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- Author or Editor: Raymond T. Pierrehumbert x

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## Abstract

We hypothesize that periods of quasi-stationary behavior in the large scales are integrally associated with an organized behavior of the synoptic scales, thus the terminology “weather regime.” To investigate our hypothesis, we extend the model of Charney and Straus (1980) to include an additional wave in the zonal direction which is highly baroclinically unstable and can interact directly with the externally forced large-scale wave. We find that such a model aperiodically vacillates between two distinct weather regime states which are not located near any of the stationary equilibria of the large-scale state; thus, we cannot ascertain the qualitative behavior of the large-scale flow in our model knowing only the large-scale equilibria and their respective stabilities to perturbations on the scale of the equilibria. The state of the model flow may remain in either one of the two regime states for several synoptic periods. During each of the two regimes, the net transports by the transient disturbances are found to have consistent, zonally inhomogeneous structure, the form of which depends upon the regime. This result implies that the transports appear as a net additional external forcing mechanism to the large-scale wave, accounting for the differences between the time-mean regime state and the stationary equilibria.

Following the analysis procedure of Frederiksen (1979), we show that the observed structure of these net transports can be accounted for by the spatial modulation of the baroclinically most unstable eigenmodes by the large-scale wave. We then consider only the tendency equations of the large-scale variables where the effects of the transients are parameterized by solving the stability problem at each time step. We find that such a dynamical system possesses two absolutely stable “regime-equilibria” which are very close in phase space to the time mean states of the regimes appearing in the full model. We then demonstrate that the instantaneous component of the transients are also capable of transferring the state of the model flow from the attractor basin of one of the stable regime-equilibria to the attractor basin of the other. Our experiments thus indicate that the transients are important in determining the qualitative behavior of both the instantaneous and time-mean components of the large-scale flow in our system, and suggest that the very different short-range climates in the atmosphere can result from entirely internal processes.

## Abstract

We hypothesize that periods of quasi-stationary behavior in the large scales are integrally associated with an organized behavior of the synoptic scales, thus the terminology “weather regime.” To investigate our hypothesis, we extend the model of Charney and Straus (1980) to include an additional wave in the zonal direction which is highly baroclinically unstable and can interact directly with the externally forced large-scale wave. We find that such a model aperiodically vacillates between two distinct weather regime states which are not located near any of the stationary equilibria of the large-scale state; thus, we cannot ascertain the qualitative behavior of the large-scale flow in our model knowing only the large-scale equilibria and their respective stabilities to perturbations on the scale of the equilibria. The state of the model flow may remain in either one of the two regime states for several synoptic periods. During each of the two regimes, the net transports by the transient disturbances are found to have consistent, zonally inhomogeneous structure, the form of which depends upon the regime. This result implies that the transports appear as a net additional external forcing mechanism to the large-scale wave, accounting for the differences between the time-mean regime state and the stationary equilibria.

Following the analysis procedure of Frederiksen (1979), we show that the observed structure of these net transports can be accounted for by the spatial modulation of the baroclinically most unstable eigenmodes by the large-scale wave. We then consider only the tendency equations of the large-scale variables where the effects of the transients are parameterized by solving the stability problem at each time step. We find that such a dynamical system possesses two absolutely stable “regime-equilibria” which are very close in phase space to the time mean states of the regimes appearing in the full model. We then demonstrate that the instantaneous component of the transients are also capable of transferring the state of the model flow from the attractor basin of one of the stable regime-equilibria to the attractor basin of the other. Our experiments thus indicate that the transients are important in determining the qualitative behavior of both the instantaneous and time-mean components of the large-scale flow in our system, and suggest that the very different short-range climates in the atmosphere can result from entirely internal processes.

## Abstract

The relative effects of dynamics and surface thermal interactions in determining the heat flux and temperature fluctuations within the lower-tropospheric portion of the Pacific storm track are quantified using the probability distribution functions (PDFs) of the temperature fluctuations and heat flux, Lagrangian passive tracer calculations, and a simple stochastic model. It is found that temperature fluctuations damp to the underlying oceanic temperature with a timescale of approximately 1 day but that dynamics still play the predominant role in determining atmospheric heat flux, due to eddy mixing lengths within the storm track of ≤ 5° latitude. These results are confirmed by the favorable comparison of the PDFs of the model-generated and observed temperature fluctuations and heat flux.

The implications of strong thermal damping in the lower troposphere are discussed and speculations are made regarding the effect of such damping upon baroclinic eddy life cycles and the general circulation.

## Abstract

The relative effects of dynamics and surface thermal interactions in determining the heat flux and temperature fluctuations within the lower-tropospheric portion of the Pacific storm track are quantified using the probability distribution functions (PDFs) of the temperature fluctuations and heat flux, Lagrangian passive tracer calculations, and a simple stochastic model. It is found that temperature fluctuations damp to the underlying oceanic temperature with a timescale of approximately 1 day but that dynamics still play the predominant role in determining atmospheric heat flux, due to eddy mixing lengths within the storm track of ≤ 5° latitude. These results are confirmed by the favorable comparison of the PDFs of the model-generated and observed temperature fluctuations and heat flux.

The implications of strong thermal damping in the lower troposphere are discussed and speculations are made regarding the effect of such damping upon baroclinic eddy life cycles and the general circulation.

## Abstract

The effect of Ekman friction on baroclinic instability is reexamined in order to address questions raised by Farrell concerning the existence of normal mode instability in the atmosphere. As the degree of meridional confinement is central to the result, a linearized two-dimensional (latitude-height) quasi-geostrophic model is used to obviate the arbitrariness inherent in choosing a channel width in one-dimensional (vertical shear only) models. The two-dimensional eigenvalue problem was solved by pseudospectral method using rational Chebyshev expansions in both vertical and meridional directions. It is concluded that the instability can be eliminated only by the combination of strong Ekman friction with weak large-scale wind shear. Estimates of Ekman friction based on a realistic boundary-layer model indicate that such conditions can prevail over land when the boundary layer is neutrally stratified. For values of Ekman friction appropriate to the open ocean, friction can reduce the growth rate of the most unstable mode by at most a factor of two but cannot eliminate the instability.

By reducing the growth rate and shifting the most unstable mode to lower zonal wavenumbers, viscous effects make the heat and momentum fluxes of the most unstable mode deeper and less meridionally confined than in the inviscid case. Nevertheless, linear theory still underestimates the penetration depth of the momentum fluxes, as compared to observations and nonlinear numerical models.

## Abstract

The effect of Ekman friction on baroclinic instability is reexamined in order to address questions raised by Farrell concerning the existence of normal mode instability in the atmosphere. As the degree of meridional confinement is central to the result, a linearized two-dimensional (latitude-height) quasi-geostrophic model is used to obviate the arbitrariness inherent in choosing a channel width in one-dimensional (vertical shear only) models. The two-dimensional eigenvalue problem was solved by pseudospectral method using rational Chebyshev expansions in both vertical and meridional directions. It is concluded that the instability can be eliminated only by the combination of strong Ekman friction with weak large-scale wind shear. Estimates of Ekman friction based on a realistic boundary-layer model indicate that such conditions can prevail over land when the boundary layer is neutrally stratified. For values of Ekman friction appropriate to the open ocean, friction can reduce the growth rate of the most unstable mode by at most a factor of two but cannot eliminate the instability.

By reducing the growth rate and shifting the most unstable mode to lower zonal wavenumbers, viscous effects make the heat and momentum fluxes of the most unstable mode deeper and less meridionally confined than in the inviscid case. Nevertheless, linear theory still underestimates the penetration depth of the momentum fluxes, as compared to observations and nonlinear numerical models.

## Abstract

This paper is a continuation of the study of the advection–diffusion problem for stratospheric flow, and deals with the probability distribution function (PDF) of gradients of a freely decaying passive tracer. Theoretical arguments are reviewed and extended showing that mixing of a weakly diffused tracer by random large-scale flows produces a tracer gradient field whose probability distribution function has “stretched exponential” tails *P*(|∇*θ*|) ∝ exp(−*b*|∇*θ*|^{
γ
}) with *γ* < 1. This contrasts with the lognormal distribution expected for advective mixing in the absence of diffusion. The non-Gaussian distribution of tracer gradients can be derived in terms of the statistics of strain rates of the random driving flow. It is shown that the tails of the gradient PDF provide information about the dissipation scale, the scale selectivity of the dissipation law, and the fluctuations of short-term strain. The gradient PDF is shown to contain information about tracer variability that is not present at all in the power spectrum of the tracer field.

To show that the predictions remain valid for the gradient statistics of passive tracers driven by the well-organized lower-stratospheric flow with mixing barriers, a series of advection–diffusion simulations of a decaying passive tracer are presented. The mixing is driven by ECMWF winds on the 420-K isentropic surface using the high-resolution finite-volume model employed in Part I of this paper. It is found that the probability distribution function of the simulated tracer gradients is indeed stretched exponential, with the stretching parameter *γ* ≈ 0.55. The largest gradients are not found in the regions of highest Lyapunov exponents, but rather in the surf-zone regions adjacent to the reservoirs of high tracer fluctuation amplitude.

## Abstract

This paper is a continuation of the study of the advection–diffusion problem for stratospheric flow, and deals with the probability distribution function (PDF) of gradients of a freely decaying passive tracer. Theoretical arguments are reviewed and extended showing that mixing of a weakly diffused tracer by random large-scale flows produces a tracer gradient field whose probability distribution function has “stretched exponential” tails *P*(|∇*θ*|) ∝ exp(−*b*|∇*θ*|^{
γ
}) with *γ* < 1. This contrasts with the lognormal distribution expected for advective mixing in the absence of diffusion. The non-Gaussian distribution of tracer gradients can be derived in terms of the statistics of strain rates of the random driving flow. It is shown that the tails of the gradient PDF provide information about the dissipation scale, the scale selectivity of the dissipation law, and the fluctuations of short-term strain. The gradient PDF is shown to contain information about tracer variability that is not present at all in the power spectrum of the tracer field.

To show that the predictions remain valid for the gradient statistics of passive tracers driven by the well-organized lower-stratospheric flow with mixing barriers, a series of advection–diffusion simulations of a decaying passive tracer are presented. The mixing is driven by ECMWF winds on the 420-K isentropic surface using the high-resolution finite-volume model employed in Part I of this paper. It is found that the probability distribution function of the simulated tracer gradients is indeed stretched exponential, with the stretching parameter *γ* ≈ 0.55. The largest gradients are not found in the regions of highest Lyapunov exponents, but rather in the surf-zone regions adjacent to the reservoirs of high tracer fluctuation amplitude.

## Abstract

In order to clarify the extent to which the two-layer model can successfully simulate the remote tropospheric response to localized stationary forcing, the structure of stationary Rossby waves in the two-layer model is compared with that in continuous models. One finds a close correspondence when the two-layer flow is supercritical in the sense of the Phillips' criterion, except for the possibility of upstream propagation in the two-layer model when the lower-layer wind is small. When the two-layer flow is subcritical, the stationary waves can be very seriously distorted. The manner in which neutral modes are spatially or temporally destabilized by damping in the two-layer model is contrasted with similar results for Charney's model.

## Abstract

In order to clarify the extent to which the two-layer model can successfully simulate the remote tropospheric response to localized stationary forcing, the structure of stationary Rossby waves in the two-layer model is compared with that in continuous models. One finds a close correspondence when the two-layer flow is supercritical in the sense of the Phillips' criterion, except for the possibility of upstream propagation in the two-layer model when the lower-layer wind is small. When the two-layer flow is subcritical, the stationary waves can be very seriously distorted. The manner in which neutral modes are spatially or temporally destabilized by damping in the two-layer model is contrasted with similar results for Charney's model.

## Abstract

The structure of stationary Rossby waves in the presence of a mean westerly zonal flow with vertical shear is examined. There is typically only one stationary vertical mode, the external mode, trapped within the troposphere. For more than one tropospheric mode to exist, we find that vertical shears must be smaller than those usually observed in extratropical latitudes. The vertical structure, horizontal wavenumber and group velocity of the external mode, and the projection onto this mode of topographic and thermal forcing are studied with continuous models (a linear shear profile as well as more realistic basic states), and a finite-differenced model with resolution and upper boundary condition similar to that used in GCMs. We point out that the rigid-lid upper boundary condition need not create artificial stationary resonances, as the artificial stationary vertical modes that are created are often horizontally evanescent.

The results are presented in a form which allows one to design the equivalent barotropic model that captures the external mode's contribution to the stationary wave field. It is found, in particular, that the wind blowing over the topography in such a barotropic model should generally be larger than the surface wind but smaller than the wind at the equivalent barotropic level. Also, the group velocity of the stationary external mode in realistic vertical shear is found to be considerably greater than that of the stationary Rossby wave in the equivalent barotropic model.

## Abstract

The structure of stationary Rossby waves in the presence of a mean westerly zonal flow with vertical shear is examined. There is typically only one stationary vertical mode, the external mode, trapped within the troposphere. For more than one tropospheric mode to exist, we find that vertical shears must be smaller than those usually observed in extratropical latitudes. The vertical structure, horizontal wavenumber and group velocity of the external mode, and the projection onto this mode of topographic and thermal forcing are studied with continuous models (a linear shear profile as well as more realistic basic states), and a finite-differenced model with resolution and upper boundary condition similar to that used in GCMs. We point out that the rigid-lid upper boundary condition need not create artificial stationary resonances, as the artificial stationary vertical modes that are created are often horizontally evanescent.

The results are presented in a form which allows one to design the equivalent barotropic model that captures the external mode's contribution to the stationary wave field. It is found, in particular, that the wind blowing over the topography in such a barotropic model should generally be larger than the surface wind but smaller than the wind at the equivalent barotropic level. Also, the group velocity of the stationary external mode in realistic vertical shear is found to be considerably greater than that of the stationary Rossby wave in the equivalent barotropic model.

## Abstract

External Rossby waves in vertical shear can be destabilized by thermal damping. They can also be destabilized by damping of potential vorticity if this damping is larger in the lower than in the upper troposphere. Results are described in detail for Charney's model. Implications for the effects of diabatic heating and mixing due to smaller scale transients on equivalent barotropic stationary or quasi-stationary long waves are discussed. It is painted out that energy or potential enstrophy budgets may indicate that transients are damping the long waves while, in fact, their presence is destabilizing these waves.

## Abstract

External Rossby waves in vertical shear can be destabilized by thermal damping. They can also be destabilized by damping of potential vorticity if this damping is larger in the lower than in the upper troposphere. Results are described in detail for Charney's model. Implications for the effects of diabatic heating and mixing due to smaller scale transients on equivalent barotropic stationary or quasi-stationary long waves are discussed. It is painted out that energy or potential enstrophy budgets may indicate that transients are damping the long waves while, in fact, their presence is destabilizing these waves.

## Abstract

Sea ice schemes with a few vertical levels are typically used to simulate the thermodynamic evolution of sea ice in global climate models. Here it is shown that these schemes overestimate the magnitude of the diurnal surface temperature cycle by a factor of 2–3 when they are used to simulate tropical ice in a Snowball earth event. This could strongly influence our understanding of Snowball termination, which occurs in global climate models when the midday surface temperature in the tropics reaches the melting point. A hierarchy of models is used to show that accurate simulation of surface temperature variation on a given time scale requires that a sea ice model resolve the *e*-folding depth to which a periodic signal on that time scale penetrates. This is used to suggest modifications to the sea ice schemes used in global climate models that would allow more accurate simulation of Snowball deglaciation.

## Abstract

Sea ice schemes with a few vertical levels are typically used to simulate the thermodynamic evolution of sea ice in global climate models. Here it is shown that these schemes overestimate the magnitude of the diurnal surface temperature cycle by a factor of 2–3 when they are used to simulate tropical ice in a Snowball earth event. This could strongly influence our understanding of Snowball termination, which occurs in global climate models when the midday surface temperature in the tropics reaches the melting point. A hierarchy of models is used to show that accurate simulation of surface temperature variation on a given time scale requires that a sea ice model resolve the *e*-folding depth to which a periodic signal on that time scale penetrates. This is used to suggest modifications to the sea ice schemes used in global climate models that would allow more accurate simulation of Snowball deglaciation.