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Michael C. Morgan

Abstract

A diagnosis of singular vector (SV) evolution (computed for the L 2 streamfunction norm) in the Eady model using potential vorticity (PV) and Eliassen–Palm (E–P) flux diagnostics is performed. In addition, a partitioning of the vertical component of the E–P flux vector based on the results of the piecewise PV inversion is introduced to better elucidate the fundamental mechanisms for SV amplification.

The initial PV structures of the Eady model SVs on both the long- and shortwave sides of the Eady model shortwave cutoff are characterized by initially upshear tilted interior PV anomalies. The results of the PV and the E–P flux diagnostics for optimal perturbations reveal a three-stage process for the SV evolution: 1) a superposition of interior PV anomalies (diagnosed by a positive vertical component of the E–P flux), 2) a subsequent intensification (characterized by maxima in the E–P flux near the boundaries) of the SV boundary potential temperature anomalies (BTAs) by winds attributed to interior PV, and 3) finally a transient or sustained mutual interaction between the BTAs (associated with a nearly nondivergent interior E–P flux). The PV inversion demonstrates that a significant fraction of the observed SV amplification may be attributed to the initially upshear tilted PV anomalies, and that the initial BTAs, while important in describing the initial SV structure, play a minimal role in the subsequent evolution.

The results of the experiments in which the Eady model SV structures were altered by removing either the BTAs or the interior PV suggest that data assimilation schemes that more heavily weight targeted surface observations to targeted tropospheric wind and temperature observations may result in a forecast correction that is either excessive (for short optimization times) or insufficient for optimization times comparable to or larger than the time required for the PV to be rendered vertical by the shear flow.

An analysis of SVs calculated about the time-varying basic state of an NWP model reveals that the SV PV initially consists of upshear tilted structures located beneath a depressed tropopause in a region of strong low-level thermal gradient (and concomitant vertical shear). The subsequent nonlinear evolution of the SV PV in the NWP model is congruous with that observed in the Eady model: The vertical superposition of SV PV, followed by amplification of a surface thermal ridge by winds attributed to the SV PV, characterize the SV amplification. These results suggest the relevance of the SV amplification mechanisms identified in the Eady model in more realistic flows, and indicate the potential utility of applying the aforementioned diagnostic techniques to understanding SV development in observed flows.

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Michael C. Morgan

Abstract

The technique of piecewise potential vorticity (PV) inversion is used to identify the nondivergent wind fields attributed to upper-, middle-, and lower-tropospheric PV anomalies in addition to the irrotational wind with the goal of diagnosing the respective wind fields’ frontogenetic potentialities. Frontogenesis is diagnosed using a piecewise separation of the Q vector into parts associated with the partitioned wind field. Partitioned geostrophic Q vectors are used to diagnose the vertical motion attributed to the upper-, middle-, and lower-tropospheric PV anomalies.

Insight gained from this new diagnostic technique is demonstrated by examining a particular case of extratropical marine cyclogenesis resulting from the interaction of an upper-tropospheric short-wave trough with a surface thermal wave. In the early stages of development, the largest contributor to surface frontogenesis was associated with winds attributed to the lower-tropospheric thermal wave. As the cyclone matured, the contributions of the upper-tropospheric PV and near-surface potential temperature anomalies to surface frontogenesis increased. Winds attributed to the upper-tropospheric PV were frontogenetical north of the thermal ridge axis and frontolytical south of the thermal ridge in the warm sector. The upper-tropospheric PV acted to amplify the thermal ridge while simultaneously narrowing the warm sector. The patterns of geostrophic Q vectors associated with the upper-tropospheric PV suggest that ascent should be favored in the narrowing surface thermal ridge. The contribution to surface frontogenesis due to lower- and middle-tropospheric PV, whose increase is imputed to latent heat release, was variable during the evolution of the cyclone—suggesting that the location of diabatically generated PV anomalies relative to frontal zones can have a significant impact on frontogenesis and associated frontal precipitation distribution. Throughout the evolution of the cyclone, the irrotational wind was frontogenetical along the warm and cold fronts with the magnitude of the irrotational frontogenesis increasing as the surface cyclone amplified.

For the case considered, partitioned geostrophic Q vector “forcing” for vertical motion revealed approximately equal contributions from the upper-tropospheric PV anomalies and the near-surface thermal perturbations.

Characteristic patterns of Q vectors and Q-vector divergence are identified and presented for cases including an upper trough interacting with a surface baroclinic zone and a propagating surface edge wave.

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Michael C. Morgan

Abstract

Analytic results and numerical experimentation reveal that a “backward” integration of the adjoint of the shallow water system linearized about a basic state at rest on an f plane is characterized by a radiation of gravity wave–like structures and the emergence of a steady adjoint state. The earlier adjoint states are linked to the prescribed adjoint state (i.e., the adjoint forcing) through the locally conserved dynamical adjoint variables of the shallow water system: the sensitivity to “balanced height” () and the sensitivity to potential vorticity (PV) . The sensitivity to balanced height is determined by the prescribed adjoint sensitivity forcings for the flow, and , and height (fluid depth), : − (g/f)(). The sensitivity to PV is diagnosed from the inversion of an elliptic operator relating the sensitivity to PV to the distribution of .

The sensitivity to PV determines the long-time (t → ∞), steady behavior of the adjoint sensitivity to height, = −(f/H 2). In the vicinity of the initial adjoint forcing, the long-time, steady-state behavior of the adjoint system (linearized about a state at rest) is characterized by nondivergent sensitivities to the flow that resemble geostrophic balance: = (1/H)/∂y and = (1/H)/∂x. The process by which this long-time, nondivergent, adjoint state emerges is termed adjoint adjustment. For the system considered, sensitivities to the ageostrophic and irrotational components of the flow vanish for the adjusted state near the prescribed adjoint forcing.

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Hyun Mee Kim and Michael C. Morgan

Abstract

A diagnosis of singular vector (SV) evolution in the Eady model for the potential enstrophy and energy norms is performed using potential vorticity (PV) inversion and Eliassen–Palm (E–P) flux diagnostics, and compared with the SV evolution for the streamfunction variance norm. The diagnostics reveal that the mechanism for SV amplification depends on the initial relative magnitudes of the interior PV and boundary temperature anomalies (BTAs). In addition, the relative magnitudes of the initial PV and BTAs are dependent on the norm chosen, the length scale of the perturbation, and the length of the optimization interval.

If the initial contribution of the PV to a given norm is larger than the contribution of the BTAs to that norm, then the SV evolution in that norm is governed by the baroclinic superposition of the interior PV followed by an amplification of the BTAs by winds attributed to the interior PV. In the other case, the mutual interaction of BTAs governs the SV evolution. The initial interior PV is most important for the energy and streamfunction variance SVs, but is less important for the potential enstrophy SVs. Excluding the longwave (i.e., wavelengths longer than the Eady instability cutoff) enstrophy norm SVs, for the shortwave SVs and for long optimization times, the importance of the initial interior PV is most apparent.

In the view of targeted observations, the sensitive regions indicated by the SV analysis can be identified with particular mechanisms for SV development. The forecast measure may be considered sensitive in some regions in the sense that the forecast measure exhibits a large response to small changes in the initial conditions in those regions. The potential enstrophy norm is identified as being dynamically sensitive at the boundaries in contrast to the energy and streamfunction variance norm in the midtroposphere. It is suggested that subjective PV diagnosis of sensitivity may be viewed as being consistent with an objective diagnosis of sensitivity using potential enstrophy norm SVs.

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Michael C. Morgan and Chih-Chieh Chen

Abstract

The structure and evolution of Eady model singular vector (SV, also referred to as optimal perturbation) streamfunction perturbations are described using a combination of two different partitions of the vector subspace describing all possible streamfunction perturbations. A modal partitioning of the SV perturbation streamfunction (expressing the SV streamfunction as a linear combination of modal structures) is used to ascribe the roles and relative importance of the continuum modes (CMs) and the discrete normal modes (NMs) in SV initial structure and subsequent evolution. In addition, a potential vorticity (PV) partitioning of the SV perturbation streamfunction into parts attributed to the SV PV and the SV boundary thermal anomalies (BTAs) is employed. The structures of the CMs and NMs are described in terms of their characteristic perturbation PV and BTAs.

Modal decomposition of the SVs reveals that for all zonal wavenumbers (k), the NMs have the largest projection coefficients (with magnitudes exceeding unity). Specifically, for k < k c, the growing NM has the largest magnitude projection coefficient, while for k > k c equally large projection coefficients are observed for the two neutral Eady modes. The fact that the magnitude of the NM projection coefficients exceeds unity necessitates the existence of structurally similar CMs to “mask” the NMs at initial time. This initial masking, which has been previously reported, is interpreted from a PV perspective as resulting from the cancellation between the NM BTAs and the BTAs associated with the CMs. For all wavenumbers, the magnitude of these NM projection coefficients increases with increasing optimization time τ opt before reaching a limiting value proportional to the mode's projectability as τ opt → ∞.

For k < k c, the lower (upper) CM BTA is of the same (opposite) sign as the interior CM PV anomaly. For k > k c, for those CMs residing between the steering levels of the two neutral Eady modes, the lower (upper) BTAs are the same (opposite) sign as the CM PV anomaly, while for those CM modes residing at other levels, the signs of the lower and upper BTAs are reversed.

For all wavenumbers, initial amplification of the SV is associated with the superposition of the interior PV anomalies. Concomitantly with the superposition of CM PV is the superposition of CM BTAs. Because of the aforementioned structure of the CM BTAs, for k < k c, the superposition of the CM BTAs represents a negative contribution to SV amplification. For k > k c, superposition of CM BTAs contributes positively to amplification, and the CM BTAs have a nondecaying streamfunction contribution nearly equivalent to the contribution from the edge waves.

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Brett T. Hoover and Michael C. Morgan

Abstract

The steering of a tropical cyclone (TC) vortex is commonly understood as the advection of the TC vortex by an “environmental wind.” In past studies, the environmental steering wind vector has been defined by the horizontal and vertical averaging of the horizontal winds in a box centered on the TC. The components of this environmental steering have been proposed as response functions to derive adjoint-derived sensitivities of TC zonal and meridional steering. The appropriateness of these response functions in adjoint sensitivity studies of TC steering is tested using a two-dimensional barotropic model and its adjoint for a 24-h forecast. It is found that these response functions do not produce sensitivities to TC steering because perturbations to the model initial conditions that change the final-time location of the TC also change the response functions in ways that have nothing to do with the steering of the TC at model verification.

An alternate response function is proposed wherein the environmental steering vector is defined as the wind averaged over the response function box attributed to vorticity outside of that box. By redefining the response functions for the zonal and meridional steering as components of this environmental steering vector, the effect of small changes to the final-time location of the TC is removed, and the resultant sensitivity gradients can be shown to truly represent the sensitivity of TC steering to perturbations of the model forecast state.

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Daryl T. Kleist and Michael C. Morgan

Abstract

A 36-h adjoint-based forecast sensitivity study of three response functions defined in the lower troposphere—average temperature in an isolated region of the upper Midwest (R 1), meridional temperature difference (R 2), and average zonal component of the wind (R 3)—is conducted with the goal of providing a synoptic and dynamic interpretation of the sensitivity gradient structure and evolution. In addition to calculating and interpreting the sensitivity gradients with respect to basic model variables along the model forecast trajectory, a technique is outlined that allows for the calculation of the sensitivity gradients with respect to variables derivable from the model state vector (including geopotential, relative vorticity, and divergence), and a method for visualizing the sensitivities with respect to the horizontal components of the wind is proposed and demonstrated.

The sensitivity of R 1 to all model and derived variables revealed that R 1 was controlled by nearly adiabatic processes associated with the addition or generation of temperature perturbations upstream of the region in which R 1 was defined. For R 2, the sensitivity gradients revealed the well-known influence of confluent horizontal flow and vertical tilting of isentropes to increase the north–south temperature gradient over the region within which R 2 was defined. The sensitivity of R 3 to the components of the horizontal wind reveals that simply adding or generating an upstream zonal wind perturbation is insufficient to change the zonal wind at 36 h as these wind perturbations upstream of the domain within which R 3 is defined are torqued by the Coriolis force as they are advected toward the domain. These results suggest adjoint-derived sensitivities of quasi-conserved response functions may be more easily interpretable than sensitivities calculated for nonconserved response functions.

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Brett T. Hoover and Michael C. Morgan

Abstract

Through the use of an adjoint model, the sensitivity of the steering of a simulated tropical cyclone (TC) to various aspects of a model forecast trajectory can be calculated. This calculation, providing a priori information about how small perturbations to the model state will impact the steering of the TC at some future time, provides a wealth of dynamical information about the importance of synoptic-scale features and associated processes to the steering of a modeled TC that is difficult or impossible to obtain by other means. Regions of strong sensitivity to cyclone steering are regions where, if errors in the model state exist, those errors would have the largest effect on TC steering at a specified time in the future. However, without a dynamical understanding of why the steering of a simulated TC is sensitive to changes in these regions, errors in the methodology of implementing an adjoint model for calculating these sensitivities may result in sensitivity gradients that do not represent sensitivity of TC steering at all, and without a strong dynamical interpretation of these sensitivities, these errors may escape notice.

An adjoint model is employed for several cases of simulated TCs in the west Pacific to determine the dynamical significance of regions for which sensitivity to TC steering is found to be particularly strong. It is found that the region of subsidence upstream of a passing midlatitude trough can play a crucial role in the development of perturbations that strongly impact a recurving TC. A dynamical interpretation of this relationship is described and tested.

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Michael C. Morgan and John W. Nielsen-Gammon

Abstract

The use of potential vorticity (PV) allows the efficient description of the dynamics of nearly balanced atmospheric flow phenomena, but the distribution of PV must be simply represented for ease in interpretation. Representations of PV on isentropic or isobaric surfaces can be cumbersome, as analyses of several surfaces spanning the troposphere must be constructed to fully apprehend the complete PV distribution.

Following a brief review of the relationship between PV and nearly balanced flows, it is demonstrated that the tropospheric PV has a simple distribution, and as a consequence, an analysis of potential temperature along the dynamic tropopause (here defined as a surface of constant PV) allows for a simple representation of the upper-tropospheric and lower-stratospheric PV. The construction and interpretation of these tropopause maps, which may be termed “isertelic” analyses of potential temperature, are described. In addition, techniques to construct dynamical representations of the lower-tropospheric PV and near-surface potential temperature, which complement these isertelic analyses, are also suggested. Case studies are presented to illustrate the utility of these techniques in diagnosing phenomena such as cyclogenesis, tropopause folds, the formation of an upper trough, and the effects of latent heat release on the upper and lower troposphere.

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Daryl T. Kleist and Michael C. Morgan

Abstract

The 24–25 January 2000 eastern United States snowstorm was noteworthy as operational numerical weather prediction (NWP) guidance was poor for lead times as short as 36 h. Despite improvements in the forecast of the surface cyclone position and intensity at 1200 UTC 25 January 2000 with decreasing lead time, NWP guidance placed the westward extent of the midtropospheric, frontogenetically forced precipitation shield too far to the east.

To assess the influence of initial condition uncertainties on the forecast of this event, an adjoint model is used to evaluate forecast sensitivities for 36- and 48-h forecasts valid at 1200 UTC 25 January 2000 using as response functions the energy-weighted forecast error, lower-tropospheric circulation about a box surrounding the surface cyclone, 750-hPa frontogenesis, and vertical motion. The sensitivities with respect to the initial conditions for these response functions are in general very similar: geographically isolated, maximized in the middle and lower troposphere, and possessing an upshear vertical tilt. The sensitivities are maximized in a region of enhanced low-level baroclinicity in the vicinity of the surface cyclone’s precursor upper trough. However, differences in the phase and structure of the gradients for the four response functions are evident, which suggests that perturbations could be constructed to alter one response function but not necessarily the others.

Gradients of the forecast error response function with respect to the initial conditions are used in an iterative procedure to construct initial condition perturbations that reduce the forecast error. These initial condition perturbations were small in terms of both spatial scale and magnitude. Those initial condition perturbations that were confined primarily to the midtroposphere grew rapidly into much larger amplitude upper-and-lower tropospheric perturbations. The perturbed forecasts were not only characterized by reduced final time forecast error, but also had a synoptic evolution that more closely followed analyses and observations.

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