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 stability of multilayer quasi-geostrophic flow

New nonlinear stability theorems are derived for disturbances to steady basic flows in the context of the multilayer quasi-geostrophic equations. These theorems are analogues of Arnol’d's second stability theorem, the latter applying to the two-dimensional Euler equations. Explicit upper bounds are obtained on both the disturbance energy and disturbance potential enstrophy in terms of the initial disturbance fields. An important feature of the present analysis is that the disturbances are allowed to have non-zero circulation. While Arnol’d's stability method relies on the energy–Casimir invariant being sign-definite, the new criteria can be applied to cases where it is sign-indefinite because of the disturbance circulations. A version of Andrews’ theorem is established for this problem, and uniform potential vorticity flow is shown to be nonlinearly stable. The special case of two-layer flow is treated in detail, with particular attention paid to the Phillips model of baroclinic instability. It is found that the short-wave portion of the marginal stability curve found in linear theory is precisely captured by the new nonlinear stability criteria.

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.

Quantifying the climatological relationship between extratropical cyclone intensity and atmospheric precursors

We introduce a novel technique in which linear regression analysis is applied to clusters of tracked cyclones to statistically assess the factors controlling cyclone development. We illustrate this technique by evaluating the differences between cyclones forming in the west and east North Atlantic (herein termed west and east Atlantic cyclones). Enhanced cyclone intensity 2 days after genesis is found to be associated with deeper upper-level troughs upstream of the cyclone center at the genesis time in both west and east Atlantic cyclones. However, whilst west Atlantic cyclones are also enhanced by the presence of strong fronts, east Atlantic cyclones are not. Instead, east Atlantic cyclones exhibit an enhancement when diabatically generated midlevel potential vorticity is present (with the enhancement being of approximately equal magnitude to that associated with the potential vorticity in the upper-level trough). This is consistent with the paradigm of latent heat release in the warm conveyor belt region playing an important role in the development of east Atlantic cyclones.

Comments on 'Balance and the slow quasimanifold: some explicit results'

The concept of slow vortical dynamics and its role in theoretical understanding is central to geophysical fluid dynamics. It leads, for example, to “potential vorticity thinking” (Hoskins et al. 1985). Mathematically, one imagines an invariant manifold within the phase space of solutions, called the slow manifold (Leith 1980; Lorenz 1980), to which the dynamics are constrained. Whether this slow manifold truly exists has been a major subject of inquiry over the past 20 years. It has become clear that an exact slow manifold is an exceptional case, restricted to steady or perhaps temporally periodic flows (Warn 1997). Thus the concept of a “fuzzy slow manifold” (Warn and Ménard 1986) has been suggested. The idea is that nearly slow dynamics will occur in a stochastic layer about the putative slow manifold. The natural question then is, how thick is this layer? In a recent paper, Ford et al. (2000) argue that Lighthill emission—the spontaneous emission of freely propagating acoustic waves by unsteady vortical flows—is applicable to the problem of balance, with the Mach number Ma replaced by the Froude number F, and that it is a fundamental mechanism for this fuzziness. They consider the rotating shallow-water equations and find emission of inertia–gravity waves at O(F2). This is rather surprising at first sight, because several studies of balanced dynamics with the rotating shallow-water equations have gone beyond second order in F, and found only an exponentially small unbalanced component (Warn and Ménard 1986; Lorenz and Krishnamurthy 1987; Bokhove and Shepherd 1996; Wirosoetisno and Shepherd 2000). We have no technical objection to the analysis of Ford et al. (2000), but wish to point out that it depends crucially on R 1, where R is the Rossby number. This condition requires the ratio of the characteristic length scale of the flow L to the Rossby deformation radius LR to go to zero in the limit F → 0. This is the low Froude number scaling of Charney (1963), which, while originally designed for the Tropics, has been argued to be also relevant to mesoscale dynamics (Riley et al. 1981). If L/LR is fixed, however, then F → 0 implies R → 0, which is the standard quasigeostrophic scaling of Charney (1948; see, e.g., Pedlosky 1987). In this limit there is reason to expect the fuzziness of the slow manifold to be “exponentially thin,” and balance to be much more accurate than is consistent with (algebraic) Lighthill emission.

Idealised simulations of sting-jet cyclones

An idealised modelling study of sting-jet cyclones is presented. Sting jets are descending mesoscale jets that occur in some extratropical cyclones and produce localised regions of strong low-level winds in the frontal fracture region. Moist baroclinic lifecycle (LC1) simulations are performed with modifications to produce cyclones resembling observed sting-jet cyclones. A sting jet exists in the idealised control cyclone with similar characteristics to the sting jet in a simulation of windstorm Gudrun (a confirmed sting-jet case). Unlike in windstorm Gudrun, a low-level layer of strong moist static stability prohibits the descent of the strong winds from above the boundary layer to the surface in the idealised case. Conditional symmetric instability (CSI) exists in the cloud head and dissipates as the sting jet leaves the cloud head and descends. The descending, initially moist, sting-jet trajectories consistently have negative or near-zero saturated moist potential vorticity but moist static stability and inertial stability, consistent with CSI release; the moist static stability becomes negative during the period of most rapid descent, by which time the air is relatively dry implying conditional instability release is unlikely. Sensitivity experiments show that the existence of the sting jet is robust to changes in the initial state, and that the initial tropospheric static stability significantly impacts the descent rate of the sting jet. Inertial and conditional instability are probably being released in the experiment with the weakest initial static stability. This suggests that sting jets can arise through the release of all three instabilities associated with negative saturated moist potential vorticity. While evaporative cooling occurs along the sting-jet trajectories, a sensitivity experiment with evaporation effects turned off shows no significant change to the wind strength or descent rate of the sting jet implying that instability release is the dominant sting-jet driving mechanism.

Balance model for equatorial long waves

Geophysical fluid models often support both fast and slow motions. As the dynamics are often dominated by the slow motions, it is desirable to filter out the fast motions by constructing balance models. An example is the quasi geostrophic (QG) model, which is used widely in meteorology and oceanography for theoretical studies, in addition to practical applications such as model initialization and data assimilation. Although the QG model works quite well in the mid-latitudes, its usefulness diminishes as one approaches the equator. Thus far, attempts to derive similar balance models for the tropics have not been entirely successful as the models generally filter out Kelvin waves, which contribute significantly to tropical low-frequency variability. There is much theoretical interest in the dynamics of planetary-scale Kelvin waves, especially for atmospheric and oceanic data assimilation where observations are generally only of the mass field and thus do not constrain the wind field without some kind of diagnostic balance relation. As a result, estimates of Kelvin wave amplitudes can be poor. Our goal is to find a balance model that includes Kelvin waves for planetary-scale motions. Using asymptotic methods, we derive a balance model for the weakly nonlinear equatorial shallow-water equations. Specifically we adopt the ‘slaving’ method proposed by Warn et al. (Q. J. R. Meteorol. Soc., vol. 121, 1995, pp. 723–739), which avoids secular terms in the expansion and thus can in principle be carried out to any order. Different from previous approaches, our expansion is based on a long-wave scaling and the slow dynamics is described using the height field instead of potential vorticity. The leading-order model is equivalent to the truncated long-wave model considered previously (e.g. Heckley & Gill, Q. J. R. Meteorol. Soc., vol. 110, 1984, pp. 203–217), which retains Kelvin waves in addition to equatorial Rossby waves. Our method allows for the derivation of higher-order models which significantly improve the representation of Rossby waves in the isotropic limit. In addition, the ‘slaving’ method is applicable even when the weakly nonlinear assumption is relaxed, and the resulting nonlinear model encompasses the weakly nonlinear model. We also demonstrate that the method can be applied to more realistic stratified models, such as the Boussinesq model.

Tracing troposphere-to-stratosphere transport above a mid-latitude deep convective system

Within the project SPURT (trace gas measurements in the tropopause region) a variety of trace gases have been measured in situ in order to investigate the role of dynamical and chemical processes in the extra-tropical tropopause region. In this paper we report on a flight on 10 November 2001 leading from Hohn, Germany (52�N) to Faro, Portugal (37�N) through a strongly developed deep stratospheric intrusion. This streamer was associated with a large convective system over the western Mediterranean with potentially significant troposphere-to-stratosphere transport. Along major parts of the flight we measured unexpectedly high NOy mixing ratios. Also H2O mixing ratios were significantly higher than stratospheric background levels confirming the extraordinary chemical signature of the probed air masses in the interior of the streamer. Backward trajectories encompassing the streamer enable to analyze the origin and physical characteristics of the air masses and to trace troposphere-to-stratosphere transport. Near the western flank of the streamer features caused by long range transport, such as tropospheric filaments characterized by sudden drops in the O3 and NOy mixing ratios and enhanced CO and H2O can be reconstructed in great detail using the reverse domain filling technique. These filaments indicate a high potential for subsequent mixing with the stratospheric air. At the south-western edge of the streamer a strong gradient in the NOy and the O3 mixing ratios coincides very well with a sharp gradient in potential vorticity in the ECMWF fields. In contrast, in the interior of the streamer the observed highly elevated NOy and H2O mixing ratios up to a potential temperature level of 365K and potential vorticity values of maximum 10 PVU cannot be explained in terms of resolved troposphere-to-stratosphere transport along the backward trajectories. Also mesoscale simulations with a High Resolution Model reveal no direct evidence for convective H2O injection up to this level. Elevated H2O mixing ratios in the ECMWF and HRM are seen only up to about tropopause height at 340 hPa and 270 hPa, respectively, well below flight altitude of about 200 hPa. However, forward tracing of the convective influence as identified by satellite brightness temperature measurements and counts of lightning strokes shows that during this part of the flight the aircraft was closely following the border of an air mass which was heavily impacted by convective activity over Spain and Algeria. This is evidence that deep convection at mid-latitudes may have a large impact on the tracer distribution of the lowermost stratosphere reaching well above the thunderstorms anvils as claimed by recent studies using cloud-resolving models.

Seasonal cycles and variability of O3 and H2O in the UT/LMS during SPURT

Airborne high resolution in situ measurements of a large set of trace gases including ozone (O3) and total water (H2O) in the upper troposphere and the lowermost stratosphere (UT/LMS) have been performed above Europe within the SPURT project. SPURT provides an extensive data coverage of the UT/LMS in each season within the time period between November 2001 and July 2003. In the LMS a distinct spring maximum and autumn minimum is observed in O3, whereas its annual cycle in the UT is shifted by 2–3 months later towards the end of the year. The more variable H2O measurements reveal a maximum during summer and a minimum during autumn/winter with no phase shift between the two atmospheric compartments. For a comprehensive insight into trace gas composition and variability in the UT/LMS several statistical methods are applied using chemical, thermal and dynamical vertical coordinates. In particular, 2-dimensional probability distribution functions serve as a tool to transform localised aircraft data to a more comprehensive view of the probed atmospheric region. It appears that both trace gases, O3 and H2O, reveal the most compact arrangement and are best correlated in the view of potential vorticity (PV) and distance to the local tropopause, indicating an advanced mixing state on these surfaces. Thus, strong gradients of PV seem to act as a transport barrier both in the vertical and the horizontal direction. The alignment of trace gas isopleths reflects the existence of a year-round extra-tropical tropopause transition layer. The SPURT measurements reveal that this layer is mainly affected by stratospheric air during winter/spring and by tropospheric air during autumn/summer. Normalised mixing entropy values for O3 and H2O in the LMS appear to be maximal during spring and summer, respectively, indicating highest variability of these trace gases during the respective seasons.

The influence of stratospheric vortex displacements and splits on surface climate

A strong link exists between stratospheric variability and anomalous weather patterns at the earth’s surface. Specifically, during extreme variability of the Arctic polar vortex termed a “weak vortex event,” anomalies can descend from the upper stratosphere to the surface on time scales of weeks. Subsequently the outbreak of cold-air events have been noted in high northern latitudes, as well as a quadrupole pattern in surface temperature over the Atlantic and western European sectors, but it is currently not understood why certain events descend to the surface while others do not. This study compares a new classification technique of weak vortex events, based on the distribution of potential vorticity, with that of an existing technique and demonstrates that the subdivision of such events into vortex displacements and vortex splits has important implications for tropospheric weather patterns on weekly to monthly time scales. Using reanalysis data it is found that vortex splitting events are correlated with surface weather and lead to positive temperature anomalies over eastern North America of more than 1.5 K, and negative anomalies over Eurasia of up to −3 K. Associated with this is an increase in high-latitude blocking in both the Atlantic and Pacific sectors and a decrease in European blocking. The corresponding signals are weaker during displacement events, although ultimately they are shown to be related to cold-air outbreaks over North America. Because of the importance of stratosphere–troposphere coupling for seasonal climate predictability, identifying the type of stratospheric variability in order to capture the correct surface response will be necessary.

The role of anomalous SST and surface fluxes over the Southeastern North Atlantic in the explosive development of windstorm Xynthia

In late February 2010 the extraordinary windstorm Xynthia crossed over Southwestern and Central Europe and caused severe damage, affecting particularly the Spanish and French Atlantic coasts. The storm was embedded in uncommon large-scale atmospheric and boundary conditions prior to and during its development, namely enhanced sea surface temperatures (SST) within the low-level entrainment zone of air masses, an unusual southerly position of the polar jet stream, and a remarkable split jet structure in the upper troposphere. To analyse the processes that led to the rapid intensification of this exceptional storm originating close to the subtropics (30°N), the sensitivity of the cyclone intensification to latent heat release is determined using the regional climate model COSMO-CLM forced with ERA-Interim data. A control simulation with observed SST shows that moist and warm air masses originating from the subtropical North Atlantic were involved in the cyclogenesis process and led to the formation of a vertical tower with high values of potential vorticity (PV). Sensitivity studies with reduced SST or increased laminar boundary roughness for heat led to reduced surface latent heat fluxes. This induced both a weaker and partly retarded development of the cyclone and a weakening of the PV-tower together with reduced diabatic heating rates, particularly at lower and mid levels. We infer that diabatic processes played a crucial role during the phase of rapid deepening of Xynthia and thus to its intensity over the Southeastern North Atlantic. We suggest that windstorms like Xynthia may occur more frequently under future climate conditions due to the warming SSTs and potentially enhanced latent heat release, thus increasing the windstorm risk for Southwestern Europe.

The dichotomous structure of the warm conveyor belt

The warm conveyor belt (WCB) of an extratropical cyclone generally splits into two branches. One branch (WCB1) turns anticyclonically into the downstream upper-level tropospheric ridge, while the second branch (WCB2) wraps cyclonically around the cyclone centre. Here, the WCB split in a typical North Atlantic cold-season cyclone is analysed using two numerical models: the Met Office Unified Model and the COSMO model. The WCB flow is defined using off-line trajectory analysis. The two models represent the WCB split consistently. The split occurs early in the evolution of the WCB with WCB1 experiencing maximum ascent at lower latitudes and with higher moisture content than WCB2. WCB1 ascends abruptly along the cold front where the resolved ascent rates are greatest and there is also line convection. In contrast, WCB2 remains at lower levels for longer before undergoing saturated large-scale ascent over the system's warm front. The greater moisture in WCB1 inflow results in greater net potential temperature change from latent heat release, which determines the final isentropic level of each branch. WCB1 also exhibits lower outflow potential vorticity values than WCB2. Complementary diagnostics in the two models are utilised to study the influence of individual diabatic processes on the WCB. Total diabatic heating rates along the WCB branches are comparable in the two models with microphysical processes in the large-scale cloud schemes being the major contributor to this heating. However, the different convective parameterisation schemes used by the models cause significantly different contributions to the total heating. These results have implications for studies on the influence of the WCB outflow in Rossby wave evolution and breaking. Key aspects are the net potential temperature change and the isentropic level of the outflow which together will influence the relative mass going into each WCB branch and the associated negative PV anomalies at the tropopause-level flow.

Maintenance of the African easterly jet

The maintenance of the African easterly jet (AEJ) has been examined using a zonally symmetric general circulation model with simple parametrizations. It is shown that the AEJ is maintained in association with two diabatically forced meridional circulations: one associated with surface fluxes and dry convection in the Saharan heat-low region and one associated with deep moist convection in the intertropical convergence zone equatorward of this. the heat-low heating, which reaches the height of the AEJ around 700 mb, is particularly important in maintaining the AEJ and its associated meridional gradients in potential vorticity. It is concluded that the mean observed AEJ results from a combination of the diabatically forced meridional circulations which maintain it and easterly waves which weaken it.