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Gregory J. Hakim

Abstract

Balance dynamics are proposed in a probabilistic framework, assuming that the state variables and the master, or control, variables are random variables described by continuous probability density functions. Balance inversion, defined as recovering the state variables from the control variables, is achieved through Bayes’ theorem. Balance dynamics are defined by the propagation of the joint probability of the state and control variables through the Liouville equation. Assuming Gaussian statistics, balance inversion reduces to linear regression of the state variables onto the control variables, and assuming linear dynamics, balance dynamics reduces to a Kalman filter subject to perfect observations given by the control variables.

Example solutions are given for an elliptical vortex in shallow water having unity Rossby and Froude numbers, which produce an outward-propagating pulse of inertia–gravity wave activity. Applying balance inversion to the potential vorticity reveals that, because potential vorticity and divergence share well-defined patterns of covariability, the inertia–gravity wave field is recovered in addition to the vortical field. Solutions for a probabilistic balance dynamics model applied to the elliptical vortex reveal smaller errors (“imbalance”) for height control compared to potential vorticity control.

Important attributes of the probabilistic balance theory include quantification of the concept of balance manifold “fuzziness,” and clear state-independent definitions of balance and imbalance in terms of the range of the probabilistic inversion operators. Moreover, the theory provides a generalization of the notion of balance that may prove useful for problems involving moist physics, chemistry, and tropical circulations.

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Gregory J. Hakim

Abstract

Observationally motivated idealized initial-value problems of cyclogenesis are studied for quasigeostrophic dynamics. The goals of this investigation are to assess the contributions of normal-mode and nonmodal growth mechanisms and the influence of nonlinearity during incipient cyclogenesis. The initial condition is represented by a coherent vortex superposed on a zero potential vorticity parallel flow. Nonlinear solutions are qualitatively in accord with observations, producing typical deepening of the surface cyclone, an asymmetry in the strength of the cyclone and anticyclone, and the formation of an upper-level front downstream from the cyclogenesis. The growth rate for the projection of the model state vector onto the most unstable mode closely approximates the linear value during the early stages of surface development. Nonlinear dynamics become important after approximately 30 h, beyond which the modal-projection growth rate declines approximately 30%.

Linear solutions accurately approximate the intensity and zonal location of the surface cyclone, as well as the asymmetry between the cyclone and upstream anticyclone. The development of the surface cyclone is explained, almost entirely, by the projection onto the growing normal modes. The growing normal modes also account for the development of a prominent ridge of high pressure that forms on the tropopause downstream from the vortex. Nonmodal processes (the complementary subset to the growing normal modes) capture the dispersion of the upper vortex but do not contribute to deepening the surface cyclone. The upper-level front is captured by the linear solutions, and results from a favorable superposition between the growing normal modes and the neutral modes. Tests reveal that surface development declines markedly for vortex length scales smaller than those of observed precursor disturbances. This effect is attributed to a reduction in the vortex projection onto the unstable normal-mode spectrum.

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Gregory J. Hakim

Abstract

Developing wave packets in the western North Pacific storm track are diagnosed observationally. An abrupt upstream edge to baroclinic wave activity over the western North Pacific facilitates comparisons between the observational results and previous theoretical predictions on the spatiotemporal evolution of an impulse disturbance. Results show that surface cyclogenesis events are preceded by a sharply peaked wave packet that originates poleward of the Himalaya Plateau and develops rapidly across the North Pacific to North America.

Composite wave-packet structure is broadly consistent with linear theory for idealized models such as Eady's. The longitude–height structure of the mature packet reveals deep growing waves with horizontal wavelengths of approximately 4000 km near the packet peak. Downstream from the peak, amplitude decays exponentially, and wavelength decreases approximately linearly to about 2500–3000 km at the leading edge. Meridional potential vorticity gradients are concentrated near the tropopause. In contrast to linear theory, the packets show an abrupt upstream edge and no evidence of upstream development. As the packet travels through the along-stream variations of the Pacific jet stream, the packet-peak and leading-edge group velocity vary. These accelerations change the packet length and suggest that the Pacific jet may act to focus the packets.

A sample of North Atlantic storm track events reveals similar results and suggests that the Atlantic storm track is often seeded by wave packets that originate over the western North Pacific Ocean. In contrast, Atlantic packets refract equatorward and become trapped on the subtropical jet to the south of Himalaya Plateau, suggesting perhaps less potential for seeding disturbances in the Pacific storm track.

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Gregory J. Hakim

Abstract

A case study is presented of an unusual cold front that affected the east coast of the United States on 22 April 1987. Noteworthy aspects of this front are its genesis behind a preexisting back-door cold front its propagation to the southwest, the shallow nature of the cold air associated with it.(<600 m), and the absence of precipitation. This front has been termed a “side-door” cold front because of the significant differences between it and typical back-door cold fronts, such as its direction of propagation and vertical structure. Weather conditions in the mid-Atlantic states changed abruptly from partly cloudy skies, light winds, and 25°–30°C surface temperatures ahead of the front to a windy (gust to 20 m s−1), low overcast regime with temperatures 10°–15°C cooler behind the front.

Analysis of satellite imagery, surface data, and sounding data revealed that the front possessed density-current structure during a portion of its lifetime. Observed front-relative flow, as well as agreement between computed and observed frontal velocity, support this conclusion. Results from this study demonstrate that the cold front evolved without the blocking and channeling effects of a major mountain range, as is typically the case for similar events around the world. It is hypothesized that the front formed in response to differential surface heating and friction along the New England coast. Similarly, the differential heating across the coastline contributed to the intensification of the front as well as to the evolution of the density-current structure later in the life history of the front. Results of this study relevant to several numerical simulations of similar events are discussed.

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Gregory J. Hakim

Abstract

The dominant vertical structures for analysis and forecast errors are estimated in midlatitudes using a small ensemble of operational analyses. Errors for fixed locations in the central North Pacific and eastern North America are selected for comparing errors in regions with relatively low and high observation density, respectively. Results for these fixed locations are compared with results for zonal wavenumber 9, which provides a representative sample of baroclinic waves. This study focuses on deviations from the ensemble mean for meridional wind and temperature at 40°N; these quantities are chosen for simplicity and because they capture dynamical and thermodynamical aspects of midlatitude baroclinic waves.

Results for the meridional wind show that analysis and forecast errors share the same dominant vertical structure as the analyses. This structure peaks near the tropopause and decays smoothly toward small values in the middle and lower troposphere. The dominant vertical structure for analysis errors exhibits upshear tilt and peaks just below the tropopause, suggesting an asymmetry in errors of the tropopause location, with a bias toward greater errors for downward tropopause displacements. The dominant vertical structure for temperature analysis errors is distinctly different from temperature analyses. Analysis errors have a sharp peak in the lower troposphere, with a secondary structure near the tropopause, whereas forecast errors and analyses show a dipole straddling the tropopause and smooth vertical structure, consistent with potential vorticity anomalies due to variance in tropopause position.

Linear regression of forecast errors onto analysis errors for the western North Pacific is used to assess the nonseparable zonal-height structure of errors and their propagation. Analysis errors near the tropopause rapidly develop into a spreading wave packet, with a group speed that matches the mean zonal wind speed of 31 m s−1. A complementary calculation for the regression of 24-h forecast errors onto analysis errors shows that forecast errors originate from analysis errors in the middle and upper troposphere. These errors rapidly expand in the vertical to span the troposphere, with a peak at the tropopause.

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Gregory J. Hakim

Abstract

The variability and predictability of axisymmetric hurricanes are determined from a 500-day numerical simulation of a tropical cyclone in statistical equilibrium. By design, the solution is independent of the initial conditions and environmental variability, which isolates the “intrinsic” axisymmetric hurricane variability.

Variability near the radius of maximum wind is dominated by two patterns: one associated primarily with radial shifts of the maximum wind, and one primarily with intensity change at the time-mean radius of maximum wind. These patterns are linked to convective bands that originate more than 100 km from the storm center and propagate inward. Bands approaching the storm produce eyewall replacement cycles, with an increase in storm intensity as the secondary eyewall contracts radially inward. A dominant time period of 4–8 days is found for the convective bands, which is hypothesized to be determined by the time scale over which subsidence from previous bands suppresses convection; a leading-order estimate based on the ratio of the Rossby radius to band speed fits the hypothesis.

Predictability limits for the idealized axisymmetric solution are estimated from linear inverse modeling and analog forecasts, which yield consistent results showing a limit for the azimuthal wind of approximately 3 days. The optimal initial structure that excites the leading pattern of 24-h forecast-error variance has largest azimuthal wind in the midtroposphere outside the storm and a secondary maximum just outside the radius of maximum wind. Forecast errors grow by a factor of 24 near the radius of maximum wind.

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Gregory J. Hakim

Abstract

Numerical experiments are performed to determine the mean state of an axisymmetric hurricane in statistical equilibrium. Most earlier studies used a damping scheme on the temperature field as a parameterization of radiative cooling, which the authors demonstrate yields storms that have little convection outside the eyewall and do not achieve statistical equilibrium. Here the effects of infrared radiation are explicitly simulated, which permits the storm to achieve radiative–convective equilibrium.

Beginning from a state of rest, a storm spontaneously develops with maximum surface wind speeds in excess of 100 m s−1 by day 10. This transient “superintense” storm weakens and is replaced by an equilibrium storm that lasts over 400 days with a time-mean maximum wind speed that compares closely with a diagnostic estimate of potential intensity (PI). The main assumptions of PI theory are found to be consistent with the properties of the equilibrium storm, but the thermodynamic cycle does not resemble a Carnot cycle, with an implied efficiency of about half that of the Carnot limit. Maximum radiative cooling is found in the midtroposphere outside the storm, where convective clouds detrain into the dry layer of storm-outflow subsidence, producing a large vertical gradient in water vapor and cloud water.

Sensitivity experiments reveal that the results are robust to changes in the prestorm thermodynamic sounding, ambient rotation, horizontal turbulent mixing, and details in the radiative heating field. Subject to the assumptions in this study, it can be concluded that 1) the undisturbed tropical atmosphere is unstable to axisymmetric hurricanes, 2) PI theory accurately bounds time-mean storm intensity (but not transient fluctuations), and 3) equilibrium storm intensity is insensitive to turbulent mixing in the radial direction.

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Gregory J. Hakim

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This paper poses and tests the hypothesis that some of the synoptic-scale and mesoscale tropopause-based disturbances that produce organized vertical motion and induce surface cyclones in the extratropical troposphere are vortexlike coherent structures. Based on the theory of nonlinear waves and vortices, tests are constructed and applied to observations of relative vorticity maxima for a 33-winter climatology at 500 hPa and three-dimensional composites for a single winter season. The method is designed to determine the following disturbance properties: nonlinearity, quasigeostrophic potential vorticity–streamfunction relationship, speed, and trapping of fluid particles. These properties are determined for four disturbance-amplitude categories, defined here in terms of 500-hPa relative vorticity.

The results show that, on average, 500-hPa relative vorticity maxima are localized monopolar vortices with length scales (radii) of approximately 500–800 km; there is a slight increase in length scale with disturbance amplitude. Nonlinearity is O(1) or greater for all amplitude categories, approaching O(10) for the strongest disturbances. Trapping of fluid particles, estimated by the presence of closed contours of potential vorticity on isentropic surfaces near the tropopause, requires greater than O(1) nonlinearity; the threshold disturbance amplitude is approximately 8 × 10−5 s−1 in the vertical component of 500-hPa relative vorticity and −8 K in anomaly tropopause potential temperature. The vortices move westward with respect to the background flow, with a slight northward drift. The observational evidence does not support an interpretation of these features in terms of modons or solitary waves.

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Chris Snyder and Gregory J. Hakim

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Singular vectors (SVs) have been applied to cyclogenesis, to initializing ensemble forecasts, and in predictability studies. Ideally, the calculation of the SVs would employ the analysis error covariance norm at the initial time or, in the case of cyclogenesis, a norm based on the statistics of initial perturbations, but the energy norm is often used as a more practical substitute.

To illustrate the roles of the choice of norm and the vertical structure of initial perturbations, an upper-level wave with no potential vorticity perturbation in the troposphere is considered as a typical cyclogenetic perturbation or analysis error, and this perturbation is then decomposed by its projection onto each energy SV. All calculations are made, for simplicity, in the context of the quasigeostrophic Eady model (i.e., for a background flow with constant vertical shear and horizontal temperature gradient). Viewed in terms of the energy SVs, the smooth vertical structure of the typical perturbation, as well as its evolution, results from strong cancellation between the growing and decaying SVs, most of which are highly structured and tilted in the vertical.

A simpler picture, involving less cancellation, follows from decomposition of the typical perturbation into SVs using an alternative initial norm, which is based on the relation between initial norms and the statistics of initial perturbations together with the empirical assumption that the initial perturbations are not dominated by interior potential vorticity. Differences between the energy SVs and those based on the alternative initial norm can be understood by noting that the energy norm implicitly assumes initial perturbations with second-order statistics given by the covariance matrix whose inverse defines the energy norm. Unlike the “typical” perturbation, perturbations with those statistics have large variance of potential vorticity in the troposphere and fine vertical structure.

Finally, a brief assessment is presented of the extent to which the upper wave, and more generally the alternative initial norm, is representative of cyclogenetic perturbations and analysis errors. There is substantial evidence supporting deep perturbations with little vertical structure as frequent precursors to cyclogenesis, but surrogates for analysis errors are less conclusive: operational midlatitude analysis differences have vertical structure similar to that of the perturbations implied by the energy norm, while short-range forecast errors and analysis errors from assimilation experiments with simulated observations are more consistent with the alternative norm.

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Brian Ancell and Gregory J. Hakim

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The sensitivity of numerical weather forecasts to small changes in initial conditions is estimated using ensemble samples of analysis and forecast errors. Ensemble sensitivity is defined here by linear regression of analysis errors onto a given forecast metric. It is shown that ensemble sensitivity is proportional to the projection of the analysis-error covariance onto the adjoint-sensitivity field. Furthermore, the ensemble-sensitivity approach proposed here involves a small calculation that is easy to implement. Ensemble- and adjoint-based sensitivity fields are compared for a representative wintertime flow pattern near the west coast of North America for a 90-member ensemble of independent initial conditions derived from an ensemble Kalman filter. The forecast metric is taken for simplicity to be the 24-h forecast of sea level pressure at a single point in western Washington State. Results show that adjoint and ensemble sensitivities are very different in terms of location, scale, and magnitude. Adjoint-sensitivity fields reveal mesoscale lower-tropospheric structures that tilt strongly upshear, whereas ensemble-sensitivity fields emphasize synoptic-scale features that tilt modestly throughout the troposphere and are associated with significant weather features at the initial time. Optimal locations for targeting can easily be determined from ensemble sensitivity, and results indicate that the primary targeting locations are located away from regions of greatest adjoint and ensemble sensitivity. It is shown that this method of targeting is similar to previous ensemble-based methods that estimate forecast-error variance reduction, but easily allows for the application of statistical confidence measures to deal with sampling error.

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