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R. T. Pierrehumbert

Abstract

Based on consideration of the perturbation enstrophy and energy equations, we have derived a general family of bounds on the growth rates of perturbations to non-parallel (vortex-like or wave-like) flow on the barotropic beta-plane, allowing for the effects of forcing, Ekman friction, and topography. The family of bounds generalizes Arnol’d's stability criterion. A number of specific applications of the family of bounds are explored. In particular, the formulas are used to demonstrate that the growth rate of the perturbations must vanish if the perturbation length-scale approaches zero or infinity. The distinction between transient and sustained growth of perturbation energy is discussed in light of our results. It is suggested that the bounds are most useful for estimating transient growth rates.

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R. T. Pierrehumbert

Abstract

We examine the factors determining whether a mountain acts as a strong barrier to an impinging flow. A primary concern is the extent to which the barrier effect is reduced when smoothed orography is used in a numerical model. These questions are addressed within the model of linear, rotating, stratified flow over topography first studied by Queney. The ground level Green's functions for this model are derived and their near- and far-field asymptotic behavior is discussed. Using the asymptotic results, the upstream deceleration is estimated as a function of Ro=U/fL and hm=Nhm/U, where U is the far upstream speed of the cross-mountain flow, f the Coriolis parameter, L the mountain width, N the Brunt-Väisälä frequency, and hm is the maximum mountain height. Effects of terrain shape are also considered. The Green's functions are evaluated numerically and used to calculate the response to a family of mountain profiles; a comparison with asymptotic results shows the latter to be very useful. It is concluded that preserving maximum terrain height of ridgelike features is superior to preserving an integrated quantity such as mountain volume. Both the Alps and the Rocky Mountains exert a pronounced barrier effect which would not generally be preserved in present numerical models using smoothed orography.

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R. T. Pierrehumbert

Abstract

We obtain simple closed-form expressions describing stratified, rotating flow over an infinite ridge in the semigeostrophic regime. The results are used to discuss the degree to which air parcels slow down as they approach a steep mountain. The critical Burger number for the appearance of infinite velocities at the mountain crest is calculated for several mountain profiles. 1t is shown that for Gaussian and bell-shaped mountains the breakdown occurs in a range in which the upstream deceleration is not pronounced.

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R. T. Pierrehumbert

Abstract

The modification of baroclinic instability by low-level orographic blocking of cold advection is considered within a simple model. In this model, the meridional flow in the lower layer of a quasi-geostrophic model is blocked by a semi-infinite knife edge barrier extending eastward from the origin. The stability problem for a vertically sheared current parallel to the barrier is solved by means of the Wiener-Hopf technique, and a number of properties of the eigenmodes are determined. The barrier is found to have an inhibitory effect on cyclogenesis, in the sense that the magnitude of the pressure perturbation in the lower layer is reduced in the vicinity of the barrier. However, the barrier also distorts the flow in such a manner as to lead to the formation of a low-level cutoff low south of the barrier, which is characterized by intense easterly winds on the barrierward side of the low. It is also found that at certain phases of the development, the lower-layer velocities have square toot singularities near the tip of the barrier, leading to a discontinuous turning of air flowing around the tip. Air impinging on the barrier from the northwest develops a split jet pattern; the splitting point moves eastward along the barrier with time.

The upper layer flow is almost unaffected near the tip of the barrier, but the pressure perturbations in both layers decay algebraically to zero sufficiently far downstream of the tip.

Despite the extreme simplicity of the model, some of the features of the solution were found to be suggestive of features found in observations and numerical simulations of cyclogenesis in the lee of the Alps.

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R. T. Pierrehumbert

Abstract

The baroclinic instability characteristics of zonally inhomogeneous basic states are examined with the intent of clarifying the factors governing the regional distribution of cyclogenesis. The vertical shear of the basic state wind is allowed to vary gradually in the zonal direction, so as to permit the representation of zonally localized regions of high baroclinicity. The resulting eigenvalue problem is solved directly by numerical means and also analytically via a WKB analysis. It was established that flows with localized baroclinicity can support two distinct classes of unstable modes, which we call “local” and “global.” The local modes have peak amplitude downstream of the point of maximum baroclinicity, decay to zero exponentially upstream and downstream of the peak and do not require zonally periodic boundary conditions for their existence. The growth rate of a local mode is equal to the absolute growth rate (in the sense of Merkine) determined locally at the point of maximum shear. The absolute growth rate decreases when the vertically averaged zonal wind is increased, in contrast with the conventional locally determined maximum growth rate. Further properties of the local modes are discussed. The global modes, on the other hand, require periodic boundary conditions for their existence and have growth rates which are sensitive to the average baroclinicity over the domain. Global modes take a much longer time than local modes to emerge from random initial conditions.

On the basis of these results, it is suggested that the locally determined absolute growth rate is a useful diagnostic for assessing the stability of inhomogeneous flow. In this connection, a tentative analysis of the results of Frederiksen on planetary wave instabilities was found to be encouraging. Although only a simple model of baroclinic instability was considered in the present work, the techniques developed can be generalized to any kind of instability provided that there is a separation in spatial scale between the eddies and the basic state. It is thus proposed that there is a general link between absolute instability and the instability of nonparallel flow.

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R. T. Pierrehumbert

Abstract

The author has reconsidered the question of the regulation of tropical sea surface temperature. This has been done in general terms through consideration of the tropical beat budget and in specific terms through consideration of an idealized radiative-dynamic model of the tropical general circulation. It is argued that evaporation on its own cannot provide an effective regulating mechanism. Clouds cannot serve as regulators unless there are substantial departures from the observed cancellation between cloud greenhouse and cloud albedo effects. In particular, it is shown that the prediction by Ramanathan and Collins of highly stable tropical climates is based on an inconsistent set of assumptions about the behavior of the atmospheric heat transports. When the heat transports are treated in a consistent manner, clouds are found to have little impact, and the tropical climate can he quite sensitive to radiative perturbations.

It is found that the main determinant of tropical climate is the clear-sky water vapor greenhouse effect averaged over the entire Tropics. In the absence of dry “radiator fins” maintained by subsidence, the tropical temperature would tend to fall into a runaway greenhouse state that could be stabilized only by heat export to the extratropics. Some speculative results on sensitivity of the climate to perturbations are presented. Determination of the relative area of dry and subsiding versus moist and convecting regions of the Tropics, and of the degree of dryness of the subsiding regions, are identified as key unsolved problems concerning the tropical climate.

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R. T. Pierrehumbert and P. Malguzzi

Abstract

We consider the response of the barotropic vorticity equation on a zonally infinite f-plane or beta-plane to a weak localized vorticity source accompanied by weak Ekman damping. By performing an expansion about the unforced, undamped problem, we derive a solvability condition determining when a slight modification of a solution to the inviscid problem can occur as a solution to the weakly forced and damped problem. This condition states simply that forcing must balance dissipation in the average along each closed streamline. Much of the degeneracy of the inviscid problem is removed by the solvability condition. The above considerations are used to show that under a fairly general set of circumstances the weakly forced system possesses a high amplitude equilibrium state (identified with blocking) and a low amplitude equilibrium state. The high amplitude response is maintained by a local nonlinear resonance phenomenon, and requires the existence of a suitable solution to the inviscid problem, such as the “modon” solution. In contrast to cases previously discussed, the multiple equilibrium mechanism we treat is not dependent on global resonance.

Explicit examples of local multiple equilibria are constructed through numerical integrations on the f-plant and on a zonally infinite beta-channel. Through introduction of a reasonable amount of meridional confinement, a high amplitude solution can be obtained on the beta-plane without the use of a small radius of deformation.

It is suggested that transient eddy fluxes may be able to play the role of the forcing required in our model. A tentative comparison with a blocking event is presented, indicating a number of problematic aspects of the theory.

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H. Yang and R. T. Pierrehumbert

Abstract

The authors have explored the factors governing upper-tropospheric relative humidity with a simple model based on isentropic mixing and condensation. Our analysis has focused on the Northern Hemisphere winter season and on the 315-K (dry) isentropic surface.

The advection–condensation model yields the following results. In the absence of moisture resupply, about half of the mass of water is lost from the isentropic surface after only 10 days, with the main brake on drying being the weak mixing between Tropics and extratropics. The moist plumes escaping from the Tropics take the form of filamentary structures, which are more numerous and space filling in the summer/Southern Hemisphere than in the winter/Northern Hemisphere. These moist plumes are accompanied by substantial importation of extratropical dry air into the Tropics. The probability distributions of midlatitude relative humidity are bimodal, with a prominent dry peak having a lognormal tail and a spike representing saturated air; the summer hemisphere has generally higher relative humidity than the winter hemisphere. When moisture is maintained by periodically resaturating the Tropics, the resulting cloud and moisture fields exhibit a fractal character, with a tendency to become less space filling with distance from the Tropics.

Some tentative comparisons with data are made, which tend to confirm the advective control of relative humidity patterns outside the Tropics. There are indications, however, that the advection-condensation model with moisture resupply only from the Tropics yields an upper troposphere that is far too dry. The authors suggest that the missing moisture is supplied from the 295-K isentropic surface via diabatic mixing arising in ascending, convecting saturated trajectories near that surface.

As found in earlier passive tracer studies, the permeable mixing barrier between the Tropics and extratropics has the potential to exert a controlling influence on the global climate.

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Y. Hu and R. T. Pierrehumbert

Abstract

The mixing of a passive tracer by realistic time-dependent stratospheric flow (European Centre for Medium-Range Weather Forecasts winds) on an isentropic surface (420 K) is studied. Simulations of the advection–diffusion problem for an initially large-scale tracer field are carried out in the limit of weak diffusivity. Owing to chaotic advection, tracer variance is cascaded to small scales, where it can be dissipated despite the weak diffusivity. The tracer fluctuations are characterized in terms of their probability distribution function (PDF). The PDFs are characterized by a Gaussian core and “fat tails,” which fall more slowly than a Gaussian, and indicate anomalously high probability of extreme concentration fluctuations. Given the nonlinearity of many chemical reactions of interest, the anomalous prevalence of extreme fluctuations could have a profound effect on reactive tracers.

Zonal variations of tracer are homogenized globally leading to a unimodal PDF. Initially meridional variations are strongly influenced by the presence of mixing barriers. Meridional gradients homogenize only within mixing zones of limited latitudinal extent, bounded by permeable mixing barriers. The PDF becomes multimodal, with distinct populations of air caused by blending of the concentration values within each mixing zone. The Tropics is a zone of weak mixing, and serves as an important repository of stratospheric tracers, which are episodically ejected into surf zones in the form of filaments bearing extreme concentration values.

The shapes of the PDFs are discussed in terms of theoretical methods developed in the context of highly idealized mixing models. It is shown that such methods retain their utility when applied to realistic stratospheric mixing. The role of the probability distribution of finite-time Lyapunov exponents for the underlying trajectory problem is highlighted. The use of conditional averages of diffusion and dissipation is also illustrated. PDFs yielded by the idealized advection–diffusion problem are found to resemble those appearing in a GCM simulation of N2O.

The theoretical arguments and numerical results imply that under the assumption that the diffusivity is set so that the dissipation scale is comparable to model resolution, the concentration PDF eventually reaches a universal shape independent of model resolution after an initial transient stage. However, the width of the distribution, or equivalently the variance of the tracer fluctuation, increases algebraically as model resolution is refined.

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R. T. Pierrehumbert and H. Yang

Abstract

The general nature of two-dimensional mixing on isentropic surfaces in the troposphere has been investigated. The daily time series of isentropic winds is obtained from a global general circulation model and is used to drive a high-resolution fully Lagrangian passive tracer model. Results are compared with the extremes of chaotic mixing by organized waves on the one hand and classical diffusion on the other and are found to lie in the middle ground. Advection by planetary-and synoptic-scale eddies generates small scales from an initially smooth tracer field exponentially fast, but given a modest degree of smoothing the tracer field evolving from a localized release rapidly attains the form of an algebraically spreading cloud. The zonal size of the cloud increases linearly with time (superdiffusively), owing to the systematic shear in the extratropical zonal jets, while the meridional spread has the square-root-of-time increase characteristic of classical diffusion. It is argued, however, that the small-scale tracer structure missing from current general circulation models and from diffusive mixing models severely compromises the fidelity with which chemical reactions and the hydrological cycle can be modeled.

Several different analyses, including a study of the spatial distribution of finite-time Lyapunov exponents, indicate the presence of a partial barrier to mixing between the tropics and extratropics. In contrast with previous studies using simpler advecting flow fields, the extratropical mixing regions appear to be zonally homogeneous, and there are no impediments to zonal homogeneization. Our calculations indicate that a zonal wave 3 tracer pattern would be mixed away in 10–15 days. If the results can be taken as indicative of potential vorticity mixing, the implication is that under normal circumstances mixing due to synoptic eddies exerts a damping effect on planetary-scale waves. Implications of the results for the general circulation of the troposphere are discussed.

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