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Abstract
Three adaptive approaches for tropical cyclone prediction are compared in this study: the conditional nonlinear optimal perturbation (CNOP) method, the first singular vector (FSV) method, and the ensemble transform Kalman filter (ETKF) method. These approaches are compared for 36-h forecasts of three northwest Pacific tropical cyclones (TCs): Matsa (2005), Nock-Ten (2004), and Morakot (2009). The sensitive regions identified by each method are obtained. The CNOPs form an annulus around the storm at the targeting time, the FSV targets areas north of the storm, and the ETKF closely targets the typhoon location itself. The sensitive results of both the CNOPs and FSV collocate well with the steering flow between the subtropical high and the TCs. Furthermore, the regions where the convection is strong are targeted by the CNOPs. Relatively speaking, the ETKF sensitive results reflect the large-scale flow.
To identify the most effective adaptive observational network, numerous probes or flights were tested arbitrarily for the ETKF method or according to the calculated sensitive regions of the CNOP and FSV methods. The results show that the sensitive regions identified by these three methods are more effective for adaptive observations than the other regions. In all three cases, the optimal adaptive observational network identified by the CNOP and ETKF methods results in similar forecast improvements in the verification region at the verification time, while the improvement using the FSV method is minor.
Abstract
Three adaptive approaches for tropical cyclone prediction are compared in this study: the conditional nonlinear optimal perturbation (CNOP) method, the first singular vector (FSV) method, and the ensemble transform Kalman filter (ETKF) method. These approaches are compared for 36-h forecasts of three northwest Pacific tropical cyclones (TCs): Matsa (2005), Nock-Ten (2004), and Morakot (2009). The sensitive regions identified by each method are obtained. The CNOPs form an annulus around the storm at the targeting time, the FSV targets areas north of the storm, and the ETKF closely targets the typhoon location itself. The sensitive results of both the CNOPs and FSV collocate well with the steering flow between the subtropical high and the TCs. Furthermore, the regions where the convection is strong are targeted by the CNOPs. Relatively speaking, the ETKF sensitive results reflect the large-scale flow.
To identify the most effective adaptive observational network, numerous probes or flights were tested arbitrarily for the ETKF method or according to the calculated sensitive regions of the CNOP and FSV methods. The results show that the sensitive regions identified by these three methods are more effective for adaptive observations than the other regions. In all three cases, the optimal adaptive observational network identified by the CNOP and ETKF methods results in similar forecast improvements in the verification region at the verification time, while the improvement using the FSV method is minor.
Abstract
Using an idealized model of a partial differential equation with parameterization “on–off” switches in the forcing term, the impacts of on–off switches on the variational data assimilation (VDA) are investigated in this paper.
It is shown that the traditional time discretization at the switches of the discrete forward model could induce awful zigzags in the associated discrete cost function (CF), which would cause the optimization to fail to work well in the VDA when using the adjoint method. In addition, it can also cause zigzag oscillations in the numerical solution of the model. A method, which is a generalization of Xu’s intermediate interpolation method, is proposed to eliminate the zigzag phenomenon. The potential merits of this treatment are examined by numerical experiments. The results show that through this treatment, the convergence in the minimization processes of the VDA is improved and the satisfactory optimization retrievals are obtained even though the adjoint models are constructed following Zou’s method from the discrete forward model with the traditional time discretization at the switches.
Abstract
Using an idealized model of a partial differential equation with parameterization “on–off” switches in the forcing term, the impacts of on–off switches on the variational data assimilation (VDA) are investigated in this paper.
It is shown that the traditional time discretization at the switches of the discrete forward model could induce awful zigzags in the associated discrete cost function (CF), which would cause the optimization to fail to work well in the VDA when using the adjoint method. In addition, it can also cause zigzag oscillations in the numerical solution of the model. A method, which is a generalization of Xu’s intermediate interpolation method, is proposed to eliminate the zigzag phenomenon. The potential merits of this treatment are examined by numerical experiments. The results show that through this treatment, the convergence in the minimization processes of the VDA is improved and the satisfactory optimization retrievals are obtained even though the adjoint models are constructed following Zou’s method from the discrete forward model with the traditional time discretization at the switches.
Abstract
This note studies the impact of horizontal diffusion (HD) on the thermohaline circulation (THC) within a modified Stommel’s box model, which was introduced by Longworth et al. HD may arise as a result of mesoscale eddies from the instability of wind-driven gyres. Focuses are on the multi-equilibriums’ existence and nonlinear stability of the THC. A nonlinear approach called conditional nonlinear optimal perturbation (CNOP) is adopted. Both numerical and analytical analyses suggest that there exists a physical mechanism, which makes HD result in more decrement of salinity difference in the meridional direction than that of temperature difference. Consequently, the effects of HD on the thermally and salinity-driven equilibriums of THC are different: HD stabilizes the former but destabilizes the latter.
Abstract
This note studies the impact of horizontal diffusion (HD) on the thermohaline circulation (THC) within a modified Stommel’s box model, which was introduced by Longworth et al. HD may arise as a result of mesoscale eddies from the instability of wind-driven gyres. Focuses are on the multi-equilibriums’ existence and nonlinear stability of the THC. A nonlinear approach called conditional nonlinear optimal perturbation (CNOP) is adopted. Both numerical and analytical analyses suggest that there exists a physical mechanism, which makes HD result in more decrement of salinity difference in the meridional direction than that of temperature difference. Consequently, the effects of HD on the thermally and salinity-driven equilibriums of THC are different: HD stabilizes the former but destabilizes the latter.
Abstract
Linear and nonlinear stability theorems for the generalized Eady model are obtained by the normal-mode method and the energy–Casimir method, respectively. The nonlinear stability criterion is optimal in the following sense: if it is destroyed, then there always exists a finite periodic zonal channel in which there is an exponentially growing normal mode. The theorems show that the “long-wave cutoff” phenomenon exists if and only if the β effect is considered in the model, and the “short-wave cutoff” phenomenon always exists both on an f plane and on a β plane.
Abstract
Linear and nonlinear stability theorems for the generalized Eady model are obtained by the normal-mode method and the energy–Casimir method, respectively. The nonlinear stability criterion is optimal in the following sense: if it is destroyed, then there always exists a finite periodic zonal channel in which there is an exponentially growing normal mode. The theorems show that the “long-wave cutoff” phenomenon exists if and only if the β effect is considered in the model, and the “short-wave cutoff” phenomenon always exists both on an f plane and on a β plane.
Abstract
A triangular T21, three-level, quasigeostrophic global spectral model was used to investigate how precursors relate to the predictability of blocking onset when a conditional nonlinear optimal perturbation approach is used. Here the authors focused on links between the optimal precursor to blocking onset and the optimally growing initial error in onset prediction.
Numerical results have shown that during the prediction of blocking events, a type-1 optimally growing initial error, which causes an overprediction of blocking onset, bears the greatest resemblance to the optimal precursor, and both are distributed primarily over the blocking and its upstream regions. A type-2 optimally growing initial error is also characterized by a similar pattern, but with the opposite sign. Further analysis reveals that a type-1 optimally growing initial error has a similar growth behavior to that of the optimal precursor, and both develop into a dipole blocking anomaly pattern with a strong positive anomaly in the north and a weak negative anomaly to the south. The evolutionary mechanism of a type-1 optimally growing initial error during blocking onset can be explained in the same manner as that of an optimal precursor triggering blocking onset. This similarity between an optimal precursor and an optimally growing initial error also suggests that targeted observations over sensitive areas may be carried out in advance to eliminate optimally growing errors (as many as possible) in the prediction of blocking onset. Thus, the improved observation network will help to better capture the spatial structure of precursors that trigger blocking onset and will increase the ability to predict blocking events.
Abstract
A triangular T21, three-level, quasigeostrophic global spectral model was used to investigate how precursors relate to the predictability of blocking onset when a conditional nonlinear optimal perturbation approach is used. Here the authors focused on links between the optimal precursor to blocking onset and the optimally growing initial error in onset prediction.
Numerical results have shown that during the prediction of blocking events, a type-1 optimally growing initial error, which causes an overprediction of blocking onset, bears the greatest resemblance to the optimal precursor, and both are distributed primarily over the blocking and its upstream regions. A type-2 optimally growing initial error is also characterized by a similar pattern, but with the opposite sign. Further analysis reveals that a type-1 optimally growing initial error has a similar growth behavior to that of the optimal precursor, and both develop into a dipole blocking anomaly pattern with a strong positive anomaly in the north and a weak negative anomaly to the south. The evolutionary mechanism of a type-1 optimally growing initial error during blocking onset can be explained in the same manner as that of an optimal precursor triggering blocking onset. This similarity between an optimal precursor and an optimally growing initial error also suggests that targeted observations over sensitive areas may be carried out in advance to eliminate optimally growing errors (as many as possible) in the prediction of blocking onset. Thus, the improved observation network will help to better capture the spatial structure of precursors that trigger blocking onset and will increase the ability to predict blocking events.
Abstract
In this paper the following question is addressed: assuming that some information is available about initial perturbations (e.g., that they belong to an ensemble, which consists of the perturbations whose magnitudes are less than a given value), how can one determine which perturbations belong to this ensemble and trigger the blocking onset? The applicability of linear singular vectors (LSVs) and conditional nonlinear optimal perturbations (CNOPs) is investigated by a T21L3 quasigeostrophic (QG) model and its tangent linear and adjoint versions. Particular attention is focused on the roles of nonlinear processes and the importance of choosing a proper objective function.
LSVs are the fastest-growing perturbations when the evolutions of the initial perturbations are well described by the tangent linear version of the nonlinear model. CNOPs are a natural generalization of LSVs into the nonlinear category, that is, the initial perturbation whose nonlinear evolution attains the maximum of the objective function at a prescribed forecast time under some initial constraint conditions. The results of this research show that in some cases for the given initial ensemble perturbations CNOPs trigger a transition to a blocking regime (whereas LSVs may not generate such a transition), which shows that nonlinear advection processes are fundamental for studying the weather regime transitions from zonal flow to blocking in the medium range.
By choosing two objective functions and investigating the resulting CNOPs, it is found that CNOPs obtained from the objective function of the blocking-index form (type-1 CNOPs) may trigger a transition to a blocking regime under some circumstances, whereas CNOPs related to the streamfunction squared norm (type-2 CNOPs) fail to yield such a transition. This demonstrates the importance of selecting a proper objective function when aiming at finding the perturbations yielding such a transition.
The mechanism of blocking onset triggered by perturbations is also explored. It is shown that the approach of type-1 CNOP remains a viable tool to capture the spatial structure of initial perturbations that trigger a blocking onset. The planetary-scale projection of the nonlinear interaction of such initial perturbations contributes to the amplification of the blocking downstream and then triggers a blocking onset.
Abstract
In this paper the following question is addressed: assuming that some information is available about initial perturbations (e.g., that they belong to an ensemble, which consists of the perturbations whose magnitudes are less than a given value), how can one determine which perturbations belong to this ensemble and trigger the blocking onset? The applicability of linear singular vectors (LSVs) and conditional nonlinear optimal perturbations (CNOPs) is investigated by a T21L3 quasigeostrophic (QG) model and its tangent linear and adjoint versions. Particular attention is focused on the roles of nonlinear processes and the importance of choosing a proper objective function.
LSVs are the fastest-growing perturbations when the evolutions of the initial perturbations are well described by the tangent linear version of the nonlinear model. CNOPs are a natural generalization of LSVs into the nonlinear category, that is, the initial perturbation whose nonlinear evolution attains the maximum of the objective function at a prescribed forecast time under some initial constraint conditions. The results of this research show that in some cases for the given initial ensemble perturbations CNOPs trigger a transition to a blocking regime (whereas LSVs may not generate such a transition), which shows that nonlinear advection processes are fundamental for studying the weather regime transitions from zonal flow to blocking in the medium range.
By choosing two objective functions and investigating the resulting CNOPs, it is found that CNOPs obtained from the objective function of the blocking-index form (type-1 CNOPs) may trigger a transition to a blocking regime under some circumstances, whereas CNOPs related to the streamfunction squared norm (type-2 CNOPs) fail to yield such a transition. This demonstrates the importance of selecting a proper objective function when aiming at finding the perturbations yielding such a transition.
The mechanism of blocking onset triggered by perturbations is also explored. It is shown that the approach of type-1 CNOP remains a viable tool to capture the spatial structure of initial perturbations that trigger a blocking onset. The planetary-scale projection of the nonlinear interaction of such initial perturbations contributes to the amplification of the blocking downstream and then triggers a blocking onset.
Abstract
Since the accuracy of the tangent linear approximation of moist physics in a mesoscale model is case dependent, the problem related to the variational data assimilation with physical “on–off” processes is studied further in both time-continuous and discrete circumstances. Two kinds of typical on–off switches represented in idealized simple models are investigated: the zero-order discontinuous switch (Type I) and the first-order discontinuous switch (Type II). The main results are as follows. For Type I: 1) For the case in which the model is time continuous, the gradient of the cost function with respect to the initial condition exists except for the threshold. 2) In the time-discrete case, there are zigzag discontinuities in the cost function, and the method that keeps the switches in the tangent linear model the same as in the forward model (called Zou's method) is able to compute the correct gradient where it exists. An optimization with this gradient might yield a local minimum rather than the global minimum if the cost function has multiple minima, however. 3) A method based on the nonlinear perturbation equation is proposed that can give the accurate gradient in the time-continuous case. 4) In the discrete case, the method of this paper is useful to obtain the global descent direction of the cost function in optimization and is helpful to find the global minimum. In addition, it still employs the adjoint model constructed by Zou's method. For Type II, Zou's method can be used for both the time-continuous and discrete cases. The importance of reducing the model error in the context of variational data assimilation with discontinuous physics is also indicated.
Abstract
Since the accuracy of the tangent linear approximation of moist physics in a mesoscale model is case dependent, the problem related to the variational data assimilation with physical “on–off” processes is studied further in both time-continuous and discrete circumstances. Two kinds of typical on–off switches represented in idealized simple models are investigated: the zero-order discontinuous switch (Type I) and the first-order discontinuous switch (Type II). The main results are as follows. For Type I: 1) For the case in which the model is time continuous, the gradient of the cost function with respect to the initial condition exists except for the threshold. 2) In the time-discrete case, there are zigzag discontinuities in the cost function, and the method that keeps the switches in the tangent linear model the same as in the forward model (called Zou's method) is able to compute the correct gradient where it exists. An optimization with this gradient might yield a local minimum rather than the global minimum if the cost function has multiple minima, however. 3) A method based on the nonlinear perturbation equation is proposed that can give the accurate gradient in the time-continuous case. 4) In the discrete case, the method of this paper is useful to obtain the global descent direction of the cost function in optimization and is helpful to find the global minimum. In addition, it still employs the adjoint model constructed by Zou's method. For Type II, Zou's method can be used for both the time-continuous and discrete cases. The importance of reducing the model error in the context of variational data assimilation with discontinuous physics is also indicated.
Abstract
Conditional nonlinear optimal perturbations (CNOPs) of a two-dimensional quasigeostrophic model are obtained numerically. The CNOP is the initial perturbation whose nonlinear evolution attains the maximum value of the cost function, which is constructed according to the physical problems of interests with physical constraint conditions. The difference between the CNOP and a linear singular vector is compared. The results demonstrate that CNOPs catch the nonlinear effects of the model on the evolutions of the initial perturbations. These results suggest that CNOPs are applicable to the study of predictability and sensitivity analysis when nonlinearity is of importance.
Abstract
Conditional nonlinear optimal perturbations (CNOPs) of a two-dimensional quasigeostrophic model are obtained numerically. The CNOP is the initial perturbation whose nonlinear evolution attains the maximum value of the cost function, which is constructed according to the physical problems of interests with physical constraint conditions. The difference between the CNOP and a linear singular vector is compared. The results demonstrate that CNOPs catch the nonlinear effects of the model on the evolutions of the initial perturbations. These results suggest that CNOPs are applicable to the study of predictability and sensitivity analysis when nonlinearity is of importance.
Abstract
In this paper, the linear stability criterion for Eady's problem is proved to be also a nonlinear stability one. Upper bounds for the disturbance potential enstrophy, the disturbance energy, and the disturbance available potential energy on the rigid lids are also established.
Abstract
In this paper, the linear stability criterion for Eady's problem is proved to be also a nonlinear stability one. Upper bounds for the disturbance potential enstrophy, the disturbance energy, and the disturbance available potential energy on the rigid lids are also established.
Abstract
Conditional nonlinear optimal perturbation (CNOP), which is a natural extension of the linear singular vector into the nonlinear regime, is proposed in this study for the determination of sensitive areas in adaptive observations for tropical cyclone prediction. Three tropical cyclone cases, Mindulle (2004), Meari (2004), and Matsa (2005), are investigated. Using the metrics of kinetic and dry energies, CNOPs and the first singular vectors (FSVs) are obtained over a 24-h optimization interval. Their spatial structures, their energies, and their nonlinear evolutions as well as the induced humidity changes are compared. A series of sensitivity experiments are designed to find out what benefit can be obtained by reductions of CNOP-type errors versus FSV-type errors. It is found that the structures of CNOPs may differ much from those of FSVs depending on the constraint, metric, and the basic state. The CNOP-type errors have larger impact on the forecasts in the verification area as well as the tropical cyclones than the FSV-types errors. The results of sensitivity experiments indicate that reductions of CNOP-type errors in the initial states provide more benefits than reductions of FSV-type errors. These results suggest that it is worthwhile to use CNOP as a method to identify the sensitive areas in adaptive observation for tropical cyclone prediction.
Abstract
Conditional nonlinear optimal perturbation (CNOP), which is a natural extension of the linear singular vector into the nonlinear regime, is proposed in this study for the determination of sensitive areas in adaptive observations for tropical cyclone prediction. Three tropical cyclone cases, Mindulle (2004), Meari (2004), and Matsa (2005), are investigated. Using the metrics of kinetic and dry energies, CNOPs and the first singular vectors (FSVs) are obtained over a 24-h optimization interval. Their spatial structures, their energies, and their nonlinear evolutions as well as the induced humidity changes are compared. A series of sensitivity experiments are designed to find out what benefit can be obtained by reductions of CNOP-type errors versus FSV-type errors. It is found that the structures of CNOPs may differ much from those of FSVs depending on the constraint, metric, and the basic state. The CNOP-type errors have larger impact on the forecasts in the verification area as well as the tropical cyclones than the FSV-types errors. The results of sensitivity experiments indicate that reductions of CNOP-type errors in the initial states provide more benefits than reductions of FSV-type errors. These results suggest that it is worthwhile to use CNOP as a method to identify the sensitive areas in adaptive observation for tropical cyclone prediction.
