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Rigorous derivation of magneto-Oberbeck-Boussinesq approximation with non-local temperature term

Piotr Gwiazda, Florian Oschmann, Aneta Wróblewska-Kamińska

TL;DR

This work rigorously justifies the magneto-Oberbeck-Boussinesq approximation for a compressible, viscous, heat-conducting, magnetically conducting fluid by analyzing the low-Mach/low-Alfvén/near-neutral stratification limit with nonlocal temperature effects. Using the relative entropy method on weak solutions of the full MHD system, the authors prove convergence to a modified OBM system where the temperature deviation satisfies a nonlocal boundary condition encoded by a function A that couples to the magnetic field perturbation B^1. The main result shows that, under well-prepared initial data, the dynamics converge to horizontal flow with a vertical or perpendicular magnetic component, and the limit system preserves div U = div B^1 = 0 along with a Boussinesq-type momentum equation and a nonlocal temperature equation. The analysis advances the mathematical foundation for OBM-type approximations in magnetized, stratified fluids and provides a framework for handling nonlocal boundary temperature effects in high-Reynolds-number regimes.

Abstract

We consider a general compressible, viscous, heat and magnetically conducting fluid described by the compressible Navier-Stokes-Fourier system coupled with induction equation. In particular, we do not assume conservative boundary conditions for the temperature and allow heating or cooling on the surface of the domain. We are interested in the mathematical analysis when the Mach, Froude, and Alfvén numbers are small, converging to zero at a specific rate. We give a rigorous mathematical justification that in the limit, in case of low stratification, one obtains a modified Oberbeck-Boussinesq-MHD system with a non-local term or a non-local boundary condition for the temperature deviation. Choosing a domain confined between parallel plates, one finds also that the flow is horizontal, and the magnetic field is perpendicular to it. The proof is based on the analysis of weak solutions to a primitive system and the relative entropy method.

Rigorous derivation of magneto-Oberbeck-Boussinesq approximation with non-local temperature term

TL;DR

This work rigorously justifies the magneto-Oberbeck-Boussinesq approximation for a compressible, viscous, heat-conducting, magnetically conducting fluid by analyzing the low-Mach/low-Alfvén/near-neutral stratification limit with nonlocal temperature effects. Using the relative entropy method on weak solutions of the full MHD system, the authors prove convergence to a modified OBM system where the temperature deviation satisfies a nonlocal boundary condition encoded by a function A that couples to the magnetic field perturbation B^1. The main result shows that, under well-prepared initial data, the dynamics converge to horizontal flow with a vertical or perpendicular magnetic component, and the limit system preserves div U = div B^1 = 0 along with a Boussinesq-type momentum equation and a nonlocal temperature equation. The analysis advances the mathematical foundation for OBM-type approximations in magnetized, stratified fluids and provides a framework for handling nonlocal boundary temperature effects in high-Reynolds-number regimes.

Abstract

We consider a general compressible, viscous, heat and magnetically conducting fluid described by the compressible Navier-Stokes-Fourier system coupled with induction equation. In particular, we do not assume conservative boundary conditions for the temperature and allow heating or cooling on the surface of the domain. We are interested in the mathematical analysis when the Mach, Froude, and Alfvén numbers are small, converging to zero at a specific rate. We give a rigorous mathematical justification that in the limit, in case of low stratification, one obtains a modified Oberbeck-Boussinesq-MHD system with a non-local term or a non-local boundary condition for the temperature deviation. Choosing a domain confined between parallel plates, one finds also that the flow is horizontal, and the magnetic field is perpendicular to it. The proof is based on the analysis of weak solutions to a primitive system and the relative entropy method.

Paper Structure

This paper contains 24 sections, 1 theorem, 138 equations.

Table of Contents

  1. Introduction
  2. Formal limit
  3. On the structure of magnetic field and heat equation
  4. Weak formulation
  5. Equation of state
  6. Transport coefficients
  7. Relative energy
  8. Relative energy inequality
  9. Main result
  10. Basic energy estimates
  11. Convergence
  12. Limiting functions
  13. Relative energy
  14. Step 1:
  15. Step 2:
  16. ...and 9 more sections

Key Result

Theorem 5.1

Let the structural hypotheses be as given above. Let $(\varrho_\varepsilon, \vartheta_\varepsilon, {\bf u}_\varepsilon, {\bf B}_\varepsilon)$ be a solution to the primitive system with boundary conditions Let the initial data be well prepared in the sense that Suppose moreover the compatibility conditions If additionally then where the functions $(\varrho^1, \vartheta^1, {\bf U}, {\bf B}^1)$

Theorems & Definitions (7)

  • Remark 2.1
  • Remark 3.1
  • Remark 4.1
  • Definition 4.2
  • Theorem 5.1
  • Remark 5.2
  • Remark 5.3