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Jeffrey M. Chagnon

Abstract

The effect of the dynamical response associated with high-frequency gravity waves on the total energy generated by imposed heating is examined in a 2D linear compressible model. The work performed by waves against a sustained forcing is defined as the dynamical resistance. The dynamical resistance is minimized and forcing efficiency maximized for basic-state and forcing configurations that yield a wave response whose phase varies minimally relative to the forcing. When generated against a forcing-relative background flow, waves that have a deep vertical scale relative to the forcing depth impose less resistance than waves of a shallow vertical scale. The efficiency of an ensemble of forcing elements is shown to differ significantly from that corresponding to an isolated forcing. If the forcing elements are all of the same sign (e.g., are all warmings), then the efficiency increases with decreasing separation between elements.

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Jeffrey M. Chagnon and Peter R. Bannon

Abstract

This study compares the response to injections of mass, heat, and momentum during hydrostatic and geostrophic adjustment in a compressible atmosphere. The sensitivity of the adjustment to these different injection types is examined at varying spatial and temporal scales through analysis of the transient evolution of the fields as well as the partitioning of total energy between acoustic waves, buoyancy waves, Lamb waves, and the steady state.

The effect of a cumulus cloud on its larger-scale environment may be represented as a vertical mass source/sink and a localized warming. To examine how the response to such injections may differ, injections of mass and heat that generate identical potential vorticity (PV) distributions and, hence, identical steady states, are compared. When the duration of the injection is very short (e.g., a minute or less), the injection of mass generates a very large acoustic wave response relative to the PV-equivalent injection of heat. However, the buoyancy wave response to these two injection types is quite similar.

The responses to injections of divergent momentum in the vertical and horizontal directions are also compared. It is shown that neither divergent momentum injection generates any PV and, thus, there is no steady-state response to these injections. The waves excited by these injections generally propagate their energy in the direction of the injection. Consequently, an injection of vertical momentum is an efficient generator of vertically propagating, horizontally trapped, high-frequency buoyancy waves. Such waves have a short time scale and are therefore very sensitive to the injection duration. Analogously, an injection of divergent horizontal momentum is an efficient generator of horizontally propagating, vertically trapped low-frequency buoyancy waves that are relatively insensitive to the injection duration. Because of this difference in the response to horizontal and vertical injections of momentum, the response to the injection of an isolated updraft differs depending on whether a compensating horizontal inflow/outflow is also specified. This additional specification of inflow/outflow helps filter acoustic waves and encourages a stronger updraft that is not removed as rapidly by the buoyancy waves. This finding is relevant to the initialization of updrafts in compressible numerical weather prediction models.

Injection of nondivergent momentum generates waves in the regions of convergence/divergence produced by the deflection of the current by Coriolis forces. The energy partitioning for such an injection is sensitive to the width and depth of the current relative to the Rossby radius of deformation, but the response is insensitive to the duration of injection for time scales shorter than several hours.

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Allison Lynn Brannan and Jeffrey M. Chagnon

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Previous case studies have noted a significant extratropical flow response to recurving Atlantic tropical cyclones (TCs), which is often linked to extreme weather events downstream. This study examines the modification of Rossby waves on the extratropical jet in response to recurving Atlantic TCs from a climatological perspective. Changes in amplitude and location of Rossby waves are identified using a wavelet decomposition technique on isentropic potential vorticity. The climatology demonstrates that recurving Atlantic TC events are capable of modifying the amplitude of the extratropical flow. Though the majority of TCs did not produce a significant, systematic modification of the extratropical flow amplitude, a subset of events were associated with a period of significant Rossby wave deamplification occurring from the time of recurvature to 48 h after recurvature, followed by a return of the Rossby wave power beginning around 96 h after recurvature. The characteristics of the TCs were not significantly associated with the resulting extratropical flow modification—a result consistent with previous western North Pacific climatologies. The nature of the extratropical flow response is most strongly tied to the average translation speed of the TC relative to the Rossby wave over the 72 h following recurvature. This study highlights the importance of investigating the extratropical flow response to recurving Atlantic TCs with regards to predictability.

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Jeffrey M. Chagnon and Peter R. Bannon

Abstract

The initial and steady-state response of a compressible atmosphere to an instantaneous, localized heat source is investigated analytically. Potential vorticity conservation removes geostrophic and hydrostatic degeneracy and provides a direct method for obtaining the steady-state solution. The heat source produces a vertical potential vorticity dipole that induces a hydrostatically and geostrophically balanced cyclone–anticyclone structure in the final state. For a typical deep mesoscale heating, the net displacements required for the adjustment to the final steady state include a small, O(100 m) ascent of the core of the heated air with weak far-field descent and a large, O(10 km) outward/inward lateral displacement at the top/base of the heating.

The heating initially generates available elastic and potential energy. The energy is then exchanged between kinetic, elastic, potential, and acoustic and gravity wave energy. In the final state, after the acoustic and gravity wave energy has dispersed, the remaining energy is partitioned between kinetic, and available potential and elastic energy. The fraction of wave energy increases with increasing horizontal wavenumber.

The effect of several vertical boundary conditions is assessed. It is shown that a rigid lid suppresses the vertical expansion of the heated layer and reduces the fraction of wave energy. The impact of the rigid lid on the steady-state solution is maximized for the horizontal wavenumber zero solution and when the heating takes place close to the rigid upper boundary.

The compressible solution is used as a prototype for comparing and evaluating several compressibility approximations: the anelastic, pseudo-incompressible, and modified-compressible approximations. The anelastic model omits the available elastic energetics entirely, but the pseudo-incompressible and modified-compressible models omit either its generation or storage. The result is an ambiguous projection of heating energy onto the remaining energy terms. The errors associated with these approximations are only significant on synoptic scales. Furthermore, the modified-compressible set does not conserve potential vorticity globally.

The initial response to the heating differs for each approximation. Although the initial compressible response consists of pressure and potential temperature anomalies confined to the heated layer, the modified-compressible atmosphere generates density and potential temperature anomalies but no pressure anomaly. The anelastic atmosphere undergoes an instantaneous acoustic adjustment in which pressure and density anomalies exist inside and outside of the heated region. The pseudo-incompressible atmosphere generates an instantaneous, net divergence characterized by a residual velocity remaining after the heating and an instantaneous pulse in the pressure and velocity fields.

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Jeffrey M. Chagnon and Peter R. Bannon

Abstract

The adjustment of a compressible, stably stratified atmosphere to sources of hydrostatic and geostrophic imbalance is investigated using a linear model. Imbalance is produced by prescribed, time-dependent injections of mass, heat, or momentum that model those processes considered “external” to the scales of motion on which the linearization and other model assumptions are justifiable. Solutions are demonstrated in response to a localized warming characteristic of small isolated clouds, larger thunderstorms, and convective systems.

For a semi-infinite atmosphere, solutions consist of a set of vertical modes of continuously varying wavenumber, each of which contains time dependencies classified as steady, acoustic wave, and buoyancy wave contributions. Additionally, a rigid lower-boundary condition implies the existence of a discrete mode—the Lamb mode— containing only a steady and acoustic wave contribution. The forced solutions are generalized in terms of a temporal Green's function, which represents the response to an instantaneous injection.

The response to an instantaneous warming with geometry representative of a small, isolated cloud takes place in two stages. Within the first few minutes, acoustic and Lamb waves accomplish an expansion of the heated region. Within the first quarter-hour, nonhydrostatic buoyancy waves accomplish an upward displacement inside of the heated region with inflow below, outflow above, and weak subsidence on the periphery—all mainly accomplished by the lowest vertical wavenumber modes, which have the largest horizontal group speed. More complicated transient patterns of inflow aloft and outflow along the lower boundary are accomplished by higher vertical wavenumber modes. Among these is an outwardly propagating rotor along the lower boundary that effectively displaces the low-level inflow upward and outward.

A warming of 20 min duration with geometry representative of a large thunderstorm generates only a weak acoustic response in the horizontal by the Lamb waves. The amplitude of this signal increases during the onset of the heating and decreases as the heating is turned off. The lowest vertical wavenumber buoyancy waves still dominate the horizontal adjustment, and the horizontal scale of displacements is increased by an order of magnitude. Within a few hours the transient motions remove the perturbations and an approximately trivial balanced state is established.

A warming of 2 h duration with geometry representative of a large convective system generates a weak but discernible Lamb wave signal. The response to the conglomerate system is mainly hydrostatic. After several hours, the only signal in the vicinity of the heated region is that of inertia–gravity waves oscillating about a nontrivial hydrostatic and geostrophic state.

This paper is the first of two parts treating the transient dynamics of hydrostatic and geostrophic adjustment. Part II examines the potential vorticity conservation and the partitioning of total energy.

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Jeffrey M. Chagnon and Suzanne L. Gray

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The structure of near-tropopause potential vorticity (PV) acts as a primary control on the evolution of extratropical cyclones. Diabatic processes such as the latent heating found in ascending moist warm conveyor belts modify PV. A dipole in diabatically generated PV (hereafter diabatic PV) straddling the extratropical tropopause, with the positive pole above the negative pole, was diagnosed in a recently published analysis of a simulated extratropical cyclone. This PV dipole has the potential to significantly modify the propagation of Rossby waves and the growth of baroclinically unstable waves. This previous analysis was based on a single case study simulated with 12-km horizontal grid spacing and parameterized convection. Here the dipole is investigated in three additional cold-season extratropical cyclones simulated in both convection-parameterizing and convection-permitting model configurations. A diabatic PV dipole across the extratropical tropopause is diagnosed in all three cases. The amplitude of the dipole saturates approximately 36 h from the time diabatic PV is accumulated. The node elevation of the dipole varies between 2 and 4 PVU (1 PVU = 106 K kg−1 m2 s−1) in the three cases, and the amplitude of the system-averaged dipole varies between 0.2 and 0.4 PVU. The amplitude of the negative pole is similar in the convection-parameterizing and convection-permitting simulations. The positive pole, which is generated by longwave radiative cooling, is weak in the convection-permitting simulations due to the small domain size, which limits the accumulation of diabatic tendencies within the interior of the domain. The possible correspondence between the diabatic PV dipole and the extratropical tropopause inversion layer is discussed.

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Jeffrey M. Chagnon and Peter R. Bannon

Abstract

This second part of a two-part study of the hydrostatic and geostrophic adjustment examines the potential vorticity and energetics of the acoustic waves, buoyancy waves, Lamb waves, and steady state that are generated following the prescribed injection of heat into an isothermal atmosphere at rest. The potential vorticity is only nonzero for the steady class and depends only on the spatial and time-integrated properties of the injection. The waves contain zero net potential vorticity, but undergo a time-dependent vorticity exchange involving latent and relative vorticities.

The energy associated with a given injection may be partitioned distinctly among the various wave classes. The characteristics of this partitioning depend on the spatiotemporal detail of the injection, as well as whether the imbalance is generated by injection of heat, mass, or momentum. Spatially, waves of a scale similar to that of the injection are preferentially excited. Temporally, an extended duration injection preferentially filters high-frequency waves. An instantaneous injection, that is, the temporal Green’s function, contains the largest proportions of the high-frequency waves.

The proportions of kinetic, available elastic, and available potential energies that are carried by the various waves are functions of the homogeneous system. For example, deep buoyancy waves of small horizontal scale primarily contain equal portions of available potential and vertical kinetic energy. The steady state contains more available potential energy than kinetic energy at small horizontal scale, and vice versa. These qualities of the wave energetics illustrate the mechanisms that characterize the physics of each wave class.

The evolution and spectral partitioning of the energetics following localized warmings identical to those in Part I are presented in order to illustrate some of these basic properties of the energetics. For example, a heating lasting longer than a few minutes does not excite acoustic waves. However, Lamb waves of wide horizontal scale can be excited by a heating of several hours. The first buoyancy waves to be filtered by an extended duration heating are those of the deepest and narrowest structure that have a frequency approaching the buoyancy frequency. The energetics of the steady state depends only on the spatial and time-integrated properties of the warming. However, the energetics and transient evolution toward a given steady state depend on the temporal properties of the warming and may differ widely.

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Ming Cai, Cory Barton, Chul-Su Shin, and Jeffrey M. Chagnon

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The continuous mutual evolution of equatorial waves and the background quasi-biennial oscillation (QBO) is demonstrated using daily NCEP–U.S. Department of Energy (DOE) reanalysis for the period from 1 January 1979 to 31 December 2010. Using a novel diagnostic technique, the phase speed, vertical tilting, and form stress of equatorial waves in the stratosphere are obtained continuously on a daily basis. The results indicate that, on top of a weak-amplitude annual-cycle signal, all of these wave properties have a pronounced QBO signal with a downward propagation that evolves continuously together with the background QBO. The analysis also highlights the potential role of wave-induced form stress in driving the QBO regime change.

Dominant waves in the equatorial stratosphere propagate very slowly relative to the ground at all times, implying that their observed intrinsic phase speed evolution follows the background QBO nearly exactly but with opposite sign, as the established theory predicts. By revealing the continuous evolution of the form stress associated with the vertically tilted waves, the new diagnostic method also demonstrates the dominance of eastward-tilted, eastward-propagating waves contributing to a deceleration of easterly flow at high altitudes, which causes a downward propagation of the easterly flow signal. Similarly, the dominance of westward-tilted, westward-propagating waves acts to reverse westerly flow to easterly flow and causes a downward propagation of westerly flow signal. The results suggest that in addition to the wave-breaking processes, such continuously alternating downward transfer of westerly and easterly angular momentum by westward-tilted, westward-propagating waves and eastward-tilted, eastward-propagating waves contributes to the wave–mean flow interaction mechanism for the QBO.

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Peter R. Bannon, Jeffrey M. Chagnon, and Richard P. James

Abstract

Numerical anelastic models solve a diagnostic elliptic equation for the pressure field using derivative boundary conditions. The pressure is therefore determined to within a function proportional to the base-state density field with arbitrary amplitude. This ambiguity is removed by requiring that the total mass be conserved in the model. This approach enables one to determine the correct temperature field that is required for the microphysical calculations. This correct, mass-conserving anelastic model predicts a temperature field that is an accurate approximation to that of a compressible atmosphere that has undergone a hydrostatic adjustment in response to a horizontally homogeneous heating or moistening. The procedure is demonstrated analytically and numerically for a one-dimensional, idealized heat source and moisture sink associated with moist convection. Two-dimensional anelastic simulations compare the effect of the new formulation on the evolution of the flow fields in a simulation of the ascent of a warm bubble in a conditionally unstable model atmosphere.

In the Boussinesq case, the temperature field is determined uniquely from the heat equation despite the fact that the pressure field can only be determined to within an arbitrary constant. Boussinesq air parcels conserve their volume, not their mass.

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