval 3 K); cross-section orientation is denoted by the line I–J in Fig. 10.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Sensitivity of *R*_{2} with respect to the distribution of 500-hPa relative vorticity (contours, interval 2 × 10^{7} s^{−1}, negative values dashed, zero contour omitted), as well as the distribution of 500-hPa relative vorticity (color filled, interval 2 × 10^{−5} s^{−1}) valid for the (a) 12-, (b) 18-, (c) 24-, and (d) 30-h forecast initialized at 0000 UTC 24 Jan 2000.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Sensitivity of *R*_{2} with respect to the distribution of 500-hPa relative vorticity (contours, interval 2 × 10^{7} s^{−1}, negative values dashed, zero contour omitted), as well as the distribution of 500-hPa relative vorticity (color filled, interval 2 × 10^{−5} s^{−1}) valid for the (a) 12-, (b) 18-, (c) 24-, and (d) 30-h forecast initialized at 0000 UTC 24 Jan 2000.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Sensitivity of *R*_{2} with respect to the distribution of 500-hPa relative vorticity (contours, interval 2 × 10^{7} s^{−1}, negative values dashed, zero contour omitted), as well as the distribution of 500-hPa relative vorticity (color filled, interval 2 × 10^{−5} s^{−1}) valid for the (a) 12-, (b) 18-, (c) 24-, and (d) 30-h forecast initialized at 0000 UTC 24 Jan 2000.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Sensitivity of *R*_{3} with respect to the distribution of 700-hPa temperature (dark contours, interval 1.5 × 10^{−15} m^{−1} s^{−1}, negative values dashed, zero contour omitted), sensitivity vectors (K m^{−2} s^{−2}; reference vector in lower left), as well as 700-hPa temperature (light dashed, interval 2°C) for (a) 18- and (b) 24-h forecast initialized at 0000 UTC 24 Jan 2000.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Sensitivity of *R*_{3} with respect to the distribution of 700-hPa temperature (dark contours, interval 1.5 × 10^{−15} m^{−1} s^{−1}, negative values dashed, zero contour omitted), sensitivity vectors (K m^{−2} s^{−2}; reference vector in lower left), as well as 700-hPa temperature (light dashed, interval 2°C) for (a) 18- and (b) 24-h forecast initialized at 0000 UTC 24 Jan 2000.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Sensitivity of *R*_{3} with respect to the distribution of 700-hPa temperature (dark contours, interval 1.5 × 10^{−15} m^{−1} s^{−1}, negative values dashed, zero contour omitted), sensitivity vectors (K m^{−2} s^{−2}; reference vector in lower left), as well as 700-hPa temperature (light dashed, interval 2°C) for (a) 18- and (b) 24-h forecast initialized at 0000 UTC 24 Jan 2000.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Forecast sensitivity with respect to the 00-h forecast of 500-hPa relative vorticity valid at 1200 UTC 23 Jan 2000 for (a) –*R*_{1} (interval 5 × 10^{4} J kg^{−1} s^{−1}), (b) *R*_{2} (interval 1 × 10^{8} s^{−1} s^{−1}), (c) *R*_{3} (interval 5 × 10^{−12} m^{−1} s^{−2}), and (d) *R*_{4} (interval 1 × 10^{−1} m s^{−2}). All sensitivity gradients are contoured with the darker line type, with negative values dashed and zero contour omitted. Also plotted in (a)–(d) is the analysis of geopotential height valid at 1200 UTC 23 Jan 2000 (light dashed, interval 60 m).

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Forecast sensitivity with respect to the 00-h forecast of 500-hPa relative vorticity valid at 1200 UTC 23 Jan 2000 for (a) –*R*_{1} (interval 5 × 10^{4} J kg^{−1} s^{−1}), (b) *R*_{2} (interval 1 × 10^{8} s^{−1} s^{−1}), (c) *R*_{3} (interval 5 × 10^{−12} m^{−1} s^{−2}), and (d) *R*_{4} (interval 1 × 10^{−1} m s^{−2}). All sensitivity gradients are contoured with the darker line type, with negative values dashed and zero contour omitted. Also plotted in (a)–(d) is the analysis of geopotential height valid at 1200 UTC 23 Jan 2000 (light dashed, interval 60 m).

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Forecast sensitivity with respect to the 00-h forecast of 500-hPa relative vorticity valid at 1200 UTC 23 Jan 2000 for (a) –*R*_{1} (interval 5 × 10^{4} J kg^{−1} s^{−1}), (b) *R*_{2} (interval 1 × 10^{8} s^{−1} s^{−1}), (c) *R*_{3} (interval 5 × 10^{−12} m^{−1} s^{−2}), and (d) *R*_{4} (interval 1 × 10^{−1} m s^{−2}). All sensitivity gradients are contoured with the darker line type, with negative values dashed and zero contour omitted. Also plotted in (a)–(d) is the analysis of geopotential height valid at 1200 UTC 23 Jan 2000 (light dashed, interval 60 m).

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Cross section of initial (a) temperature perturbation (shaded, interval 0.15 K) and PV perturbation (contours, interval 0.05 PVU), as well as (b) meridional wind perturbation (shaded, interval 0.1 m s^{−1}) and PV perturbation (contours, interval 0.05 PVU), valid at 1200 UTC 23 Jan 2000. For the contoured variables, the negative values are dashed and the zero line has been omitted. Cross-section orientation is shown by line G–H in Fig. 7b.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Cross section of initial (a) temperature perturbation (shaded, interval 0.15 K) and PV perturbation (contours, interval 0.05 PVU), as well as (b) meridional wind perturbation (shaded, interval 0.1 m s^{−1}) and PV perturbation (contours, interval 0.05 PVU), valid at 1200 UTC 23 Jan 2000. For the contoured variables, the negative values are dashed and the zero line has been omitted. Cross-section orientation is shown by line G–H in Fig. 7b.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Cross section of initial (a) temperature perturbation (shaded, interval 0.15 K) and PV perturbation (contours, interval 0.05 PVU), as well as (b) meridional wind perturbation (shaded, interval 0.1 m s^{−1}) and PV perturbation (contours, interval 0.05 PVU), valid at 1200 UTC 23 Jan 2000. For the contoured variables, the negative values are dashed and the zero line has been omitted. Cross-section orientation is shown by line G–H in Fig. 7b.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Evolved temperature perturbation (shaded, interval 0.5 K) for (a) 6- and (b) 12-h forecasts initialized at 1200 UTC 23 Jan 2000. Also plotted is the evolved PV perturbation valid for the (a) 6-h (contours, interval 0.15 PVU) and (b) 12-h (contours, interval 0.15 PVU) forecasts initialized at 1200 UTC 23 Jan 2000. For the PV perturbations, the negative values are dashed and the zero contour has been omitted. Cross-section orientation is shown by line G–H in Fig. 7b.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Evolved temperature perturbation (shaded, interval 0.5 K) for (a) 6- and (b) 12-h forecasts initialized at 1200 UTC 23 Jan 2000. Also plotted is the evolved PV perturbation valid for the (a) 6-h (contours, interval 0.15 PVU) and (b) 12-h (contours, interval 0.15 PVU) forecasts initialized at 1200 UTC 23 Jan 2000. For the PV perturbations, the negative values are dashed and the zero contour has been omitted. Cross-section orientation is shown by line G–H in Fig. 7b.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Evolved temperature perturbation (shaded, interval 0.5 K) for (a) 6- and (b) 12-h forecasts initialized at 1200 UTC 23 Jan 2000. Also plotted is the evolved PV perturbation valid for the (a) 6-h (contours, interval 0.15 PVU) and (b) 12-h (contours, interval 0.15 PVU) forecasts initialized at 1200 UTC 23 Jan 2000. For the PV perturbations, the negative values are dashed and the zero contour has been omitted. Cross-section orientation is shown by line G–H in Fig. 7b.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

As in Fig. 5, except for the 48-h forecast initialized at 1200 UTC 23 Jan 2000 using the perturbed analysis. Line K–L in (b) denotes orientation of cross section for (c).

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

As in Fig. 5, except for the 48-h forecast initialized at 1200 UTC 23 Jan 2000 using the perturbed analysis. Line K–L in (b) denotes orientation of cross section for (c).

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

As in Fig. 5, except for the 48-h forecast initialized at 1200 UTC 23 Jan 2000 using the perturbed analysis. Line K–L in (b) denotes orientation of cross section for (c).

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

MSLP (black contours, interval 4 hPa) and 700–950-hPa-layer-averaged relative humidity (green contours, interval 15% starting at 60%) and vertical velocity omega (color filled, interval 2 × 10^{−1} Pa s^{−1}, legend at bottom) for control and perturbed forecasts at (a), (b) 24-, (c), (d) 30-, and (e), (f) 36-h initialized at 1200 UTC 23 Jan 2000.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

MSLP (black contours, interval 4 hPa) and 700–950-hPa-layer-averaged relative humidity (green contours, interval 15% starting at 60%) and vertical velocity omega (color filled, interval 2 × 10^{−1} Pa s^{−1}, legend at bottom) for control and perturbed forecasts at (a), (b) 24-, (c), (d) 30-, and (e), (f) 36-h initialized at 1200 UTC 23 Jan 2000.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

MSLP (black contours, interval 4 hPa) and 700–950-hPa-layer-averaged relative humidity (green contours, interval 15% starting at 60%) and vertical velocity omega (color filled, interval 2 × 10^{−1} Pa s^{−1}, legend at bottom) for control and perturbed forecasts at (a), (b) 24-, (c), (d) 30-, and (e), (f) 36-h initialized at 1200 UTC 23 Jan 2000.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

(a) Forecasted accumulated precipitation (interval 10 mm) for final 24-h period of the (a) control and (b) perturbed 48-h forecasts valid at 1200 UTC 25 Jan 2000.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

(a) Forecasted accumulated precipitation (interval 10 mm) for final 24-h period of the (a) control and (b) perturbed 48-h forecasts valid at 1200 UTC 25 Jan 2000.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

(a) Forecasted accumulated precipitation (interval 10 mm) for final 24-h period of the (a) control and (b) perturbed 48-h forecasts valid at 1200 UTC 25 Jan 2000.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Difference between 24-h perturbed and control forecasts of (a) 900-hPa horizontal divergence (interval 2 × 10^{−5} s^{−1}), (b) precipitable water (interval 2 mm), and (c) 800–900-hPa-layer potential vorticity (interval 0.3 PVU) valid at 1200 UTC 24 Jan 2000.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Difference between 24-h perturbed and control forecasts of (a) 900-hPa horizontal divergence (interval 2 × 10^{−5} s^{−1}), (b) precipitable water (interval 2 mm), and (c) 800–900-hPa-layer potential vorticity (interval 0.3 PVU) valid at 1200 UTC 24 Jan 2000.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

Difference between 24-h perturbed and control forecasts of (a) 900-hPa horizontal divergence (interval 2 × 10^{−5} s^{−1}), (b) precipitable water (interval 2 mm), and (c) 800–900-hPa-layer potential vorticity (interval 0.3 PVU) valid at 1200 UTC 24 Jan 2000.

Citation: Monthly Weather Review 133, 11; 10.1175/MWR3023.1

^{1}

Strictly speaking, the adjoint is defined for the TLM (the nonlinear model linearized about a model forecast trajectory). For brevity, however, we will refer to the adjoint as the adjoint of the NWP model.

^{3}

In this work, the elements of **C** are defined as described in Zou et al. [1997, Eq. (2.124)].

^{4}

All sensitivity gradients for *R*_{1} have been multiplied by −1, for a direct comparison with the sensitivity gradients for the other response functions, as a decrease in the value of *R*_{1} should be associated with increases in the values of the other three response functions.

^{5}

Details of this calculation may be found in Kleist and Morgan (2005).

^{6}

A motivation for and interpretation of sensitivity vectors is described in Kleist and Morgan (2005).