pith. sign in

arxiv: 2411.02291 · v4 · submitted 2024-11-04 · ⚛️ physics.ao-ph · physics.flu-dyn

Emergent vorticity asymmetry of one and two-layer shallow water system captured by a next-order balanced model

Ryan Sh\`iji\'e D\`u , K. Shafer Smith This is my paper

Pith reviewed 2026-05-23 17:32 UTC · model grok-4.3

classification ⚛️ physics.ao-ph physics.flu-dyn
keywords shallow waterquasi-geostrophicvorticity asymmetrypotential vorticitybalanced modelsbaroclinic instabilityRossby numberturbulence
0
0 comments X

The pith

A next-order extension of quasi-geostrophic balance captures the negative vorticity skew that emerges in shallow-water turbulence.

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

The standard quasi-geostrophic equations produce symmetric vorticity evolution, yet full shallow-water dynamics generate an asymmetric, negatively skewed distribution even in balanced flow. The paper constructs SWQG^{+1} by extending the diagnostic relations that recover ageostrophic velocity and height from potential vorticity to the next order in Rossby number. In one layer this yields a potential-based formulation; the same relations are then written for two layers. Freely decaying turbulence runs show the model reproduces the observed skew, while baroclinically unstable jet runs reproduce both the skew and the finite divergence that appears at strain-driven fronts. Because the model remains strictly balanced, it filters gravity waves by construction and therefore isolates the mechanism responsible for the asymmetry.

Core claim

The SWQG^{+1} system, which retains a single prognostic variable (potential vorticity) and diagnoses all other fields from it via next-order relations, reproduces the negatively skewed vorticity statistics of one-layer shallow-water decaying turbulence and the vorticity asymmetry together with finite divergence at strain-driven fronts in two-layer baroclinically unstable jets, while the classical quasi-geostrophic model does not.

What carries the argument

Next-order diagnostic relations that recover ageostrophic velocity, height, and divergence directly from potential vorticity, expressed through a potential formulation for one layer and generalized to multiple layers.

If this is right

  • The model supplies a single balanced equation set that includes finite divergence at fronts without explicit gravity-wave dynamics.
  • The same diagnostic structure applies without change to one-layer and two-layer configurations.
  • Numerical integrations remain stable on the same grids used for classical quasi-geostrophic runs.
  • The potential formulation yields diagnostic expressions that can be inverted by standard elliptic solvers.

Where Pith is reading between the lines

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

  • The same next-order relations could be tested against observed front widths in satellite altimetry to check whether the predicted divergence matches real ocean fronts.
  • Because the model stays balanced at higher Rossby numbers than classical QG, it may serve as an intermediate bridge for regimes where full primitive-equation runs remain expensive.
  • Extension to three or more layers would follow the same algebraic pattern already written for two layers.

Load-bearing premise

The next-order diagnostic relations derived from the shallow-water equations remain accurate and stable for the turbulent regimes tested without requiring additional filtering or introducing spurious instabilities.

What would settle it

A side-by-side time series of domain-integrated vorticity skewness from SWQG^{+1} and from the full shallow-water equations in the freely decaying one-layer turbulence case; systematic divergence between the two curves would falsify the claim that the model captures the asymmetry.

Figures

Figures reproduced from arXiv: 2411.02291 by K. Shafer Smith, Ryan Sh\`iji\'e D\`u.

Figure 1
Figure 1. Figure 1: Vorticity ({𝜀}𝜁/ 𝑓 ) fields (top) and height ({𝜀/𝐵𝑢}ℎ/𝐻) fields (bottom) for 𝜀 = 0.1 at time 𝑡/𝑇 = 200 from the shallow water simulation (left) and SWQG+1(right). remarkably well during the entire time series. They lie within 1/ √ 10 times the ensemble standard deviation from each other for most of the time series. We use this range since it is the standard scaling of error of a Monte-Carlo estimate of the… view at source ↗
Figure 2
Figure 2. Figure 2: Left: time series of vorticity (𝜁) skewness for the 𝜀 = 0.1 simulations from the shallow water model as well as SWQG+1. The lighter lines are the individual ensemble members. The darker lines are the ensemble mean, and the 1/ √ 10 of the ensemble standard deviation is the filled color around the mean. Right: the vorticity (𝜁) skewness at 𝑡/𝑇 = 200 for 𝜀 = 0.01, 0.03, 0.05, 0.07, 0.1, 0.12. The error bar is… view at source ↗
Figure 3
Figure 3. Figure 3: The same as Figure [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Left: the time series of the total energy, EKE, and APE ( [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Time series of the total energy at the QG level ( [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The initial jets’ nondimensional velocity in the upper ( [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The vorticity field of the evolution of the jets for the shallow water model (left) [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Vorticity skewness of two-layer shallow water (left) and SWQG [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The first local maxima of vorticity skewness of a set of simulations with varying Rossby numbers. strain field 𝑈 𝑀 = 𝛼𝑥, 𝑉 𝑀 = −𝛼𝑦, (3.13) which can be captured by a horizontal streamfunction Φ 𝑀 = −𝛼𝑥𝑦. (3.14) The strain field has no imprint on the PV if we take the rigid-lid limit of SWQG+1. That is, we ignore all terms that are divided by 𝑔. All inversions are not affected by the strain field except for… view at source ↗
Figure 10
Figure 10. Figure 10: Divergence of a cold filament driven by a barotropic strain, modeled by a [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
read the original abstract

The turbulent evolution of the shallow water system exhibits asymmetry in vorticity. This emergent phenomenon can be classified as "balanced", that is, it is not due to the inertial-gravity wave modes. The Quasi-Geostrophic (QG) system, the canonical model for balanced motion, has a symmetric evolution of vorticity, thus misses this phenomenon. Here we present a next-order-in-Rossby extension of QG, QG$^{+1}$, in the shallow water context. We recapitulate the derivation of the model in one-layer shallow water grounded in physical principles and provide a new formulation using "potentials". Then, the multi-layer extension of the SWQG$^{+1}$ model is formulated for the first time. The SWQG$^{+1}$ system is still balanced in the sense that there is only one prognostic variable, potential vorticity (PV), and all other variables are diagnosed from PV. It filters out inertial gravity waves by design. This feature is attractive for modeling the dynamics of balanced motions that dominate transport in geophysical systems. The diagnostic relations connect ageostrophic physical variables and extend the massively useful geostrophic balance. Simulations of these systems in classical set-ups provide evidence that SWQG$^{+1}$ captures the vorticity asymmetry in the shallow water system. Simulations of freely decaying turbulence in one-layer show that SWQG$^{+1}$ can capture the negatively skewed vorticity, and simulations of the nonlinear evolution of a baroclinically unstable jet show that it can capture vorticity asymmetry and finite divergence of strain-driven fronts.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 4 minor

Summary. The paper introduces SWQG^{+1}, a next-order-in-Rossby-number balanced extension of the quasi-geostrophic (QG) model for one- and two-layer shallow-water systems. Derived from the shallow-water equations using physical principles and a new potential formulation, the model retains a single prognostic variable (potential vorticity) while diagnosing ageostrophic fields; it is shown via simulations to reproduce the negatively skewed vorticity of freely decaying turbulence and the vorticity asymmetry plus finite divergence at strain-driven fronts in a baroclinically unstable jet, phenomena absent from standard QG.

Significance. If the diagnostic relations remain accurate in the tested turbulent regimes, the model supplies a computationally efficient balanced framework that extends the practical reach of QG to include emergent asymmetry and frontogenesis, with direct relevance to transport and balanced dynamics in geophysical fluids.

minor comments (4)
  1. [Derivation] § on one-layer derivation: the transition from the standard QG balance to the next-order diagnostic relations for divergence and ageostrophic velocity should include an explicit statement of the truncation error order to confirm consistency with the Rossby-number expansion.
  2. [Numerical results] Simulation sections: the time series or histograms of vorticity skewness should report the number of independent realizations or ensemble size used to establish that the negative skew is statistically robust rather than a single-run feature.
  3. [Multi-layer extension] Multi-layer formulation: the interface conditions and vertical structure of the potentials are introduced without an accompanying table of symbols; adding such a table would improve readability of the two-layer extension.
  4. [Figures] Figure captions for the jet simulation: the color scale for divergence should be stated explicitly so that the reported 'finite divergence' can be compared quantitatively with the parent shallow-water run.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work on the SWQG^{+1} model and for recommending minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives the SWQG^{+1} diagnostic relations directly from the shallow-water equations expanded to next order in Rossby number, using a potential formulation that remains a single prognostic PV equation with all other fields diagnosed. The vorticity asymmetry is an emergent feature observed in independent full shallow-water simulations; the model is then tested against those simulations in decaying turbulence and baroclinic jet cases. No step reduces the target asymmetry to a fitted parameter, self-definition, or self-citation chain; the derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; the model rests on the standard shallow-water equations and the assumption that a next-order Rossby expansion remains balanced. No free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption Shallow-water dynamics admit a consistent next-order balanced reduction that filters inertia-gravity waves while retaining ageostrophic corrections to vorticity evolution.
    Invoked to justify the QG^{+1} construction and its multi-layer extension.

pith-pipeline@v0.9.0 · 5821 in / 1110 out tokens · 41155 ms · 2026-05-23T17:32:56.635204+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

67 extracted references · 67 canonical work pages

  1. [1]

    , " * write output.state after.block = add.period write newline

    ENTRY address author booktitle chapter edition editor howpublished institution journal key month note number organization pages publisher school series title type volume year eprint label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence ...

    work page

  2. [2]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize " " * FUNCT...

    work page

  3. [3]

    Allen, J. S. , Barth, J. A. & Newberger, P. A. 1990 On Intermediate Models for Barotropic Continental Shelf and Slope Flow Fields . Part I : Formulation and Comparison of Exact Solutions . Journal of Physical Oceanography 20 (7), 1017--1042

    work page 1990

  4. [4]

    Chaos: An Interdisciplinary Journal of Nonlinear Science 4 (2), 163--175

    Arai, Masazumi & Yamagata, Toshio 1994 Asymmetric evolution of eddies in rotating shallow water . Chaos: An Interdisciplinary Journal of Nonlinear Science 4 (2), 163--175

    work page 1994

  5. [5]

    , Ruuth, Steven J

    Ascher, Uri M. , Ruuth, Steven J. & Spiteri, Raymond J. 1997 Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations . Applied Numerical Mathematics 25 (2), 151--167

    work page 1997

  6. [6]

    Barth, J. A. , Allen, J. S. & Newberger, P. A. 1990 On Intermediate Models for Barotropic Continental Shelf and Slope Flow Fields . Part II : Comparison of Numerical Model Solutions in Doubly Periodic Domains . Journal of Physical Oceanography 20 (7), 1044--1076

    work page 1990

  7. [7]

    & Merlis, Timothy M

    Bembenek, Eric , Straub, David N. & Merlis, Timothy M. 2020 Effects of Moisture in a Two-Layer Model of the Midlatitude Jet Stream . Journal of the Atmospheric Sciences 77 (1), 131--147

    work page 2020

  8. [8]

    Tellus 7 (1), 27--49

    Bolin, Bert 1955 Numerical Forecasting with the Barotropic Model . Tellus 7 (1), 27--49

    work page 1955

  9. [9]

    & Haidvogel, Dale B

    Bretherton, Francis P. & Haidvogel, Dale B. 1976 Two-dimensional turbulence above topography . Journal of Fluid Mechanics 78 (1), 129--154

    work page 1976

  10. [10]

    , Pauluis, Olivier & Gerber, Edwin P

    Brown, Marguerite L. , Pauluis, Olivier & Gerber, Edwin P. 2023 Scaling for Saturated Moist Quasi-Geostrophic Turbulence . Journal of the Atmospheric Sciences -1 (aop)

    work page 2023

  11. [11]

    , Vasil, Geoffrey M

    Burns, Keaton J. , Vasil, Geoffrey M. , Oishi, Jeffrey S. , Lecoanet, Daniel & Brown, Benjamin P. 2020 Dedalus: A flexible framework for numerical simulations with spectral methods . Physical Review Research 2 (2), 023068

    work page 2020

  12. [12]

    & Frederiksen, Jorgen S

    Carnevale, George F. & Frederiksen, Jorgen S. 1987 Nonlinear stability and statistical mechanics of flow over topography . Journal of Fluid Mechanics 175 , 157--181

    work page 1987

  13. [13]

    1955 The Use of the Primitive Equations of Motion in Numerical Prediction

    Charney, J. 1955 The Use of the Primitive Equations of Motion in Numerical Prediction . Tellus 7 (1), 22--26

    work page 1955

  14. [14]

    & Marshall, D

    Chassignet, E. & Marshall, D. 2008 Gulf Stream separation in numerical ocean models . Geophysical Monograph Series 177

    work page 2008

  15. [15]

    , Schlax, Michael G

    Chelton, Dudley B. , Schlax, Michael G. , Samelson, Roger M. & de Szoeke , Roland A. 2007 Global observations of large oceanic eddies . Geophysical Research Letters 34 (15)

    work page 2007

  16. [16]

    & Polvani, Lorenzo M

    Cho, James Y-K. & Polvani, Lorenzo M. 1996 The emergence of jets and vortices in freely evolving, shallow-water turbulence on a sphere . Physics of Fluids 8 (6), 1531--1552

    work page 1996

  17. [17]

    Journal of Fluid Mechanics 971 , A2

    Chouksey, Manita , Eden, Carsten , Masur, G \"o kce Tuba & Oliver, Marcel 2023 A comparison of methods to balance geophysical flows . Journal of Fluid Mechanics 971 , A2

    work page 2023

  18. [18]

    Cloke, Paul & Cullen, M. J. P. 1994 A semi-geostrophic ocean model with outcropping . Dynamics of Atmospheres and Oceans 21 (1), 23--48

    work page 1994

  19. [19]

    Archive for Rational Mechanics and Analysis 156 (3), 241--273

    Cullen, Mike & Gangbo, Wilfrid 2001 A Variational Approach for the 2- Dimensional Semi-Geostrophic Shallow Water Equations . Archive for Rational Mechanics and Analysis 156 (3), 241--273

    work page 2001

  20. [20]

    Journal of Physical Oceanography 16 (1), 132--143

    Cushman-Roisin , Benoit 1986 Frontal Geostrophic Dynamics . Journal of Physical Oceanography 16 (1), 132--143

    work page 1986

  21. [21]

    & Tang, Benyang 1989 Geostrophic Regimes and Geostrophic Turbulence Beyond the Radius of Deformation

    Cushman-Roisin , B. & Tang, Benyang 1989 Geostrophic Regimes and Geostrophic Turbulence Beyond the Radius of Deformation . In Elsevier Oceanography Series \/ (ed. J. C. J. Nihoul & B. M. Jamart ) , Mesoscale/ Synoptic Coherent Structures in Geophysical Turbulence , vol. 50 , pp. 51--74 . Elsevier

    work page 1989

  22. [22]

    Journal of Physical Oceanography 20 (1), 97--113

    Cushman-Roisin , Benoit & Tang, Benyang 1990 Geostrophic Turbulence and Emergence of Eddies beyond the Radius of Deformation . Journal of Physical Oceanography 20 (1), 97--113

    work page 1990

  23. [23]

    Bulletin of the American Mathematical Society 51 (4), 527--580

    De Philippis, Guido & Figalli, Alessio 2014 The Monge -- Amp \`e re equation and its link to optimal transportation . Bulletin of the American Mathematical Society 51 (4), 527--580

    work page 2014

  24. [24]

    & Ingersoll, Andrew P

    Dowling, Timothy E. & Ingersoll, Andrew P. 1989 Jupiter's Great Red Spot as a Shallow Water System . Journal of the Atmospheric Sciences 46 (21), 3256--3278

    work page 1989

  25. [25]

    Dritschel, D. G. & McIntyre, M. E. 2008 Multiple Jets as PV Staircases : The Phillips Effect and the Resilience of Eddy-Transport Barriers . Journal of the Atmospheric Sciences 65 (3), 855--874

    work page 2008

  26. [26]

    Shafer & Bühler, Oliver 2024 Next-order balanced model captures submesoscale physics and statistics, arXiv:arXiv: 2408.03422

    Dù, Ryan Shìjié , Smith, K. Shafer & Bühler, Oliver 2024 Next-order balanced model captures submesoscale physics and statistics, arXiv:arXiv: 2408.03422

    work page arXiv 2024

  27. [27]

    , Samelson, R

    Early, Jeffrey J. , Samelson, R. M. & Chelton, Dudley B. 2011 The Evolution and Propagation of Quasigeostrophic Ocean Eddies * . Journal of Physical Oceanography 41 (8), 1535--1555

    work page 2011

  28. [28]

    Proceedings of the National Academy of Sciences 117 (9), 4491--4497

    Gallet, Basile & Ferrari, Raffaele 2020 The vortex gas scaling regime of baroclinic turbulence . Proceedings of the National Academy of Sciences 117 (9), 4491--4497

    work page 2020

  29. [29]

    AGU Advances 2 (3), e2020AV000362

    Gallet, Basile & Ferrari, Raffaele 2021 A Quantitative Scaling Theory for Meridional Heat Transport in Planetary Atmospheres and Oceans . AGU Advances 2 (3), e2020AV000362

    work page 2021

  30. [30]

    Gill, A. E. , Green, J. S. A. & Simmons, A. J. 1974 Energy partition in the large-scale ocean circulation and the production of mid-ocean eddies . Deep Sea Research and Oceanographic Abstracts 21 (7), 499--528

    work page 1974

  31. [31]

    & Montgomery, Michael T

    Graves, Lee Paul , McWilliams, James C. & Montgomery, Michael T. 2006 Vortex evolution due to straining: A mechanism for dominance of strong, interior anticyclones . Geophysical & Astrophysical Fluid Dynamics 100 (3), 151--183

    work page 2006

  32. [32]

    & Smith, K

    Grooms, Ian , Majda, Andrew J. & Smith, K. Shafer 2015 Stochastic superparameterization in a quasigeostrophic model of the Antarctic Circumpolar Current . Ocean Modelling 85 , 1--15

    work page 2015

  33. [33]

    Haynes, P. H. & McIntyre, M. E. 1987 On the Evolution of Vorticity and Potential Vorticity in the Presence of Diabatic Heating and Frictional or Other Forces . Journal of the Atmospheric Sciences 44 (5), 828--841

    work page 1987

  34. [34]

    & Larichev, Vitaly D

    Held, Isaac M. & Larichev, Vitaly D. 1996 A scaling theory for horizontally homogeneous, baroclinically unstable flow on a beta plane . Journal of the Atmospheric Sciences 53 (7), 946--963

    work page 1996

  35. [35]

    1975 The Geostrophic Momentum Approximation and the Semi-Geostrophic Equations

    Hoskins, Brian J. 1975 The Geostrophic Momentum Approximation and the Semi-Geostrophic Equations . Journal of the Atmospheric Sciences 32 (2), 233--242

    work page 1975

  36. [36]

    1991 Towards a PV- view of the general circulation

    Hoskins, Brian J. 1991 Towards a PV- view of the general circulation . Tellus A: Dynamic Meteorology and Oceanography 43 (4), 27--36

    work page 1991

  37. [37]

    & Bretherton, Francis P

    Hoskins, Brian J. & Bretherton, Francis P. 1972 Atmospheric Frontogenesis Models : Mathematical Formulation and Solution . Journal of the Atmospheric Sciences 29 (1), 11--37

    work page 1972

  38. [38]

    , Draghici, I

    Hoskins, Brian J. , Draghici, I. & Davies, H. C. 1978 A new look at the -equation . Quarterly Journal of the Royal Meteorological Society 104 (439), 31--38

    work page 1978

  39. [39]

    Hoskins, B. J. , McIntyre, M. E. & Robertson, A. W. 1985 On the use and significance of isentropic potential vorticity maps . Quarterly Journal of the Royal Meteorological Society 111 (470), 877--946

    work page 1985

  40. [40]

    , Pal \'o czy, A

    LaCasce, J.H. , Pal \'o czy, A. & Trodahl, M. 2024 Vortices over bathymetry . Journal of Fluid Mechanics 979 , A32

    work page 2024

  41. [41]

    Journal of the Atmospheric Sciences 69 (4), 1405--1426

    Lambaerts, Julien , Lapeyre, Guillaume & Zeitlin, Vladimir 2012 Moist versus Dry Baroclinic Instability in a Simplified Two-Layer Atmospheric Model with Condensation and Latent Heat Release . Journal of the Atmospheric Sciences 69 (4), 1405--1426

    work page 2012

  42. [42]

    & Held, I

    Lapeyre, G. & Held, I. M. 2004 The Role of Moisture in the Dynamics and Energetics of Turbulent Baroclinic Eddies . Journal of the Atmospheric Sciences 61 (14), 1693--1710

    work page 2004

  43. [43]

    Climate Dynamics 54 (1), 575--598

    Lucarini, Valerio & Gritsun, Andrey 2020 A new mathematical framework for atmospheric blocking events . Climate Dynamics 54 (1), 575--598

    work page 2020

  44. [44]

    & Hell, Momme C

    Lutsko, Nicholas J. & Hell, Momme C. 2021 Moisture and the Persistence of Annular Modes . Journal of the Atmospheric Sciences 78 (12), 3951--3964

    work page 2021

  45. [45]

    Cambridge University Press

    Majda, Andrew & Wang, Xiaoming 2006 Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows \/ . Cambridge University Press

    work page 2006

  46. [46]

    Maltrud, M. E. & Vallis, G. K. 1991 Energy spectra and coherent structures in forced two-dimensional and beta-plane turbulence . Journal of Fluid Mechanics 228 , 321--342

    work page 1991

  47. [47]

    Journal of the Atmospheric Sciences 50 (12), 1792--1818

    Marshall, John & Molteni, Franco 1993 Toward a Dynamical Understanding of Planetary-Scale Flow Regimes . Journal of the Atmospheric Sciences 50 (12), 1792--1818

    work page 1993

  48. [48]

    , Snyder, Chris & Rotunno, Richard 1999 The Next-Order Corrections to Quasigeostrophic Theory

    Muraki, David J. , Snyder, Chris & Rotunno, Richard 1999 The Next-Order Corrections to Quasigeostrophic Theory . Journal of the Atmospheric Sciences 56 (11), 1547--1560

    work page 1999

  49. [49]

    Pizzo, Nick & Salmon, Rick 2024 Exact planetary waves and jet streams

    work page 2024

  50. [50]

    Polvani, L. M. , McWilliams, J. C. , Spall, M. A. & Ford, R. 1994 The coherent structures of shallow-water turbulence: Deformation -radius effects, cyclone/anticyclone asymmetry and gravity-wave generation . Chaos: An Interdisciplinary Journal of Nonlinear Science 4 (2), 177--186

    work page 1994

  51. [51]

    1996 Potential vorticities in semi-geostrophic theory

    Roulstone, Ian & Sewell, Michael J. 1996 Potential vorticities in semi-geostrophic theory . Quarterly Journal of the Royal Meteorological Society 122 (532), 983--992

    work page 1996

  52. [52]

    Geophysical & Astrophysical Fluid Dynamics 15 (1), 167--211

    Salmon, Rick 1980 Baroclinic instability and geostrophic turbulence . Geophysical & Astrophysical Fluid Dynamics 15 (1), 167--211

    work page 1980

  53. [53]

    2023 Two-dimensional turbulence above topography: Vortices and potential vorticity homogenization

    Siegelman, Lia & Young, William R. 2023 Two-dimensional turbulence above topography: Vortices and potential vorticity homogenization . Proceedings of the National Academy of Sciences 120 (44), e2308018120

    work page 2023

  54. [54]

    & Ingersoll, Andrew P

    Siegelman, Lia , Young, William R. & Ingersoll, Andrew P. 2022 Polar vortex crystals: Emergence and structure . Proceedings of the National Academy of Sciences 119 (17), e2120486119

    work page 2022

  55. [55]

    Smith, K. S. , Boccaletti, G. , Henning, C. C. , Marinov, I. , Tam, C. Y. , Held, I. M. & Vallis, G. K. 2002 Turbulent diffusion in the geostrophic inverse cascade . Journal of Fluid Mechanics 469 , 13--48

    work page 2002

  56. [56]

    & Stechmann, Samuel N

    Smith, Leslie M. & Stechmann, Samuel N. 2017 Precipitating Quasigeostrophic Equations and Potential Vorticity Inversion with Phase Changes . Journal of the Atmospheric Sciences 74 (10), 3285--3303

    work page 2017

  57. [57]

    & McWilliams, James C

    Solodoch, Aviv , Stewart, Andrew L. & McWilliams, James C. 2021 Formation of Anticyclones above Topographic Depressions . Journal of Physical Oceanography 51 (1), 207--228

    work page 2021

  58. [58]

    & Mcwilliams, James C

    Spall, Michael A. & Mcwilliams, James C. 1992 Rotational and gravitational influences on the degree of balance in the shallow-water equations . Geophysical & Astrophysical Fluid Dynamics 64 (1-4), 1--29

    work page 1992

  59. [59]

    Srinivasan, Kaushik & Young, W. R. 2014 Reynolds Stress and Eddy Diffusivity of - Plane Shear Flows . Journal of the Atmospheric Sciences 71 (6), 2169--2185

    work page 2014

  60. [60]

    Journal of Physical Oceanography 27 (8), 1743--1769

    Stammer, Detlef 1997 Global Characteristics of Ocean Variability Estimated from Regional TOPEX / POSEIDON Altimeter Measurements . Journal of Physical Oceanography 27 (8), 1743--1769

    work page 1997

  61. [61]

    Thiry, Louis , Li, Long , M \'e min, Etienne & Roullet, Guillaume 2024 Quasi-geostrophic projection of rotating shallow-water equations

    work page 2024

  62. [62]

    Proceedings of the National Academy of Sciences 98 (22), 12346--12350

    Turkington, Bruce , Majda, Andrew , Haven, Kyle & DiBattista, Mark 2001 Statistical equilibrium predictions of jets and spots on Jupiter . Proceedings of the National Academy of Sciences 98 (22), 12346--12350

    work page 2001

  63. [63]

    Journal of Atmospheric and Oceanic Technology 32 (1), 177--184

    Ubelmann, Clement , Klein, Patrice & Fu, Lee-Lueng 2015 Dynamic Interpolation of Sea Surface Height and Potential Applications for Future High-Resolution Altimetry Mapping . Journal of Atmospheric and Oceanic Technology 32 (1), 177--184

    work page 2015

  64. [64]

    1996 Potential vorticity inversion and balanced equations of motion for rotating and stratified flows

    Vallis, Geoffrey K. 1996 Potential vorticity inversion and balanced equations of motion for rotating and stratified flows . Quarterly Journal of the Royal Meteorological Society 122 (529), 291--322

    work page 1996

  65. [65]

    2017 Atmospheric and Oceanic Fluid Dynamics : Fundamentals and Large-Scale Circulation \/ , 2nd edn

    Vallis, Geoffrey K. 2017 Atmospheric and Oceanic Fluid Dynamics : Fundamentals and Large-Scale Circulation \/ , 2nd edn. Cambridge: Cambridge University Press

    work page 2017

  66. [66]

    , Bokhove, O

    Warn, T. , Bokhove, O. , Shepherd, T. G. & Vallis, G. K. 1995 Rossby number expansions, slaving principles, and balance dynamics . Quarterly Journal of the Royal Meteorological Society 121 (523), 723--739

    work page 1995

  67. [67]

    Oxford, New York: Oxford University Press

    Zeitlin, Vladimir 2018 Geophysical Fluid Dynamics : Understanding (Almost) Everything with Rotating Shallow Water Models\/ . Oxford, New York: Oxford University Press

    work page 2018