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Relativistic turbulence features two flow patterns, one reproducing classical Kolmogorov scaling and another supported by rapidly decaying modes.

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2026-06-29 20:59 UTC pith:UCT5LM3P

load-bearing objection The paper claims two decay rates per mode in relativistic Israel-Stewart turbulence under a c-scaling assumption that recovers Navier-Stokes, but without explicit forms or derivations the central result stays unverified. the 3 major comments →

arxiv 2605.25329 v1 pith:UCT5LM3P submitted 2026-05-25 physics.flu-dyn hep-ph

Quantum field approach to relativistic turbulence

Esteban Calzetta This is my paper
classification physics.flu-dyn hep-ph
keywords relativistic fluidsturbulenceIsrael-Stewart theoryKolmogorov scalingentropy cascadehydrodynamics
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper applies field theory techniques to model turbulence in relativistic fluids using an Israel-Stewart framework. It shows that fluid modes act as overdamped oscillators with two decay rates for finite speed of light. One rate matches the classical Kolmogorov value while the other becomes very large as the speed of light increases. This leads to two possible turbulence states, with the classical one emerging when fast modes are suppressed. The work identifies how the alternative state can maintain an entropy cascade through specific scaling.

Core claim

For finite speed of light, each mode of the fluid behaves as an overdamped oscillator with two decaying rates, one that converges to the K41 value and another that diverges when c approaches infinity. There are therefore two basic flow patterns, one where the fast decaying modes are absent and which reproduces Kolmogorov turbulence, and another made only of fast decaying modes that can sustain an entropy cascade.

What carries the argument

The Israel-Stewart model supplemented by a Cattaneo-Maxwell equation for the viscous stress tensor, with parameters scaled so the infinite speed of light limit yields incompressible Navier-Stokes flow.

Load-bearing premise

The parameters in the fluid model are chosen to scale with the speed of light so that the limit of infinite speed of light recovers the incompressible Navier-Stokes equations.

What would settle it

Numerical simulation of the relativistic fluid equations showing whether mode decay rates split into one approaching the classical value and one increasing without bound as the speed of light is raised.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The slow-decay pattern reproduces standard Kolmogorov turbulence.
  • The fast-decay pattern can support an entropy cascade via appropriate scaling relations.
  • Relativistic corrections introduce an additional class of flow behaviors absent in the non-relativistic limit.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This structure suggests that relativistic turbulence may exhibit distinct dissipation channels not captured by classical models.
  • Simulations varying the speed of light could reveal transitions between the two patterns.
  • Field theory methods might enable new calculations of correlation functions in relativistic turbulent flows.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 1 minor

Summary. The paper applies quantum field theory methods to relativistic turbulence using the Israel-Stewart framework, where energy-momentum conservation is supplemented by a Cattaneo-Maxwell relaxation equation for the viscous tensor. It assumes model parameters scale with the speed of light c such that the c→∞ limit recovers incompressible Navier-Stokes equations. The central claim is that each fluid mode behaves as an overdamped oscillator with two decay rates (one converging to the K41 value, one diverging as c→∞), yielding two flow patterns: one reproducing Kolmogorov turbulence (fast modes absent) and one consisting only of fast decaying modes that can sustain an entropy cascade via specified scaling relations.

Significance. If the scaling assumption and resulting dispersion relations were explicitly derived and verified, the work could provide a useful bridge between relativistic hydrodynamics and non-relativistic turbulence phenomenology, particularly by identifying distinct decay channels and an entropy-cascade regime. The manuscript currently contains no derivations, explicit functional forms for the c-scaling, dispersion relations, or numerical/analytic checks, so the significance cannot be assessed beyond the conceptual outline.

major comments (3)
  1. [Abstract] Abstract: The scaling of model parameters (relaxation time, viscosity, heat conductivity) with c is asserted to recover the incompressible Navier-Stokes equations as c→∞, but no explicit functional form is supplied; this assumption is load-bearing for both the two-decay-rate claim and the identification of the K41 limit.
  2. [Abstract] Abstract: The statement that 'each mode of the fluid behaves as an overdamped oscillator with two decaying rates' is presented without derivation of the dispersion relation, linearization of the Israel-Stewart equations, or explicit solution for the decay rates as functions of c and wavenumber.
  3. [Abstract] Abstract: The claim that one flow pattern 'reproduces Kolmogorov turbulence' and that the fast-mode pattern sustains an entropy cascade via 'scaling relations' is asserted without showing that the derived decay rates or mode amplitudes are consistent with the K41 spectrum or with the entropy production equation under the stated scaling.
minor comments (1)
  1. [Abstract] Abstract contains multiple typographical errors: 'representtive' → 'representative', 'peed' → 'speed', 'repreduces' → 'reproduces'.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their detailed report and constructive criticism. Our manuscript is a concise conceptual outline applying quantum field theory methods to relativistic turbulence in the Israel-Stewart model, with the goal of identifying distinct decay channels in the non-relativistic limit. We address each major comment below and indicate where revisions will be made.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The scaling of model parameters (relaxation time, viscosity, heat conductivity) with c is asserted to recover the incompressible Navier-Stokes equations as c→∞, but no explicit functional form is supplied; this assumption is load-bearing for both the two-decay-rate claim and the identification of the K41 limit.

    Authors: We agree that the explicit functional forms are not supplied in the current version. The scaling is chosen to ensure the standard non-relativistic limit of relativistic hydrodynamics is recovered, but providing the concrete expressions (e.g., relaxation time scaling as 1/c² or similar) would make the load-bearing assumption transparent. We will include these explicit forms in a revised manuscript. revision: yes

  2. Referee: [Abstract] Abstract: The statement that 'each mode of the fluid behaves as an overdamped oscillator with two decaying rates' is presented without derivation of the dispersion relation, linearization of the Israel-Stewart equations, or explicit solution for the decay rates as functions of c and wavenumber.

    Authors: The manuscript focuses on the physical consequences rather than the intermediate algebra. The overdamped-oscillator behavior follows directly from linearizing the Israel-Stewart equations around a uniform background and solving the resulting characteristic equation for the modes. We will add a short appendix or section deriving the dispersion relation and the two decay rates (one approaching the K41 scaling, one diverging with c) to address this concern. revision: yes

  3. Referee: [Abstract] Abstract: The claim that one flow pattern 'reproduces Kolmogorov turbulence' and that the fast-mode pattern sustains an entropy cascade via 'scaling relations' is asserted without showing that the derived decay rates or mode amplitudes are consistent with the K41 spectrum or with the entropy production equation under the stated scaling.

    Authors: The identification of the K41 limit rests on one decay rate converging to the expected Kolmogorov value while the fast modes are projected out; the entropy-cascade regime is indicated by the scaling relations that keep the fast-mode contribution to entropy production finite. We acknowledge that explicit verification of spectral consistency and direct substitution into the entropy equation is not performed. A partial revision will outline these consistency arguments using the derived rates, while a full numerical or analytic check may require additional work beyond the present scope. revision: partial

Circularity Check

0 steps flagged

No significant circularity; model assumption defines the limit without reducing derived results to inputs by construction

full rationale

The paper states an explicit modeling assumption on parameter scaling with c to recover the incompressible NS limit, then derives finite-c mode behavior (overdamped oscillators, two decay rates) by linearizing the Israel-Stewart equations under that assumption. This is a standard setup for a relativistic extension, not a self-definitional loop or fitted input renamed as prediction. No self-citations, uniqueness theorems, or ansatzes are invoked as load-bearing in the provided text. The central claims are consequences of the equations rather than equivalent to the scaling assumption itself.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; the scaling assumption with c and the choice of Israel-Stewart as representative model are the main unexamined inputs.

free parameters (1)
  • scaling of model parameters with c
    Chosen so that c→∞ recovers incompressible Navier-Stokes; no explicit values given.
axioms (1)
  • domain assumption Israel-Stewart framework supplements energy-momentum conservation with a Cattaneo-Maxwell equation for the viscous part
    Taken as the representative model for relativistic real fluids.

pith-pipeline@v0.9.1-grok · 5694 in / 1129 out tokens · 25213 ms · 2026-06-29T20:59:02.556337+00:00 · methodology

0 comments
read the original abstract

The goal of this work is apply field theory methods to discuss turbulence in relativistic real fluids. We shalltake as representtive model an Israel-Stewart framework, where the conservation laws for the energy-momentum tensor are supplemented by a Cattaneo-Maxwell equation for its viscous part, which relaxes to its Landau-Lifshitz value. We assume the parameters of the model scale with the peed of light $c$ in such a way that as $c\to\infty$ the fluid becomes an incompressible fluid obeying the Navier-Stokes equations. We find that for finite $c$ each mode of the fluid behaves as an overdamped oscillator with two decaying rates, one that converges to the K41 value and another that diverges when $c\to\infty$. There are therefore two basic flow patterns, one where the fast decaying modes are absent, and which repreduces Kolmogorov turbulence, and another made only of fast decaying modes. We point out the scaling relations that allow the latter flow pattern to sustain an entropy cascade.

discussion (0)

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Cited by 1 Pith paper

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Reference graph

Works this paper leans on

121 extracted references · 6 canonical work pages · cited by 1 Pith paper

  1. [1]

    Krommes, Physics Reports 283 (1997) 5-48

    Systematic statistical theories of plasma turbulence and intermittency: current status and future prospects John A. Krommes, Physics Reports 283 (1997) 5-48

    1997

  2. [2]

    Rezzolla and O

    L. Rezzolla and O. Zanotti,Relativistic Hydrodynamics(Oxford University Press, Oxford, 2013)

    2013

  3. [3]

    Romatschke and U

    P. Romatschke and U. Romatschke,Relativistic fluid dynamics in and out equilibrium, Cambridge University Press (2019)

    2019

  4. [4]

    Khachatryan,Modified Kolmogorov Wave Turbulence in QCD matched onto Bottom-up Thermal- ization, Nucl

    V . Khachatryan,Modified Kolmogorov Wave Turbulence in QCD matched onto Bottom-up Thermal- ization, Nucl. Phys. A810, 109 (2008)

    2008

  5. [5]

    Stefan Floerchinger and Urs Achim Wiedemann,Fluctuations around Bjorken flow and the onset of turbulent phenomena, JHEP11(2011)100

    2011

  6. [6]

    M. E. Carrington and A. Rheban,Perturbative and Nonperturbative Kolmogorov Turbulence in a Gluon Plasma, The European Physical Journal C 71, 1787 (2011)

    2011

  7. [7]

    Berges, K

    J. Berges, K. Boguslavski, S. Schlichting, and R. Venugopalan,Turbulent Thermalization Process in Heavy Ion Collisions at Ultrarelativistic Energies, Phys. Rev. D 89, 074011 (2014)

    2014

  8. [8]

    Fukushima,Turbulent pattern formation and diffusion in the early-time dynamics in the relativistic heavy-ion collision, Phys.Rev

    K. Fukushima,Turbulent pattern formation and diffusion in the early-time dynamics in the relativistic heavy-ion collision, Phys.Rev. C89, 024907 (2014)

    2014

  9. [9]

    M. C. Abraao York, A. Kurkela, E. Lu and G. D. Moore,UV Cascade in Classical Yang-Mills via Kinetic Theory, Phys. Rev. D 89, 074036 (2014)

    2014

  10. [10]

    Gra ˜na and E

    M. Gra ˜na and E. Calzetta,Reheating and Turbulence, Phys. Rev. D 65, 063522 (2002)

    2002

  11. [11]

    Micha and I

    R. Micha and I. Tkachev,Relativistic Turbulence: A Long Way from Preheating to Equilibrium, Phys. Rev. Lett. 90, 121301 (2003). 21

    2003

  12. [12]

    Micha and I

    R. Micha and I. Tkachev,Turbulent thermalization, Phys. Rev. D 70, 043538 (2004)

    2004

  13. [13]

    Jos ´e Crespo and H. P. de Oliveira, Aspects of wave turbulence in preheating, JCAP06(2014)006

    2014

  14. [14]

    Crespo and H.P

    J.A. Crespo and H.P. de Oliveira, Aspects of wave turbulence in preheating. Part II. Rebirth of the nonminimal coupled models, JCAP12(2019)045

    2019

  15. [15]

    Niksa, M

    P. Niksa, M. Schlederer and G. Sigl,Gravitational waves produced by compressible MHD turbulence from cosmological phase transitions, Class. Quant. Grav.35(2018) 144001

    2018

  16. [16]

    Brandenburg and A

    A. Brandenburg and A. Nordlund,Astrophysical turbulence modeling, Rept. Prog. Phys. 74, 046901 (2011)

    2011

  17. [17]

    P. D. Lasky, M. F. Bennett and A. Melatos,Stochastic gravitational wave background from hydrody- namic turbulence in differentially rotating neutron stars, Phys. Rev. D 87, 063004 (2013)

    2013

  18. [18]

    Andrade, C

    T. Andrade, C. Pantelidou, J. Sonner and B. Withers,Driven black holes: from Kolmogorov scaling to turbulent wakes, arXiv:1912.00032 (2019)

    work page arXiv 1912

  19. [19]

    Saha, Abhisek ; Sanyal, Soma , Anisotropic turbulence in relativistic plasmas , The European Physical Journal Plus, V olume 137, Issue 9, article id.1074 (2022)

    2022

  20. [20]

    E. A. Spiegel (Editor),The Theory of Turbulence: Subrahmanyan Chandrasekhar’s 1954 Lectures Lect. Notes Phys. 810 (Springer, Dordrecht 2011)

    1954

  21. [21]

    A. S. Monin and A. M. Yaglom,Statistical Fluid Mechanics, vol. 1, ed. J. Lumley. (MIT Press, Cambridge, MA., (1971))

    1971

  22. [22]

    A. S. Monin and A. M. Yaglom,Statistical Fluid Mechanics, vol. 2, ed. J. Lumley. (MIT Press, Cambridge, MA., (1975))

    1975

  23. [23]

    Frisch,Turbulence(Cambridge U

    U. Frisch,Turbulence(Cambridge U. P., (1995))

    1995

  24. [24]

    Jonathan Zrake, Andrew MacFadyen, Spectral and Intermittency Properties of Relativistic Turbu- lence, 2013 ApJ 763 L12

    2013

  25. [25]

    L10, 5 pp

    Radice, David; Rezzolla, Luciano, Universality and Intermittency in Relativistic Turbulent Flows of a Hot Plasma, The Astrophysical Journal Letters, V olume 766, Issue 1, article id. L10, 5 pp. (2013)

    2013

  26. [26]

    Cruz-Osorio, A

    Megale, R. ; Cruz-Osorio, A. ; Ficarra, G. ; Imbrogno, M. ; Meringolo, C. ; Primavera, L. ; Rezzolla, L. ; Servidio, S., Measuring the properties of homogeneous turbulence in curved space-times, Mon Not R Astron Soc (2025) 2011-2023

    2025

  27. [27]

    Celora, M

    T. Celora, M. J. Hatton, I. Hawke, N. Andersson, Covariant approach to relativistic large-eddy simu- lations: Lagrangian filtering, Phys. Rev. D 110, 123040 (2024)

    2024

  28. [28]

    Comer, Marcus J

    Thomas Celora, Nils Andersson, Ian Hawke, Greg L. Comer, Marcus J. Hatton, Higher-level large- eddy filtering strategy for general relativistic fluid simulations, Phys. Rev. D 110, 123039 (2024)

    2024

  29. [29]

    Kraichnan, Inertial Ranges in Two-Dimensional Turbulence, Phys

    Robert H. Kraichnan, Inertial Ranges in Two-Dimensional Turbulence, Phys. Fluids 10, 1417 (1967)

    1967

  30. [30]

    Waleffe,The Nature of Triad Interactions in Homogeneous Turbulence, Phys

    F. Waleffe,The Nature of Triad Interactions in Homogeneous Turbulence, Phys. of Fluids A4(1992) 350

    1992

  31. [31]

    G. L. Eyink and Th. D. Drivas,Cascades and Dissipative Anomalies in Compressible Fluid Turbu- lence, Phys. Rev. X 8, 011022 (2018). 22

    2018

  32. [32]

    G. L. Eyink,Cascades and Dissipative Anomalies in Nearly Collisionless Plasma Turbulence, Phys. Rev. X 8, 041020 (2018)

    2018

  33. [33]

    Alexakis and L

    A. Alexakis and L. Biferale,Cascades and transitions in turbulent flows, Physics Reports 767-769, 1 (2018)

    2018

  34. [34]

    G. L. Eyink and T. D. Drivas,Cascades and Dissipative Anomalies in Relativistic Fluid Turbulence, Phys. Rev. X8, 011023 (2018)

    2018

  35. [35]

    Israel,Nonstationary irreversible thermodynamics: A causal relativistic theory, Ann

    W. Israel,Nonstationary irreversible thermodynamics: A causal relativistic theory, Ann. Phys. (NY) 100(1976) 310

    1976

  36. [36]

    Israel and J

    W. Israel and J. M. Stewart,Thermodynamics of nonstationary and transient effects in a relativistic gas, Phys. Lett. A58(1976) 213

    1976

  37. [37]

    Israel and M

    W. Israel and M. Stewart,Transient relativistic thermodynamics and kinetic theory, Ann. Phys. (NY) 118(1979) 341

    1979

  38. [38]

    Israel and M

    W. Israel and M. Stewart,On transient relativistic thermodynamics and kinetic theory. II, Proc. R. Soc. London, Ser A365(1979) 43

    1979

  39. [39]

    Israel and J

    W. Israel and J. M. Stewart,Progress in relativistic thermodynamics and electrodynamics of contin- uous media, inGeneral Relativity and Gravitation, edited by A. Held, Plenum, New York,2(1980) 491

    1980

  40. [40]

    Israel,Covariant fluid mechanics and thermodynamics: An introduction, inRelativistic Fluid Dynamics, edited by A

    W. Israel,Covariant fluid mechanics and thermodynamics: An introduction, inRelativistic Fluid Dynamics, edited by A. M. Anile and Y . Choquet-Bruhat, Springer (1988) New York USA

    1988

  41. [41]

    J. C. Maxwell,On the dynamical theory of gases, Philos. Trans. Soc. London157(1867) 49

  42. [42]

    Cattaneo,Sulla conduzione del calore, Atti del Semin

    C. Cattaneo,Sulla conduzione del calore, Atti del Semin. Mat. e Fis. Univ. Modena3(1948) 3

    1948

  43. [43]

    Cattaneo,Sur la propagation de la chaleur en relativit ´e, C

    C. Cattaneo,Sur la propagation de la chaleur en relativit ´e, C. R. Acad. Sci. Paris247(1958) 431

    1958

  44. [44]

    D. D. Joseph and L. Preziosi,Heat waves, Rev. Mod. Phys.61(1989) 41; Addendum, Rev. Mod. Phys.62(1990) 375

    1989

  45. [45]

    L. D. Landau and E. M. Lifshitz,Fluid Mechanics, Pergamon Press eds (1959) Oxford, UK

    1959

  46. [46]

    Hiscock and L

    W. Hiscock and L. Lindblom, Stability and Causality in Dissipative Relativistic Fluids, Ann. Phys. (NY) 151, 466 (1983)

    1983

  47. [47]

    Mitra, Causality and stability analysis of first-order field redefinition in relativistic hydrodynamics from kinetic theory, Phys

    S. Mitra, Causality and stability analysis of first-order field redefinition in relativistic hydrodynamics from kinetic theory, Phys. Rev. C 105, 054910 (2022)

    2022

  48. [48]

    de Brito, G

    C. de Brito, G. Rocha and G. Denicol, Hydrodynamic theories for a system of weakly self-interacting classical ultra-relativistic scalar particles: causality and stability , arXiv:2311.07272

    work page arXiv

  49. [49]

    Hoult and P

    R. Hoult and P. Kovtun, Causality and classical dispersion relations, Phys. Rev. D 109 (2024) 4, 046018

    2024

  50. [50]

    Pu, Stability and causality criteria in linear mode analysis: stability means causality, Phys

    D-L Wang and S. Pu, Stability and causality criteria in linear mode analysis: stability means causality, Phys. Rev. D 109, L031504 (2024)

    2024

  51. [51]

    Gavassino, M

    L. Gavassino, M. Disconzi and J. Noronha, Dispersion relations alone cannot guarantee causality, Phys. Rev. Lett. 132, 162301 (2024). 23

    2024

  52. [52]

    Bhattacharyya, S

    S. Bhattacharyya, S. Mitra and S. Roy, Causality and Stability in relativistic hydrodynamic theory - a choice to be endured, Phys. Lett. B 856, 138918 (2024)

    2024

  53. [53]

    Rev D113, 023530 (2026)

    Juan Pablo El ´ıa, Lucas Cantarutti, Esteban Calzetta and Nahuel Mir´on-Granese, Hydrodynamic mod- els of reheating, Phys. Rev D113, 023530 (2026)

    2026

  54. [54]

    E. A. Novikov, Random force method in turbulence theory, J. Exptl. Theoret. Phys. (U.S.S.R.) 44, 2159-2168 (1963) (Eng. trans. Sov. Phys. JETP 17, 1449 (1965))

    1963

  55. [55]

    E. A. Novikov, Functionals and the random-force method in turbulence theory, J. Exptl. Theoret. Phys. (U.S.S.R.) 47, 1919-1926 (1964) (Eng. trans. Sov. Phys. JETP 20, 1290 (1965))

    1919

  56. [56]

    Poursina, Ali; Pourjamal, Ali; Bozorg, Ali, Stochastic Approach to Turbulence: A Comprehensive Review, arXiv:2211.12918 (2022)

    work page arXiv 2022

  57. [57]

    Calzetta, Energy spectrum of non-Newtonian turbulence, Phys

    E. Calzetta, Energy spectrum of non-Newtonian turbulence, Phys. Rev. Fluids 10, 054607 (2025)

    2025

  58. [58]

    Milton van Dyke, Perturbation methods in fluid mechanics, The Parabolic Press (1975)

    1975

  59. [59]

    L. D. Landau and E. M. Lifshitz,Statistical Mechanics, part II, Pergamon Press eds (1959) Oxford, UK

    1959

  60. [60]

    Esteban Calzetta, Fully developed relativistic turbulence, Phys. Rev. D 103, 056018 (2021)

    2021

  61. [61]

    P. C. Martin, E. D. Siggia and H. A. Rose,Statistical Dynamics of Classical Systems, Phys. Rev. A8 (1973) 423

    1973

  62. [62]

    De Dominicis,Techniques de renormalisation de la th ´eorie des champs et dynamique des ph´enom`enes critiques, J

    J. De Dominicis,Techniques de renormalisation de la th ´eorie des champs et dynamique des ph´enom`enes critiques, J. Physique Coll.37(1976) C1

    1976

  63. [63]

    De Dominicis and P

    C. De Dominicis and P. C. Martin,Energy spectra of certain randomly-stirred fluids, Phys. Rev. A19 (1979) 419

    1979

  64. [64]

    de Dominicis and I

    C. de Dominicis and I. Giardina,Random fields and spin glasses: a field theory approach(Cambridge University Press, Cambridge, England, 2006)

    2006

  65. [65]

    Forster, D

    D. Forster, D. R. Nelson and M. J. Stephen,Large-distance and long-time properties of a randomly stirred fluid, Phys. Rev. A16(1977) 732

    1977

  66. [66]

    S Gauthier, M-E Brachet and J-D Fournier, Testing field-theoretical methods on a classical cubic equation with stochastic driving, J. Phys. A: Math. Gen. 14, 2969 (1981)

    1981

  67. [67]

    R Phythian, The functional formalism of classical statistical dynamics, J. Phys. A: Math. Gen., V ol. 10, No. 5, 777 (1977)

    1977

  68. [68]

    V . I. Belinicher and V .S. L’vov, A scale-invariant theory of fully developed hydrodynamic turbulence, Zh. Eksp. Teor. fiz. 93, 533 (1987) (Engl. Trans. Sov. Phys. JETP 66, 303 (1987))

    1987

  69. [69]

    spherical model

    G. L. Eyink, Large-Nlimit of the “spherical model” of turbulence, Phys. Rev. E49, 3990 (1994)

    1994

  70. [70]

    G. L. Eyink,Turbulence NoiseJ. Stat. Phys. 1,83(1996) 955

    1996

  71. [71]

    W. D. Thacker, A path integral for turbulence in incompressible fluids, Journal of Mathematical Physics 38, 300 (1997)

    1997

  72. [72]

    Krommes, Fundamental statistical descriptions of plasma turbulence in magnetic fields, Physics Reports 360 (2002) 1–352

    John A. Krommes, Fundamental statistical descriptions of plasma turbulence in magnetic fields, Physics Reports 360 (2002) 1–352. 24

    2002

  73. [73]

    Formulation of the Theory of Turbulence in an Incompressible Fluid.Ann

    Wyld, H.W., Jr. Formulation of the Theory of Turbulence in an Incompressible Fluid.Ann. Phys. 1961,14, 143–165

    1961

  74. [74]

    L. L. Lee, A Formulation of the Theory of Isotropic Hydromagnetic Turbulence in an Incompressible Fluid, Ann. Phys. 32, 292 (1965)

    1965

  75. [75]

    Berera, M

    A. Berera, M. Salewski and W. D. McComb, Eulerian field-theoretic closure formalisms for fluid turbulence, Phys. Rev. E 87, 013007 (2013)

    2013

  76. [76]

    Mir ´on-Granese, A

    N. Mir ´on-Granese, A. Kandus and E. Calzetta,Nonlinear fluctuations in relativistic causal fluids, JHEP 07, 064 (2020)

    2020

  77. [77]

    Mir ´on Granese, A

    N. Mir ´on Granese, A. Kandus and E. Calzetta, Field Theory Approaches to Relativistic Hydrody- namics. Entropy 24, 1790 (2022)

    2022

  78. [78]

    ORSZAG, Analytical theories of turbulence, J

    STEVEN A. ORSZAG, Analytical theories of turbulence, J. Fluid Mech. (1970), vol. 41, part 2, pp. 363-386

    1970

  79. [79]

    Kraichnan, An almost-Markovian Galilean-invariant turbulence model, J

    R. Kraichnan, An almost-Markovian Galilean-invariant turbulence model, J. Fluid Mech. 47, part 3, 513, (1971)

    1971

  80. [81]

    Tsinober,An informal conceptual introduction to turbulence(Springer, Dordretch, 2009)

    A. Tsinober,An informal conceptual introduction to turbulence(Springer, Dordretch, 2009)

    2009

Showing first 80 references.