Search Results

You are looking at 1 - 10 of 10 items for :

  • Author or Editor: Lloyd J. Shapiro x
  • Journal of the Atmospheric Sciences x
  • Refine by Access: All Content x
Clear All Modify Search
Lloyd J. Shapiro

Abstract

A three-layer multinested numerical model is used to evaluate the asymmetric evolution of a hurricane and its interaction with the large-scale environment. The model uses a compressible fluid in isentropic coordinates. In 72 h the hurricane vortex on a beta plane moves northwest at an average speed of 2.4 m s−1. In the presence of a westerly zonal wind in the upper model layer, the hurricane on an f plane moves to the southeast at an average speed of 0.9 m s−1.A series of experiments establishes that the southeastward drift in the presence of westerly shear is primarily due to the southward isentropic gradient of background potential vorticity (PV) in the middle model layer that is associated with the background temperature field. The cyclonic circulation advects low PV air southward on the west side of the vortex, inducing a negative isentropic PV anomaly to the southwest. This anomaly is associated with a wind field that advects the vortex to the southeast, just as the northward isentropic gradient of PV due to the beta effect advects the hurricane to the northwest. The northward gradient of background PV in the upper layer has little effect on the motion. The westerly wind advects upper-layer low PV outside the vortex core to the east, inducing an anticyclonic anomaly that tends to advect the middle-layer vortex to the north; this tendency is secondary to the motion. The role of vertical transports of momentum due to cumulus convection on the hurricane motion is also evaluated.

Results are presented that generalize the homogenization of asymmetric absolute vorticity and oscillation in relative angular momentum (RAM) found on the beta plane in a previous study with a barotropic model. Outside the vortex core and within ∼350 km of the center, the asymmetries reach a near-steady state. The middle-layer asymmetry is associated with a PV gradient that neutralizes the background gradient due to planetary vorticity or environmental temperature, thereby insulating the symmetric vortex from distortion. Horizontal fluxes in the presence of the planetary vorticity gradient tend to counteract the development of strong anticyclonic total RAM within a large circle about the vortex center.

Full access
Lloyd J. Shapiro

Abstract

The nonlinear evolution of a barotropic Rossby wave in a nonuniform basic state is studied numerically. The simulations are designed to isolate and clarify the role of advective nonlinearities in the development process. The model is barotropic, nondivergent and inviscid, on a beta-plane. The steady basic flow is zonally and meridionally nonuniform, and is maintained by a specified steady vorticity source. The wave propagates through an isolated inhomogeneity and interacts with the basic flow. Nonlinear effects are isolated by suppressing the nonlinear terms in the equations.

Two sets of experiments have been carried out. In the first the basic state is an isolated steady vortex embedded in a uniform easterly flow. A single plane wave propagates through the isolated vortex inhomogeneity. In the second the basic state is a zonally varying unstable easterly jet. The inhomogeneity is an isolated region of enhanced instability of the jet. The linearly most unstable wave mode is allowed to evolve to finite amplitude in a uniform region of the jet. The wave then propagates through the isolated region of enhanced instability.

It is found that advective nonlinearities enhance the development of the waves evolving in a nonuniform environment by allowing more effective use of sources of vorticity associated with the inhomogeneity. The nonlinearities allow fluid parcels to move more slowly and/or more directly through the vorticity source. The results are compared with both the observed development of tropical storms and previous theoretical results.

Full access
Lloyd J. Shapiro

Abstract

An investigation is made of the role of the translation of a hurricane in determining the distribution of boundary layer winds and in the organization of convection. A slab boundary layer model of constant depth is used to analyze the steady flow under a specified translating symmetric vortex in gradient balance. A truncated spectral formulation is used, including asymmetries through wavenumber 2. The role of linear and nonlinear asymmetric effects in the determination of the boundary layer response is diagnosed. These effects am relevant to relatively slowly and rapidly translating hurricanes, respectively.

The analysis is compared to observations of Hurricanes Frederic of 1979 and Allen of 1980, as well as to other observational and theoretical cures. Allen's translation speed was approximately twice that of Frederic. It is found that the simple boundary layer formulation simulates the qualitative features of the wind field observed in Frederic. The distribution of convection in Frederic and Allen compares favorably with boundary layer convergence diagnosed from the model.

Full access
Lloyd J. Shapiro

Abstract

A triggering mechanism is presented for the transformation of a wave in the easterlies to an intensifying tropical depression. Thermodynamic processes appear to be of secondary importance at this early stage of tropical storm formation. A development criterion is presented that measures the importance of nonlinear vorticity advection for the dynamics of the wave disturbance. If the contributions of the nonlinearities become significant then formation of an intensifying depression is hypothesized. The hypothesis allows one to predict the tune and place of tropical.storm development. Both climatology and the 1975 hurricane season are analyzed in order to test the theory for Atlantic easterly waves. The development criterion is found to have predictive ability in anticipating tropical storms during August and September 1975, several days prior to development.

Full access
Lloyd J. Shapiro

Abstract

The role of potential vorticity (PV) asymmetries in the evolution of a tropical cyclone is investigated using a three-layer model that includes boundary layer friction, surface moisture fluxes, and a convergence-based convective parameterization. In a benchmark experiment, a symmetric vortex is first spun up on an f plane for 24 h. The symmetric vortex has a realistic structure, including a local PV maximum inside its radius of maximum wind (RMW). A weak azimuthal-wavenumber 2 PV asymmetry confined to the lower two layers of the model is then added to the vortex near the RMW. After an additional 2 h (for a total 26-h simulation), the asymmetric PV anomaly produces changes in the symmetric vortex that have significant differences from those in dry experiments with the present model or previous barotropic studies. A diagnosis of the contributions to changes in the symmetric wind tendency due to the asymmetry confirm the dominance of horizontal eddy fluxes at early times. The barotropic eddy kick provided by the anomaly lasts ∼2 h, which is the damping timescale for the disturbance.

Additional experiments with an imposed isolated double-PV anomaly are made. Contrary to expectation from the dry experiments or barotropic studies, based on arguments involving “wave activity,” moving the anomaly closer to the center of the vortex or farther out does not change the overall evolution of the symmetric vortex. The physical mechanism responsible for the differences between the barotropic studies and those including moist physics as well as for the robustness of the response is established using a budget for the asymmetric vorticity. It is shown that the interactions between the asymmetries and the symmetric hurricane vortex at early times depend on realistic features of the model hurricane and not on interactions between the asymmetries and the boundary layer, which possibly depend on the convective parameterization. In particular, the changes in the symmetric wind tendency due to the asymmetry can be most simply explained by a combination of horizontal advection and damping of wave activity. In conjunction with horizontal advection and damping, the reversal of the radial vorticity gradient associated with the local PV maximum constrains the asymmetries to reduce the symmetric vorticity near the RMW. The location of the PV maximum controls the response to the extent that moving the PV anomaly radially inward or outward has no qualitative effect on the results. The longer-term evolution of the vortex is more problematic and may depend on the convective parameterization used.

Full access
J. Dominique Möller and Lloyd J. Shapiro

Abstract

While previous idealized studies have demonstrated the importance of asymmetric atmospheric features in the intensification of a symmetric tropical cyclone vortex, the role of convectively generated asymmetries in creating changes in the azimuthally averaged cyclone is not well understood. In the present study the full-physics nonhydrostatic fifth-generation Pennsylvania State University–National Center for Atmospheric Research (PSU–NCAR) Mesoscale Model (MM5) is used to evaluate the influence of such asymmetries. Rather than adding winds and temperatures in balance with a specified potential vorticity (PV) asymmetry, or temperature perturbations themselves, to a symmetric vortex as in previous studies, a diabatic heating asymmetry is imposed on a spunup model hurricane. The impact of short-duration eyewall-scale monochromatic azimuthal wavenumber diabatic heating on the short- and long-term evolution of the azimuthally averaged vortex is evaluated, and a tangential wind budget is made to determine the mechanisms responsible for the short-term impact.

It is found that the small eddy kick created by the additional diabatic heating asymmetry leads to a substantially amplified long-term change in the azimuthally averaged vortex, with episodes of strong relative weakening and strengthening following at irregular intervals. This behavior is diabatically controlled. It is also found that the symmetric secondary circulation can be active in creating short-term changes in the vortex, and is not simply a passive response as in previous studies with dry physics. A central conclusion of the study is that the structure of the spunup hurricane vortex, in particular preexisting asymmetric features, can have a substantial influence on the character of the response to an additional diabatic heating asymmetry. The results also imply that a small change in the factors that control convective activity will have a substantial lasting consequence for the intensification of a hurricane.

Full access
Lloyd J. Shapiro and Katsuyuki V. Ooyama

Abstract

A barotropic, primitive equation (shallow water) model is used on the beta plane to investigate the influence of divergence, total relative angular momentum (RAM) and advective nonlinearities on the evolution of a hurricane-like vortex. The multinested numerical model is based on the spectral application of a finite element representation. The undisturbed fluid depth is taken to be 1 km. Scaling of the vorticity equation, in conjunction with a Bessel function spectral decomposition, indicates that divergence should have a very small effect on the hurricane motion. Simulations with an initially symmetric cyclonic vortex in a resting environment confirm this analysis, and contradict previous published studies on the effect of divergence in a barotropic model.

During a 120 h simulation the cyclonic vortex develops asymmetries that have an influence far from the initial circulation. The total RAM within a large circle centered on the vortex decreases with time, and then oscillates about zero. For circles with radii ≲ 1000 km, the total RAM approaches, but does not reach, zero. An angular momentum budget indicates that the horizontal angular momentum flux tends to counteract the net Coriolis torque on the vortex. If the total RAM of the initial symmetric vortex is zero, the weak far-field asymmetries are essentially eliminated. The motion of the vortex is not, however, related to the RAM in any simple way.

Within a few days the near-vortex asymmetries reach a near-steady state. The Asymmetric Absolute vorticity (AAV) is nearly uniform within ∼350 km of the vortex center. The homogenization of AAV, which occurs within the closed vortex gyre, is likely due to shearing by the symmetric wind, combined with removal of energy at the smallest scales. The homogenization effectively neutralizes the planetary beta effect, as well as the vorticity associated with an environmental wind.

Full access
Lloyd J. Shapiro and Huch E. Willoughby

Abstract

Eliassen's (1951) diagnostic technique is used to calculate the secondary circulation induced by point sources of heat and momentum in balanced, hurricane-like vortices. Scale analysis reveals that such responses are independent of the horizontal scale of the vortex. Analytic solutions for the secondary circulation are readily obtained in idealized barotropic vortices, but numerical methods are required for more realistic barotropic and baroclinic vortices. For sources near the radius of maximum wind, the local, two-dimensional, streamfunction dipole response of Eliassen is modified by both the spatial variations of the vortex structure and the influences of boundary conditions.

The secondary flow advects mean-flow buoyancy and angular momentum and thus leads to a slow evolution of the vortex structure. In weak systems (maximum tangential wind <35 m s−1), the restraining influences of structure and boundaries lengthen the time scale of the vortex evolution. In stronger vortices, the horizontal scale of the response is smaller, the restraining influences are less important, and the evolution is faster. When the maximum wind exceeds 35 m s−1, recirculation of air within the vortex core tends to form an eye.

The most rapid temporal changes in tangential wind lie inside the eye, where the horizontal gradients of angular momentum are strongest. In most cases, the tangential wind increases most rapidly just inside the radius of maximum wind and decreases near the central axis of the vortex. This effect leads to contraction of the wind maximum as the vortex intensifies. The present results are compared with observations and other theoretical mutts.

Full access
Lloyd J. Shapiro and Michael T. Montgomery

Abstract

A three-dimensional balance formulation for rapidly rotating vortices, such as hurricanes, is presented. The asymmetric balance (AB) theory represents a new mathematical framework for studying the slow evolution of rapidly rotating fluid systems. The AB theory is valid for large Rossby number; it makes no formal restriction on the magnitude of the divergence or vertical advection, which need not be small. The AB is an ordered expansion in the square of the ratio of orbital to inertial frequencies, the square of a local Rossby number. The approximation filters gravity and inertial waves from the system. Advantage is taken of the weak asymmetries near the vortex care as well as the tendency for low azimuthal wavenumber asymmetries to dominate. Linearization about a symmetric balanced vortex allows the three-dimensional asymmetric dynamics to be deduced properly. The AB formulation has a geopotential tendency equation with a three-dimensional elliptic operator. The AB system has a uniformly valid continuation to nonlinear quasigeostrophic theory in the environment. It includes the full inertial dynamics of the vortex core, and reduces to Eliassen's formulation for purely axi-symmetric flow. It has a full set of conservation laws on fluid parcels analogous to those for primitive equations, including conservation of potential temperature, potential vorticity, three-dimensional vorticity, and energy. A weakly nonlinear extension of the formulation in the near-vortex region is presented. Appropriate physical applications for the AB system, as well as its limitations, are discussed.

Full access
Michael T. Montgomery and Lloyd J. Shapiro

Abstract

A generalized Charney–Stern theorem for rapidly rotating (large Rossby number) baroclinic vortices, such as hurricanes, is derived based on the asymmetric balance (AB) approximation. In the absence of dissipative processes, a symmetrically stable baroclinic vortex is shown to be exponentially stable to nonaxisymmetric perturbations if a generalized potential vorticity gradient on theta surfaces remains single signed throughout the vortex. The generalized potential vorticity gradient involves the sum of an interior potential vorticity gradient associated with the symmetric vortex and surface contributions associated with the vertical shear of the tangential wind. The AB stability formulation is then shown to yield Fjortoft's theorem as a corollary.

In the modem view of shear instabilities the theorems admit simple interpretation. The Charney–Stern theorem represents a necessary condition for the existence of counterpropagating Rossby waves associated with the radial potential vorticity gradient, while Fjortoft's theorem represents a necessary condition for these waves to phase lock and grow in strength. Potential application of these results as well as limitations of the slow-manifold approach are briefly discussed.

Full access