In this study, the structures and growth rates of singular vectors (SVs) for Typhoon Usagi were investigated using different moist physics and norms. The fifth-generation Pennsylvania State University–National Center for Atmospheric Research Mesoscale Model (MM5) and its tangent linear and adjoint models with a Lanczos algorithm were used to calculate SVs over a 36-h period. The moist physics used for linear (i.e., tangent linear and adjoint model) integrations is large-scale precipitation, and the norms used are dry and moist total energy (TE) norms. Overall, moist physics in linear integrations and a moist TE norm increase the growth rates of SVs and cause smaller horizontal structures and vertical distributions closer to the lower boundary. With a dry TE norm, the SV energy distributions show similar (dissimilar) large- (small-) scale horizontal SV structures for experiments, regardless of physics. The SVs with moist linear physics and a moist TE norm have maximum horizontal energy structures near the typhoon center. With a small weighting on the moisture term in the moist TE norm, both the remote and nearby influences on the TC are indicated by the horizontal SV energy distributions. The kinetic energy shows the largest contributions to the vertical SV TE distributions in most of the experiments, except for the largest moisture (potential energy) contributions to the SV TE at the final (initial) time in the moist TE norm (dry and weighted moist TE norms at uppermost levels). In contrast, the SV vorticity distributions show more consistent structures among experiments with different linear physics and norms, implying that, in terms of the rotational component of the wind field, the SVs are not sensitive to the choice of moist physics and norms. Given large-scale precipitation as the linear moist physics, the SV energy structures and growth rate with a moist TE norm show the largest difference when compared with those with other norms.
Adaptive (or targeted) observation strategies have been applied to high-impact weather events to identify regions where additional observations have the potential to significantly improve weather forecasts. These regions may be considered “sensitive” in the sense that changes to the initial conditions in these regions are expected to have a larger effect on a particular measure of forecast skill than changes in other regions (Kim et al. 2004). Singular vectors (SVs) are the fastest-growing perturbations during a specified time period (i.e., the optimization interval) for a given norm and basic state (Kim and Morgan 2002). SVs have been used to detect regions of large sensitivity to small perturbations for the purpose of making adaptive observations (e.g., Palmer et al. 1998), and are categorized as a dynamics-based adaptive observation strategy because SVs are calculated based on the dynamics information of the flow (Kim et al. 2004).
The sensitive regions indicated by the SVs depend on the norm chosen; lower to midtroposphere for the energy and streamfunction variance norm (e.g., Mukougawa and Ikeda 1994; Buizza and Palmer 1995; Hartmann et al. 1995; Hoskins et al. 2000; Morgan 2001; Kim and Morgan 2002), upper and lower boundaries for the potential enstrophy norm (e.g., Kim and Morgan 2002), and near tropopause for the analysis error covariance metric (AECM; e.g., Barkmeijer et al. 1998; Hamill et al. 2003). The AECM is the most appropriate norm at the initial time to calculate SVs for adaptive observations because in a tangent linear framework AECM SVs evolve into the eigenvectors of the forecast error covariance matrix at a later time (e.g., Ehrendorfer and Tribbia 1997). The AECM SVs have been used to construct initial perturbations for ensemble forecasts (e.g., Barkmeijer et al. 1998, 1999; Hamill et al. 2003) and used for adaptive observations (e.g., Petersen and Thorpe 2007; Rabier et al. 2008). SVs with a total energy (TE) norm that is the first-order approximation to the AECM (Palmer et al. 1998) have also been tested for adaptive observations (e.g., Bergot 1999; Bergot et al. 1999; Buizza and Montani 1999; Gelaro et al. 1999; Langland et al. 1999; Montani et al. 1999; Peng and Reynolds 2005, 2006; Majumdar et al. 2006).
By using moist diabatic physics for nonlinear model (NLM), tangent linear model (TLM), and adjoint model integrations, the influences of moist diabatic processes and corresponding moist norms on SVs have been studied for extratropical cyclones (e.g., Ehrendorfer et al. 1999; Coutinho et al. 2004; Hoskins and Coutinho 2005) and tropical cyclones (TCs) (Barkmeijer et al. 2001). Ehrendorfer et al. (1999) demonstrated that the SVs calculated with moist physics show different structures from, and faster growth than, SVs calculated with dry adiabatic physics, and therefore inclusion of moist physics and the use of an appropriate norm is necessary to reveal all the structures that might grow rapidly in a moist primitive equation model. Coutinho et al. (2004) confirmed that the inclusion of moist physics in basic-state, TLM, and adjoint integrations makes the horizontal scales of SVs smaller and enhances the growth of SVs. Barkmeijer et al. (2001) also concluded that moist diabatic processes and a moist TE norm that includes the specific humidity are necessary to reveal the essential SV features in a tropical environment, and further showed that SVs using moist physics are useful for targeting TCs. These targeted diabatic SVs were used for generating initial perturbations in the European Centre for Medium-Range Weather Forecasts (ECMWF) ensemble prediction system for TC prediction (Puri et al. 2001). Peng and Reynolds (2006) showed that inclusion of moist physics (i.e., large-scale precipitation) in the SV calculation yields similar patterns to those obtained from dry SVs, but moist SVs have relatively stronger sensitivities in the vicinity of the TC.
Nevertheless, SVs with moist physics and norm have not been fully compared with SVs with dry physics and norm for TCs. Moreover, because SVs with moist physics tend to have smaller structures than SVs with dry physics (Ehrendorfer et al. 1999) and tend to reside near the TC (Barkmeijer et al. 2001), the sensitive regions detected by SVs with moist physics and norm may be different from those detected by SVs with dry physics and norm. Because of the remote environmental effects (e.g., Peng and Reynolds 2005, 2006; Majumdar et al. 2006), the SVs with dry physics and norm usually indicate the sensitive regions in remote locations from the TC center (e.g., midlatitude trough), as well as in the vicinity of the TC center.
In this study, the influence of moist diabatic physics and the choice of norm for the SVs of a TC is investigated, and SVs with moist diabatic physics and norms are compared with SVs with dry physics and norm. Section 2 describes the methodology and experimental framework, section 3 contains a case description, and the results of the moist physics and norm effects on SVs are presented in section 4. Section 5 includes a linearity evaluation, and section 6 contains a summary and discussion.
2. Methodology and experimental framework
a. Singular vectors
The calculation of SVs essentially involves selecting an initial disturbance, with the constraint that the initial disturbance has a unit amplitude in a specified norm and evolves to have maximum amplitude in a specified norm after some finite optimization time, t = τopt.1 In this study, we choose the initial and final norms to be the same. We define the amplitude of the state vector2 (x′) in the specific norm 𝗖:
where the inner product is denoted by 〈,〉, 𝗠 is the TLM of the NLM, and 𝗖 is the matrix operator appropriate to the specific norm. In (1), the state vector at the initial time is assumed to evolve linearly. The constrained optimization problem seeks to maximize the Rayleigh quotient λ2 (the amplification factor):
at time t = τopt, where 𝗣 is a local projection operator that defines the state vector as zero outside a given domain3 (Buizza 1994). By defining the local projection operator, the amplitude of the state vector with norm 𝗖 at the optimization time is maximized over a specific region. The maximum of this ratio is realized when x′(0) is the leading SV of TLM 𝗠 for the 𝗖 norm, that is x′(0) satisfies
The generalized eigenvalue problem in (3) can be reduced to an ordinary eigenvalue problem by multiplying both sides of (3) with the inverse of the square root of 𝗖. Then a Lanczos-type algorithm (e.g., Ehrendorfer and Errico 1995; Kim 2003; Kim et al. 2004) is used to solve for x′(0) in (3).
To investigate the effect of different norms on the growth and structure of SVs, dry and moist TE norms were used. The dry TE is defined according to Zou et al. (1997) as
where Ed is the dry TE in a nonhydrostatic model; u′, υ′, and w′ are the zonal, meridional, and vertical wind perturbations, respectively; θ′ is the potential temperature perturbation; p′ is the pressure perturbation; N, θ, ρ, and cs are Brunt–Väisälä frequency, potential temperature, density, and speed of sound, respectively, at the reference level; and x, y, and σ denote zonal, meridional, and vertical coordinate, respectively.
where Em is the moist TE in a nonhydrostatic model; and L, cp, Tr, and q′ are latent heat of condensation per unit mass, specific heat at constant pressure, reference temperature, and the perturbed mixing ratio, respectively. According to Ehrendorfer et al. (1999), the parameter ωq is added to the moisture term to control the relative contributions of each component to the moist TE norm. Three tests were performed to compare the norm effects on SV amplitude and structures: dry TE norm, moist TE norm without weighting on the moisture term in (5) (i.e., ωq = 1), and moist TE norm with weighting (i.e., ωq = 0.1). The reason for setting ωq = 0.1 is explained in section 2c.
Because the vertical wind perturbation can be calculated explicitly in the nonhydrostatic model, (4) and (5) have the vertical wind perturbation in the TE norm. Other studies of TE SVs (e.g., Ehrendorfer et al. 1999; Barkmeijer et al. 2001; Peng and Reynolds 2005, 2006; Majumdar et al. 2006) do not have an explicit vertical wind perturbation term in the TE norm because these studies used a hydrostatic model.
b. Model and physical processes
To calculate SVs, this study used the fifth-generation Pennsylvania State University–National Center for Atmospheric Research Mesoscale Model (MM5), together with the MM5 tangent linear and adjoint modeling system (Zou et al. 1997) and a Lanczos algorithm. The model domain for this study is a 50 × 50 horizontal grid (centered at 33°N, 133°E), with a 100-km horizontal resolution and 14 evenly spaced sigma levels in the vertical from the surface to 50 hPa. The model’s initial and lateral boundary conditions are taken from the National Centers for Environmental Prediction (NCEP) final analysis (FNL; 1° × 1° global grid). Physical parameterizations used in the simulation include the Grell convective scheme, a bulk aerodynamic formulation of the planetary boundary layer, the simple cooling radiation scheme, horizontal and vertical diffusion, dry convective adjustment, large-scale precipitation, and explicit treatment of cloud water, rain, snow, and ice. Physical parameterizations of NLM, TLM, and adjoint model integrations4 in the specific simulations are shown in Table 1. The same parameterizations were used in the TLM and adjoint model integrations for each experiment. EXP1 (EXP2) was designed to calculate SVs without (with) linear moist physics on the full physics basic state. For EXP1, the dry TE norm was used to calculate SVs. For EXP2, three norms were tested and denoted as EXP2_d for dry TE norm, EXP2_m for moist TE norm with ωq = 1, and EXP2_mw for moist TE norm with ωq = 0.1. EXP2_d was designed to investigate the only effect of the moist physics with the same norm as EXP1. EXP2_m was designed to see the effect of the moist physics on SVs measured in a moist TE norm. EXP2_mw was designed to investigate the relative effect of the dry and moist components in the moist TE norm.
c. Weights of the moisture term in the moist TE norm
The weight of the moisture term in (5) is determined from a relative ratio between analysis uncertainties (e.g., analysis error variances) of wind and moisture. Instead of using exact AECM or analysis error variance norms (e.g., Barkmeijer et al. 1998; Gelaro et al. 2002; Errico et al. 2003; Errico et al. 2004; Reynolds et al. 2007) to calculate SVs, a relative weight between wind and moisture terms in (5) was approximated from the analysis uncertainties because the norm effects on SV growth and structures are investigated in the frame of TE norm in this study. The analysis uncertainties of wind and moisture variables were determined from every 6-h difference between NCEP FNL and NCEP–Department of Energy (DOE) reanalysis data (Kanamitsu et al. 2002), for the 36-h period over the model domain. As in Errico et al. (2004), the NCEP FNL and NCEP–DOE analyses were considered to be random realizations of uncertain analyses.
Figure 1a shows the vertical distributions of the standard deviations of the analysis uncertainties for winds and specific humidity. While the standard deviation of the analysis uncertainties of specific humidity has a maximum at the lower troposphere and decreases with height above the maximum level, the standard deviations of the analysis uncertainties of zonal and meridional winds have maxima at the upper troposphere, similar to Errico et al. (2004). Based on values in Fig. 1a, the ratio between the zonal wind and moisture components in (5) is calculated for the lower troposphere below 550 hPa (Fig. 1b). Even though the ratio varies with respect to vertical model levels, the average ratio throughout the lower levels is around 0.1, which implies that the moisture component is around 10 times larger than the zonal wind component in the lower troposphere. In contrast, the moisture component is small in the upper troposphere because the source of moisture is close to the surface (not shown).
Based on Fig. 1, the moisture component is kept similar in amplitude to the wind component in the lower troposphere below 550 hPa (i.e., ωq = 0.1) and is set to nil (i.e., ωq = 0) in the upper troposphere above 550 hPa for EXP2_mw. These constraints on moisture in the vertical are used in EXP2_mw to investigate the moisture effect in the moist TE norm on SV growth and structure. With ωq = 0.1, the moisture contribution to the moist TE norm for EXP2_mw is 1.32% at the initial time and 11.07% (11.69%) at the final time for the whole model domain (verification region), as shown in Table 2. In contrast, the experiment with a moist TE norm without weighting (EXP2_m) shows large contributions from the moisture term to the moist TE norm at the final time (Table 2). By applying ωq = 0.1 to the moisture term, the moisture contributions to the moist TE norm are smaller than the contributions of the remaining terms at the initial and final times, similar to Ehrendorfer et al. (1999).
3. Case description
The best track of Typhoon Usagi from the Regional Specialized Meteorological Center (RSMC) Tokyo Typhoon Center and the model-predicted track of the experiments are shown in Fig. 2. Typhoon Usagi, the fifth typhoon in 2007, moved westward after its formation northeast of Guam at 0600 UTC 29 July 2007, then moved northwestward and intensified after 0000 UTC 30 July. Between 0000 UTC 1 and 0000 UTC 2 August, the central pressure of Usagi reached its minimum of 945 hPa. Usagi was located to the southeast of the midlatitude trough and southwest of the subtropical high from 0000 to 1200 UTC 1 August (Figs. 3a,b). At 0000 UTC 2 August, Usagi moved up to 30°N and approached Kyushu Island (Fig. 3c). Usagi made landfall on Kyushu Island between 0600 and 1200 UTC 2 August. After 1200 UTC 2 August, Usagi recurved northeastward along the edge of a subtropical high located to the east of Usagi (Fig. 3d), then finally experienced an extratropical transition (Fig. 3e). Numerical experiments were performed during the 36 h from 0000 UTC 1 August, when the intensity of Usagi reached its maximum, to 1200 UTC 2 August, when Usagi weakened upon landing on Kyushu Island.
4. Moist physics and norm effects on singular vectors
a. Amplification factors
The amplification factors [i.e., singular values in (3)] of 50 SVs from each experiment are shown in Fig. 4a. Among the experiments, EXP1 shows the smallest amplification factors. By adding moist physics for linear integrations (EXP2), the amplification factors of SVs increase. The growth rates are in increasing order from EXP1, EXP2_d, EXP2_mw to EXP2_m. The amplification factors of the SVs for EXP2_m are much larger than those for other experiments using moist linear physics (i.e., EXP2_d and EXP2_mw), implying that the moist TE norm significantly increases the growth rate of SVs. Therefore, the effect of using moist linear physics on amplification factors is largest when the moist TE norm is used to calculate the SVs.
The ratios of amplification factors between the second (fourth) SV and first SV are 0.468 (0.292), 0.575 (0.267), 0.941 (0.686), and 0.563 (0.321) for EXP1, EXP2_d, EXP2_m, and EXP2_mw, respectively. In contrast with other spectra that have steeper slopes for the leading four to five SVs, the singular value spectrum of EXP2_m shows a rather even distribution (Fig. 4b), implying that a relatively large number of SVs may be necessary to explain the general structure and behavior of the SVs of EXP2_m. The even distribution of EXP2_m may be attributed to the many small structures associated with SVs computed with the moist linear physics and moist TE norm.
Similar to previous studies that have used linear moist physics (i.e., large-scale precipitation) on the full physics basic state (e.g., Ehrendorfer et al. 1999; Coutinho et al. 2004) for extratropical cyclones, moist physics enhance the growth of SVs. However, apparent increases in the SV growth rate, relative to the SV growth rate with dry linear physics and dry TE norm, are obtained when the moist TE norm is used to calculate SVs in combination with the moist physics in linear integrations.
b. Horizontal structure and evolution
1) SV energy distribution
The vertically integrated SV energy fields for EXP1 superimposed with the mean sea level pressure (MSLP) are shown in Fig. 5. The SV energy field is calculated to combine each component of the SV with a different unit into a single SV field with a unit of energy (J kg−1). For EXP1, Fig. 5a shows the leading SV at the initial time that results in the largest TE inside of the verification region in Fig. 5c. The major sensitive regions, indicated by the leading SV, are located from 500 to 1000 km south of Typhoon Usagi (Fig. 5a). In addition, there are minor sensitivities over the midlatitudes and the edge of the subtropical high. The leading SV at the final time of EXP1 is located in the verification region (Fig. 5c).
To investigate the properties of other SVs, the composite of the first to third SVs was calculated,5 and the formula of the vertically integrated energy composite of the first–third SVs is
where i and j denote grid points in the x and y directions, respectively; λ12 and λn2 are singular values for the first and nth SV, respectively; and Sijn denotes the nth vertically integrated SV energy field. While the vertically integrated energy distributions of the composite SV of EXP1 are overall similar to those of the leading SV shown in Fig. 5a, the major sensitive regions of the composite SV are over the midlatitudes, where the first SV has minor sensitivities (Fig. 5b). Individual horizontal structures of the second and third SV show that the large- (small-) scale6 sensitivity distributions are similar (different) for each SV (not shown), which implies that the small-scale SV structures have more variability than do the large-scale SV structures. The composite SV of EXP1 at the final time is also located in the verification region (Fig. 5d).
The vertically integrated energy distributions of the leading SV of EXP2 are shown in Fig. 6. In EXP2_d, the major sensitive regions indicated by the leading SV are located around 500 km to the southeast and northwest of TC center (Fig. 6a). Similar to EXP1, minor sensitivities over the midlatitudes and the edge of the subtropical high are also indicated. The evolved leading SV of EXP2_d shows very similar structures to those of EXP1 (Fig. 6d). With the moist TE norm (EXP2_m), the major sensitive regions over the northeast part of the TC center and the minor sensitive regions in the midlatitudes are indicated by the leading SV at the initial time (Fig. 6b), and some of the evolved leading SV structures are not located in the verification region (Fig. 6e). The initial and evolved leading SVs for EXP2_mw show very similar sensitivity distributions to those of EXP2_d (Figs. 6c,f). The vertically integrated energy distributions of composite SVs of EXP2 are shown in Fig. 7. While the large-scale SV distributions of the composite SV are very similar to those of the leading SV shown in Fig. 6, for EXP2_d and EXP2_mw (Figs. 7a,c), the major sensitive regions indicated by small-scale features of the composite SV are slightly different from those of the leading SV. For EXP2_mw, the magnitude of the composite (leading) SV is larger (smaller) in midlatitude than that near the TC, which implies that the second and third SVs have maximum magnitudes in the midlatitude.
Except for in EXP2_m, the large-scale features of SVs in EXP1 and EXP2 are similar. Thus, large-scale structures of the initial SVs do not change that much, regardless of the moist physics options for linear integrations, that is, whether the linear physics are dry or simple moist (e.g., large-scale precipitation). Small-scale features of SVs for EXP1 and EXP2 are different [e.g., large sensitivities in midlatitude trough (north to the TC center) in Fig. 5b (Figs. 7a,c)], implying that the small-scale features are controlled more by the moist physics. For EXP2_m, the effects of the midlatitude system are much less than those near the TC. In fact, the effect of the midlatitude systems on the SVs may be more clearly revealed by using either dry linear physics with dry TE norm (e.g., Fig. 5b) or moist linear physics with the weighted moist TE norm (e.g., Fig. 7c). As indicated in the amplification factors for each experiment, the dominant component to the horizontal SV energy distributions is the moist TE norm used to measure SVs with moist linear physics (i.e., large-scale precipitation).
2) SV vorticity and divergence distributions
To show the sensitivity to wind fields similar to Wu et al. (2007) and Kleist and Morgan (2005), vorticity and divergence fields of the TE SVs were vertically integrated with L2 norm and superimposed with MSLP for EXP1 (Fig. 8). In EXP1, the major sensitive regions indicated by the vorticity field of the leading SV are located from 500 to 1000 km south and southeast of TC center (Fig. 8a). The minor sensitivities are elongated from approximately 500 to 600 km north of the TC center to the midlatitudes (Fig. 8a). The evolved SV vorticity is located in the verification region and shows a dipole structure, with the larger magnitude on the right side of the TC center (Fig. 8c). The sensitive regions indicated by the divergence component of the leading SV are confined horizontally in small regions, and located around 700 km southeast and northwest of the TC center, with relatively small traces of sensitivities over the midlatitudes (Fig. 8b). The evolved SV divergence is mostly located in the small regions over the TC center (Fig. 8d). The relative magnitude of the SV divergence component is much less than the magnitude of the SV vorticity component.
In EXP2, the major and minor sensitive regions indicated by the vorticity components of the leading SV at the initial and final times are quite similar to those of EXP1, showing large sensitivities south to southeast and north to northeast of the TC center close to the edge of the subtropical ridge (Fig. 9). Compared to the TE distributions of the SVs in Fig. 6, in which the SV energy distribution of EXP2_m (Fig. 6b) is quite different from that of EXP2_d (Fig. 6a) and EXP2_mw (Fig. 6c), the vorticity distributions of the leading SVs (Figs. 9a–c) are not particularly sensitive to the choice of norms. This difference in the sensitivity patterns and magnitudes [cf. similar values of the SV vorticity fields in Figs. 9a–c and large differences in the TE values of EXP2_m (Fig. 6b) from EXP2_d (Fig. 6a) and EXP2_mw (Fig. 6c)] of the SVs to norms in the TE and vorticity distributions is mainly caused by the temperature component of the TE (not shown), which implies that the temperature (rotational wind) component of the TE has a large (small) variation depending on the choice of norms. The large variation in the temperature component of the TE for each experiment in EXP2 may be associated with the different realization of the moisture effect by the moist physics in different norms. The SV divergence distributions have a little variation in each experiment, but are in general similar to each other in EXP2 (Fig. 10). The relative magnitude of the SV divergence component is much less than the magnitude of the SV vorticity component, similar to in EXP1.
c. Total energy vertical profile
Vertical SV energy distributions at the initial and final times for EXP1 and EXP2 are shown in Fig. 11. In EXP1, while the maximum TE of the initial leading SV is located in the mid- to lower troposphere, with another peak in the upper boundary (Fig. 11a), the TE of the evolved leading SV is distributed throughout the troposphere (Fig. 11b). For the initial leading SV, the kinetic energy (KE) explains most of the TE throughout most of the troposphere, except in the upper troposphere (Fig. 11a). For the evolved leading SV, the KE dominates the potential energy (PE) throughout the troposphere (Fig. 11b). To avoid the shallow upper-tropospheric SVs discussed in Barkmeijer et al. (2001), the TE of the evolved SVs at the upper three levels was set to zero (Fig. 11b). The composite SV profiles for EXP1 at the initial and final times are shown in Figs. 11c,d. The initial composite SV has a maximum in the lower troposphere, implying that the second or third SVs may also have maxima in the lower troposphere. Except for that, the general characteristics of the composite SV are similar to those of the leading SV.
In EXP2, the maxima of the initial leading SVs for the dry and weighted moist TE norms are located in the mid- to upper troposphere, with another peak in the upper boundary (Figs. 11e,m). Maxima of the evolved leading SVs for dry and weighted moist TE norms are located in the lower troposphere with most of the moisture (Figs. 11f,n). The maxima of the composite SVs at the initial and final times for the dry TE norm show vertical distributions similar to those of the leading SV (cf. Figs. 11g,h with 11e,f). Relative to EXP1, EXP2_d revealed that the moist linear physics shift the initial time SV maximum from the lower to midtroposphere (Figs. 11a,c) to the mid- to upper troposphere (Figs. 11e,g). The initial SV maximum in the mid- to upper troposphere for EXP2_d (Figs. 11e,g) may be associated with large negative SV vertical wind perturbation (i.e., downward motion) and positive SV temperature perturbation (i.e., warming) in the mid- to upper troposphere near the wall of the TC center shown in Kim and Jung (2009). These large SVs in the vertical wind and temperature components in the mid- to upper troposphere are associated with the moist physics (i.e., large-scale precipitation) effects in the linear calculation, and likely lead to the increase in energy of the evolved SV at the final time.
With the moist TE norm, the maxima of the leading and composite SVs at the initial and final times are located in the lower troposphere (Figs. 11i–l). In contrast to the small contribution from the moisture to the TE of the leading and composite SVs at the initial time (Figs. 11i,k), the moisture components dominate the TE of the leading and composite SVs at the final time (Figs. 11j,l). Different from the initial SVs of EXP1, EXP2_d, and EXP2_mw, the initial SVs of EXP2_m have a considerably larger PE in the lower levels (Figs. 11i,k), which is consistent with the large contribution of the temperature component to different horizontal SV TE distributions at the initial time in Figs. 6 and 7. The maxima of the composite initial and evolved SVs for EXP2_mw are located in the lower troposphere, which implies that the second and third SVs have maxima in the lower troposphere (Figs. 11o,p). In contrast to the dominant moisture component in the final TE in EXP2_m, the PE and moisture component are considerable, with the largest KE in EXP2_mw (Fig. 11p).
Except for EXP2_m SVs at the final time and SVs for EXP1, EXP2_d, and EXP2_mw at the uppermost levels at the initial time, the KE shows the largest contributions to the TE of the SVs.
5. Linearity evaluation
For the linearization of moist diabatic processes, discontinuous on–off processes and conditionals are indicated as a main reason to degrade tangent linearity (Vukicevic and Errico 1993; Mahfouf 1999; Errico and Raeder 1999). Nevertheless, moist linearization is favorable for adjoint integrations because more realistic adjoint-based sensitivities (e.g., adjoint sensitivity or SVs) can be obtained using moist physics for linear integrations (Mahfouf 1999; Errico and Raeder 1999; Ehrendorfer et al. 1999; Coutinho et al. 2004).
To verify the linearity assumption on which the SV calculation is based, 36-h linear and nonlinear evolutions of SV perturbations were compared with specific measures (Table 3). The measures used were the ratio of linearly and nonlinearly evolved perturbation magnitude (Zou et al. 1997) and correlation. Instead of using infinitesimal perturbations, SVs (i.e., finite perturbations) were used for the linearity test in this study because the linearity of finite perturbations has more practical interests, as mentioned in Errico and Raeder (1999). The maximum amplitudes of the initial perturbations chosen for the nonlinear evolution were 4 m s−1 for u′ and υ′, 2 K for T ′, and 1 g kg−1 for q′, which are the typical sizes for analysis error as mentioned in Errico and Raeder (1999), Errico et al. (2004), and Mahfouf (1999). Similar to in Errico et al. (2004), once the SVs are calculated for each experiment, initial perturbations for nonlinear evolution are determined by applying a single scaling factor that brings all SV fields below the maximum amplitudes mentioned above.
For the first and third SVs, in terms of ratio, EXP2 shows a generally better agreement than EXP1 between the linearly and nonlinearly evolved wind and temperature components of SVs (Table 3). However, the better linearity of EXP2 relative to EXP1 does not hold for the second SV. The ratio of specific humidity is smaller than the ratio of winds for most of the experiments that utilize the moist TE norm (i.e., EXP2_m and EXP2_mw), which implies that the tangent linear growth of the moisture component is much larger than the nonlinear growth of the moisture component. The small ratio of the specific humidity for EXP2_m further suggests that the large singular values of EXP2_m in Fig. 4 are associated with this very large linear growth of the moisture component.
The linear and nonlinear time evolutions of the moist TE norm, the dry component, and the moist component for the leading SV of EXP2 (Fig. 12) were computed to further investigate the moisture effect in linearity issues. The same perturbations used for Table 3 were used as the initial perturbations for the nonlinear evolution. To compare the linear and nonlinear evolutions, the TE of the nonlinear evolutions was normalized to have amplitude of 1 at the initial time. For EXP2_m, the SV grows linearly by a factor of 10 in the first hour, subsequently grows by a factor of 100 in the subsequent 20 h, and then decreases slowly. While the moist component of TE dominates the dry TE after the first 6 h of the evolution, the dry TE of the SV shows rather smooth growth by a factor of 20 during the 36-h integration (Fig. 12a). The dominant moist component at 36 h is also shown in Fig. 11j. In contrast, the leading SV for EXP2_m, which grows nonlinearly by a factor of 60 during the 36 h (Fig. 12b), has a similar magnitude for the dry and moist TE at the final time (Fig. 12b). This implies that the SV perturbations imposed on the basic state at the initial time result in similar magnitudes of dry and moist TE at the final time. Rapid growth of the SV during the first hour is also noticed for the nonlinear evolution in EXP2_m (Fig. 12b). The large moist TE of the linearly evolved SV (Fig. 12a), when compared to that of nonlinearly evolved SV (Fig. 12b), is consistent with the small ratio of q in Table 3. In terms of nonlinear growth, SV perturbations of the typical analysis error size may result in energy partitioning between dry and moist energy, different from in Fig. 11j.
For EXP2_mw, the linear and nonlinear SV growth structures are similar overall, with growth rates of a factor of 55–65 during the 36 h (Figs. 12c,d). The dry TE components of the SV always dominate the moist components of the TE throughout the evolutions in EXP2_mw, indicating that the SV growth and evolution with this weighting are mostly controlled by the dry component of TE (Figs. 12c,d) as intended in determining the weighting factor. The weighted moist field has a smaller difference between the linear and nonlinear evolution, which is also consistent with a relatively larger ratio and coefficient of q for EXP2_mw than those for EXP2_m in Table 3. The linear and nonlinear time evolutions of the second and third SVs differs a lot (not shown), which is also consistent with the even smaller ratio and correlation for the second and third SVs than those for the first SV shown in Table 3. The rapid growth of the SV TE during the first hour in any case (Fig. 12) may be associated with the initial adjustment of unbalanced SV perturbations.
Compared to the ratio that shows large variations for the dry (i.e., u, υ, and T) and moist (i.e., q) components, the correlation between the linearly and nonlinearly evolved SVs shows relatively small variation between the dry and moist components, which implies that the degree of resemblance between linearly and nonlinearly evolved SVs may not differ much for dry and moist components. To compare SV structures, the linearly and nonlinearly evolved zonal wind component of the leading SVs at 500 hPa at the initial time for EXP1 and EXP2_m are compared and shown in Fig. 13. The EXP1 (EXP2_m) is shown because it represents an experiment using dry (moist) linear physics with the dry (moist) TE norm. The nonlinearly evolved leading SVs of EXP1 and EXP2_m are similar to the linearly evolved leading SVs, with similar maximum magnitudes and slight phase differences (Figs. 13b,c,e,f). The EXP2_d and EXP2_mw also show similar linearly and nonlinearly evolved leading SV structures (not shown). Despite the slight phase and magnitude differences in the major features of EXP1 and EXP2_m, the major evolved structures in the verification region at the final time are quite similar. For the leading SV, the winds and moisture (temperature) component of the SV at different levels show similar (slightly dissimilar) nonlinearly and linearly evolved structures in the verification region (not shown). For the second and third SVs, the pattern and magnitude of the nonlinearly and linearly evolved SV structures for most of the variables in the verification region are less similar to each other than those of the first SV (not shown), consistent with the smaller correlations for the second and third SVs shown in Table 3, which implies that the linearity is degraded for the second and third SVs.
6. Summary and discussion
In this study, the horizontal structures, vertical distributions, and growth rates of SVs for Typhoon Usagi were investigated using different moist physics and norms. The MM5, its tangent linear, and adjoint models with a Lanczos algorithm were used to calculate SVs, which maximize the tropospheric TE over a region including land and ocean near Korea and Japan during a 36-h period. Dry and moist diabatic physics (i.e., large-scale precipitation) in MM5 tangent linear and adjoint models, with the full physics nonlinear basic state, and dry and moist TE norms were used to investigate the effect of moist physics and norms on SVs of the TC. The linearity assumption associated with the SV calculation was also examined with the chosen norms and physics for basic state and linear integrations.
There are two difficulties in using moist physics in the context of finite-time optimal perturbations (i.e., SVs) (Ehrendorfer et al. 1999). One is determining an appropriate measure for the magnitude of the SVs and the other is evaluating the tangent linear approximation used to calculate the SVs. The moist TE norm is designed by combining the dry TE norm (Zou et al. 1997) and the moisture term in Ehrendorfer et al. (1999) for calculating SVs in the nonhydrostatic MM5. According to Ehrendorfer et al. (1999) and Errico et al. (2004), the weighting of the moisture term is used to investigate the relative effects of the dry and moist components in the moist TE norm.
Generally, moist physics in linear (i.e., TLM and adjoint) integrations and a moist TE norm increase the growth rate of SVs. The increase of the SV growth rate is much larger when using the moist TE norm than when using other norms (i.e., dry or weighted moist TE norm) or moist physics (i.e., large-scale precipitation) in linear integrations, which implies that the moist TE norm is a dominant component to the increase in SV growth rate. The SV energy distributions with moist linear physics and moist TE norm show horizontal structures with smaller scales when compared with the SVs with the dry and moist linear physics with the dry and weighted moist TE norm. Except for SVs with the moist TE norm, large-scale features of other SVs are similar. This implies that large-scale structures of the initial SVs do not depend much on the linear physics options if the linear physics are dry or simple moist (e.g., large-scale precipitation). On the other hand, small-scale features of SVs show large variation depending on the moist physics and norms. Similar (dissimilar) large (small) scale horizontal SV structures with dry or moist physics are consistent with the comparison results for different adaptive observation strategies in Majumdar et al. (2006). For SVs with moist linear physics and moist TE norm without weighting, the effects of the midlatitude system are much less than those of TC itself, which implies that the effect of the midlatitude systems on the SVs may not be revealed well enough for these moist physics and norm configurations. The dynamic interpretation of the large magnitude of SVs near the TC center, midlatitude system, and edge of the subtropical ridge are provided in a companion paper by Kim and Jung (2009).
As mentioned in Ehrendorfer et al. (1999), changing the weights of the moisture term in the moist TE norm leads to different sensitivity structures. With a small weighting on the moisture term in the moist TE norm, horizontal SV structures become similar to those of the dry TE norm. Both the remote and nearby influences on the TC are detected by the SVs (or possibly adjoint sensitivities) with moist linear physics and a weighted moist TE norm. As indicated in the amplification factors for each experiment, the dominant component in the horizontal SV energy distributions is the moist TE norm used to measure SVs with moist physics.
In contrast to the horizontal SV energy distributions, the SV vorticity distributions show more consistent structures among experiments, implying that, in terms of the rotational component of the wind field, the SVs are not sensitive to the choice of physics and norms. All the SV vorticity distributions denote large sensitivities in the southern and northern parts of the TC center, close to the edge of the subtropical ridge. The magnitude of the SV divergence component is much less than that of the SV vorticity component. Moist linear physics and moist TE norm make the vertical structure of the initial SVs close to the lower boundary, causing considerable PE in the lower troposphere. Except for the evolved SVs with the moist TE norm and the initial SVs with the dry and weighted moist TE norms at uppermost levels, the KE shows the largest contribution to the TE of the SVs.
The linearity evaluation indicates that the major evolved SV structures in the verification region at the final time are overall similar for most of the leading SV components, even though there are slight phase and magnitude differences between linearly and nonlinearly evolved leading SVs. For the second and third SVs, the linearity degraded. The initial SV perturbations used for linearity evaluation grow very rapidly over a very short period from the initial time for any experiment, which may be associated with the initial adjustment of unbalanced SV perturbations. These unbalanced initial SV perturbations may not be associated with the analysis error as shown in Isaksen et al. (2005) and Caron et al. (2007). Further studies are necessary to fully reveal the relevance of the SV perturbation and analysis error, which is beyond the scope of this study. In addition, because all the results in this study are based on the use of simple moist linear physics (i.e., large-scale precipitation), the use of different moist linear physics (e.g., moist convection and explicit moist microphysics) will provide more insight to SV growth and structures of TCs.
The authors thank two anonymous reviewers for their valuable comments. This study was supported by the Korea Meteorological Administration Research and Development Program under Grant CATER 2006-2102. This research was partially supported by the Brain Korea 21 (BK21) program.
Corresponding author address: Hyun Mee Kim, Department of Atmospheric Sciences, Yonsei University, Shinchon-dong 134, Seodaemun-ku, Seoul 120-749, South Korea. Email: email@example.com
This article included in the Targeted Observations, Data Assimilation, and Tropical Cyclone Predictability special collection.
In this study, the optimization time τopt is the final time.
The state vector in this study refers to the perturbation.
In this study, the regions of nonzero projection operator defined are referred to as the verification region.
For brevity, the basic-state integrations using NLM may be mentioned as the nonlinear integrations. In contrast, TLM and adjoint integrations may be mentioned as the linear integrations.
Experiments using different numbers of SVs to construct the composite SV show that the composite SV structures do not change much by including more SVs, because of the weighting of the amplification factors of the individual SV by that of the leading SV in (6). The composite SVs including the leading three, five, and seven SVs show very similar horizontal and vertical structures (not shown), implying that the leading three SVs are key members to construct the composite SV.
In this study, the overall horizontal SV structures encompassing around 30°–40° in longitude and 20°–30° in latitude may be referred to as the large scale. The individual horizontal SV structures encompassing less than 10° in both longitude and latitude may be referred to as the small scale. For example, major sensitivities located from 500 to 1000 km south of the TC center in Fig. 5a are referred to as small-scale SV structures.