scholarly journals On Linear Instability of Atmospheric Quasi-hydrostatic Equations in Response to Small Shortwave Perturbations

2021 ◽  
Vol 42 (9) ◽  
pp. 2237-2256
Author(s):  
X. Xu ◽  
R. I. Nigmatulin
Keyword(s):  
Global Climate ◽  
Linear Instability ◽  
Exact Equation ◽  
Sound Waves ◽  
Smooth Functions ◽  
Limited Class ◽  
The Cauchy Problem ◽  
Hydrostatic Approximation ◽  
The Stability ◽  
Approach Zero

Abstract A set of 3-dimensional atmospheric-dynamics equations with quasi-hydrostatic approximation is proposed and justified with the practical goal to optimize atmospheric modelling at scales ranging from meso meteorology to global climate. Sound waves are filtered by applying the quasi-hydrostatic approximation. In the closed system of hydro/thermodynamic equations, the inertial forces are negligibly small compared to gravity forces, and the asymptotically exact equation for vertical velocity is obtained. Investigation of the stability of solutions to this system in response to small shortwave perturbations has shown that solutions have the property of shortwave instability. There are situations when the increment of the perturbation amplitude tends to infinity, corresponding to absolute instability. It means that the Cauchy problem for such equations may be ill-posed. Its formulation can become conditionally correct if solutions are sought in a limited class of sufficiently smooth functions whose Fourier harmonics tend to zero reasonably quickly when the wavelengths of the perturbations approach zero. Thus, the numerical scheme for the quasi-hydrostatic equations using the finite-difference method requires an adequately selected pseudo-viscosity to eliminate the instability caused by perturbations with wavelengths of the order of the grid size. The result is useful for choosing appropriate vertical and horizontal grid sizes for modelling to avoid shortwave instability associated with the property of the system of equations. Implementation of pseudo-viscosities helps to smoothen or suppress the perturbations that occur during modelling.

scholarly journals Quasi-hydrostatic equations for climate models and the study on linear instability

2020 ◽  
Author(s):  
Robert Nigmatulin ◽  
Xiulin Xu
Keyword(s):  
Vertical Velocity ◽  
Climate Models ◽  
Global Climate ◽  
Momentum Conservation ◽  
Linear Instability ◽  
Atmospheric Dynamics ◽  
Exact Equation ◽  
Sound Waves ◽  
Smooth Functions ◽  
Practical Result

Abstract. An advanced “quasi-hydrostatic approximation” of 3-dimensional atmospheric-dynamics equations is proposed and justified with the practical goal to optimize atmospheric modelling at scales ranging from meso meteorology to global climate. For the vertically quasi-hydrostatic flow with inertial forces negligibly small compared to gravity forces, the asymptotically exact equation for vertical velocity is obtained. In the closed system of hydro/thermodynamic equations, the pressure is determined by the total air mass above, so that mass instead of pressure is considered as a dependent variable. In such a system, the sound waves are filtered, though the horizontal inertia forces are taken into account in the horizontal momentum conservation equations. The major practical result is an asymptotically exact equation for vertical velocity in the quasi-hydrostatic system of the atmospheric dynamics equations. Investigation of the stability of solutions to the system in response to small shortwave perturbations has shown that solutions have the property of shortwave instability. There are situations when the growth rate of the perturbation amplitude tends to infinity. It means that the Cauchy problem for such equations may be ill-posed. Its formulation can become conditionally correct if solutions are sought in a limited class of sufficiently smooth functions whose Fourier harmonics tend to zero reasonably quickly when the wavelengths of the perturbations approach zero. Thus, the numerical scheme for the quasi-hydrostatic equations using the finite-difference method requires an adequately selected pseudo-viscosity to eliminate the instability caused by perturbations with wavelengths of the order of the grid size. The result is useful for choosing appropriate vertical and horizontal grid sizes for modelling to avoid shortwave instability associated with the property of the system of equations. Implementation of pseud-viscosities helps to smoothen or suppress the perturbations that occur during modelling.

Any large solution of a non-linear heat conduction equation becomes monotonic in time

1991 ◽  
Vol 118 (1-2) ◽  
pp. 13-20 ◽  
Author(s):  
Victor A. Galaktionov ◽  
Sergey A. Posashkov
Keyword(s):  
Parabolic Equation ◽  
Initial Function ◽  
Heat Conduction Equation ◽  
Stationary Solutions ◽  
Classical Solutions ◽  
Smooth Functions ◽  
Large Solution ◽  
Additional Hypothesis ◽  
The Cauchy Problem ◽  
Degenerate Parabolic

SynopsisIn this paper we prove a certain monotonicity in time of non-negative classical solutions of the Cauchy problem for the quasilinear uniformly parabolic equation u1 = (ϕ(u))xx + Q(u) in wT = (0, T] × R1 with bounded sufficiently smooth initial function u(0, x) = uo(x)≧0 in Rl. We assume that ϕ(u) and Q(u) are smooth functions in [0, +∞) and ϕ′(u) >0, Q(u) > 0 for u > 0. Under some additional hypothesis on the growth of Q(u)ϕ′(u) at infinity, it is proved that if u(to, xo) becomes sufficiently large at some point (to, xo) ∈ wT, then ut(t, x0) ≧0 for all t ∈ [t0, T]. The proof is based on the method of intersection comparison of the solution with the set of the stationary solutions of the same equation. Some generalisations of this property for a quasilinear degenerate parabolic equation are discussed.

scholarly journals The Cauchy problem for the Finsler heat equation

2020 ◽  
Vol 13 (3) ◽  
pp. 257-278 ◽  
Author(s):  
Goro Akagi ◽  
Kazuhiro Ishige ◽  
Ryuichi Sato
Keyword(s):  
Cauchy Problem ◽  
Linear Operator ◽  
Heat Equation ◽  
Laplace Operator ◽  
Sufficient Condition ◽  
Smooth Functions ◽  
Dual Norm ◽  
The Cauchy Problem

AbstractLet H be a norm of {\mathbb{R}^{N}} and {H_{0}} the dual norm of H. Denote by {\Delta_{H}} the Finsler–Laplace operator defined by {\Delta_{H}u:=\operatorname{div}(H(\nabla u)\nabla_{\xi}H(\nabla u))}. In this paper we prove that the Finsler–Laplace operator {\Delta_{H}} acts as a linear operator to {H_{0}}-radially symmetric smooth functions. Furthermore, we obtain an optimal sufficient condition for the existence of the solution to the Cauchy problem for the Finsler heat equation\partial_{t}u=\Delta_{H}u,\quad x\in\mathbb{R}^{N},\,t>0,where {N\geq 1} and {\partial_{t}:=\frac{\partial}{\partial t}}.

scholarly journals Remarks on the Systems of Semilinear Fractional Rayleigh-Stokes Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Le Dinh Long
Keyword(s):  
Cauchy Problem ◽  
Fourier Analysis ◽  
Stokes Equation ◽  
Stokes Equations ◽  
Newtonian Fluids ◽  
System Of Equations ◽  
Main Technique ◽  
The Cauchy Problem ◽  
The Stability

In this paper, we study the Cauchy problem for a system of Rayleigh-Stokes equations. In this system of equations, we use derivatives in the classical Riemann-Liouville sense. This system has many applications in some non-Newtonian fluids. We obtained results for the existence, uniqueness, and frequency of the solution. We discuss the stability of the solutions and find the solution spaces. Our main technique is to use the Banach mapping theorem combined with some techniques in Fourier analysis.

scholarly journals The linear instability of the stratified plane Couette flow

2018 ◽  
Vol 853 ◽  
pp. 205-234 ◽  
Author(s):  
Giulio Facchini ◽  
Benjamin Favier ◽  
Patrice Le Gal ◽  
Meng Wang ◽  
Michael Le Bars
Keyword(s):  
Reynolds Number ◽  
Stability Analysis ◽  
Numerical Simulations ◽  
Couette Flow ◽  
Vertical Direction ◽  
Stability Diagram ◽  
Linear Instability ◽  
Plane Couette Flow ◽  
Horizontal Shear ◽  
The Stability

We present the stability analysis of a plane Couette flow which is stably stratified in the vertical direction orthogonal to the horizontal shear. Interest in such a flow comes from geophysical and astrophysical applications where background shear and vertical stable stratification commonly coexist. We perform the linear stability analysis of the flow in a domain which is periodic in the streamwise and vertical directions and confined in the cross-stream direction. The stability diagram is constructed as a function of the Reynolds number $Re$ and the Froude number $Fr$, which compares the importance of shear and stratification. We find that the flow becomes unstable when shear and stratification are of the same order (i.e. $Fr\sim 1$) and above a moderate value of the Reynolds number $Re\gtrsim 700$. The instability results from a wave resonance mechanism already known in the context of channel flows – for instance, unstratified plane Couette flow in the shallow-water approximation. The result is confirmed by fully nonlinear direct numerical simulations and, to the best of our knowledge, constitutes the first evidence of linear instability in a vertically stratified plane Couette flow. We also report the study of a laboratory flow generated by a transparent belt entrained by two vertical cylinders and immersed in a tank filled with salty water, linearly stratified in density. We observe the emergence of a robust spatio-temporal pattern close to the threshold values of $Fr$ and $Re$ indicated by linear analysis, and explore the accessible part of the stability diagram. With the support of numerical simulations we conclude that the observed pattern is a signature of the same instability predicted by the linear theory, although slightly modified due to streamwise confinement.

Study on Laminar Turbulent Transition in Square Arrayed Rod Bundles

2021 ◽  
Author(s):  
Carolina S. B. Dutra ◽  
Elia Merzari
Keyword(s):  
Turbulent Flows ◽  
Flow Behavior ◽  
Spectral Element ◽  
Linear Instability ◽  
Rod Bundles ◽  
Rod Bundle ◽  
Turbulent Transition ◽  
Spatio Temporal ◽  
The Stability ◽  
Laminar Turbulent Transition

Abstract The study of coolant flow behavior in rod bundles is of relevance to the design of nuclear reactors. Although laminar and turbulent flows have been researched extensively, there are still gaps in understanding the process of laminar-turbulent transition. Such a process may involve the formation of a gap vortex street as the consequence of a related linear instability. In the present work, a parametric study was performed to analyze the spatially developing turbulence in a simplified geometry setting. The geometry includes two square arrayed rod bundle subchannels with periodic boundary conditions in the cross-section. The pitch-to-diameter ratios range from 1.05 to 1.20, and the length of the domain was selected to be 100 diameters. No-slip condition at the wall, and inlet-outlet configuration were employed. Then, to investigate the stability of the flow, the Reynolds number was varied from 250 to 3000. The simulations were carried out using the spectral-element code Nek5000, with a Direct Numerical Simulation (DNS) approach. Data were analyzed to examine this Spatio-temporal developing instability. In particular, we evaluate the location of onset and spatial growth of the instability.

scholarly journals On the Construction of Large Algebras Not Contained in the Image of the Borel Map

2019 ◽  
Vol 75 (1) ◽  
Author(s):  
Céline Esser ◽  
Gerhard Schindl
Keyword(s):  
Sequence Space ◽  
Analytic Functions ◽  
General Setting ◽  
Smooth Functions ◽  
The Real ◽  
Cauchy Product ◽  
Real Analytic Functions ◽  
Borel Map ◽  
Real Analytic ◽  
The Stability

AbstractThe Borel map $$j^{\infty }$$j∞ takes germs at 0 of smooth functions to the sequence of iterated partial derivatives at 0. It is well known that the restriction of $$j^{\infty }$$j∞ to the germs of quasianalytic ultradifferentiable classes which are strictly containing the real analytic functions can never be onto the corresponding sequence space. In a recent paper the authors have studied the size of the image of $$j^{\infty }$$j∞ by using different approaches and worked in the general setting of quasianalytic ultradifferentiable classes defined by weight matrices. The aim of this paper is to show that the image of $$j^{\infty }$$j∞ is also small with respect to the notion of algebrability and we treat both the Cauchy product (convolution) and the pointwise product. In particular, a deep study of the stability of the considered spaces under the pointwise product is developed.

scholarly journals Linear stability of boundary layers under solitary waves

2014 ◽  
Vol 761 ◽  
pp. 62-104 ◽  
Author(s):  
Joris C. G. Verschaeve ◽  
Geir K. Pedersen
Keyword(s):  
Linear Stability ◽  
Solitary Waves ◽  
Reynolds Numbers ◽  
Quantitative Agreement ◽  
Linear Instability ◽  
Navier Stokes ◽  
Acceleration Region ◽  
Critical Reynolds Numbers ◽  
Tollmien Schlichting ◽  
The Stability

AbstractIn the present treatise, the stability of the boundary layer under solitary waves is analysed by means of the parabolized stability equation. We investigate both surface solitary waves and internal solitary waves. The main result is that the stability of the flow is not of parametric nature as has been assumed in the literature so far. Not only does linear stability analysis highlight this misunderstanding, it also gives an explanation why Sumer et al. (J. Fluid Mech., vol. 646, 2010, pp. 207–231), Vittori & Blondeaux (Coastal Engng, vol. 58, 2011, pp. 206–213) and Ozdemir et al. (J. Fluid Mech., vol. 731, 2013, pp. 545–578) each obtained different critical Reynolds numbers in their experiments and simulations. We find that linear instability is possible in the acceleration region of the flow, leading to the question of how this relates to the observation of transition in the acceleration region in the experiments by Sumer et al. or to the conjecture of a nonlinear instability mechanism in this region by Ozdemir et al. The key concept for assessment of instabilities is the integrated amplification which has not been employed for this kind of flow before. In addition, the present analysis is not based on a uniformization of the flow but instead uses a fully nonlinear description including non-parallel effects, weakly or fully. This allows for an analysis of the sensitivity with respect to these effects. Thanks to this thorough analysis, quantitative agreement between model results and direct numerical simulation has been obtained for the problem in question. The use of a high-order accurate Navier–Stokes solver is primordial in order to obtain agreement for the accumulated amplifications of the Tollmien–Schlichting waves as revealed in this analysis. An elaborate discussion on the effects of amplitudes and water depths on the stability of the flow is presented.

scholarly journals Generalized-Fractional Tikhonov-Type Method for the Cauchy Problem of Elliptic Equation

Mathematics ◽  
2020 ◽  
Vol 8 (1) ◽  
pp. 48
Author(s):  
Hongwu Zhang ◽  
Xiaoju Zhang
Keyword(s):  
Cauchy Problem ◽  
A Priori ◽  
Type Equation ◽  
Conditional Stability ◽  
Elliptic Type ◽  
Type Method ◽  
The Cauchy Problem ◽  
Ill Posed ◽  
The Stability ◽  
Regularized Solution

This article researches an ill-posed Cauchy problem of the elliptic-type equation. By placing the a-priori restriction on the exact solution we establish conditional stability. Then, based on the generalized Tikhonov and fractional Tikhonov methods, we construct a generalized-fractional Tikhonov-type regularized solution to recover the stability of the considered problem, and some sharp-type estimates of convergence for the regularized method are derived under the a-priori and a-posteriori selection rules for the regularized parameter. Finally, we verify that the proposed method is efficient and acceptable by making the corresponding numerical experiments.

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