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Abstract
The question of convective (i.e., spatial) instability of baroclinic waves on an f-plane is studied in the context of the two-layer model. The viscous and inviscid marginal curves for linear convective instability are obtained. The finite-amplitude problem shows that when dissipation is O(1) it acts to stabilize the waves that are of Eady type. For very small dissipation the weakly nonlinear analysis reveals that at low frequencies, contrary to what is known to occur in the temporal problem, in addition to the baroclinic component a barotropic correction to the “mean” flow is generated by the nonlinearities, and spatial equilibration occurs provided the ratio of shear to mean flow does not exceed some critical value. In the same limit, the slightly dissipative nonlinear dynamics reveals the presence of large spatial vacillations immediately downstream of the source, even if asymptotically (i.e., very far away from the source) the amplitudes are found to reach steady values. No case of period doubling or aperiodic behavior was found. The results obtained seem to be qualitatively independent of the form chosen to model the dissipation.
Abstract
The question of convective (i.e., spatial) instability of baroclinic waves on an f-plane is studied in the context of the two-layer model. The viscous and inviscid marginal curves for linear convective instability are obtained. The finite-amplitude problem shows that when dissipation is O(1) it acts to stabilize the waves that are of Eady type. For very small dissipation the weakly nonlinear analysis reveals that at low frequencies, contrary to what is known to occur in the temporal problem, in addition to the baroclinic component a barotropic correction to the “mean” flow is generated by the nonlinearities, and spatial equilibration occurs provided the ratio of shear to mean flow does not exceed some critical value. In the same limit, the slightly dissipative nonlinear dynamics reveals the presence of large spatial vacillations immediately downstream of the source, even if asymptotically (i.e., very far away from the source) the amplitudes are found to reach steady values. No case of period doubling or aperiodic behavior was found. The results obtained seem to be qualitatively independent of the form chosen to model the dissipation.
Abstract
The behavior of an isolated vortex perturbed by topographically forced Rossby waves is studied using the method of Contour Dynamics. For a single-contour vortex a distinct forcing threshold exists above which the wave breaks in a dynamically significant way, leading to a disruption of the vortex. This breaking is distinguished from the process of weak filamentary breaking described by Dritschel and classified here as microbreaking; the latter occurs in nondivergent flow even at very small forcing amplitudes but does not affect the vortex in a substantial manner. In cases with finite Rossby deformation radius (comparable with the vortex radius) neither breaking nor microbreaking occurs below the forcing threshold. In common with previous studies using high-resolution spectral models, the vortex is not diluted by intrusion of outside air, except during remerger with a secondary vortex shed previously from the main vortex during a breaking event. The kinematics of the breaking process and of the vortex interior and the morphology of material ejected from the vortex are described. When the Rossby radius is finite there is substantial mixing in the deep interior of the vortex, even when the vortex is only mildly disturbed. Implications for the stratospheric polar vortex are discussed.
Abstract
The behavior of an isolated vortex perturbed by topographically forced Rossby waves is studied using the method of Contour Dynamics. For a single-contour vortex a distinct forcing threshold exists above which the wave breaks in a dynamically significant way, leading to a disruption of the vortex. This breaking is distinguished from the process of weak filamentary breaking described by Dritschel and classified here as microbreaking; the latter occurs in nondivergent flow even at very small forcing amplitudes but does not affect the vortex in a substantial manner. In cases with finite Rossby deformation radius (comparable with the vortex radius) neither breaking nor microbreaking occurs below the forcing threshold. In common with previous studies using high-resolution spectral models, the vortex is not diluted by intrusion of outside air, except during remerger with a secondary vortex shed previously from the main vortex during a breaking event. The kinematics of the breaking process and of the vortex interior and the morphology of material ejected from the vortex are described. When the Rossby radius is finite there is substantial mixing in the deep interior of the vortex, even when the vortex is only mildly disturbed. Implications for the stratospheric polar vortex are discussed.
Abstract
The three-dimensional nature of breaking Rossby waves in the polar wintertime stratosphere is studied using an idealized global primitive equation model. The model is initialized with a well-formed polar vortex, characterized by a latitudinal band of steep potential vorticity (PV) gradients. Planetary-scale Rossby waves are generated by varying the topography of the bottom boundary, corresponding to undulations of the tropopause. Such topographically forced Rossby waves then propagate up the edge of the vortex, and their amplification with height leads to irreversible wave breaking.
These numerical experiments highlight several nonlinear aspects of stratospheric dynamics that are beyond the reach of both isentropic two-dimensional models and fully realistic GCM simulations. They also show that the polar vortex is contorted by the breaking Rossby waves in a surprisingly wide range of shapes.
With zonal wavenumber-1 forcing, wave breaking usually initiates as a deep helical tongue of PV that is extruded from the polar vortex. This tongue is often observed to roll up into deep isolated columns, which, in turn, may be stretched and tilted by horizontal and vertical shears. The wave amplitude directly controls the depth of the wave breaking region and the amount of vortex erosion. At large forcing amplitudes, the wave breaking in the middle/lower portions of the vortex destroys the PV gradients essential for vertical propagation, thus shielding the top of the vortex from further wave breaking.
The initial vertical structure of the polar vortex is shown to play an important role in determining the characteristics of the wave breaking. Perhaps surprisingly, initially steeper PV gradients allow for stronger vertical wave propagation and thus lead to stronger erosion. Vertical wind shear has the notable effect of tilting and stretching PV structures, and thus dramatically accelerating the downscale stirring. An initial decrease in vortex area with increasing height (i.e., a conical shape) leads to focusing of wave activity, which amplifies the wave breaking. This effect provides a geometric interpretation of the “preconditioning” that often precedes a stratospheric sudden warming event. The implications for stratospheric dynamics of these and other three-dimensional vortex properties are discussed.
Abstract
The three-dimensional nature of breaking Rossby waves in the polar wintertime stratosphere is studied using an idealized global primitive equation model. The model is initialized with a well-formed polar vortex, characterized by a latitudinal band of steep potential vorticity (PV) gradients. Planetary-scale Rossby waves are generated by varying the topography of the bottom boundary, corresponding to undulations of the tropopause. Such topographically forced Rossby waves then propagate up the edge of the vortex, and their amplification with height leads to irreversible wave breaking.
These numerical experiments highlight several nonlinear aspects of stratospheric dynamics that are beyond the reach of both isentropic two-dimensional models and fully realistic GCM simulations. They also show that the polar vortex is contorted by the breaking Rossby waves in a surprisingly wide range of shapes.
With zonal wavenumber-1 forcing, wave breaking usually initiates as a deep helical tongue of PV that is extruded from the polar vortex. This tongue is often observed to roll up into deep isolated columns, which, in turn, may be stretched and tilted by horizontal and vertical shears. The wave amplitude directly controls the depth of the wave breaking region and the amount of vortex erosion. At large forcing amplitudes, the wave breaking in the middle/lower portions of the vortex destroys the PV gradients essential for vertical propagation, thus shielding the top of the vortex from further wave breaking.
The initial vertical structure of the polar vortex is shown to play an important role in determining the characteristics of the wave breaking. Perhaps surprisingly, initially steeper PV gradients allow for stronger vertical wave propagation and thus lead to stronger erosion. Vertical wind shear has the notable effect of tilting and stretching PV structures, and thus dramatically accelerating the downscale stirring. An initial decrease in vortex area with increasing height (i.e., a conical shape) leads to focusing of wave activity, which amplifies the wave breaking. This effect provides a geometric interpretation of the “preconditioning” that often precedes a stratospheric sudden warming event. The implications for stratospheric dynamics of these and other three-dimensional vortex properties are discussed.
Abstract
The weak temperature gradient (WTG) approximation is applied to simple shallow-water models of the Hadley circulation. While it is difficult to formally justify the use of the WTG approximation for this problem, the derived WTG solutions are shown to agree well with numerical solutions of the full equations and to converge to the traditional angular momentum conserving (AMC) solutions in the inviscid limit. Heuristic arguments are given to explain this. The WTG method also provides semianalytical solutions in the case of nonvanishing viscosity, in contrast to the AMC solutions, which are strictly inviscid.
Abstract
The weak temperature gradient (WTG) approximation is applied to simple shallow-water models of the Hadley circulation. While it is difficult to formally justify the use of the WTG approximation for this problem, the derived WTG solutions are shown to agree well with numerical solutions of the full equations and to converge to the traditional angular momentum conserving (AMC) solutions in the inviscid limit. Heuristic arguments are given to explain this. The WTG method also provides semianalytical solutions in the case of nonvanishing viscosity, in contrast to the AMC solutions, which are strictly inviscid.
Abstract
This paper examines the nature and robustness of internal stratospheric variability, namely the variability resulting from the internal dynamics of the stratosphere itself, as opposed to that forced by external sources such as the natural variability of the free troposphere. Internal stratospheric variability arises from the competing actions of radiative forcing, which under perpetual winter conditions strengthens the polar vortex, and planetary wave breaking, which weakens it. The results from a stratosphere-only model demonstrate that strong internal stratospheric variability, consisting of repeated sudden warming-type events, exists over a wide range of realistic radiative and wave forcing conditions, and is largely independent of other physical and numerical parameters. In particular, the coherent form of the variability persists as the number of degrees of freedom is increased, and is therefore not an artifact of severe model truncation. Various diagnostics, including three-dimensional representations of the potential vorticity, illustrate that the variability is determined by the vertical structure of the vortex and the extent to which upward wave propagation is favored or inhibited. In this paper, the variability arising from purely internal stratosphere dynamics is isolated by specifying thermal and wave forcings that are completely time independent. In a second paper, the authors investigate the relative importance of internal and external variability by considering time-dependent wave forcing as a simple representation of tropospheric variability.
Abstract
This paper examines the nature and robustness of internal stratospheric variability, namely the variability resulting from the internal dynamics of the stratosphere itself, as opposed to that forced by external sources such as the natural variability of the free troposphere. Internal stratospheric variability arises from the competing actions of radiative forcing, which under perpetual winter conditions strengthens the polar vortex, and planetary wave breaking, which weakens it. The results from a stratosphere-only model demonstrate that strong internal stratospheric variability, consisting of repeated sudden warming-type events, exists over a wide range of realistic radiative and wave forcing conditions, and is largely independent of other physical and numerical parameters. In particular, the coherent form of the variability persists as the number of degrees of freedom is increased, and is therefore not an artifact of severe model truncation. Various diagnostics, including three-dimensional representations of the potential vorticity, illustrate that the variability is determined by the vertical structure of the vortex and the extent to which upward wave propagation is favored or inhibited. In this paper, the variability arising from purely internal stratosphere dynamics is isolated by specifying thermal and wave forcings that are completely time independent. In a second paper, the authors investigate the relative importance of internal and external variability by considering time-dependent wave forcing as a simple representation of tropospheric variability.
Abstract
Although possibly the simplest model for the atmospheres of the giant planets, the turbulent forced-dissipative shallow-water system in spherical geometry has not, to date, been investigated; the present study aims to fill this gap. Unlike the freely decaying shallow-water system described by Cho and Polvani, equilibrium states in the forced-dissipative system are highly dependent on details of the forcing and the dissipation. For instance, it is found that for a given equilibrated energy level, the steadiness of zonal jets depends crucially on the balance between forcing and dissipation.
With long (up to 100 000 days) high-resolution (T170) calculations, the dependence of the equilibrium states on Rossby number Ro and Rossby deformation radius LD is explored, for the case when the dissipation takes the form of hypodiffusion (acting predominantly at large scales) and the random forcing at small scales is δ correlated in time. When LD is large compared to the planetary radius, zonal jets are verified to scale closely with the Rhines scale over a wide range of Ro; furthermore, the jets at the equator are found to be both prograde and retrograde with approximately equal likelihood. As LD is decreased, the equatorial jets become increasingly and consistently retrograde, in agreement with the freely decaying turbulence results. Also, the regime recently discussed by Theiss, where zonal jets are confined to low latitudes, is illustrated to emerge robustly in the limit of small LD . Finally, specific calculations with parameter values typical of the giant planets are presented, confirming many of the earlier results obtained in the freely decaying case.
Abstract
Although possibly the simplest model for the atmospheres of the giant planets, the turbulent forced-dissipative shallow-water system in spherical geometry has not, to date, been investigated; the present study aims to fill this gap. Unlike the freely decaying shallow-water system described by Cho and Polvani, equilibrium states in the forced-dissipative system are highly dependent on details of the forcing and the dissipation. For instance, it is found that for a given equilibrated energy level, the steadiness of zonal jets depends crucially on the balance between forcing and dissipation.
With long (up to 100 000 days) high-resolution (T170) calculations, the dependence of the equilibrium states on Rossby number Ro and Rossby deformation radius LD is explored, for the case when the dissipation takes the form of hypodiffusion (acting predominantly at large scales) and the random forcing at small scales is δ correlated in time. When LD is large compared to the planetary radius, zonal jets are verified to scale closely with the Rhines scale over a wide range of Ro; furthermore, the jets at the equator are found to be both prograde and retrograde with approximately equal likelihood. As LD is decreased, the equatorial jets become increasingly and consistently retrograde, in agreement with the freely decaying turbulence results. Also, the regime recently discussed by Theiss, where zonal jets are confined to low latitudes, is illustrated to emerge robustly in the limit of small LD . Finally, specific calculations with parameter values typical of the giant planets are presented, confirming many of the earlier results obtained in the freely decaying case.
Abstract
The linear and nonlinear dynamics of layers of anomalously high potential vorticity (PV) are studied in detail. It is well known that PV layers are subject to slow, balanced, mixed barotropic–baroclinic instabilities. In this paper, it is shown that, in addition, PV layers are subject to a Kelvin–Helmholtz instability, operating on much smaller spatial and faster temporal scales.
For simplicity, spatially infinite layers of uniform anomalous PV are considered. Such layers are characterized by two key parameters: the ratio Δq of their anomalous PV to the background PV, and the angle α between the layer and the direction of the ambient stratification gradient (in suitably scaled coordinates). It is found that Kelvin–Helmholtz appears, for certain values of α, whenever Δq > 8.
Of notable interest is the case of an initially vertical PV layer embedded in a weak ambient shear flow: for sufficiently large Δq, once the PV layer is tilted past a critical angle, Kelvin–Helmholtz instability becomes possible. It is argued that the breakdown of PV layers due to a Kelvin–Helmholtz instability induced by ambient shear might be an important systematic mechanism leading to irreversible mixing during stratosphere–troposphere exchange events. This is discussed in the context of an example of Kelvin–Helmholtz instability observed near a tropopause fold.
Abstract
The linear and nonlinear dynamics of layers of anomalously high potential vorticity (PV) are studied in detail. It is well known that PV layers are subject to slow, balanced, mixed barotropic–baroclinic instabilities. In this paper, it is shown that, in addition, PV layers are subject to a Kelvin–Helmholtz instability, operating on much smaller spatial and faster temporal scales.
For simplicity, spatially infinite layers of uniform anomalous PV are considered. Such layers are characterized by two key parameters: the ratio Δq of their anomalous PV to the background PV, and the angle α between the layer and the direction of the ambient stratification gradient (in suitably scaled coordinates). It is found that Kelvin–Helmholtz appears, for certain values of α, whenever Δq > 8.
Of notable interest is the case of an initially vertical PV layer embedded in a weak ambient shear flow: for sufficiently large Δq, once the PV layer is tilted past a critical angle, Kelvin–Helmholtz instability becomes possible. It is argued that the breakdown of PV layers due to a Kelvin–Helmholtz instability induced by ambient shear might be an important systematic mechanism leading to irreversible mixing during stratosphere–troposphere exchange events. This is discussed in the context of an example of Kelvin–Helmholtz instability observed near a tropopause fold.
Abstract
An accurate assessment of the role of solar variability is a key step toward a proper quantification of natural and anthropogenic climate change. To this end, climate models have been extensively used to quantify the solar contribution to climate variability. However, owing to the large computational cost, the bulk of modeling studies to date have been performed without interactive stratospheric photochemistry: the impact of this simplification on the modeled climate system response to solar forcing remains largely unknown. Here this impact is quantified by comparing the response of two model configurations, with and without interactive ozone chemistry. Using long integrations, robust surface temperature and precipitation responses to an idealized irradiance increase are obtained. Then, it is shown that the inclusion of interactive stratospheric chemistry significantly reduces the surface warming (by about one-third) and the accompanying precipitation response. This behavior is linked to photochemically induced stratospheric ozone changes, and their modulation of the surface solar radiation. The results herein suggest that neglecting stratospheric photochemistry leads to a sizable overestimate of the surface response to changes in solar irradiance. This has implications for simulations of the climate in the last millennium and geoengineering applications employing irradiance changes larger than those observed over the 11-yr sunspot cycle, where models often use simplified treatments of stratospheric ozone that are inconsistent with the imposed solar forcing.
Abstract
An accurate assessment of the role of solar variability is a key step toward a proper quantification of natural and anthropogenic climate change. To this end, climate models have been extensively used to quantify the solar contribution to climate variability. However, owing to the large computational cost, the bulk of modeling studies to date have been performed without interactive stratospheric photochemistry: the impact of this simplification on the modeled climate system response to solar forcing remains largely unknown. Here this impact is quantified by comparing the response of two model configurations, with and without interactive ozone chemistry. Using long integrations, robust surface temperature and precipitation responses to an idealized irradiance increase are obtained. Then, it is shown that the inclusion of interactive stratospheric chemistry significantly reduces the surface warming (by about one-third) and the accompanying precipitation response. This behavior is linked to photochemically induced stratospheric ozone changes, and their modulation of the surface solar radiation. The results herein suggest that neglecting stratospheric photochemistry leads to a sizable overestimate of the surface response to changes in solar irradiance. This has implications for simulations of the climate in the last millennium and geoengineering applications employing irradiance changes larger than those observed over the 11-yr sunspot cycle, where models often use simplified treatments of stratospheric ozone that are inconsistent with the imposed solar forcing.
Abstract
It has been suggested that changes in the atmospheric circulation caused by anthropogenic forcings are highly uncertain, owing to the large natural variability intrinsic to the system. Here, to assess the statistical significance of such changes for the midlatitude, large-scale atmospheric circulation of the Southern Hemisphere, a new 40-member ensemble of integrations, from 1920 to 2080, of the Community Earth System Model, version 5, is analyzed together with a companion 1800-yr-long preindustrial control integration of the same fully coupled model. For simplicity, only the latitudinal position and the strength of the zonal-mean eddy-driven jet are considered. Given the large year-to-year variability of these jet properties, this paper focuses on their decadal averages, which reflect the more slowly varying climate state. The analysis herein reveals that the forced response in such decadal averages easily emerges from the natural variability, with only a few model integrations typically needed to establish statistical significance. In particular, a forced summertime poleward shift of the jet in the latter part of the twentieth century and a strengthening of the jet during the twenty-first century in all seasons of the year are found. Contrasting these with changes in the southern annular mode, this confirms earlier studies demonstrating that such a mode is unable to distinguish different structural changes in the midlatitude jet.
Abstract
It has been suggested that changes in the atmospheric circulation caused by anthropogenic forcings are highly uncertain, owing to the large natural variability intrinsic to the system. Here, to assess the statistical significance of such changes for the midlatitude, large-scale atmospheric circulation of the Southern Hemisphere, a new 40-member ensemble of integrations, from 1920 to 2080, of the Community Earth System Model, version 5, is analyzed together with a companion 1800-yr-long preindustrial control integration of the same fully coupled model. For simplicity, only the latitudinal position and the strength of the zonal-mean eddy-driven jet are considered. Given the large year-to-year variability of these jet properties, this paper focuses on their decadal averages, which reflect the more slowly varying climate state. The analysis herein reveals that the forced response in such decadal averages easily emerges from the natural variability, with only a few model integrations typically needed to establish statistical significance. In particular, a forced summertime poleward shift of the jet in the latter part of the twentieth century and a strengthening of the jet during the twenty-first century in all seasons of the year are found. Contrasting these with changes in the southern annular mode, this confirms earlier studies demonstrating that such a mode is unable to distinguish different structural changes in the midlatitude jet.
Abstract
The practical question of whether the classical spectral transform method, widely used in atmospheric modeling, can be efficiently implemented on inexpensive commodity clusters is addressed. Typically, such clusters have limited cache and memory sizes. To demonstrate that these limitations can be overcome, the authors have built a spherical general circulation model dynamical core, called BOB (“Built on Beowulf”), which can solve either the shallow water equations or the atmospheric primitive equations in pressure coordinates.
That BOB is targeted for computing at high resolution on modestly sized and priced commodity clusters is reflected in four areas of its design. First, the associated Legendre polynomials (ALPs) are computed “on the fly” using a stable and accurate recursion relation. Second, an identity is employed that eliminates the storage of the derivatives of the ALPs. Both of these algorithmic choices reduce the memory footprint and memory bandwidth requirements of the spectral transform. Third, a cache-blocked and unrolled Legendre transform achieves a high performance level that resists deterioration as resolution is increased. Finally, the parallel implementation of BOB is transposition-based, employing load-balanced, one-dimensional decompositions in both latitude and wavenumber.
A number of standard tests is used to compare BOB's performance to two well-known codes—the Parallel Spectral Transform Shallow Water Model (PSTSWM) and the dynamical core of NCAR's Community Climate Model CCM3. Compared to PSTSWM, BOB shows better timing results, particularly at the higher resolutions where cache effects become important. BOB also shows better performance in its comparison with CCM3's dynamical core. With 16 processors, at a triangular spectral truncation of T85, it is roughly five times faster when computing the solution to the standard Held–Suarez test case, which involves 18 levels in the vertical. BOB also shows a significantly smaller memory footprint in these comparison tests.
Abstract
The practical question of whether the classical spectral transform method, widely used in atmospheric modeling, can be efficiently implemented on inexpensive commodity clusters is addressed. Typically, such clusters have limited cache and memory sizes. To demonstrate that these limitations can be overcome, the authors have built a spherical general circulation model dynamical core, called BOB (“Built on Beowulf”), which can solve either the shallow water equations or the atmospheric primitive equations in pressure coordinates.
That BOB is targeted for computing at high resolution on modestly sized and priced commodity clusters is reflected in four areas of its design. First, the associated Legendre polynomials (ALPs) are computed “on the fly” using a stable and accurate recursion relation. Second, an identity is employed that eliminates the storage of the derivatives of the ALPs. Both of these algorithmic choices reduce the memory footprint and memory bandwidth requirements of the spectral transform. Third, a cache-blocked and unrolled Legendre transform achieves a high performance level that resists deterioration as resolution is increased. Finally, the parallel implementation of BOB is transposition-based, employing load-balanced, one-dimensional decompositions in both latitude and wavenumber.
A number of standard tests is used to compare BOB's performance to two well-known codes—the Parallel Spectral Transform Shallow Water Model (PSTSWM) and the dynamical core of NCAR's Community Climate Model CCM3. Compared to PSTSWM, BOB shows better timing results, particularly at the higher resolutions where cache effects become important. BOB also shows better performance in its comparison with CCM3's dynamical core. With 16 processors, at a triangular spectral truncation of T85, it is roughly five times faster when computing the solution to the standard Held–Suarez test case, which involves 18 levels in the vertical. BOB also shows a significantly smaller memory footprint in these comparison tests.
