An amplitude-evolution equation for linearly unstable modes in stratified shear flows

1982 ◽  
Vol 117 ◽  
pp. 457-471 ◽  
Author(s):  
L. Engevik
Keyword(s):  
Evolution Equation ◽  
Couette Flow ◽  
Shear Flows ◽  
Temporal Evolution ◽  
Amplitude Equation ◽  
Second Order ◽  
Buoyancy Frequency ◽  
Density Profiles ◽  
Long Wave ◽  
Second Order In Time

A nonlinear amplitude equation of second order in time, which governs the temporal evolution of linearly unstable modes in stratified shear flows, is derived. It applies to a class of flows with continuous velocity and density profiles, and two examples of such flows are studied.One of the flows that is studied is the stratified Couette flow with the buoyancy frequency equal to Qy2, where Q is a constant and y the vertical co-ordinate. The nonlinear amplitude equation is studied for various values of Q.For the Garcia flow the nonlinear amplitude equation for the long-wave modes is evaluated, and it is compared with the corresponding equation in the Kelvin–Helmholtz flow, which has been found previously.

scholarly journals Inhomogeneous vortex tangles in counterflow superfluid turbulence: flow in convergent channels

2016 ◽  
Vol 7 (2) ◽  
pp. 130-149 ◽  
Author(s):  
Lidia Saluto ◽  
Maria Stella Mongioví
Keyword(s):  
Evolution Equation ◽  
Dimensional Analysis ◽  
Unit Volume ◽  
Vortex Formation ◽  
Second Order ◽  
Line Density ◽  
Superfluid Turbulence ◽  
Turbulence Flow ◽  
Vortex Diffusion

Abstract We investigate the evolution equation for the average vortex length per unit volume L of superfluid turbulence in inhomogeneous flows. Inhomogeneities in line density L andincounterflowvelocity V may contribute to vortex diffusion, vortex formation and vortex destruction. We explore two different families of contributions: those arising from asecondorder expansionofthe Vinenequationitself, andthose whichare notrelated to the original Vinen equation but must be stated by adding to it second-order terms obtained from dimensional analysis or other physical arguments.

Vortices evolution in ideal (M)HD

2021 ◽  
Author(s):  
José Roberto Canivete Cuissa ◽  
Oskar Steiner
Keyword(s):  
Evolution Equation ◽  
Solar Atmosphere ◽  
Vortex Dynamics ◽  
Shear Flows ◽  
Dynamical Equation ◽  
Space Plasma ◽  
Optimal Criterion ◽  
Swirling Strength ◽  
Vortex Tubes

<p>Vortices and vortex tubes are ubiquitous in the solar atmosphere and space plasma. In order to identify vortices and to study their evolution, we seek a suitable mathematical criterium for which a dynamical equation exists. So far, the only option available is given by the vorticity, which however is not the optimal criterion since it can be biased by shear flows. Therefore, we look at another criterion, the swirling strength, for which we found an evolution equation, which we suggest as a novel tool for the analysis of vortex dynamics in (magneto-)hydrodynamics. We highlight a few results obtained by applying the swirling strength and its dynamical equation to simulations of the solar atmosphere.</p>

Evolving inextensible and elastic curves with clamped ends under the second-order evolution equation in ℝ2

2018 ◽  
Vol 3 (1) ◽  
pp. 14-18 ◽  
Author(s):  
Chun-Chi Lin ◽  
Yang-Kai Lue
Keyword(s):  
Evolution Equation ◽  
Convergent Subsequence ◽  
Second Order ◽  
Plane Curves ◽  
Smooth Solutions ◽  
Fixed Position ◽  
Elastic Curves ◽  
Long Time Existence ◽  
Long Time ◽  
Time Existence

Abstract For any given C2-smooth initial open curves with fixed position and fixed tangent at the boundary points, we obtain the long-time existence of smooth solutions under the second-order evolution of plane curves. Moreover, the asymptotic limit of a convergent subsequence is an inextensible elastica.

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