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Melinda S. Peng
and
Carolyn A. Reynolds

Abstract

Singular vector (SV) sensitivity, calculated using the adjoint model of the U.S. Navy Operation Global Atmosphere Prediction System (NOGAPS), is used to study the dynamics associated with tropical cyclone evolution. For each model-predicted tropical cyclone, SVs are constructed that optimize perturbation energy within a 20° by 20° latitude/longitude box centered on the 48-h forecast position of the cyclone. The initial SVs indicate regions where the 2-day forecast of the storm is very sensitive to changes in the analysis. Composites of the SVs for straight-moving cyclones and non-straight-moving cyclones that occurred in the Northern Hemisphere during its summer season in 2003 are examined. For both groups, the initial-time SV sensitivity exhibits a maximum within an annulus approximately 500 km from the center of the storms, in the region where the potential vorticity gradient of the vortex first changes sign. In the azimuthal direction, the composite initial-time SV maximum for the straight-moving group is located in the rear right quadrant with respect to the storm motion. The composite based on the non-straight-moving cyclones does not have a preferred quadrant in the vicinity of the storms and has larger amplitude away from the cyclones compared with the straight-moving storms, indicating more environmental influence on these storms. For both groups, the maximum initial sensitive areas are collocated with regions of flow moving toward the storm.

While the initial SV maximum is located where the potential vorticity gradient changes sign, the final SV maximum is located where the potential vorticity gradient is a maximum. Examinations of individual cases demonstrate how SV sensitivity can be used to identify specific environmental influences on the storms. The relationship between the SV sensitivity and the potential vorticity is discussed. The results support the utility of SVs in applications to phenomena beyond midlatitude baroclinic systems.

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Melinda S. Peng
and
R. T. Williams

Abstract

This note compares the error distributions for three transformation formulae between temporal growth rate and spatial growth rate with the linearized barotropic vorticity equation. The sech2 and the tanh basic-state profiles are used for illustration. The transformation which uses the phase velocity gives a moderate error which does not have a strong dependence on the growth rate. The formulae derived by Gaster, and later by Nayfeh and Padhye, which employ the group velocity, have errors that are a function of the ratio of the spatial growth rate to the wavenumber. The errors from their formulae are small when the ratio is small, but the errors increase with the ratio so that all three transformation formulae give similar errors when the ratio is of order one. Nayfeh and Padhye's formula is rederived for the barotropic vorticity equation with a procedure which shows that ratio of growth rate to wavenumber must be small for accuracy.

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Melinda S. Peng
and
R. T. Williams

Abstract

Spatial baroclinic instability in a mean flow with slow streamwise variation is studied with the quasi-geostrophic two-layer model. The two-scale expansion technique which was employed by Peng and Williams is used in this study. The zero-order terms give the local spatial instability solution. The next order terms determine the correction to the local solution due to the streamwise variation of the mean flow. It is found that this correction is not negligible when the β effect is large and the vertical shear is small. The results are explained with the lag effect, which was discussed by Peng and Williams. The lag effect occurs when the local solution changes its structure substantially in the streamwise direction. When the vertical shear is large or when the β effect is small, the ratio between the disturbances of the two layers is nearly uniform in the streamwise direction, even though the shear changes substantially. Thus, only a small lag effect is experienced by a disturbance as it propagates, and the streamwise effect is unimportant. The dependence of the vertical structure on the basic flow variation and other parameters is analyzed.

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Melinda S. Peng
and
R. T. Williams

Abstract

A two-scale expansion technique is used to study the barotropic instability of basic flows with slow streamwise variation. Disturbances in nonparallel flow possess properties that differ from those calculated from parallel flow theory. The difference, which is obtained at higher order in the parameter that measures the nonparallelism, depends on the first derivative of the parallel flow properties with respect to the streamwise direction. This higher order correction shifts the spatial growth rate profile for the nonparallel flow downstream relative to the spatial growth rate profile for parallel flow. These results are compared with a previous numerical study by Tupaz, Williams and Chang and some of their conclusions are modified.

Physically, the difference in the spatial instability for parallel and nonparallel flow is subject to two combined effects. The first is the lag effect discussed by Tupaz et al., which causes the disturbance structure to lag the parallel-flow solution structure in regions where the mean flow changes rapidly downstream. This causes the downstream shifting of the nonparallel growth rate profile. The second is related to the phase speed difference between the parallel and nonparallel flows. If the disturbance propagates faster than predicted by the parallel flow theory, the local spatial growth rate will be smaller than that calculated by the parallel flow and vice versa.

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Melinda S. Peng
and
R. T. Williams

Abstract

A linear nondivergent barotropic model is developed to obtain the asymmetric circulation associated with a vortex moving on the β-plane. The total system is transformed to a coordinate system moving with the vortex. The direction and speed of movement is specified from full nonlinear model results. Two wavenumber one gyres are obtained from the asymmetric vorticity equation. The inner gyres move in the azimuthal direction whose maximum amplitude is located at the radius of maximum wind. These inner gyres are associated either with the unstable mode or the neutral mode depending on the resolution of the model. The outer gyres, whose orientations are always along the track direction specified by the movement, correspond to the β-gyres obtained in the nonlinear numerical model. The strength of the inner gyres is much larger than the strength of the outer gyres. For the steady state solution with high finite difference resolution, only the inner gyres are present. In a steady state solution, the outer β-gyres can be isolated by modifying the inner part of the basic wind profile or by reducing the resolution of the mode. In a time dependent solution, the inner gyres will not form if there is no discrete mode existing in the free model system. The outer β-gyres thus obtained have the correct orientation and magnitude when compared to the solutions of the full nonlinear model. These solutions can be used as a tool for bogusing the vortex into a numerical hurricane forecast model.

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Melinda S. Peng
,
Bao-Fong Jeng
, and
R. T. Williams

Abstract

The effect of planetary vorticity gradient (beta) and the presence of a uniform mean flow on the intensification of tropical cyclones are studied using a limited-area primitive equation model. The most intense storm evolves on a constant-f plane with zero-mean flow and its structure is symmetric with respect to the vortex center. The presence of an environmental flow induces an asymmetry in a vortex due to surface friction. When f varies the vortex is distorted by the beta gyres. Fourier analysis of the wind field shows that a deepening cyclone is associated with a small asymmetry in the low-level wavenumber-one wind field. A small degree of asymmetry in the wind field allows a more symmetric distribution of the surface fluxes and low-level moisture convergence. On the other hand, a weakening or nonintensifying cyclone is associated with a larger asymmetry in its wavenumber-one wind field. This flow pattern generates asymmetric moisture convergence and surface fluxes and a phase shift may exist between their maxima. The separation of the surface flux maximum and the lateral moisture convergence reduces precipitation and inhibits the development of the tropical cyclone. Since the orientation of the asymmetric circulation induced by beta is in the southeast to northwest direction, the asymmetry induced by a westerly flow partially cancels the beta effect asymmetry while that of an easterly flow enhances it. Therefore, in a variable-f environment, westerly flows are more favorable for tropical cyclone intensification than easterly flows of the same speed.

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Xuyang Ge
,
Tim Li
,
Yuqing Wang
, and
Melinda S. Peng

Abstract

The three-dimensional (3D) Rossby wave energy dispersion of a tropical cyclone (TC) is studied using a baroclinic primitive equation model. The model is initialized with a symmetric vortex on a beta plane in an environment at rest. The vortex intensifies while becoming asymmetric and moving northwestward because of the beta effect. A synoptic-scale wave train forms in its wake a few days later. The energy-dispersion-induced Rossby wave train has a noticeable baroclinic structure with alternating cyclonic–anticyclonic–cyclonic (anticyclonic–cyclonic–anticyclonic) circulations in the lower (upper) troposphere.

A key feature associated with the 3D wave train development is a downward propagation of the relative vorticity and kinetic energy. Because of the vertical differential inertial stability, the upper-level wave train develops faster than the lower-level counterpart. The upper anticyclonic circulation rapidly induces an intense asymmetric outflow jet in the southeast quadrant, and then further influences the lower-level Rossby wave train. On one hand, the outflow jet exerts an indirect effect on the lower-level wave train strength through changing TC intensity and structure. On the other hand, it triggers downward energy propagation that further enhances the lower-level Rossby wave train. A sudden removal of the diabatic heating may initially accelerate the energy dispersion through the increase of the radius of maximum wind and the reduction of the lower-level inflow. The latter may modulate the group velocity of the Rossby wave train through the Doppler shift effect. The 3D numerical results illustrate more complicated Rossby wave energy dispersion characteristics than 2D barotropic dynamics.

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Shang-Wu Li
,
Melinda S. Peng
, and
R. T. Williams

Abstract

The objective of this study is to investigate mountain effects on a frontal system in three dimensions. The frontal system is developed from the most unstable Eady wave in a baroclinic state without a mountain. The developed frontal system is then introduced into a new model domain that contains mountains with different sizes, shapes, and orientations. In general, it is found that the cold front experiences a weakening on the upwind slope and strengthening on the downwind slope of a mountain. The locations of these upwind and downwind sides are determined by the horizontal winds associated with the front. Before the front reaches a mountain, the prevailing wind impinging on the mountain is the prefrontal southwesterly. After the front reaches the top of the mountain, the impinging wind shifts to be the postfrontal northwesterly. Therefore, mountain-induced fronto-genetic forcing by these winds varies spatially as the front passes the mountain. When the front moves down the slope, it speeds up and the frontal deformation is then caused by the strong advection over the northern part of the mountain. After the front has moved away from the mountain, its original horizontal structure and location are restored. The frontogenetic forcing is dominated mainly by the convergence-divergence associated with the flow over the mountain. The front experiences major intensification when it is in the leeside convergence zone. As the front moves farther downstream, it enters the divergence zone and its intensity is reduced. When the front has moved away from the influence of the mountain, its intensity returns approximately to its original level irrespective of the mountain's size and shape. The postfrontal winds contribute to the strong convergence, which causes enhanced lee frontogenesis. For an east-west oriented elliptic mountain that resembles the Alps, the Ieeside downslope wind induced by the postfrontal flow is toward the south instead of toward the east as in the other cases. Therefore, the front moves with an average speed that is the same as the front with no mountain. In this case, the front also has a net increase in its intensity for the same period of integration. Simulations with this mountain profile compare favorably with many observed phenomena near the Alps. Overall, the most important factor that determines the net effect of the mountain on a front is its orientation relative, to the front.

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R. T. Williams
,
Melinda S. Peng
, and
D. A. Zankofski

Abstract

The hydrostatic Boussinesq equations are used to simulate the passage of fronts over a two-dimensional mountain in a cyclic domain. The fronts are forced by a confluent, periodic deformation field that moves with the uniform mean flow over the mountain. The initial conditions are selected to give a cold front confined to the lower part of the domain. Fourth-order diffusion terms are included in the numerical model to control energy cascade to the grid size scale. A numerical frontogenesis experiment with no topography produces a realistic surface front in about two days. Numerical solutions for flow over the mountain with no front are found by integrating the equations from the initial conditions, which are semigeostrophic steady-state solutions. Various mountains are considered that have the same height but different widths. The numerical solutions for wide mountains remain close to the semigeostrophic initial conditions, while for narrower mountains vertically propagating waves and a hydraulic jump develop on the lee side of the mountain. The frontal solution and the mountain solution are combined to produce the initial conditions for the basic experiments. The numerical solutions show reduced frontogenesis on the upwind slope and increased frontogenesis on the lee slope. This behavior is caused by the mountain-forced divergence on the upwind side and convergence on the lee side in agreement with the semigeostrophic solution of Zehnder and Bannon. Further experiments with no deformation forcing are carried out to correspond to the semigeostrophic passive scalar studies of Blumen and Gross. A passive scalar that represents the perturbation potential temperature is advected with the mountain solution. The frontal scale, based on the tracer field, increases on the upwind side until it reaches a maximum at the top and then decreases on the lee side, back to its original value as the front moves away from the mountain. The numerical solutions for the interactive potential temperature field have a similar behavior, although some additional blocking effects are present. For the narrower mountains the frontal structure is distorted by the gravity waves on the lee side of the mountain. These solutions resemble those of Schumann for smaller-scale mountains.

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Eric A. Hendricks
,
Wayne H. Schubert
,
Yu-Han Chen
,
Hung-Chi Kuo
, and
Melinda S. Peng

Abstract

A forced shallow-water model is used to understand the role of diabatic and frictional effects in the generation, maintenance, and breakdown of the hurricane eyewall potential vorticity (PV) ring. Diabatic heating is parameterized as an annular mass sink of variable width and magnitude, and the nonlinear evolution of tropical storm–like vortices is examined under this forcing. Diabatic heating produces a strengthening and thinning PV ring in time due to the combined effects of the mass sink and radial PV advection by the induced divergent circulation. If the forcing makes the ring thin enough, then it can become dynamically unstable and break down into polygonal asymmetries or mesovortices. The onset of barotropic instability is marked by simultaneous drops in both the maximum instantaneous velocity and minimum pressure, consistent with unforced studies. However, in a sensitivity test where the heating is proportional to the relative vorticity, universal intensification occurs during barotropic instability, consistent with a recent observational study. Friction is shown to help stabilize the PV ring by reducing the eyewall PV and the unstable-mode barotropic growth rate. The radial location and structure of the heating is shown to be of critical importance for intensity variability. While it is well known that it is critical to heat in the inertially stable region inside the radius of maximum winds to spin up the hurricane vortex, these results demonstrate the additional importance of having the net heating as close as possible to the center of the storm, partially explaining why tropical cyclones with very small eyes can rapidly intensify to high peak intensities.

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