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36 - Four–quark correlators
- from Part VIII - QCD two–point functions
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- QCD as a Theory of Hadrons
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- 04 November 2022
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- 09 February 2023, pp 371-377
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Scaling in Rayleigh–Bénard convection
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- Journal of Fluid Mechanics / Volume 956 / 10 February 2023
- Published online by Cambridge University Press:
- 09 February 2023, A34
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We consider the Nusselt–Rayleigh number problem of Rayleigh–Bénard convection and make the hypothesis that the velocity and thermal boundary layer widths,
$\delta _u$ and 
$\delta _T$, in the absence of a strong mean flow are controlled by the dissipation scales of the turbulence outside the boundary layers and, therefore, are of the order of the Kolmogorov and Batchelor scales, respectively. Under this assumption, we derive 
$Nu \sim Ra^{1/3}$ in the high 
$Ra$ limit, independent of the Prandtl number, 
$\delta _T/L \sim Ra^{-1/3}$ and 
$\delta _u/L \sim Ra^{-1/3} Pr^{1/2}$, where 
$L$ is the height of the convection cell. The scaling relations are valid as long as the Prandtl number is not too far from unity. For 
$Pr \sim 1$, we make a more general ansatz, 
$\delta _u \sim \nu ^{\alpha }$, where 
$\nu$ is the kinematic viscosity and assume that the dissipation scales as 
$\sim u^3/L$, where 
$u$ is a characteristic turbulent velocity. Under these assumptions we show that 
$Nu \sim Ra^{\alpha /(3-\alpha )}$, implying that 
$Nu \sim Ra^{1/5}$ if 
$\delta _u$ were scaling as in a Blasius boundary layer and 
$Nu \sim Ra^{1/2}$ (with some logarithmic correction) if it were scaling as in a standard turbulent shear boundary layer. It is argued that the boundary layers will retain the intermediate scaling 
$\alpha = 3/4$ in the limit of high 
$Ra$.
41 - Lattice gauge theory
- from Part IX - QCD non–perturbative methods
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- QCD as a Theory of Hadrons
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- 04 November 2022
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- 09 February 2023, pp 396-408
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4 - The supersymmetry algebra
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- Weak Scale Supersymmetry
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- 05 November 2022
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- 09 February 2023, pp 41-48
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1 - Classical kinks
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- Kinks and Domain Walls
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- 04 November 2022
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- 09 February 2023, pp 1-19
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9 - Highlights of hadron production
- from III - Particle production
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- Hadrons and Quark–Gluon Plasma
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- 04 November 2022
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- 09 February 2023, pp 159-186
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48 - Theoretical foundations
- from Part X - QCD spectral sum rules
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- QCD as a Theory of Hadrons
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- 04 November 2022
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- 09 February 2023, pp 491-498
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3 - The Wess–Zumino model
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- Weak Scale Supersymmetry
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- 05 November 2022
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- 09 February 2023, pp 23-40
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29 - Tjnaf(Cebaf)
- from Part 5 - Future directions
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- Electron Scattering for Nuclear and Nucleon Structure
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- 05 November 2022
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- 09 February 2023, pp 263-271
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13 - Triggers
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- Introduction to Experimental Particle Physics
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- 05 November 2022
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- 09 February 2023, pp 303-324
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2 - The Material Derivative: The First Step to the Navier–Stokes Equations
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- A Student's Guide to the Navier-Stokes Equations
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- 10 March 2023
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- 09 February 2023, pp 32-59
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Appendix A - Units, numbers and conventions
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- Kinks and Domain Walls
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- 04 November 2022
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- 09 February 2023, pp 154-154
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31 - Future directions
- from Part 5 - Future directions
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- Electron Scattering for Nuclear and Nucleon Structure
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- 05 November 2022
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- 09 February 2023, pp 278-287
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35 - Baryonic correlators
- from Part VIII - QCD two–point functions
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- QCD as a Theory of Hadrons
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- 04 November 2022
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- 09 February 2023, pp 364-370
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8 - Periodic magnetic channels
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- Principles of Magnetostatics
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- 04 November 2022
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- 09 February 2023, pp 197-210
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3 - Phenomenology of the Standard Model
- from Part 1 - Effective Field Theory: The Standard Model, Supersymmetry, Unification
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- Supersymmetry and String Theory
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- 10 November 2022
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- 09 February 2023, pp 27-62
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13 - Micro–pattern gaseous detectors
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- Gaseous Radiation Detectors
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- 04 November 2022
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- 09 February 2023, pp 365-398
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7 - The Standard Model
- from Part II - The Standard Model
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- Advanced Concepts in Particle and Field Theory
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- 04 November 2022
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- 09 February 2023, pp 251-290
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References
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- Hadrons and Quark–Gluon Plasma
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- 04 November 2022
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- 09 February 2023, pp 371-388
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Acknowledgements
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- QCD as a Theory of Hadrons
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- 04 November 2022
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- 09 February 2023, pp xxxii-xxxii
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