Barotropic instability of a zonal jet on the sphere: From non-divergence through quasi-geostrophy to shallow water

2020 ◽  
Author(s):  
Ofer Shamir ◽  
Nathan Paldor ◽  
Chaim Garfinkel
Keyword(s):  
Shallow Water ◽  
Growth Rates ◽  
Numerical Solutions ◽  
Barotropic Instability ◽  
Geostrophic Balance ◽  
Deep Layers ◽  
The Mean ◽  
Barotropic Vorticity ◽  
Theoretical Considerations ◽  
Crucial Parameter

<p>Two common approximations to the full Shallow Water Equations (SWEs) are non-divergence and quasi-geostrophy, and the degree to which these approximations lead to biases in numerical solutions are explored using the testbed of barotropic instability. Specifically, we examine the linear stability of strong polar and equatorial jets and compare the growth rates obtained from the SWEs along with those obtained from the Non-Divergent barotropic vorticity (ND) equation and the Quasi-Geostrophic (QG) equation. The main result of this paper is that the depth over which a layer is barotropically unstable is a crucial parameter in controlling the growth rate of small-amplitude perturbations and this dependence is completely lost in the ND equation and is overly weak in the QG system. Only for depths of 30 km or more are the growth rates predicted by the ND and QG systems a good approximation to those of the SWEs, and such a convergence for deep layers can be explained using theoretical considerations. However, for smaller depths, the growth rates predicted by the SWEs become smaller than those of the ND and QG systems and for depths of between 5 and 10 km they can be smaller by more than 50%. For polar jets, and for depths below 2 km the mean height in geostrophic balance with the strong zonal jet becomes negative and hence the barotropic instability problem is ill-defined. While in the SWEs an equatorial jet becomes stable for layer depths smaller than ~3-4 km, in the QG and ND approximation it is unstable for layer depths down to 1 km. These results may have implications for the importance of barotropic instability in Earth's upper stratosphere and perhaps also other planets such as Venus.</p>

Growth of fingers at an unstable diffusing interface in a porous medium or Hele-Shaw cell

1969 ◽  
Vol 39 (3) ◽  
pp. 477-495 ◽  
Author(s):  
R. A. Wooding
Keyword(s):  
Porous Medium ◽  
Numerical Solutions ◽  
Hyperbolic Equations ◽  
Saturated Porous Medium ◽  
Mean Amplitude ◽  
Wave Number ◽  
Two Fluids ◽  
Averaged Equations ◽  
The Mean ◽  
Hele Shaw Cell

Waves at an unstable horizontal interface between two fluids moving vertically through a saturated porous medium are observed to grow rapidly to become fingers (i.e. the amplitude greatly exceeds the wavelength). For a diffusing interface, in experiments using a Hele-Shaw cell, the mean amplitude taken over many fingers grows approximately as (time)2, followed by a transition to a growth proportional to time. Correspondingly, the mean wave-number decreases approximately as (time)−½. Because of the rapid increase in amplitude, longitudinal dispersion ultimately becomes negligible relative to wave growth. To represent the observed quantities at large time, the transport equation is suitably weighted and averaged over the horizontal plane. Hyperbolic equations result, and the ascending and descending zones containing the fronts of the fingers are replaced by discontinuities. These averaged equations form an unclosed set, but closure is achieved by assuming a law for the mean wave-number based on similarity. It is found that the mean amplitude is fairly insensitive to changes in wave-number. Numerical solutions of the averaged equations give more detailed information about the growth behaviour, in excellent agreement with the similarity results and with the Hele-Shaw experiments.

Highly Loaded Low-Pressure Turbine: Design, Numerical, and Experimental Analysis

2010 ◽  
Author(s):  
J. T. Schmitz ◽  
S. C. Morris ◽  
R. Ma ◽  
T. C. Corke ◽  
J. P. Clark ◽  
...  
Keyword(s):  
High Speed ◽  
Numerical Solutions ◽  
Pressure Ratio ◽  
Low Pressure ◽  
Boundary Layer Transition ◽  
Layer Transition ◽  
Free Stream Turbulence ◽  
Low Pressure Turbine ◽  
The Mean ◽  
Highly Loaded

The performance and detailed flow physics of a highly loaded, transonic, low-pressure turbine stage has been investigated numerically and experimentally. The mean rotor Zweifel coefficient was 1.35, with dh/U2 = 2.8, and a total pressure ratio of 1.75. The aerodynamic design was based on recent developments in boundary layer transition modeling. Steady and unsteady numerical solutions were used to design the blade geometry as well as to predict the design and off-design performance. Measurements were acquired in a recently developed, high-speed, rotating turbine facility. The nozzle-vane only and full stage characteristics were measured with varied mass flow, Reynolds number, and free-stream turbulence. The efficiency calculated from torque at the design speed and pressure ratio of the turbine was found to be 90.6%. This compared favorably to the mean line target value of 90.5%. This paper will describe the measurements and numerical solutions in detail for both design and off-design conditions.

Study of the Mechanisms of Vortex Variability in the Lofoten Basin Based on the Energy Analysis

2021 ◽  
Vol 37 (3) ◽  
Author(s):  
V. S. Travkin ◽  
◽  
T. V. Belonenko ◽  
Keyword(s):  
Kinetic Energy ◽  
Potential Energy ◽  
Baroclinic Instability ◽  
Positive Trend ◽  
Barotropic Instability ◽  
Available Potential Energy ◽  
Conversion Rates ◽  
Order Of Magnitude ◽  
The Mean ◽  
Mean Kinetic Energy

Purpose. The Lofoten Basin is one of the most energetic zones of the World Ocean characterized by high activity of mesoscale eddies. The study is aimed at analyzing different components of general energy in the basin, namely the mean kinetic and vortex kinetic energy calculated using the integral of the volume of available potential and kinetic energy of the Lofoten Vortex, as well as variability of these characteristics. Methods and Results. GLORYS12V1 reanalysis data for the period 2010–2018 were used. The mean kinetic energy and the eddy kinetic one were analyzed; and as for the Lofoten Vortex, its volume available potential and kinetic energy were studied. The mesoscale activity of eddies in winter is higher than in summer. Evolution of the available potential energy and kinetic energy of the Lofoten Vortex up to the 1000 m horizon was studied. It is shown that the vortex available potential energy exceeds the kinetic one by an order of magnitude, and there is a positive trend with the coefficient 0,23⋅1015 J/year. It was found that in the Lofoten Basin, the intermediate layer from 600 to 900 m made the largest contribution to the potential energy, whereas the 0–400 m layer – to kinetic energy. The conversion rates of the mean kinetic energy into the vortex kinetic one and the mean available potential energy into the vortex available potential one (barotropic and baroclinic instability) were analyzed. It is shown that the first type of transformation dominates in summer, while the second one is characterized by its increase in winter. Conclusions. The vertical profile shows that the kinetic energy of eddies in winter is higher than in summer. The available potential energy of a vortex is by an order of magnitude greater than the kinetic energy. An increase in the available potential energy is confirmed by a significant positive trend and by a decrease in the vortex Burger number. The graphs of the barotropic instability conversion rate demonstrate the multidirectional flows in the vortex zone with the dipole structure observed in a winter period, and the tripole one – in summer. The barotropic instability highest intensity is observed in summer. The baroclinic instability is characterized by intensification of the regime in winter that is associated with weakening of stratification in this period owing to winter convection.

scholarly journals Ultrasound Surveillance of Ectatic Abdominal Aortas

2008 ◽  
Vol 90 (6) ◽  
pp. 477-482 ◽  
Author(s):  
S Devaraj ◽  
SR Dodds
Keyword(s):  
Growth Rate ◽  
Growth Rates ◽  
Screening Programme ◽  
T Test ◽  
Aortic Aneurysms ◽  
Aortic Diameter ◽  
Ultrasound Screening ◽  
The Mean ◽  
Clinically Significant

INTRODUCTION Some studies have considered abdominal aortas of 2.6–2.9 cm diameter (ectatic aortas) at age 65 years as being abnormal and have recommended surveillance, whereas others have considered these normal and surveillance unnecessary. It is, therefore, not clear how to manage patients with an initial aortic diameter between 2.6–2.9 cm detected at screening. The aim of this study was to evaluate growth rates of ectatic aortas detected on initial ultrasound screening to determine if any developed into clinically significant abdominal aortic aneurysms (AAAs; > 5.0 cm) and clarify the appropriate surveillance intervals for these patients. PATIENTS AND METHODS Data were obtained from a prospective AAA screening programme which commenced in 1992. The group of patients with initial aortic diameters of 2.6–2.9 cm with a minimum of 1-year follow-up were included in this study (Group 2). This was further divided into two subgroups (Groups 3a and 3b) based on a minimum follow-up interval obtained from outcome analysis. Mean growth rate was calculated as change in aortic diameter with time. The comparison of growth rates in Groups 3a and 3b was performed using the t-test. The number and proportion of AAAs that expanded to ≥ 3.0 cm and ≥ 5.0 cm in diameter were also calculated. RESULTS Out of 999 patients with AAA ≥ 2.6 cm with minimum 1-year follow-up, 358 (36%) were classified as ectatic aortas (2.6–2.9 cm) at initial ultrasound screening with the mean growth rate of 1.69 mm/year (95% CI, 1.56–1.82 mm/year) with a mean follow-up of 5.4 years. Of these 358 ectatic aortas, 314 (88%) expanded into ≥ 3.0 cm, 45 (13%) expanded to ≥ 5.0 cm and only 8 (2%) expanded to ≥ 5.5 cm over a mean follow-up of 5.4 years (range, 1–14 years). No ectatic aortas expanded to ≥ 5.0 cm within the first 4 years of surveillance. Therefore, the minimum follow-up interval was set at 4 years and this threshold was then used for further analysis. The mean growth rate in Group 3a (< 5.0 cm at last scan) was 1.33 mm/year (95% CI, 1.23–1.44 mm/year) with a mean follow-up of 7 years compared to Group 3b (≥ 5.0 cm at last scan) with the mean growth rate of 3.33 mm/year (95% CI 3.05–3.61 mm/year) and a mean follow-up of 8 years. The comparison of mean growth rates between Groups 3a and 3b is statistically significant (t-test; T = 13.00; P < 0.001). CONCLUSIONS One-third of patients undergoing AAA screening will have ectatic aortas (2.6–2.9 cm) and at least 13% of these will expand to a size of ≥ 5.0 cm over a follow-up of 4–14 years. A threshold diameter of 2.6 cm for defining AAAs in a screening programme is recommended and ectatic aortas detected at age 65 years can be re-screened at 4 years after the initial scan. A statistically significant difference was found in the growth rates of ectatic aortas with minimum 4 years follow-up, expanding to ≥ 5.0 cm compared to those less than 5.0 cm at last surveillance scan. Further studies are required to test the hypothesis of whether growth rate over the first 4 years of surveillance will identify those who are most likely to expand to a clinically significant size (> 5.0 cm).

scholarly journals Consistent Weighted Average Flux of Well-Balanced TVD-RK Discontinuous Galerkin Method for Shallow Water Flows

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Thida Pongsanguansin ◽  
Montri Maleewong ◽  
Khamron Mekchay
Keyword(s):  
Shallow Water ◽  
Discontinuous Galerkin ◽  
Numerical Solutions ◽  
Order Approximation ◽  
Weighted Average ◽  
Flat Bottom ◽  
Flux Function ◽  
Dg Method ◽  
Average Flux ◽  
Steady Solutions

A well-balanced scheme with total variation diminishing Runge-Kutta discontinuous Galerkin (TVD-RK DG) method for solving shallow water equations is presented. Generally, the flux function at cell interface in the TVD-RK DG scheme is approximated by using the Harten-Lax-van Leer (HLL) method. Here, we apply the weighted average flux (WAF) which is higher order approximation instead of using the HLL in the TVD-RK DG method. The consistency property is shown. The modified well-balanced technique for flux gradient and source terms under the WAF approximations is developed. The accuracy of numerical solutions is demonstrated by simulating dam-break flows with the flat bottom. The steady solutions with shock can be captured correctly without spurious oscillations near the shock front. This presents the other flux approximations in the TVD-RK DG method for shallow water simulations.

scholarly journals Effects of Matric and Osmotic Priming Treatments on Broccoli Seed Germination

1996 ◽  
Vol 121 (3) ◽  
pp. 423-429 ◽  
Author(s):  
Lewis W. Jett ◽  
Gregory E. Welbaum ◽  
Ronald D. Morse
Keyword(s):  
Polyethylene Glycol ◽  
Seed Germination ◽  
Root Growth ◽  
Growth Rates ◽  
Germination Rate ◽  
Membrane Integrity ◽  
Seed Vigor ◽  
Radicle Emergence ◽  
The Mean ◽  
Peg 8000

Priming, a controlled-hydration treatment followed by redrying, improves the germination and emergence of seeds from many species. We compared osmotic and matric priming to determine which was the most effective treatment for improving broccoli seed germination and to gain a greater understanding of how seed vigor is enhanced by priming. Broccoli (Brassica oleracea L. var. italica) seeds were osmotically primed in polyethylene glycol (PEG 8000) at -1.1 MPa or matrically primed in a ratio of 1.0 g seed:0.8 g synthetic calcium silicate (Micro-Cel E):1.8 ml water at -1.2 MPa. In the laboratory, germination rates and root lengths were recorded from 5 to 42C and 10 to 35C, respectively. Broccoli seeds germinated poorly at >35C. Root growth after germination was more sensitive to temperatures >30C and <15C than radicle emergence. Matric and osmotic priming increased germination rate in the laboratory, greenhouse, and field. However, matric priming had a greater effect on germination and root growth rates from 15 to 30C. Neither priming treatment affected minimum or maximum germination or root growth temperatures. Both priming treatments decreased the mean thermal time for germination by >35%. The greater germination performance of matrically primed seeds was most likely the result of increased oxygen availability during priming, increased seed Ca content, or improved membrane integrity.

A Balanced Approximation of the One-Layer Shallow-Water Equations on a Sphere

2009 ◽  
Vol 66 (6) ◽  
pp. 1735-1748 ◽  
Author(s):  
W. T. M. Verkley
Keyword(s):  
Shallow Water ◽  
Potential Vorticity ◽  
Shallow Water Equations ◽  
Rossby Waves ◽  
Coriolis Parameter ◽  
Beta Plane ◽  
Propagating Waves ◽  
Barotropic Vorticity Equation ◽  
Barotropic Vorticity ◽  
The One

Abstract A global version of the equivalent barotropic vorticity equation is derived for the one-layer shallow-water equations on a sphere. The equation has the same form as the corresponding beta plane version, but with one important difference: the stretching (Cressman) term in the expression of the potential vorticity retains its full dependence on f 2, where f is the Coriolis parameter. As a check of the resulting system, the dynamics of linear Rossby waves are considered. It is shown that these waves are rather accurate approximations of the westward-propagating waves of the second class of the original shallow-water equations. It is also concluded that for Rossby waves with short meridional wavelengths the factor f 2 in the stretching term can be replaced by the constant value f02, where f0 is the Coriolis parameter at ±45° latitude.

Do abiotic mechanisms determine interannual variability in length-at-age of juvenile Arcto-Norwegian cod?

2002 ◽  
Vol 59 (1) ◽  
pp. 57-65 ◽  
Author(s):  
Geir Ottersen ◽  
Kristin Helle ◽  
Bjarte Bogstad
Keyword(s):  
Growth Rates ◽  
Gadus Morhua ◽  
Barents Sea ◽  
Water Masses ◽  
Inverse Relation ◽  
Lower Growth ◽  
Density Dependent ◽  
The Barents Sea ◽  
The Mean ◽  
Dependent Growth

For the large Arcto-Norwegian stock of cod (Gadus morhua L.) in the Barents Sea, year-to-year variability in growth is well documented. Here three hypotheses for the observed inverse relation between abundance and the mean length-at-age of juveniles (ages 1–4) are suggested and evaluated. Based on comprehensive data, we conclude that year-to-year differences in length-at-age are mainly determined by density-independent mechanisms during the pelagic first half year of the fishes' life. Enhanced inflow from the southwest leads to an abundant cohort at the 0-group stage being distributed farther east into colder water masses, causing lower postsettlement growth rates. We can not reject density-dependent growth effects related to variability in food rations, but our data do not suggest this to be the main mechanism. Another hypothesis suggests that lower growth rates during periods of high abundance are a result of density-dependent mechanisms causing the geographic range of juveniles to extend eastwards into colder water masses. This is rejected mainly because year-to-year differences in mean length are established by age 2, which is too early for movements over large distances.

Quantification of Bioerosion by Sphaerechinus Granularis on ‘Coralugène’ Concretions of the Western Mediterranean

1997 ◽  
Vol 77 (2) ◽  
pp. 565-568 ◽  
Author(s):  
Stephane Sartoretto ◽  
Patrice Francour
Keyword(s):  
Shallow Water ◽  
Small Diameter ◽  
Western Mediterranean ◽  
Biological Agent ◽  
High Abundance ◽  
Shallow Waters ◽  
Mean Diameter ◽  
The Mean ◽  
The Mediterranean

Sphaerechinus granularis (Echinodermata: Echinidea) is involved in the erosion of ‘coralligène’ concretions in the Mediterranean. In shallow water (10 m), a high abundance of this species (>20 ind 25 m−2) is associated with small diameter individuals (56·7 ±7·7 mm). In deep clean waters (>40 m), the abundance is lower (<1 ind 25 m−2) and the mean diameter is higher (86·0±9·3 mm). Daily erosion of Corallinaceae by this species is related to the urchin diameter (r=0.87). Local variations in urchin abundance and diameter influence the amount of CaCO3 eroded annually. In shallow waters, the eroded CaCO3 mass reaches 210 g m−2 y−1 vs 16 g m−2 y−1 in coralligène concretions in deep clean waters. Sphaerechinus granularis is an important biological agent which substantially erodes the Mediterranean coralligène concretions.

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