Stirring and mixing in two-dimensional divergent flow

While stirring and mixing properties in the stratosphere are reasonably well understood in the context of balanced (slow) dynamics, as is evidenced in numerous studies of chaotic advection, the strongly enhanced presence of high-frequency gravity waves in the mesosphere gives rise to a significant unbalanced (fast) component to the flow. The present investigation analyses result from two idealized shallow-water numerical simulations representative of stratospheric and mesospheric dynamics on a quasi-horizontal isentropic surface. A generalization of the Hua–Klein Eulerian diagnostic to divergent flow reveals that velocity gradients are strongly influenced by the unbalanced component of the flow. The Lagrangian diagnostic of patchiness nevertheless demonstrates the persistence of coherent features in the zonal component of the flow, in contrast to the destruction of coherent features in the meridional component. Single-particle statistics demonstrate t2 scaling for both the stratospheric and mesospheric regimes in the case of zonal dispersion, and distinctive scaling laws for the two regimes in the case of meridional dispersion. This is in contrast to two-particle statistics, which in the mesospheric (unbalanced) regime demonstrate a more rapid approach to Richardson’s t3 law in the case of zonal dispersion and is evidence of enhanced meridional dispersion.

Zonal-mean dynamics of extended recoveries from stratospheric sudden warmings

The recovery of the Arctic polar vortex following stratospheric sudden warmings is found to take upward of 3 months in a particular subset of cases, termed here polar-night jet oscillation (PJO) events. The anomalous zonal-mean circulation above the pole during this recovery is characterized by a persistently warm lower stratosphere, and above this a cold midstratosphere and anomalously high stratopause, which descends as the event unfolds. Composites of these events in the Canadian Middle Atmosphere Model show the persistence of the lower-stratospheric anomaly is a result of strongly suppressed wave driving and weak radiative cooling at these heights. The upper-stratospheric and lower-mesospheric anomalies are driven immediately following the warming by anomalous planetary-scale eddies, following which, anomalous parameterized nonorographic and orographic gravity waves play an important role. These details are found to be robust for PJO events (as opposed to sudden warmings in general) in that many details of individual PJO events match the composite mean. Azonal-mean quasigeostrophic model on the sphere is shown to reproduce the response to the thermal and mechanical forcings produced during a PJO event. The former is well approximated by Newtonian cooling. The response can thus be considered as a transient approach to the steady-state, downward control limit. In this context, the time scale of the lower-stratospheric anomaly is determined by the transient, radiative response to the extended absence of wave driving. The extent to which the dynamics of the wave-driven descent of the stratopause can be considered analogous to the descending phases of the quasi-biennial oscillation (QBO) is also discussed.

The angular momentum constraint on climate sensitivity and downward influence in the middle atmosphere

It is shown that under reasonable assumptions, conservation of angular momentum provides a strong constraint on gravity wave drag feedbacks to radiative perturbations in the middle atmosphere. In the time mean, radiatively induced temperature perturbations above a given altitude z cannot induce changes in zonal mean wind and temperature below z through feedbacks in gravity wave drag alone (assuming an unchanged gravity wave source spectrum). Thus, despite the many uncertainties in the parameterization of gravity wave drag, the role of gravity wave drag in middle-atmosphere climate perturbations may be much more limited than its role in climate itself. This constraint limits the possibilities for downward influence from the mesosphere. In order for a gravity wave drag parameterization to respect the momentum constraint and avoid spurious downward influence, any nonzero parameterized momentum flux at a model lid must be deposited within the model domain, and there must be no zonal mean sponge layer. Examples are provided of how violation of these conditions leads to spurious downward influence. For planetary waves, the momentum constraint does not prohibit downward influence, but it limits the mechanisms by which it can occur: in the time mean, downward influence from a radiative perturbation can only arise through changes in reflection and meridional propagation properties of planetary waves.

Uncertainties and assessments of chemistry-climate models of the stratosphere

In recent years a number of chemistry-climate models have been developed with an emphasis on the stratosphere. Such models cover a wide range of time scales of integration and vary considerably in complexity. The results of specific diagnostics are here analysed to examine the differences amongst individual models and observations, to assess the consistency of model predictions, with a particular focus on polar ozone. For example, many models indicate a significant cold bias in high latitudes, the “cold pole problem”, particularly in the southern hemisphere during winter and spring. This is related to wave propagation from the troposphere which can be improved by improving model horizontal resolution and with the use of non-orographic gravity wave drag. As a result of the widely differing modelled polar temperatures, different amounts of polar stratospheric clouds are simulated which in turn result in varying ozone values in the models. The results are also compared to determine the possible future behaviour of ozone, with an emphasis on the polar regions and mid-latitudes. All models predict eventual ozone recovery, but give a range of results concerning its timing and extent. Differences in the simulation of gravity waves and planetary waves as well as model resolution are likely major sources of uncertainty for this issue. In the Antarctic, the ozone hole has probably reached almost its deepest although the vertical and horizontal extent of depletion may increase slightly further over the next few years. According to the model results, Antarctic ozone recovery could begin any year within the range 2001 to 2008. The limited number of models which have been integrated sufficiently far indicate that full recovery of ozone to 1980 levels may not occur in the Antarctic until about the year 2050. For the Arctic, most models indicate that small ozone losses may continue for a few more years and that recovery could begin any year within the range 2004 to 2019. The start of ozone recovery in the Arctic is therefore expected to appear later than in the Antarctic.

Extended Canadian middle atmosphere model: zonal-mean climatology and physical parameterizations

This paper describes the energetics and zonal-mean state of the upward extension of the Canadian Middle Atmosphere Model, which extends from the ground to ~210 km. The model includes realistic parameterizations of the major physical processes from the ground up to the lower thermosphere and exhibits a broad spectrum of geophysical variability. The rationale for the extended model is to examine the nature of the physical and dynamical processes in the mesosphere/lower thermosphere (MLT) region without the artificial effects of an imposed sponge layer which can modify the circulation in an unrealistic manner. The zonal-mean distributions of temperature and zonal wind are found to be in reasonable agreement with observations in most parts of the model domain below ~150 km. Analysis of the global-average energy and momentum budgets reveals a balance between solar extreme ultraviolet heating and molecular diffusion and a thermally direct viscous meridional circulation above 130 km, with the viscosity coming from molecular diffusion and ion drag. Below 70 km, radiative equilibrium prevails in the global mean. In the MLT region between ~70 and 120 km, many processes contribute to the global energy budget. At solstice, there is a thermally indirect meridional circulation driven mainly by parameterized nonorographic gravity-wave drag. This circulation provides a net global cooling of up to 25 K d^-1.

Averaging, slaving and balance dynamics in a simple atmospheric model

We report numerical results from a study of balance dynamics using a simple model of atmospheric motion that is designed to help address the question of why balance dynamics is so stable. The non-autonomous Hamiltonian model has a chaotic slow degree of freedom (representing vortical modes) coupled to one or two linear fast oscillators (representing inertia-gravity waves). The system is said to be balanced when the fast and slow degrees of freedom are separated. We find adiabatic invariants that drift slowly in time. This drift is consistent with a random-walk behaviour at a speed which qualitatively scales, even for modest time scale separations, as the upper bound given by Neishtadt’s and Nekhoroshev’s theorems. Moreover, a similar type of scaling is observed for solutions obtained using a singular perturbation (‘slaving’) technique in resonant cases where Nekhoroshev’s theorem does not apply. We present evidence that the smaller Lyapunov exponents of the system scale exponentially as well. The results suggest that the observed stability of nearly-slow motion is a consequence of the approximate adiabatic invariance of the fast motion.

On the nature of large-scale mixing in the stratosphere and mesosphere

Studies of tracer transport in the stratosphere have shown that adiabatic quasi-horizontal tracer evolution is controlled primarily by the large-scale low-frequency component of the flow. This behavior is consistent with the concept of chaotic advection, wherein the Eulerian velocity field is spatially coherent and temporally quasi-regular on timescales over which the Lagrangian evolution is chaotic. In this study, winds from a middle atmosphere general circulation model (the Canadian Middle Atmosphere Model) are used to compare and contrast the nature of tracer evolution in the stratosphere and mesosphere. It is found that the concept of chaotic advection is relevant in the stratosphere but not in the mesosphere. The explanation for this behavior is the increased strength of gravity wave activity in the mesosphere as compared with the stratosphere, which leads to shallower kinetic energy spectra on synoptic scales and a much shorter Eulerian correlation time. The shallower kinetic energy spectra imply that tracer evolution in the mesosphere is spectrally local, in contrast with the spectrally nonlocal regime that prevails in the stratosphere. This means that tracer advection calculations in the mesosphere are controlled primarily by the gravity wave spectrum and are intrinsically resolution dependent.

Sponge layer feedbacks in middle-atmosphere models

Middle-atmosphere models commonly employ a sponge layer in the upper portion of their domain. It is shown that the relaxational nature of the sponge allows it to couple to the dynamics at lower levels in an artificial manner. In particular, the long-term zonally symmetric response to an imposed extratropical local force or diabatic heating is shown to induce a drag force in the sponge that modifies the response expected from the “downward control” arguments of Haynes et al. [1991]. In the case of an imposed local force the sponge acts to divert a fraction of the mean meridional mass flux upward, which for realistic parameter values is approximately equal to exp(−Δz/H), where Δz is the distance between the forcing region and the sponge layer and H is the density scale height. This sponge-induced upper cell causes temperature changes that, just below the sponge layer, are of comparable magnitude to those just below the forcing region. In the case of an imposed local diabatic heating, the sponge induces a meridional circulation extending through the entire depth of the atmosphere. This circulation causes temperature changes that, just below the sponge layer, are of opposite sign and comparable in magnitude to those at the heating region. In both cases, the sponge-induced temperature changes are essentially independent of the height of the imposed force or diabatic heating, provided the latter is located outside the sponge, but decrease exponentially as one moves down from the sponge. Thus the effect of the sponge can be made arbitrarily small at a given altitude by placing the sponge sufficiently high; e.g., its effect on temperatures two scale heights below is roughly at the 10% level, provided the imposed force or diabatic heating is located outside the sponge. When, however, an imposed force is applied within the sponge layer (a highly plausible situation for parameterized mesospheric gravity-wave drag), its effect is almost entirely nullified by the sponge-layer feedback and its expected impact on temperatures below largely fails to materialize. Simulations using a middle-atmosphere general circulation model are described, which demonstrate that this sponge-layer feedback can be a significant effect in parameter regimes of physical interest. Zonally symmetric (two dimensional) middle-atmosphere models commonly employ a Rayleigh drag throughout the model domain. It is shown that the long-term zonally symmetric response to an imposed extratropical local force or diabatic heating, in this case, is noticeably modified from that expected from downward control, even for a very weak drag coefficient

On Hamiltonian balanced dynamics and the slowest invariant manifold

The concept of a slowest invariant manifold is investigated for the five-component model of Lorenz under conservative dynamics. It is shown that Lorenz's model is a two-degree-of-freedom canonical Hamiltonian system, consisting of a nonlinear vorticity-triad oscillator coupled to a linear gravity wave oscillator, whose solutions consist of regular and chaotic orbits. When either the Rossby number or the rotational Froude number is small, there is a formal separation of timescales, and one can speak of fast and slow motion. In the same regime, the coupling is weak, and the Kolmogorov–Arnold-Moser theorem is shown to apply. The chaotic orbits are inherently unbalanced and are confined to regions sandwiched between invariant tori consisting of quasi-periodic regular orbits. The regular orbits generally contain free fast motion, but a slowest invariant manifold may be geometrically defined as the set of all slow cores of invariant tori (defined by zero fast action) that are smoothly related to such cores in the uncoupled system. This slowest invariant manifold is not global; in fact, its structure is fractal; but it is of nearly full measure in the limit of weak coupling. It is also nonlinearly stable. As the coupling increases, the slowest invariant manifold shrinks until it disappears altogether. The results clarify previous definitions of a slowest invariant manifold and highlight the ambiguity in the definition of “slowness.” An asymptotic procedure, analogous to standard initialization techniques, is found to yield nonzero free fast motion even when the core solutions contain none. A hierarchy of Hamiltonian balanced models preserving the symmetries in the original low-order model is formulated; these models are compared with classic balanced models, asymptotically initialized solutions of the full system and the slowest invariant manifold defined by the core solutions. The analysis suggests that for sufficiently small Rossby or rotational Froude numbers, a stable slowest invariant manifold can be defined for this system, which has zero free gravity wave activity, but it cannot be defined everywhere. The implications of the results for more complex systems are discussed.

Orographic drag associated with lee waves trapped at an inversion

The drag produced by 2D orographic gravity waves trapped at a temperature inversion and waves propagating in the stably stratified layer existing above are explicitly calculated using linear theory, for a two-layer atmosphere with neutral static stability near the surface, mimicking a well-mixed boundary layer. For realistic values of the flow parameters, trapped lee wave drag, which is given by a closed analytical expression, is comparable to propagating wave drag, especially in moderately to strongly non-hydrostatic conditions. In resonant flow, both drag components substantially exceed the single-layer hydrostatic drag estimate used in most parametrization schemes. Both drag components are optimally amplified for a relatively low-level inversion and Froude numbers Fr ≈ 1. While propagating wave drag is maximized for approximately hydrostatic flow, trapped lee wave drag is maximized for l_2 a = O(1) (where l_2 is the Scorer parameter in the stable layer and a is the mountain width). This roughly happens when the horizontal scale of trapped lee waves matches that of the mountain slope. The drag behavior as a function of Fr for l_2 H = 0.5 (where H is the inversion height) and different values of l2a shows good agreement with numerical simulations. Regions of parameter space with high trapped lee wave drag correlate reasonably well with those where lee wave rotors were found to occur in previous nonlinear numerical simulations including frictional effects. This suggests that trapped lee wave drag, besides giving a relevant contribution to low-level drag exerted on the atmosphere, may also be useful to diagnose lee rotor formation.

On the 'downward control' of extratropical diabatic circulations by eddy-induced mean zonal forces

The situation considered is that of a zonally symmetric model of the middle atmosphere subject to a given quasi-steady zonal force F̄, conceived to be the result of irreversible angular momentum transfer due to the upward propagation and breaking of Rossby and gravity waves together with any other dissipative eddy effects that may be relevant. The model's diabatic heating is assumed to have the qualitative character of a relaxation toward some radiatively determined temperature field. To the extent that the force F̄ may be regarded as given, and the extratropical angular momentum distribution is realistic, the extratropical diabatic mass flow across a given isentropic surface may be regarded as controlled exclusively by the F̄ distribution above that surface (implying control by the eddy dissipation above that surface and not, for instance, by the frequency of tropopause folding below). This “downward control” principle expresses a critical part of the dynamical chain of cause and effect governing the average rate at which photochemical products like ozone become available for folding into, or otherwise descending into, the extratropical troposphere. The dynamical facts expressed by the principle are also relevant, for instance, to understanding the seasonal-mean rate of upwelling of water vapor to the summer mesopause, and the interhemispheric differences in stratospheric tracer transport. The robustness of the principle is examined when F̄ is time-dependent. For a global-scale, zonally symmetric diabatic circulation with a Brewer-Dobson-like horizontal structure given by the second zonally symmetric Hough mode, with Rossby height HR = 13 km in an isothermal atmosphere with density scale height H = 7 km, the vertical partitioning of the unsteady part of the mass circulation caused by fluctuations in F̄ confined to a shallow layer LF̄ is always at least 84% downward. It is 90% downward when the force fluctuates sinusoidally on twice the radiative relaxation timescale and 95% if five times slower. The time-dependent adjustment when F̄ is changed suddenly is elucidated, extending the work of Dickinson (1968), when the atmosphere is unbounded above and below. Above the forcing, the adjustment is characterized by decay of the meridional mass circulation cell at a rate proportional to the radiative relaxation rate τr−1 divided by {1 + (4H2/HR2)}. This decay is related to the boundedness of the angular momentum that can be taken up by the finite mass of air above LF̄ without causing an ever-increasing departure from thermal wind balance. Below the forcing, the meridional mass circulation cell penetrates downward at a speed τr−1 HR2/H. For the second Hough mode, the time for downward penetration through one density scale height is about 6 days if the radiative relaxation time is 20 days, the latter being representative of the lower stratosphere. At any given altitude, a steady state is approached. The effect of a rigid lower boundary on the time-dependent adjustment is also considered. If a frictional planetary boundary layer is present then a steady state is ultimately approached everywhere, with the mass circulation extending downward from LF̄ and closing via the boundary layer. Satellite observations of temperature and ozone are used in conjunction with a radiative transfer scheme to estimate the altitudes from which the lower stratospheric diabatic vertical velocity is controlled by the effective F̄ in the real atmosphere. The data appear to indicate that about 80% of the effective control is usually exerted from below 40 km but with significant exceptions up to 70 km (in the high latitude southern hemispheric winter). The implications for numerical modelling of chemical transport are noted.

The effect of non-uniform radiative damping on the zonal-mean dynamics of the extratropical middle atmosphere

The effect of spatial and temporal variations in the radiative damping rate on the response to an imposed forcing or diabatic heating is examined in a zonal-mean model of the middle atmosphere. Attention is restricted to the extratropics, where a linear approach is viable. It is found that regions with weak radiative damping rates are more sensitive in terms of temperature to the remote influence of the diabatic circulation. The delay in the response in such regions can mean that ‘downward’ control is not achieved on seasonal time-scales. A seasonal variation in the radiative damping rate modulates the evolution of the response and leaves a transient-like signature in the annual mean temperature field. Several idealized examples are considered, motivated by topical questions. It is found that wave drag outside the polar vortex can significantly affect the temperatures in its interior, so that high-latitude, high-altitude gravity-wave drag is not the only mechanism for warming the southern hemisphere polar vortex. Diabatic mass transport through the 100 hPa surface is found to lag the seasonal evolution of the wave drag that drives the transport, and thus cannot be considered to be in the downward control regime. On the other hand, the seasonal variation of the radiative damping rate is found to make only a weak contribution to the annual mean temperature increase that has been observed above the ozone hole. Copyright © 2002 Royal Meteorological Society.

Runge-Kutta IMEX schemes for the Horizontally Explicit/Vertically Implicit (HEVI) solution of wave equations

Many operational weather forecasting centres use semi-implicit time-stepping schemes because of their good efficiency. However, as computers become ever more parallel, horizontally explicit solutions of the equations of atmospheric motion might become an attractive alternative due to the additional inter-processor communication of implicit methods. Implicit and explicit (IMEX) time-stepping schemes have long been combined in models of the atmosphere using semi-implicit, split-explicit or HEVI splitting. However, most studies of the accuracy and stability of IMEX schemes have been limited to the parabolic case of advection–diffusion equations. We demonstrate how a number of Runge–Kutta IMEX schemes can be used to solve hyperbolic wave equations either semi-implicitly or HEVI. A new form of HEVI splitting is proposed, UfPreb, which dramatically improves accuracy and stability of simulations of gravity waves in stratified flow. As a consequence it is found that there are HEVI schemes that do not lose accuracy in comparison to semi-implicit ones. The stability limits of a number of variations of trapezoidal implicit and some Runge–Kutta IMEX schemes are found and the schemes are tested on two vertical slice cases using the compressible Boussinesq equations split into various combinations of implicit and explicit terms. Some of the Runge–Kutta schemes are found to be beneficial over trapezoidal, especially since they damp high frequencies without dropping to first-order accuracy. We test schemes that are not formally accurate for stiff systems but in stiff limits (nearly incompressible) and find that they can perform well. The scheme ARK2(2,3,2) performs the best in the tests.

Wave activity in the tropical tropopause layer in seven reanalysis and four chemistry climate model data sets

Sub-seasonal variability including equatorial waves significantly influence the dehydration and transport processes in the tropical tropopause layer (TTL). This study investigates the wave activity in the TTL in 7 reanalysis data sets (RAs; NCEP1, NCEP2, ERA40, ERA-Interim, JRA25, MERRA, and CFSR) and 4 chemistry climate models (CCMs; CCSRNIES, CMAM, MRI, and WACCM) using the zonal wave number-frequency spectral analysis method with equatorially symmetric-antisymmetric decomposition. Analyses are made for temperature and horizontal winds at 100 hPa in the RAs and CCMs and for outgoing longwave radiation (OLR), which is a proxy for convective activity that generates tropopause-level disturbances, in satellite data and the CCMs. Particular focus is placed on equatorial Kelvin waves, mixed Rossby-gravity (MRG) waves, and the Madden-Julian Oscillation (MJO). The wave activity is defined as the variance, i.e., the power spectral density integrated in a particular zonal wave number-frequency region. It is found that the TTL wave activities show significant difference among the RAs, ranging from ∼0.7 (for NCEP1 and NCEP2) to ∼1.4 (for ERA-Interim, MERRA, and CFSR) with respect to the averages from the RAs. The TTL activities in the CCMs lie generally within the range of those in the RAs, with a few exceptions. However, the spectral features in OLR for all the CCMs are very different from those in the observations, and the OLR wave activities are too low for CCSRNIES, CMAM, and MRI. It is concluded that the broad range of wave activity found in the different RAs decreases our confidence in their validity and in particular their value for validation of CCM performance in the TTL, thereby limiting our quantitative understanding of the dehydration and transport processes in the TTL.

First results from a 3-dimensional middle atmosphere model

Initial results are presented from a middle atmosphere extension to a version of the European Centre For Medium Range Weather Forecasting tropospheric model. The extended version of the model has been developed as part of the UK Universities Global Atmospheric Modelling Project and extends from the ground to approximately 90 km. A comprehensive solar radiation scheme is included which uses monthly averaged climatological ozone values. A linearised infrared cooling scheme is employed. The basic climatology of the model is described; the parametrization of drag due to orographically forced gravity waves is shown to have a dramatic effect on the simulations of the winter hemisphere.