Saturation mechanism and generated viscosity in gravito-turbulent accretion disks

2020 ◽  
Vol 640 ◽  
pp. A53
Author(s):  
L. Löhnert ◽  
S. Krätschmer ◽  
A. G. Peeters
Keyword(s):  
Growth Rate ◽  
Numerical Simulations ◽  
Accretion Disks ◽  
Gravitational Instability ◽  
Turbulent Viscosity ◽  
Radial Distance ◽  
Maximum Growth ◽  
Wave Number ◽  
Radiative Cooling ◽  
Mixing Length

Here, we address the turbulent dynamics of the gravitational instability in accretion disks, retaining both radiative cooling and irradiation. Due to radiative cooling, the disk is unstable for all values of the Toomre parameter, and an accurate estimate of the maximum growth rate is derived analytically. A detailed study of the turbulent spectra shows a rapid decay with an azimuthal wave number stronger than ky−3, whereas the spectrum is more broad in the radial direction and shows a scaling in the range kx−3 to kx−2. The radial component of the radial velocity profile consists of a superposition of shocks of different heights, and is similar to that found in Burgers’ turbulence. Assuming saturation occurs through nonlinear wave steepening leading to shock formation, we developed a mixing-length model in which the typical length scale is related to the average radial distance between shocks. Furthermore, since the numerical simulations show that linear drive is necessary in order to sustain turbulence, we used the growth rate of the most unstable mode to estimate the typical timescale. The mixing-length model that was obtained agrees well with numerical simulations. The model gives an analytic expression for the turbulent viscosity as a function of the Toomre parameter and cooling time. It predicts that relevant values of α = 10−3 can be obtained in disks that have a Toomre parameter as high as Q ≈ 10.

Non-linear capillary instability of a liquid jet

1968 ◽  
Vol 33 (1) ◽  
pp. 151-163 ◽  
Author(s):  
Man-Chuen Yuen
Keyword(s):  
Growth Rate ◽  
Maximum Growth ◽  
Wave Number ◽  
Liquid Jet ◽  
Initial Amplitude ◽  
Dimensionless Wave Number ◽  
Capillary Instability ◽  
Linear Effect ◽  
Non Linear ◽  
Linearized Theory

A third-order theory has been developed to study capillary instability of a liquid jet. The result shows that the asymmetrical development of an initially sinusoidal wave is a non-linear effect with generation of higher harmonics as well as feedback into the fundamental. The growth of the surface wave is found to depend explicitly on the dimensionless initial amplitude of the disturbance and the dimensionless wave-number k of the wave. For the same initial disturbance, the wave is found to have a maximum growth rate at k = 0·7 in agreement with the linearized theory. For the same wave-number, the growth is proportional to the initial amplitude of the disturbance. The cut-off wave-number and the fundamental frequency (or growth rate for the unstable case) of the wave for a given k are found to be different from the linearized theory. Furthermore, at the cut-off wave-number, the present theory shows the disturbance experiences a growth which is proportional to t2. The excellent agreement between Donnelly & Glaberson's experiment and Rayleigh's linearized theory is found to be due to their method of measurement.

Longitudinal waves in a perpendicular collisionless plasma shock: I. Cold ions

1970 ◽  
Vol 4 (4) ◽  
pp. 739-751 ◽  
Author(s):  
S. Peter Gary ◽  
J. J. Sanderson
Keyword(s):  
Magnetic Field ◽  
Growth Rate ◽  
Collisionless Plasma ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Wave Number ◽  
Zero Magnetic Field ◽  
Linear Dispersion ◽  
Electrostatic Waves ◽  
Cold Ions

This paper considers electrostatic waves in a Vlasov plasma of unmagnetized ions and magnetized, Maxwellian electrons. The linear dispersion relation is derived for waves in a perpendicular shock such that the most important sources of instability are the E × B and ∇B electron drifts. For the case of cold ions, propagation perpendicular to the applied magnetic field, and the E × B drift alone, a numerical analysis of frequency vs. wave-number is presented. The effects of the ∇B drift are also considered, and it is shown that the maximum growth rate can be larger than the maximum growth rate for the zero magnetic field ion acoustic instabifity under comparable conditions.

scholarly journals Fourth-order nonlinear evolution equations for surface gravity waves in the presence of a thin thermocline

1997 ◽  
Vol 39 (2) ◽  
pp. 214-229 ◽  
Author(s):  
Sudebi Bhattacharyya ◽  
K. P. Das
Keyword(s):  
Growth Rate ◽  
Fourth Order ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Maximum Growth ◽  
Single Equation ◽  
Maximum Growth Rate ◽  
Nonlinear Evolution Equations ◽  
Wave Number ◽  
Coupled Equations

AbstractTwo coupled nonlinear evolution equations correct to fourth order in wave steepness are derived for a three-dimensional wave packet in the presence of a thin thermocline. These two coupled equations are reduced to a single equation on the assumption that the space variation of the amplitudes takes place along a line making an arbitrary fixed angle with the direction of propagation of the wave. This single equation is used to study the stability of a uniform wave train. Expressions for maximum growth rate of instability and wave number at marginal stability are obtained. Some of the results are shown graphically. It is found that a thin thermocline has a stabilizing influence and the maximum growth rate of instability decreases with the increase of thermocline depth.

Three-dimensional instabilities of oscillatory equatorial zonal shear flows

2009 ◽  
Vol 623 ◽  
pp. 59-74 ◽  
Author(s):  
ANDREI NATAROV ◽  
KELVIN J. RICHARDS
Keyword(s):  
Growth Rate ◽  
Three Dimensional ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Oscillatory Shear ◽  
Resonant Excitation ◽  
Wave Number ◽  
Zonal Flows ◽  
Vertical Scale ◽  
Zonal Wave Number

In this paper, we investigate the linear stability of oscillating zonal flows on the equatorial β-plane in the presence of fully three-dimensional disturbances. To exclude inflection point effects, we focus on the simplest case of a linear meridional shear with time-mean and oscillating components. For purely oscillatory background flows we find that in addition to resonant excitation of ‘additive’ type that occurs in the zonally invariant case, resonant excitation of ‘difference’ type is also possible. For flows with an oscillatory shear superimposed on an unstable time-mean shear it is shown that while the oscillatory shear has a stabilizing influence on disturbances with a small zonal wave number k, at higher k the effect of the oscillating shear diminishes and can even be destabilizing. Overall, a small oscillatory shear tends to reduce the fastest growth rate in the system and pushes the dominant k to higher values. Calculation of dominant zonal and vertical modes shows that the zonally asymmetric modes dominate a large portion of the parameter space, especially at high time-mean background shear and low oscillatory shear. As a result, the dominant vertical mode can have a somewhat larger vertical scale than in the zonally invariant case. At intermediate values of the time-mean shear the growth rate is relatively flat with respect to the zonal mode number, with maximum growth rate occurring in bands of high and low k. We have uncovered a rich assortment of vertical and zonal modes which are likely to play a role in the nonlinear evolution of equatorial flows.

Jeans’ Gravitational Instability of a Thermally Conducting, Unbounded, Partially Ionized Plasma

2006 ◽  
Vol 61 (5-6) ◽  
pp. 275-280 ◽  
Author(s):  
Shaista Shaikh ◽  
Aiyub Khan ◽  
Prem Kumar Bhatia
Keyword(s):  
Growth Rate ◽  
Gravitational Instability ◽  
The Other ◽  
Wave Number ◽  
Finite Conductivity ◽  
Jeans Criterion ◽  
Partially Ionized ◽  
Thermally Conducting ◽  
Limiting Case ◽  
Oblique Magnetic Field

The gravitational instability of an infinitely extending homogenous, partially ionized plasma, permeated by an oblique magnetic field, has been studied to investigate the effects of Hall currents, finite conductivity, viscosity, collision with neutrals and thermal conductivity on the growth rate of the disturbance. The dispersion relation obtained has been solved numerically, and it has been found that Hall currents and collision with neutrals have a destabilizing influence on the growth rate while the other mechanisms reinforce the gravitational instability. Jeans’ criterion, derived within a purely hydrodynamic framework, has been rediscussed along a nonextensive kinetic theory. A new Jeans’ criterion was deduced, which depends on the nonextensive parameter q and the standard Jeans’ wave number is recovered in the limiting case q = 1

A note on growth curves of rabbit lines selected on growth rate or litter size

1993 ◽  
Vol 57 (2) ◽  
pp. 332-334 ◽  
Author(s):  
A. Blasco ◽  
E. Gómez
Keyword(s):  
Growth Rate ◽  
Litter Size ◽  
Growth Curves ◽  
Live Weight ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Standard Errors ◽  
Adult Weight ◽  
Weight Growth ◽  
Using Data

Two synthetic lines of rabbits were used in the experiment. Line V, selected on litter size, and line R, selected on growth rate. Ninety-six animals were randomly collected from 48 litters, taking a male and a female each time. Richards and Gompertz growth curves were fitted. Sexual dimorphism appeared in the line V but not in the R. Values for b and k were similar in all curves. Maximum growth rate took place in weeks 7 to 8. A break due to weaning could be observed in weeks 4 to 5. Although there is a remarkable similarity of the values of all the parameters using data from the first 20 weeks only, the higher standard errors on adult weight would make 30 weeks the preferable time to take data for live-weight growth curves.

Reassessment of Maximum Growth Rates for C3 and C4 Crops

1978 ◽  
Vol 14 (1) ◽  
pp. 1-5 ◽  
Author(s):  
J. L. Monteith
Keyword(s):  
Growth Rate ◽  
Growth Rates ◽  
Growing Season ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Crop Growth ◽  
Close Inspection ◽  
Similar Difference ◽  
Photosynthetic Efficiencies

SUMMARYFigures for maximum crop growth rates, reviewed by Gifford (1974), suggest that the productivity of C3 and C4 species is almost indistinguishable. However, close inspection of these figures at source and correspondence with several authors revealed a number of errors. When all unreliable figures were discarded, the maximum growth rate for C3 stands fell in the range 34–39 g m−2 d−1 compared with 50–54 g m−2 d−1 for C4 stands. Maximum growth rates averaged over the whole growing season showed a similar difference: 13 g m−2 d−1 for C3 and 22 g m−2 d−1 for C4. These figures correspond to photosynthetic efficiencies of approximately 1·4 and 2·0%.

scholarly journals Growth of GaN from elemental Gallium and Ammonia via Modified Sandwich Growth Technique

2004 ◽  
Vol 831 ◽  
Author(s):  
E. Berkman ◽  
R. Collazo ◽  
R. Schlesser ◽  
Z. Sitar
Keyword(s):  
Growth Rate ◽  
Gas Flow ◽  
Transport Model ◽  
Theoretical Calculations ◽  
Substrate Surface ◽  
Maximum Growth ◽  
Direct Reaction ◽  
Pure Nitrogen ◽  
Flow Rates ◽  
Reactor Pressure

ABSTRACTGallium nitride (GaN) films were grown on (0001) sapphire substrates at 1050°C by controlled evaporation of gallium (Ga) metal and reaction with ammonia (NH3) at a total reactor pressure of 800 Torr. Pure nitrogen (N2) was flowed directly above the molten Ga source to prevented direct reaction between the molten Ga and ammonia, which causes Ga spattering and GaN crust formation. At the same time, this substantially enhanced the Ga transport to the substrate. A simple mass-transport model based on total reactor pressure, gas flow rates and source temperature was developed and verified. The theoretical calculations and growth rate measurements at different ammonia flow rates and reactor pressures showed that the maximum growth rate was controlled by transport of both Ga species and reactive ammonia to the substrate surface.

Dynamic behavior of a stochastic SIRS model with two viruses

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiandong Zhao ◽  
Tonghua Zhang ◽  
Zhixia Han
Keyword(s):  
Growth Rate ◽  
Asymptotic Stability ◽  
Global Existence ◽  
Numerical Simulations ◽  
Positive Solution ◽  
Sufficient Conditions ◽  
Environmental Noise ◽  
Sirs Model ◽  
The Moment ◽  
Disease Free

AbstractTo study the effect of environmental noise on the spread of the disease, a stochastic Susceptible, Infective, Removed and Susceptible (SIRS) model with two viruses is introduced in this paper. Sufficient conditions for global existence of positive solution and stochastically asymptotic stability of disease-free equilibrium in the model are given. Then, it is shown that the positive solution is stochastically ultimately bounded and the moment average in time of the positive solution is bounded. Our results mean that the environmental noise suppresses the growth rate of the individuals and drives the disease to extinction under certain conditions. Finally, numerical simulations are given to illustrate our main results.

Interpretation of Experimental Data with Regard to the Activated Sludge Model No.1 and Calibration of the Model for Municipal Wastewater Treatment Plants

1992 ◽  
Vol 25 (6) ◽  
pp. 167-183 ◽  
Author(s):  
H. Siegrist ◽  
M. Tschui
Keyword(s):  
Experimental Data ◽  
Growth Rate ◽  
Activated Sludge ◽  
Municipal Wastewater ◽  
Wastewater Treatment Plants ◽  
Treatment Plant ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Municipal Wastewater Treatment ◽  
Activated Sludge Model

The wastewater of the municipal treatment plants Zürich-Werdhölzli (350000 population equivalents), Zürich-Glatt (110000), and Wattwil (20000) have been characterized with regard to the activated sludge model Nr.1 of the IAWPRC task group. Zürich-Glatt and Wattwil are partly nitrifying treatment plants and Zürich-Werdhölzli is fully nitrifying. The mixing characteristics of the aeration tanks at Werdhölzli and Glatt were determined with sodium bromide as a tracer. The experimental data were used to calibrate hydrolysis, heterotrophic growth and nitrification. Problems arising by calibrating hydrolysis of the paniculate material and by measuring oxygen consumption of heterotrophic and nitrifying microorganisms are discussed. For hydrolysis the experimental data indicate first-order kinetics. For nitrification a maximum growth rate of 0.40±0.07 d−1, corresponding to an observed growth rate of 0.26±0.04 d−1 was calculated at 10°C. The half velocity constant found for 12 and 20°C was 2 mg NH4-N/l. The calibrated model was verified with experimental dam of me Zürich-Werdhölzli treatment plant during ammonia shock load.

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