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Abstract
Singular vector (SV) sensitivity, calculated using the adjoint model of the U.S. Navy Operation Global Atmosphere Prediction System (NOGAPS), is used to study the dynamics associated with tropical cyclone evolution. For each model-predicted tropical cyclone, SVs are constructed that optimize perturbation energy within a 20° by 20° latitude/longitude box centered on the 48-h forecast position of the cyclone. The initial SVs indicate regions where the 2-day forecast of the storm is very sensitive to changes in the analysis. Composites of the SVs for straight-moving cyclones and non-straight-moving cyclones that occurred in the Northern Hemisphere during its summer season in 2003 are examined. For both groups, the initial-time SV sensitivity exhibits a maximum within an annulus approximately 500 km from the center of the storms, in the region where the potential vorticity gradient of the vortex first changes sign. In the azimuthal direction, the composite initial-time SV maximum for the straight-moving group is located in the rear right quadrant with respect to the storm motion. The composite based on the non-straight-moving cyclones does not have a preferred quadrant in the vicinity of the storms and has larger amplitude away from the cyclones compared with the straight-moving storms, indicating more environmental influence on these storms. For both groups, the maximum initial sensitive areas are collocated with regions of flow moving toward the storm.
While the initial SV maximum is located where the potential vorticity gradient changes sign, the final SV maximum is located where the potential vorticity gradient is a maximum. Examinations of individual cases demonstrate how SV sensitivity can be used to identify specific environmental influences on the storms. The relationship between the SV sensitivity and the potential vorticity is discussed. The results support the utility of SVs in applications to phenomena beyond midlatitude baroclinic systems.
Abstract
Singular vector (SV) sensitivity, calculated using the adjoint model of the U.S. Navy Operation Global Atmosphere Prediction System (NOGAPS), is used to study the dynamics associated with tropical cyclone evolution. For each model-predicted tropical cyclone, SVs are constructed that optimize perturbation energy within a 20° by 20° latitude/longitude box centered on the 48-h forecast position of the cyclone. The initial SVs indicate regions where the 2-day forecast of the storm is very sensitive to changes in the analysis. Composites of the SVs for straight-moving cyclones and non-straight-moving cyclones that occurred in the Northern Hemisphere during its summer season in 2003 are examined. For both groups, the initial-time SV sensitivity exhibits a maximum within an annulus approximately 500 km from the center of the storms, in the region where the potential vorticity gradient of the vortex first changes sign. In the azimuthal direction, the composite initial-time SV maximum for the straight-moving group is located in the rear right quadrant with respect to the storm motion. The composite based on the non-straight-moving cyclones does not have a preferred quadrant in the vicinity of the storms and has larger amplitude away from the cyclones compared with the straight-moving storms, indicating more environmental influence on these storms. For both groups, the maximum initial sensitive areas are collocated with regions of flow moving toward the storm.
While the initial SV maximum is located where the potential vorticity gradient changes sign, the final SV maximum is located where the potential vorticity gradient is a maximum. Examinations of individual cases demonstrate how SV sensitivity can be used to identify specific environmental influences on the storms. The relationship between the SV sensitivity and the potential vorticity is discussed. The results support the utility of SVs in applications to phenomena beyond midlatitude baroclinic systems.
Abstract
Computational models based on discrete dynamical equations are a successful way of approaching the problem of predicting or forecasting the future evolution of dynamical systems. For linear and mildly nonlinear models, the solutions of the numerical algorithms on which they are based converge to the analytic solutions of the underlying differential equations for small time steps and grid sizes. In this paper, the authors investigate the time step sensitivity of three nonlinear atmospheric models of different levels of complexity: the Lorenz equations, a quasigeostrophic (QG) model, and a global weather prediction system (NOGAPS). It is illustrated here how, for chaotic systems, numerical convergence cannot be guaranteed forever. The time of decoupling of solutions for different time steps follows a logarithmic rule (as a function of time step) similar for the three models. In regimes that are not fully chaotic, the Lorenz equations are used to illustrate how different time steps may lead to different model climates and even different regimes. A simple model of truncation error growth in chaotic systems is proposed. This model decomposes the error onto its stable and unstable components and reproduces well the short- and medium-term behavior of the QG model truncation error growth, with an initial period of slow growth (a plateau) before the exponential growth phase. Experiments with NOGAPS suggest that truncation error can be a substantial component of total forecast error of the model. Ensemble simulations with NOGAPS show that using different time steps may be a simple and natural way of introducing an important component of model error in ensemble design.
Abstract
Computational models based on discrete dynamical equations are a successful way of approaching the problem of predicting or forecasting the future evolution of dynamical systems. For linear and mildly nonlinear models, the solutions of the numerical algorithms on which they are based converge to the analytic solutions of the underlying differential equations for small time steps and grid sizes. In this paper, the authors investigate the time step sensitivity of three nonlinear atmospheric models of different levels of complexity: the Lorenz equations, a quasigeostrophic (QG) model, and a global weather prediction system (NOGAPS). It is illustrated here how, for chaotic systems, numerical convergence cannot be guaranteed forever. The time of decoupling of solutions for different time steps follows a logarithmic rule (as a function of time step) similar for the three models. In regimes that are not fully chaotic, the Lorenz equations are used to illustrate how different time steps may lead to different model climates and even different regimes. A simple model of truncation error growth in chaotic systems is proposed. This model decomposes the error onto its stable and unstable components and reproduces well the short- and medium-term behavior of the QG model truncation error growth, with an initial period of slow growth (a plateau) before the exponential growth phase. Experiments with NOGAPS suggest that truncation error can be a substantial component of total forecast error of the model. Ensemble simulations with NOGAPS show that using different time steps may be a simple and natural way of introducing an important component of model error in ensemble design.
Abstract
The sensitivity of tropical cyclogenesis and subsequent intensification is explored by applying small perturbations to the initial state in the presence of organized mesoscale convection and synoptic-scale forcing using the adjoint and tangent linear models for the Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS). The forward, adjoint, and tangent linear models are used to compare and contrast predictability characteristics for the disturbance that became Typhoon Nuri and a nondeveloping organized convective cluster in the western Pacific during The Observing System Research and Predictability Experiment (THORPEX) Pacific Asian Regional Campaign (T-PARC) and the Tropical Cyclone Structure-2008 (TCS-08) experiments.
The adjoint diagnostics indicate that the intensity (e.g., maximum surface wind speed, minimum surface pressure) of a tropical disturbance is very sensitive to perturbations in the moisture and temperature fields and to a lesser degree the wind fields. The highest-resolution adjoint simulations (grid increment of 13 km) indicate that the most efficient intensification is through moistening in the lower and middle levels and heating in banded regions that are coincident with vorticity maxima in the initial state. Optimal adjoint perturbations exhibit rapid growth for the Nuri case and only modest growth for the nondeveloping system. The adjoint results suggest that Nuri was near the threshold for development, indicative of low predictability. The low-level sensitivity maximum and tendency for optimal perturbation growth to extend vertically through the troposphere are consistent with a “bottom up” development process of TC genesis, although a secondary midlevel sensitivity maximum is present as well. Growth originates at small scales and projects onto the scale of the vortex, a manifestation of perturbations that project onto organized convection embedded in regions of cyclonic vorticity.
Abstract
The sensitivity of tropical cyclogenesis and subsequent intensification is explored by applying small perturbations to the initial state in the presence of organized mesoscale convection and synoptic-scale forcing using the adjoint and tangent linear models for the Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS). The forward, adjoint, and tangent linear models are used to compare and contrast predictability characteristics for the disturbance that became Typhoon Nuri and a nondeveloping organized convective cluster in the western Pacific during The Observing System Research and Predictability Experiment (THORPEX) Pacific Asian Regional Campaign (T-PARC) and the Tropical Cyclone Structure-2008 (TCS-08) experiments.
The adjoint diagnostics indicate that the intensity (e.g., maximum surface wind speed, minimum surface pressure) of a tropical disturbance is very sensitive to perturbations in the moisture and temperature fields and to a lesser degree the wind fields. The highest-resolution adjoint simulations (grid increment of 13 km) indicate that the most efficient intensification is through moistening in the lower and middle levels and heating in banded regions that are coincident with vorticity maxima in the initial state. Optimal adjoint perturbations exhibit rapid growth for the Nuri case and only modest growth for the nondeveloping system. The adjoint results suggest that Nuri was near the threshold for development, indicative of low predictability. The low-level sensitivity maximum and tendency for optimal perturbation growth to extend vertically through the troposphere are consistent with a “bottom up” development process of TC genesis, although a secondary midlevel sensitivity maximum is present as well. Growth originates at small scales and projects onto the scale of the vortex, a manifestation of perturbations that project onto organized convection embedded in regions of cyclonic vorticity.
Abstract
An exact closed form expression for the infinite time analysis and forecast error covariances of a Kalman filter is used to investigate how the locations of fixed observing platforms such as radiosonde stations affect global distributions of analysis and forecast error variance. The solution pertains to a system with no model error, time-independent nondefective unstable dynamics, time-independent observation operator, and time-independent observation error covariance. As far as the authors are aware, the solutions are new. It is shown that only nondecaying normal modes (eigenvectors of the dynamics operator) are required to represent the infinite time error covariance matrices. Consequently, once a complete set of nondecaying eigenvectors has been obtained, the solution allows for the rapid assessment of the error-reducing potential of any observational network that bounds error variance.
Atmospherically relevant time-independent basic states and their corresponding tangent linear propagators are obtained with the help of a (T21L3) quasigeostrophic global model. The closed form solution allows for an examination of the sensitivity of the error variances to many different observing configurations. It is also feasible to determine the optimal location of one additional observation given a fixed observing network, which, through repetition, can be used to build effective observing networks.
Effective observing networks result in error variances several times smaller than other types of networks with the same number of column observations, such as equally spaced or land-based networks. The impact of the observing network configuration on global error variance is greater when the observing network is less dense. The impact of observations at different pressure levels is also examined. It is found that upper-level observations are more effective at reducing globally averaged error variance, but midlevel observations are more effective at reducing forecast error variance at and downstream of the baroclinic regions associated with midlatitude jets.
Abstract
An exact closed form expression for the infinite time analysis and forecast error covariances of a Kalman filter is used to investigate how the locations of fixed observing platforms such as radiosonde stations affect global distributions of analysis and forecast error variance. The solution pertains to a system with no model error, time-independent nondefective unstable dynamics, time-independent observation operator, and time-independent observation error covariance. As far as the authors are aware, the solutions are new. It is shown that only nondecaying normal modes (eigenvectors of the dynamics operator) are required to represent the infinite time error covariance matrices. Consequently, once a complete set of nondecaying eigenvectors has been obtained, the solution allows for the rapid assessment of the error-reducing potential of any observational network that bounds error variance.
Atmospherically relevant time-independent basic states and their corresponding tangent linear propagators are obtained with the help of a (T21L3) quasigeostrophic global model. The closed form solution allows for an examination of the sensitivity of the error variances to many different observing configurations. It is also feasible to determine the optimal location of one additional observation given a fixed observing network, which, through repetition, can be used to build effective observing networks.
Effective observing networks result in error variances several times smaller than other types of networks with the same number of column observations, such as equally spaced or land-based networks. The impact of the observing network configuration on global error variance is greater when the observing network is less dense. The impact of observations at different pressure levels is also examined. It is found that upper-level observations are more effective at reducing globally averaged error variance, but midlevel observations are more effective at reducing forecast error variance at and downstream of the baroclinic regions associated with midlatitude jets.
Abstract
This paper investigates the nature of model error in complex deterministic nonlinear systems such as weather forecasting models. Forecasting systems incorporate two components, a forecast model and a data assimilation method. The latter projects a collection of observations of reality into a model state. Key features of model error can be understood in terms of geometric properties of the data projection and a model attracting manifold. Model error can be resolved into two components: a projection error, which can be understood as the model’s attractor being in the wrong location given the data projection, and direction error, which can be understood as the trajectories of the model moving in the wrong direction compared to the projection of reality into model space. This investigation introduces some new tools and concepts, including the shadowing filter, causal and noncausal shadow analyses, and various geometric diagnostics. Various properties of forecast errors and model errors are described with reference to low-dimensional systems, like Lorenz’s equations; then, an operational weather forecasting system is shown to have the same predicted behavior. The concepts and tools introduced show promise for the diagnosis of model error and the improvement of ensemble forecasting systems.
Abstract
This paper investigates the nature of model error in complex deterministic nonlinear systems such as weather forecasting models. Forecasting systems incorporate two components, a forecast model and a data assimilation method. The latter projects a collection of observations of reality into a model state. Key features of model error can be understood in terms of geometric properties of the data projection and a model attracting manifold. Model error can be resolved into two components: a projection error, which can be understood as the model’s attractor being in the wrong location given the data projection, and direction error, which can be understood as the trajectories of the model moving in the wrong direction compared to the projection of reality into model space. This investigation introduces some new tools and concepts, including the shadowing filter, causal and noncausal shadow analyses, and various geometric diagnostics. Various properties of forecast errors and model errors are described with reference to low-dimensional systems, like Lorenz’s equations; then, an operational weather forecasting system is shown to have the same predicted behavior. The concepts and tools introduced show promise for the diagnosis of model error and the improvement of ensemble forecasting systems.
Abstract
In this study, the leading singular vectors (SVs), which are the fastest-growing perturbations (in a linear sense) to a given forecast, are used to examine and classify the dynamic relationship between tropical cyclones (TCs) and synoptic-scale environmental features that influence their evolution. Based on the 72 two-day forecasts of the 18 western North Pacific TCs in 2006, the SVs are constructed to optimize perturbation energy within a 20° × 20° latitude–longitude box centered on the 48-h forecast position of the TCs using the Navy Operational Global Atmospheric Prediction System (NOGAPS) forecast and adjoint systems. Composite techniques are employed to explore these relationships and highlight how the dominant synoptic-scale features that impact TC forecasts evolve on seasonal time scales.
The NOGAPS initial SVs show several different patterns that highlight the relationship between the TC forecast sensitivity and the environment during the western North Pacific typhoon season in 2006. In addition to the relation of the SV maximum to the inward flow region of the TC, there are three patterns identified where the local SV maxima collocate with low-radial-wind-speed regions. These regions are likely caused by the confluence of the flow associated with the TC itself and the flow from other synoptic systems, such as the subtropical high and the midlatitude jet. This is the new finding beyond the previous NOGAPS SV results on TCs. The subseasonal variations of these patterns corresponding to the dynamic characteristics are discussed. The SV total energy vertical structures for the different composites are used to demonstrate the contributions from kinetic and potential energy components of different vertical levels at initial and final times.
Abstract
In this study, the leading singular vectors (SVs), which are the fastest-growing perturbations (in a linear sense) to a given forecast, are used to examine and classify the dynamic relationship between tropical cyclones (TCs) and synoptic-scale environmental features that influence their evolution. Based on the 72 two-day forecasts of the 18 western North Pacific TCs in 2006, the SVs are constructed to optimize perturbation energy within a 20° × 20° latitude–longitude box centered on the 48-h forecast position of the TCs using the Navy Operational Global Atmospheric Prediction System (NOGAPS) forecast and adjoint systems. Composite techniques are employed to explore these relationships and highlight how the dominant synoptic-scale features that impact TC forecasts evolve on seasonal time scales.
The NOGAPS initial SVs show several different patterns that highlight the relationship between the TC forecast sensitivity and the environment during the western North Pacific typhoon season in 2006. In addition to the relation of the SV maximum to the inward flow region of the TC, there are three patterns identified where the local SV maxima collocate with low-radial-wind-speed regions. These regions are likely caused by the confluence of the flow associated with the TC itself and the flow from other synoptic systems, such as the subtropical high and the midlatitude jet. This is the new finding beyond the previous NOGAPS SV results on TCs. The subseasonal variations of these patterns corresponding to the dynamic characteristics are discussed. The SV total energy vertical structures for the different composites are used to demonstrate the contributions from kinetic and potential energy components of different vertical levels at initial and final times.
Abstract
An analysis of the surface latent heat flux (SLHF) influence on the accumulated precipitation associated with an idealized extratropical cyclone using the Coupled Ocean–Atmosphere Mesoscale Prediction System is presented. There are two distinct precipitation regions found to be strongly influenced by the SLHF, referred to as the primary maximum and the cold-frontal precipitation. A substantial reduction by approximately 30 mm (35%) and 15 mm (75%) is observed in the accumulated precipitation of the two regions, respectively, when the SLHF is eliminated domain wide at 96 h—the starting time of the most rapid cyclone deepening. The source of this reduction is investigated by systematically controlling the SLHF in three cyclone sectors, which are the low-latitude, baroclinic, and high-latitude sectors. The precipitation in the primary maximum is most strongly controlled by the baroclinic sector, which experiences strong upward SLHF due to its dry environmental air. In contrast, the precipitation in the cold-frontal zone is most strongly controlled by the low-latitude sector, which experiences only a moderate amount of upward SLHF into the already warm and moist boundary layer air. The results underscore the crucial role of SLHF and boundary layer processes in precipitation predictions and demonstrate the need for accurate forecasts of air–sea temperature contrast, surface level winds, and moisture to properly simulate air–sea interactions.
Abstract
An analysis of the surface latent heat flux (SLHF) influence on the accumulated precipitation associated with an idealized extratropical cyclone using the Coupled Ocean–Atmosphere Mesoscale Prediction System is presented. There are two distinct precipitation regions found to be strongly influenced by the SLHF, referred to as the primary maximum and the cold-frontal precipitation. A substantial reduction by approximately 30 mm (35%) and 15 mm (75%) is observed in the accumulated precipitation of the two regions, respectively, when the SLHF is eliminated domain wide at 96 h—the starting time of the most rapid cyclone deepening. The source of this reduction is investigated by systematically controlling the SLHF in three cyclone sectors, which are the low-latitude, baroclinic, and high-latitude sectors. The precipitation in the primary maximum is most strongly controlled by the baroclinic sector, which experiences strong upward SLHF due to its dry environmental air. In contrast, the precipitation in the cold-frontal zone is most strongly controlled by the low-latitude sector, which experiences only a moderate amount of upward SLHF into the already warm and moist boundary layer air. The results underscore the crucial role of SLHF and boundary layer processes in precipitation predictions and demonstrate the need for accurate forecasts of air–sea temperature contrast, surface level winds, and moisture to properly simulate air–sea interactions.
Abstract
The influence of the surface latent and surface sensible heat flux on the development and interaction of an idealized extratropical cyclone (termed “primary”) with an upstream cyclone (termed “upstream”) using the Navy’s Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS) is analyzed. The primary cyclone develops from an initial perturbation to a baroclinically unstable jet stream, while the upstream cyclone results from Rossby wave dispersion at the surface where a bottom-up style development occurs. The intensity of the upstream cyclone is strongly enhanced by surface latent heat fluxes and, to a lesser degree, by surface sensible heat fluxes. Forward trajectories initiated from the postfrontal sector of the primary cyclone travel south of the upstream anticyclone and feed into the atmospheric river and warm conveyor belt region of the upstream cyclone. Substantial moistening of this airstream is a result of upward surface latent heat flux present in both the primary cyclone’s postfrontal sector and along the southern flank of the anticyclone. Backward trajectories initiated from the same region show that these air parcels originate from a broad area north of both the anticyclone and the primary cyclone in the lower troposphere. The airstream identified represents a new pathway through which dry, descending air that is preconditioned through surface moistening enhances the development of an upstream cyclone through diabatically generated potential vorticity.
Abstract
The influence of the surface latent and surface sensible heat flux on the development and interaction of an idealized extratropical cyclone (termed “primary”) with an upstream cyclone (termed “upstream”) using the Navy’s Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS) is analyzed. The primary cyclone develops from an initial perturbation to a baroclinically unstable jet stream, while the upstream cyclone results from Rossby wave dispersion at the surface where a bottom-up style development occurs. The intensity of the upstream cyclone is strongly enhanced by surface latent heat fluxes and, to a lesser degree, by surface sensible heat fluxes. Forward trajectories initiated from the postfrontal sector of the primary cyclone travel south of the upstream anticyclone and feed into the atmospheric river and warm conveyor belt region of the upstream cyclone. Substantial moistening of this airstream is a result of upward surface latent heat flux present in both the primary cyclone’s postfrontal sector and along the southern flank of the anticyclone. Backward trajectories initiated from the same region show that these air parcels originate from a broad area north of both the anticyclone and the primary cyclone in the lower troposphere. The airstream identified represents a new pathway through which dry, descending air that is preconditioned through surface moistening enhances the development of an upstream cyclone through diabatically generated potential vorticity.
Abstract
An analysis of the influence and sensitivity of moisture in an idealized two-dimensional moist semigeostrophic frontogenesis model is presented. A comparison between a dry (relative humidity RH = 0%) version and a moist (RH = 80%) version of the model demonstrates that the impact of moisture is to increase frontogenesis, strengthen the transverse circulation (u ag, w), generate a low-level potential-vorticity anomaly, and develop a low-level jet. The idealized model is compared with a real case simulated with the full-physics three-dimensional Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS) model, establishing good agreement and thereby confirming that the idealized model retains the essential physical processes relevant for improving understanding of midlatitude frontogenesis. Optimal perturbations of mixing ratio are calculated to quantify the circulation response of the model through the computation of singular vectors, which determines the fastest-growing modes of a linearized version of the idealized model. The vertical velocity is found to respond strongly to initial-condition mixing-ratio perturbations such that small changes in moisture lead to large changes in the ascent. The progression of physical processes responsible for this nonlinear growth is (in order) jet/front transverse circulation → moisture convergence ahead of the front → latent heating at mid- to low elevations → reduction in static stability ahead of the front → strengthening of the transverse circulation, and the feedback cycle repeats. Together, these physical processes represent a pathway by which small perturbations of moisture can have a strong impact on a forecast involving midlatitude frontogenesis.
Abstract
An analysis of the influence and sensitivity of moisture in an idealized two-dimensional moist semigeostrophic frontogenesis model is presented. A comparison between a dry (relative humidity RH = 0%) version and a moist (RH = 80%) version of the model demonstrates that the impact of moisture is to increase frontogenesis, strengthen the transverse circulation (u ag, w), generate a low-level potential-vorticity anomaly, and develop a low-level jet. The idealized model is compared with a real case simulated with the full-physics three-dimensional Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS) model, establishing good agreement and thereby confirming that the idealized model retains the essential physical processes relevant for improving understanding of midlatitude frontogenesis. Optimal perturbations of mixing ratio are calculated to quantify the circulation response of the model through the computation of singular vectors, which determines the fastest-growing modes of a linearized version of the idealized model. The vertical velocity is found to respond strongly to initial-condition mixing-ratio perturbations such that small changes in moisture lead to large changes in the ascent. The progression of physical processes responsible for this nonlinear growth is (in order) jet/front transverse circulation → moisture convergence ahead of the front → latent heating at mid- to low elevations → reduction in static stability ahead of the front → strengthening of the transverse circulation, and the feedback cycle repeats. Together, these physical processes represent a pathway by which small perturbations of moisture can have a strong impact on a forecast involving midlatitude frontogenesis.
