scholarly journals Ekman-inertial instability

2021 ◽  
Author(s):  
Nicolas Grisouard ◽  
Varvara E Zemskova
Keyword(s):  
Growth Rate ◽  
Stokes Problem ◽  
Initial Perturbation ◽  
Initial Response ◽  
Initial Growth ◽  
Mean Flow ◽  
Viscous Effects ◽  
Coriolis Parameter ◽  
Initial Flow ◽  
Inertial Instability

<p>We report on an instability arising in sub-surface, laterally sheared geostrophic flows. When the lateral shear of a horizontal flow in geostrophic balance has a sign opposite to the Coriolis parameter and exceeds it in magnitude, embedded perturbations are subjected to inertial instability, albeit modified by viscosity. When the perturbation arises from the surface of the fluid, the initial response is akin to a Stokes problem, with an initial flow aligned with the initial perturbation. The perturbation then grows quasi-inertially, rotation deflecting the velocity vector, which adopts a well-defined angle with the mean flow, and viscous stresses, transferring horizontal momentum downward. The combination of rotational and viscous effects in the dynamics of inertial instability prompts us to call this process “Ekman-inertial instability.” While the perturbation initially grows super-inertially, the growth rate then becomes sub-inertial, eventually tending back to the inertial value. The same process repeats downward as time progresses. Ekman-inertial transport aligns with the asymptotic orientation of the flow and grows exactly inertially with time once the initial disturbance has passed. Because of the strongly super-inertial initial growth rate, this instability might compete favourably against other instabilities arising in ocean fronts.</p>

The nonlinear evolution of zonally symmetric equatorial inertial instability

2003 ◽  
Vol 474 ◽  
pp. 245-273 ◽  
Author(s):  
STEPHEN D. GRIFFITHS
Keyword(s):  
Long Memory ◽  
Initial Conditions ◽  
Zonal Flow ◽  
Nonlinear Regime ◽  
Mean Flow ◽  
Coriolis Parameter ◽  
Weakly Nonlinear ◽  
Flow Change ◽  
Inertial Instability ◽  
The Mean

The inertial instability of equatorial shear flows is studied, with a view to understanding observed phenomena in the Earth's stratosphere and mesosphere. The basic state is a zonal flow of stratified fluid on an equatorial β-plane, with latitudinal shear. The simplest self-consistent model of the instability is used, so that the basic state and the disturbances are zonally symmetric, and a vertical diffusivity provides the scale selection. We study the interaction between the inertial instability, which takes the form of periodically varying disturbances in the vertical, and the mean flow, where ‘mean’ is a vertical mean.The weakly nonlinear regime is investigated analytically, for flows with an arbitrary dependence on latitude. An amplitude equation of the form dA/dt = A−k2A∫[mid ]A[mid ]2dt is derived for the disturbances, and the evolving stability properties of the mean flow are discussed. In the final steady state, the disturbances vanish, but there is a persistent mean flow change that stabilizes the flow. However, the magnitude of the mean flow change depends strongly on the initial conditions, so that the system has a long memory. The analysis is extended to include the effects of Rayleigh friction and Newtonian cooling, destroying the long-memory property.A more strongly nonlinear regime is investigated with the help of numerical simulations, extending the results up to the point where the instability leads to density contour overturning. The instability is shown to lead to a homogenization of fQ¯ around the initially unstable region, where f is the Coriolis parameter, and Q¯ is the vertical mean of the potential vorticity. As the instability evolves, the line of zero Q¯ moves polewards, rather than equatorwards as might be expected from a simple self-neutralization argument.

Note on weakly nonlinear stability theory of a free mixing layer

1987 ◽  
Vol 409 (1837) ◽  
pp. 351-367 ◽  
Keyword(s):  
Growth Rate ◽  
Nonlinear Evolution Equation ◽  
Linear Growth ◽  
Nonlinear Evolution ◽  
Second Harmonic ◽  
Mixing Layer ◽  
Linear Growth Rate ◽  
Mean Flow ◽  
Initial Flow ◽  
Weakly Nonlinear

This paper is concerned with an analysis of the derivation of a nonlinear evolution equation governing the dynamics of a weakly unstable pertur­bation in a mixing layer. It is demonstrated that the Landau constant has no universal character and is determined by the degree of supercriticality of perturbation, whose measure is represented by the linear growth rate of instability γ (which is proportional to the departure from neutral wavenumber). It is shown that the interaction of the fundamental harmonic of the perturbation with the associated distortion of a mean flow is the dominant nonlinear effect. It is this that makes the main contribution to the Landau constant rather than the interaction with the second harmonic as was thought previously. We examine the régimes of both a viscous and a non-stationary critical layer and calculate the Landau constant. It is shown that, except for the case when the supercriticality is very small, γ « v ( v is the inverse of the Reynolds number), the nonlinearity cannot substantially influence the exponential growth of the perturbation predicted by linear theory. When γ « v , however, the nonlinear time exceeds the time of viscous spreading of the initial flow and for a correct formulation of the problem, one has, as was first pointed out by Huerre, to introduce here an artificial force field.

The limiting form of inertial instability in geophysical flows

2008 ◽  
Vol 605 ◽  
pp. 115-143 ◽  
Author(s):  
STEPHEN D. GRIFFITHS
Keyword(s):  
Growth Rate ◽  
Asymptotic Series ◽  
Normal Modes ◽  
Perturbation Expansion ◽  
Vertical Component ◽  
Phase Speed ◽  
Coriolis Parameter ◽  
Uniform Shear ◽  
Inertial Instability ◽  
Schrödinger Perturbation

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 wavenumber k and vertical wavenumber m. 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 of k and m. 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 of fQ, where f is the Coriolis parameter and Q is the vertical component of the absolute vorticity. The limiting solutions are near-inertial waves or inertial instabilities, according to whether the minimum value of fQ is positive or negative.We focus on the latter case, and investigate how the growth rate and structure of the solutions changes with m and k. 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 when k is 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-zero k, for both a hyperbolic-tangent velocity profile on an f-plane, and a uniform shear flow on an equatorial β-plane.

Tilting at wave beams: a new perspective on the St. Andrew’s Cross

2017 ◽  
Vol 830 ◽  
pp. 660-680 ◽  
Author(s):  
T. Kataoka ◽  
S. J. Ghaemsaidi ◽  
N. Holzenberger ◽  
T. Peacock ◽  
T. R. Akylas
Keyword(s):  
Wave Field ◽  
Finite Length ◽  
Line Source ◽  
Cutoff Frequency ◽  
Three Dimensional ◽  
Resonance Phenomenon ◽  
Buoyancy Frequency ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Viscous Effects

The generation of internal gravity waves by a vertically oscillating cylinder that is tilted to the horizontal in a stratified Boussinesq fluid of constant buoyancy frequency, $N$, is investigated. This variant of the widely studied horizontal configuration – where a cylinder aligned with a plane of constant gravitational potential induces four wave beams that emanate from the cylinder, forming a cross pattern known as the ‘St. Andrew’s Cross’ – brings out certain unique features of radiated internal waves from a line source tilted to the horizontal. Specifically, simple kinematic considerations reveal that for a cylinder inclined by a given angle $\unicode[STIX]{x1D719}$ to the horizontal, there is a cutoff frequency, $N\sin \unicode[STIX]{x1D719}$, below which there is no longer a radiated wave field. Furthermore, three-dimensional effects due to the finite length of the cylinder, which are minor in the horizontal configuration, become a significant factor and eventually dominate the wave field as the cutoff frequency is approached; these results are confirmed by supporting laboratory experiments. The kinematic analysis, moreover, suggests a resonance phenomenon near the cutoff frequency as the group-velocity component perpendicular to the cylinder direction vanishes at cutoff; as a result, energy cannot be easily radiated away from the source, and nonlinear and viscous effects are likely to come into play. This scenario is examined by adapting the model for three-dimensional wave beams developed in Kataoka & Akylas (J. Fluid Mech., vol. 769, 2015, pp. 621–634) to the near-resonant wave field due to a tilted line source of large but finite length. According to this model, the combination of three-dimensional, nonlinear and viscous effects near cutoff triggers transfer of energy, through the action of Reynolds stresses, to a circulating horizontal mean flow. Experimental evidence of such an induced mean flow near cutoff is also presented.

Rotational and Quasiviscous Cold Flow Models for Axisymmetric Hybrid Propellant Chambers

2010 ◽  
Vol 132 (10) ◽  
Author(s):  
Joseph Majdalani ◽  
Michel Akiki
Keyword(s):  
Rocket Engine ◽  
Navier Stokes ◽  
Mathematical Treatment ◽  
Hybrid Rocket ◽  
Mean Flow ◽  
Viscous Effects ◽  
Navier Stokes Equation ◽  
Flow Models ◽  
Hybrid Rocket Engine ◽  
Wall Injection

In this work, we present two simple mean flow solutions that mimic the bulk gas motion inside a full-length, cylindrical hybrid rocket engine. Two distinct methods are used. The first is based on steady, axisymmetric, rotational, and incompressible flow conditions. It leads to an Eulerian solution that observes the normal sidewall mass injection condition while assuming a sinusoidal injection profile at the head end wall. The second approach constitutes a slight improvement over the first in its inclusion of viscous effects. At the outset, a first order viscous approximation is constructed using regular perturbations in the reciprocal of the wall injection Reynolds number. The asymptotic approximation is derived from a general similarity reduced Navier–Stokes equation for a viscous tube with regressing porous walls. It is then compared and shown to agree remarkably well with two existing solutions. The resulting formulations enable us to model the streamtubes observed in conventional hybrid engines in which the parallel motion of gaseous oxidizer is coupled with the cross-streamwise (i.e., sidewall) addition of solid fuel. Furthermore, estimates for pressure, velocity, and vorticity distributions in the simulated engine are provided in closed form. Our idealized hybrid engine is modeled as a porous circular-port chamber with head end injection. The mathematical treatment is based on a standard similarity approach that is tailored to permit sinusoidal injection at the head end.

Abstract 180: The Growth Rate of Early DWI Lesions is Highly Variable and Associated with Penumbral Salvage and Clinical Outcomes Following Endovascular Reperfusion

Stroke ◽  
2013 ◽  
Vol 44 (suppl_1) ◽  
Author(s):  
Hayley M Wheeler ◽  
Michael Mlynash ◽  
Manabu Inoue ◽  
Aaryani Tipirneni ◽  
John Liggins ◽  
...  
Keyword(s):  
Growth Rate ◽  
Acute Stroke ◽  
Clinical Outcomes ◽  
Endovascular Therapy ◽  
Symptom Onset ◽  
Growth Rates ◽  
Initial Growth ◽  
Stroke Patients ◽  
Final Volume ◽  
Favorable Clinical Response

Background: The degree of variability in the rate of early DWI expansion has not been well characterized. We hypothesized that Target Mismatch patients with slowly expanding DWI lesions have more penumbral salvage and better clinical outcomes following endovascular reperfusion than Target Mismatch patients with rapidly expanding DWI lesions. Methods: This substudy of DEFUSE 2 included all patients with a clearly established time of symptom onset. The initial DWI growth rate was determined from the baseline scan by assuming a volume 0 ml just prior to symptom onset. Target Mismatch patients who achieved reperfusion (>50% reduction in PWI after endovascular therapy), were categorized into tertiles according to their initial DWI growth rates. For each tertile, penumbral salvage (comparison of final volume to the volume of PWI (Tmax > 6 sec)/ DWI mismatch prior to endovascular therapy), favorable clinical response, and good functional outcome (see figure for definitions) were calculated. We also compared the growth rate in patients with the Target mismatch vs. Malignant Profile. Results: 64 patients were eligible for this study. Target mismatch patients (n=44) had initial growth rates (range 0 to 43 ml/hr, median of 3 ml/hr) that were significantly less than the growth rates in Malignant profile (n=7) patients (12 to 92 ml/hr, median 39 ml/hr; p < 0.001). In Target mismatch patients who achieved reperfusion (n=30), slower early DWI growth rates were associated with better clinical outcomes (p<0.05) and a trend toward more penumbral salvage (n=27, p=0.137). Conclusions: The growth rate of early DWI lesions in acute stroke patients is highly variable; Malignant profile patients have higher growth rates than other MRI profiles. Among Target Mismatch patients, a slower rate of DWI growth is associated with a greater degree of penumbral salvage and improved clinical outcomes following endovascular reperfusion.

Suppression of Baroclinic Instabilities in Buoyancy-Driven Flow over Sloping Bathymetry

2017 ◽  
Vol 47 (1) ◽  
pp. 49-68 ◽  
Author(s):  
Robert D. Hetland
Keyword(s):  
Growth Rate ◽  
Linear Instability ◽  
Linear Stability Theory ◽  
Buoyancy Frequency ◽  
Bottom Slope ◽  
Coriolis Parameter ◽  
Plume Front ◽  
Instability Theory ◽  
Instability Growth ◽  
Flow Configurations

AbstractBaroclinic instabilities are ubiquitous in many types of geostrophic flow; however, they are seldom observed in river plumes despite strong lateral density gradients within the plume front. Supported by results from a realistic numerical simulation of the Mississippi–Atchafalaya River plume, idealized numerical simulations of buoyancy-driven flow are used to investigate baroclinic instabilities in buoyancy-driven flow over a sloping bottom. The parameter space is defined by the slope Burger number S = Nf−1α, where N is the buoyancy frequency, f is the Coriolis parameter, and α is the bottom slope, and the Richardson number Ri = N2f2M−4, where M2 = |∇Hb| is the magnitude of the lateral buoyancy gradients. Instabilities only form in a subset of the simulations, with the criterion that SH ≡ SRi−1/2 = Uf−1W−1 = M2f−2α 0.2, where U is a horizontal velocity scale and SH is a new parameter named the horizontal slope Burger number. Suppression of instability formation for certain flow conditions contrasts linear stability theory, which predicts that all flow configurations will be subject to instabilities. The instability growth rate estimated in the nonlinear 3D model is proportional to ωImaxS−1/2, where ωImax is the dimensional growth rate predicted by linear instability theory, indicating that bottom slope inhibits instability growth beyond that predicted by linear theory. The constraint SH 0.2 implies a relationship between the inertial radius Li = Uf−1 and the plume width W. Instabilities may not form when 5Li > W; that is, the plume is too narrow for the eddies to fit.

scholarly journals Initial perturbations based on Ensemble Transfrom Kalman Filter with rescaling method for ensemble forecast

2021 ◽  
Author(s):  
Jingzhuo Wang ◽  
Jing Chen ◽  
Hanbin Zhang ◽  
Hua Tian ◽  
Yining Shi
Keyword(s):  
Growth Rate ◽  
Kalman Filter ◽  
Weather Forecasting ◽  
Initial Perturbation ◽  
Ensemble Forecast ◽  
Ensemble Prediction ◽  
Forecast Errors ◽  
Model Uncertainties ◽  
Ensemble Prediction System ◽  
Probabilistic Forecasts

AbstractEnsemble forecast is a method to faithfully describe initial and model uncertainties in a weather forecasting system. Initial uncertainties are much more important than model uncertainties in the short-range numerical prediction. Currently, initial uncertainties are described by Ensemble Transform Kalman Filter (ETKF) initial perturbation method in Global and Regional Assimilation and Prediction Enhanced System-Regional Ensemble Prediction System (GRAPES-REPS). However, an initial perturbation distribution similar to the analysis error cannot be yielded in the ETKF method of the GRAPES-REPS. To improve the method, we introduce a regional rescaling factor into the ETKF method (we call it ETKF_R). We also compare the results between the ETKF and ETKF_R methods and further demonstrate how rescaling can affect the initial perturbation characteristics as well as the ensemble forecast skills. The characteristics of the initial ensemble perturbation improve after applying the ETKF_R method. For example, the initial perturbation structures become more reasonable, the perturbations are better able to explain the forecast errors at short lead times, and the lower kinetic energy spectrum as well as perturbation energy at the initial forecast times can lead to a higher growth rate of themselves. Additionally, the ensemble forecast verification results suggest that the ETKF_R method has a better spread-skill relationship, a faster ensemble spread growth rate and a more reasonable rank histogram distribution than ETKF. Furthermore, the rescaling has only a minor impact on the assessment of the sharpness of probabilistic forecasts. The above results all suggest that ETKF_R can be effectively applied to the operational GRAPES-REPS.

scholarly journals A Spatio-Temporal Investigation of the Growth Rate of COVID-19 Incidents in Ohio, USA, Early in the Pandemic

2021 ◽  
Vol 121 (2) ◽  
pp. 33-47
Author(s):  
Alessandro M. Selvitella ◽  
Liam Carolan ◽  
Justin Smethers ◽  
Christopher Hernandez ◽  
Kathleen L. Foster
Keyword(s):  
Growth Rate ◽  
Transmission Rate ◽  
Time Lag ◽  
Early Period ◽  
Mitigation Strategies ◽  
Initial Growth ◽  
School Closures ◽  
Policy Makers ◽  
Susceptible Population ◽  
Spatio Temporal

Understanding the initial growth rate of an epidemic is important for epidemiologists and policy makers as it can impact their mitigation strategies such as school closures, quarantines, or social distancing. Because the transmission rate depends on the contact rate of the susceptible population with infected individuals, similar growth rates might be experienced in nearby geographical areas. This research determined the growth rate of cases and deaths associated with COVID-19 in the early period of the 2020 pandemic in Ohio, United States. The evolution of cases and deaths was modeled through a Besag-York-Molliè model with linear- and power-type deterministic time dependence. The analysis showed that the growth rate of the time component of the model was subexponential in both cases and deaths once the time-lag across counties of the appearance of the first COVID-19 case was considered. Moreover, deaths in the northeast counties in Ohio were strongly related to the deaths in nearby counties.

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