Symmetric Instability Research Articles

Diffusion‐Controlled, Self‐Organized Growth of Symmetric Wrinkling Patterns

Symmetric instability patterns in the UV-ozone crosslinked surface of a polystyrene film grow radially outward from local defect sites in the presence of solvent vapor. The diffusion kinetics of the solvent governs the pattern morphology by establishing stress fields that favor the orientation of wrinkles in a dendritic-like spoke pattern (left) or in a target pattern consisting of concentric rings (right). Detailed facts of importance to specialist readers are published as ”Supporting Information”. Such documents are peer-reviewed, but not copy-edited or typeset. They are made available as submitted by the authors. Please note: The publisher is not responsible for the content or functionality of any supporting information supplied by the authors. Any queries (other than missing content) should be directed to the corresponding author for the article.

On the equilibration of a symmetrically unstable front via a secondary shear instability

The equilibration of a symmetrically unstable density front is examined using linear stability theory and nonlinear numerical simulations. The initial state, chosen to approximate conditions in the surface ocean, consists of a weakly stratified mixed layer above a strongly stratified thermocline. Each layer has a uniform horizontal density gradient and a velocity field in thermal wind balance. The potential vorticity (PV) in the mixed layer is negative, indicating conditions favourable for symmetric instability. Once the instability reaches finite amplitude, a secondary Kelvin–Helmholtz (K-H) instability forms. Linear theory accurately predicts the time and the wavenumber at which the secondary instability occurs. Following the secondary instability, small-scale turbulence injects positive PV into the mixed layer from the thermocline and from the upper boundary, resulting in a rapid equilibration of the flow as the PV is brought back to zero. While the physical parameters used in this study correspond to typical conditions near a surface ocean front, many of the conclusions apply to symmetric instabilities in the atmosphere.

The influence of solar wind on extratropical cyclones – Part 2: A link mediated by auroral atmospheric gravity waves?

Abstract. Cases of mesoscale cloud bands in extratropical cyclones are observed a few hours after atmospheric gravity waves (AGWs) are launched from the auroral ionosphere. It is suggested that the solar-wind-generated auroral AGWs contribute to processes that release instabilities and initiate slantwise convection thus leading to cloud bands and growth of extratropical cyclones. Also, if the AGWs are ducted to low latitudes, they could influence the development of tropical cyclones. The gravity-wave-induced vertical lift may modulate the slantwise convection by releasing the moist symmetric instability at near-threshold conditions in the warm frontal zone of extratropical cyclones. Latent heat release associated with the mesoscale slantwise convection has been linked to explosive cyclogenesis and severe weather. The circumstantial and statistical evidence of the solar wind influence on extratropical cyclones is further supported by a statistical analysis of high-level clouds (<440 mb) extracted from the International Satellite Cloud Climatology Project (ISCCP) D1 dataset. A statistically significant response of the high-level cloud area index (HCAI) to fast solar wind from coronal holes is found in mid-to-high latitudes during autumn-winter and in low latitudes during spring-summer. In the extratropics, this response of the HCAI to solar wind forcing is consistent with the effect on tropospheric vorticity found by Wilcox et al. (1974) and verified by Prikryl et al. (2009). In the tropics, the observed HCAI response, namely a decrease in HCAI at the arrival of solar wind stream followed by an increase a few days later, is similar to that in the northern and southern mid-to-high latitudes. The amplitude of the response nearly doubles for stream interfaces associated with the interplanetary magnetic field BZ component shifting southward. When the IMF BZ after the stream interface shifts northward, the autumn-winter effect weakens or shifts to lower (mid) latitudes and no statistically significant response is found at low latitudes in spring-summer. The observed effect persists through years of low and high volcanic aerosol loading. The similarity of the response in mid-to-high and low latitudes, the lack of dependence upon aerosol loading, and the enhanced amplitude of the effect when IMF BZ component shifts southward, favor the proposed AGW link over the atmospheric electric circuit (AEC) mechanism (Tinsley et al., 1994). The latter requires the presence of stratospheric aerosols for a significant effect and should produce negative and positive cloud anomalies in mid-to-high and low latitudes, respectively. However, if the requirement of aerosols for the AEC mechanism can be relaxed, the AGW and AEC mechanisms should work in synergy at least in mid-to-high latitudes.

Symmetric Stability of Zonal Flow under Full-Component Coriolis Force -Effect of the Horizontal Component of the Planetary Vorticity-

The effect of the horizontal component fH of the planetary vorticity on the symmetric stability of zonal flow is investigated using the linearized Boussinesq equations on the f -plane. It is shown that, as in the case of neglecting fH, the stability under full-component Coriolis force is determined by the sign of the potential vorticity. It is also revealed that even in such a generalized situation, the movement associated with the symmetric instability can be decomposed into two independent motions, i.e., the buoyancy oscillation (or instability) modified by the Coriolis force and the inertial oscillation (or instability) modified by the buoyancy. The squared product of their frequencies remains proportional to the potential vorticity of the zonal flow. Meanwhile, the horizontal component of the planetary vorticity is found to exhibit both stabilizing and destabilizing effects, although there is a wide range of stable regions that are not affected by fH. The existence of fH also causes an asymmetry such that the stability changes depending on the sign of the vertical shear of the zonal flow, even if the Richardson number and the dimensionless relative vorticity are maintained constant.

Degenerate Modes of Gravitational, Barotropic and Baroclinic Instabilities

In normal mode analysis, the unstable mode grows as an exponential function of time, and the stable mode oscillates as a trigonometric function of time. At a critical point of the relevant parameter space, the mode becomes marginally unstable. The marginally unstable mode is a degenerate mode of growing and decaying modes, as shown by Xu (2007) in the symmetric instability. In this note, we examine marginally unstable modes of gravitational, barotropic and baroclinic instabilities. As in the case of symmetric instability, the marginally unstable modes grow as a linear function of time. In the case of gravitational instability, the linear growth of the marginally unstable mode is rather trivial. On the other hand, in the cases of barotropic and baroclinic instabilities, the linear growth is not trivial, and can be qualitatively explained in terms of the PV thinking.

Buoyancy-Forced Downwelling in Boundary Currents

Abstract The issue of downwelling resulting from surface buoyancy loss in boundary currents is addressed using a high-resolution, nonhydrostatic numerical model. It is shown that the net downwelling is determined by the change in the mixed layer density along the boundary. For configurations in which the density on the boundary increases in the direction of Kelvin wave propagation, there is a net downwelling within the domain. For cases in which the density decreases in the direction of Kelvin wave propagation, cooling results in a net upwelling within the domain. Symmetric instability within the mixed layer drives an overturning cell in the interior, but it does not contribute to the net vertical motion. The net downwelling is determined by the geostrophic flow toward the boundary and is carried downward in a very narrow boundary layer of width E1/3, where E is the Ekman number. For the calculations here, this boundary layer is O(100 m) wide. A simple model of the mixed layer temperature that balances horizontal advection with surface cooling is used to predict the net downwelling and its dependence on external parameters. This model shows that the net sinking rate within the domain depends not only on the amount of heat loss at the surface but also on the Coriolis parameter, the mixed layer depth (or underlying stratification), and the horizontal velocity. These results indicate that if one is to correctly represent the buoyancy-forced downwelling in general circulation models, then it is crucial to accurately represent the velocity and mixed layer depth very close to the boundary. These results also imply that processes that lead to weak mixing within a few kilometers of the boundary, such as ice formation or freshwater runoff, can severely limit the downwelling forced by surface cooling, even if there is strong heat loss and convection farther offshore.

Decoupling of Balanced and Unbalanced Motions and Inertia–Gravity Wave Emission: Small versus Large Rossby Numbers

Abstract This paper provides a brief review of recent results on decoupling of fast [inertia–gravity wave (IGW)] and slow (vortex) motions at small Rossby numbers obtained in the framework of the geostrophic adjustment of localized perturbations. Special attention is paid to the IGW emission and its interpretation in the context of “spontaneous imbalance.” Several mechanisms that lead to spontaneous IGW emission and, thus, to violations of fast–slow splitting at large Rossby numbers are reviewed: Lighthill radiation, symmetric/inertial instability, and ageostrophic shear (Rossby–Kelvin) instability. New results on the saturation of symmetric instability and on the existence of Rossby–Kelvin instability in continuously stratified fluid are presented.

The Slope of Moist Symmetric Instability with Water Loading

Abstract Idealized numerical experiments, supported by analytic considerations, are performed to determine the preferred direction of symmetric instability when water loading is considered. It is concluded that the most unstable direction is tangent to a surface of neutral buoyancy, which can be defined numerically from the water content of lifted air, and coincides with the tangent to saturated isentropes only when all condensed water is precipitated out, consistent with the thermodynamic approximations made in the definition of equivalent potential temperature. In more common situations, when part or all of the condensed water is retained in the cloud, the orientation of symmetrically unstable normal modes is much more slanted toward the horizontal, to the point that regions of the atmosphere, diagnosed as unstable from the consideration of equivalent potential temperature and vorticity, can in fact be stable.

Non‐normal growth in symmetric shear flow

AbstractAn analysis of symmetric instability from the perspective of Generalized Stability Theory is presented. For Richardson number smaller than one, the optimal growth exceeds that predicted by normal mode analysis yielding potentially a much faster generation of slantwise convection. In both normal and non‐normal evolution, the parcel trajectory remains close to the isentropes and energy growth results primarily from the vertical Reynolds stress term in the energy equation. The large non‐normal growth obtained results from the optimal perturbations having parcel trajectories in the mean shear plane, in the initial stage, that maximize the growth by Reynolds stress. This plane is perpendicular to the plane of the isentropes and the absolute momentum isolines, usually associated with slantwise convection. For Richardson number larger than unity, transient growth results primarily from the meridional heat flux term in the energy equation, however this growth is relatively small. Copyright © 2008 Royal Meteorological Society

Numerical study of two-dimensional moist symmetric instability

Abstract. The 2-D version of the non-hydrostatic fully compressible model MOLOCH developed at ISAC-CNR was used in idealized set-up to study the start-up and finite amplitude evolution of symmetric instability. The unstable basic state was designed by numerical integration of the equation which defines saturated equivalent potential vorticity qe*. We present the structure and growth rates of the linear modes both for a supersaturated initial state ("super"-linear mode) and for a saturated one ("pseudo"-linear mode) and the modifications induced on the base state by their finite amplitude evolution.

The limiting form of inertial instability in geophysical flows

The instability of a rotating, stratified flow with arbitrary horizontal cross-stream shear is studied, in the context of linear normal modes with along-stream wavenumberkand vertical wavenumberm. A class of solutions are developed which are highly localized in the horizontal cross-stream direction around a particular streamline. A Rayleigh–Schrödinger perturbation analysis is performed, yielding asymptotic series for the frequency and structure of these solutions in terms ofkandm. The accuracy of the approximation improves as the vertical wavenumber increases, and typically also as the along-stream wavenumber decreases. This is shown to correspond to a near-inertial limit, in which the solutions are localized around the global minimum offQ, wherefis the Coriolis parameter andQis the vertical component of the absolute vorticity. The limiting solutions are near-inertial waves or inertial instabilities, according to whether the minimum value offQis positive or negative.We focus on the latter case, and investigate how the growth rate and structure of the solutions changes withmandk. Moving away from the inertial limit, we show that the growth rate always decreases, as the inertial balance is broken by a stabilizing cross-stream pressure gradient. We argue that these solutions should be described as non-symmetric inertial instabilities, even though their spatial structure is quite different to that of the symmetric inertial instabilities obtained whenkis equal to zero.We use the analytical results to predict the growth rates and phase speeds for the inertial instability of some simple shear flows. By comparing with results obtained numerically, it is shown that accurate predictions are obtained by using the first two or three terms of the perturbation expansion, even for relatively small values of the vertical wavenumber. Limiting expressions for the growth rate and phase speed are given explicitly for non-zerok, for both a hyperbolic-tangent velocity profile on anf-plane, and a uniform shear flow on an equatorial β-plane.

Investigation of stability characteristics of cold‐season convective precipitation events by utilizing the growth rate parameter

The seldom utilized growth rate parameter (σ2), which predicts how rapidly a small‐amplitude disturbance will grow in a conditional symmetrically unstable environment, was applied to study the stability characteristics of convective precipitation case studies across the central United States during the winter seasons of 2003–2004 and 2004–2005. The goals were to improve our understanding of how the environment becomes destabilized over a relatively short period of time, as well as to determine approximately where and when elevated thunderstorms are likely to develop. The comprehensive evaluation comprised a case study example and summary of statistics obtained by tabulations at the initiation site and spatial compositing of all case studies identified. The doubling time for the convection (the time required for a convective element to achieve twice its current depth) was found to be on the order of 1.3 h, which is consistent with the typical timescale for moist slantwise convection resulting from the release of conditional symmetric instability. The development of cold‐season precipitation with lightning (i.e., thundersnow) and any associated banding was correctly and most accurately predicted from trends in plots of σ2 analyzed at the level at which the highest significant growth rates occurred. While this naturally varied from one event to the next, the average elevation tended to be close to 650 hPa. Furthermore, a term‐by‐term diagnosis of the mathematical expression for the growth rate was determined to be quite useful as another means of identifying the type of instability released within instances of wintertime convection. By calculating the individual contributions to the growth rate and observing whether a positive or negative response was obtained, the nature of the stability regime present was also ascertained. The inclusion of a set of non‐thundering snowstorms helped to substantiate the assumption that atmospheres are less stable and more susceptible to vertical motions during those occasions where lightning develops in wintertime scenarios. Thus an outlook for elevated, cold‐season thunderstorms can be more accurately issued by identifying regions where reduced values of equivalent potential vorticity (i.e., small symmetric stability or instability) are collocated with estimates of high σ2 (i.e., where small‐scale slantwise perturbations will grow). Given the overall success, it is hoped that some of the conclusions established by this work will be implemented routinely in an operational environment and provide forecasters an additional, essential tool in dealing with nowcasting situations of hazardous winter weather events.

The impact of deformation strain on the formation of banded clouds in idealized modelling experiments

AbstractExperiments are performed using an idealized version of an operational forecast model to determine the impact on banded frontal clouds of the strength of deformational forcing, low‐level baroclinicity, and model representation of convection. Line convection is initiated along the front, and slantwise bands extend from the top of the line‐convection elements into the cold air. This banding is attributed primarily to ΔM adjustment. The cross‐frontal spreading of the cold pool generated by the line convection leads to further triggering of upright convection in the cold air that feeds into these slantwise bands. Secondary low‐level bands form later in the simulations; these are attributed to the release of conditional symmetric instability. Enhanced deformation strain leads to earlier onset of convection and more coherent line convection. A stronger cold pool is generated, but its speed is reduced relative to that seen in experiments with weaker deformational strain, because of inhibition by the strain field. Enhanced low‐level baroclinicity leads to the generation of more inertial instability by line convection (for a given capping height of convection), and consequently greater strength of the slantwise circulations formed by ΔM adjustment. These conclusions are based on experiments without a convective‐parametrization scheme. Experiments using the standard or a modified scheme for this model demonstrate known problems with the use of this scheme at the awkward 4 km grid length used in these simulations. Copyright © 2008 Royal Meteorological Society

Combustion Instability and Emission Control by Pulsating Fuel Injection

Open-loop control methodologies were used to suppress symmetric and helical thermoacoustic instabilities in an experimental low-emission swirl-stabilized gas-turbine combustor. The controllers were based on fuel (or equivalence ratio) modulations in the main premixed combustion (premixed fuel injection (PMI)) or, alternatively, in the secondary pilot fuel. PMI included symmetric and asymmetric fuel injection. The symmetric instability mode responded to symmetric excitation only when the two frequencies matched. The helical fuel injection affected the symmetric mode only at frequencies that were much higher than that of the instability mode. The asymmetric excitation required more power to obtain the same amount of reduction as that required by symmetric excitation. Unlike the symmetric excitation, which destabilized the combustion when the modulation amplitude was excessive, the asymmetric excitation yielded additional suppression as the modulation level increased. The NOx emissions were reduced to a greater extent by the asymmetric modulation. The second part of the investigation dealt with the control of low frequency symmetric instability and high frequency helical instability by the secondary fuel injection in a pilot flame. Adding a continuous flow of fuel into the pilot flame controlled both instabilities. However, modulating the fuel injection significantly decreased the amount of necessary fuel. The reduced secondary fuel resulted in a reduced heat generation by the pilot diffusion flame and therefore yielded lower NOx emissions. The secondary fuel pulsation frequency was chosen to match the time scales typical to the central flow recirculation zone, which stabilizes the flame in the burner. Suppression of the symmetric mode pressure oscillations by up to 20dB was recorded. High frequency instabilities were suppressed by 38dB, and CO emissions reduced by using low frequency modulations with 10% duty cycle.

Degenerate Modes of Symmetric Instability

A basic ow, which is symmetric in one of the horizontal direction (say, y direction), is unstable for y-symmetric disturbances when the basic potential vorticity is negative (so called symmetric instability). The temporal dependence of the growing modes is exponential, that is, eσt. In the case of vanishing growth rate, that is, σ → 0, there emerge degenerate modes as shown by Xu (2007). Under the thermal wind-balanced initial condition, Xu (2007) showed that the temporal dependence of the degenerate modes is linear, that is, they grow as a linear function of time. In this note, we show that in general the degener ate modes grow quadratically in time, and that in particular they are reduced to the result of Xu (2007), (i.e., linear growth in time) under the thermal-wind-balanced initial conditions.

The major research advances of mesoscale weather dynamics in China since 2003

This paper reviews the main theoretical progress of mesoscale weather dynamics since 2003, including: (1) The dynamic mechanisms of balanced and unbalanced flow are applied to study the genesis and development problems of mesoscale circulation. The symmetric instability and transverse-wave instability are analyzed in line and vortex atmosphere convection, and further research has been done on nonlinear convective symmetric instability. The interaction between forced convection and unstable convection and the wave characteristics of mesoscale motion are also discussed. (2) Intermediate atmosphere dynamic boundary layer models are developed. The complicated nonlinear interaction is analyzed theoretically between the atmospheric boundary layer and the free atmosphere. The structure of the topography boundary layer, atmospheric frontogenesis, the structure and circulation of the low-level front and other boundary layer dynamic problems are discussed. (3) The formation and development of meso-β-scale rainstorms under the background of the East-Asia atmosphere circulation are diagnosed with the variation of MPV (moist potential vorticity) anomalies. And some physical vectors are modified and applied in the moist atmosphere.

A Northern Hemispheric climatology of indices for clear air turbulence in the tropopause region derived from ERA40 reanalysis data

Clear air turbulence (CAT) has relevance for aviation but also for cross‐tropopause transport of chemical constituents. This study presents a 44‐year climatology (from 1958 to 2001) of indicators for CAT for the Northern Hemisphere tropopause on the basis of reanalysis data (ERA40) from the European Centre for Medium‐Range Weather Forecasts. Small Richardson numbers (Ri), negative squared Brunt‐Väisälä (N2) and negative potential vorticity (PV) values are used as indicators for Kelvin‐Helmholtz, hydrostatic, and symmetric instability, respectively. Additionally, an empirical indicator (TI) is applied. All indicators have winter frequency maxima for CAT over the North American east and west coasts. Other local maxima are found over the western part of the North Atlantic and North Pacific, the Himalayas, central Europe, and eastern China. In summer, frequencies are smaller, except for N2, and patterns are shifted poleward. Frequencies of TI and PV depend strongly on jet position, whereas Ri has moderate dependence. N2 frequencies are largest over land and are not markedly influenced by jets. The frequency maxima relative to the jets differ with a northern maximum for TI, a southern maximum for PV, and one for Ri that is essentially along the jet axis. For the 44‐year period pronounced nonlinear trends are identified with an increase of 40–90% over the North Atlantic, United States, and European sector. For the two phases of the North Atlantic Oscillation and the Pacific/North American flow pattern the interannual variability of CAT, indicated by TI and PV, is significant (in contrast to Ri and especially N2).

Vertical Motions in Precipitation Bands in Three Winter Cyclones

Abstract The University of Alabama in Huntsville Mobile Integrated Profiling System 915-MHz profiler was deployed in January and February of 2004 to measure vertical air velocities in finescale precipitation bands in winter cyclones. The profiler was placed to sample the “wraparound” quadrant of three winter cyclones in the central and southern United States, and it obtained high-resolution measurements of the vertical structure of a series of bands in each storm. The data revealed bands that were up to 6 km deep, 10–50 km wide, and spaced about 5–20 km apart. Measurements of vertical air motion w within these bands were retrieved from the Doppler spectra using the lower-bound method, adapted to account for the effects of spectral broadening caused by the horizontal wind, wind shear, and turbulence. Derived vertical air motions ranged from −4.3 to 6.7 m s−1, with an uncertainty of about ±0.6 m s−1. Approximately 29% of the 1515 total derived values were negative, 35% exceeded 1 m s−1, and 9% exceeded 2.0 m s−1. These values are consistent with studies in the Pacific Northwest, except that more extreme values were observed in one band than have been previously reported. There was a high correlation between values of signal-to-noise ratio (SNR) and w within each band (0.60 ≤ r ≤ 0.85), in the composite of bands from each cyclone (0.59 ≤ r ≤ 0.79), and in the overall analysis (r = 0.68). The strongest updrafts were typically between 2.0 and 4.0 m s−1 and were located near the center of each band in regions of high SNR. Regions of downdrafts within the bands had maximum values between −1.0 and −4.3 m s−1 and were typically located along the edges of the bands in regions of low SNR. These results are consistent with snow growth and sublimation processes. The magnitudes of the vertical velocities in the core of the bands were comparable to theoretical predictions for moist symmetric instability (MSI) under inviscid conditions but would appear to be somewhat larger than expected for MSI when turbulent mixing is considered, suggesting that other instabilities, such as potential instability, may have contributed to the band development in these storms.

Banded Convection Caused by Frontogenesis in a Conditionally, Symmetrically, and Inertially Unstable Environment

Abstract Several east–west-oriented bands of clouds and light rain formed on 20 July 2005 over eastern Montana and the Dakotas. The cloud bands were spaced about 150 km apart, and the most intense band was about 20 km wide and 300 km long, featuring areas of maximum radar reflectivity factor of about 50 dBZ. The cloud bands formed poleward of an area of lower-tropospheric frontogenesis, where air of modest convective available potential energy was being lifted. During initiation and maintenance of the bands, mesoscale regions of dry symmetric and inertial instability were present in the region of the bands, suggesting a possible mechanism for the banding. Interpretation of the extant instabilities in the region of the bands was sensitive to the methodology to assess the instability. The release of these instabilities produced circulations with enough vertical motion to lift parcels to their lifting condensation level, resulting in the observed cloud bands. A high-resolution, numerical weather prediction model demonstrated that forecasting these types of events in such real-time models is possible, although the timing, evolution, and spacing of the bands were not faithfully reproduced. This case is compared to two previous cases in the literature where banded convection was associated with a combination of conditional, symmetric, and inertial instability.

Instability of bounded gas‐particle fluidized beds

AbstractSteady flows of gas and particles in unbounded‐fluidized beds have been shown to lose stability to localized voidage disturbances. An insight is provided into the linear stability of bounded gas‐particle flows using a kinetic theory approach. The stability analysis is carried out for two illustrative sets corresponding to fluidized systems having walls acting as sinks and sources of fluctuation energy. Instabilities are characterized by computing leading eigenvalue profiles and dominant eigenfunction contour maps. Results show two types of dominant instability patterns (symmetric and antisymmetric) for flow in a vertical duct, both of which propagate through the bed in the form of traveling waves at speeds comparable to that of the solid phase. Moreover, model predictions show that increasing solids fraction and decreasing particle inelasticity will suppress the wavy disturbance. The physical mechanism of instability formation is further investigated using a term‐by‐term method of analysis. Results show that instability modes can be suppressed if solid phase inertia or inelastic particle collisions are eliminated from the momentum and pseudo‐thermal energy balance equations. Finally, the symmetric instabilities were shown to develop into bubble‐like structures while the antisymmetric instabilities develop into streamer‐like structures. © 2007 American Institute of Chemical Engineers AIChE J, 2007