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Stability Analysis of Thermohaline Convection With a Time-Varying Shear Flow Using the Lyapunov Method

Kalin Kochnev, Chang Liu

TL;DR

This work tackles the stability of thermohaline convection under a periodic time-varying shear by analyzing a linear time-varying system with a Lyapunov-based approach. It formulates a Lyapunov function using a periodic weighting matrix $P(t)$ and an LMI to bound the growth rate $\overline{\lambda}$, discretizing in time with forward Euler and comparing results to direct simulations and Floquet theory. The results show that, given enough temporal discretization points, the Lyapunov-bound growth rate converges to the values obtained from numerical simulations and Floquet analysis, while the eigendecomposition of $P(t)$ reveals the instantaneous instability directions driven mainly by temperature fluctuations. The framework offers formal stability guarantees for time-varying systems and provides a path to extend to nonlinear and non-periodic settings, with explicit trade-offs in computational cost highlighted. Overall, the Lyapunov method is demonstrated as a viable alternative to Floquet analysis for linear time-periodic stability questions in geophysical flows, with potential for broader application.

Abstract

This work identifies instabilities and computes the growth rate of a linear time-varying system using the Lyapunov method. The linear system describes cold fresh water on top of hot salty water with a periodically time-varying background shear flow. We employ a time-dependent weighting matrix to construct a Lyapunov function candidate, and the resulting linear matrix inequalities formulation is discretized in time using the forward Euler method. As the number of temporal discretization points increases, the growth rate predicted from the Lyapunov method or the Floquet theory will converge to the same value as that obtained from numerical simulations. We also use the Lyapunov method to analyze the instantaneous principal direction of instabilities and compare the computational resources required by the Lyapunov method, numerical simulations, and the Floquet theory.

Stability Analysis of Thermohaline Convection With a Time-Varying Shear Flow Using the Lyapunov Method

TL;DR

This work tackles the stability of thermohaline convection under a periodic time-varying shear by analyzing a linear time-varying system with a Lyapunov-based approach. It formulates a Lyapunov function using a periodic weighting matrix and an LMI to bound the growth rate , discretizing in time with forward Euler and comparing results to direct simulations and Floquet theory. The results show that, given enough temporal discretization points, the Lyapunov-bound growth rate converges to the values obtained from numerical simulations and Floquet analysis, while the eigendecomposition of reveals the instantaneous instability directions driven mainly by temperature fluctuations. The framework offers formal stability guarantees for time-varying systems and provides a path to extend to nonlinear and non-periodic settings, with explicit trade-offs in computational cost highlighted. Overall, the Lyapunov method is demonstrated as a viable alternative to Floquet analysis for linear time-periodic stability questions in geophysical flows, with potential for broader application.

Abstract

This work identifies instabilities and computes the growth rate of a linear time-varying system using the Lyapunov method. The linear system describes cold fresh water on top of hot salty water with a periodically time-varying background shear flow. We employ a time-dependent weighting matrix to construct a Lyapunov function candidate, and the resulting linear matrix inequalities formulation is discretized in time using the forward Euler method. As the number of temporal discretization points increases, the growth rate predicted from the Lyapunov method or the Floquet theory will converge to the same value as that obtained from numerical simulations. We also use the Lyapunov method to analyze the instantaneous principal direction of instabilities and compare the computational resources required by the Lyapunov method, numerical simulations, and the Floquet theory.

Paper Structure

This paper contains 11 sections, 1 theorem, 19 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Methods
  4. Lyapunov stability analysis
  5. Numerical simulation
  6. Floquet Theory
  7. Results
  8. Lyapunov Method Accurately Predicts the Growth Rate
  9. Eigendecomposition of $\boldsymbol{P}(t)$ Provides Insights into Time-dependent Instabilities
  10. Computational Resources Requirement of the Lyapunov Method
  11. Conclusion and Future Work

Key Result

Theorem 1

Given a linear time-varying system $\dot{\boldsymbol{\psi}}=\boldsymbol{A}(t)\boldsymbol{\psi}$ in eq:ode. If we can find a continuously differentiable Hermitian matrix $\boldsymbol{P}(t)\in \mathbb{C}^{5\times 5}$ and $\overline{\lambda}$ by solving then $\|\boldsymbol{\psi}(t)\|_2\leq C e^{\overline{\lambda} t}\|\boldsymbol{\psi}(0)\|_2$, where $C$ is a constant.

Figures (5)

  • Figure 1: The linear time-varying model analyzed here describes cold fresh water on top of hot salty water with a time-varying shear flow $\overline{u}(z,t)$. The background shear $\overline{u}(z,t)=A_U(t)z$, temperature $\overline{T}(z)=-z$, and salinity $\overline{S}(z)=-R_\rho z$ vary linearly over $z$.
  • Figure 2: $\ln(e)/2$ over $t$ from numerical simulations ($$) of system in \ref{['eq:ode']} and a least squares fitted $\ln(e)/2=\underline{\lambda}t+B$ ($$). The results are associated with wavenumbers $(k, m_0)=(0.179, 3.01 \times 10^{-3})$ and the initial condition producing the largest growth rate among 50 simulations with random initial conditions.
  • Figure 3: Stability analysis of the linear time-varying system in \ref{['eq:ode']} over different $(k, m_0)$ using three different methods in § \ref{['sec:methods']}. (a) $\text{log}_{10}(\overline{\lambda})$ using the Lyapunov method in § \ref{['subsec:lyapunov']} with $n = 400$, (b) $\text{log}_{10}(\overline{\lambda})$ using the Lyapunov method in § \ref{['subsec:lyapunov']} with $n=800$, (c) $\text{log}_{10}(\underline{\lambda})$ using numerical simulations as described in Section \ref{['subsec:dns']}, and (d) $\text{log}_{10}(\lambda_F)$ using the Floquet theory in § \ref{['subsec:floquet']} with $n=2000$.
  • Figure 4: The convergence of the Lyapunov method and the Floquet theory for $(k, m_0)=(0.179, 3 \times 10^{-3})$ as increasing $n$, the number of the temporal discretization points. The thick black line shows the growth rate obtained by numerical simulations in §\ref{['subsec:dns']}.
  • Figure 5: Eigendecomposition of $\boldsymbol{P}(t)$ obtained from the Lyapunov method for $(k, m_{0})=(0.179, 3\times 10^{-3})$ with $n=2000$. (a) The two largest eigenvalues $\mu_1[\boldsymbol{P}(t)]$ and $\mu_2[\boldsymbol{P}(t)]$ over time with the background shear flow $A_U(t)$. (b) Eigenvector components corresponding to the dominant eigenvalue over time. Note that $u$ and $v$ components are nearly zero.

Theorems & Definitions (2)

  • Theorem 1
  • proof