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.

Nonlinear stability of Eady's model

A nonlinear stability theorem is established for Eady's model of baroclinic flow. In particular, the Eady basic state is shown to be nonlinearly stable (for arbitrary shear) provided (Δz)/(Δy) > 2(5)^1/2f/(πN),where Δz is the height of the domain, Δy the channel width, f the Coriolis parameter, and N the buoyancy frequency. When this criterion is satisfied, explicit bounds can be derived on the disturbance potential enstrophy, the disturbance energy, and the disturbance available potential energy on the rigid lids, which are expressed in terms of the initial disturbance fields. The disturbances are completely general (with nonzero potential vorticity) and are not assumed to be of small amplitude. The results may be regarded as an extension of Arnol'd's second nonlinear stability theorem to continuously stratified quasigeostrophic baroclinic flow.

Nonlinear saturation of baroclinic instability: part III: bounds on the energy

Rigorous upper bounds are derived on the saturation amplitude of baroclinic instability in the two-layer model. The bounds apply to the eddy energy and are obtained by appealing to a finite amplitude conservation law for the disturbance pseudoenergy. These bounds are to be distinguished from those derived in Part I of this study, which employed a pseudomomentum conservation law and provided bounds on the eddy potential enstrophy. The bounds apply to conservative (inviscid, unforced) flow, as well as to forced-dissipative flow when the dissipation is proportional to the potential vorticity. Bounds on the eddy energy are worked out for a general class of unstable westerly jets. In the special case of the Phillips model of baroclinic instability, and in the limit of infinitesimal initial eddy amplitude, the bound states that the eddy energy cannot exceed ϵβ2/6F where ϵ = (U − Ucrit)/Ucrit is the relative supercriticality. This bound captures the essential dynamical scalings (i.e., the dependence on ϵ, β, and F) of the saturation amplitudes predicted by weakly nonlinear theory, as well as exhibiting remarkable quantitative agreement with those predictions, and is also consistent with heuristic baroclinic adjustment estimates.

Nonlinear saturation of baroclinic instability: part II: continuously stratified fluid

Rigorous upper bounds are derived that limit the finite-amplitude growth of arbitrary nonzonal disturbances to an unstable baroclinic zonal flow in a continuously stratified, quasi-geostrophic, semi-infinite fluid. Bounds are obtained bath on the depth-integrated eddy potential enstrophy and on the eddy available potential energy (APE) at the ground. The method used to derive the bounds is essentially analogous to that used in Part I of this study for the two-layer model: it relies on the existence of a nonlinear Liapunov (normed) stability theorem, which is a finite-amplitude generalization of the Charney-Stern theorem. As in Part I, the bounds are valid both for conservative (unforced, inviscid) flow, as well as for forced-dissipative flow when the dissipation is proportional to the potential vorticity in the interior, and to the potential temperature at the ground. The character of the results depends on the dimensionless external parameter γ = f02ξ/β0N2H, where ξ is the maximum vertical shear of the zonal wind, H is the density scale height, and the other symbols have their usual meaning. When γ ≫ 1, corresponding to “deep” unstable modes (vertical scale ≈H), the bound on the eddy potential enstrophy is just the total potential enstrophy in the system; but when γ≪1, corresponding to ‘shallow’ unstable modes (vertical scale ≈γH), the eddy potential enstrophy can be bounded well below the total amount available in the system. In neither case can the bound on the eddy APE prevent a complete neutralization of the surface temperature gradient which is in accord with numerical experience. For the special case of the Charney model of baroclinic instability, and in the limit of infinitesimal initial eddy disturbance amplitude, the bound states that the dimensionless eddy potential enstrophy cannot exceed (γ + 1)2/24&gamma2h when γ ≥ 1, or 1/6;&gammah when γ ≤ 1; here h = HN/f0L is the dimensionless scale height and L is the width of the channel. These bounds are very similar to (though of course generally larger than) ad hoc estimates based on baroclinic-adjustment arguments. The possibility of using these kinds of bounds for eddy-amplitude closure in a transient-eddy parameterization scheme is also discussed.

Nonlinear saturation of baroclinic instability: part I: the two-layer model

A rigorous bound is derived which limits the finite-amplitude growth of arbitrary nonzonal disturbances to an unstable baroclinic zonal flow within the context of the two-layer model. The bound is valid for conservative (unforced) flow, as well as for forced-dissipative flow that when the dissipation is proportional to the potential vorticity. The method used to derive the bound relies on the existence of a nonlinear Liapunov (normed) stability theorem for subcritical flows, which is a finite-amplitude generalization of the Charney-Stern theorem. For the special case of the Philips model of baroclinic instability, and in the limit of infinitesimal initial nonzonal disturbance amplitude, an improved form of the bound is possible which states that the potential enstrophy of the nonzonal flow cannot exceed ϵβ2, where ϵ = (U − Ucrit)/Ucrit is the (relative) supereriticality. This upper bound turns out to be extremely similar to the maximum predicted by the weakly nonlinear theory. For unforced flow with ϵ < 1, the bound demonstrates that the nonzonal flow cannot contain all of the potential enstrophy in the system; hence in this range of initial supercriticality the total flow must remain, in a certain sense, “close” to a zonal state.

Application of the direct Liapunov method to the problem of symmetric stability in the atmosphere

The problem of symmetric stability is examined within the context of the direct Liapunov method. The sufficient conditions for stability derived by Fjørtoft are shown to imply finite-amplitude, normed stability. This finite-amplitude stability theorem is then used to obtain rigorous upper bounds on the saturation amplitude of disturbances to symmetrically unstable flows.By employing a virial functional, the necessary conditions for instability implied by the stability theorem are shown to be in fact sufficient for instability. The results of Ooyama are improved upon insofar as a tight two-sided (upper and lower) estimate is obtained of the growth rate of (modal or nonmodal) symmetric instabilities.The case of moist adiabatic systems is also considered.

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.

Nonlinear wave-activity conservation laws and Hamiltonian structure for the two-dimensional anelastic equations

Exact, finite-amplitude, local wave-activity conservation laws are derived for disturbances to steady flows in the context of the two-dimensional anelastic equations. The conservation laws are expressed entirely in terms of Eulerian quantities, and have the property that, in the limit of a small-amplitude, slowly varying, monochromatic wave train, the wave-activity density A and flux F, when averaged over phase, satisfy F = cgA where cg is the group velocity of the waves. For nonparallel steady flows, the only conserved wave activity is a form of disturbance pseudoenergy; when the steady flow is parallel, there is in addition a conservation law for the disturbance pseudomomentum. The above results are obtained not only for isentropic background states (which give the so-called “deep form” of the anelastic equations), but also for arbitrary background potential-temperature profiles θ0(z) so long as the variation in θ0(z) over the depth of the fluid is small compared with θ0 itself. The Hamiltonian structure of the equations is established in both cases, and its symmetry properties discussed. An expression for available potential energy is also derived that, for the case of a stably stratified background state (i.e., dθ0/dz > 0), is locally positive definite; the expression is valid for fully three-dimensional flow. The counterparts to results for the two-dimensional Boussinesq equations are also noted.

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.

A spectral view of nonlinear fluxes and stationary-transient interaction in the atmosphere

Nonlinear spectral transfers of kinetic energy and enstrophy, and stationary-transient interaction, are studied using global FGGE data for January 1979. It is found that the spectral transfers arise primarily from a combination, in roughly equal measure, of pure transient and mixed stationary-transient interactions. The pure transient interactions are associated with a transient eddy field which is approximately locally homogeneous and isotropic, and they appear to be consistently understood within the context of two-dimensional homogeneous turbulence. Theory based on spatial wale separation concepts suggests that the mixed interactions may be understood physically, to a first approximation, as a process of shear-induced spectral transfer of transient enstrophy along lines of constant zonal wavenumber. This essentially conservative enstrophy transfer generally involves highly nonlocal stationary-transient energy conversions. The observational analysis demonstrates that the shear-induced transient enstrophy transfer is mainly associated with intermediate-scale (zonal wavenumber m > 3) transients and is primarily to smaller (meridional) scales, so that the transient flow acts as a source of stationary energy. In quantitative terms, this transient-eddy rectification corresponds to a forcing timescale in the stationary energy budget which is of the same order of magnitude as most estimates of the damping timescale in simple stationary-wave models (5 to 15 days). Moreover, the nonlinear interactions involved are highly nonlocal and cover a wide range of transient scales of motion.

Time development of small disturbances to plane Couette Flow

The question of linear sheared-disturbance evolution in constant-shear parallel flow is here reexamined with regard to the temporary-amplification phenomenon noted first by Orr in 1907. The results apply directly to Rossby waves on a beta-plane, and are also relevant to the Eady model of baroclinic instability. It is shown that an isotropic initial distribution of standing waves maintains a constant energy level throughout the shearing process, the amplification of some waves being precisely balanced by the decay of the others. An expression is obtained for the energy of a distribution of disturbances whose wavevectors lie within a given angular wedge and an upper bound derived. It is concluded that the case for ubiquitous amplification made in recent studies may have been somewhat overstated: while carefully-chosen individual Fourier components can amplify considerably before they decay. a general distribution will tend to exhibit little or no amplification.

Large-scale two-dimensional turbulence in the atmosphere

Global FGGE data are used to investigate several aspects of large-scale turbulence in the atmosphere. The approach follows that for two-dimensional, nondivergent turbulent flows which are homogeneous and isotropic on the sphere. Spectra of kinetic energy, enstrophy and available potential energy are obtained for both the stationary and transient parts of the flow. Nonlinear interaction terms and fluxes of energy and enstrophy through wavenumber space are calculated and compared with the theory. A possible method of parameterizing the interactions with unresolved scales is considered. Two rather different flow regimes are found in wavenumber space. The high-wavenumber regime is dominated by the transient components of the flow and exhibits, at least approximately, several of the conditions characterizing homogeneous and isotropic turbulence. This region of wavenumber space also displays some of the features of an enstrophy-cascading inertial subrange. The low-wavenumber region, on the other hand, is dominated by the stationary component of the flow, exhibits marked anisotropy and, in contrast to the high-wavenumber regime, displays a marked change between January and July.

Lower-stratospheric radiative damping and polar-night jet oscillation events

The effect of stratospheric radiative damping time scales on stratospheric variability and on stratosphere–troposphere coupling is investigated in a simplified global circulation model by modifying the vertical profile of radiative damping in the stratosphere while holding it fixed in the troposphere. Perpetual-January conditions are imposed, with sinusoidal topography of zonal wavenumber 1 or 2. The depth and duration of the simulated sudden stratospheric warmings closely track the lower-stratospheric radiative time scales. Simulations with the most realistic profiles of radiative damping exhibit extended time-scale recoveries analogous to polar-night jet oscillation (PJO) events, which are observed to follow sufficiently deep stratospheric warmings. These events are characterized by weak lower-stratospheric winds and enhanced stability near the tropopause, which persist for up to 3 months following the initial warming. They are obtained with both wave-1 and wave-2 topography. Planetary-scale Eliassen–Palm (EP) fluxes entering the vortex are also suppressed, which is in agreement with observed PJO events. Consistent with previous studies, the tropospheric jets shift equatorward in response to the warmings. The duration of the shift is closely correlated with the period of enhanced stability. The magnitude of the shift in these runs, however, is sensitive only to the zonal wavenumber of the topography. Although the shift is sustained primarily by synoptic-scale eddies, the net effect of the topographic form drag and the planetary-scale fluxes is not negligible; they damp the surface wind response but enhance the vertical shear. The tropospheric response may also reduce the generation of planetary waves, further extending the stratospheric dynamical time scales.

Southern annular mode dynamics in observations and models, Part I: The influence of climatological zonal wind biases in a comprehensive GCM

A common bias among global climate models (GCMs) is that they exhibit tropospheric southern annular mode (SAM) variability that is much too persistent in the Southern Hemisphere (SH) summertime. This is of concern for the ability to accurately predict future SH circulation changes, so it is important that it be understood and alleviated. In this two-part study, specifically targeted experiments with the Canadian Middle Atmosphere Model (CMAM) are used to improve understanding of the enhanced summertime SAM persistence. Given the ubiquity of this bias among comprehensive GCMs, it is likely that the results will be relevant for other climate models. Here, in Part I, the influence of climatological circulation biases on SAM variability is assessed, with a particular focus on two common biases that could enhance summertime SAM persistence: the too-late breakdown of the Antarctic stratospheric vortex and the equatorward bias in the SH tropospheric midlatitude jet. Four simulations are used to investigate the role of each of these biases in CMAM. Nudging and bias correcting procedures are used to systematically remove zonal-mean stratospheric variability and/or remove climatological zonal wind biases. The SAM time-scale bias is not alleviated by improving either the timing of the stratospheric vortex breakdown or the climatological jet structure. Even in the absence of stratospheric variability and with an improved climatological circulation, the model time scales are biased long. This points toward a bias in internal tropospheric dynamics that is not caused by the tropospheric jet structure bias. The underlying cause of this is examined in more detail in Part II of this study.