scholarly journals The influence of front strength on the development and equilibration of symmetric instability. Part 1. Growth and saturation

2021 ◽  
Vol 926 ◽  
Author(s):  
A.F. Wienkers ◽  
L.N. Thomas ◽  
J.R. Taylor
Keyword(s):  
Stability Analysis ◽  
Shear Instability ◽  
Nonlinear Stability Analysis ◽  
Coriolis Parameter ◽  
Weakly Nonlinear ◽  
Thermal Wind ◽  
Inertial Instability ◽  
Slantwise Convection ◽  
The Stability ◽  
Symmetric Instability

Submesoscale fronts with large horizontal buoyancy gradients and $O(1)$ Rossby numbers are common in the upper ocean. These fronts are associated with large vertical transport and are hotspots for biological activity. Submesoscale fronts are susceptible to symmetric instability (SI) – a form of stratified inertial instability which can occur when the potential vorticity is of the opposite sign to the Coriolis parameter. Here, we use a weakly nonlinear stability analysis to study SI in an idealised frontal zone with a uniform horizontal buoyancy gradient in thermal wind balance. We find that the structure and energetics of SI strongly depend on the front strength, defined as the ratio of the horizontal buoyancy gradient to the square of the Coriolis frequency. Vertically bounded non-hydrostatic SI modes can grow by extracting potential or kinetic energy from the balanced front and the relative importance of these energy reservoirs depends on the front strength and vertical stratification. We describe two limiting behaviours as ‘slantwise convection’ and ‘slantwise inertial instability’ where the largest energy source is the buoyancy flux and geostrophic shear production, respectively. The growing linear SI modes eventually break down through a secondary shear instability, and in the process transport considerable geostrophic momentum. The resulting breakdown of thermal wind balance generates vertically sheared inertial oscillations and we estimate the amplitude of these oscillations from the stability analysis. We finally discuss broader implications of these results in the context of current parameterisations of SI.

scholarly journals The influence of front strength on the development and equilibration of symmetric instability. Part 2. Nonlinear evolution

2021 ◽  
Vol 926 ◽  
Author(s):  
A.F. Wienkers ◽  
L.N. Thomas ◽  
J.R. Taylor
Keyword(s):  
Primary Source ◽  
Inertial Oscillations ◽  
Turbulent Dissipation ◽  
Weakly Nonlinear ◽  
Thermal Wind ◽  
Background Density ◽  
Inertial Instability ◽  
Slantwise Convection ◽  
Symmetric Instability ◽  
Energy Pathways

In Part 1 (Wienkers, Thomas & Taylor, J. Fluid Mech., vol. 926, 2021, A6), we described the theory for linear growth and weakly nonlinear saturation of symmetric instability (SI) in the Eady model representing a broad frontal zone. There, we found that both the fraction of the balanced thermal wind mixed down by SI and the primary source of energy are strongly dependent on the front strength, defined as the ratio of the horizontal buoyancy gradient to the square of the Coriolis frequency. Strong fronts with steep isopycnals develop a flavour of SI we call ‘slantwise inertial instability’ by extracting kinetic energy from the background flow and rapidly mixing down the thermal wind profile. In contrast, weak fronts extract more potential energy from the background density profile, which results in ‘slantwise convection.’ Here, we extend the theory from Part 1 using nonlinear numerical simulations to focus on the adjustment of the front following saturation of SI. We find that the details of adjustment and amplitude of the induced inertial oscillations depend on the front strength. While weak fronts develop narrow frontlets and excite small-amplitude vertically sheared inertial oscillations, stronger fronts generate large inertial oscillations and produce bore-like gravity currents that propagate along the top and bottom boundaries. The turbulent dissipation rate in these strong fronts is large, highly intermittent and intensifies during periods of weak stratification. We describe each of these mechanisms and energy pathways as the front evolves towards the final adjusted state, and in particular focus on the effect of varying the dimensionless front strength.

Spatio-temporal structure of the ‘fully developed’ transitional flow in a symmetric wavy channel. Linear and weakly nonlinear stability analysis

2014 ◽  
Vol 764 ◽  
pp. 250-276 ◽  
Author(s):  
S. Blancher ◽  
Y. Le Guer ◽  
K. El Omari
Keyword(s):  
Stability Analysis ◽  
Laminar Flow ◽  
Steady Flow ◽  
Nonlinear Stability ◽  
Temporal Structure ◽  
Main Stream ◽  
Nonlinear Stability Analysis ◽  
Wavy Channel ◽  
Weakly Nonlinear ◽  
Spatio Temporal

AbstractThis work addresses the transition from 2D steady to 2D unsteady laminar flow for a fully developed regime in a symmetric wavy channel geometry. We investigate the existence and characteristics of the spatio-temporal structure of the fully developed unsteady laminar flow for those particular geometries for which the steady flow presents a periodic variation of the main stream velocity component. We perform a 2D global linear stability analysis of the fully developed steady laminar flow, and we show that, for all the geometries studied, the transition is triggered by a Hopf bifurcation associated with the breaking of the symmetries and the invariance of the steady flow. Critical Reynolds numbers, most unstable modes and their characteristics are presented for large ranges of the geometric parameters, namely wavenumber${\it\alpha}$from 0.3 to 5 and amplitude from 0 (straight channel) to 0.5. We show that it is possible to define geometries for which the wavenumber is proportional to the most unstable mode wavenumber for the critical Reynolds number. From this modal study we address a weakly nonlinear stability analysis with a view to obtaining the Landau coefficient$g$, and then the sub- or supercritical nature of the first bifurcation characterising the transition. We show that a critical geometric amplitude beyond which the first bifurcation is supercritical is associated with each geometric wavenumber.

A Nonlinear Stability Analysis for Difference Equations Using Semi-Implicit Mobility

1977 ◽  
Vol 17 (01) ◽  
pp. 79-91 ◽  
Author(s):  
D.W. Peaceman
Keyword(s):  
Porous Media ◽  
Stability Analysis ◽  
Finite Difference ◽  
Nonlinear Stability ◽  
Difference Equations ◽  
Nonlinear Stability Analysis ◽  
Time Step ◽  
Two Phase ◽  
Linearized Stability ◽  
The Stability

Abstract The usual linearized stability analysis of the finite-difference solution for two-phase flow in porous media is not delicate enough to distinguish porous media is not delicate enough to distinguish between the stability of equations using semi-implicit mobility and those using completely implicit mobility. A nonlinear stability analysis is developed and applied to finite-difference equations using an upstream mobility that is explicit, completely implicit, or semi-implicit. The nonlinear analysis yields a sufficient (though not necessary) condition for stability. The results for explicit and completely implicit mobilities agree with those obtained by the standard linearized analysis; in particular, use of completely implicit mobility particular, use of completely implicit mobility results in unconditional stability. For semi-implicit mobility, the analysis shows a mild restriction that generally will not be violated in practical reservoir simulations. Some numerical results that support the theoretical conclusions are presented. Introduction Early finite-difference, Multiphase reservoir simulators using explicit mobility were found to require exceedingly small time steps to solve certain types of problems, particularly coning and gas percolation. Both these problems are characterized percolation. Both these problems are characterized by regions of high flow velocity. Coats developed an ad hoc technique for dealing with gas percolation, but a more general and highly successful approach for dealing with high-velocity problems has been the use of implicit mobility. Blair and Weinaug developed a simulator using completely implicit mobility that greatly relaxed the time-step restriction. Their simulator involved iterative solution of nonlinear difference equations, which considerably increased the computational work per time step. Three more recent papers introduced the use of semi-implicit mobility, which proved to be greatly superior to the fully implicit method with respect to computational effort, ease of use, and maximum permissible time-step size. As a result, semi-implicit mobility has achieved wide use throughout the industry. However, this success has been pragmatic, with little or no theoretical work to justify its use. In this paper, we attempt to place the use of semi-implicit mobility on a sounder theoretical foundation by examining the stability of semi-implicit difference equations. The usual linearized stability analysis is not delicate enough to distinguish between the semi-implicit and completely implicit difference equation. A nonlinear stability analysis is developed that permits the detection of some differences between the stability of difference equations using implicit mobility and those using semi-implicit mobility. DIFFERENTIAL EQUATIONS The ideas to be developed may be adequately presented using the following simplified system: presented using the following simplified system: horizontal, one-dimensional, two-phase, incompressible flow in homogeneous porous media, with zero capillary pressure. A variable cross-section is included so that a variable flow velocity may be considered. The basic differential equations are (1) (2) The total volumetric flow rate is given by (3) Addition of Eqs. 1 and 2 yields =O SPEJ P. 79

Weakly Nonlinear Stability Analysis of a Thin Liquid Film With Condensation Effects During Spin Coating

2009 ◽  
Vol 131 (10) ◽  
Author(s):  
C. K. Chen ◽  
M. C. Lin
Keyword(s):  
Liquid Film ◽  
Spin Coating ◽  
Multiple Scales ◽  
Landau Equation ◽  
Thin Liquid Film ◽  
Kinematic Model ◽  
Circular Disk ◽  
Nonlinear Stability Analysis ◽  
Weakly Nonlinear ◽  
The Stability

This paper investigates the stability of a thin liquid film with condensation effects during spin coating. A generalized nonlinear kinematic model is derived by the long-wave perturbation method to represent the physical system. The weakly nonlinear dynamics of a film flow are studied by the multiple scales method. The Ginzburg–Landau equation is determined to discuss the necessary conditions of the various states of the critical flow states, namely, subcritical stability, subcritical instability, supercritical stability, and supercritical explosion. The study reveals that decreasing the rotation number and the radius of the rotating circular disk generally stabilizes the flow.

Effects of three-dimensionality on instability and turbulence in a frontal zone

2015 ◽  
Vol 784 ◽  
pp. 252-273 ◽  
Author(s):  
Eric Arobone ◽  
Sutanu Sarkar
Keyword(s):  
Stability Analysis ◽  
Potential Energy ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Frontal Zone ◽  
Gravitational Instability ◽  
Three Dimensional ◽  
Computational Domain ◽  
Coriolis Parameter ◽  
Symmetric Instability

Linear stability analysis and direct numerical simulation are used to investigate the evolution of a symmetrically unstable uniform frontal zone. Simulations in a three-dimensional computational domain capable of resolving near-symmetric currents develop strong nonlinearities without the emergence of pure symmetric instability. Linear stability analysis demonstrates that for $ft>1$ ( $f$ is the Coriolis parameter and $t$ denotes time) the flow generates strongly asymmetric structures which become nearly symmetric when $ft\gg 1$. Unlike the currents generated during pure symmetric instability, near-symmetric instability generates currents that do not align with isopycnals. This greatly modifies their energetics and evolution, leading to regions of the flow that are unstable to gravitational instability and energized by the reservoir of available potential energy. A high-resolution simulation demonstrates the flow evolution from near-symmetric currents to secondary shear-convective instabilities and finally, through tertiary instabilities, to fully three-dimensional turbulence. The effect of this sequence of instabilities is quantified through velocity and vorticity statistics as well as budgets for turbulent kinetic and potential energy. It is not until $ft\sim 10$ that the energy source for fluctuations is primarily shear, in contrast to the purely symmetric instability which draws its energy exclusively from shear production.

Nonlinear Stability Analysis of Thin-Walled Pier

2013 ◽  
Vol 361-363 ◽  
pp. 1251-1254
Author(s):  
Xiao Mei Dong
Keyword(s):  
Stability Analysis ◽  
Nonlinear Stability ◽  
Shell Element ◽  
Nonlinear Stability Analysis ◽  
Nonlinearity Effect ◽  
Updated Lagrangian Formulation ◽  
Thin Walled ◽  
Displacement Theory ◽  
The Stability ◽  
Updated Lagrangian

Shell element was used to simulate thin-walled piers. Mander constitutive model was adopted for analysis about the material nonlinearity. By finite displacement theory the geometric nonlinearity effect was reckoned in stability analysis based on Updated Lagrangian formulation. Nonlinear stability analysis during different construction stages indicates that the stability of pier in cantilever stage is weakest. Considered the dual non-linearity, the stability coefficient descends distinctly.

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