Path 8: Geophysical and Rotating/Stratified Turbulence

Base Camp 8.1: Rapidly Rotating Turbulence and Inertial Waves

Stepping Stones: Effect of background rotation (Coriolis force) – tendency to two-dimensionalize turbulence (energy cascade slows in vertical direction); formation of columnar vortices (Taylor columns); inertial waves and wave–turbulence interactions; Rossby number as governing parameter (turbulence vs. waves dominance); experiments in rotating tanks (e.g. Hopfinger's vortex arrays).

• Hopfinger & van Heijst (1993), "Vortices in Rotating Fluids" – A landmark ARFM article reviewing experiments on rotating turbulence. They describe how even moderate rotation produces long-lived columnar vortices aligned with the rotation axis, and how the cascade of energy shifts toward the horizontal planes (leading to an inverse cascade of energy for strong rotation). Fundamental processes like the merger of like-signed vortices are inhibited by rotation, and small-scale isotropy is broken (anisotropic turbulence). This pedagogical review is widely cited for insights into how rotation modifies turbulent structure (intuitive to intermediate).
• Cambon & Jacquin (1989), "Spectral Approach to Non-linear Wave–Turbulence Interactions in Rotating Flows" – A theoretical paper analyzing how inertial waves (oscillations due to rotation) interact with turbulent eddies. They showed that non-linear resonances between triads are restricted by rotation, effectively draining energy transfer to certain modes. This partly explains the slow-down of the forward cascade. Their spectral closures predict energy anisotropy consistent with experiments. Recommended for those seeking a deeper theoretical understanding of rotating turbulence (rigorous).
• Baroud et al. (2002), "Anisotropy in Rotating Turbulence" – An experiment using particle tracking velocimetry in a rotating water tank to quantify anisotropy. They measured velocity structure functions parallel and perpendicular to the rotation axis and observed that for strong rotation (low Rossby), scaling exponents differ, confirming theoretical expectations of two-dimensionalization. This paper provides concrete data and is often cited as experimental evidence of how rotation alters turbulence scaling (intermediate).
• Davidson (2013), Turbulence in Rotating, Stratified and Electrically Conducting Fluids – This book (by the author of Turbulence: An Introduction) specifically addresses the complexities of geophysical-style turbulence. For rotation, Davidson offers an accessible explanation of topics like Taylor columns, the Proudman–Taylor theorem (constraint of rotation making flows 2D), and how energy spectra evolve as $E(k_\perp, k_\parallel)$ with $k_\parallel$ (parallel to rotation) being suppressed. It's often recommended for its clear physical reasoning applied to rotating and magnetohydrodynamic turbulence (intuitive to intermediate).
• Moisy et al. (2011), "Decay Laws, Anisotropy and Cyclone–Anticyclone Asymmetry in Decaying Rotating Turbulence" – A detailed study on how freely decaying turbulence behaves under rotation. They found that anticyclonic vortices (spinning opposite to the background rotation) tend to dominate because cyclonic ones dissipate faster – a striking asymmetry observed in atmosphere/ocean as well. They also measured how kinetic energy decays with time under different rotation rates. This paper is a rich source of data and physical interpretation for rotating decaying turbulence (intermediate).

Base Camp 8.2: Stably Stratified Turbulence and Internal Gravity Waves

Stepping Stones: Effect of stable density stratification (buoyancy) – turbulence suppressed in vertical (pancake vortices, layer formation); Froude number as key parameter; internal gravity waves propagating on density gradients; energy partition into wave modes vs. vortical modes; stratified turbulence phenomenology (strong stratification leads to quasi-2D layered turbulence with an upscale buoyancy-range cascade); applications to atmosphere (tropopause, stable boundary layers) and ocean (thermocline turbulence).

• Riley & Lindborg (2008), "Stratified Turbulence: A Possible Interpretation of Some Geophysical Measurements" – A significant paper that outlines the concept of a stratified turbulence inertial range. They suggest that in strongly stratified flows, there exists a distinct scaling regime (between isotropic Kolmogorov and wave-dominated regimes) where horizontal spectra of kinetic energy follow $\sim k_h^{-5/3}$. This interpretation helped explain atmospheric observations of spectra. The paper is quite accessible despite tackling complex field data, making it a key resource (intermediate).
• Ivey, Winters & Koseff (2008), "Density Stratification Effects in Turbulence" – An Annual Review article reviewing laboratory and numerical studies of stratified turbulence. It discusses how turbulence is altered when the buoyancy Reynolds number ($Re_b$) is low vs. high, the emergence of layer-and-cascade structure (thin turbulent layers separated by quiescent stable layers), and mixing efficiency (how much kinetic energy goes into irreversible potential energy increase). This article is frequently recommended as an entry point to stratified turbulence, summarizing both the phenomenology and practical implications for oceanography and atmospheric science (intermediate).
• Lilly (1983), "Stratified Turbulence and the Mesoscale Spectrum of Atmospheric Motions" – A visionary paper by D.K. Lilly linking stratified turbulence to the observed $-5/3$ kinetic energy spectrum in the upper atmosphere (mesoscales). Lilly argued that patches of stratified turbulence could be responsible for the so-called "mesoscale bump". This was one of the first attempts to reconcile observations with turbulence theory and is cited as an early influence for later formal stratified turbulence theories (intuitive to intermediate).
• Brethouwer et al. (2007), "Scaling Analysis and Simulation of Strongly Stratified Turbulence" – Combined analysis and DNS to propose that strongly stratified flows fall into two regimes depending on $Re_b$. At sufficiently high $Re_b$, turbulence can survive even under strong stratification (with a forward cascade in horizontal planes), whereas at low $Re_b$ the flow transitions to a quasi-2D internal wave field. Their simulations showed the formation of horizontal layers and an $E(k_h)\sim k_h^{-5/3}$ horizontal spectrum for energetic turbulence. This paper is more technical but is central in modern stratified turbulence literature (rigorous).
• Zilitinkevich et al. (2008), "Turbulence and Equilibrium States in Stably Stratified Flows" – A geophysically oriented paper discussing when turbulence can exist in very stable stratification (like polar night boundary layers). They introduce the concept of a "maximum sustainable turbulence" and how turbulence can collapse into wave-dominated regimes. It blends fundamental ideas with observed behaviors in the atmosphere and is valuable for those interested in real-world stratified flows and their parameterization in models (intermediate).

Base Camp 8.3: Geophysical Turbulence – Atmosphere, Ocean, and Planetary Flows

Stepping Stones: Quasi-geostrophic turbulence (turbulence on a rotating planet dominated by Coriolis and pressure gradient balance – leads to 2D-like inverse cascade and formation of jets, e.g. zonal flows on Jupiter or oceanic jets); $\beta$-effect and Rhines scale (how planetary vorticity gradient halts the inverse cascade at a certain scale, forming zonal jets); ocean mixed layer vs. deep interior (interaction of stratified and rotating effects); atmospheric turbulence in the boundary layer vs. free atmosphere (surface-driven vs. radiatively driven).

• Vallis (2017), Atmospheric and Oceanic Fluid Dynamics – A comprehensive graduate text which includes chapters on geostrophic turbulence. Vallis explains how the combination of rotation and stratification (and often a planetary vorticity gradient, $\beta$) yields flows very different from laboratory turbulence. Key concepts like the Rhines scale (where the inverse cascade is arrested by wave dispersion, leading to zonal jets) are clearly presented. Vallis uses both physical arguments and simple models (quasi-geostrophic equations), making it a top recommendation on forums for understanding large-scale turbulence in nature (intermediate to rigorous).
• Salmon (1980), "Baroclinic Instability and Geostrophic Turbulence" – A foundational paper linking baroclinic instabilities in the atmosphere (which drive weather systems) to geostrophic turbulence theory. Salmon used quasi-geostrophic two-layer models to show how an inverse cascade of energy can coexist with a forward cascade of potential enstrophy, generalizing the 2D turbulence concepts to baroclinic flows. This is a keystone in understanding atmospheric turbulence at planetary scales (rigorous).
• McWilliams (1984), "Emergence of Isolated Coherent Vortices in Turbulent Flow" – A well-known JFM paper where McWilliams' simulations of 2D (and quasi-geostrophic) turbulence on a beta-plane showed spontaneous emergence of coherent vortices (analogous to Gulf Stream rings or Jupiter's Great Red Spot). This work demonstrated that turbulent flows can self-organize into long-lived structures on a rotating planet. It's a striking example often cited to emphasize how geophysical turbulence can be both turbulent and organized (intermediate).
• Charney (1971), "Geostrophic Turbulence" – The brief but classic paper by J.G. Charney that applied Kolmogorov ideas to geophysical flows. Charney argued that in a barotropic atmosphere, there is a downscale cascade of enstrophy and an inverse upscale transfer of energy, constrained by $\beta$. This laid the theoretical groundwork for understanding the observed kinetic energy spectra in the atmosphere (which Charney predicted to have a $-3$ slope in the upper inertial range, later refined by Nastrom & Gage's data). Though short, this paper is historically important (intuitive to intermediate).
• Sagaut & Cambon (2008), Homogeneous Turbulence Dynamics – This advanced monograph includes specialized chapters on rotating, stratified, and magnetohydrodynamic turbulence. For geophysical turbulence, it provides theoretical treatments (e.g. structure functions on a $\beta$-plane, spectral transfers with rotation/stratification) with mathematical rigor. It's useful for readers who want to see the unification of turbulence theory with geophysical specifics under one formalism (rigorous).

(Paths 8.1–8.3 extend our knowledge to the huge scales of oceans and atmospheres. They show turbulence entwined with waves and global forces, illustrating both the universality and adaptability of turbulence concepts. Lastly, Path 9 will take us to the forefront of how data and computation are driving new turbulence insights.)

What to Upload Next

To continue our deep exploration, it's recommended to gather key original sources and textbooks for each base camp. Below is a prioritized list of PDFs (5–6 each) grouped by base camp:

Base Camp 8.1 (Rotating)

• J. P. Hopfinger & G. J. F. van Heijst (1993) – ARFM
• P. Bartello et al. (1994) – JFM on breakdown of rotating turbulence
• P. H. Rhines (1975) – JFM on geostrophic turbulence (introduces Rhines scale concept)
• L. M. Smith & F. Waleffe (1999) – Phys. Fluids on transfer in rotating flows
• S. Galtier (2003) – ApJ on weak inertial wave turbulence (if interested in wave-dominated theory)

Base Camp 8.2 (Stratified)

• J. J. Riley & E. Lindborg (2008) – J. Atmos. Sci. "Stratified turbulence…"
• H. J. S. Fernando & J. C. Weil (2010) – ARFM on stable boundary layers
• D. M. Staquet & J. Sommeria (2002) – ARFM on internal gravity waves and turbulence
• M. Brethouwer et al. (2007) – JFM DNS paper
• O. M. Phillips (1972) – "Turbulence in a strongly stratified fluid" (classic)

Base Camp 8.3 (Geophysical flows)

• I. M. Held (1999) – "Horizontal Turbulence" (notes on quasi-2D atmospheric turbulence)
• G. K. Vallis (2017) – textbook chapters on barotropic/baroclinic turbulence
• J. C. McWilliams (2006) – Fundamentals of Geophysical Fluid Dynamics (turbulence sections)
• W. Young & P. Rhines (1982) – J. Atmos. Sci. on spectral transfers with β-effect
• K. S. Smith (2004) – JFM on oceanic inverse cascades and eddy scales