Math 2.0

Astronomy and astrophysics Definition

Mixing length theory

Builds on Schwarzschild criterion for convectionHydrostatic equilibrium

Definition

Consider a convective layer of a star in hydrostatic equilibrium. Let PP, ρ\rho and TT be its pressure, density and temperature, gg the local gravity and cPc_P the specific heat at constant pressure. Let HP=P/(ρg)H_P = P / (\rho g) be the pressure scale height and δ=−(∂ln⁡ρ/∂ln⁡T)P\delta = -(\partial \ln \rho / \partial \ln T)_P. Let ∇=dln⁡T/dln⁡P\nabla = d \ln T / d \ln P be the gradient of the surroundings and ∇e\nabla_e the gradient followed by a moving blob, with ∇>∇e\nabla > \nabla_e. Mixing length theory with parameter α>0\alpha > 0 models convection by blobs that travel a mean distance ℓ=αHP\ell = \alpha H_P before they dissolve. The distance ℓ\ell is the mixing length, and the theory sets, as averages over the blobs:

  1. the temperature excess of a blob, ΔT/T=(∇−∇e) ℓ/(2HP)\Delta T / T = (\nabla - \nabla_e) \, \ell / (2 H_P);
  2. the blob speed, v2=gδ(∇−∇e) ℓ2/(8HP)v^2 = g \delta (\nabla - \nabla_e) \, \ell^2 / (8 H_P);
  3. the convective flux, Fconv=ρcPv ΔTF_{\mathrm{conv}} = \rho c_P v \, \Delta T.

Remarks

Combining the three clauses gives Fconv=ρcPTgδ ℓ242 HP−3/2(∇−∇e)3/2.F_{\mathrm{conv}} = \rho c_P T \sqrt{g \delta} \, \frac{\ell^2}{4 \sqrt{2}} \, H_P^{-3/2} (\nabla - \nabla_e)^{3/2}. The numerical factors 1/21/2 and 1/81/8 come from averaging over blobs that have travelled between 00 and ℓ\ell. Other versions of the theory use other factors, which change the fitted value of α\alpha. Example: in the deep convective envelope of the Sun the density is high. So a tiny excess ∇−∇e\nabla - \nabla_e already carries the whole luminosity, and ∇\nabla is very close to the adiabatic gradient. Non-example: in a layer with ∇<∇ad\nabla < \nabla_{\mathrm{ad}}, stable by the Schwarzschild criterion, a displaced blob returns, and the model gives no convective flux. Solar models usually need α\alpha between about 1.51.5 and 22.

Reading path

To understand this page from the ground up, read these 88 pages in order. The foundations come first.

  1. Brunt–Väisälä frequency concept
  2. Derivative concept
  3. Differential equation concept
  4. Euclidean space object
  5. Euclidean vector concept
  6. Euler equations concept
  7. Function concept
  8. Interval concept
  9. Lane–Emden equation concept
  10. Limit point of a set definition
  11. Local maximum definition
  12. Ordinary differential equation concept
  13. Partition of an interval definition
  14. Poisson's equation concept
  15. Sequence definition
  16. Velocity concept
  17. Fréchet derivative definition
  18. Cross product definition
  19. Ordinary differential equation definition
  20. Function of several variables concept
  21. Vector-valued function concept
  22. Lipschitz condition definition
  23. Classical mechanics concept
  24. Limit of a function definition
  25. Local minimum definition
  26. Mesh of a partition definition
  27. Riemann sum definition
  28. Real sequence definition
  29. Subsequence definition
  30. Limit of a function of several variables concept
  31. Momentum concept
  32. Partial derivative concept
  33. Solution of an ordinary differential equation definition
  34. Inertial frame of reference concept
  35. Continuity at a point definition
  36. Derivative definition
  37. Riemann integrable function definition
  38. Bounded sequence definition
  39. Limit of a sequence definition
  40. Monotone sequence definition
  41. Continuity in R^n definition
  42. Differentiable function of several variables definition
  43. Partial derivative definition
  44. Initial value problem definition
  45. Newton's laws of motion statement
  46. Continuous function definition
  47. Antiderivative definition
  48. Riemann integral definition
  49. Fermat's theorem on stationary points statement
  50. Convergent sequence definition
  51. Monotone convergence theorem for sequences statement
  52. Every sequence has a monotone subsequence statement
  53. Differentiable on an open set definition
  54. Gradient definition
  55. Higher partial derivatives definition
  56. Picard–Lindelöf hypotheses definition
  57. Intermediate value theorem statement
  58. Definite integral definition
  59. Bolzano–Weierstrass theorem statement
  60. Iterated derivative definition
  61. Picard operator definition
  62. Additivity of the integral over intervals statement
  63. Boundedness theorem statement
  64. Monotonicity of the integral statement
  65. C^k function characterised by continuous derivatives statement
  66. Integral equation form of an initial value problem statement
  67. An iterate of the Picard operator is a contraction statement
  68. Extreme value theorem statement
  69. Vector field definition
  70. Picard–Lindelöf theorem statement
  71. Mean value theorem for integrals statement
  72. Rolle's theorem statement
  73. Conservative vector field definition
  74. Mean value theorem statement
  75. First fundamental theorem of calculus statement
  76. Classical mechanics definition
  77. Potential energy definition
  78. Momentum definition
  79. Second fundamental theorem of calculus statement
  80. Kinetic energy definition
  81. Angular momentum definition
  82. Fundamental theorem of calculus statement
  83. Central force definition
  84. Virial theorem statement
  85. Newton's law of universal gravitation statement
  86. Hydrostatic equilibrium statement
  87. Schwarzschild criterion for convection statement
  88. Mixing length theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Mixing_length_model
About this page
TagMS1Q8E9YS4
URIm2://MS1Q8E9YS4
Kinddefinition
WritingW0 stub
MSC85A15, 85A30
DefinesMixing length theory
UsesSchwarzschild criterion for convection, Hydrostatic equilibrium
Changed3 November 2026

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