Spatially amplifying modes of the Charney baroclinic-instability problem

1986 ◽  
Vol 170 ◽  
pp. 293-317 ◽  
Author(s):  
R. T. Pierrehumbert
Keyword(s):  
Broad Class ◽  
Baroclinic Instability ◽  
Surface Wind ◽  
Stationary Wave ◽  
Vertical Shear ◽  
Coriolis Parameter ◽  
Spatial Instability ◽  
Wide Range ◽  
Amplification Rate ◽  
Zero Frequency

We determine the circumstances under which baroclinic instability in the Charney model subjected to localized time-periodic forcing manifests itself as a wavetrain that oscillates at the source frequency and amplifies in space with distance from the source; analytical and numerical results describing the salient characteristics of such waves are presented. The spatially amplifying disturbance is a hitherto unsuspected part of the response to a pulsating source, and coexists with the more familiar neutral Rossby wavetrains; it is likely to play a role in a wide range of atmospheric and oceanic phenomena.The central results rely on a careful application of a causality criterion due to Briggs. These results illustrate a practical means of attacking spatial instability problems, which can be applied to a broad class of systems besides the one at hand. We have found that the Charney problem with positive vertical shear is not absolutely unstable, so long as the wind at the ground is non-negative. This implies that spatial instability and forced stationary-wave problems are well posed in an open domain under typical atmospheric circumstances.The amplifying waves appear on the downstream side of the source, have eastward (downstream) phase propagation and have wavelengths that increase monotonically with decreasing frequency, becoming infinite at zero frequency. When the surface wind is not too large, the spatial amplification rate has a single maximum near the frequency ωm= (f/N)Uz, wherefis the Coriolis parameter,Nis the stability frequency andUzis the vertical shear; the rate approaches zero at zero frequency and asymptotes algebraically to zero at large frequency for any positive surface wind. Distinct Charney and Green modes do not appear until the surface wind is made very large. The amplification rate at ωmbecomes infinite as surface wind approaches zero, suggesting a mechanism for the concentration of eddy activity.We also discuss the relationship of these results to the structure of low- and high-frequency atmospheric variability.

scholarly journals Instabilities of continuously stratified zonal equatorial jets in a periodic channel model

2002 ◽  
Vol 20 (5) ◽  
pp. 729-740 ◽  
Author(s):  
S. Masina
Keyword(s):  
Channel Model ◽  
Baroclinic Instability ◽  
Phase Speed ◽  
Maximum Velocity ◽  
Equatorial Region ◽  
Eastern Pacific ◽  
Vertical Shear ◽  
Coriolis Parameter ◽  
Vertical Scale ◽  
Surface Jet

Abstract. Several numerical experiments are performed in a nonlinear, multi-level periodic channel model centered on the equator with different zonally uniform background flows which resemble the South Equatorial Current (SEC). Analysis of the simulations focuses on identifying stability criteria for a continuously stratified fluid near the equator. A 90 m deep frontal layer is required to destabilize a zonally uniform, 10° wide, westward surface jet that is symmetric about the equator and has a maximum velocity of 100 cm/s. In this case, the phase velocity of the excited unstable waves is very similar to the phase speed of the Tropical Instability Waves (TIWs) observed in the eastern Pacific Ocean. The vertical scale of the baroclinic waves corresponds to the frontal layer depth and their phase speed increases as the vertical shear of the jet is doubled. When the westward surface parabolic jet is made asymmetric about the equator, in order to simulate more realistically the structure of the SEC in the eastern Pacific, two kinds of instability are generated. The oscillations that grow north of the equator have a baroclinic nature, while those generated on and very close to the equator have a barotropic nature.  This study shows that the potential for baroclinic instability in the equatorial region can be as large as at mid-latitudes, if the tendency of isotherms to have a smaller slope for a given zonal velocity, when the Coriolis parameter vanishes, is compensated for by the wind effect.Key words. Oceanography: general (equatorial oceanography; numerical modeling) – Oceanography: physics (fronts and jets)

scholarly journals Transient Dynamics of Nonsymmetric Perturbations versus Symmetric Instability in Baroclinic Zonal Shear Flows

2010 ◽  
Vol 67 (9) ◽  
pp. 2972-2989 ◽  
Author(s):  
G. R. Mamatsashvili ◽  
V. S. Avsarkisov ◽  
G. D. Chagelishvili ◽  
R. G. Chanishvili ◽  
M. V. Kalashnik
Keyword(s):  
Shear Flows ◽  
Stable Stratification ◽  
Linear Operators ◽  
Vertical Shear ◽  
Transient Dynamics ◽  
Transient Growth ◽  
Coriolis Parameter ◽  
Linear Dynamics ◽  
Wide Range ◽  
Symmetric Instability

Abstract The linear dynamics of symmetric and nonsymmetric perturbations in unbounded zonal inviscid flows with a constant vertical shear of velocity, when a fluid is incompressible and density is stably stratified along the vertical and meridional directions, is investigated. A small–Richardson number Ri ≲ 1 and large–Rossby number Ro ≳ 1 regime is considered, which satisfies the condition for symmetric instability. Specific features of this dynamics are closely related to the nonnormality of linear operators in shear flows and are well interpreted in the framework of the nonmodal approach by tracing the linear dynamics of spatial Fourier harmonics (Kelvin modes) of perturbations in time. The roles of stable stratification, the Coriolis parameter, and vertical shear in the dynamics of perturbations are analyzed. Classification of perturbations into two types or modes—vortex (i.e., quasigeostrophic balanced motions) and inertia–gravity wave—is made according to the value of potential vorticity. The emerging picture of the (linear) transient dynamics for these two modes at Ri ≲ 1 and Ro ≳ 1 indicates that vortex mode perturbations are able to gain basic flow energy and undergo exponential transient amplification and in this process generate inertia–gravity waves. Transient growth of the vortex mode and, consequently, the effectiveness of the wave generation both increase with decreasing Ri and increasing Ro. This linear coupling of perturbation modes is, in general, specific to shear flows but is not fully appreciated yet. A parallel analysis of the transient dynamics of nonsymmetric perturbations versus symmetric instability is also presented. It is shown that the nonnormality-induced transient growth of nonsymmetric perturbations can prevail over the symmetric instability for a wide range of Ri and Ro. The current analysis suggests that the dynamical activity of fronts and jet streaks at Ri ≲ 1 and Ro ≳ 1 should be determined by nonsymmetric perturbations rather than by symmetric ones, as was accepted in earlier papers. It is noteworthy that the transient growth of perturbations is asymmetric in the wavenumber space—the constant phase plane of maximally amplified perturbations is inclined in a direction northeast to the zonal one and the inclination angle is different for different Ri and Ro.

scholarly journals Continental Shelf Baroclinic Instability. Part II: Oscillating Wind Forcing

2016 ◽  
Vol 46 (2) ◽  
pp. 569-582 ◽  
Author(s):  
K. H. Brink ◽  
H. Seo
Keyword(s):  
Flow Field ◽  
Continental Shelf ◽  
Large Scale ◽  
Baroclinic Instability ◽  
Wind Forcing ◽  
Develop Model ◽  
Bottom Slope ◽  
Coriolis Parameter ◽  
Wide Range ◽  
Eddy Field

AbstractContinental shelf baroclinic instability energized by fluctuating alongshore winds is treated using idealized primitive equation numerical model experiments. A spatially uniform alongshore wind, sinusoidal in time, alternately drives upwelling and downwelling and so creates highly variable, but slowly increasing, available potential energy. For all of the 30 model runs, conducted with a wide range of parameters (varying Coriolis parameter, initial stratification, bottom friction, forcing period, wind strength, and bottom slope), a baroclinic instability and subsequent eddy field develop. Model results and scalings show that the eddy kinetic energy increases with wind amplitude, forcing period, stratification, and bottom slope. The dominant alongshore length scale of the eddy field is essentially an internal Rossby radius of deformation. The resulting depth-averaged alongshore flow field is dominated by the large-scale, periodic wind forcing, while the cross-shelf flow field is dominated by the eddy variability. The result is that correlation length scales for alongshore flow are far greater than those for cross-shelf velocity. This scale discrepancy is qualitatively consistent with midshelf observations by Kundu and Allen, among others.

scholarly journals Baroclinic Stationary Waves in Aquaplanet Models

2011 ◽  
Vol 68 (5) ◽  
pp. 1023-1040 ◽  
Author(s):  
Giuseppe Zappa ◽  
Valerio Lucarini ◽  
Antonio Navarra
Keyword(s):  
Dispersion Relation ◽  
Baroclinic Instability ◽  
Surface Wind ◽  
Low Frequency ◽  
Phase Speed ◽  
Stationary Wave ◽  
Instability Wave ◽  
Stationary Waves ◽  
Frequency Wave ◽  
Inverse Energy Cascade

Abstract An aquaplanet model is used to study the nature of the highly persistent low-frequency waves that have been observed in models forced by zonally symmetric boundary conditions. Using the Hayashi spectral analysis of the extratropical waves, the authors find that a quasi-stationary wave 5 belongs to a wave packet obeying a well-defined dispersion relation with eastward group velocity. The components of the dispersion relation with k ≥ 5 baroclinically convert eddy available potential energy into eddy kinetic energy, whereas those with k < 5 are baroclinically neutral. In agreement with Green’s model of baroclinic instability, wave 5 is weakly unstable, and the inverse energy cascade, which had been previously proposed as a main forcing for this type of wave, only acts as a positive feedback on its predominantly baroclinic energetics. The quasi-stationary wave is reinforced by a phase lock to an analogous pattern in the tropical convection, which provides further amplification to the wave. It is also found that the Pedlosky bounds on the phase speed of unstable waves provide guidance in explaining the latitudinal structure of the energy conversion, which is shown to be more enhanced where the zonal westerly surface wind is weaker. The wave’s energy is then trapped in the waveguide created by the upper tropospheric jet stream. In agreement with Green’s theory, as the equator-to-pole SST difference is reduced, the stationary marginally stable component shifts toward higher wavenumbers, while wave 5 becomes neutral and westward propagating. Some properties of the aquaplanet quasi-stationary waves are found to be in interesting agreement with a low frequency wave observed by Salby during December–February in the Southern Hemisphere so that this perspective on low frequency variability, apart from its value in terms of basic geophysical fluid dynamics, might be of specific interest for studying the earth’s atmosphere.

scholarly journals Planetary–Geometric Constraints on Isopycnal Slope in the Southern Ocean

2015 ◽  
Vol 45 (12) ◽  
pp. 2991-3004 ◽  
Author(s):  
Daniel C. Jones ◽  
Takamitsu Ito ◽  
Thomas Birner ◽  
Andreas Klocker ◽  
David Munday
Keyword(s):  
Southern Ocean ◽  
Large Scale ◽  
Baroclinic Instability ◽  
Surface Wind ◽  
Relative Vorticity ◽  
Geometric Constraints ◽  
Coriolis Parameter ◽  
Sector Model ◽  
Surface Wind Stress ◽  
Flux Parameterization

AbstractOn planetary scales, surface wind stress and differential buoyancy forcing act together to produce isopycnal surfaces that are relatively flat in the tropics/subtropics and steep near the poles, where they tend to outcrop. Tilted isopycnals in a rapidly rotating fluid are subject to baroclinic instability. The turbulent, mesoscale eddies generated by this instability have a tendency to homogenize potential vorticity (PV) along density surfaces. In the Southern Ocean (SO), the tilt of isopycnals is largely maintained by competition between the steepening effect of surface forcing and the flattening effect of turbulent, spatially inhomogeneous eddy fluxes of PV. Here quasigeostrophic theory is used to investigate the influence of a planetary–geometric constraint on the equilibrium slope of tilted density/buoyancy surfaces in the SO. If the meridional gradients of relative vorticity and PV are small relative to β, then quasigeostrophic theory predicts ds/dz = β/f0 = cot(ϕ0)/a, or equivalently r ≡ |∂zs/(β/f0)| = 1, where f is the Coriolis parameter, β is the meridional gradient of f, s is the isopycnal slope, ϕ0 is a reference latitude, a is the planetary radius, and r is the depth-averaged criticality parameter. It is found that the strict r = 1 condition holds over specific averaging volumes in a large-scale climatology. A weaker r = O(1) condition for depth-averaged quantities is generally satisfied away from large bathymetric features. The r = O(1) constraint is employed to derive a depth scale to characterize large-scale interior stratification, and an idealized sector model is used to test the sensitivity of this relationship to surface wind forcing. Finally, the possible implications for eddy flux parameterization and for the sensitivity of SO circulation/stratification to changes in forcing are discussed.

scholarly journals Stability Analysis of the Labrador Current

2014 ◽  
Vol 44 (2) ◽  
pp. 445-463 ◽  
Author(s):  
Sören Thomsen ◽  
Carsten Eden ◽  
Lars Czeschel
Keyword(s):  
Stability Analysis ◽  
Growth Rates ◽  
Baroclinic Instability ◽  
Cross Flow ◽  
Vertical Shear ◽  
Maximal Growth ◽  
Phase Velocities ◽  
Labrador Current ◽  
Lateral Mixing ◽  
Weak Stratification

Abstract Mooring observations and model simulations point to an instability of the Labrador Current (LC) during winter, with enhanced eddy kinetic energy (EKE) at periods between 2 and 5 days and much less EKE during other seasons. Linear stability analysis using vertical shear and stratification from the model reveals three dominant modes of instability in the LC: 1) a balanced interior mode with along-flow wavelengths of about 30–45 km, phase velocities of 0.3 m s−1, maximal growth rates of 1 day−1, and surface-intensified but deep-reaching amplitudes; 2) a balanced shallow mode with along-flow wavelengths of about 0.3–1.5 km, phase velocities of 0.55 m s−1, about 3 times larger growth rates, but amplitudes confined to the mixed layer (ML); and 3) an unbalanced symmetric mode with the largest growth rates, vanishing phase speeds, and along-flow structure, and very small cross-flow wavelengths, also confined to the ML. Both balanced modes are akin to baroclinic instability but operate at moderate-to-small Richardson numbers Ri with much larger growth rates as for the quasigeostrophic limit of Ri ≫ 1. The interior mode is found to be responsible for the instability of the LC during winter. Weak stratification and enhanced vertical shear due to local buoyancy loss and the advection of convective water masses from the interior result in small Ri within the LC and up to 3 times larger growth rates of the interior mode in March compared to summer and fall conditions. Both the shallow and the symmetric modes are not resolved by the model, but it is suggested that they might also play an important role for the instability in the LC and for lateral mixing.

Parameterization of Mixed Layer Eddies. Part I: Theory and Diagnosis

2008 ◽  
Vol 38 (6) ◽  
pp. 1145-1165 ◽  
Author(s):  
Baylor Fox-Kemper ◽  
Raffaele Ferrari ◽  
Robert Hallberg
Keyword(s):  
Mixed Layer ◽  
Mixed Layer Depth ◽  
Baroclinic Instability ◽  
Finite Amplitude ◽  
Climate Impacts ◽  
Surface Mixed Layer ◽  
Rotation Rates ◽  
Wide Range ◽  
Horizontal Density Gradient ◽  
Mixed Layers

Abstract Ageostrophic baroclinic instabilities develop within the surface mixed layer of the ocean at horizontal fronts and efficiently restratify the upper ocean. In this paper a parameterization for the restratification driven by finite-amplitude baroclinic instabilities of the mixed layer is proposed in terms of an overturning streamfunction that tilts isopycnals from the vertical to the horizontal. The streamfunction is proportional to the product of the horizontal density gradient, the mixed layer depth squared, and the inertial period. Hence restratification proceeds faster at strong fronts in deep mixed layers with a weak latitude dependence. In this paper the parameterization is theoretically motivated, confirmed to perform well for a wide range of mixed layer depths, rotation rates, and vertical and horizontal stratifications. It is shown to be superior to alternative extant parameterizations of baroclinic instability for the problem of mixed layer restratification. Two companion papers discuss the numerical implementation and the climate impacts of this parameterization.

scholarly journals A Two-Cool-Season Wind Profiler–Based Analysis of Westward-Directed Gap Flow through the Columbia River Gorge

2019 ◽  
Vol 147 (12) ◽  
pp. 4653-4680 ◽  
Author(s):  
Paul J. Neiman ◽  
Daniel J. Gottas ◽  
Allen B. White
Keyword(s):  
Source Region ◽  
Surface Wind ◽  
Columbia River ◽  
Vertical Shear ◽  
Early Event ◽  
Flow Depth ◽  
Flow Strength ◽  
Freezing Rain ◽  
Gap Flow ◽  
The Mean

Abstract This observational study of westward-directed gap flows through the Columbia River Gorge uses three radar wind profilers during two winter seasons between October 2015 and April 2017, with a focus on the gap-exit region at Troutdale, Oregon. Of the 92 gap-flow events identified at Troutdale, the mean duration was 38.5 h, the mean gap-jet speed was 12 m s−1, and the mean gap-flow depth was 570 m MSL. The mean gap-jet height and gap-flow depth were situated below the top of the inner gorge, while a maximum depth of 1087 m MSL was contained within the gorge’s outer-wall rim. The mean gap-flow depth was deepest in the cold-air source region east of the gorge and decreased westward to the coast. Strong gap-flow events were longer lived, deeper, and capped by stronger vertical shear than their weak counterparts, and strong (weak) events were forced primarily by a cold-interior anticyclone (offshore cyclone). Deep gap-flow events were longer lived, stronger, and had weaker capping vertical shear than shallow events, and represented a combination of gap-flow and synoptic forcing. Composite temporal analysis shows that gap-flow strength (depth) was maximized midevent (early event), freezing rain was most prevalent during the second half of the event, and accumulated precipitation was greatest late-event. Gap-flow events tended to begin (end) during the evening (morning) hours and were most persistent in January. Surface wind gusts and snow occurrences around Portland, Oregon, were associated primarily with the deepest gap flows, whereas freezing rain occurred predominantly during shallow gap flows.

scholarly journals Stochastic cycle selection in active flow networks

2016 ◽  
Vol 113 (29) ◽  
pp. 8200-8205 ◽  
Author(s):  
Francis G. Woodhouse ◽  
Aden Forrow ◽  
Joanna B. Fawcett ◽  
Jörn Dunkel
Keyword(s):  
Broad Class ◽  
Generic Model ◽  
Rate Theory ◽  
Flow Networks ◽  
Complex Flow ◽  
Slime Molds ◽  
Arterial Blood ◽  
Wide Range ◽  
Theoretical And Numerical Analysis ◽  
Active Flow

Active biological flow networks pervade nature and span a wide range of scales, from arterial blood vessels and bronchial mucus transport in humans to bacterial flow through porous media or plasmodial shuttle streaming in slime molds. Despite their ubiquity, little is known about the self-organization principles that govern flow statistics in such nonequilibrium networks. Here we connect concepts from lattice field theory, graph theory, and transition rate theory to understand how topology controls dynamics in a generic model for actively driven flow on a network. Our combined theoretical and numerical analysis identifies symmetry-based rules that make it possible to classify and predict the selection statistics of complex flow cycles from the network topology. The conceptual framework developed here is applicable to a broad class of biological and nonbiological far-from-equilibrium networks, including actively controlled information flows, and establishes a correspondence between active flow networks and generalized ice-type models.

scholarly journals ETHOS – an effective parametrization and classification for structure formation: the non-linear regime at z ≳ 5

2020 ◽  
Vol 498 (3) ◽  
pp. 3403-3419
Author(s):  
Sebastian Bohr ◽  
Jesús Zavala ◽  
Francis-Yan Cyr-Racine ◽  
Mark Vogelsberger ◽  
Torsten Bringmann ◽  
...  
Keyword(s):  
Power Spectrum ◽  
Structure Formation ◽  
Parameter Space ◽  
Broad Class ◽  
High Redshift ◽  
Acoustic Oscillations ◽  
Effective Parameters ◽  
Matter Power Spectrum ◽  
Wide Range ◽  
Non Linear

ABSTRACT We propose two effective parameters that fully characterize galactic-scale structure formation at high redshifts (z ≳ 5) for a variety of dark matter (DM) models that have a primordial cutoff in the matter power spectrum. Our description is within the recently proposed ETHOS framework and includes standard thermal warm DM (WDM) and models with dark acoustic oscillations (DAOs). To define and explore this parameter space, we use high-redshift zoom-in simulations that cover a wide range of non-linear scales from those where DM should behave as CDM (k ∼ 10 h Mpc−1), down to those characterized by the onset of galaxy formation (k ∼ 500 h Mpc−1). We show that the two physically motivated parameters hpeak and kpeak, the amplitude and scale of the first DAO peak, respectively, are sufficient to parametrize the linear matter power spectrum and classify the DM models as belonging to effective non-linear structure formation regions. These are defined by their relative departure from cold DM (kpeak → ∞) and WDM (hpeak = 0) according to the non-linear matter power spectrum and halo mass function. We identify a region where the DAOs still leave a distinct signature from WDM down to z = 5, while a large part of the DAO parameter space is shown to be degenerate with WDM. Our framework can then be used to seamlessly connect a broad class of particle DM models to their structure formation properties at high redshift without the need of additional N-body simulations.

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