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- Author or Editor: Mark R. Schoeberl x
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Abstract
Several simple modifications of the Lindzen-Kuo Gaussian elimination algorithm for solving elliptic differential equations are derived. These modifications greatly decrease the auxiliary memory requirements with only some increase in computation, thus making this method suitable for solving general elliptic equations over large grids.
Abstract
Several simple modifications of the Lindzen-Kuo Gaussian elimination algorithm for solving elliptic differential equations are derived. These modifications greatly decrease the auxiliary memory requirements with only some increase in computation, thus making this method suitable for solving general elliptic equations over large grids.
Abstract
A β-plane model of the stratosphere is used to investigate the planetary-wave amplitude vacillations first reported by Holton and Maw (1976). This model differs from theirs in allowing more horizontal modes.
For low surface wave amplitudes, a new class of solutions is found which exhibits a stationary, partially reflecting critical line at steady state. The critical line equilibrates at lower altitudes as the wave forcing is increased. Vacillating solutions occur when the steady state critical line occurs near the lower boundary.
The maximum wave amplitude and the maximum steady-state wave amplitude found in the model are in the ratio of 2:1, in good agreement with theoretical predictions. The maximum wave amplitude never exceeds 2200 gpm which is quite close to the saturation limit predicted by Schoeberl (1982a).
An analysis of the statistics of slowly and rapidly vacillating flows shows that both the wave and zonal mean variances are important in determining the time mean, zonal mean dynamics of the upper stratosphere.
Abstract
A β-plane model of the stratosphere is used to investigate the planetary-wave amplitude vacillations first reported by Holton and Maw (1976). This model differs from theirs in allowing more horizontal modes.
For low surface wave amplitudes, a new class of solutions is found which exhibits a stationary, partially reflecting critical line at steady state. The critical line equilibrates at lower altitudes as the wave forcing is increased. Vacillating solutions occur when the steady state critical line occurs near the lower boundary.
The maximum wave amplitude and the maximum steady-state wave amplitude found in the model are in the ratio of 2:1, in good agreement with theoretical predictions. The maximum wave amplitude never exceeds 2200 gpm which is quite close to the saturation limit predicted by Schoeberl (1982a).
An analysis of the statistics of slowly and rapidly vacillating flows shows that both the wave and zonal mean variances are important in determining the time mean, zonal mean dynamics of the upper stratosphere.
Abstract
A simple equation for the Lagrangian-mean flow induced by damped planetary waves is derived. The flow computed for stationary planetary waves of a β-plane is found to be generally poleward and downward during winter and appears to be about twice as strong as the diabatic circulation in the lower stratosphere. An important factor in determining the high-latitude Lagrangian-mean flow field is the subtropical jet stream which blocks planetary wave propagation toward the equatorial regions.
Computations using two types of Lagrangian-mean boundary conditions at the surface show that incorrect orographic forcing distorts the Lagrangian-mean How up to three scale heights or more above ground.
Abstract
A simple equation for the Lagrangian-mean flow induced by damped planetary waves is derived. The flow computed for stationary planetary waves of a β-plane is found to be generally poleward and downward during winter and appears to be about twice as strong as the diabatic circulation in the lower stratosphere. An important factor in determining the high-latitude Lagrangian-mean flow field is the subtropical jet stream which blocks planetary wave propagation toward the equatorial regions.
Computations using two types of Lagrangian-mean boundary conditions at the surface show that incorrect orographic forcing distorts the Lagrangian-mean How up to three scale heights or more above ground.
Abstract
The wave-mean flow interaction has been computed near an energy-absorbing, baroclinic, planetary wave critical line tilted at an arbitrary angle from the vertical. This problem is a generalization of the critical line interaction problems studied by Matsuno and Nakamura (1979) and Schoeberl (1980).
A tilted critical line can directly tap the eddy heat transport of a Rossby wave and produce a singular rate of change in the zonally averaged temperature at the critical line. This implies that sudden stratospheric warmings may not always require an induced Eulerian-mean secondary circulation to create significant temperature changes in the zonally averaged flow as suggested by Matsuno (1971). Strong Lagrangian-mean motion also exists along the critical line if it is not perfectly vertical. These results are discussed with application to the 1976/77 sudden warming.
Abstract
The wave-mean flow interaction has been computed near an energy-absorbing, baroclinic, planetary wave critical line tilted at an arbitrary angle from the vertical. This problem is a generalization of the critical line interaction problems studied by Matsuno and Nakamura (1979) and Schoeberl (1980).
A tilted critical line can directly tap the eddy heat transport of a Rossby wave and produce a singular rate of change in the zonally averaged temperature at the critical line. This implies that sudden stratospheric warmings may not always require an induced Eulerian-mean secondary circulation to create significant temperature changes in the zonally averaged flow as suggested by Matsuno (1971). Strong Lagrangian-mean motion also exists along the critical line if it is not perfectly vertical. These results are discussed with application to the 1976/77 sudden warming.
Abstract
The balance of potential enstrophy and its relationship to vacillation cycles and the sudden warming is studied for a β-channel model of the stratosphere. It is shown that the mean flow cannot be steady in the presence of large-amplitude quasi-geostrophic waves [∼1–0.25 geopotential kilometers (gpkm)] when any dissipation is present, and the maximum wave amplitude allowed is ∼2 gpkm.
If wave forcing (transience plus dissipation) is artificially maintained, the mean flow decelerates slowly at first then explosively as the potential vorticity gradient of the basic state is wiped out over the channel. This process is called wave saturation. The initial phase of the explosive deceleration resembles both the observed and modeled mean flow evolution during a sudden stratospheric warming. A simple vacillation model based upon thew ideas shows remarkable similarity to the results of Holton and Mass (1976) and Davies (1981).
Abstract
The balance of potential enstrophy and its relationship to vacillation cycles and the sudden warming is studied for a β-channel model of the stratosphere. It is shown that the mean flow cannot be steady in the presence of large-amplitude quasi-geostrophic waves [∼1–0.25 geopotential kilometers (gpkm)] when any dissipation is present, and the maximum wave amplitude allowed is ∼2 gpkm.
If wave forcing (transience plus dissipation) is artificially maintained, the mean flow decelerates slowly at first then explosively as the potential vorticity gradient of the basic state is wiped out over the channel. This process is called wave saturation. The initial phase of the explosive deceleration resembles both the observed and modeled mean flow evolution during a sudden stratospheric warming. A simple vacillation model based upon thew ideas shows remarkable similarity to the results of Holton and Mass (1976) and Davies (1981).
Abstract
A relation between the statistics of large-scale waves and the mean flow is derived from the potential enstrophy equations integrated over an isobaric surface. The difference between time-averaged zonal-mean state and the radiative-dynamical equilibrium state due to the symmetric circulation is determined by three components: the steady wave enstrophy, the variance in the wave enstrophy and the variance mean flow enstrophy. With some simplifications, the relationship between these components can be used to estimate the maximum amplitude for Rossby waves derived from a statistical data set. We obtain an upper limit of ∼1200 gpm for a wave disturbance with a meridional scale of ∼1800 km. If the Rossby wave amplitudes are observed near that upper limit, then the wave energy spectrum should exhibit a −5 power law.
The three enstrophy components are estimated for a parameterized model of wave–mean flow interaction at a single level. We find that the steady wave enstrophy, the wave enstrophy variance and the mean enstrophy variance all are within a factor of 2 of each other with the wave variance being the largest. These results suggest that attempts to model the time-mean stratospheric structure in winter, using only the time-mean stationary wave forcing of the mean flow, may not be successful.
Abstract
A relation between the statistics of large-scale waves and the mean flow is derived from the potential enstrophy equations integrated over an isobaric surface. The difference between time-averaged zonal-mean state and the radiative-dynamical equilibrium state due to the symmetric circulation is determined by three components: the steady wave enstrophy, the variance in the wave enstrophy and the variance mean flow enstrophy. With some simplifications, the relationship between these components can be used to estimate the maximum amplitude for Rossby waves derived from a statistical data set. We obtain an upper limit of ∼1200 gpm for a wave disturbance with a meridional scale of ∼1800 km. If the Rossby wave amplitudes are observed near that upper limit, then the wave energy spectrum should exhibit a −5 power law.
The three enstrophy components are estimated for a parameterized model of wave–mean flow interaction at a single level. We find that the steady wave enstrophy, the wave enstrophy variance and the mean enstrophy variance all are within a factor of 2 of each other with the wave variance being the largest. These results suggest that attempts to model the time-mean stratospheric structure in winter, using only the time-mean stationary wave forcing of the mean flow, may not be successful.
Abstract
A linear nonhydrostatic model of gravity waves forced by a bell-shaped ridge is used to investigate the penetration of mountain waves into the stratosphere and mesosphere during winter and fall. Gravity waves with horizontal scales less than 30 km are found to be trapped near the tropopause and the stratopause in regions of strong winds. The effect of trapping these modes produces a disturbance whose structure broadens with height. In the mesosphere the disturbance appears 20–40 km downstream from the forcing depending on the strength of the intervening winds.
Wavebreaking associated with the mountain wave is predicted in the lower stratosphere as a result of wave superposition; no individual harmonic reaches breaking amplitude. In the mesosphere, wave breakdown is more prevalent, and the disturbance spectrum is relatively more monochromatic as a result of the filtering of the shorter scale modes by the lower atmosphere.
Abstract
A linear nonhydrostatic model of gravity waves forced by a bell-shaped ridge is used to investigate the penetration of mountain waves into the stratosphere and mesosphere during winter and fall. Gravity waves with horizontal scales less than 30 km are found to be trapped near the tropopause and the stratopause in regions of strong winds. The effect of trapping these modes produces a disturbance whose structure broadens with height. In the mesosphere the disturbance appears 20–40 km downstream from the forcing depending on the strength of the intervening winds.
Wavebreaking associated with the mountain wave is predicted in the lower stratosphere as a result of wave superposition; no individual harmonic reaches breaking amplitude. In the mesosphere, wave breakdown is more prevalent, and the disturbance spectrum is relatively more monochromatic as a result of the filtering of the shorter scale modes by the lower atmosphere.
Abstract
A steady WKB model of gravity wave propagation including convective adjustment is used to investigate approximations used in various gravity-wave parameterization schemes. First, it is shown that estimates of the wave breaking height assuming a single horizontal wavenumber gravity wave can lead to errors if the topography is not sinusoidal. Second, the model results show that the assumption that wave growth ceases with the onset of convection or shear instability is an oversimplification. Since convection appears first over a very limited spatial region of the wave field, the wave is initially unaffected by turbulent mixing. However, when the convection zone spreads over a large portion of the wave field the amplitude is constrained. Estimates of the heat flux by breaking gravity waves are used to develop a simple parameterization of the vertical diffusion in terms of the Reynolds stress.
Abstract
A steady WKB model of gravity wave propagation including convective adjustment is used to investigate approximations used in various gravity-wave parameterization schemes. First, it is shown that estimates of the wave breaking height assuming a single horizontal wavenumber gravity wave can lead to errors if the topography is not sinusoidal. Second, the model results show that the assumption that wave growth ceases with the onset of convection or shear instability is an oversimplification. Since convection appears first over a very limited spatial region of the wave field, the wave is initially unaffected by turbulent mixing. However, when the convection zone spreads over a large portion of the wave field the amplitude is constrained. Estimates of the heat flux by breaking gravity waves are used to develop a simple parameterization of the vertical diffusion in terms of the Reynolds stress.
Abstract
A fully nonlinear numerical model of the point jet barotropic instability is used to test and confirm the hypothesis that the magnitude of the wave vorticity does not exceed the magnitude of the initial sheer. This result arises directly from the local conservation of vorticity following a parcel and the fact that unstable waves are principally confined to the region where the zonal mean vorticity can be smoothed by the wave so as to eliminate the instability.
Comparisons are made between fully nonlinear and quasi-linear models of the point jet instability and their tracer transport properties. Differences become particularly evident after wave saturation. The most important effect neglected by the wave-mean flow model appears to be the advection of wave vorticity by the most unstable mode. However, as equilibration of the instability proceeds, the globally averaged properties of both models are found to be similar.
Abstract
A fully nonlinear numerical model of the point jet barotropic instability is used to test and confirm the hypothesis that the magnitude of the wave vorticity does not exceed the magnitude of the initial sheer. This result arises directly from the local conservation of vorticity following a parcel and the fact that unstable waves are principally confined to the region where the zonal mean vorticity can be smoothed by the wave so as to eliminate the instability.
Comparisons are made between fully nonlinear and quasi-linear models of the point jet instability and their tracer transport properties. Differences become particularly evident after wave saturation. The most important effect neglected by the wave-mean flow model appears to be the advection of wave vorticity by the most unstable mode. However, as equilibration of the instability proceeds, the globally averaged properties of both models are found to be similar.
Abstract
Lincizen's model of gravity wave breaking is shown to be inconsistent with the process of convective adjustment and associated turbulent outbreak. The K-theory turbulent diffusion model used by Lindzen implies a spatially uniform turbulent field which is not in agreement with the fact that gravity wave saturation and the associated convection produce turbulence only in restricted zones. The Lindzen model may be corrected to some extent by taking the turbulent Prandtl number for a diffusion acting on the wave itself to he very large. The eddy diffusion coefficients computed by Lindzen then become a factor of 2 larger and eddy transports of heat and constituents by wave fields vanish to first order.
Abstract
Lincizen's model of gravity wave breaking is shown to be inconsistent with the process of convective adjustment and associated turbulent outbreak. The K-theory turbulent diffusion model used by Lindzen implies a spatially uniform turbulent field which is not in agreement with the fact that gravity wave saturation and the associated convection produce turbulence only in restricted zones. The Lindzen model may be corrected to some extent by taking the turbulent Prandtl number for a diffusion acting on the wave itself to he very large. The eddy diffusion coefficients computed by Lindzen then become a factor of 2 larger and eddy transports of heat and constituents by wave fields vanish to first order.
