pith. sign in

arxiv: 2606.10386 · v1 · pith:G5AHKEQFnew · submitted 2026-06-09 · ⚛️ physics.flu-dyn

Baroclinic wave dynamics in the Ekman-free rotating rectangular annulus with localized forced plume

Shivam Swarnakar , Ayan Kumar Banerjee , Sridhar Balasubramanian , Amitabh Bhattacharya This is my paper

Pith reviewed 2026-06-27 12:04 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords baroclinic wavesrotating annulusforced plumeRichardson numberRossby numberturbulent heat fluxnumerical simulationEkman-free flow
0
0 comments X

The pith

Mixing-length argument predicts bulk turbulent heat flux scales as Ri_0 to the power of negative one half in rotating baroclinic annulus flows.

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

The paper reports numerical simulations of an Ekman-free rotating rectangular annulus with a localized forced heated plume under bi-directional temperature gradients. It identifies geostrophic-hydrostatic balance as the leading-order state and tracks the transition of baroclinic waves from mode m=2 to m=3 as Rossby number decreases, alongside a shift in plume morphology from laterally swept to sustained columnar structures as source Richardson number falls. A mixing-length argument derives a scaling for the turbulent heat flux that anticipates an order-of-magnitude increase from Ri_0=99 to Ri_0=1, and this prediction matches the simulation data. The results are summarized in a regime map showing that, within the explored range, Ri_0 controls the plume regime while Ro selects the dominant wave mode.

Core claim

In the Ekman-free rotating rectangular annulus with localized forced plume, baroclinic waves of mode m=2 at Ro=0.3 transition to m=3 at lower Ro consistent with contraction of the Eady deformation radius, while plume morphology transitions from weak laterally-swept structures at Ri_0=99 to sustained columnar plumes at Ri_0 less than or equal to 4; a mixing-length argument yields bulk turbulent heat flux proportional to Ri_0 to the power of negative one half, and a regime map in the (Ri_0, Ro) plane shows the plume-regime and wave-selection problems are approximately separable.

What carries the argument

The regime map in the (Ri_0, Ro) plane, together with the mixing-length scaling for turbulent heat flux, that establishes approximate separability of plume and wave problems.

If this is right

  • Turbulent heat flux increases by an order of magnitude when source Richardson number drops from 99 to 1.
  • Plume entrainment coefficient Gamma(z) exhibits opposite rotational sensitivities at low and high Ri_0, organized by a local plume Rossby number.
  • A Hopf-bifurcated vacillating state appears at Ri_0=99 and Ro=0.1.
  • Wave mode selection follows the contraction of the Eady deformation radius L_rho = NH/f as Ro decreases.

Where Pith is reading between the lines

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

  • The observed separability suggests that plume and wave dynamics could be tuned independently when modeling larger geophysical flows.
  • The Ri_0^{-1/2} scaling for heat flux may be tested directly in laboratory experiments using similar forced-plume configurations.
  • Extending the rectangular setup to include weak Ekman layers could reveal how boundary effects modify the bulk scaling.

Load-bearing premise

The rectangular annulus geometry isolates the Ekman-free bulk dynamics of the cylindrical baroclinic annulus without significant boundary-layer effects from the full cylindrical setup.

What would settle it

A simulation or experiment at an intermediate Ri_0 value between 1 and 99 that produces a turbulent heat flux scaling different from the predicted Ri_0 to the power of negative one half would falsify the mixing-length argument.

Figures

Figures reproduced from arXiv: 2606.10386 by Amitabh Bhattacharya, Ayan Kumar Banerjee, Shivam Swarnakar, Sridhar Balasubramanian.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p027_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p029_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p034_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p037_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p040_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p042_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p045_14.png] view at source ↗
read the original abstract

We report numerical simulations of a rotating rectangular annulus that isolates the Ekman-free bulk of the cylindrical baroclinic annulus, subjected to bi-directional temperature gradients imposed by a uniformly cooled inner wall and a localized forced heated plume at the outer bottom. The finite-volume OpenFOAM solver is employed across combinations of source Richardson number $Ri_0 = 99, 4, 1$ and Rossby number $Ro = 0.3, 0.1, 0.07$. A non-dimensional scaling of the governing equations identifies geostrophic-hydrostatic balance as the leading-order bulk state, a result confirmed a posteriori by the $x$ and $z-$momentum budgets. Baroclinic waves of mode $m=2$ at $Ro=0.3$ transition to $m=3$ as $Ro$ decreases, consistent with the contraction of the Eady deformation radius $L_\rho = NH/f$; Complex Empirical Orthogonal Function (CEOF) analysis characterizes the wave regime and detects a Hopf-bifurcated vacillating state at $Ri_0 = 99,~Ro = 0.1$. The plume morphology, classified through the Morton length scale and source flux-balance parameter, transitions from weak, laterally-swept structures at $Ri_0 = 99$ to sustained columnar plumes traversing the full baroclinic depth at $Ri_0 \leq 4$. The plume entrainment coefficient $\Gamma(z)$ shows opposite rotational sensitivities at low and high $Ri_0$, which we organize through a local plume Rossby number $Ro_p = w/(2\Omega b)$. A mixing-length argument predicts a bulk turbulent heat flux $\overline{u'T'} \propto Ri_0^{-1/2}$, anticipating an order-of-magnitude enhancement from $Ri_0 = 99$ to $Ri_0 = 1$, in agreement with the simulations. A regime map in the $(Ri_0, Ro)$ plane reveals that, within the explored range, the plume-regime and wave-selection problems are approximately separable: $Ri_0$ sets the plume regime while $Ro$ selects the dominant baroclinic wave mode.

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

2 major / 2 minor

Summary. The manuscript reports numerical simulations using OpenFOAM of baroclinic wave dynamics in a rotating rectangular annulus with a localized forced plume at the outer bottom and cooled inner wall. It identifies geostrophic-hydrostatic balance via momentum budgets, documents transitions in baroclinic wave modes (m=2 to m=3) with decreasing Rossby number Ro and a vacillating state via CEOF analysis, classifies plume morphology transitions with source Richardson number Ri_0 using Morton length scale, derives a mixing-length scaling predicting bulk turbulent heat flux \overline{u'T'} \propto Ri_0^{-1/2} that agrees with simulations across Ri_0=99 to 1, and constructs a regime map in (Ri_0, Ro) showing approximate separability of plume and wave regimes.

Significance. This study offers a simplified numerical setup to isolate Ekman-free interior dynamics of the classic rotating annulus experiment with localized thermal forcing. The confirmation of leading-order balances, characterization of wave vacillation, and the proposed mixing-length heat flux scaling, if robust, could provide useful benchmarks and insights for understanding plume-wave interactions in rotating stratified flows relevant to oceanography and atmospheric science. The separability in the regime map is a notable organizational result.

major comments (2)
  1. [Abstract (mixing-length argument and plume morphology classification)] The mixing-length argument yields \overline{u'T'} \propto Ri_0^{-1/2} and is reported to agree with simulations from Ri_0=99 to Ri_0=1. However, the plume morphology transitions at Ri_0 ≤ 4 from weak laterally-swept structures to sustained full-depth columnar plumes (as classified by Morton length scale and source flux-balance parameter). This transition likely modifies the vertical and horizontal scales entering the mixing-length estimate, yet the scaling is applied uniformly without regime-specific adjustment or re-derivation. It is unclear if the reported agreement holds generally or is influenced by the limited discrete sampling of Ri_0 values.
  2. [Abstract (momentum budgets)] The x and z-momentum budgets are said to confirm geostrophic-hydrostatic balance a posteriori. To strengthen this central claim, explicit quantification of the residual terms (e.g., their magnitude relative to the leading terms across the domain and parameter space) would be helpful, particularly near the plume where local imbalances might occur.
minor comments (2)
  1. The notation for the entrainment coefficient Γ(z) and local plume Rossby number Ro_p = w/(2Ω b) could be clarified with explicit definitions or a dedicated notation table if not already present.
  2. Figure captions or methods text should specify the grid resolution and any convergence tests used in the finite-volume simulations to allow assessment of numerical accuracy.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful review and positive assessment of the work. Below we respond point-by-point to the two major comments.

read point-by-point responses

  1. Referee: [Abstract (mixing-length argument and plume morphology classification)] The mixing-length argument yields \overline{u'T'} \propto Ri_0^{-1/2} and is reported to agree with simulations from Ri_0=99 to Ri_0=1. However, the plume morphology transitions at Ri_0 ≤ 4 from weak laterally-swept structures to sustained full-depth columnar plumes (as classified by Morton length scale and source flux-balance parameter). This transition likely modifies the vertical and horizontal scales entering the mixing-length estimate, yet the scaling is applied uniformly without regime-specific adjustment or re-derivation. It is unclear if the reported agreement holds generally or is influenced by the limited discrete sampling of Ri_0 values.

    Authors: The mixing-length scaling is derived from bulk considerations in which the source Richardson number sets the characteristic turbulent velocity and length scales that enter the heat-flux estimate; the derivation itself does not presuppose a particular plume morphology. The simulations nevertheless show that the predicted Ri_0^{-1/2} dependence continues to describe the measured bulk flux even after the morphology transition at Ri_0 ≤ 4. This empirical agreement across the sampled points (Ri_0 = 99, 4, 1) suggests that any change in vertical or horizontal scales is already incorporated into the effective mixing length that the argument employs. We acknowledge that only three discrete values were examined and that a denser sampling would be desirable for generality. In the revised manuscript we will add a short paragraph discussing the applicability of the scaling across the documented plume transition and noting the limited sampling as a caveat. revision: partial

  2. Referee: [Abstract (momentum budgets)] The x and z-momentum budgets are said to confirm geostrophic-hydrostatic balance a posteriori. To strengthen this central claim, explicit quantification of the residual terms (e.g., their magnitude relative to the leading terms across the domain and parameter space) would be helpful, particularly near the plume where local imbalances might occur.

    Authors: We agree that explicit quantification of the residuals would strengthen the central claim. The existing momentum-budget diagnostics already contain the full term-by-term decomposition; we will therefore add a new figure (or panel) that reports the domain-averaged and plume-region magnitudes of the residual terms normalized by the leading geostrophic and hydrostatic contributions, for each (Ri_0, Ro) combination. This will be included in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

Mixing-length scaling presented as independent theoretical prediction; no reduction to simulation inputs by construction

full rationale

The paper derives a mixing-length argument that predicts \overline{u'T'} \propto Ri_0^{-1/2} and reports agreement with simulations as validation. No quoted step shows the scaling being fitted to the same data or reduced by definition to simulation outputs. Regime transitions and wave selections are simulation observations, but the scaling claim remains a separate theoretical estimate. No self-citation chains, ansatzes smuggled via prior work, or uniqueness theorems are invoked in the provided text. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The paper relies on standard non-dimensional parameters (Ri0, Ro) varied as inputs and domain assumptions common to rotating stratified flows; no new free parameters or invented entities are introduced beyond the simulation controls.

free parameters (2)
  • Ri_0
    Source Richardson number used as control parameter across discrete values
  • Ro
    Rossby number used as control parameter across discrete values
axioms (2)
  • domain assumption Geostrophic-hydrostatic balance is the leading-order bulk state
    Derived from non-dimensional scaling and confirmed a posteriori by momentum budgets
  • domain assumption Rectangular annulus isolates Ekman-free bulk dynamics
    Stated as the modeling choice to represent interior cylindrical annulus flow

pith-pipeline@v0.9.1-grok · 5960 in / 1441 out tokens · 33725 ms · 2026-06-27T12:04:15.225719+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

44 extracted references · 1 linked inside Pith

  1. [1]

    npj Climate and Atmospheric Science , volume=

    Seasonal predictability of baroclinic wave activity , author=. npj Climate and Atmospheric Science , volume=. 2021 , publisher=

    2021

  2. [2]

    Philosophical Transactions of the Royal Society of London

    An experimental study of thermal convection in a rotating liquid , author=. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences , volume=. 1958 , publisher=

    1958

  3. [3]

    Experiments in fluids , volume=

    PIV-and LDV-measurements of baroclinic wave interactions in a thermally driven rotating annulus , author=. Experiments in fluids , volume=. 2011 , publisher=

    2011

  4. [4]

    Nonlinear Processes in Geophysics , volume=

    An experimental study of regime transitions in a differentially heated baroclinic annulus with flat and sloping bottom topographies , author=. Nonlinear Processes in Geophysics , volume=. 2014 , publisher=

    2014

  5. [5]

    Nonlinear Processes in Geophysics , volume=

    Experiments on transitions of baroclinic waves in a differentially heated rotating annulus , author=. Nonlinear Processes in Geophysics , volume=. 2005 , publisher=

    2005

  6. [6]

    arXiv preprint arXiv:1403.2715 , year=

    Benchmarking in a rotating annulus: a comparative experimental and numerical study of baroclinic wave dynamics , author=. arXiv preprint arXiv:1403.2715 , year=

    Pith/arXiv arXiv

  7. [7]

    Journal of Fluid Mechanics , volume=

    A combined laboratory and numerical study of heat transport by baroclinic eddies and axisymmetric flows , author=. Journal of Fluid Mechanics , volume=. 2003 , publisher=

    2003

  8. [8]

    Modeling atmospheric and oceanic flows: Insights from laboratory experiments and numerical simulations , pages=

    General circulation of planetary atmospheres: insights from rotating annulus and related experiments , author=. Modeling atmospheric and oceanic flows: Insights from laboratory experiments and numerical simulations , pages=. 2014 , publisher=

    2014

  9. [9]

    Journal of Fluid Mechanics , volume=

    Baroclinic annulus waves , author=. Journal of Fluid Mechanics , volume=. 1971 , publisher=

    1971

  10. [10]

    Experiments in Fluids , volume=

    A rotating annulus driven by localized convective forcing: a new atmosphere-like experiment , author=. Experiments in Fluids , volume=. 2017 , publisher=

    2017

  11. [11]

    AIP Advances , volume=

    Experimental study of rotating convection in the presence of bi-directional thermal gradients with localized heating , author=. AIP Advances , volume=. 2018 , publisher=

    2018

  12. [12]

    Physics of Fluids , volume=

    Numerical study of rotating convection with bi-directional temperature gradients , author=. Physics of Fluids , volume=. 2023 , publisher=

    2023

  13. [13]

    Journal of Heat Transfer , volume=

    Investigation of heat transfer characteristics in a rotating convection system with bi-directional thermal gradients , author=. Journal of Heat Transfer , volume=. 2021 , publisher=

    2021

  14. [14]

    Atmosphere , volume=

    New Laboratory Experiments to Study the Large-Scale Circulation and Climate Dynamics , author=. Atmosphere , volume=. 2023 , publisher=

    2023

  15. [15]

    2015 , publisher=

    Fluid mechanics , author=. 2015 , publisher=

    2015

  16. [16]

    Computers in physics , volume=

    A tensorial approach to computational continuum mechanics using object-oriented techniques , author=. Computers in physics , volume=. 1998 , publisher=

    1998

  17. [17]

    Journal of computational physics , volume=

    Solution of the implicitly discretised fluid flow equations by operator-splitting , author=. Journal of computational physics , volume=. 1986 , publisher=

    1986

  18. [18]

    International Journal of Climatology: A Journal of the Royal Meteorological Society , volume=

    Empirical orthogonal functions and related techniques in atmospheric science: A review , author=. International Journal of Climatology: A Journal of the Royal Meteorological Society , volume=. 2007 , publisher=

    2007

  19. [19]

    Journal of Atmospheric Sciences , volume=

    Thermal convection in a rotating fluid subject to a horizontal temperature gradient: spatial and temporal characteristics of fully developed baroclinic waves , author=. Journal of Atmospheric Sciences , volume=

  20. [20]

    Quarterly Journal of the Royal Meteorological Society , volume=

    An investigation of the structure of baroclinic waves using three-level streak photography , author=. Quarterly Journal of the Royal Meteorological Society , volume=. 1972 , publisher=

    1972

  21. [21]

    Journal of Atmospheric Sciences , volume=

    Wave dispersion in a rotating, differentially heated cylindrical annulus of fluid , author=. Journal of Atmospheric Sciences , volume=

  22. [22]

    Geophysical & Astrophysical Fluid Dynamics , volume=

    On the transition between axisymmetric and non-axisymmetric flow in a rotating liquid annulus subject to a horizontal temperature gradient: Hysteresis effects at moderate taylor number and baroclinic waves beyond the eady cut-off at high taylor number , author=. Geophysical & Astrophysical Fluid Dynamics , volume=. 1978 , publisher=

    1978

  23. [23]

    Journal of Atmospheric Sciences , volume=

    Thermal convection in a rotating annulus of liquid: effect of viscosity on the transition between axisymmetric and non-axisymmetric flow regimes , author=. Journal of Atmospheric Sciences , volume=

  24. [24]

    Journal of Fluid Mechanics , volume=

    Numerical integration of the three-dimensional Navier-Stokes equations for incompressible flow , author=. Journal of Fluid Mechanics , volume=. 1969 , publisher=

    1969

  25. [25]

    Journal of the Meteorological Society of Japan

    A comparison of laboratory experiments and numerical simulations of steady baroclinic waves produced in a differentially heated rotating fluid annulus , author=. Journal of the Meteorological Society of Japan. Ser. II , volume=. 1989 , publisher=

    1989

  26. [26]

    Quarterly Journal of the Royal Meteorological Society , volume=

    A comparison of laboratory measurements and numerical simulations of baroclinic wave flows in a rotating cylindrical annulus , author=. Quarterly Journal of the Royal Meteorological Society , volume=. 1985 , publisher=

    1985

  27. [27]

    Quarterly Journal of the Royal Meteorological Society , volume=

    A combined laboratory and numerical study of fully developed steady baroclinic waves in a cylindrical annulus , author=. Quarterly Journal of the Royal Meteorological Society , volume=. 1981 , publisher=

    1981

  28. [28]

    Environmental Fluid Mechanics , volume=

    Role of dispersed particles on the dynamics of an umbrella cloud of a forced plume in a linearly stratified environment , author=. Environmental Fluid Mechanics , volume=. 2018 , publisher=

    2018

  29. [29]

    Physics of Fluids , volume=

    Axisymmetric Study of Convection in Rotating Annulus in the Presence of Localized Heating , author=. Physics of Fluids , volume=. 2024 , month=

    2024

  30. [30]

    Proceedings of the Royal Society of London, Series A , volume=

    Turbulent gravitational convection from maintained and instantaneous sources , author=. Proceedings of the Royal Society of London, Series A , volume=. 1956 , publisher=

    1956

  31. [31]

    Annual Review of Fluid Mechanics , volume=

    Turbulent plumes in nature , author=. Annual Review of Fluid Mechanics , volume=. 2010 , publisher=

    2010

  32. [32]

    Atmosphere-Ocean , volume=

    Turbulent plumes in stratified environments: a review of recent work , author=. Atmosphere-Ocean , volume=. 2008 , publisher=

    2008

  33. [33]

    1979 , publisher=

    Mixing in Inland and Coastal Waters , author=. 1979 , publisher=

    1979

  34. [34]

    Journal of the Hydraulics Division, ASCE , volume=

    Two-dimensional buoyant jets in stratified fluid , author=. Journal of the Hydraulics Division, ASCE , volume=

  35. [35]

    Journal of Hydraulic Engineering , volume=

    Numerical simulation of vertical buoyant wall jet discharged into a linearly stratified environment , author=. Journal of Hydraulic Engineering , volume=. 2018 , publisher=

    2018

  36. [36]

    Physics of Fluids , volume=

    Development of a point plume in the presence of background rotation , author=. Physics of Fluids , volume=. 1998 , publisher=

    1998

  37. [37]

    Journal of Physical Oceanography , volume=

    Unsteady, turbulent convection into a homogeneous, rotating fluid, with oceanographic applications , author=. Journal of Physical Oceanography , volume=. 1994 , publisher=

    1994

  38. [38]

    Journal of Geophysical Research: Oceans , volume=

    Experiments on baroclinic vortex shedding from hydrothermal plumes , author=. Journal of Geophysical Research: Oceans , volume=. 1991 , publisher=

    1991

  39. [39]

    Journal of Fluid Mechanics , volume=

    The generation of coherent vortices by line plumes in rotating stratified fluid , author=. Journal of Fluid Mechanics , volume=. 1999 , publisher=

    1999

  40. [40]

    Journal of Fluid Mechanics , volume=

    Turbulent fountains in a stratified fluid , author=. Journal of Fluid Mechanics , volume=. 1998 , publisher=

    1998

  41. [41]

    Journal of Fluid Mechanics , volume=

    Investigations of round vertical turbulent buoyant jets , author=. Journal of Fluid Mechanics , volume=. 1988 , publisher=

    1988

  42. [42]

    Journal of Fluid Mechanics , volume=

    Lazy plumes , author=. Journal of Fluid Mechanics , volume=. 2005 , publisher=

    2005

  43. [43]

    Journal of Hydraulic Research , volume=

    Submerged turbulent jets in stagnant linearly stratified fluids , author=. Journal of Hydraulic Research , volume=. 1988 , publisher=

    1988

  44. [44]

    Journal of Fluid Mechanics , volume=

    Turbulent entrainment: the development of the entrainment assumption, and its application to geophysical flows , author=. Journal of Fluid Mechanics , volume=. 1986 , publisher=

    1986