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Abstract
Previous simulations of dipole vortices propagating through rotating, stratified fluid have revealed small-scale inertia–gravity waves that are embedded within the dipole near its leading edge and are approximately stationary relative to the dipole. The mechanism by which these waves are generated is investigated, beginning from the observation that the dipole can be reasonably approximated by a balanced quasigeostrophic (QG) solution. The deviations from the QG solution (including the waves) then satisfy linear equations that come from linearization of the governing equations about the QG dipole and are forced by the residual tendency of the QG dipole (i.e., the difference between the time tendency of the QG solution and that of the full primitive equations initialized with the QG fields). The waves do not appear to be generated by an instability of the balanced dipole, as homogeneous solutions of the linear equations amplify little over the time scale for which the linear equations are valid. Linear solutions forced by the residual tendency capture the scale, location, and pattern of the inertia–gravity waves, although they overpredict the wave amplitude by a factor of 2. There is thus strong evidence that the waves are generated as a forced linear response to the balanced flow. The relation to and differences from other theories for wave generation by balanced flows, including those of Lighthill and Ford et al., are discussed.
Abstract
Previous simulations of dipole vortices propagating through rotating, stratified fluid have revealed small-scale inertia–gravity waves that are embedded within the dipole near its leading edge and are approximately stationary relative to the dipole. The mechanism by which these waves are generated is investigated, beginning from the observation that the dipole can be reasonably approximated by a balanced quasigeostrophic (QG) solution. The deviations from the QG solution (including the waves) then satisfy linear equations that come from linearization of the governing equations about the QG dipole and are forced by the residual tendency of the QG dipole (i.e., the difference between the time tendency of the QG solution and that of the full primitive equations initialized with the QG fields). The waves do not appear to be generated by an instability of the balanced dipole, as homogeneous solutions of the linear equations amplify little over the time scale for which the linear equations are valid. Linear solutions forced by the residual tendency capture the scale, location, and pattern of the inertia–gravity waves, although they overpredict the wave amplitude by a factor of 2. There is thus strong evidence that the waves are generated as a forced linear response to the balanced flow. The relation to and differences from other theories for wave generation by balanced flows, including those of Lighthill and Ford et al., are discussed.
Abstract
Results from homogeneous, isotropic turbulence suggest that predictability behavior is linked to the slope of a flow’s kinetic energy spectrum. Such a link has potential implications for the predictability behavior of atmospheric models. This article investigates these topics in an intermediate context: a multilevel quasigeostrophic model with a jet and temperature perturbations at the upper surface (a surrogate tropopause). Spectra and perturbation growth behavior are examined at three model resolutions. The results augment previous studies of spectra and predictability in quasigeostrophic models, and they provide insight that can help interpret results from more complex models. At the highest resolution tested, the slope of the kinetic energy spectrum is approximately 
Abstract
Results from homogeneous, isotropic turbulence suggest that predictability behavior is linked to the slope of a flow’s kinetic energy spectrum. Such a link has potential implications for the predictability behavior of atmospheric models. This article investigates these topics in an intermediate context: a multilevel quasigeostrophic model with a jet and temperature perturbations at the upper surface (a surrogate tropopause). Spectra and perturbation growth behavior are examined at three model resolutions. The results augment previous studies of spectra and predictability in quasigeostrophic models, and they provide insight that can help interpret results from more complex models. At the highest resolution tested, the slope of the kinetic energy spectrum is approximately
Abstract
Normal modes of a linear vertical shear (Eady shear) are studied within the linearized primitive equations for a rotating stratified fluid above a rigid lower boundary. The authors' interest is in modes having an inertial critical layer present at some height within the flow. Below this layer, the solutions can be closely approximated by balanced edge waves obtained through an asymptotic expansion in Rossby number. Above, the solutions behave as gravity waves. Hence these modes are an example of a spatial coupling of balanced motions to gravity waves.
The amplitude of the gravity waves relative to the balanced part of the solutions is obtained analytically and numerically as a function of parameters. It is shown that the waves are exponentially small in Rossby number. Moreover, their amplitude depends in a nontrivial way on the meridional wavenumber. For modes having a radiating upper boundary condition, the meridional wavenumber for which the gravity wave amplitude is maximal occurs when the tilts of the balanced edge wave and gravity waves agree.
Abstract
Normal modes of a linear vertical shear (Eady shear) are studied within the linearized primitive equations for a rotating stratified fluid above a rigid lower boundary. The authors' interest is in modes having an inertial critical layer present at some height within the flow. Below this layer, the solutions can be closely approximated by balanced edge waves obtained through an asymptotic expansion in Rossby number. Above, the solutions behave as gravity waves. Hence these modes are an example of a spatial coupling of balanced motions to gravity waves.
The amplitude of the gravity waves relative to the balanced part of the solutions is obtained analytically and numerically as a function of parameters. It is shown that the waves are exponentially small in Rossby number. Moreover, their amplitude depends in a nontrivial way on the meridional wavenumber. For modes having a radiating upper boundary condition, the meridional wavenumber for which the gravity wave amplitude is maximal occurs when the tilts of the balanced edge wave and gravity waves agree.
Abstract
In the course of adapting a nonhydrostatic cloud model [or primitive-equation model (PE)] for simulations of large-scale baroclinic waves, we have encountered systematic discrepancies between the PE solutions and those of the semigeostrophic (SG) equations. Direct comparisons using identical, uniform potential vorticity jets show that 1) the linear modes of the PE have distinctively different structure than the SG modes; 2) at finite amplitude, the PE pressure field develops lows that are deeper, and highs that are weaker, than in the SG solution; and 3) the nonlinear PE wave produces a characteristic “cyclonic wrapping” of the temperature contours on both horizontal boundaries and has an associated “bent-back” frontal structure at the surface, while in the SG solutions (for this particular basic state jet) there is an equal tendency to pull temperature contours anticyclonically around highs and cyclonically around lows. An analysis of the vorticity and potential vorticity equations for small Rossby number reveals that the SG model errs in its treatment of terms involving the ageostrophic vorticity. Simulations based on an equation set that includes the leading-order dynamical contributions of the ageostrophic vorticity agree more closely with the PE simulations.
Abstract
In the course of adapting a nonhydrostatic cloud model [or primitive-equation model (PE)] for simulations of large-scale baroclinic waves, we have encountered systematic discrepancies between the PE solutions and those of the semigeostrophic (SG) equations. Direct comparisons using identical, uniform potential vorticity jets show that 1) the linear modes of the PE have distinctively different structure than the SG modes; 2) at finite amplitude, the PE pressure field develops lows that are deeper, and highs that are weaker, than in the SG solution; and 3) the nonlinear PE wave produces a characteristic “cyclonic wrapping” of the temperature contours on both horizontal boundaries and has an associated “bent-back” frontal structure at the surface, while in the SG solutions (for this particular basic state jet) there is an equal tendency to pull temperature contours anticyclonically around highs and cyclonically around lows. An analysis of the vorticity and potential vorticity equations for small Rossby number reveals that the SG model errs in its treatment of terms involving the ageostrophic vorticity. Simulations based on an equation set that includes the leading-order dynamical contributions of the ageostrophic vorticity agree more closely with the PE simulations.
Abstract
A comparative analysis of simulations of baroclinic waves with and without surface drag is presented, with particular reference to surface features. As in recent studies, the present simulations show that, compared to simulations with no drag, those with surface drag are less inclined to develop a secluded warm sector, and that drag weakens the warm front while the cold front remains strong. The authors demonstrate that analogous effects occur when Ekman pumping is used in nonlinear quasigeostrophic numerical simulations of unstable baroclinic waves in a channel. However, since the quasigeostrophic model produces symmetric highs and lows in the unstable baroclinic wave, the cold and warm fronts are therefore also symmetric and hence equally affected by the Ekman pumping. The different effect that friction has on the warm front with respect to the cold front in the primitive-equation simulations is fundamentally related to the tendency for the lows to be strong and narrow and the highs weak and broad, and for the warm front to form just north of, and extend eastward from, the low, while the cold front extends between the high and the low. The authors’ thesis is that the Ekman pumping associated with the low, at the location where the warm front would form in the absence of surface friction, acts to resist the formation of the warm front, while the cold front, positioned between the high and the low where Ekman pumping associated with the baroclinic wave is weak, is therefore relatively unaffected.
Given the weakness of Ekman pumping associated with the baroclinic wave in the vicinity of the incipient cold front, the present simulations indicate that cold frontogenesis occurs in the drag case in much the same way as in the no-drag case. Present analysis shows that the horizontal advection creating the cold front is a combination of geostrophic and ageostrophic effects. A portion of the ageostrophic frontogenesis is a response to geostrophic frontogenesis, as in the case without surface drag; however with surface drag, a significant portion of the cross-front ageostrophic flow is due to the Ekman layer associated with the front itself.
Abstract
A comparative analysis of simulations of baroclinic waves with and without surface drag is presented, with particular reference to surface features. As in recent studies, the present simulations show that, compared to simulations with no drag, those with surface drag are less inclined to develop a secluded warm sector, and that drag weakens the warm front while the cold front remains strong. The authors demonstrate that analogous effects occur when Ekman pumping is used in nonlinear quasigeostrophic numerical simulations of unstable baroclinic waves in a channel. However, since the quasigeostrophic model produces symmetric highs and lows in the unstable baroclinic wave, the cold and warm fronts are therefore also symmetric and hence equally affected by the Ekman pumping. The different effect that friction has on the warm front with respect to the cold front in the primitive-equation simulations is fundamentally related to the tendency for the lows to be strong and narrow and the highs weak and broad, and for the warm front to form just north of, and extend eastward from, the low, while the cold front extends between the high and the low. The authors’ thesis is that the Ekman pumping associated with the low, at the location where the warm front would form in the absence of surface friction, acts to resist the formation of the warm front, while the cold front, positioned between the high and the low where Ekman pumping associated with the baroclinic wave is weak, is therefore relatively unaffected.
Given the weakness of Ekman pumping associated with the baroclinic wave in the vicinity of the incipient cold front, the present simulations indicate that cold frontogenesis occurs in the drag case in much the same way as in the no-drag case. Present analysis shows that the horizontal advection creating the cold front is a combination of geostrophic and ageostrophic effects. A portion of the ageostrophic frontogenesis is a response to geostrophic frontogenesis, as in the case without surface drag; however with surface drag, a significant portion of the cross-front ageostrophic flow is due to the Ekman layer associated with the front itself.
Abstract
Using a primitive equation (PE) model, we revisit two canonical flows that were previously studied using a semigeostrophic equation (SG) model. In a previous paper, the authors showed that the PE and the SG models can have significantly different versions of the large-scale dynamics—here they report on the implications of this difference for frontogenesis. The program for the study of frontogenesis developed by B. J. Hoskins and collaborators is followed to show how, in the PE version of the canonical cases, the surface warm front develops before the cold front, and why the upper-level front is a long, nearly continuous feature going from ridge to trough. The frontogenesis experienced by an air parcel is computed following the parcel to illustrate better the mechanisms involved. As the present calculations are carried out longer than most previous ones, the relation of the upper frontogenesis to the formation of the upper-level “cutoff” cyclone is also examined. Trajectory and three-dimensional graphical analyses show, with respect to the latter, the extreme distortions of the isentropic surfaces and mixing-induced variations in the potential vorticity field.
Abstract
Using a primitive equation (PE) model, we revisit two canonical flows that were previously studied using a semigeostrophic equation (SG) model. In a previous paper, the authors showed that the PE and the SG models can have significantly different versions of the large-scale dynamics—here they report on the implications of this difference for frontogenesis. The program for the study of frontogenesis developed by B. J. Hoskins and collaborators is followed to show how, in the PE version of the canonical cases, the surface warm front develops before the cold front, and why the upper-level front is a long, nearly continuous feature going from ridge to trough. The frontogenesis experienced by an air parcel is computed following the parcel to illustrate better the mechanisms involved. As the present calculations are carried out longer than most previous ones, the relation of the upper frontogenesis to the formation of the upper-level “cutoff” cyclone is also examined. Trajectory and three-dimensional graphical analyses show, with respect to the latter, the extreme distortions of the isentropic surfaces and mixing-induced variations in the potential vorticity field.
Abstract
A nonhydrostatic numerical model is used to simulate two-dimensional frontogenesis forced by either horizontal deformation or shear. Both inviscid frontogenesis prior to frontal collapse and frontogenesis with horizontal diffusion following collapse are considered. The numerical solutions generally agree well with semigeostrophic (SG) theory, though differences can be substantial for intense fronts. Certain deviations from SG that have been previously discussed in the literature area, upon closer examination, associated with spurious gravity waves produced by inadequate resolution or by the initialization of the numerical model. Even when spurious waves are eliminated, however, significant deviations from SG still exist: gravity waves are emitted by the frontogenesis when the cross-front scale becomes sufficiently small, and higher-order corrections to SG may also be present. In the postcollapse solutions (where they are most prominent), the emitted waves are stationary with respect to the front and lead to a band of increased low-level ascent just ahead of the surface front. It is suggested here that, when small, the deviations from SG arise as the linear forced response to the cross-front accelerations neglected by SG.
Abstract
A nonhydrostatic numerical model is used to simulate two-dimensional frontogenesis forced by either horizontal deformation or shear. Both inviscid frontogenesis prior to frontal collapse and frontogenesis with horizontal diffusion following collapse are considered. The numerical solutions generally agree well with semigeostrophic (SG) theory, though differences can be substantial for intense fronts. Certain deviations from SG that have been previously discussed in the literature area, upon closer examination, associated with spurious gravity waves produced by inadequate resolution or by the initialization of the numerical model. Even when spurious waves are eliminated, however, significant deviations from SG still exist: gravity waves are emitted by the frontogenesis when the cross-front scale becomes sufficiently small, and higher-order corrections to SG may also be present. In the postcollapse solutions (where they are most prominent), the emitted waves are stationary with respect to the front and lead to a band of increased low-level ascent just ahead of the surface front. It is suggested here that, when small, the deviations from SG arise as the linear forced response to the cross-front accelerations neglected by SG.
Abstract
In this study, a one-step-ahead ensemble Kalman smoother (EnKS) is introduced for the purposes of parameter estimation. The potential for this system to provide new constraints on the surface-exchange coefficients of momentum (Cd ) and enthalpy (Ck ) is then explored using a series of observing system simulation experiments (OSSEs). The surface-exchange coefficients to be estimated within the data assimilation system are highly uncertain, especially at high wind speeds, and are well known to be important model parameters influencing the intensity and structure of tropical cyclones in numerical simulations. One major benefit of the developed one-step-ahead EnKS is that it allows for simultaneous updates of the rapidly evolving model state variables found in tropical cyclones using a short assimilation window and a long smoother window for the parameter updates, granting sufficient time for sensitivity to the parameters to develop. Overall, OSSEs demonstrate potential for this approach to accurately constrain parameters controlling the amplitudes of Cd and Ck , but the degree of success in recovering the truth model parameters varies throughout the tropical cyclone life cycle. During the rapid intensification phase, rapidly growing errors in the model state project onto the parameter updates and result in an overcorrection of the parameters. After the rapid intensification phase, however, the parameters are correctly adjusted back toward the truth values. Last, the relative success of parameter estimation in recovering the truth model parameter values also has sensitivity to the ensemble size and smoothing forecast length, each of which are explored.
Significance Statement
Large uncertainty in the surface-exchange coefficients of momentum and heat/moisture exists for hurricane conditions. This is a problem because the numerical weather model predictions of hurricane intensity and storm structure are sensitive to the surface-exchange coefficient values used. In this study we use data assimilation, or the relationships estimated between the surface-exchange coefficients and forecasted observations, to constrain uncertainty in the model’s surface-exchange coefficient values. More specifically, an approach to limit both the rapidly growing errors associated with the hurricane itself and the hurricane’s accumulated response to the surface-exchange coefficient values is presented. Overall, this approach has potential to accurately estimate the surface-exchange coefficients, but the success depends on the number of forecast realizations used and how rapidly the hurricane is changing.
Abstract
In this study, a one-step-ahead ensemble Kalman smoother (EnKS) is introduced for the purposes of parameter estimation. The potential for this system to provide new constraints on the surface-exchange coefficients of momentum (Cd ) and enthalpy (Ck ) is then explored using a series of observing system simulation experiments (OSSEs). The surface-exchange coefficients to be estimated within the data assimilation system are highly uncertain, especially at high wind speeds, and are well known to be important model parameters influencing the intensity and structure of tropical cyclones in numerical simulations. One major benefit of the developed one-step-ahead EnKS is that it allows for simultaneous updates of the rapidly evolving model state variables found in tropical cyclones using a short assimilation window and a long smoother window for the parameter updates, granting sufficient time for sensitivity to the parameters to develop. Overall, OSSEs demonstrate potential for this approach to accurately constrain parameters controlling the amplitudes of Cd and Ck , but the degree of success in recovering the truth model parameters varies throughout the tropical cyclone life cycle. During the rapid intensification phase, rapidly growing errors in the model state project onto the parameter updates and result in an overcorrection of the parameters. After the rapid intensification phase, however, the parameters are correctly adjusted back toward the truth values. Last, the relative success of parameter estimation in recovering the truth model parameter values also has sensitivity to the ensemble size and smoothing forecast length, each of which are explored.
Significance Statement
Large uncertainty in the surface-exchange coefficients of momentum and heat/moisture exists for hurricane conditions. This is a problem because the numerical weather model predictions of hurricane intensity and storm structure are sensitive to the surface-exchange coefficient values used. In this study we use data assimilation, or the relationships estimated between the surface-exchange coefficients and forecasted observations, to constrain uncertainty in the model’s surface-exchange coefficient values. More specifically, an approach to limit both the rapidly growing errors associated with the hurricane itself and the hurricane’s accumulated response to the surface-exchange coefficient values is presented. Overall, this approach has potential to accurately estimate the surface-exchange coefficients, but the success depends on the number of forecast realizations used and how rapidly the hurricane is changing.
Abstract
The potential for storm surge to cause extensive property damage and loss of life has increased urgency to more accurately predict coastal flooding associated with landfalling tropical cyclones. This work investigates the sensitivity of coastal inundation from storm tide (surge + tide) to four hurricane parameters—track, intensity, size, and translation speed—and the sensitivity of inundation forecasts to errors in forecasts of those parameters. An ensemble of storm tide simulations is generated for three storms in the Gulf of Mexico, by driving a storm surge model with best track data and systematically generated perturbations of storm parameters from the best track. The spread of the storm perturbations is compared to average errors in recent operational hurricane forecasts, allowing sensitivity results to be interpreted in terms of practical predictability of coastal inundation at different lead times. Two types of inundation metrics are evaluated: point-based statistics and spatially integrated volumes. The practical predictability of surge inundation is found to be limited foremost by current errors in hurricane track forecasts, followed by intensity errors, then speed errors. Errors in storm size can also play an important role in limiting surge predictability at short lead times, due to observational uncertainty. Results show that given current mean errors in hurricane forecasts, location-specific surge inundation is predictable for as little as 12–24 h prior to landfall, less for small-sized storms. The results also indicate potential for increased surge predictability beyond 24 h for large storms by considering a storm-following, volume-integrated metric of inundation.
Abstract
The potential for storm surge to cause extensive property damage and loss of life has increased urgency to more accurately predict coastal flooding associated with landfalling tropical cyclones. This work investigates the sensitivity of coastal inundation from storm tide (surge + tide) to four hurricane parameters—track, intensity, size, and translation speed—and the sensitivity of inundation forecasts to errors in forecasts of those parameters. An ensemble of storm tide simulations is generated for three storms in the Gulf of Mexico, by driving a storm surge model with best track data and systematically generated perturbations of storm parameters from the best track. The spread of the storm perturbations is compared to average errors in recent operational hurricane forecasts, allowing sensitivity results to be interpreted in terms of practical predictability of coastal inundation at different lead times. Two types of inundation metrics are evaluated: point-based statistics and spatially integrated volumes. The practical predictability of surge inundation is found to be limited foremost by current errors in hurricane track forecasts, followed by intensity errors, then speed errors. Errors in storm size can also play an important role in limiting surge predictability at short lead times, due to observational uncertainty. Results show that given current mean errors in hurricane forecasts, location-specific surge inundation is predictable for as little as 12–24 h prior to landfall, less for small-sized storms. The results also indicate potential for increased surge predictability beyond 24 h for large storms by considering a storm-following, volume-integrated metric of inundation.
