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## Abstract

Non-quasigeostrophic effects in unstable baroclinic waves, defined as differences between solutions in primitive equation and quasigeostrophic models due to terms in the governing equations that are neglected in the quasigeostrophic model, are examined. It is determined that these non-quasigeostrophic effects produce significant asymmetries in a baroclinic wave, as compared to the symmetry that would be expected if that wave were governed by the quasigeostrophic equations, even when the Rossby number is relatively small. These asymmetries result in significant differences in some of the eddy fluxes in a primitive equation model as compared to a quasigeostrophic model. This was found to be especially true for *u*â€²*v*&prime

## Abstract

Non-quasigeostrophic effects in unstable baroclinic waves, defined as differences between solutions in primitive equation and quasigeostrophic models due to terms in the governing equations that are neglected in the quasigeostrophic model, are examined. It is determined that these non-quasigeostrophic effects produce significant asymmetries in a baroclinic wave, as compared to the symmetry that would be expected if that wave were governed by the quasigeostrophic equations, even when the Rossby number is relatively small. These asymmetries result in significant differences in some of the eddy fluxes in a primitive equation model as compared to a quasigeostrophic model. This was found to be especially true for *u*â€²*v*&prime

## Abstract

The linear instability of two zonal mean flows, one computed by a general circulation model and the other corresponding to the observed winter zonal mean flow, is presented. For these calculations, we utilize a numerical primitive equation model, where the spherical geometry of the earth has been retained.

By comparing the waves predicted by linear theory with the eddies that appear in the general circulation model, it is determined that significant discrepancies exist. For the wavenumber range 1 through 15, the linear theory predicts the maximum growth rate to be for wavenumbers 12-15. The wavenumbers that dominate the intermediate-scale transient eddies in the general circulation model are much longer (5â€“7). In addition, linear theory predicts the maximum amplitude of the geopotential perturbation for wavenumbers 5â€“7 to be near the earth's surface, while in the general circulation model, the maximum amplitude of this quantity for wavenumbers 5â€“7 is at the tropopause level. Also, the phase speed of wavenumbers 7â€“9 in the general circulation model is considerably faster. that it is for the corresponding waves predicted by linear theory.

It is determined that these discrepancies also exist for wavenumbers 7â€“15 in the real atmosphere. It is concluded that these discrepancies must be due to some nonlinear process.

## Abstract

The linear instability of two zonal mean flows, one computed by a general circulation model and the other corresponding to the observed winter zonal mean flow, is presented. For these calculations, we utilize a numerical primitive equation model, where the spherical geometry of the earth has been retained.

By comparing the waves predicted by linear theory with the eddies that appear in the general circulation model, it is determined that significant discrepancies exist. For the wavenumber range 1 through 15, the linear theory predicts the maximum growth rate to be for wavenumbers 12-15. The wavenumbers that dominate the intermediate-scale transient eddies in the general circulation model are much longer (5â€“7). In addition, linear theory predicts the maximum amplitude of the geopotential perturbation for wavenumbers 5â€“7 to be near the earth's surface, while in the general circulation model, the maximum amplitude of this quantity for wavenumbers 5â€“7 is at the tropopause level. Also, the phase speed of wavenumbers 7â€“9 in the general circulation model is considerably faster. that it is for the corresponding waves predicted by linear theory.

It is determined that these discrepancies also exist for wavenumbers 7â€“15 in the real atmosphere. It is concluded that these discrepancies must be due to some nonlinear process.

## Abstract

The nonlinear interaction between a single zonal wave and the zonal mean flow is simulated with a primitive equation model. It is determined that as the wave evolves and modifies the zonal flow, the wave growth rate diminishes more rapidly near the earth's surface than it does aloft, allowing the upper portions of the wave to grow to a larger amplitude than the surface disturbance. The more rapid reduction in growth rate near the surface is accomplished primarily by an increase of static stability. It is proposed that this mechanism accounts for some of the differences in wave structures between linear baroclinic instability theory (where the, maximum amplitudes of the geopotential perturbation for wavenumbers 5-7 is at the earth's surface) and the eddies in a general circulation model (where the maximum amplitude for wavenumbers 5â€“7 is at the tropopause) that were noted by Gall (1976).

In addition, the amount of kinetic energy within an individual wave at the time that the increase of kinetic energy ceases through nonlinear interaction with the zonal mean flow is a function of wavenumber. This is because the short wavelengths are primarily surface disturbances, and therefore the increase of the zonal mean static stability near the earth's surface by the wave and frictional dissipation cause wave growth to cease sooner than it does for the long waves which extend up to the tropopause. It is this mechanism that may explain why the short wavelengths do not dominate the general circulation statistics, even though the shorter wavelengths were found by Gall to have the highest growth rates.

## Abstract

The nonlinear interaction between a single zonal wave and the zonal mean flow is simulated with a primitive equation model. It is determined that as the wave evolves and modifies the zonal flow, the wave growth rate diminishes more rapidly near the earth's surface than it does aloft, allowing the upper portions of the wave to grow to a larger amplitude than the surface disturbance. The more rapid reduction in growth rate near the surface is accomplished primarily by an increase of static stability. It is proposed that this mechanism accounts for some of the differences in wave structures between linear baroclinic instability theory (where the, maximum amplitudes of the geopotential perturbation for wavenumbers 5-7 is at the earth's surface) and the eddies in a general circulation model (where the maximum amplitude for wavenumbers 5â€“7 is at the tropopause) that were noted by Gall (1976).

In addition, the amount of kinetic energy within an individual wave at the time that the increase of kinetic energy ceases through nonlinear interaction with the zonal mean flow is a function of wavenumber. This is because the short wavelengths are primarily surface disturbances, and therefore the increase of the zonal mean static stability near the earth's surface by the wave and frictional dissipation cause wave growth to cease sooner than it does for the long waves which extend up to the tropopause. It is this mechanism that may explain why the short wavelengths do not dominate the general circulation statistics, even though the shorter wavelengths were found by Gall to have the highest growth rates.

## Abstract

The effects of released latent heat on the development of baroclinic waves are explored using two numerical experiments in which these waves are allowed to grow from small perturbations on a flow that initially was zonally constant. In one experiment the effects of released latent heat were excluded; in the other, these effects were included and the initial zonally constant flow was considered saturated everywhere.

In the moist experiment the growth rates of all wavelengths were found to be significantly increased over the corresponding growth rates in the dry experiment. However, the wavelength of maximum growth rate (wavenumber 15) was the same in both the dry and moist experiments.

At the time of maximum development of wavenumber 15, the kinetic energy structure in the moist experiment was quite different from that in the dry experiment. In the moist experiment there was a distinct double maximum in the vertical; while in the dry experiment most of the kinetic energy of wavenumber 15 was near the earth's surface. In this respect wavenumber 15 in the moist experiment more nearly resembles the corresponding wave observed in a full general circulation model than does the wave in the dry experiment. These differences in wavenumber 15 in the moist and dry experiments result from the convective adjustment process in the model, which tends to increase the warm-temperature perturbation of the baroclinic wave at mid-tropospheric levels, while reducing it at the earth's surface.

A comparison of the structure of the longer waves in the moist and dry experiments showed that they did not differ significantly.

## Abstract

The effects of released latent heat on the development of baroclinic waves are explored using two numerical experiments in which these waves are allowed to grow from small perturbations on a flow that initially was zonally constant. In one experiment the effects of released latent heat were excluded; in the other, these effects were included and the initial zonally constant flow was considered saturated everywhere.

In the moist experiment the growth rates of all wavelengths were found to be significantly increased over the corresponding growth rates in the dry experiment. However, the wavelength of maximum growth rate (wavenumber 15) was the same in both the dry and moist experiments.

At the time of maximum development of wavenumber 15, the kinetic energy structure in the moist experiment was quite different from that in the dry experiment. In the moist experiment there was a distinct double maximum in the vertical; while in the dry experiment most of the kinetic energy of wavenumber 15 was near the earth's surface. In this respect wavenumber 15 in the moist experiment more nearly resembles the corresponding wave observed in a full general circulation model than does the wave in the dry experiment. These differences in wavenumber 15 in the moist and dry experiments result from the convective adjustment process in the model, which tends to increase the warm-temperature perturbation of the baroclinic wave at mid-tropospheric levels, while reducing it at the earth's surface.

A comparison of the structure of the longer waves in the moist and dry experiments showed that they did not differ significantly.

## Abstract

Two numerical models have been constructed and used to investigate the formation of secondary vortices in axisymmetrically forced rotating flows. The vortex flow examined is that developed in a laboratory vortex simulator where secondary vortices have been produced and extensively studied. The first numerical model generated a collection of steady-state, axisymmetric, two-dimensional vortex flows for a range of swirl ratios. The second model tested those flows for instability by simulating the behavior of small-amplitude, linear perturbations superimposed on the flows: amplification of the perturbations indicated instability, whereas damping indicated stability.

The results of the instability study show that the vortex is stable for the lowest swirl ratios but that, above a certain value, instability persists indefinitely. The most rapidly growing wavenumber shifts steadily with increasing swirl from 1 to approximately 5 in the swirl range investigated. Growth rates were found to be high enough for secondary vortices to form in the laboratory simulator in just a few seconds. The perturbation fields were found to have a helical tilt and to be centered near the radius of maximum vertical vorticity in the axisymmetric vortex. They propagated in the same azimuthal direction as the rotation of the axisymmetric flow. These linear results we consistent with observed laboratory behavior, as well as with a full three-dimensional numerical multiple-vortex simulation by Rotunno. From this, it was concluded that linear theory is capable of explaining many important aspects of secondary vortices in the simulator.

At the higher swirl ratios, the perturbation received most of its energy from the deformation of the axisymmetric flow due to the radial distribution of azimuthal velocity, while for low swirl, the primary source was from the radial distribution of the vertical velocity. No other component of the axisymmetric vortex ever contributed more than about 25% of these terms.

## Abstract

Two numerical models have been constructed and used to investigate the formation of secondary vortices in axisymmetrically forced rotating flows. The vortex flow examined is that developed in a laboratory vortex simulator where secondary vortices have been produced and extensively studied. The first numerical model generated a collection of steady-state, axisymmetric, two-dimensional vortex flows for a range of swirl ratios. The second model tested those flows for instability by simulating the behavior of small-amplitude, linear perturbations superimposed on the flows: amplification of the perturbations indicated instability, whereas damping indicated stability.

The results of the instability study show that the vortex is stable for the lowest swirl ratios but that, above a certain value, instability persists indefinitely. The most rapidly growing wavenumber shifts steadily with increasing swirl from 1 to approximately 5 in the swirl range investigated. Growth rates were found to be high enough for secondary vortices to form in the laboratory simulator in just a few seconds. The perturbation fields were found to have a helical tilt and to be centered near the radius of maximum vertical vorticity in the axisymmetric vortex. They propagated in the same azimuthal direction as the rotation of the axisymmetric flow. These linear results we consistent with observed laboratory behavior, as well as with a full three-dimensional numerical multiple-vortex simulation by Rotunno. From this, it was concluded that linear theory is capable of explaining many important aspects of secondary vortices in the simulator.

At the higher swirl ratios, the perturbation received most of its energy from the deformation of the axisymmetric flow due to the radial distribution of azimuthal velocity, while for low swirl, the primary source was from the radial distribution of the vertical velocity. No other component of the axisymmetric vortex ever contributed more than about 25% of these terms.

## Abstract

The flow generated by a general circulation model is zonally averaged at 1-day intervals, and, for each of the axisymmetric flows obtained, the linear growth-rate spectrum for baroclinic waves is computed using a linear model. The time sequence of these linear growth-rate spectra is compared to the corresponding time sequence of kinetic-energy spectra for the flow of the general circulation model, in order to-test the hypothesis that the baroclinic instability of the axisymmetric flow is an important factor in determining variations of the amplitudes of the midlatitude eddies. No significant correlation is found between the linear growth rate of a particular wavenumber and the kinetic energy of that wave at some later time.

## Abstract

The flow generated by a general circulation model is zonally averaged at 1-day intervals, and, for each of the axisymmetric flows obtained, the linear growth-rate spectrum for baroclinic waves is computed using a linear model. The time sequence of these linear growth-rate spectra is compared to the corresponding time sequence of kinetic-energy spectra for the flow of the general circulation model, in order to-test the hypothesis that the baroclinic instability of the axisymmetric flow is an important factor in determining variations of the amplitudes of the midlatitude eddies. No significant correlation is found between the linear growth rate of a particular wavenumber and the kinetic energy of that wave at some later time.

## Abstract

In the absence of momentum diffusion (viscous or turbulent) in a steady state axially bounded vortex, such as that produced by a Ward-type tornado simulator, a two-celled vortex configuration in which the inner cell is entirely stagnant is to be expected. Therefore, diffusion is directly responsible or at least highly influential in producing other vortex phenomena, including a single-cell structure, a vortex breakdown, a central downdraft, and a subhydrostatic central surface pressure. The means by which these are brought about by diffusion are discussed in the context of the tornado simulator. Some of the given arguments are extended in a more speculative manner to atmospheric vortices.

## Abstract

In the absence of momentum diffusion (viscous or turbulent) in a steady state axially bounded vortex, such as that produced by a Ward-type tornado simulator, a two-celled vortex configuration in which the inner cell is entirely stagnant is to be expected. Therefore, diffusion is directly responsible or at least highly influential in producing other vortex phenomena, including a single-cell structure, a vortex breakdown, a central downdraft, and a subhydrostatic central surface pressure. The means by which these are brought about by diffusion are discussed in the context of the tornado simulator. Some of the given arguments are extended in a more speculative manner to atmospheric vortices.

## Abstract

A simple vertically-integrated axisymmetric model is used to Calculate axisymmetric flows for different swirl ratios(*s*) in tornado simulators. Thew axisymmetric states are then tested for stability using a primitive-equation linear model where the waves have both an azimuthal and a vertical wavenumber.

For S high enough for there to be a central downdraft in the axisymmetric vortex, the vortex is unstable; otherwise it is stable. For relatively low*s* only azimuthal waves 1 and 2 are unstable, with wave 1 most unstable at low*s* followed by 2 at somewhat higher*s* As S is further increased, the most unstable wave shifts to 4, then 5, and so forth. With some tuning, the model predicts the transitions from 0â€“1 and 1â€“2 secondary vortices to occur at about the observed value of*s* Vertical wavelength are about 3 m, but they increase with increase*s*

There are two modes of instability: one in which only waves 1 or 2 are unstable and which appears at low*s* and a second mode where waves 4, 5 or 6 are most unstable and which appears at high*s* These two modes am distinguished mostly by their energetics. Mode 1 receives most of its energy from the radial shear of the vertical wind, while mode 2 receives most of its energy from the radial shear of the tangential wind. In mode 1, all the amplitude of the horizontal streamfunction is contained inside the tangential wind maximum, while in mode 2 much of the amplitude is outside the tangential wind maximum.

## Abstract

A simple vertically-integrated axisymmetric model is used to Calculate axisymmetric flows for different swirl ratios(*s*) in tornado simulators. Thew axisymmetric states are then tested for stability using a primitive-equation linear model where the waves have both an azimuthal and a vertical wavenumber.

For S high enough for there to be a central downdraft in the axisymmetric vortex, the vortex is unstable; otherwise it is stable. For relatively low*s* only azimuthal waves 1 and 2 are unstable, with wave 1 most unstable at low*s* followed by 2 at somewhat higher*s* As S is further increased, the most unstable wave shifts to 4, then 5, and so forth. With some tuning, the model predicts the transitions from 0â€“1 and 1â€“2 secondary vortices to occur at about the observed value of*s* Vertical wavelength are about 3 m, but they increase with increase*s*

There are two modes of instability: one in which only waves 1 or 2 are unstable and which appears at low*s* and a second mode where waves 4, 5 or 6 are most unstable and which appears at high*s* These two modes am distinguished mostly by their energetics. Mode 1 receives most of its energy from the radial shear of the vertical wind, while mode 2 receives most of its energy from the radial shear of the tangential wind. In mode 1, all the amplitude of the horizontal streamfunction is contained inside the tangential wind maximum, while in mode 2 much of the amplitude is outside the tangential wind maximum.

## Abstract

A simple model of flow through a tornado vortex simulator is described. This model assumes a very simple distribution in the vertical of the radial and tangential components of the wind, consistent with the flow found in the simulator. With these assumptions, and with careful attention to the distribution of pressure in the lower and upper portions of the chamber, the axisymmetric equations can be reduced to one-dimensional equations.

The model illustrates that all interesting dynamics of the vortex, such as the development of the downdraft and the expansion of the core, are a result of the pressure distribution in the upper part of the chamber. In this model, this pressure distribution is caused by a slow radial spreading with height of the vorticity of the vortex, due to diffusion processes.

The model is shown to provide a realistic distribution of observed velocity fields in the simulator, including the downdraft at the center.

The dependence of the vertical velocity distribution on swirl ratio is shown and the details explained. An equation that predicts the core radius as a function of swirl ratio is given; it appears superior to previous similar equations.

Finally, predictions of the minimum pressure as a function of swirl ratio given by the model are presented. It is suggested that the observed minimum pressure curve shows two regimes, turbulent at high swirl ratio and nonturbulent at low swirl ratio. It is shown why the pressure in a turbulent vortex is much higher than in a nonturbulent vortex at the same swirl ratio and volume flow rate, and why this explains the observed curve.

## Abstract

A simple model of flow through a tornado vortex simulator is described. This model assumes a very simple distribution in the vertical of the radial and tangential components of the wind, consistent with the flow found in the simulator. With these assumptions, and with careful attention to the distribution of pressure in the lower and upper portions of the chamber, the axisymmetric equations can be reduced to one-dimensional equations.

The model illustrates that all interesting dynamics of the vortex, such as the development of the downdraft and the expansion of the core, are a result of the pressure distribution in the upper part of the chamber. In this model, this pressure distribution is caused by a slow radial spreading with height of the vorticity of the vortex, due to diffusion processes.

The model is shown to provide a realistic distribution of observed velocity fields in the simulator, including the downdraft at the center.

The dependence of the vertical velocity distribution on swirl ratio is shown and the details explained. An equation that predicts the core radius as a function of swirl ratio is given; it appears superior to previous similar equations.

Finally, predictions of the minimum pressure as a function of swirl ratio given by the model are presented. It is suggested that the observed minimum pressure curve shows two regimes, turbulent at high swirl ratio and nonturbulent at low swirl ratio. It is shown why the pressure in a turbulent vortex is much higher than in a nonturbulent vortex at the same swirl ratio and volume flow rate, and why this explains the observed curve.

## Abstract

The reason for existence of two separate unstable modes, previously described by Gall for flows in vortex simulators, is explored. When the energy equation for an unstable disturbance is considered, it is clear that the most unstable wave must be centered inside the maximum in the vertical vorticity of the basic state if this vorticity has a radial distribution that is triangular-shaped and this triangle is near the center of the vortex. When this vorticity is at large radius, the most unstable wave can be centered near or even outside the basic-state vorticity maximum. This suggests different modes when the triangular profile of vorticity is near or far from the center and that the transition from one to another mode should be gradual. These notions are verified by a careful analysis of the stability properties of the triangular-shaped vorticity profile.

It is shown that the triangular-shaped vorticity profile closely resembles the vorticity distribution in the vortex simulator after a downdraft has been established along the centerline of the vortex. In fact, the stability properties of these triangular profiles closely resemble the stability properties of the simulated vortex when the scale of the triangular profile is comparable to the vorticity distribution in the simulators.

Square-shaped vorticity profiles, which have been considered in the past, have significantly different stability properties, as compared to the triangular profiles. In particular, there is only one unstable mode, and the instability is extinguished as the vorticity region approaches the center of the basic vortex. The reason for this is easily explained by considering the perturbation energy equation.

## Abstract

The reason for existence of two separate unstable modes, previously described by Gall for flows in vortex simulators, is explored. When the energy equation for an unstable disturbance is considered, it is clear that the most unstable wave must be centered inside the maximum in the vertical vorticity of the basic state if this vorticity has a radial distribution that is triangular-shaped and this triangle is near the center of the vortex. When this vorticity is at large radius, the most unstable wave can be centered near or even outside the basic-state vorticity maximum. This suggests different modes when the triangular profile of vorticity is near or far from the center and that the transition from one to another mode should be gradual. These notions are verified by a careful analysis of the stability properties of the triangular-shaped vorticity profile.

It is shown that the triangular-shaped vorticity profile closely resembles the vorticity distribution in the vortex simulator after a downdraft has been established along the centerline of the vortex. In fact, the stability properties of these triangular profiles closely resemble the stability properties of the simulated vortex when the scale of the triangular profile is comparable to the vorticity distribution in the simulators.

Square-shaped vorticity profiles, which have been considered in the past, have significantly different stability properties, as compared to the triangular profiles. In particular, there is only one unstable mode, and the instability is extinguished as the vorticity region approaches the center of the basic vortex. The reason for this is easily explained by considering the perturbation energy equation.